SPACAR User Manual dr. ir. R. G. K. M. Aarts, dr. ir. J. P. Meijaard and prof. dr. ir. J. B. Jonker 2009 Edition, March 1, 2009 Report No. WA-1180 Table of contents Preface iii 1 The SPACAR program 1.1 Introduction . . . . . . . . . . . . . . . . . 1.2 SPACAR & MATLAB . . . . . . . . . . . 1.3 SPAVISUAL . . . . . . . . . . . . . . . . 1.4 SPASIM and SIMULINK . . . . . . . . . . 1.5 Perturbation method and modal techniques . . . . . . 2 Keywords 2.1 Introduction . . . . . . . . . . . . . . . . . . 2.2 Kinematics . . . . . . . . . . . . . . . . . . 2.3 Dynamics . . . . . . . . . . . . . . . . . . . 2.4 Inverse dynamics (setpoint generation) . . . . 2.4.1 Trajectory generation . . . . . . . . . 2.4.2 Nominal inputs and reference outputs 2.5 Linearization . . . . . . . . . . . . . . . . . 2.6 Non-linear simulation of manipulator control 2.7 Visualization and animation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Examples 3.1 Planar sliding bar . . . . . . . . . . . . . . . . . 3.2 Planar slider–crank mechanism . . . . . . . . . . 3.3 Cardan-joint mechanism . . . . . . . . . . . . . 3.4 Planar four-bar mechanism . . . . . . . . . . . . 3.5 Rotating mass–spring system . . . . . . . . . . . 3.6 Cantilever beam in Euler buckling . . . . . . . . 3.7 Cantilever beam subject to concentrated end force 3.8 Short beam . . . . . . . . . . . . . . . . . . . . 3.9 Lateral buckling of cantilever beam . . . . . . . 3.10 State-variable and output equations . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 10 10 12 . . . . . . . . . 15 15 16 24 31 31 36 38 42 44 . . . . . . . . . . 47 47 49 57 60 63 66 68 71 73 76 ii Table of contents 3.11 3.12 3.13 3.14 Rigid spatial manipulator mechanism . . Flexible spatial manipulator mechanism . Chord-driven underactuated robotic finger Tricycle . . . . . . . . . . . . . . . . . . A SPACAR installation B SPACAR error messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 89 90 93 99 101 C MATLAB tutorial 103 C.1 Basic MATLAB graphics commands . . . . . . . . . . . . . . . . . . . . . . . 103 C.2 Quitting and saving the workspace . . . . . . . . . . . . . . . . . . . . . . . . 105 References 107 Preface This is the 2009 edition of the manual that describes the use of the SPACAR-package in a MATLAB / SIMULINK environment. This software is being developed at the Laboratory of Mechanical Automation of the Department of Engineering Technology, University of Twente, and is partly based on work carried out at the Department of Engineering Mechanics, Delft University of Technology. This manual accompanies the 2009 UT-release of SPACAR. With respect to the previous editions of this manual new keywords have been included reflecting changes in the software. In particular, systems with non-holonomic constraints resulting from the rolling contact of wheels can now be modelled. As before, the SPAVISUAL manual is integrated. The examples are updated to show the use of SPAVISUAL. The references to sections and examples in the lecture notes [1] are updated for the 2005 edition of these lecture notes. They may be only approximate for other editions. The visualisation tool SPAVISUAL has been implemented by Jan Bennik, who also provided the description of the keywords and the examples for the part of the manual related to this tool. Corrections of errors, suggestions for improvements and other comments are welcome. March 1, 2009, dr. ir. R. G. K. M. Aarts (Email: [email protected]), dr. ir. J. P. Meijaard and prof. dr. ir. J. B. Jonker. iii iv Preface 1 The SPACAR program 1.1 Introduction The computer program SPACAR is based on the non-linear finite element theory for multi-degree of freedom mechanisms as described in Jonker’s lecture notes on the Dynamics of Machines and Mechanisms [1]. The program is capable of analysing the dynamics of planar and spatial mechanisms and manipulators with flexible links and treats the general case of coupled large displacement motion and small elastic deformation. The motion can be simulated by solving the complete set of non-linear equations of motion or by using the so-called perturbation method. The computational efficiency of the latter method can be improved further by applying modal techniques. In this chapter an outline of the SPACAR package for use with MATLAB and SIMULINK is given in the next sections. For instance, for the design mechanical systems involving automatic controls (such as robotic manipulators), interfaces with MATLAB [2] are provided for open-loop system analyses, Section 1.2. Open-loop and closed-loop simulations can be carried out with blocks from a SIMULINK library, Section 1.4. A special visualization tool, SPAVISUAL, is described in Section 1.3. Additional tools are available for using the perturbation method and the modal techniques in SIMULINK (Section 1.5). Installation notes for SPACAR are given in Appendix A. A graphical user interface (GUI) for generating input files for spatial systems is under development and will be available in near future. People interested in rigid planar mechanisms may consider the use of the commercially available package SAM by ARTAS [4]. It has a nice graphical interface for the definition of mechanisms and it provides more elements than SPACAR. 1.2 SPACAR & MATLAB The SPACAR program system for use in the MATLAB environment contains five modules, which obtain their input from format-free user-supplied data. In the following a short description of every module will be given. The functional connections between the modules are illustrated in Fig. 1.1. 1 2 Chapter 1. The SPACAR program mechanism connectivity configuration DOFs q = (xm , em ) KIN dynamic properties forces (f m , σ m ) trajectory path velocity profile (xpo , ẋpo ) mode 1 DYN mode 2 INVDYN mode 3 LINEAR mode 4 mode 7 STATIO LINEAR mode 8 mode 9 Figure 1.1. Functional relations between modules in available in the MATLAB environment. KIN SPACAR . DOFs ( q, q̇) m m joint variables (em 0 , ė , σ 0 ) nominal inputs (u0 ) reference outputs (y 0 ) State space matrices Linearized equation Eigen frequencies Buckling loads State space matrices The indicated modes are is the kinematics module that analyses the configuration of the mechanism. The kinematic properties of the motion are specified by the geometric transfer functions. The following steps are provided by the KIN-module: 1. Definition of the mechanism connectivity, the configuration and the degrees of freedom (DOFs), q = (x(m) , e(m) ). 2. System preparation. 3. Calculation of the geometric transfer functions. DYN is the dynamics module that generates the equations of motion and performs numerical integration in the forward dynamic analysis (in the so-called mode=1 of SPACAR). Furthermore, it generates and solves the equations for the kinetostatic analysis. is the inverse manipulator dynamics module that performs the inverse kinematics and dynamics (mode=2) and generates the setpoints for the simulation of manipulator motion with closed-loop control in SIMULINK (see Sect. 1.4). The system inputs, represented by the nominal input vector u0 , are to be varied by the control system actuators. The system outputs, represented by the reference output vector y 0 , consist of the coordinates to be monitored by control sensors. Coordinates that are not measured may be added to check the performance of the manipulator in the simulation. INVDYN computes stationary solutions of autonomous systems. Stationary solutions are solutions in which the vector of dynamic degrees of freedom q d has a constant value. This can represent a static equilibrium configuration or a state of steady motion. STATIO is a forward dynamics stage for the generation of linearized equations and state space matrices. It can be used in different modes as described below. LINEAR Section 1.2. SPACAR & MATLAB 3 In mode=4 the LINEAR module is an extension of the forward dynamic analysis (mode=1) where coefficient matrices of the linearized equations are calculated as functions of the set of degrees of freedom q. If there are only holonomic deformations in a system, the linearized equations are generated in the form: (x)T M̄ 0 δq̈ + [C 0 + D 0 ] δ q̇ + [K 0 + N 0 + G0 ] δq = DF 0 (e)T δf − DF 0 δσ a , (1.1) where M̄ 0 is the reduced mass matrix, C 0 the velocity sensitivity matrix, D 0 the damping matrix, K 0 the structural stiffness matrix, and N 0 and G0 are the dynamic and geometric stiffness matrices respectively. External and internal driving forces are represented by the vectors δf and δσ a , respectively. In addition, if input and output vectors δu and δy are defined also the linearized state equations and output equations are computed (see mode=9). In mode=3 locally linearized models are generated about a predefined nominal trajectory where the output data (setpoints) from the inverse dynamics module (i.e. a previous mode=2 run) are used. In addition to the coefficient matrices, a complete state space system is generated and written to a so-called ltv file (see Sect. 1.5). In the case of a flexible mechanism additional degrees of freedom describing the elastic behaviour of the mechanism have to be included in the dynamic models (both mode=2 and 3). At this stage in the so-called “rigidified” model, these flexibilities are prescribed zero, i.e. εm i ≡ 0. In mode=7 eigenvalues (frequencies) and corresponding eigenvectors of the state space matrix A are computed for a static equilibrium configuration or a state of steady motion. The associated frequency equation of the undamped system is given by dd dd dd det −ωi2 M̄ 0 + K dd = 0, 0 + N 0 + G0 (1.2) where the quantities ωi are the natural frequencies of the system. In mode=8 a linear buckling analysis is carried out for a static equilibrium configuration or a state of steady motion. Critical load parameters λi are determined by solving the eigenvalue problem: dd det(K dd 0 + λi G0 ) = 0, (1.3) where the load multipliers satisfy f i = λi f 0 . (1.4) dd Here, K dd 0 is the structural stiffness matrix and G0 is the geometric stiffness matrix due to the reference load f 0 giving rise to the reference stresses σ 0 . f i represents the bucking load that corresponds with λi . In addition, directional nodal compliances are computed. In mode=9 linearized equations for control system analysis are computed for a static equilibrium configuration or a state of steady motion and are generated in the form: h dd i h i dd dd dd dd d d M̄ 0 δq̈ d + C dd 0 + D 0 δ q̇ + K 0 + N 0 + G0 δq = B 0 δu, where h (x)T B 0 = Dqd F 0 is the input matrix and h (e)T | − Dqd F 0 dr dr dr dr | − K dr | − M̄ 0 | − C dr 0 + N 0 + G0 0 + D0 , δq̈ rT , δ q̇ rT , δq rT δu = δf (c,m)T , δσ (m,c)T a iT (1.5) i (1.6) (1.7) 4 Chapter 1. The SPACAR program is the input vector. The vectors δq̈ rT , δ q̇ rT , δq rT represent the prescribed (input) accelerations, velocities and displacements respectively. The linearized equations can be transformed into the linearized state space form: δ ż = Aδz + Bδu, δy = Cδz + Dδu, (1.8) where A is the state matrix, B the input matrix, C the output matrix and D the feed through matrix. The state vector δz is defined by δz = [δq dT , δ q̇ dT ]T , where δq d is the vector of dynamic degrees of freedom. The matrices B, C and D depend on the chosen input vector δu and the output vector δy. Details of the linearization are discussed in Chapter 12 of the lecture notes. Systems with non-holonomic deformations For systems with non-holonomic deformations arising from wheel elements, the above description has to be modified in several respects. Only mode=0, mode=1, mode=4, mode=7 and mode=9 are supported. The state vector consists of the coordinates describing the configuration, q k , and the velocity coordinates, q̇ d . The configuration coordinates are split in coordinates whose derivatives are velocity coordinates and coordinates that have no corresponding velocity coordinates; the latter are called kinematic coordinates. The dynamic equations consist of two parts, the kinematic differential equations defining the derivatives of the configuration coordinates and the equations of motion defining the time derivatives of the velocity coordinates. The linearized equations have the form " I O O M̄ 0 #" # " −B k0 −Ak0 δ q̇ k d + K 0 + N 0 + G0 C 0 + D 0 δq̈ #" # " # 0 δq k , (1.9) (e)T (x)T d = δ q̇ DF 0 δf − DF 0 δσ a where Ak0 and B k0 are kinematic matrices. For ordinary systems, B k0 is a zero matrix and Ak0 is an identity matrix. For mode=7, a stationary solution is first obtained with the module STATIO and the eigenvalues are obtained by solving the characteristic equation " # Iλ − B k0 −Ak0 = 0. det K 0 + N 0 + G0 M̄ 0 λ + C 0 + D 0 (1.10) For mode=9, the linearized state equations are obtained as in equation (1.8), with the difference that the variations in the states are now δz = [δq kT , δ q̇ dT ]T . Definition of a mechanism model A model of a mechanism must be defined in an input file of file type (or file name extension) dat. This input file consists of a number of keywords with essential and optional parameters. The input file can be generated with any text editor. In Chapter 2 the meaning of the keywords and their parameters is discussed in detail. In the examples in Chapter 3 complete input files are presented. Section 1.2. SPACAR & MATLAB 5 Running SPACAR in the MATLAB environment Once the mechanism is defined and this information is saved to a dat input file, SPACAR can be activated with the MATLAB command >> spacar(mode,’filename’) Here, mode indicates the type of computation as shown in Fig. 1.1. filename is the name of the input file, without the extension .dat. The filename is limited to 20 characters from the set “0”–“9”, “a”–“z”, “A”–“Z” and “ ”, so it can not include drive or path specifications. The linearization with mode=3 needs data from a previous inverse dynamics computation. To that end the specified filename is truncated with at least one character at the right until a valid output data file is found. So e.g. spacar(3,’testlin’) can use data from an earlier spacar(2,’test’) computation. If no data file can be found in this way the linearization is aborted. During the computation a plot of the mechanism is shown in a separate window. While the simulation is running an Abort button is activated in the plot area. Pressing this button will terminate the simulation (possibly after some delay). To speed up the computation, the plot can be disabled by specifying the mode with a minus sign, e.g. mode=-2 for an inverse dynamics computation without a continually updated plot. The plotting utility spadraw can also be used after the simulation to visualize the results, see page 9. During the computations the results are stored in one or more data files and in MATLAB arrays. A log file is always created when SPACAR starts processing the input dat file. This log file contains an analysis of the input and possible errors and warnings. It is described in more detail on page 8. Some errors in the input file do not lead to an early termination of the SPACAR computation, but nevertheless give unusable results. Therefore it is advisable to check the log file for unexpected messages. All other data files are so-called SPACAR binary data files (SBF), which implies that these are in a binary format and cannot be easily read by a user. Therefore, utilities are provided to read and modify data in these files, see page 9. Depending on the mode up to three binary output files may be created. For all modes a SPACAR binary data file with filename identical to the input file and extension sbd is written. The contents of this file are also stored in MATLAB arrays, that are of course immediately available in the MATLAB workspace e.g. to be visualized with the standard MATLAB graphics commands, such as plot (see e.g. Chapter 3 and Appendix C). The following variables are created or overwritten: mode SPACAR mode number ndof number of DOFs including rheonomic ones nddof number of dynamic DOFs nkdof number of configuration coordinates including rheonomic ones nkddof number of configuration coordinates nx number of coordinates ne number of deformation parameters nxp number of fixed, calculable, input, dynamic and kinematic coordinates nep number of fixed, calculable, input, dynamic and kinematic deformation parameters lnp location matrix for the nodes *1 6 Chapter 1. The SPACAR program le ln it rxyz time x xd xdd fx fxgrav fxtot e ed edd sig dec dxc de dx d2e d2x xcompl location matrix for the elements connection matrix for the nodes in the elements list of element types initial orientations of elements time column vector coordinates (nodal displacements) nodal velocities nodal accelerations prescribed nodal forces/moments gravity nodal forces/moments reaction forces/moments generalized deformations velocities of generalized deformations accelerations of generalized deformations generalized stress resultants first order geometric transfer function for the deformations DF c(e) first order geometric transfer function for the coordinates DF c(x) first order geometric transfer function for the deformations DF (e) first order geometric transfer function for the coordinates DF (x) second order geometric transfer function for the deformations D2 F (e) second order geometric transfer function for the coordinates D2 F (x) location vector for directional nodal compliances *1 *2 *2 *2 *3 *3 *3 *3 *3 *3 *4 Notes: ∗1 The two location matrices provide information to find the location of a specific quantity in the data matrices: lnp le location matrix for the nodes. The matrix element lnp(i,j) denotes the location of the jth coordinate (j=1..4) of node i. location matrix for the elements. The matrix element le(i,j) denotes the location of the jth generalized deformation (j=1..6) of element i. The locations of undefined or unused coordinates and deformations equal zero. For example, the x- and y-coordinates of node 7 can be shown as function of time in a graph by typing >> plot(time,x(:,lnp(7,1:2))) and the first generalized stresses in elements 1, 2 and 3 can be plotted by typing >> plot(time,sig(:,le(1:3,1))) Obviously, storage in the x, xd, xdd, fx, e, ed, edd and sig matrices is like x(t,k) where t is the time step and k ranges from 1 to nx for x, xd, xdd and fx, fxtot and from 1 to ne for e, ed, edd and sig, respectively. Section 1.2. SPACAR & MATLAB 7 ∗2 The variables ln, it and rxyz are mainly intended for internal use in the drawing tool spadraw. More user-friendly information is available in the log file, page 8. ∗3 The (large) variables dec, dxc, de, dx, d2e and d2x are only created if the parameters of the LEVELLOG are set accordingly, Sect. 2.2. ∗4 After a linearization run (mode=8) directional nodal compliances (inverse stiffnesses) are computed. Using the location matrix, xcompl(lnp(i,j)) gives this quantity for the jth coordinate (j=1..4) of node i. After a linearization run (mode=3, 4, 7, 8 or 9) the coefficient matrices are stored in a SPACAR binary matrix file with extension sbm. Besides nnom (see infra) and time, the accompanying MATLAB matrices are: