Joint Torque Reduction of a Three Dimensional Redundant

Joint Torque Reduction of a Three Dimensional Redundant
Sensors 2012, 12, 6869-6892; doi:10.3390/s120606869
OPEN ACCESS
sensors
ISSN 1424-8220
www.mdpi.com/journal/sensors
Article
Joint Torque Reduction of a Three Dimensional Redundant
Planar Manipulator
Samer Yahya 1,*, Mahmoud Moghavvemi 1,2 and Haider Abbas F. Almurib 3
1
2
3
Center of Research in Applied Electronics (CRAE), University of Malaya, Kuala Lumpur 50603,
Malaysia; E-Mail: [email protected]
Faculty of Electrical and Computer Engineering, University of Tehran, P.O. Box 14399-57131,
Tehran, Iran
Department of Electrical & Electronic Engineering, University of Nottingham Malaysia,
Jalan Broga, Semenyih 43500, Malaysia; E-Mail: [email protected]
* Author to whom correspondence should be addressed; E-Mail: [email protected];
Tel.: +60-172-841-560.
Received: 1 April 2012; in revised form: 2 May 2012 / Accepted: 22 May 2012 /
Published: 25 May 2012
Abstract: Research on joint torque reduction in robot manipulators has received
considerable attention in recent years. Minimizing the computational complexity of torque
optimization and the ability to calculate the magnitude of the joint torque accurately will
result in a safe operation without overloading the joint actuators. This paper presents
a mechanical design for a three dimensional planar redundant manipulator with the
advantage of the reduction in the number of motors needed to control the joint angle, leading
to a decrease in the weight of the manipulator. Many efforts have been focused on decreasing
the weight of manipulators, such as using lightweight joints design or setting the actuators at
the base of the manipulator and using tendons for the transmission of power to these joints.
By using the design of this paper, only three motors are needed to control any n degrees of
freedom in a three dimensional planar redundant manipulator instead of n motors. Therefore
this design is very effective to decrease the weight of the manipulator as well as the number of
motors needed to control the manipulator. In this paper, the torque of all the joints are
calculated for the proposed manipulator (with three motors) and the conventional three
dimensional planar manipulator (with one motor for each degree of freedom) to show the
effectiveness of the proposed manipulator for decreasing the weight of the manipulator and
minimizing driving joint torques.
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Keywords: redundant manipulator; dynamics; robot; rotary encoders; joint torques reduction
1. Introduction
Theoretically, for a structure of the robot manipulator one actuator can be mounted on each link to
drive the next link via a speed reduction unit, but actuators and speed reducers installed on the distal end
become the load for actuators installed on the proximal end of a manipulator, resulting in a bulky and
heavy system [1]. To reduce the weight and the inertia of a robot manipulator, many mechanisms have
been proposed so far to remove the weight restriction. Some reported by [2,3] include:
(a) Lightweight joint design based on a special rotary joint [4–6]
(b) Provision of a powerful slider at the base to bear as much required driving force as possible [7]
(c) The parallel mechanism is another method to reduce the mass and inertia of the manipulator [8].
A typical parallel manipulator consists of a moving platform that is connected with a fixed base
by several limbs. Generally, the number of degrees of freedom of a parallel manipulator is equal
to the number of its limbs. The actuators are usually mounted on or near the base, which
contributes to reduce the inertia of manipulators, and
(d) Concentration of the actuators at the base and transmission of the power to each joint through
tendons or a special transmission mechanism [2,3,9]. This mechanism allows the actuators to be
situated remotely on the manipulator base, allowing the manipulator to be made more lightweight
and compact.
For a serial manipulator, direct kinematics are fairly straightforward, whereas inverse kinematics
becomes very difficult. Reference [10] proposes a fused smart sensor network to estimate the forward
kinematics of an industrial robot, while reference [11] measures the range data with respect to the robot
base frame using the robot forward kinematics and the optical triangulation principle. The inverse
kinematics problem is much more interesting and its solution is more useful, but one of the difficulties of
inverse kinematics is that when a manipulator is redundant, it is anticipated that the inverse kinematics
has an infinite number of solutions. This implies that, for a given location of the manipulator’s
end-effector, it is possible to induce a self-motion of the structure without changing the location of the
end-effector. In this paper we depend on our prior works [12,13] which present a new method to solve
the problem of multi-solutions of a three dimensinal planar redundant manipulator. Because this paper
explains the dynamic of the manipulator and not its kinematics, the inverse kinematics methods
will not be explained here. For more details about the inverse kinematics of redundant manipulators,
our works [14–16] can be checked.
It is mentioned earlier that the proposed manipulator could be used to reduce the weight of the
manipulator which yields to a decrease in the size (power) of the motors used to control the manipulator.