m0 reduced mass matrix M 0 *5 b0 input matrix B 0 *5, *6 c0 velocity sensitivity matrix C 0 *5 d0 damping matrix D 0 *5 k0 structural stiffness matrix K 0 *5 n0 geometric stiffness matrix N 0 *5 g0 geometric stiffness matrix G0 *5 ak0 kinematic matrix Ak0 *5 bk0 kinematic matrix B k0 *5 Notes: ∗5 Storage of the time-varying matrices is in a row for each time step, so in m0(t,k) index t is the time step and k ranges from 1 to ndof×ndof. To restore the matrix structure at some time step type e.g. reshape(m0(t,:),ndof,ndof)’. ∗6 Only available for mode=4 and mode=9. In mode=2, 3, 4 and 9 a so-called ltv file is created. The contents of this file varies and is not automatically imported to the MATLAB workspace. From a mode=2 run the following data are available (the names identitify the data used in the file; data marked with “*” are available at each time step): NNOM number of (actuator) inputs NY number of outputs T time * U0 nominal input for the desired motion * Y0 reference output of the desired motion * In the addition the linearization runs yield additional setpoints, state space matrices and other data in the ltv file (not all data are always present): NNOM number of (actuator) inputs NX number of states (2×ndof) NU number of inputs (length of U0) NY number of outputs (length of Y0) NRBM number of rigid body DOFs NYS number of outputs with 2nd order expression 8 Chapter 1. The SPACAR program NYSI index array for outputs with 2nd order expression DFT direct feedthrough flag (D6=0) X0 initial state vector T time * A state space system matrix * B state space input matrix * C state space output matrix * D state space direct feedthrough matrix * G second order output tensor * M0 mass matrix M 0 * C0B combined damping matrix C 0 + D 0 * K0B combined stiffness matrix K 0 + N 0 + G0 * SIG0 generalized stress resultants * The getss tool can be used to read the state space matrices from the ltv file, see page 9. Other utilities are available to use parts of these data in a SIMULINK environment, e.g. to read setpoints or to simulate a linear time-varying (LTV) system (see Sect. 1.4). The log file The log file contains an analysis of the input and possible errors and warnings that are encountered. The error and warning messages are explained in more detail in Appendix B. The other output can be separated into a number of blocks. The first lines indicate the version and release date of the software and a copyright note. Next the lines from the input file read by the KIN module are shown (not showing comments present in the input file), see also Sect. 2.2. From the analysis is written: • The elements used in this model. The deformations of all elements are shown with the internal numbers according to the le array and the classification of each deformation: O = fixed, C = calculable and M = DOF. • The nodal point information with the internal numbers of the coordinates according to the lnp array and the classification as above. • A list showing the degrees of freedom, in which dynamic degrees of freedom are indicated. • The condition number of the part of the difference matrix that has to be inverted, which shows how well the degrees of freedom have been chosen. The DYN module reads the next data block and processed input lines are shown. From the analysis we get • The numbers NEO, NEMM, NEM and NEC indicating the numbers of deformations in each class as explained in the lecture notes [1]. • The numbers NXO, NXC, NXMM and NXM indicating the numbers of position coordinates in each class as explained in the lecture notes [1]. • The stiffness, damping and mass of the elements. Section 1.2. SPACAR & MATLAB 9 • The nodal point forces, mass and gyroscopic terms. • The total mass of the system. The zeroth, first, second and third order transfer functions are shown next, each for the position parameters and deformation parameters, respectively. The amount of output can be controlled by the keyword OUTLEVEL in the input file. Next for a forward analysis (mode=1 and mode=4) the name of the integrator and accuracy settings are shown. Finally a list with all time steps and the number of internal iterations are given. For an inverse dynamics analysis the trajectories and input/output definitions (see also Sect. 2.4) are read and analysed. In case of mode=3 the name of the data file of the previous mode=2 is shown. In case of mode=7 the eigenvalues (frequencies) and normalized eigenvectors of the state system matrix are shown. In case of mode=8 load multipliers and normalized buckling modes are presented. In addition the vector of directional nodal compliances is shown. SPACAR binary data files Some utilities are available to show, check, load or replace the data in SPACAR binary data files (SBF). These are files with extensions sbd, sbm and ltv. checksbf checks and shows the contents of a SPACAR binary data file. The output for each variable is the name (“Id”), the type (1 for integer, 2 for real, 3 for text) and the size (number of rows and columns). First the “header” variables are shown with their values. Long vectors may be truncated. Between TDEF and TDAT the time-varying data are given. The number of time steps is equal to the number of rows specified for TDEF. getfrsbf extracts a variable from a SPACAR binary data file. The “Id” must be specified and for time-varying data the time step as well. repinsbf replaces the value of a variable in a SPACAR binary data file. The “Id” must be specified and for time-varying data the time step as well. loadsbd loads all data from a SPACAR binary data (sbd) file into MATLAB’s workspace. loadsbm loads all data from a SPACAR binary matrix data (sbm) file into MATLAB’s workspace. getss loads the state space matrices at one time instant from a SPACAR ltv file into a state space system in MATLAB’s workspace. combsbd combines data from two or more SPACAR binary data (sbd) files into a single output file. The specified output file is overwritten without a warning. spadraw is the plotting utility used internally by SPACAR. It can also be used to visualize results after a simulation has been completed. For all utilities additional online help is available by typing help command at the MATLAB prompt. 10 Chapter 1. The SPACAR program Limitations The SPACAR package has some built-in limitations on the size of the manipulators that can be analysed. Table 1.1 shows the limits for the so-called “Student version” that can be downloaded as describes in Appendix A. In case your requirements are larger, you need to contact the authors. The licence for the freely downloadable software is time limited. Maximum number of coordinates/deformations Maximum number of DOFs Maximum number of elements/nodal points Maximum number of inputs Maximum number of outputs 175 20 50 12 25 Table 1.1. Built-in limitations of the “Student version” of the SPACAR package. 1.3 SPAVISUAL SPAVISUAL is the visualization tool for SPACAR . It can visualize deformation, vibration and buckling modes. SPAVISUAL shows beams, trusses, and hinges in 2-D as well as in 3-D. It works with default settings which can be adjusted by the user. The only input of SPAVISUAL is a filename. This file has to be a .dat file which has been analysed with SPACAR. This is necessary because SPAVISUAL needs the .sbd files for the deformation modes and also the .sbm files for the vibration and the buckling modes. There are some keywords that can adjust the default settings. These keywords are listed in section 2.7. SPAVISUAL is a stand-alone function in MATLAB . To run SPAVISUAL the user has to type the next command. >> spavisual(’filename’) Here filename refers to the .dat-file that is executed by SPACAR. 1.4 SPASIM and SIMULINK The behaviour of a manipulator mechanism with e.g. closed-loop control can be simulated using SIMULINK. The closed-loop simulation is defined as the problem of computing the actual trajectory of e.g. the manipulator tip with controlled actuation of the motion. Tracking errors with respect to a nominal prescribed trajectory can be calculated. Figure 1.2 shows an overview of a typical simulation scheme. The simulation is characterized by the inverse dynamics stage, based on a rigid link model and a forward dynamic stage. At the forward dynamics stage the tracking behaviour of the manipulator system is studied. In the case of flexible manipulators additional generalized coordinates (εm i ) describing the elastic behaviour of the manipulator links can be used in the dynamic system. The block diagram in Fig. 1.3 shows a typical closed-loop simulation in more detail. Blocks are used from the SPACAR SIMULINK library spacar_lib that is part of the SPACAR package. These blocks are front-ends to so-called S-functions in SIMULINK [3]. The following blocks are provided: Section 1.4. SPASIM and SIMULINK 11 Rigid or rigidified link model prescribed trajectory Flexible link model hp µ actual trajectory xp0 xp read setpoints INVDYN analyses simulation y read coeff. M 0 ,C 0 ,K 0 control parameters matrices LINEAR MATLAB u0 y0 − SIMULINK simulation Figure 1.2. Typical overview with MATLAB analyses and a SIMULINK simulation. Read Unom u0 Read Yref y0 read setpoints & coeff. matrices Read M0 + + − control system δu u actuator model + SPASIM ėm σ e,m mechanism m e model sensor model y S-function control parameters e.g. M-function Figure 1.3. Block diagram of a typical closed-loop simulation in SIMULINK. The left blocks read setpoints and coefficient matrices stored in data files during previous SPACAR analyses (Fig. 1.1). 1. SPASIM: the non-linear open-loop model of the manipulator with its actuators and sensors. It operates in a way comparable to the forward dynamics mode in SPACAR as discussed for the MATLAB interface in Sect. 1.2. The mechanism is defined in an input data file of file type dat. The filename of the input file must be specified. An output log file is written. Note that in a SIMULINK simulation the integration is determined by the SIMULINK environment, e.g. the kind of solver, the step size and tolerances. The degrees of freedom of the mechanism and their first time derivatives are the “states” of the SPACAR S-function. The dimensions of the input and output vectors are determined from the input file and should match the requirements of the other SIMULINK blocks they are connected to. 2. LTV: simulation of a linear time-varying system as defined in an ltv file, see Sect. 1.5. 12 Chapter 1. The SPACAR program 3. Setpoint U0: reads the nominal input from an ltv file with setpoints generated e.g. with mode=2 or mode=3. The filename must be specified. The setpoints are interpolated between the specified time steps. The interpolation method can be chosen from: Stepwise, Linear (default) and Spline. The block has no input and the dimension of the output vector equals the number of nominal inputs found in the file. 4. Setpoint Sigma0: reads σ 0 from an ltv file generated with e.g. mode=3, see Sect. 1.5. 5. Reference Y0: reads the reference output from an ltv data file with setpoints. The filename must be specified. Interpolation is as above. This block has no input and the dimension of the output vector equals the number of reference outputs found in the file. 6. Times M0: reads the square reduced mass matrix M 0 from an ltv file generated with e.g. mode=3. The output of the block equals the input of the block is multiplied with the mass matrix. The filename must be specified. In the case not the full dimension of M 0 in the ltv is used, the reduced dimension has to be specified. All elements of M 0 are interpolated linearly (default) or stepwise. The dimension of the output vector equals the dimension of the input vector. In the block diagram in Fig. 1.3 the output vector y of the SPASIM block is compared with the reference output vector y 0 . The difference of these vectors is the input of the control system. The state matrices can be used to develop and tune a controller of any type (e.g. linear, non-linear, discrete, continuous) by means of the available software tools in MATLAB and SIMULINK . The output of the controller δu is added to the nominal input vector u0 to actuate the mechanism. An example is discussed in Sect. 3.11. When using blocks from the SPACAR SIMULINK library spacar_lib note the following: • Using any of the LTV, Setpoint U0, Setpoint Sigma0, Reference Y0 and Times M0 blocks at times beyond the last time step found in the data file may lead to unexpected results. • In the current version of the software all spasim blocks in a block diagram should refer to the same input filename. Analogously, all LTV, Setpoint U0, Setpoint Sigma0, Reference Y0 and Times M0 must use the same ltv file. 1.5 Perturbation method and modal techniques For systems with a larger number of degrees of freedom the required computer time for a SPASIM simulation may be unacceptable, in particular when high eigenfrequencies play a role. Then the perturbation method may provide a numerically efficient solution strategy. Consider e.g. the motion of the flexible manipulator depicted in Fig. 1.2. In the case the flexibility is taken into account, the generalized coordinates or degrees of freedom can be written as " # em q= m , ε (1.11) Section 1.5. Perturbation method and modal techniques 13 where em represent the large relative displacements and rotations and εm are the flexible deformation parameters. Due to the flexibility the actual trajectory motion will deviate from the prescribed motion. If the deviations are small compared with the large scale motion, then the (small) vibrational motion of the manipulator can be modelled as a first-order perturbation δq of the nominal rigid link motion q 0 by writing for the degrees of freedom q = q 0 + δq. (1.12) The perturbation method involves two steps: 1. Compute nominal rigid link motion q 0 from the non-linear equations of motion with all flexible deformation parameters εm ≡ 0. This analysis will also provide the nominal input u0 of the manipulator necessary to carry out the nominal motion and the generalized stress resultants (Langrange multipliers) σ εm 0 of the rigidified deformations, i.e. the flexible deformations that are prescribed as zero. 2. Compute the vibrational motion δq from linearized equations of motion M̄ 0 δq̈ + C̄ 0 δ q̇ + K̄ 0 δq = σ 0 , (1.13) where M̄ 0 is the reduced mass matrix, C̄ 0 includes the velocity sensitivity and damping matrices and all stiffness matrices are combined into K̄ 0 . The right-hand side equals " # δu , σ0 = σ εm 0 (1.14) where δu = u − u0 is the actual control action u minus the nominal input u0 . The previously computed generalized stress resultants σ εm 0 are now applied as internal excitation forces. To solve the linearized equations of motion (1.13) these are expressed as a linear time varying (LTV) system. A SPACAR mode=3 run generates time-varying state space matrices that are well suited for this purpose. Then a typical SPACAR analysis and linearized simulation procedure is as follows: • Use e.g. an inverse dynamics run (mode=2) to define the nominal motion for the rigidified manipulator. Inputs and outputs of the system may be specified. • Next the system is linearized with a mode=3 call. The system is analysed along the nominal path computed previously. The elastic deformations are defined with INPUTE commands. Inputs and outputs must be specified. • Finally the linearized simulation can be run with a SIMULINK model of which a typical example is shown in Fig. 1.4. In comparison with the non-linear simulation of Fig. 1.3 the spasim block is replaced by an LTV block that uses the linearized equations of motion. Note that now only the differences compared to the nominal motion are computed. Only the difference δu of the manipulator’s input compared to the nominal input is needed. In addition, the generalized stress resultants σ εm 0 are part of the input of the LTV block. 14 Chapter 1. The SPACAR program u0 u + y0 + - controller + y + δu σ εm 0 LTV Figure 1.4. Block diagram of a typical closed-loop simulation in perturbation method. δy SIMULINK based on the In addition to the above outlined standard implementation some further extensions are provided. It is possible to include the effect of proportional controller gain, i.e. a proportional control matrix K p , into the stiffness matrix K̄ 0 . Of course, in that case this part of the control action should no longer be included in the controller in the block scheme. This approach offers advantages when subsequently a modal analysis is applied to the linear time varying state space system. Such an analysis discriminates quasi-static behaviour of the system, low-frequency vibrational modes and high-frequency vibrational modes. Mostly the latter do not significantly affect the output of the system while they can have a detrimental effect on the computational efficiency, even for a linearized system. With a modal analysis it is possible to eliminate these high-frequency modes. A more profound description of the latter two techniques is currently outside the scope of this manual. 2 Keywords 2.1 Introduction In this chapter the user is informed about the creation of correct input data for the software package SPACAR. The input must have a specific form. Behind a number of permitted keywords the user supplies a list of arguments. The arguments behind a keyword are well defined. Each module of SPACAR, except mode=4 of LINEAR, has its own list of available keywords. They form blocks that are separated by the following pair of keywords: END HALT The final closure of the input is effected by: END END The first block contains the kinematic data. The input of the mechanism model (by means of keywords) is treated in the “Kinematics” section 2.2. A second block of input is reserved for the dynamics module. The keywords for this block are presented in the “Dynamics” section 2.3. The solution of inverse dynamics problems demands additional input for the trajectory description and for the definition of the input and output vectors u0 and y 0 . Trajectory keywords and system keywords are treated in the “Inverse dynamics” section 2.4. The keywords for the linearization of mode=3, mode=4 and mode=9 are given in the “Linearization” section 2.5. At the end of the file custom settings for SPAVISUAL can be added. The keywords for SPAVISUAL are presented in section 2.7. The simulation of mechanisms using SIMULINK is controlled by the keywords described in the “Simulation” section 2.6. Some general remarks: • Keywords and arguments can be separated by one or more spaces, tabs or line breaks. • Lines must not contain more than 160 characters. 15 16 Chapter 2. Keywords • Any text in a line following a #, % or ; is treated as a comment. • All input is case insensitive. • Data read from the input file are echoed in the log file, after the comments have been removed and all text is transformed into upper case (capitals). • Angles are always specified in radians. • For some commands, such as XF and STARTDE, not all arguments have to be specified. Default values are zero unless otherwise specified. 2.2 Kinematics A kinematic mechanism model can be built up with finite elements by letting them have nodal points in common. The nodal coordinates of the finite elements are described by position and orientation coordinates. Therefore, two types of nodes are distinguished: position or transx lational nodes, denoted by ~p for node p, and orientation or rotational nodes denoted by p. The nodes, nodal coordinates, and deformation parameters for the truss, beam, planar bearing, hinge, pinbody (rigid beam) planar belt (gear) element and wheel elements are summarized in Table 2.1. Usually, the convention is made that node p of an element is assigned to the lower number of the element nodes, and that node q is assigned to the higher node number. The interconnections between the elements are accomplished by indicating common nodes between the elements. For instance, with a pin-joint connection only the translational nodes are shared. In case of a hinge-joint connection only the rotational nodes are shared whereas translational coordinates can either be shared or unshared. When elements are rigidly connected to each other, both the translational and rotational nodes are shared, see Fig. 2.1. It can be observed from Table 2.1 that a truss element and a hinge element do not have common nodal types and therefore cannot be connected to each other. pin-joint hinge-joint rigid-joint Figure 2.1. Joint connections between finite elements. In the first block of the kinematics module either two-dimensional (planar) or three-dimensional (spatial) elements can be specified. In the second block the initial configuration of the mechanism is specified. In the third block the coordinates and generalized deformations are divided into four groups, depending on the boundary conditions: 1. 2. 3. 4. fixed prescribed coordinates (supports) dependent, calculable deformations prescribed, time-dependent coordinates dynamic degrees of freedom Section 2.2. Kinematics 17 end node p x ~ end node q x ~ generalized deformations planar beam xp φp xq φq ε1 , ε2 , ε3 PLTRUSS planar truss xp – xq – ε1 PLTOR planar hinge – φp – φq ε1 PLBEAR planar bearing xp φp xq φq ε1 , ε2 , ε3 PLPINBOD planar pinbody xp φp xq – ε1 , ε2 PLRBEAM planar rigid beam xp φp xq – ε1 , ε2 PLWHEEL planar wheel xp φp – φq ε̇1 , ε̇2 PLBELT planar belt (gear) xp φp xq φq ε1 BEAM spatial beam xp λp xq λq ε1 , ε2 , ε3 , ε4 , ε5 , ε6 TRUSS spatial truss xp – xq – ε1 HINGE spatial hinge – λp – λq ε1 , ε2 , ε3 spatial pinbody xp λp xq – ε1 , ε2 , ε3 RBEAM spatial rigid beam xp λp xq – ε1 , ε2 , ε3 WHEEL spatial disk wheel xp λp xq – ε1 , ε2 , ε3 , ε4 , ε̇5 , ε̇6 TWHEEL spatial torus wheel xp λp xq – ε1 , ε2 , ε3 , ε4 , ε̇5 , ε̇6 keyword type PLBEAM PINBODY Table 2.1. Nodes, nodal coordinates and deformation parameters for the planar and spatial truss, beam, bearing, hinge, pinbody, belt (gear) and wheel elements. For the keywords in the third block it is important to remark that there are no keywords to fix a deformation or to release a coordinate. These are the default settings. So a deformation is fixed unless a RLSE, INPUTE or DYNE keyword specifies otherwise. Similarly, a coordinate is calculable unless a FIX, INPUTX or DYNX keyword specifies otherwise. For systems with non-holonomic deformations, dependent coordinates or deformations can be specified as generalized configuration coordinates by the keywords KINX and KINE; these are called the kinematic generalized coordinates and the corresponding velocities are not dynamic degrees of freedom. With the keywords of the fourth optional block, the calculation of some non-linear terms in the expressions for the deformations of planar or spatial beams can be suppressed and geometric properties for PINBODY elements and their cognates (rigid beam, planar pinbody, planar rigid beam) can be specified. The keyword in the fifth section is not really a kinematic keyword as it sets the level of output from the program. 18 Chapter 2. Keywords KEYWORDS KINEMATICS 1 PLBEAM PLTRUSS PLTOR PLBEAR PLPINBOD PLRBEAM PLWHEEL PLBELT BEAM TRUSS HINGE PINBODY RBEAM WHEEL TWHEEL Planar beam element Planar truss element Planar hinge element Planar bearing element (not supported !!) Planar pinbody element Planar rigid beam element Planar wheel element Planar belt (gear) element Beam element Truss element Hinge element Spatial pinbody element Spatial rigid beam element Spatial disk wheel element Spatial torus wheel element X Specification of the initial Cartesian nodal positions FIX RLSE INPUTX INPUTE DYNX DYNE KINX KINE Support coordinates x0 Calculable deformations ec Prescribed DOF xm Prescribed DOF em Dynamic DOF xm Dynamic DOF em Configuration coordinate xk Configuration coordinate ek LDEFORM Suppresses the calculation of non-linear elastic strains of a beam element, due to possibly large curvatures and twists of the elastic line. Defines the orientations of the generalized deformations for the PINBODY elements and cognates. Defines the undeformed reference distances for the PINBODY elements and cognates. 2 3 4 ORPINBOD DRPINBOD 5 OUTLEVEL Sets the level of output generated in the log file and in the SPACAR binary data (sbd) file. The parameters for these keywords are listed below. {∗i} refers to note i listed at the end of the keywords. Section 2.2. Kinematics PLBEAM PLTRUSS PLTOR PLPINBOD PLRBEAM PLWHEEL PLBELT BEAM [ [ TRUSS HINGE 1 2 3 4 5 1 2 3 1 2 3 1 2 3 4 1 2 3 4 1 2 3 4 5 6–7 1 2 3 4 5 6 7 1 2 3 4 5 6–8 6/9 1 2 3 1 2 3 4–6 19 element number first position node first orientation node second position node second orientation node element number first position node second position node element number first orientation node second orientation node element number first position node first orientation node second position node element number first position node first orientation node second position node element number position node first orientation node, yaw angle second orientation node, spin angle wheel radius initial direction of the spin axis, i.e. the y ′ -axis element number first position node first orientation node second position node second orientation node first pulley/base circle radius second pulley/base circle radius element number first position node first orientation node second position node second orientation node initial direction of the principal y ′ -axis of the beam crosssection ] {∗1} torsion–elongation coupling parameter ft ] {∗1} element number first position node second position node element number first orientation node second orientation node initial direction of the x′ -axis of rotation {∗2} 20 Chapter 2. Keywords [ 1 2 3 4 5–7 [ 1 2 3 4 5–7 [ 1 2 3 4 5–7 1 2 3 4 5 6 7–9 1 2 3 4 PINBODY RBEAM WHEEL TWHEEL X element number first position node first orientation node second position node initial direction of the principal y ′ -axis of the beam crosssection ] {∗3} element number first position node first orientation node second position node initial direction of the principal y ′ -axis of the beam crosssection ] {∗1} element number first position node first orientation node second position node initial direction of the spin axis, i.e. the z ′ -axis element number first position node first orientation node second position node wheel radius in equatorial plane transverse wheel radius initial direction of the spin axis, i.e. the z ′ -axis position node number x1 -coordinate x2 -coordinate x3 -coordinate ] {∗4} Section 2.2. Kinematics FIX [ RLSE [ INPUTX [ INPUTE [ DYNX [ DYNE [ KINX [ KINE [ 1 2– 1 2– 1 2– 1 2– 1 2– 1 2– 1 2– 1 2– LDEFORM ORPINBOD 1 1 DRPINBOD 2–10 1 2 3 [ 4 [ 1 2 OUTLEVEL NOTES: 21 node number coordinate number (1, 2, 3 or 4) ] {∗5} element number deformation mode coordinate number (1, {∗6} node number coordinate number (1, 2, 3 or 4) ] {∗5} element number deformation mode coordinate number (1, {∗6} node number coordinate number (1, 2, 3 or 4) ] {∗5} element number deformation mode coordinate number (1, {∗6} node number coordinate number (1, 2, 3 or 4) ] {∗5} element number deformation mode coordinate number (1, {∗6} 2, 3, 4, 5 or 6) ] 2, 3, 4, 5 or 6) ] 2, 3, 4, 5 or 6) ] 2, 3, 4, 5 or 6) ] BEAM element number PINBODY, RBEAM, PLPINBOD or PLRBEAM element number direction vectors {∗7} PINBODY, RBEAM, PLPINBOD or PLRBEAM element number undeformed projection of xq − xp on the first direction vector undeformed projection of xq − xp on the second direction vector undeformed projection of xq − xp on the third direction vector for spatial elements ] level of output in log file {∗8} level of output in the SPACAR binary data (sbd) file] {∗8} 22 Chapter 2. Keywords ∗1 The direction vector lies in the local x′ y ′ -plane of the beam element. If no direction is specified, the local direction vector is chosen as the standard basis vector that makes the largest angle with axis of the beam; in case of a draw, the vector with the highest index is chosen. The torsion–elongation coupling parameter takes into account the shortening of the beam due to torsion, such that for a twisted, axially unloaded beam the axial strain is − 12 ft α2 , where α is the specific twist of the beam. For thin-walled open cross-sections, ft = (Iy′ + Iz′ )/A, but it may have a different value, or even be negative, for solid crosssections. ∗2 The local y ′ and z ′ unit vectors are chosen as follows. First, the standard basis vector with the largest angle with the hinge axis is chosen; in case of a draw, the vector with the highest index is chosen. Then the local y ′ is chosen in the direction of the cross product of the local x′ -direction with this basis vector. The local z ′ -direction is chosen so as to complete an orthogonal right-handed coordinate system. ∗3 If no direction is specified, directions initially aligned with the global coordinate axes are chosen; otherwise the line connecting the translational nodes is chosen as the local x′ direction and the specified vector is in the local x′ y ′ -plane. The directions used are made orthonormal. The directions can also be specified with the keyword ORPINBOD. ∗4 The specification of the initial positions with the keyword X is only required for non-zero position-coordinates. The initial orientations cannot be chosen freely. ∗5 If the keywords INPUTX, DYNX, FIX and KINX are used without an explicit specification of the coordinate, all (independent) coordinates will be marked as degrees of freedom or supports. This means that x1 , x2 (and x3 ) are marked for position nodes and β or λ1 , λ2 and λ3 for orientation nodes. If more than one coordinate is specified, each of the specified coordinates is chosen as a degree of freedom or a support. ∗6 If the keywords INPUTE, DYNE, RLSE and KINE are used without an explicit specification of the deformation mode coordinate, all deformation mode coordinates will be marked as degrees of freedom or released. If more than one deformation mode coordinate is specified, each of the specified coordinates is chosen as a degree of freedom or as released. ∗7 There are four distinct cases, two for the planar elements and two for the spatial elements. For the planar elements, if two numbers are specified, this is the direction of the local x′ -axis and an orthogonal y ′ -direction is found by rotating by a right angle in the positive direction and the directions are normalized; if four numbers are specified, these are taken as the direction vectors in the local x′ - and y ′ -directions as they are. For the spatial elements, if six numbers are specified, these are taken as the direction of the x′ -axis and a direction in the local x′ y ′ -plane, which are made orthonormal and completed by a local z ′ -axis; if nine numbers are specified, these are taken as the three direction vectors as they are. Section 2.2. Kinematics 23 ∗8 Both parameters for the output level are integers of which the values are the sum of the desired outputs. A value of 0 implies the least output; an output level of −1 means maximum output; to obtain multiple outputs, the specified values for the parameters should be added. For the first parameter for the log file are defined: 0 Default: All “normal” output. 1 Additional output of the first order geometric transfer functions in de and dx. 2 Additional output of the second order geometric transfer functions in d2e and d2x for mode=4, 7, 8 and 9. 4 Additional output of the third order geometric transfer functions in d3e and d3x for mode=4, 7, 8 and 9. 8 Additional output of the derivative of the global deformation function for mode=4, 7, 8 and 9. For the second parameter (SPACAR binary data (sbd) file) are defined: 0 Default for all modes except mode=7, 8 and 9: All “normal” output. 1 Default for mode=7, 8 and 9: Additional output of the first order geometric transfer functions in de and dx. 2 Additional output of the second order geometric transfer functions in d2e and d2x. 3 Additional output of the first and second order geometric transfer functions (a combination of 1 and 2). 24 2.3 Chapter 2. Keywords Dynamics With the keywords of the dynamics module the following blocks of information can be supplied. Blocks 1 and 2 are optional. If deformable elements have been defined in the kinematics, block 3 has to be filled, lest the stiffness and damping are zero. If the motion is not prescribed by trajectories, block 4 has to be used to define the input motion. Finally with the keywords from the 5th block miscellaneous settings can be adjusted. KEYWORDS DYNAMICS 1 2 XM EM XGYRO MEE Inertia specification of lumped masses Inertia specification of distributed element masses Inertia specification of gyrostat User-defined mass put into M (e,e) XF External force specification of the mechanism in nodes Specification of MATLAB M-file for user functions with input for forces and stresses USERSIG 3 ESTIFF ESIG EDAMP Specification of elastic constants Specification of preloaded state Specification of viscous damping coefficients TIMESTEP INPUTX INPUTE STARTDX STARTDE USERINP Duration and number of time steps Specification of simple time functions for the prescribed degrees of freedom Specification of initial values for the dynamic degrees of freedom Specification of MATLAB M-file for user functions with input for the degrees of freedom GRAVITY INTEGRAT ERROR ITERSTEP Specification of the gravitational acceleration vector Selection of integrator Specification of error tolerances for the integrator Specification of number of iterations and steps and error tolerance for static calculations in modes 7, 8 and 9 4 5 Section 2.3. Dynamics 25 The parameters required with these keywords are listed below. {∗i} refers to note i listed at the end of the keywords. 1 2 XM [ 3 4 5 6 7 1 2 3 [ 4 [ 5 [ 6 EM 1 2 3 4 XGYRO MEE [ XF [ USERSIG 5 6 1 2 3 4 3/5 1 2 3-5 1 node number concentrated mass for position nodes; rotational inertia I for planar orientation nodes; for spatial orientation nodes, the inertia components Jxx {∗1} Jxy {∗1} Jxz {∗1} Jyy {∗1} Jyz {∗1} Jzz {∗1} element number mass per unit of length rotational inertia Jx′ x′ per unit of length for spatial beam; {∗2} rotational inertia J per unit of length for planar beam; {∗2} angle over which the belt is initially wound over the first pulley for a planar belt ] rotational inertia Jy′ y′ per unit of length for spatial beam; {∗2} angle over which the belt is initially wound over the second pulley for a planar belt ] rotational inertia Jz′ z′ per unit of length for spatial beam ] {∗2} rotational inertia Jy′ z′ per unit of length for spatial beam ] {∗2} node number Ω1 components of absolute angular rotor velocity (free Ω2 rotor motion) or components of constant angular rotor Ω3 velocity relative to the carrier body (prescribed rotor motion) rotor inertia J type of rotor motion (0: free, 1: prescribed) first element number deformation coordinate of first element second element number deformation coordinate of second element ] entry in the mass matrix M (e,e) {∗3} node number forces dual with the 1st nodal coordinate forces dual with the 2nd , 3rd and 4th nodal coordinate ] Name of the MATLAB M-file with user functions with forces and stresses {∗4} 26 Chapter 2. Keywords 1 2 ESTIFF [ 3 [ 4 [ [ [ 5 6 7 1 2– 1 2 [ 3 [ 4 [ 5 ESIG EDAMP element number EA for beam, truss and belt elements S1 = St for hinge elements S1 , first stiffness coefficient for pinbody and cognates GIt for spatial beam EI for planar beam S2 , second stiffness coefficient for pinbody and cognates ] {∗5} EIy′ for spatial beam EI/(GAk) for planar beam S3 , third stiffness coefficient for pinbody and cognates ] {∗5} EIz′ for spatial beam ] {∗5} EIy′ /(GAkz′ ) for spatial beam ] {∗5} EIz′ /(GAky′ ) for spatial beam ] {∗5} element number preloaded generalized stresses {∗5} element number Ed A, longitudinal damping for beam, truss and belt elements Sd1 , torsional damping for hinge elements Sd1 , first damping coefficient for pinbody and cognates Gd It , torsional damping for beam elements Ed I, bending damping for planar beams Sd2 , second damping coefficient for pinbody and cognates ] {∗5} Ed Iy′ , bending damping in y ′ -direction for spatial beams Sd3 , third damping coefficient for pinbody and cognates ] {∗5} Ed Iz′ , bending damping in z ′ -direction for spatial beam ] {∗5} Section 2.3. Dynamics 1 2 1 2 3 4 5 1 2 TIMESTEP INPUTX INPUTE 3 4 5 1 2 3 4 1 2 3 4 1 STARTDX STARTDE USERINP GRAVITY [ INTEGRAT ERROR ITERSTEP 27 1 2 3 1 2 1 2 1 2 3 length of time period number of time steps node number (position or orientation node) {∗6} coordinate number (1, 2, 3 or 4) start value {∗7} start rate acceleration (constant) element number {∗8} deformation mode coordinate number (1, 2, 3, 4, 5 or 6) {∗9} start value {∗10} start rate acceleration (constant) node number coordinate number (1, 2, 3 or 4) start value {∗7} start rate element number deformation mode coordinate number (1, 2, 3, 4, 5 or 6) start value {∗10} start rate Name of the MATLAB M-file with user defined input functions {∗11} x-component of the acceleration of gravity y-component of the acceleration of gravity z-component of the acceleration of gravity ] Specify integrator type {∗12} Step size or initial step size Absolute error for the integrator Relative error for the integrator {∗13} maximal number of iterations in calculating a stationary solution (default value 10) number of load steps (default value 4) error tolerance (default value 5.0E–7) NOTES: ∗1 The inertia components are related to the global coordinate system (x, y, z) in the initial configuration. The tensor components are needed, so Jxy , etc., represent the negative of the products of inertia. ∗2 The distributed moments of inertia are lumped to the orientation nodes of the beam elements. They represent the mass moments of inertia of the cross-section of the beam, so Jx′ y′ and Jx′ z′ are zero. 28 Chapter 2. Keywords ∗3 The keyword MEE is used to add a fixed mass coupled to deformation mode coordinates. If all five numbers are specified, the mass is placed as a coupling between the two deformation mode coordinates; if three numbers are specified, the mass is placed on the diagonal. ∗4 The (required) parameter of the USERSIG keyword is the name of a MATLAB M-file without the extension .m and with a maximum filename length of 8 characters. The calling syntax of the M-script is function [time,sig,f]=pushsig(t,ne,le,e,ep,nx,lnp,x,xp); The input parameters are the time t and a list of variables that store the instantaneous values of the same quantities as are represented by the corresponding variables in the SPACAR binary data, see the overview on page 5. The script should return (again) time t, user defined stresses sig and user defined nodal forces fx. Either sig or fx or both may be empty in the case no stresses and/or forces are prescribed. Otherwise each row in sig and/or fx should define one stress value or force component at the specified time t. Three columns should be provided with 1. The element number (e) or the node number (x). 