To show the effectiveness of the proposed manipulator in reducing the torques of its motors the inverse
dynamic of the manipulator has been calculated mathematically. The inverse dynamic model provides
the joint torques in terms of the joint positions, velocities and accelerations. For robot design, the
inverse dynamic model is used to compute the actuator torques, which are needed to achive a desired
motion [17]. Several approaches have been proposed to model the dynamics of robots. The most
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frequently employed in robotics are the Lagrange formulation and the Newton-Euler formulation.
Because the Lagrange formulation is conceptually simple and systematic [18], it has been used in this
paper. The Lagrange formulation provides a description of the relationship between the joint actuator
forces and the motion of the mechanism, and fundamentally operates on the kinetic and potential energy
in the system [19].
The work presented in this paper is based on our previous work [14], which presents a mechanical
design for a three dimensional planar redundant manipulator, which guarantees to decrease the weight of
the manipulator by decreasing the number of motors needed to control it. Because the inverse kinematics
model gives an infinite number of solutions for a redundant manipulator, consequently, secondary
performance criteria can be optimized [17], such as avoiding singular configurations and minimizing
driving joint torques. Reference [14] studied the kinematics of the manipulator of this paper and showed
in details its ability to avoid singular configurations. A comparison of the manipulability index values
and the manipulability ellipsoids for the manipulator is made with the manipulability index values and
the manipulability ellipsoids of the PUMA arm to show the effectiveness of using the proposed
manipulator to avoid singularity. In this paper, the dynamics of this manipulator are explained in detail.
The contribution of this work is to explain the ability of this manipulator for joint torque minimization.
The links and motors mass distribution is studied for both the proposed (with three motors) and
conventional manipulators (six motors). The driving joint torques have been studied for the proposed
manipulator for each joint and the results are compared with the results of the conventional manipulators
to show the effectiveness of this manipulator for minimizing driving joint torques.
2. The Mechanical Design of the Manipulator
To control the motion of the end-effector of the manipulator shown of Figure 1(a), all the motors of
the manipulator should be controlled. For example, to control a five links planar redundant manipulator
with the ability to rotate the entire manipulator around its vertical axis, the six motors (five motors for
each joint angle and one motor to rotate the entire manipulator around its vertical axis) of the
manipulator should be controlled. Using the method of our papers [12,13], the configuration of the
manipulator will have three angles to be controlled instead of n angles. Figure 1(b) shows the
configuration of the manipulator when there are just three angles that need to be controlled.
Because the end-effector can follow any desired path by controlling three angles (θ1, θ2 and θ3) only,
therefore instead of using a motor for each joint angle, three motors can be used for controlling the
manipulator. This means that for any number of degrees of freedom three dimensional planar redundant
manipulators, the weight of the links will be significantly decreased using the proposed design. To make
the manipulator capable of moving in a three dimensional work space, one motor will control the value
of θ1—this means controlling the rotation of the entire manipulator around the vertical axis. This motor
is situated in such a way as to rotate the base of the manipulator around the z-axis. The second motor
controls the value of θ2, which means the rotation of the entire manipulator with its configuration. The
motor is situated at the base. The third motor controls the value of θ3 and this motor is situated on the first
link. This motor will rotate the second link of manipulator about the second axis, and because all the next
links should rotate about their axes by the same angle θ3 therefore, there is no need to use motor for each
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joint angle, but the rotation of the second motor will be transferred to the next joints using gears boxes.
Figure 2 shows the mechanism of the proposed manipulator.
Figure 1. (a) A three dimensional planar redundant manipulator configuration; (b) A three
dimensional planar redundant manipulator configuration using the method of [12,13].
(xtp,ytp,ztp)
θn+1
z-axis
(xtp,ytp,ztp)
z-axis
θ3
ln
s
ztp
θ1
θ2
l3 θ 3
l2
y-axis
θ3
l1
(0,0)
xtp
ytp
ztp
θ3
l1
(0,0)
s
l3 θ4
l2
x-axis
ln
θ1
x-axis
(a)
θ2
y-axis
xtp
ytp
(b)
Figure 2. The manipulator used in experiments [14]. The draft of the manipulator using the
SolidWorks software (left). The mechanical design of the manipulator (right).
Second
motor
Third
motor
First motor
Elaborating further, the second motor is connected to the first link using a worm gear to control the
angle θ2. Figure 3 shows the position of the second motor.
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Figure 3. The design of the second joint angle (first link with second motor) of the
manipulator [14]. The draft of the second joint angle using the SolidWorks software (left).
The mechanical design of the second joint angle (right).
Second
motor
gear
1st
link
worm
(driver)
The third motor is connected to the second link using a worm gear for the same reasons it was used
with the first link. Controlling the third motor means controlling the angle between the first link and the
second link i.e., the angle θ3. Figure 4 shows the position of the third motor.
Figure 4. The design of the third joint angle (second link with third motor) of the
manipulator [14]. The draft of the third joint angle using the SolidWorks software (top left).