2. The deformation mode number (e) or the coordinate number (x). 3. The current value of the stress or force component. Two more columns can be provided, which specify the diagonal elements of the stiffness and damping matrices, respectively, coresponding to the stress or force component. ∗5 Unspecified values for the stiffness and damping are assumed to be zero by default. The meaning of the variables is: E, elasticity modulus (Young’s modulus); G = E/(2 + 2ν), shear modulus; ν, Poisson’s ratio; Ed , damping modulus in Kelvin–Voigt model; Gd , shear damping modulus in Kelvin–Voigt model; A, cross-sectional area; I (Iy′ , Iz′ ), second area moment (about y ′ -axis and z ′ -axis); It , Saint-Venant’s torsion constant; k (ky′ and kz′ ), shear correction factor (in y ′ -direction and z ′ -direction). The shear correction factors are about 0.85; a table of values for various cross-sections can be found in [5]. The generalized stresses are calculated according to the Kelvin–Voigt model as follows. All first stresses are calculated as σ1 = S1 ε1 + Sd1 ε̇1 + σ0 , where S1 = EA/l0 and Sd1 = Ed A/l0 for the truss and beam elements, where l0 is the undeformed length of the element, and the first stiffness and damping coefficients as defined in the input for the other types of elements. σ0 is the preload defined by the keyword ESIG. For hinge and pinbody elements, the other stresses are calculated in an analogous way. For a planar beam element, the bending stresses are calculated as " # " S2 σ2 4 + Φ −2 + Φ = σ3 1 + Φ −2 + Φ 4 + Φ #" # " Sd2 ε2 4 + Φ −2 + Φ + ε3 1 + Φ −2 + Φ 4 + Φ #" ε̇2 ε̇3 # , where S2 = EI/l03 , Φ = 12EI/(GAkl02 ) and Sd2 = Ed I/l03 . For a spatial beam element, the torsional stress is calculated as σ2 = S2 ε2 + Sd2 ε̇2 , where S2 = GIt /l03 and Sd2 = Section 2.3. Dynamics 29 Gd It /l03 . For bending along the local y ′ - and z ′ -axes, the stresses are, analogous to the planar case, " # " #" ε̇3 ε̇4 # # " #" ε̇5 ε̇6 # # " #" Sd3 ε3 4 + Φz −2 + Φz + ε4 1 + Φz −2 + Φz 4 + Φz # " #" Sd4 ε5 4 + Φy −2 + Φy + ε6 1 + Φy −2 + Φy 4 + Φy S3 4 + Φz −2 + Φz σ3 = σ4 1 + Φz −2 + Φz 4 + Φz and " S4 4 + Φy −2 + Φy σ5 = σ6 1 + Φy −2 + Φy 4 + Φy , where S3 = EIy′ /l03 , Φz = 12EIy′ /(GAkz′ l02 ), Sd3 = Ed Iy′ /l03 , S4 = EIz′ /l03 , Φy = 12EIz′ /(GAky′ l02 ), and Sd4 = Ed Iz′ /l03 . To all stress components, a preload can be added by the key word ESIG. ∗6 In a mode=7, 8 or 9 run a (deformed) mechanism configuration is computed which corresponds with the specified nodal position. ∗7 The default position start value for INPUTX and STARTDX is the value specified by the kinematic keyword X, which has a default zero. ∗8 Stiffness and damping properties of the corresponding element are not used for the dynamic computations. In a mode=7, 8 or 9 run a (deformed) mechanism configuration is computed which corresponds with the specified element deformation. ∗9 Rotational deformations are defined in radians. ∗10 Note that the keyword X defines an initial configuration in which the deformations are zero. (An exception is an element for which the keyword DRPINBOD has been used.) A start value defined with INPUTE or STARTDE defines a deformation with respect to the initial configuration. ∗11 The (required) parameter of the USERINP keyword is the name of a MATLAB M-file without the extension .m and with a maximum filename length of 8 characters. The calling syntax of the M-script is function [t,e,x]=mymotion(t,is); The input parameters are the time t and time step number is. The script should return (again) time t, prescribed deformations e and prescribed coordinates x. Either e or x may be empty in the case no deformations or coordinates are prescribed. Otherwise each row in e and/or x should define one deformation or coordinate at the specified time t. Five columns should be provided with 1. The element number (e) or the node number (x). 2. The deformation mode number (e) or the coordinate number (x). 3. The current value of the deformation (e) or coordinate (x). 4. The current rate of the deformation (ė) or velocity (ẋ). 30 Chapter 2. Keywords 5. The current acceleration of the deformation (ë) or coordinate (ẍ). The user has to assure the correctness of the derivatives. SPACAR does not carry out any checks, but the results depend heavily on these derivatives. ∗12 Available integrator types are: 0 Default: Shampine–Gordon. 130 Explicit third-order Runge–Kutta, fixed step size. 135 Explicit third-order Runge–Kutta, variable step size. 140 Explicit fourth-order Runge–Kutta, fixed step size. 155 Explicit fifth-order Runge–Kutta, variable step size. 220 Explicit Runge-Kutta for second-order systems, second-order accurate, fixed step size. 225 Explicit Runge–Kutta for second-order systems, second-order accurate, variable step size. 310 Semi-implicit Runge–Kutta–Rosenberg, first-order accurate, fixed step size. 320 Semi-implicit Runge–Kutta–Rosenberg, second-order accurate, fixed step size. 330 Semi-implicit Runge–Kutta–Rosenberg, third-order accurate, fixed step size. 410 Singly diagonally implicit Runge–Kutta (implicit Euler), first-order accurate, fixed step size. 420 Singly diagonally implicit Runge–Kutta, second-order accurate, fixed step size. 430 Singly diagonally implicit Runge–Kutta, third-order accurate, fixed step size. Change this only if you know what you are doing. ∗13 The error tolerances are used for integration methods with a variable step size in the integrators of type 0, 135, 155 and 255. For the integrators of type 410, 420 and 430, the absolute tolerance is used as the tolerance for the modified Newton–Raphson iteration. Section 2.4. Inverse dynamics (setpoint generation) 2.4 31 Inverse dynamics (setpoint generation) For clarity the keywords for the inverse dynamics including the generation of setpoints are discussed in two subsections. In the input file keywords from both subsections must be combined into one part, so there should be no END/HALT pair in between. 2.4.1 Trajectory generation There are three essential keyword blocks: KEYWORDS TRAJECTORY GENERATION 1 TRAJECT Trajectory header; the given trajectory number is valid for all keywords before the next TRAJECT. TROT TRANS Definition of the actual trajectory: the number and type of DOFs determine which keywords and how many of them have to be specified: TROT, TRANS, and TRCIRL for nodes and TRE for elements (maximum of 6). 2 TRCIRL TRE 3 USERTRAJ Trajectory defined by a user function. TRTIME Definition of trajectory time and number of time steps. and there are two blocks of optional keywords: 1 2 TREPMAX TRVMAX TRFRONT Specification of velocity profile: rise time and maximum velocity. Specification of acceleration front for each velocity profile. TRM TRF Specification of extra masses and forces on the end-effector. 32 Chapter 2. Keywords The trajectories can be constructed in two ways: with a user function or with built-in profiles. The latter are defined below and are of course limited to (combinations of) the built-in profiles. On the other hand, practically any input can be generated with user functions. This feature is activated by defining exactly one TRAJECT with the USERTRAJ keyword. The (required) parameter is the name of an MATLAB M-script that is to be called. With TRTIME the total trajectory time and the number of time steps must be specified. The calling syntax of the Mscript is exactly equal to that of the M-script for the USERINP keyword, see page 29. Alternatively, one can use the built-in trajectory profiles. The next scheme shows in more detail the combination possibilities of the setpoint generation keywords. Essential keywords are accompanied by a number of optional keywords placed between brackets. Other optional keywords than those mentioned are not allowed for that specific essential keyword. TRAJECT TRE TROT [TREPMAX TRFRONT] [TRVMAX TRM TRFRONT] TRANS TRCIRL [TRVMAX TRM TRF TRFRONT] [TRVMAX TRM TRF TRFRONT] TRTIME - The way to follow through the scheme is almost fully dictated by the number and type of degrees of freedom. Each trajectory is defined for the same DOF and therefore runs through the same branch of the scheme. Only TRANS and TRCIRL may be changed into one another after each trajectory. Section 2.4. Inverse dynamics (setpoint generation) 33 At this stage it is useful to mention the way in which degrees of freedom are declared: Position and orientation coordinates are declared as DOF by input-command INPUTX node-number component-number Deformation mode coordinates are declared as DOF by input-command INPUTE element-number component-number (INPUTX and INPUTE are “kinematic keywords”, Sect. 2.2). So, degrees of freedom are declared separately. For generation of setpoints in relative coordinates (such as joint angles), each INPUTE in the kinematics input prepares one TRE in the setpoint generation input (only the first relative coordinate per element is allowed as input for the setpoint generation). For the positions and orientations the situation is more complex because a trajectory in two or three dimensions is defined on node level, not on coordinate level. The keywords TROT, TRANS and TRCIRL prescribe the motion of one node: keyword TROT TRANS TRCIRL description rotation about a fixed axis in space translation along a straight line translation along a circle segment node type and type number 2-D orientation 1 3-D orientation 4 2-D position 2 3-D position 3 2-D position 2 3-D position 3 DOF φ φ, h1 , h2 , h3 x1 , x2 x1 , x2 , x3 x1 , x2 x1 , x2 , x3 For the administration of trajectories two numbers are of main importance: the trajectory number and the node or element number. The trajectory number has to be given once after TRAJECT, node numbers or element numbers follow immediately after all other keywords. In this way information about the path, the velocity profile and additional loads can be grouped and worked up by node/element number. Taking as starting point the type of DOF the picture becomes: DOF ELEMENT e1 NODE xi PATH TRE TROT TRANS TRCIRL VELOCITY PROFILE LOADS TREPMAX TRFRONT TRVMAX TRFRONT TRM TRVMAX TRFRONT TRM TRF TRVMAX TRFRONT TRM TRF 34 Chapter 2. Keywords The parameters required with these keywords are listed below. {∗i} refers to note i listed at the end of the keywords. TRAJECT 1 trajectory number TROT 1 2 3 4 5 1 2 3 4 1 2-3 4-5 2-4 5-7 1 2 1 node number (orientation node) total angle in rad h1 -coordinate of fixed rotation axis h2 -coordinate h3 -coordinate ] node number (position node) x1 -coordinate of end position x2 -coordinate x3 -coordinate ] node number (position node) 2D: c1 and c2 coordinates of circle centre point {∗1} 2D: b1 and b2 coordinates of circle end point {∗1} 3D: c1 , c2 and c3 coordinates of circle centre point {∗1} 3D: b1 , b2 and b3 coordinates of circle end point {∗1} element number total displacement (relative angle or elongation) name of M-script {∗2} 1 2 3 total time for the trajectory number of time steps number of intermediate time steps ] {∗3} 1 2 3 1 2 3 1 2 element number rise time (period of acceleration) extreme velocity ] {∗4} node number (position or orientation node) rise time (period of acceleration) extreme value of the velocity ] {∗4} node or element number acceleration front type {∗5} 1 2 3 4 5 6 7 1 2 3 4 node number (position or orientation node) extra mass (m, I or Jxx ) Jxy Jxz Jyy Jyz Jzz ] {∗6} node number (position node) f1 -coordinate of external force f2 -coordinate f3 -coordinate ] [ TRANS [ TRCIRL TRE USERTRAJ TRTIME [ TREPMAX [ TRVMAX [ TRFRONT TRM [ TRF [ Section 2.4. Inverse dynamics (setpoint generation) 35 NOTES: ∗1 The positions of the parameters of keyword TRCIRL are different in 2-D and in 3-D cases. Places 2–5 are used for 2-D, places 2–7 for 3-D. Note that the “endpoint” of the circle cannot be taken literally, as it is over-determined. The second point defines a line through the centre on which the circle ends. ∗2 See the note for the USERINP keyword on page 29. ∗3 The keyword TRTIME has an optional third argument that influences the meaning of the second argument: 2 arguments 3 arguments 1 total trajectory time total trajectory time 2 number of time steps number of time steps for an extended analysis 3 number of time steps within the previous step For three arguments the total number of time steps is a multiplication of the last two arguments. In intermediate points a standard analysis is done. ∗4 The keywords TRVMAX and TREPMAX have an optional third argument to express the extreme velocity (creation of a zero-acceleration period). If no extreme is given it can be calculated from the total time and path length. The second argument contains the risetime. The period of deceleration is calculated from the (a) total time, (b) rise time, (c) total path length, (d) extreme velocity. In this way the velocity profile is fully determined. For asymmetrical velocity profiles the rise time can be calculated too. To indicate the symmetry of the profile the second argument is given a dummy argument: a non-positive value. The default velocity profile is: symmetrical without constant velocity period. ∗5 The keyword TRFRONT has a second argument for the type of acceleration and deceleration function of time. There are three types of fronts: 0 – constant acceleration 1 – sine function (half period) 2 – quadratic sine function (half period) The default velocity front has a constant acceleration (type 0). ∗6 The keyword TRM has only for 3-D orientation nodes a real list of parameters. For 2-D orientation and position nodes one mass parameter is sufficient. In the 3-D case six values determine the symmetric rotational inertia matrix: Jxx Jxy Jxz Jyy Jyz Jzz 123 45 6 36 Chapter 2. Keywords 2.4.2 Nominal inputs (u0 ) and reference outputs (y 0 ) The nominal input vector u0 and the reference output vector y 0 are defined in the following blocks. These blocks are optional, but omitting one or both blocks means that no input and/or output vectors are defined and hence no setpoints for that input and/or output vector are generated and written to the ltv file. KEYWORDS NOMINAL INPUT VECTOR u0 (mode=2) 1 Specification of actuator elements. Specification of actuated nodes. NOMS NOMF KEYWORDS REFERENCE OUTPUT VECTOR y 0 (mode=2) 2 REFE REFEP REFEDP REFX REFXP REFXDP Specification of the deformation parameters to be sensed. The same, first time derivative. The same, second time derivative. Specification of the nodal coordinates to be sensed. The same, first time derivative. The same, second time derivative. The parameters for these keywords are listed below. {∗i} refers to note i listed at the end of the keywords. NOMS NOMF REFE REFEP REFEDP REFX REFXP REFXDP 1 2 3 1 2 3 nominal input number {∗1} element number deformation parameter number (1, 2, 3, 4, 5 or 6) nominal input number {∗1} node number coordinate number (1, 2, 3, or 4) 1 2 3 1 2 3 reference output number {∗1, 2} element number deformation parameter number (1, 2, 3, 4, 5 or 6) reference output number {∗1, 2} node number coordinate number (1, 2, 3, or 4) NOTES: ∗1 The nominal input numbers and reference output numbers are the positions of the specified input or output in the input and output vectors, respectively. Section 2.4. Inverse dynamics (setpoint generation) 37 ∗2 The keywords REFES and REFXS that are defined for the linearization module (Sect. 2.5) are accepted as well and do not give errors. Their meaning and usage is identical to the normal keywords REFE and REFX, respectively. 38 2.5 Chapter 2. Keywords Linearization As mentioned in Sect. 1.2 the module LINEAR is a forward dynamics stage for the generation of linearized equations of motion and state space matrices that can be used in two different modes. mode=4 is basically an extension of the forward dynamic analysis of mode=1. No further keywords are required to obtain the coefficient matrices of the linearized equations as functions of the set of dynamic degrees of freedom q d . These matrices are stored in a SPACAR binary matrix data file with extension sbm. This file can be loaded with the utility loadsbm. If input and output vectors δu and δy are defined, also the linearized state equations and output equations are computed (see mode=9). Linearization in mode=3 is around a predefined nominal trajectory and takes place after that trajectory has been generated in an inverse dynamics run (mode=2). The set of DOFs used in the inverse dynamics computation represent the actuator joint coordinates em . In case of a flexible manipulator mechanism additional DOFs εm i ≡ 0 describing the elastic behaviour of the mechanism links should be included in the dynamic model (both in mode=2 and mode=3). Clearly, the mechanisms used in both runs have to be closely related. If the manipulation task is prescribed in terms of relative DOFs (TRE) the list of keywords is identical with those used in the inverse dynamics run (mode=2). If the manipulation task is prescribed as a motion of some nodal points (triads) (TROT, TRANS, TRCIRL) then the corresponding RLSE command of the actuators should be replaced by INPUTE commands in the kinematic block. In the software some checks are carried out to verify that data from the inverse dynamics run can be reasonably used during the linearization. The nominal input vector u0 and the reference output vector y 0 are again defined in the following blocks. These blocks are optional, but as before omitting one or both blocks means that no input and/or output vectors are defined and hence no state space matrices can be generated and written to the ltv file. The keywords are similar to the input and output keywords in Sect. 2.4.2. In the output ltv file of a mode=3 run the setpoints of the input and output vector are stored in the same way as for a mode=2 run. In addition the state space matrices for the linearized equations of motion (Sect. 1.5) are generated. Obviously, the input matrix B and output matrix C depend on the chosen input and output vectors. In a usual state space system the output vector is computed from a linear expression. In the case a larger accuracy is required, SPACAR can be instructed to use a second order expression. This feature is available for all deformation parameters and coordinates (not for the time derivatives) with the keywords REFES and REFXS. The use of these keywords will generate elements in the output reference vector that are the same as the elements from REFE and REFX, respectively. Also the associated row in the output matrix C is the same, but in addition a tensor denoted G in the ltv file is computed with the second order geometric transfer function. Linearization in mode=7, 8 and 9 is around a pre-computed static equilibrium configuration, or a state of steady motion. In addition in mode=9 the state space matrix A, the input matrices B 0 and B, the output matrix C and the feedthrough matrix D are calculated. Obviously, the matrices B 0 , B, C and D depend on the chosen input and output vectors δu and δy respectively. These vectors are again defined in the blocks on page 40. These blocks are optional, but as before omitting one or both blocks means that no input and/or output vectors are defined and hence no state space matrices can be generated and written to the ltv-file. Section 2.5. Linearization 39 KEYWORDS NOMINAL INPUT VECTOR u0 (mode=3) 1 Specification of actuator elements. Specification of actuated nodes. NOMS NOMF KEYWORDS REFERENCE OUTPUT VECTOR y 0 (mode=3) 2 REFE REFES REFEP REFEDP REFX REFXS REFXP REFXDP Specification of the deformation parameters to be sensed. The same, with second order expression. The same, first time derivative. The same, second time derivative (see note). Specification of the nodal coordinates to be sensed. The same, with second order expression. The same, first time derivative. The same, second time derivative (see note). Note: Specifying second derivatives in the output vector implies an algebraic coupling between input and output, i.e. a non-zero state space matrix D. This is currently not implemented and the keywords REFEDP and REFXDP are ignored for the linearization. The parameters for these keywords are listed below. {∗i} refers to note i listed at the end of the keywords. NOMS NOMF REFE REFES REFEP REFEDP REFX REFXS REFXP REFXDP 1 2 3 1 2 3 nominal input number {∗1} element number deformation parameter number (1, 2, 3, 4, 5 or 6) nominal input number {∗1} node number coordinate number (1, 2, 3, or 4) 1 2 3 reference output number {∗1} element number deformation parameter number (1, 2, 3, 4, 5 or 6) 1 2 3 reference output number {∗1} node number coordinate number (1, 2, 3, or 4) NOTES: ∗1 The nominal input numbers and output numbers are the positions of the specified input or output in the input and output vectors, respectively. 40 Chapter 2. Keywords KEYWORDS INPUT VECTOR δu (mode=4,9) 1 INPUTS INPUTF INE INEP INEDP INX INXP INXDP Specification of input stresses. Specification of input forces. Specification of input deformation parameters. The same, first time derivative. The same, second time derivative. Specification of input nodal coordinates. The same, first time derivative. The same, second time derivative. KEYWORDS OUTPUT VECTOR δy (mode=4,9) 2 OUTS OUTF OUTE OUTEP OUTEDP OUTX OUTXP OUTXDP Specification of output stresses. Specification of output forces. Specification of output deformation parameters. The same, first time derivative. The same, second time derivative. Specification of output nodal coordinates. The same, first time derivative. The same, second time derivative (see note). The parameters for these keywords are listed below. {∗i} refers to note i listed at the end of the keywords. INPUTS {∗2} INE INEP {∗3} INEDP INPUTF {∗4} INX INXP {∗5} INXDP OUTS {∗6} OUTE OUTEP {∗7} OUTEDP OUTF {∗8} OUTX OUTXP {∗9} OUTXDP 1 2 3 1 2 3 1 2 3 1 2 3 input number {∗1} element number deformation parameter number (1, 2, 3, 4, 5 or 6) input number {∗1} node number coordinate number (1, 2, 3, or 4) output number {∗1} element number deformation parameter number (1, 2, 3, 4, 5 or 6) output number {∗1} node number coordinate number (1, 2, 3, or 4) Section 2.5. Linearization 41 NOTES: ∗1 The input numbers and output numbers are the positions of the specified inputs or outputs in the input and output vectors, respectively. ∗2 Associated with dynamic DOFs e(m,d) or dependent coordinates e(c) . ∗3 Associated with prescribed deformations e(m,r) . For INE, only holonomic deformations are allowed. ∗4 Associated with calculable coordinates x(c) or dynamic DOFs x(m,d) . ∗5 Associated with prescribed nodal coordinates x(m,r) . ∗6 Can be associated with prescribed deformations e(o) or e(m,r) , but can also be associated with the free types, in which case the output stress is calculated from the constitutive equations and, possibly, from the input stress. ∗7 Associated with calculable deformations e(c) or dynamic DOFs e(m,d) or prescibed deformations e(m,r) . ∗8 Associated with prescribed nodal coordinates x(o) or x(m,r) . ∗9 Associated with calculable coordinates x(c) or dynamic DOFs x(m,d) or prescribed coordinates x(m,r) . 42 2.6 Chapter 2. Keywords Non-linear simulation of manipulator control To simulate the behaviour of a manipulator with a control system the SPACAR program is also accessible as an “S-function” block SPASIM from SIMULINK. SIMULINK treats this block like a non-linear state-space system which has a state vector z, an input vector u and an output vector y. Each of these vectors has a well-defined meaning in the SPACAR block: the states correspond to the degrees of freedom and their first time derivatives. The input and output are coupled to actuators and coordinates as specified by keywords in the SPACAR input data file (see below). In the SIMULINK graphical user interface the input and output vectors must be coupled to other blocks, e.g. the control system. The states are used internally in SIMULINK and are usually not available to the user. That implies that any coordinate or deformation parameter that is used for control purposes or is monitored in a graph must be included in the output vector y (block 2). KEYWORDS INPUT VECTOR u (SPASIM) 1 INPUTS INPUTF Specification of actuator elements. Specification of actuated nodes. KEYWORDS OUTPUT VECTOR y (SPASIM) 2 OUTE OUTEP OUTEDP OUTX OUTXP OUTXDP Specification of the deformation parameters to be sensed. The same, first time derivative. The same, second time derivative. Specification of the nodal coordinates to be sensed. The same, first time derivative. The same, second time derivative. Section 2.6. Non-linear simulation of manipulator control 43 The parameters for these keywords are listed below. {∗i} refers to note i listed at the end of the keywords. INPUTS INPUTF OUTE OUTEP OUTEDP OUTX OUTXP OUTXDP 1 2 3 1 2 3 input number {∗1} element number deformation parameter number (1, 2, 3, 4, 5 or 6) input number {∗1} node number coordinate number (1, 2, 3, or 4) 1 2 3 1 2 3 output number {∗1} element number deformation parameter number (1, 2, 3, 4, 5 or 6) output number {∗1} node number coordinate number (1, 2, 3, or 4) NOTES: ∗1 The input numbers and output numbers are the positions of the specified input or output in the input and output vectors, respectively. They need not be identical to the nominal input vector and reference output vector specified during the generation of setpoints (see Sect. 2.4.2 and/or Sect. 2.5), but for a quite straightforward comparison it is convenient to use, at least partially, the same numbering scheme. 44 2.7 Chapter 2. Keywords Visualization and animation To adjust the default settings of SPAVISUAL the user can type VISUALIZATION after the last two END commands in the .dat file. All commands after the command VISUALIZATION are read by SPAVISUAL as an adjustment on the default settings. Adjusts the height and the width of a beam element. Adjusts the radius and the length of the hinge element. Sets the visualization of the truss element on or off. Adjusts the transparency of the elements. Selects the vibration modes. Selects the buckling modes. Sets the amplitude of the vibration or buckling modes. Sets recordmovie on or off, the movie is saved as an .avi file in the workspace. MOVIENAME Sets the name of the recorded movie. UNDEFORMED Sets the visualization of the undeformed mechanism on or off. VIBREND Sets the period of the sine function for the vibration mode. STEPLINE Sets the size of the line elements that are used to draw the elements. STEPVIBRATION Sets the number of steps in the vibration visualization. LIGHT Sets the light on or off. JOINTS Sets the joints on or off. TRAJECTVIS Sets the trajectory on or off. TRAJECTNODE Selects the node for the trajectory. BEAMVIS HINGEVIS TRUSSVIS TRANSPARENCY VIBRATIONMODE BUCKLINGMODE ENLARGEFACTOR RECORDMOVIE Section 2.7. Visualization and animation 45 The parameters for these keywords are listed below. KEYWORD DESCRIPTION BEAMVIS {∗1} size of all beam elements in the local y’ direction size of all beam elements in the local z’ direction element number size of the element for the local y’ direction size of the element for the local z’ direction element number length of the hinge radius of the hinge sets the visualization of the truss elements on (1) or off (0) adjusts the transparency between (0) and (1) switch between the 10 lowest vibration modes {∗2} switch between the 10 buckling modes {∗2} amplitude of the vibration or buckling modes sets record movie on (1) or off (0) filename for the recorded movie sets the visualization of the initial (un-deformed) mechanism configuration on (1) or off (0) sets the period of the sin function for the vibration mode sets the size of the line elements that are used to draw the elements sets the number of intermediate steps in the vibration visualization sets the light source on (1) or off (0) sets the joints on (1) or off (0) sets the trajectory on (1) or off (0) selects the node for the trajectory 1 2 BEAMVIS {∗1} 1 2 3 HINGEVIS 1 2 3 TRUSSVIS 1 TRANSPARENCY 1 VIBRATIONMODE 1 BUCKLINGMODE 1 ENLARGEFACTOR 1 RECORDMOVIE 1 MOVIENAME 1 UNDEFORMED 1 VIBREND 1 STEPLINE 1 STEPVIBRATION 1 LIGHT JOINTS TRAJECTVIS TRAJETCNODE 1 1 1 1 DEFAULT SETTINGS 0.006 0.006 0.006 0.006 0.003 0.009 on 1 1 1 1 off filename on 2π {∗3} 0.2 π 10 {∗3} off on off 1 Notes ∗1 The BEAMVIS command has two variations. The one with only two parameters adjusts all beam elements. The variant with three parameters can be used to adjust only a single beam element. ∗2 Only the lowest vibration and buckling modes are available with a maximum of 10 modes. ∗3 Only numerical values are allowed, no symbols or functions. 46 Chapter 2. Keywords 3 Examples The data files used to run the examples in this chapter can be downloaded from the SPACAR web site, see Appendix A. 3.1 Planar sliding bar In example 4.3.1 of the lecture notes [1] the sliding bar of Fig. 3.1 is described. A rigid bar pq of length 2 m is suspended from two sliders. The bar is driven by the condition xp − vt = 0, where v =| v | is the constant horizontal velocity component of point p. Thus ẋp = v and √ p q q ẍ = 0. We want to compute ẏ and ÿ for 0 ≤ t ≤ 2 3 s and v = 1 m/s. q √ The position y q can be computed easily from the symbolic expression y q = 4 − ( 3 − xp )2 , so q y = q √ 4 − ( 3 − t)2 . y q β p β = 30o 1 v x √ 3 Figure 3.1. Sliding bar. 47 48 Chapter 3. Examples Differentiating once and twice with respect to the time t yields √ 4 − 3+t q , ÿ q = . ẏ = − q q √ √ √ 1 + 2 3t − t2 1 + 2 3t − t2 −1 − 2 3t + t2 The mechanism has one degree of freedom and there is only one element. This is the planar truss element denoted by 1 that connects nodal points 1 and 2 in the following SPACAR input file (slider.dat): PLTRUSS 1 1 2 X X 1 2 0. 0. 1.7321 1. FIX FIX 1 2 2 1 INPUTX 1 1 INPUTX 1 1 TIMESTEP 3.4641 END HALT 0. 1. 0. 100 END END Both symbolic and numeric results are shown in Figs. 3.2 and 3.3 with the Matlab commands >> t=time; 2 0 1.5 −0.5 −1 1 −1.5 2 (d/dt) y(2) [m/s ] 0 −2 2 (d/dt) y(2) [m/s] 0.5 −2.5 −0.5 −3 −1 −3.5 −1.5 −2 −4 0 0.5 1 1.5 2 2.5 3 3.5 time [s] Figure 3.2. Vertical velocity ẏ q of the sliding bar. −4.5 0 0.5 1 1.5 2 2.5 3 3.5 time [s] Figure 3.3. Acceleration ÿ q of the sliding bar. Section 3.2. Planar slider–crank mechanism 49 >> plot(t,[xd(:,lnp(2,2)) ... -(-3ˆ(1/2)+t)./(1+2*3ˆ(1/2)*t-t.ˆ2).ˆ(1/2)]) >> grid >> xlabel(’time [s]’) >> ylabel(’(d/dt) y(2) [m/s]’) >> figure >> plot(t,[xdd(:,lnp(2,2)) ... 4./(-1-2*3ˆ(1/2)*t+t.ˆ2)./(1+2*3ˆ(1/2)*t-t.ˆ2).ˆ(1/2)]) >> grid >> xlabel(’time [s]’) >> ylabel(’(d/dt)ˆ2 y(2) [m/sˆ2]’) Obviously, in both graphs the symbolic and numeric results are practically identical, which illustrates the good agreement between both solutions. Note that in this example no masses are defined. There are no dynamic degrees of freedom either, so effectively only a kinematic problem is solved. 3.2 Planar slider–crank mechanism The slider–crank mechanism is frequently applied as a subsystem in the design of a mechanism. It finds its applications in combustion engines, compressors and regulators. Figure 3.4 presents a slider–crank mechanism for which three dynamics computations have to be carried out. In the first problem (case 1, see also example 5.7.2 in the lecture notes [1]), the crank and the connecting rod are assumed to be rigid. In the second computation (case 2), the connecting rod is shorter but still somewhat longer than the crank. In case 3, the flexibility of the connecting rod with the dimensions of case 1 is taken into account; see also example 8.3.1 in the lecture notes [1]. Case 1 First of all, the nodal coordinates must be specified. In the initial configuration, the crank and the connecting rod are horizontal. The crank length is 0.15 m, the length of the connecting rod y x x ~3, 4 , 5 B 1m 2m ω0 x x ~1, 2 ~6, 7 A x C Figure 3.4. Planar slider–crank mechanism. 50 Chapter 3. Examples is 0.30 m. For the dynamic analysis the following parameters are needed. The connecting rod has a circular cross-section with diameter d = 6 mm. The mass density is ρ = 7.87 · 103 kg/m3 and the Young’s modulus is E = 2.1 · 1011 N/m2 . Consequently, the mass per unit length is 0.2225 kg/m and its total mass ms = 0.06675 kg. The mass of the sliding block or plunger C is given by mC = 21 ms = 0.033375 kg. The crank is driven at a constant angular velocity ω0 = 150 rad/s. The total simulation should comprise two crank rotations. Node B must be defined as a single translational node and a double rotational node, since the rotations of the slider and the crank are not the same. The mass of the crank is taken as zero. An input file (crank.dat) describing this case is: PLBEAM PLBEAM 1 2 X X X 1 3 6 FIX FIX INPUTX 1 6 2 1 3 2 5 3 6 0.00 0.15 0.45 4 7 0. 0. 0. 2 1 END HALT XM EM 6 2 INPUTX 2 TIMESTEP 0.033375 0.2225 1 0. 0.1 150. 100 0. END END The initial configuration of case 1 is depicted in Fig. 3.5. The horizontal position, velocity and acceleration of the sliding block as function of time are given in Figs. 3.6–3.8. The driving moment in node 2 versus time is shown in Fig. 3.9 and the supporting forces acting on the sliding block are presented in Fig. 3.10. The MATLAB commands used to plot these results are: >> >> >> >> >> >> >> >> >> plot(time,x(:,lnp(6,1))) grid xlabel(’time [s]’) ylabel(’x(6) [m]’) plot(time,xd(:,lnp(6,1))) grid xlabel(’time [s]’) ylabel(’(d/dt) x(6) [m/s]’) Section 3.2. Planar slider–crank mechanism 51 0.5 0.2 0.45 0.15 0.4 0.1 0.35 1 0 1 3 2 6 x(6) [m] 0.05 0.3 0.25 -0.05 0.2 -0.1 0.15 -0.15 0.1 -0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.01 0.02 0.03 0.04 0.45 Figure 3.5. Case 1: Initial configuration of the slider–crank mechanism. 0.05 time [s] 0.06 0.07 0.08 0.09 0.1 Figure 3.6. Case 1: Horizontal position of the sliding block. 30 3000 2000 20 1000 0 (d/dt)2 x(6) [m/s2] (d/dt) x(6) [m/s] 10 0 −10 −1000 −2000 −3000 −4000 −20 −5000 −30 0 0.01 0.02 0.03 0.04 0.05 time [s] 0.06 0.07 0.08 0.09 0.1 Figure 3.7. Case 1: Horizontal velocity of the sliding block. −6000 0 0.01 0.02 0.03 0.04 0.05 time [s] 0.06 0.07 0.08 0.09 0.1 Figure 3.8. Case 1: Horizontal acceleration of the sliding block. 40 100 80 30 60 20 40 Fx(6), Fy(6) [N] M(2) [Nm] 10 0 −10 20 0 −20 −40 −20 −60 −30 −40 −80 0 0.01 0.02 0.03 0.04 0.05 time [s] 0.06 0.07 0.08 0.09 0.1 Figure 3.9. Case 1: Driving moment in rotational node 2. −100 0 0.01 0.02 0.03 0.04 0.05 time [s] 0.06 0.07 0.08 0.09 0.1 Figure 3.10. Case 1: Supporting forces on the sliding block. 52 Chapter 3. Examples >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> plot(time,xdd(:,lnp(6,1))) grid xlabel(’time [s]’) ylabel(’(d/dt)ˆ2 x(6) [m/sˆ2]’) plot(time,fxtot(:,lnp(2,1))) grid xlabel(’time [s]’) ylabel(’M(2) [Nm]’) plot(time,fxtot(:,lnp(6,1:2))) grid xlabel(’time [s]’) ylabel(’Fx(6), Fy(6) [N]’) Case 2 The input file of case 1 (page 50) is modified to account for the shortened connecting rod. Only the initial position of node 6 in the second block of the kinematic definition has to be changed: X 6 0.35 0. The initial configuration of case 2 is depicted in Fig. 3.11. The horizontal position, velocity and acceleration of the sliding block as a function of time are given in Figs. 3.12–3.14. The driving moment in node 2 versus time is shown in Fig. 3.15 and the supporting forces acting on the sliding block are presented in Fig. 3.16. The MATLAB commands used to plot these results are the same as in case 1 (page 50). Case 3 To take the flexibility of the connecting rod into account with a reasonable accuracy the planar beam element used for this rod (see Fig. 3.4) is split into two parts. One translational node and one rotational node are inserted and the numbers of the nodes in the sliding block C are changed. The bending stiffness of the connecting rod is computed using the moment of inertia I = πd4 /64. The input file (crankfl.dat) is now: Section 3.2. Planar slider–crank mechanism 53 0.4 0.15 0.35 0.1 0.3 0.05 1 0 1 3 2 6 x(6) [m] 0.25 0.2 0.15 -0.05 0.1 -0.1 0.05 0 -0.15 0 0.05 0.1 0.15 0.2 0.25 0.3 30 6000 20 4000 10 2000 0 −2000 −20 −4000 0.01 0.02 0.03 0.04 0.05 time [s] 0.06 0.07 0.08 0.09 0.1 −6000 40 200 30 150 20 100 10 50 0 −50 −20 −100 −30 −150 0.01 0.02 0.03 0.04 0.05 time [s] 0.06 0.07 0.08 0.04 0.09 0.1 Figure 3.15. Case 2: Driving moment in rotational node 2. 0.05 time [s] 0.06 0.07 0.08 0.09 0.1 0 0.01 0.02 0.03 0.04 0.05 time [s] 0.06 0.07 0.08 0.09 0.1 0 −10 0 0.03 Figure 3.14. Case 2: Horizontal acceleration of the sliding block. Fx(6), Fy(6) [N] M(2) [Nm] Figure 3.13. Case 2: Horizontal velocity of the sliding block. −40 0.02 0 −10 0 0.01 Figure 3.12. Case 2: Horizontal position of the sliding block. (d/dt)2 x(6) [m/s2] (d/dt) x(6) [m/s] Figure 3.11. Case 2: Initial configuration of the slider–crank mechanism. −30 0 0.35 −200 0 0.01 0.02 0.03 0.04 0.05 time [s] 0.06 0.07 0.08 0.09 0.1 Figure 3.16. Case 2: Supporting forces on the sliding block. 54 Chapter 3. Examples PLBEAM PLBEAM PLBEAM 1 2 3 X X X X 1 3 6 8 FIX FIX INPUTX DYNE DYNE 1 8 2 2 3 1 3 6 2 5 7 3 6 8 4 7 9 0.000 0.150 0.300 0.450 2 1 2 2 0.000 0.000 0.000 0.000 3 3 END HALT XM EM EM 8 2 3 0.033375 0.2225 0.2225 ESTIFF ESTIFF 2 3 0.000000 0.000000 INPUTX TIMESTEP STARTDE STARTDE STARTDE STARTDE 2 1 0.000000 0.100000 100 2 0.000000 3 0.000000 2 0.000000 3 0.000000 2 2 3 3 13.359623 13.359623 150.000000 0.000000 0.000000 0.000000 0.000000 0.000000 END END The second-order contributions of the bending deformations on the elongation (Eq. (6.4.22) in the lecture notes) are taken into account. The initial configuration of case 3 is depicted in Fig. 3.17. The horizontal acceleration of the sliding block as function of time is given in Fig. 3.18. The bending of the slider, given by (2) (3) v = 21 (ε3 + ε2 ), as function of the crank angle φ2 , is presented in Fig. 3.19. The MATLAB commands used to plot these results are: >> >> >> >> >> plot(time,xdd(:,lnp(8,1))) grid xlabel(’time [s]’) ylabel(’(d/dt)ˆ2 x(8) [m/sˆ2]’) Section 3.2. Planar slider–crank mechanism 55 3000 0.2 2000 0.15 1000 0.1 1 0 1 3 2 6 3 8 (d/dt)2 x(8) [m/s2] 0 0.05 −1000 −2000 -0.05 −3000 -0.1 −4000 −5000 -0.15 −6000 -0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Figure 3.17. Case 3: Initial configuration of the slider–crank mechanism. 0.01 0.02 0.03 0.04 0.05 time [s] 0.06 0.07 0.08 0.09 0.1 Figure 3.18. Case 3: Horizontal acceleration of the sliding block. −3 5 0 0.45 −5 x 10 2 x 10 0 4 −2 3 x(8)−x0(6) [m] v=(eps(2,3)+eps(3,2))/2 [m] −4 2 1 0 −6 −8 −10 −1 −12 −2 −14 −3 −4 −16 0 2 4 6 8 phi(2) [rad] 10 12 14 16 Figure 3.19. Case 3: Bending of the flexible connecting rod (elements 2 and 3). >> >> >> >> −18 0 0.01 0.02 0.03 0.04 0.05 time [s] 0.06 0.07 0.08 0.09 0.1 Figure 3.20. Case 3: Difference in the horizontal position of the sliding block compared to case 1. plot(x(:,lnp(2,1)),(e(:,le(2,3))+e(:,le(3,2)))/2) grid xlabel(’phi(2) [rad]’) ylabel(’v=(eps(2,3)+eps(3,2))/2 [m]’) Figure 3.20 shows the (small) vibration of the sliding block due to the bending by comparing its position with the rigid simulation of case 1 (Fig. 3.6). 56 Chapter 3. Examples Nodal points for the planar slider-crank mechanism node 1 node 2 node 3 node 4,5,7 node 6 node 8 node 9 node type T R T R T T R x-coordinate 0 0.15 0.45 0.30 y-coordinate 0 0 0 0 BC-type x 1 2 2 2 BC-type y 1 2 1 2 BC-type φ 3 2 2 φ0 0 ω0 = φ̇ 150 ω̇ 0 forces/moment 0 0 0 0 0 0 0 mass/inertia 0 0 0 0 0.033 0 0 T=translational, R=rotational, BC=boundary condition The numbers of the BC-type refers to the numbers of the groups mentioned on page 16. Elements for the planar slider-crank mechanism element 1 element 2 element 2a, 3 element type beam beam beam T-nodes 1, 3 3, 6 3, 8 / 8, 6 R-nodes 2, 4 5, 7 5, 9 / 9, 7 type e1 1 1 1 type e2 1 1 4 (e2 )0 0 (ė2 )0 0 type e3 1 1 4 mass per length 0.2225 0.2225 0.2225 EA 5.65 · 106 5.65 · 106 5.65 · 106 EI 13.4 13.4 13.4 damping 0 0 0 T=translational, R=rotational Section 3.3. Cardan-joint mechanism 3.3 57 Cardan-joint mechanism In section 11.1 of the lecture notes [1] a cardan joint is described. Cardan joints, also known as Hooke’s joints, have been used as a shaft coupling in a wide range of machinery, which includes locomotive as well as automotive drive lines. A drive line connected by a Cardan joint may exhibit torsional oscillations due to fluctuating angular velocity ratios inherent in such systems. (4) e1 (1) e1 (3) e1 (2) e1 (a) (b) Figure 3.21. Schematic of Cardan joint system. Figure 3.21a shows a one-degree of freedom shaft system incorporating a Cardan joint. The Cardan joint is modelled by four spatial hinge elements as shown in Figure 3.21b. The rotating shaft axes having an angular misalignment of β = 45◦ is driven at a constant angular speed ė(1) = Ωin . The quantities e(1) and e(4) represent the input and output angles of the hinge elements 1mand 4m , respectively. The essential behaviour of the joint can be simulated with the following input file (cardansimp.dat): HINGE HINGE HINGE HINGE 1 2 3 4 1 2 3 4 FIX FIX INPUTE RLSE RLSE RLSE 1 5 1 2 3 4 1 1 1 1 END -1. 0. 0. 0. -1. 0. 0. 0. -1. 0.707 -0.707 0. HALT INPUTE TIMESTEP END 2 3 4 5 END 1 1 0. 1.0 100 6.28 0. 58 Chapter 3. Examples Note that in the initial configuration, the input shaft is rotated by a right angle with respect to the configuration in Figure 3.21. However, the visualization of this simulation is quite poor. This can be improved by adding some beams to the input and output rotational nodes numbers 2 and 4, respectively. The complete input file (cardan.dat) becomes: HINGE HINGE HINGE HINGE 1 2 3 4 1 2 3 4 2 3 4 5 BEAM BEAM BEAM BEAM 5 6 7 8 6 7 6 11 2 8 4 12 FIX FIX FIX 1 5 6 X X X X X 6 7 9 11 13 0. 1. 1. -0.707 -0.707 INPUTE 1 1 RLSE RLSE RLSE 2 3 4 1 1 1 INPUTE 1 1 TIMESTEP 1.0 100 -1. 0. 0. 0. -1. 0. 0. 0. -1. 0.707 -0.707 0. 7 9 11 13 8 10 12 14 0. 0. 0. 0.707 0.707 0. 0. 0.707 0.707 1. 1. 0.707 0.707 0. 0. 0. 0. 0. 0. 0.15 0. 0.15 END HALT 0. 6.28 0. END END The initial configuration of this mechanism is shown in Fig. 3.22. Figures 3.23, 3.24 and 3.25 (1) show the zeroth, first and second order geometric transfer functions from input e1 to output (4) e1 , respectively. The MATLAB commands to plot these data are: >> plot(e(:,le(1,1)),e(:,le(1,1)), e(:,le(1,1)),e(:,le(4,1))) Section 3.3. Cardan-joint mechanism 59 6 5 [rad] 4 e (1) 4 −0.8 −0.6 3 −0.4 0.1 −0.2 0 −0.1 2 0 0.2 0 1 0.4 0.2 0.6 0.4 0.8 0.6 0 1 0.8 0 1 2 3 4 5 6 e(1) [rad] 1 Figure 3.22. Initial configuration of the cardan joint. Figure 3.23. Zeroth order geometric transfer function for the cardan joint. 0.8 1.4 0.6 1.3 0.4 1.2 0.2 (1) (1) 1 DF(e1 ) D F(e ) 1.5 0 2 1.1 1 −0.2 0.9 −0.4 0.8 −0.6 0.7 0 1 2 3 4 5 6 e(1) [rad] 1 Figure 3.24. First order geometric transfer function for the cardan joint. >> >> >> >> >> >> >> >> >> >> >> >> >> −0.8 0 1 2 3 4 5 6 e(1) [rad] 1 Figure 3.25. Second order geometric transfer function for the cardan joint. grid xlabel(’e_1ˆ{(1)} [rad]’) ylabel(’e_4ˆ{(1)} [rad]’) plot(e(:,le(1,1)),ed(:,le(4,1))./ed(:,le(1,1))) grid xlabel(’e_1ˆ{(1)} [rad]’) ylabel(’DF(e_1ˆ{(1)})’) plot(e(:,le(1,1)),edd(:,le(4,1))./(ed(:,le(1,1)).ˆ2)) grid xlabel(’e_1ˆ{(1)} [rad]’) ylabel(’Dˆ2F(e_1ˆ{(1)})’) 60 Chapter 3. Examples 3.4 Planar four-bar mechanism In examples 5.7.1 and 12.4.1 of the lecture notes [1] the planar four-bar mechanism of Fig. 3.26 is analysed analytically. The mechanism has one degree of freedom. The mechanism is modelled by four rigid truss elements, denoted by 1, 2, 4 and 5, which are joined together at their nodal points to form a rhombus. As Fig. 3.26 implies, these four bars are set at right angles y g m ~4 4m 5m 3m k ~2 ~3 1m 2m ~1 x Figure 3.26. Four-bar mechanism. to one another. The diagonal element 3 represents a spring with stiffness k = EA/l0 . A concentrated mass m is attached to node 4. The deformation parameter e3 has been chosen as the generalized coordinate. The equation of motion is √ më3 + 2m(ė3 )2 + ke3 = mg . (3.1) Using the coefficient matrices from the lecture notes, the linearized equation of motion is √ √ √ mδë3 + 2 2mė3 δ ė3 + (k − 2mg + 2 2më3 + 5m(ė3 )2 )δe3 = 0 . (3.2) These results can also be obtained numerically from a SPACAR analysis. E.g. with numerical values for m = 1, g = 10 and k = √ 1 and initial conditions e3 = 0 and ė3 = 1 the acceleration is according to Eq. (3.1) ë3 = 10 − 2 = 8.59. A SPACAR input file (fourbar.dat) for this case is: PLTRUSS PLTRUSS PLTRUSS PLTRUSS PLTRUSS 1 2 3 4 5 X 1 1 1 2 2 3 2 3 3 4 4 0. 0. Section 3.4. Planar four-bar mechanism X X X 2 3 4 FIX FIX DYNE 1 4 3 -0.7071 0.7071 0. 61 0.7071 0.7071 1.4142 1 1 END HALT XM XF ESTIFF STARTDE 4 4 3 3 1 1. 0. -10. 1.4142 0. 1. END END In a MATLAB session we get (the literal text of the session is modified somewhat to get a more compact presentation): >> spacar(1,’fourbar’) >> e(le(3,1)) ans = 0 >> ed(le(3,1)) ans = 1 >> edd(le(3,1)) ans = 8.5858 Substituting the numerical values of the parameters into the linearized equation of motion Eq. (3.2) gives √ √ √ √ δë3 + 2 2δ ė3 + (1 − 10 2 + 2 2(10 − 2) + 5)δe3 = 0 , (3.3) or δë3 + 2.83 δ ė3 + 16.14 δe3 = 0 . (3.4) The stiffness term is a combination of K0 = √ k=1 G0 = √2ke3 = 0 N 0 = 2g + ė3 )2 − k/m e3 = 15.14 where the solution of Eq. 3.1 √ ë3 = g − 2(ė3 )2 − k/m e3 has been used. In a MATLAB session we get: (3.5) (3.6) 62 Chapter 3. Examples >> spacar(4,’fourbar’) >> m0 m0 = 1.0000 >> c0 c0 = 2.8285 >> k0 k0 = 1 >> n0 n0 = 15.1423 >> g0 g0 = 0 Section 3.5. Rotating mass–spring system 3.5 63 Rotating mass–spring system Figure 3.27. Rotating mass–spring system. Consider the system shown in Fig. 3.27. A smooth horizontal tube containing masses m1 and m2 connected with springs k1 = EA1 /l1 and k2 = EA2 /l2 is mounted on a rotating shaft. The shaft rotates at constant angular speed φ̇. The unstretched lengths of the springs are denoted by l1 and l2 . The equations of motion in terms of the generalized coordinates r1 and r2 are " m1 0 0 m2 #" # " r̈1 m1 φ̇2 r1 − k1 (r1 − l1 ) + k2 (r2 − r1 − l2 ) = r̈2 m2 φ̇2 r2 − k2 (r2 − r1 − l2 ) # (3.7) The stationary solution (r01 , r02 ) is obtained by substituting ṙ1 = ṙ2 = r̈1 = r̈2 = 0 " k1 + k2 − m1 φ̇2 −k2 −k2 k2 − m2 φ̇2 #" # " # k l − k2 l2 r01 , = 11 k2 l2 r02 (3.8) from which the stationary configuration (r01 , r02 ) is obtained analytically as r01 = −(m2 k1 l1 − m2 k2 l2 )φ̇2 + k1 k2 l1 m1 m2 φ̇4 − (k2 m2 + k2 m1 + k1 m2 )φ̇2 + k1 k2 (3.9) r02 = −m1 φ̇2 k2 l2 + k1 k2 (l1 + l2 ) m1 m2 φ̇4 − (k2 m2 + k2 m1 + k1 m2 )φ̇2 + k1 k2 (3.10) This result can also be obtained numerically from a SPACAR analysis. E.g. with the following numerical values: l1 l2 m1 m2 = = = = 0.10 m 0.15 m 0.80 kg 0.50 kg k1 k2 φ̇ = 1.3 kN/m = 0.7 kN/m = 10 rad/s A SPACAR input file (massspring.dat) describing this case is : 64 Chapter 3. Examples PLBEAM 1 1 2 3 4 PLBEAM 2 3 4 5 6 PLTRUSS 3 1 5 X 1 0. 0. X 3 0.1 0. X 5 0.25 0. FIX 1 INPUTX 2 1 DYNE 1 1 DYNE 3 1 RLSE 2 1 END HALT XM 2 1. XM 3 0.8 XM 5 0.5 ESTIFF 1 130. ESTIFF 2 105. INPUTX 2 1 0.0 10.0 END END In a MATLAB session we find for the stationary configuration (r01 ) and (r02 ) in agreement with Eqs. (3.9) and (3.10): >> spacar(7,’massspring’) >> x(lnp(3,1)) ans = 0.1184 >> x(lnp(5,1)) ans = 0.2891 The linearized equations of motion in terms of the dynamic degrees of freedom are: " m1 0 0 m2 #" # " k + k2 − m1 φ̇2 −k2 δr̈1 + 1 δr̈2 −k2 k2 − m2 φ̇2 #" # " # 0 δr1 = 0 δr2 (3.11) Section 3.5. Rotating mass–spring system 65 The associated frequency equation is given by: det − ωi 2 " # " k + k2 − m1 φ̇2 −k2 m1 0 + 1 0 m2 −k2 k2 − m2 φ̇2 #! = 0, (3.12) where the quantities ωi are the natural frequencies of the system. In a MATLAB session we obtain: >> spacar(7,’massspring’) >> m0 m0 = 0.8000 0 0 0.5000 >> k0 k0 = 2000 -700 -700 700 >> n0 n0 = -80.0000 0 -0.0000 -50.0000 The complex eigenvalues and associated eigenvectors can be found in the log file: Complex eigenvalues and normalised system matrix Notation (real : imaginary) Eigenvalue numbers 1 to 4 ( 0.00000E+00 : +/-5.55511E+01) Eigenvector numbers 1 to 4 ( 0.0141650 : 0.0000000) ( -0.0111041 : 0.0000000) ( 0.0000000 : +/-0.7868804) ( 0.0000000 : -/+0.6168430) eigenvectors of the state-space ( 0.00000E+00 : +/-2.47806E+01) ( ( ( ( 0.0177403 0.0362089 0.0000000 0.0000000 : : : : 0.0000000) 0.0000000) +/-0.4396164) +/-0.8972801) From the eigenvalues numbers in this table we find ω1 = 24.78 rad/s and ω2 = 55.55 rad/s. 66 Chapter 3. Examples 3.6 Cantilever beam in Euler buckling l F 3 4 1 2 1 2 5 6 Figure 3.28. Cantilever beam loaded axially by a force F at the free end. Consider a slender cantilever beam or column, with suppressed rotation of the free end, loaded axially by a force F . The smallest load that produces buckling is called the critical or Euler load Fcr . For a load equal to or greater than the critical load, the beam is unstable. The bent shape shown represents the buckling mode. Euler’s theoretical buckling load for the above beam end conditions is Fth = π 2 EI/l2 , where EI is the flexural rigidity and l the length of the beam. This result can also be obtained numerically from a SPACAR analysis, e.g. with the following numerical values, l = 1, EI = 1, F0 = 1. The beam is modelled by two equal planar beam elements as shown in Figure 3.28. A SPACAR input file (column2.dat) for this case is: PLBEAM 1 1 2 3 4 PLBEAM 2 3 4 5 6 X 1 0.0 0.0 X 3 0.5 0.0 X 5 1.0 0.0 FIX 1 FIX 2 FIX 6 DYNX DYNX DYNX RLSE RLSE 3 4 5 1 2 END HALT EM 1 1. EM 2 1. 2 1 2 2 3 2 3 Section 3.6. Cantilever beam in Euler buckling 67 ESTIFF 1 0. 1. ESTIFF 2 0. 1. XF 5 -1.0 0.0 END END In a MATLAB session we obtain: >> spacar(8,’column2’) >> edit column2.log Load multipliers and normalized buckling modes Load multiplier no 1 to 3 9.94384680E+00 4.00000000E+01 1.28722820E+02 Buckling mode nro 1 to 3 0.2596610869 -1.0000000000 -0.0519056301 0.8141747968 0.0000000000 0.9932416764 0.5193221738 0.0000000000 -0.1038112603 Hence, we find a load multiplier λ1 = Fcr /F0 = 9.944. Since F0 = 1 we have Fcr /Fth = 9.944/π 2 = 1.0075. 68 3.7 Chapter 3. Examples Cantilever beam subject to concentrated end force F l 1 2 3 4 5 Figure 3.29. Cantilever beam loaded by a concentrated force at the free end. Consider a slender cantilever beam with a circular cross-section of diameter d = 1 cm and length l = 10 m. The material properties for this example are EI = 102 Nm2 . The beam is subdivided into 5 planar finite elements as shown in Fig. 3.29. A point force F of 14 N is applied along the vertical axis at the free end of the beam. It generates an elastic deformation as shown in the figure. The deformation is reached in ten steps of loading. For each step the residual vector converges in 4 Newton–Raphson iterations with an accuracy equal to 0.5E − 6. A SPACAR input file (plbeam5.dat) for this case is: PLBEAM PLBEAM PLBEAM PLBEAM PLBEAM X 1 X 3 X 5 1 2 3 4 5 1 3 5 7 9 2 4 6 8 10 0. 0. 1.666 0. 3.333 0. 3 5 7 9 11 4 6 8 10 12 Section 3.7. Cantilever beam subject to concentrated end force X 7 5. 0. X 9 6.666 0. X 11 10.00 0. FIX 1 FIX 2 DYNE DYNE DYNE DYNE DYNE 1 2 3 4 5 2 2 2 2 2 3 3 3 3 3 1 2 3 4 5 0.0 0.0 0.0 0.0 0.0 END HALT EM EM EM EM EM 1 2 3 4 5 1. 1. 1. 1. 1. ESTIFF ESTIFF ESTIFF ESTIFF ESTIFF 102.0 102.0 102.0 102.0 102.0 XF 11 0.0 -14 END END In a MATLAB session we get: >> spacar(8,’plbeam5’) >> x(lnp(11,1)) ans = 3.6808 (theoretically, 3.8109) >> x(lnp(11,2)) ans = -8.4897 (theoretically, -8.4044) >> xcompl(lnp(11,1)) ans = 69 70 Chapter 3. Examples 0.4859 (undeformed configuration, 0.) >> xcompl(lnp(11,2)) ans = 0.0633 (undeformed configuration, 3.268) To show the usefulness of SPAVISUAL the first three free vibration modes (no external loads) and buckling modes (axially loaded by an end force) are displayed for the cantilever beam of this example in figures 3.30 to 3.35. Figure 3.30. First vibration mode for a cantilever beam with 5 elements, ω1 = 0.355131 rad/s (theoretically, 0.355100 rad/s). Figure 3.31. First buckling mode for a cantilever beam with 5 elements, Fcr1 = 2.516776 N (theoretically, 2.516749 N). Figure 3.32. Second vibration mode for a cantilever beam with 5 elements, ω2 = 2.2266 rad/s (theoretically, 2.22537 rad/s). Figure 3.33. Second buckling mode for a cantilever beam with 5 elements, Fcr2 = 22.715 N (theoretically, 22.651 N). Figure 3.34. Third vibration mode for a cantilever beam with 5 elements, ω3 = 6.25198 rad/s (theoretically, 6.23111 rad/s). Figure 3.35. Third buckling mode for a cantilever beam with 5 elements, Fcr3 = 64.798 N (theoretically, 62.919 N). Section 3.8. Short beam 3.8 71 Short beam l F y h Figure 3.36. Short Timoshenko beam loaded in shear. In this example the influence of shear deformation on the behaviour of short beams is studied. A square plate is loaded in shear in its plane as shown in Figure 3.36. The beam has unit height, h, length, l, and Young’s modulus, E, and a small unit width, t. With Poisson’s ratio ν = 0.27, the shear correction value is k = 10(1 + ν)/(12 + 11ν) = 0.8484. The deflection, if shear deflection is taken into account, is y= 2(1 + ν)F l F l3 + 12EI kEth (3.13) with I = th3 /12. So the compliance is l3 2(1 + ν)l 2(1 + ν) y = + =1+ = 3.9940. F 12EI kEth k (3.14) The moment of inertia per unit of length is J = ρth3 /12. An input file in which the beam is modelled by two planar beams of equal length (shear2.dat) in this case is: PLBEAM 1 1 2 3 4 PLBEAM 2 3 4 5 6 X 1 0.0 0.0 X 3 0.5 0.0 X 5 1.0 0.0 FIX 1 FIX 2 FIX 6 FIX 5 1 RLSE 1 RLSE 2 DYNX 3 72 Chapter 3. Examples DYNX 4 DYNX 5 2 END HALT EM 1 1.0 EM 2 1.0 ESTIFF 1 ESTIFF 2 0.0833333333 0.0833333333 1.0 0.0833333333 0.2495 1.0 0.0833333333 0.2495 ITERSTEP 10 1 0.000000000001 END END In a MATLAB session, the compliances and eigenfrequencies can be found as follows >> spacar(8,’shear2’) >> xcompl(lnp(5,2)) ans = 3.9940 >> spacar(7,’shear2’) >> type shear2.log ... Eigenvalue numbers 5 to 8 ( 0.00000E+00 : +/-2.54107E+00) Eigenvector numbers 5 to 8 ( 0.0000000 : 0.0000000) ( -0.1544078 : 0.0000000) ( 0.2825588 : 0.0000000) ( 0.1744141 : 0.0000000) ( 0.0000000 : 0.0000000) ( 0.0000000 : -/+0.3923612) ( 0.0000000 : +/-0.7180016) ( 0.0000000 : +/-0.4431985) ( 0.00000E+00 : +/-7.95645E-01) ( ( ( ( ( ( ( ( 0.0000000 0.4065978 0.2566711 0.6173727 0.0000000 0.0000000 0.0000000 0.0000000 : : : : : : : : 0.0000000) 0.0000000) 0.0000000) 0.0000000) 0.0000000) +/-0.3235074) +/-0.2042190) +/-0.4912094) ... The compliance based on thin plate theory is 3.8822 m/N, so the approximation with a short beam overrates the compliance by about 3%. If the shear flexibility were nor included, the compliance would be 1.0 m/N. The lowest numerical eigenfrequency, ω1 = 0.795645 rad/s, compares well with a value from plate theory, ω1,pl = 0.7987 rad/s. If shear flexibility nor rotational inertia is included, the first numerical eigenfrequency is 1.6168 rad/s. Section 3.9. Lateral buckling of cantilever beam 3.9 73 Lateral buckling of cantilever beam In this example lateral buckling is considered of a cantilever beam with a narrow rectangular cross-section which is loaded by a transverse force Fkipp at its free end √ in the direction of the larger flexural rigidity. The theoretical buckling load is Fth = 4.013 (EISt )/l2 , where EI is the smaller flexural rigidity, St the torsional rigidity and l the length of the beam. For numerical analysis, the beam is divided into four equal spatial beam elements in which the second-order terms in the bending deformations are included in the analysis. In a MATLAB session we get: >>spacar(8,’lateral4’) >>spavisual(’lateral4’) An input file (lateral4.dat) describing this case is: BEAM BEAM BEAM BEAM X X X X X 1 3 5 7 9 1 2 3 4 1 3 5 7 DYNE DYNE DYNE DYNE 0.00 0.25 0.50 0.75 1.00 1 2 3 4 2 2 2 2 2 4 6 8 3 4 5 6 7 8 9 10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5 5 5 5 0. 0. 0. 0. 1. 1. 1. 1. 6 6 6 6 FIX 1 FIX 2 OUTLEVEL 0 1 END HALT EM EM EM EM 1 2 3 4 1.0 1.0 1.0 1.0 ESTIFF ESTIFF ESTIFF ESTIFF 1 2 3 4 0.0033 0.0033 0.0033 0.0033 0.0 0.0 0.0 0.0 2.0 2.0 2.0 2.0 0.0 0.0 0.0 0.0 1.0 1.0 1.0 1.0 0. 0. 0. 0. 74 Chapter 3. Examples XF 9 0.0 0.0 -1.0 END END VISUALIZATION BUCKLINGMODE 1 TRANSPERANCY 0.9 BEAMVIS 0.01 0.1 LIGHT 1 STEPLINE 0.01 ENLARGEFACTOR 0.04 The 3D-visualization of this file is presented in figure 3.37. The buckling load found is 5.7619 N, whereas the theoretical value is 5.6752 N. If the warping is constrained at the clamped end, the q first element is effectively shorter for torsion by a distance b (1 + ν)/24, where b is the height of the beam, here b = 0.2 m, and ν is Poisson’s ratio, here q ν = 0. The torsional stiffness of the first beam element now increases with a factor l/(l − b 2/3) = 1.19517. The input line for the stiffness of the first beam element now becomes ESTIFF 1 0.0 2.39034 0.0 1.0 The critical load is now increased to 6.1694 N. Section 3.9. Lateral buckling of cantilever beam 75 F Figure 3.37. Cantilever beam lateral buckling (buckling mode 1). 76 Chapter 3. Examples 3.10 State-variable and output equations k5 7 9 5 y 4 l4 5 b1 m2 k2 3 l3 1 2 x 1 3 2 Figure 3.38. Lever system. Find the state-space-variable and output equations for the system shown in Fig. 3.38. The input is the displacement δx7 of the left end of spring k2 = EA2 /l2 ; it affects the mass m2 through spring k5 = EA5 /l5 and the lever, which is modelled by the planar beam elements 3 and 4. The lever has a fixed pivot at node 5 and is assumed to be massless yet rigid. Its angular orientation is small so that only horizontal motion need be considered. We will select δx2 and δ ẋ2 as state variables, with δx7 being the input and reaction force δfx 5 as output. With these definitions the state variable and output equations are then: " h # " δ ẋ2 0 1 2 = δ v̇ −k2 /m2 −b1 /m2 | i {z A h δfx 5 = −k2 (1 + l3 /l4 ) | {z C #" } # " # i h δx2 0 7 δx + δ ẋ2 −(k2 /m2 )(l4 /l3 ) | 0 # " i δx2 2 } δ ẋ {z } B h + −k2 l3 /l4 (1 + l3 /l4 ) | {z D ih } (3.15) i δx7 , (3.16) which have the desired form. These results can also be obtained numerically from a SPACAR analysis. E.g. with numerical values for m2 = 1, b1 = Ed A1 /l1 = 5, k2 = k5 = 1000 and l4 /l3 = 2. A SPACAR input file (lever.dat) for this case is: PLTRUSS PLTRUSS PLBEAM PLBEAM PLTRUSS 1 2 3 4 5 1 2 3 5 7 X 1 0.0 0.0 2 3 4 5 6 6 7 8 9 Section 3.10. State-variable and output equations X X X X X 2 3 5 7 9 1.0 2.0 2.0 2.0 3.0 FIX FIX FIX FIX 1 2 2 5 9 DYNX 2 INPUTX RLSE 1 RLSE 2 RLSE 5 0.0 0.0 2.0 3.0 3.0 1 7 1 1 1 1 END HALT XM 2 1.0 ESTIFF 2 1000. ESTIFF 5 1000. EDAMP 1 5 END HALT INX 1 7 1 OUTF 1 5 1 END END In a MATLAB session we get: >> spacar(9,’lever’) >> A=getfrsbf(’lever.ltv’, ’A’, 1) A = 0 -1000 1 -5 >> B=getfrsbf(’lever.ltv’, ’B’, 1) 77 78 Chapter 3. Examples B = 0 -2000 >> C=getfrsbf(’lever.ltv’, ’C’, 1) C = -3000 0 >> D=getfrsbf(’lever.ltv’, ’D’, 1) D = -6000 The state-space matrices can also be obtained with the command getss(’lever’). A Bode diagram (Figure 3.39) can be made by the command >> bode(getss(’lever’)) Bode Diagram 100 90 80 Magnitude (dB) 70 60 50 40 30 20 10 0 360 Phase (deg) 315 270 225 180 −1 10 0 10 1 10 Frequency (rad/sec) 2 10 Figure 3.39. Bode diagram for the lever system. 3 10 Section 3.11. Rigid spatial manipulator mechanism 3.11 79 Rigid spatial manipulator mechanism Figure 3.40 gives an example of a simplified manipulator. The prescribed motion of the endeffector C is represented by the coordinates xC , y C and z C as functions of time. z v 1.76 m/s 3 4 B 5 y 1, 2 II 0.0 0.2 0.4 1.0 1.2 C 1.4 t I x Figure 3.40. Spatial manipulator mechanism, trajectory and velocity profile of the endeffector. The end-effector must follow the straight trajectory from point I to point II. Three trajectories are distinguished: Initially the manipulator is at rest for 0.2 s. Next, during 1.0 s, the motion is carried out according to the velocity profile in Fig. 3.40 with constant acceleration and deceleration during the first and final 0.2 s. Finally the manipulator is at rest again. The motion of manipulator is determined by the rotation of three hinges. Hinge 1 enables rotations about the z-axis, while hinge 2 enables motions perpendicular to the xy-plane. Hinge 3 takes care of motions in the same plane wherein hinge 2 is active. The hinges are driven by internal actuators. For control purposes we assume that sensors are available that measure the rotations and the speed of rotation of the hinges. The manipulator consists of two beams, elements 4 and 5, which are equal in length: l4 = l5 = 0.7 m. The distributed mass per length is ρ4 = 4 kg/m for element 4 and ρ5 = 2 kg/m for element 5. The concentrated masses in nodes B and C are 10 kg and 30 kg respectively. The effect of gravity is accounted for by applying external forces mi g in negative z-direction, where g = 10 m/s2 . Inverse dynamics problem First the inverse dynamics problem is analysed. Figure 3.44 shows the velocity components of the end-effector that are computed for the trajectory defined in the input file. The position and acceleration components of the end-effector are shown Fig. 3.43 and Fig. 3.45, respectively. The following input file (robotinv.dat) is used (SPACAR mode=2): 80 Chapter 3. Examples HINGE HINGE BEAM HINGE BEAM X X X 4 5 8 1 2 4 3 5 1 2 4 6 5 2 3 3 5 6 7 7 8 9 0. 0. 0.268 0. 0.536 0. FIX FIX INPUTX INPUTX INPUTX RLSE RLSE RLSE 1 4 8 8 8 1 2 3 0 0 0 -1 0 1 0 -1 0 1 1 0 0 0 0 0. 0.6467 0. 5 8 4 5 1 5 8 1 2 3 1 1 1 10. 30. 4. 2. 0. 0. 0. 1.3 0. 1 1 1 2 2 1 3 3 1 1 2 3 1 2 3 1 2 3 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 1 2 3 1 2 3 END END 0. -14. 0. -121. 0. -307. END HALT TRAJECT 1 TRANS 8 TRTIME 0.2 TRAJECT 2 TRANS 8 TRVMAX 8 TRFRONT 8 TRTIME 1.0 NOMS NOMS NOMS REFE 1 REFE 2 REFE 3 REFEP 4 REFEP 5 REFEP 6 REFEDP 7 REFEDP 8 REFEDP 9 REFX 10 REFX 11 REFX 12 REFXP 13 REFXP 14 REFXP 15 END HALT XM XM EM EM XF XF XF TRAJECT 3 TRANS 8 0. TRTIME 0.2 20 0.536 0. 20 0. 0. 0.2 0. 100 0. 1.3 1.76 VISUALIZATION BEAMVIS 0.01 0.01 HINGEVIS 1 0.01 0.03 HINGEVIS 2 0.01 0.03 HINGEVIS 3 0.01 0.03 LIGHT 1 TRANSPARENCY 0.6 TRAJECT 1 TRAJECTNODE 8 The inverse dynamics analysis yields the stresses that have to be applied at the hinges and the deformations of the hinges. Fig. 3.42 shows the stresses. Figures 3.46 and 3.47 show the deformations which are the relative rotations of the hinges, and the first time derivatives, respectively. Clearly, to accomplish the quite simple trajectory of the end-effector of this non- Section 3.11. Rigid spatial manipulator mechanism Figure 3.41. SPAVISUAL 81 output for the spatial manipulator mechanism. linear mechanism rather complicated functions for the rotation of the hinges are needed. Note that the input file defines the inputs and outputs that will be used in a SIMULINK simulation. The nominal inputs are computed to accomplish the deformations of the hinges. The outputs include the six sensor signals with the rotations and the speeds of rotation of the hinges. Nine more outputs are defined to obtain extra information on the performance of the manipulator: the acceleration of the rotation of the hinges and position and velocity of the end-effector. At the end of the file visualization settings for SPAVISUAL are defined. In figure 3.41 the output of SPAVISUAL is presented. 82 Chapter 3. Examples 600 1.4 500 1.2 400 u0(1), u0(2), u0(3) [N/m] 1 x,y,z(8) [m] 300 200 0.8 0.6 100 0.4 0 0.2 −100 −200 0 0.2 0.4 0.6 0.8 1 1.2 0 1.4 0 0.2 0.4 0.6 time [s] 0.8 1 1.2 1.4 time [s] Figure 3.42. Stresses to be applied at the hinges (u0 ). Figure 3.43. effector. 2 Position coordinates of the end- 10 8 1.5 6 4 2 (d/dt) x,y,z(8) [m/s ] 0.5 2 0 2 (d/dt) x,y,z(8) [m/s] 1 −2 0 −4 −6 −0.5 −8 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 −10 0 0.2 0.4 0.6 time [s] Figure 3.44. Velocity components of the endeffector. 1.5 3 (d/dt) e1(1),e1(2),e1(3) [rad/s] 4 1 e1(1),e1(2),e1(3) [rad] 1 1.2 1.4 Figure 3.45. Acceleration components of the endeffector. 2 0.5 0 −0.5 −1 0.8 time [s] 2 1 0 −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time [s] Figure 3.46. Deformations (relative rotations) of hinges 1, 2 and 3. −2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time [s] Figure 3.47. Velocities of deformation of hinges 1, 2 and 3. Section 3.11. Rigid spatial manipulator mechanism 83 Rotational nodes for the spatial manipulator node 1 node 2 node 3 node 6 node 7 node 9 type 2 2 2 2 2 2 λ0 2 2 2 2 2 2 λ1 1 2 2 2 2 2 λ2 1 2 2 2 2 2 λ3 1 2 2 2 2 2 forces 0 0 0 0 0 0 I∗∗ 0 0 0 0 0 0 Translational nodes for the spatial manipulator node 4 node 5 node 8 type 1 1 1 x-coordinate 0 0.268 0.536 y-coordinate 0 0 0 z-coordinate 0 0.647 0 BC-type x 1 2 3 x0 0.536 ẋ0 0 BC-type y 1 2 3 y0 0 ẏ0 0 BC-type z 1 2 3 z0 0 ż0 0 force x 0 0 0 force y 0 0 0 force z (∗) -14 -121 -307 mass 0 10 30 (∗) including the element masses element type T-nodes R-nodes x local y-axis y local y-axis z local z-axis type e1 type e2 type e3 type e4 type e5 type e6 Elements for the spatial manipulator element 1 element 2 element 3 element 4 hinge hinge hinge beam 4, 5 1, 2 2, 3 6, 7 3, 6 0 0 0 0 0 -1 -1 1 1 0 0 0 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 T=translational, R=rotational element 5 beam 5, 8 7, 9 0 1 0 1 1 1 1 1 1 84 Chapter 3. Examples Linearization In one of the next sections the design of a closed-loop controller for this manipulator will be discussed. This controller depends on parameters derived from the linearized equations of motion. Therefore, a linearization is needed in terms of the DOFs corresponding to the actuator joints. An input file (robotinvlin.dat) for this analysis (SPACAR mode=3) is: HINGE HINGE BEAM HINGE BEAM X X X 4 5 8 1 2 4 3 5 1 2 4 6 5 2 3 3 5 6 7 7 8 9 0. 0. 0.268 0. 0.536 0. FIX FIX INPUTE INPUTE INPUTE 0 0 0 -1 0 1 0 -1 0 1 0. 0.6467 0. 1 4 1 1 2 1 3 1 END HALT XM XM EM EM XF XF XF 5 8 4 5 1 5 8 10. 30. 4. 2. 0. 0. 0. 1 0 0 0 0 NOMS NOMS NOMS 1 1 1 2 2 1 3 3 1 REFE 1 REFE 2 REFE 3 REFEP 4 REFEP 5 REFEP 6 REFEDP 7 REFEDP 8 REFEDP 9 REFX 10 REFX 11 REFX 12 REFXP 13 REFXP 14 REFXP 15 1 2 3 1 2 3 1 2 3 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 1 2 3 1 2 3 END END 0. -14. 0. -121. 0. -307. END HALT Note that the setpoints are read from the sbd data file of which the name is the longest substring of the name of the input file name robotinvlin. The file from the previous inverse dynamics run robotinv is a likely candidate. Section 3.11. Rigid spatial manipulator mechanism 85 Open-loop simulation The behaviour of the manipulator mechanism without feed-back control is simulated using SIMULINK for the open-loop configuration of Fig. 3.48. Two blocks from the SPACAR library spacar_lib are used to read the Setpoint U0 and Reference Y0 data, respectively, from the inverse dynamics run (file name robotinv). In this open-loop configuration the nominal input is fed directly into the SPASIM block (also available in the library). In the input file robotsim for this block the actual inputs and outputs are identical to the previously defined inputs and outputs. HINGE HINGE BEAM HINGE BEAM X X X 4 5 8 1 2 4 3 5 1 2 4 6 5 2 3 3 5 6 7 7 8 9 0. 0. 0.268 0. 0.536 0. FIX FIX DYNE DYNE DYNE 0 0 0 -1 0 1 0 -1 0 1 0. 0.6467 0. 1 4 1 1 2 1 3 1 END HALT XM XM EM EM XF XF XF 5 8 4 5 1 5 8 10. 30. 4. 2. 0. 0. 0. 1 0 0 0 0 INPUTS 1 INPUTS 2 INPUTS 3 OUTE 1 OUTE 2 OUTE 3 OUTEP 4 OUTEP 5 OUTEP 6 OUTEDP 7 OUTEDP 8 OUTEDP 9 OUTX 10 OUTX 11 OUTX 12 OUTXP 13 OUTXP 14 OUTXP 15 1 2 3 1 2 3 1 2 3 1 2 3 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 1 2 3 END END 0. -14. 0. -121. 0. -307. END HALT The other blocks in the block diagram are standard SIMULINK blocks and are used to export data to workspace and to display results on the screen. The “Selector” blocks select only specified components from an input vector. They are e.g. used to select only the first three components of the output vector (deformations of the hinges) as displaying all components makes the graphs unreadable. SIMULINK ’s ode45 solver is used with a relative tolerance of 10−5 , an absolute tolerance of 10−8 and a maximum time step of 0.01 s. With these parameters the simulation of the motion from t = 0.0 s to t = 1.5 s is completed after 172 time steps. The size of many time steps is 86 Chapter 3. Examples robotinvlin 3 15 3 15 robotsim SPASIM Setpoint U0 unom 3 Selector 3 Selector E Scope E y 15 Y To Workspace Unom To Workspace 15 15 15 Selector 3 Selector Ytip robotinvlin 15 15 Selector Eref Reference Y0 Scope dYtip Selector 3 1.5 Scope Eref Display Time t yref 15 Clock Time To Workspace Yref To Workspace Figure 3.48. Block diagram for an open-loop simulation of the motion of the manipulator mechanism using SIMULINK. 2 1.4 1.2 1.5 (1) 0.8 0.5 x,y,z(8) [m] e1(1),e1(2),e1(3) [rad] 1 e1 1 (3) e1 y (8) x(8) 0.6 0.4 0 (2) e1 0.2 −0.5 0 z (8) −1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time [s] Figure 3.49. Deformation of the hinges of spatial manipulator mechanism in an open-loop simulation. −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time [s] Figure 3.50. Position of the end-effector of spatial manipulator mechanism in an open-loop simulation. dictated by the specified maximum value. The results from the simulation are plotted using the MATLAB commands; >> plot(t,yref(:,1),’r’,t,yref(:,2),’g’,... t,yref(:,3),’b’,t,y(:,1),’r:’,... t,y(:,2),’g:’,t,y(:,3),’b:’) >> plot(t,yref(:,10),’r’,t,yref(:,11),’g’,... t,yref(:,12),’b’,t,y(:,10),’r:’,... t,y(:,11),’g:’,t,y(:,12),’b:’) Figures 3.49 and 3.50 show the deformation of the hinges and the position coordinates of the end-effector from this simulation. The solid lines are the reference data (yref) and the dotted lines are from the actual simulation (y). Clearly, small errors during the integration lead to relatively large position errors at the end of the motion. The error can be decreased by increasing the integration accuracy, e.g. by enlarging the number of computed setpoints. More reliable results can be obtained by applying feedback control, as will be discussed next. Section 3.11. Rigid spatial manipulator mechanism 87 Closed-loop simulation The block diagram of Fig. 3.48 is extended with a feedback controller as shown in Fig. 3.51. A feedback signal is computed by a controller that is implemented as a subsystem block and a multiplication with the reduced mass matrix M 0 . The subsystem assumes that the input is a vector with both δe and δ ė. These are the differences in joint positions and velocities which are computed by comparing the actual motion and the nominal output. The output of the subsystem is K p δe + K v δ ė with well chosen matrices K p and K v (see e.g. the lecture notes [1]). This output is multiplied with the time-dependent 3 × 3 reduced mass matrix M 0 using a block from the spacar_lib library. Finally the nominal input vector u0 is added as a feedforward signal. robotinvlin 3 Unom 3 Setpoint U0 unom 3 u Unom To Workspace y 15 U To Workspace Y To Workspace 3 6 6 Omega = 28 beta = 0.85 Kp Kv control 6 3 robotinvlin times M0 3 3 15 3 15 robotsim SPASIM Selector 6 6 Selector E + Ed Selector Selector E 3 Scope E yref 15 1.5 Yref To Workspace robotinvlin Reference Y0 15 15 Selector 6 Selector Eref + Edref 6 Selector Display Time 3 Selector Eref 15 Scope Eref 15 15 Selector t 3 Selector Ytip Clock Time To Workspace Scope dYtip Figure 3.51. Block diagram for a closed-loop simulation of the motion of the manipulator mechanism using SIMULINK. Most signals are vectors and the numbers indicate the size of the vectors. The motion is simulated with the same parameters as in the open-loop simulation (see page 85). In this case the actual size of the variable time step is somewhat smaller and after 183 time steps the simulation is completed. The differences between the prescribed and actual trajectory is much smaller in this case as is illustrated in Figs. 3.52 and 3.53. The maximum error of the tip position is less than 1 mm which is better than 0.1%. Figs. 3.54 and 3.55 show the feedforward part (u0 ) and feedback part (u − u0 ) of the input applied to the manipulator, respectively. Clearly, the larger contribution is from the feedforward part. The size of the feedback part is smaller and relatively large correction are only applied during limited periods of time. However, as is clear from this example, this feedback is essential to keep the manipulator on track. The simulation for 1.5 s now requires 182 time steps, which is only slightly more than in the open-loop simulation. However, the simulation takes much more time which is caused by the occurrence of a so-called algebraic loop in the block diagram. The reason for this algebraic loop is the presence of the joint accelerations in the output vector of the spasim block, as accelerations depend algebraically on the input torques. These accelerations are only exported to the workspace and are not used in the feedback loop so there is no real algebraic loop. Unfortunately, SIMULINK has no means to detect this. If you are not interested in the accelerations, they can easily be removed from the output vector and the simulation speed will increase significantly. 88 Chapter 3. Examples −4 −3 x 10 1.5 3 x 10 2 (8) (2) e1 0.5 − (2) e1,ref 0 0 −0.5 (1) (1) e1 − e1,ref −1 x(8) − xref 1 x−xref,y−yref,z−zref(8) [m] e1(1)−e1ref(1),e1(2)−e1ref(2),e1(3)−e1ref(3) [rad] 1 (8) z (8) − zref −1 −2 −3 −4 (8) y (8) − yref −1.5 (3) e1 −2 −2.5 0 0.2 0.4 0.6 0.8 − (3) e1,ref −5 −6 1 1.2 −7 1.4 time [s] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time [s] Figure 3.52. Error in the deformation of the hinges of spatial manipulator mechanism in a closed-loop simulation. Figure 3.53. Position error of the end-effector of spatial manipulator mechanism in a closed-loop simulation. 40 600 (1) u(1) − u0 500 30 400 (3) u0 20 (2) u0 200 u−u0(1,2,3) [N/m] u0(1,2,3) [N/m] 300 10 100 0 0 (1) u0 −100 −200 0 0.2 0.4 −10 0.6 0.8 1 1.2 1.4 time [s] Figure 3.54. Input applied to the manipulator: feedforward part (u0 ). −20 (3) u(3) − u0 0 0.2 0.4 (2) u(2) − u0 0.6 0.8 1 1.2 1.4 time [s] Figure 3.55. Input applied to the manipulator: feedback part (u − u0 ). Section 3.12. Flexible spatial manipulator mechanism 3.12 Flexible spatial manipulator mechanism To be added ... 89 90 Chapter 3. Examples 3.13 Chord-driven underactuated robotic finger proximal phalanx first pulley second pulley distal phalanx F M third pulley Figure 3.56. Robotic finger. In order to illustrate the use of the planar belt/gear element, a model for a chord-driven underactuated robotic finger with two phalanges, as in a thumb, is considered; see Fig. 3.56. The distal phalanx can rotate with respect to the proximal phalanx and the proximal phalanx can rotate with respect to the palm, which is assumed to be immobile. A pulley is rigidly connected to the distal phalanx, centred at its rotation point, which carries a chord that is slung over a second pulley centred at the rotation point of the proximal phalanx, but free to rotate. The chord is finally attached to a third pulley, which is driven by a motor. The finger is underactuated, for it has two degrees of freedom, but a single motor. The degrees of freedom are chosen as the relative rotation angle between the two phalanges and the rotation angle of the motor pulley. A force F = 10 N acts near the tip of the finger, which is balanced by a moment M = 1.5 Nm delivered by the motor. Owing to the choice of the dimensions of the radii of the pulleys and the lengths of the phalanges, this is an equilibrium position. y 2 1 3 x 6 M 5 4 F Figure 3.57. Finite element model for the robotic finger. The finite element model is shown in Fig. 3.57. The input file, finger.dat, is PLBEAM PLTOR PLBEAM PLBELT 1 2 3 4 1 4 3 8 2 3 4 5 5 6 7 9 1 10 -0.015 -0.015 Section 3.13. Chord-driven underactuated robotic finger 91 PLBELT 5 1 10 3 5 -0.015 -0.009 PLRBEAM 6 6 7 11 FIX 1 1 2 FIX 8 1 2 DYNX 9 1 DYNE 2 1 X 1 0 0 X 3 0.06 0.0 X 6 0.15 0.0 X 8 0 -0.06 X 11 0.15 -0.015 END HALT EM 3 1.0 EM 5 1.0 STARTDX 9 1 0.0 0.0 STARTDE 2 1 0.0 0.0 XF 11 0.0 10.0 XF 9 -1.5 TIMESTEP 0.2 100 END END VISUALIZATION VIBRATIONMODE 1 ENLARGEFACTOR 0.2 By running SPACAR with mode=7, it can be checked that the initial position is indeed an equilibrium position, but it is unstable. One mode has an eigenvalue equal to zero, and in the corresponding mode shape, the distal phalanx remains horizontal. The other mode has an exponentially increasing or decaying motion and is shown in Figure 3.58. (Note that the current version of SPAVISUAL does not draw the pulleys.) A simulation with an initial perturbation, specified by STARTDX 9 1 0.001 0.04 STARTDE 2 1 0.004 0.16 shows this unstable behaviour. Note that the initial perturbation is approximately in the direction of the eigenvector corresponding to the unstable eigenvalue. 92 Chapter 3. Examples Figure 3.58. Unstable mode. Section 3.14. Tricycle 3.14 93 Tricycle y x λ z x Figure 3.59. Tricycle. A tricycle has two rear wheels on a common axle and a front wheel in a fork that can rotate about a steering axis with respect to the rear frame. The steering axis can be vertical (λ = 0) or be inclined; see Fig. 3.59. The input file for the planar version, trike2.dat, is: PLRBEAM 1 1 2 3 PLWHEEL 2 3 2 4 0.3 0.0 1.0 PLRBEAM 3 1 2 5 PLWHEEL 4 5 2 6 0.3 0.0 1.0 PLRBEAM 5 1 2 7 PLTOR 6 2 8 PLRBEAM 7 7 8 9 PLWHEEL 8 9 8 10 0.25 0.0 1.0 X 1 0.3 0.0 X 3 0.0 0.35 X 5 0.0 -0.35 X 7 1.05 0.0 X 9 1.00 0.0 RLSE 4 2 DYNE 6 1 KINX 1 1 2 KINX 2 1 DYNX 4 1 KINX 6 1 KINX 10 1 END 94 Chapter 3. Examples HALT XM 1 80.0 XM 2 5.0 XM 3 2.0 XM 4 0.1 XM 5 2.0 XM 6 0.1 XM 8 0.025 XM 9 1.5 XM 10 0.05 STARTDX 4 1 0.0 10.0 STARTDE 6 1 0.5 0.0 TIMESTEP 1.0 100 END END The origin of the coordinate system is initially located at the centre of the rear axle, with the x-axis pointing in the forward direction and the y-axis pointing to the left. The centre of mass of the frame is at a distance of 0.3 m in front of the rear axle. The rear wheels, elements 2 and 4, have a radius of 0.3 m and are connected to the centre of mass of the frame by two rigid beams, elements 1 and 3. Another rigid beam, element 5, connects the centre of mass of the frame to the steering head, where the hinge, element 6, makes the connection to the front fork. The rigid beam 7 represents the rigid connection between the steering head and the the front wheel, element 8, with radius 0.25 m, which is conncted to the front fork. All wheels can rotate freely about their spin axis. The frame and the wheels have mass, but the front fork is assumed to be massless. The system has two degrees of freedom: the rotation angle of the left rear wheel and the steering angle are chosen as generalized coordinates. The lateral slip of the right rear wheel is released, because otherwise the system whould be overconstrained. The other five slips at the wheels are prescribed as zero to impose the non-holonomic constraints of pure rolling. Five kinematic coordinates are defined as the two position coordinates and the yaw angle for the rear frame and the two rotation angles at the other wheels. The moments of inertia at the nodes 4, 6 and 10 are the moments of inertia about the spin axes of the wheels. The stationary motion and the linearized equations can be found by running SPACAR with mode=7. It appears that there are seven eigenvalues equal to zero, with eigenvectors which correspond to the three rotations of the wheels and the two positions and yaw angle of the rear frame, and a change in the forward velocity. The other two eigenvalues are real and negative, corresponding to exponentially decaying motion. A simulation can be made with mode=1. A three-dimensional model of the same tricycle is (file trike3v.dat) RBEAM HINGE WHEEL RBEAM HINGE WHEEL RBEAM HINGE 1 2 3 4 5 6 7 8 1 2 2 4 3 4 1 2 2 7 6 7 1 2 2 10 3 5 6 8 9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 Section 3.14. Tricycle 95 RBEAM 9 9 10 11 0.0 1.0 0.0 HINGE 10 10 12 0.0 1.0 0.0 WHEEL 11 11 12 13 0.0 1.0 0.0 HINGE 12 14 2 0.0 0.0 1.0 X 1 0.3 0.0 0.9 X 3 0.0 0.35 0.3 X 5 0.0 0.35 0.0 X 6 0.0 -0.35 0.3 X 8 0.0 -0.35 0.0 X 9 1.05 0.0 0.3 X 11 1.00 0.0 0.25 X 13 1.00 0.0 0.0 FIX 14 RLSE 12 2 3 RLSE 6 6 DYNE 2 1 DYNE 8 1 KINE 5 1 KINE 10 1 KINE 12 1 KINX 1 1 KINX 1 2 END HALT GRAVITY 0.0 0.0 -9.81 XM 1 80.0 XM 2 2.0 0.0 0.0 3.0 0.0 5.0 XM 3 2.0 XM 4 0.0 0.0 0.0 0.1 0.0 0.0 XM 6 2.0 XM 7 0.0 0.0 0.0 0.1 0.0 0.0 XM 11 1.5 XM 12 0.025 0.0 0.0 0.05 0.0 0.025 STARTDE 2 1 0.0 10.0 STARTDE 8 1 -0.5 0.0 TIMESTEP 1.0 100 END END Note that hinges (elements 2, 5 and 10) are used to connect the wheels to the rigid beams and an additional hinge, element 12, is introduced in order to make the yaw angle available. With mode=7, the same eigenvalues are found as for the planar model. If variables are saved from the run with the planar model, it will be seen that the results of a simulation are very nearly the same. The three-dimensional model has the advantage that the normal forces at the wheels are calculated, which are the first components of the stress of the wheel elements (Fig 3.60): >> plot(time,sig(:,le(3,1)),’k-’,time,sig(:,le(6,1)),’k:’, ... time,sig(:,le(11,1)),’k--’); grid on 96 Chapter 3. Examples 0 −100 normal force [N] −200 −300 −400 −500 −600 0 0.1 0.2 0.3 0.4 0.5 time [s] 0.6 0.7 0.8 0.9 1 Figure 3.60. Normal forces at road contact points. The fully drawn line is for the left rear wheel, the dotted line for the right rear wheel and the dashed line for the front wheel. >> xlabel(’time [s]’) >> ylabel(’normal force [N]’) It is seen that all forces are negative, which means that the normal force is compressive, as it should be. The normal force in the right rear wheel is initially much higher than the corresponding force at the left rear wheel. Because the wheel planes remain perpendicular to the road surface, the third components of the stresses are also equal to the normal force at the road. Lateral forces are in the second components, as well as in the sixth in a scaled version. The fourth components of the stresses are zero, as they should be. The fifth components represent scaled longitudinal tyre forces at the contact points. A model with an inclined steering axis, as shown in Fig. 3.59, is in the input file trike3i.dat, which differs from trike3v.dat in the definition of the hinge at the steering head, element 8, and the position of node 9, ... HINGE 8 2 10 -0.30901699437495 ... X 9 ... 0.96 0.0 0.3 0.0 0.95105651629515 Section 3.14. Tricycle 97 0 −100 normal force [N] −200 −300 −400 −500 −600 0 0.1 0.2 0.3 0.4 0.5 time [s] 0.6 0.7 0.8 0.9 1 Figure 3.61. Normal forces at road contact points. The fully drawn line is for the left rear wheel, the dotted line for the right rear wheel and the dashed line for the front wheel. Note that for this case, the normal force in the right rear wheel is initially just compressive (Figure 3.61), so for a slightly higher speed, the wheel would lose contact with the ground. This loss of contact cannot directly be included in the model. The front wheel no longer stays perpendicular to the road surface, so the first and third stress components are no longer equal. Morreover, the rotation angle of the hinge with element number 12 is not exactly equal to the yaw angle in this case. 98 Chapter 3. Examples A SPACAR installation Prerequisites Before installing SPACAR on a computer system it is advisable to check that the system is suitable of running the software and to have MATLAB installed. This SPACAR version has been developed and tested with MATLAB 7.0.4 and SIMULINK 6.2 (Release 14SP2). It is expected to work with any modern version of MATLAB/SIMULINK since R12, but in case of problems we can offer only limited support. The system requirements depend heavily on the version of MATLAB you are using. Consult the accompanying Installation Guide or check The Mathworks. You may expect that SPACAR will run on any Microsoft 32-bit Windows PC on which MATLAB/SIMULINK are running. Only the base systems of MATLAB and SIMULINK are required to run SPACAR, but additional toolboxes like the Control System Toolbox may be helpful to develop and analyse control systems. The installation of SPACAR uses less than 4 MB extra disk space. The SPACAR files are stored in ZIP-archives or, in Microsoft Windows XP, a compressed folder. In Windows XP you can easily open such archives, but of course you may chose to use your favourite unzipper. The ZIP-archives can be downloaded from http://www.wa.ctw.utwente.nl/Software/SPACAR/. In addition to the software there is a ZIP-archive with the data files that are used for the examples in Chapter 3. Installation First of all, you should create a subdirectory e.g. \Matlab\Toolbox\Spacar. Next, you extract the files from the SPACAR software ZIP-archive spacar2007_bin.zip into this subdirectory. There are three types of files: • Files with the extension .dll are the actual executables of the SPACAR package. The original SPACAR-code (not provided) is written in C and FORTRAN 77, compiled and linked into so-called MEX-modules, that are executables for use within the MATLABenvironment. The following files must exist: 99 100 Appendix A. SPACAR installation checksbf.dll loadsbm.dll spacar.dll combsbd.dll ltv.dll spacntrl.dll getfrsbf.dll mrltv.dll spasim.dll loadsbd.dll repinsbf.dll • Files with extension .m are the MATLAB-files necessary to use the SPACAR program. The following file must exist: spadraw.m Other .m-files provide help text for the MEX-modules. These files are: checksbf.m loadsbd.m repinsbf.m combsbd.m loadsbm.m spacar.m getfrsbf.m ltv.m spacntrl.m getss.m mrltv.m spasim.m • Files with extension .mdl are SIMULINK models. There is only one file which is actually a library from which the SPACAR modules for use in SIMULINK can be copied: spacar_lib.mdl The (optional) data files from spadata.zip can be extracted in a separate working directory. The files in the SPACAR subdirectory should be in the MATLAB path when MATLAB is running. There are two ways to accomplish this: 1. Make sure that the SPACAR subdirectory is the local directory. You can verify this by typing pwd. If necessary, change your local directory by typing cd \Matlab\Toolbox\Spacar or whatever directory you chose to store your files. 2. Another possibility is to change the settings of the MATLAB environment by adding the SPACAR subdirectory to the MATLAB path. This modification is either temporary or permanent. The path can be modified from the pulldown menu with File|Set Path..., or by using the MATLAB commands path or addpath. Now you are ready to run SPACAR in MATLAB and SIMULINK. B SPACAR error messages An analysis with SPACAR in MATLAB or a simulation with SPASIM in SIMULINK can suffer from errors. These errors can be divided into fatal errors that cause an immediate terminations and less severe errors which may report unexpected conditions in the log file, while the calculation continues. Most fatal error have a clear error message: • SPACAR requires 2 or 3 input arguments, SPACAR requires no output argument, CCONST must be 1 x N or N x 1 vector, CCONST contains too many parameters, MODE has an invalid value and FILENAME contains illegal characters indicate an incorrect call of SPACAR from MATLAB. The last error can also occur in SPASIM ( SIMULINK ). • Wrong number of input arguments, Flag must be a scalar variable, Too many output arguments, Time must be a scalar variable, State vector of wrong size, Input vector of wrong sizeand Not a valid flag number indicate an incorrect call of SPASIM from SIMULINK and should not occur during normal operation. • ERROR opening file ... means that SPACAR can not open the specified file for output. • ERROR opening existing file ... means that a file from a previous run is not found. • ERROR in subroutine DINVOE is caused by an error in the dynamics input, see Sect. 2.3. 101 102 Appendix B. SPACAR error messages • PREPTR: Illegal velocity profile is reported when no valid velocity profile can be determined. • Can not determine valid and existing input file names from ... means that no mode=2 output data file with extension sbd matching the current (mode=3) data file can be found. • Mechanisms are different, Configuration mismatch LE and Configuration mismatch LNP arise from an error during the comparison between a the configuration used in a (previous) mode=2 run and the current mode=3 run. • ERROR in subroutine ORDE0: IFLAG = 2 and ERROR in subroutine ORDE0: No convergence indicate problems with the zeroth order iteration. In SPASIM this may be avoided by setting or decreasing the maximum time step of SIMULINK’s solver. • ERROR in subroutine SOLDYN is usually caused by a singular mass matrix. • PRPARE: NUMBER OF NXC NOT EQUAL TO NEO+NEM is caused by an ill-defined mechanism. • ERROR in subroutine PRPARE: Too many ... means that the mechanism that is defined is too large to be handled by the SPACAR version you are using, see Table 1.1 on page 10. Simplify the mechanism or contact the authors. The messages written to the log file may be self-explanatory, but also a somewhat cryptic messages “ERROR OR POSSIBLE ERROR CODED: <code> ITEM: <number>” can occur. The <code> is related to a procedure in the software. Typical examples are: • INVOERi input for the kinematics (Sect. 2.2). • SINVOERi input for the inverse dynamics (setpoint generation) (Sect. 2.4). • LIMVOEi input for the linearization (Sect. 2.5). • SIMVOEi input for the non-linear simulation of manipulator control (Sect. 2.6). • PREPTR.. trajectory data processing. Note that errors in the input file are often reported one line later than the actual error position. C MATLAB tutorial C.1 Basic MATLAB graphics commands MATLAB provides a variety of functions for displaying data. This section describes some of these functions. For a complete survey of graphics functions available in MATLAB we refer to the official MATLAB documentation [2] or to the online help utility. Elementary plotting functions The following list summarizes the functions that produce basic line plots of data. These functions differ only in the way they scale the plot axes. Each accepts input in the form of vectors or matrices and automatically scales the axes to accommodate the input data. • plot – creates a plot of vectors or columns of matrices. • loglog – creates a plot using logarithmic scales for both axes. • semilogx – creates a plot using a logarithmic scale for the x-axis and a linear scale for the y-axis. • semilogy – creates a plot using a linear scale for the x-axis and a logarithmic scale for the y-axis. You can add titles, axis labels, grid lines, and text to your graph using • title – adds a title to the graph. • xlabel – adds a label to the x-axis. • ylabel – adds a label to the y-axis. • text – displays a text string at a specified location. • gtext – places text on the graph using the mouse. • grid – turns on/off grid lines. 103 104 Appendix C. MATLAB tutorial Creating a plot If y is a vector, plot(y) produces a linear graph of the elements of y versus the index of the elements of y. If you specify two vectors as arguments, plot(x,y) produces a graph of y versus x. Line styles, markers, and color You can pass a character string as an argument to the plot function in order to specify various line styles, plot symbols, and colors. In the statement plot(x,y,s) s is a 1-, 2-, or 3-character string (delineated by single quotes) constructed from the characters in the following table: Symbol y m c r g b w k Color yellow magenta cyan red green blue white black Symbol . o x + * : -. -- Linestyle point circle x-mark plus star solid dotted dashdot dashed For example, plot(x,y,’c+’) plots a cyan plus symbol at each data point. If you do not specify a color, the plot function automatically uses the colors in the above table. For one line, the default is yellow because this is the most visible color on a black background. For multiple lines, the plot function cycles through the first six colors in the table. Adding lines to an existing graph You can add lines to an existing graph using the hold command. When you set hold to on, MATLAB does not remove the existing lines; instead it adds the new lines to the current axes. It may, however, rescale the axes if the new data fall outside the range of the previous data. For example: plot(f1) hold on plot(f2,’--’) plot(f3,’-.’) hold off These statements produce a graph displaying three plots. Appendix C. MATLAB tutorial 105 Creating hardcopy of MATLAB figures You can make a hardcopy of a figure from the figure’s menu (File|Print...) or by pressing Ctrl+P. Output to several graphics formats can be carried out as well (File|Export...). Alternatively, MATLAB’s print command can be used at the MATLAB command prompt. E.g. you can generate PostScript output of the contents the current MATLAB figure window. The print command sends the output directly to your default printer or writes it to the specified file, if you supply a filename. You can also specify the type of PostScript file. Supported types include • PostScript (-dps) • Color PostScript (-dpsc) • Encapsulated PostScript (-deps) • Encapsulated color PostScript (-depsc) For example, the statement print dataplot -deps saves the contents of the current figure window as Encapsulated PostScript in the file called dataplot.eps. Depending on your MATLAB installation other graphics formats are supported, try help print. C.2 Quitting and saving the workspace To quit MATLAB, type quit or exit. Terminating a MATLAB session deletes the variables in the workspace. Before quitting, you can save the workspace for later use by typing save This command saves all variables in a file on disk named matlab.mat. The next time MATLAB is invoked, you can execute load to restore the workspace from matlab.mat. You can use save and load with other filenames, or to save only selected variables. The command save temp stores the current variables in the file named temp.mat. The command save temp X saves only variable X, while save temp X,Y,Z saves X, Y, and Z. load temp retrieves all the variables from the file named temp.mat. 106 Appendix C. MATLAB tutorial References [1] Jonker, J. B., Dynamics of Machines and Mechanisms, A Finite Element Approach, Lecture notes, Department of Mechanical Engineering, University of Twente, vakcode 113130, October 2001. [2] The Math Works Inc., Getting Started with MATLAB, version 7, Revised for MATLAB 7.1 (Release 14SP3), September 2005. [3] The Math Works Inc., SIMULINK — Getting Started, version 6, New for SIMULINK 6.3 (Release 14SP3), September 2005. [4] SAM, Version 4.2, 5.0 or 5.1, ARTAS - Engineering Software, The Netherlands, http://www.artas.nl/, 2001–2005. [5] Cowper, G. R., “The shear coefficient in Timoshenko’s beam theory”, ASME Journal of Applied Mechanics 33 (1966), pp. 335–340. 107

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