The draft of the whole manipulator using the SolidWorks software (top right). The mechanical
design of the third joint angle (bottom).
2nd link
arm
planetary
gear
bevel
gear 2
bevel
gear 1
gear
(wheel)
worm
1st link
Third motor
The mechanism of the third link is shown in Figure 5. The same mechanism of the second link is
used; the only one difference is that instead of using s worm as a driver and s wheel gear as a driven, two
bevel gears are used. The same mechanism of the third link can be used with the next links. The last link
has the mechanism shown in the Figure 6. For further details of the mechanical design of the
manipulator, our reference [14] can be checked.
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Figure 5. The design of the fourth joint angle (third link) of the manipulator [14]. The draft
of the fourth joint angle using the SolidWorks software (top left). The draft of the whole
manipulator using the SolidWorks software (top right). The mechanical design of the fourth
joint angle (bottom).
Figure 6. The last joint of the manipulator.
the last
link
To ensure that all the links move at the same joint angle, the ratio between the bevel gears of each
planetary gear should be equal to one. This means the bevel gears of each planetary gear should have the
same diameter and number of teeth. If this arm is fixed, we get:
w1
N
=− 2
w2
N1
(1)
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where w is the angular velocity of gear and N is the number of teeth of gear. In our manipulator, it is
noted that the first gear is fixed while the second gear and the arm are rotating. It is desired that both the
arm and the second gear have the same angular velocity. Because the arm is not stationary, then we
cannot use the previous equation. i.e., the mechanism is not an ordinary gear train but a planetary gear
train. To convert this planetary gear train to an ordinary gear train, it is assumed that the arm is stationary
while a first gear has an angular velocity and not fixed. This means that:
w1′ = w1 − w a
(2)
wa′ = wa − wa = 0
(3)
And because the second gear will continue rotating with the same angular velocity, then:
w2′ = w2
(4)
Now the Equation (1) can be rewritten as follows:
w − wa
w1′
N
=− 2 = 1
w2′
N1
w2
(5)
For our manipulator it is desired to move both the arm and the second gear by the same angular
velocity w which means:
N
(0) − w
=− 2
w
N1
(6)
N1 = N 2
To make the manipulator to have the ability to move in a three dimensional work space, a motor is
added to the base of the manipulator to make the whole manipulator capable of rotating around the
z-axis. This motor controls θ1. Figure 7 shows the mechanism of the first motor.
Figure 7. The mechanism of the first motor.
To calculate the transformation matrix of the manipulator, the draft of the manipulator shown in
Figure 8, is used. The corresponding link parameters of the manipulator are shown in Table 1. Where l1,
l2, …, l5 are the length of the links, while d1 is the offset between the origin and the end-effector.
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Figure 8. The manipulator used in experiments.
z-axis
l4
l5
target
point
l3
y-axis
l2
x-axis
origin
l1
d1
Table 1. Link parameters of the manipulator.
i
1
2
3
4
5
6
α
90
0
0
0
0
0
a
0
l1
l2
l3
l4
l5
d
0
d1
0
0
0
0
θ
θ1
θ2
θ3
θ4
θ5
θ6
From the links parameters shown in Table 1 and using Equation (7) which defines the transformation
matrix T for the links [1], we compute the individual transformations for each link:
⎡ cos θ
i
⎢
⎢ sin θ
i −1
i
Ti = ⎢⎢
0
⎢
⎢⎣
0
− sin θ i cos α i
cos θ i cos α i
sin α i
0
sin θ i sin α i
− cos θ i sin α i
cos α i
0
a i cos θ i
a i sin θ i
d
1
i
⎤
⎥
⎥
⎥
⎥
⎥
⎥⎦
(7)
where ci = cos(θi) and si = sin(θi).
⎡c1 0 s1
⎢s 0 − c
1
0
T1 = ⎢ 1
⎢0 1 0
⎢
⎣0 0 0
⎡c 4
⎢s
3
T4 = ⎢ 4
⎢0
⎢
⎣0
− s4
c4
0
0
0⎤
⎡c 2
⎥
⎢s
0⎥ , 1
T2 = ⎢ 2
⎢0
0⎥
⎥
⎢
1⎦
⎣0
0 l3 c4 ⎤
⎡c 5
⎥
⎢s
0 l3 s 4 ⎥ , 4
T5 = ⎢ 5
⎢0
1 0 ⎥
⎥
⎢
0 1 ⎦
⎣0
− s2
c2
0
0
− s5
c5
0
0
0 l1c2 ⎤
⎡c 3
⎥
⎢s
0 l1 s 2 ⎥ , 2
T3 = ⎢ 3
1 d1 ⎥
⎢0
⎥
⎢
0 1 ⎦
⎣0
0 l 4 c5 ⎤
⎡c 6
⎥
⎢s
0 l 4 s5 ⎥ , 5
T6 = ⎢ 6
⎢0
1 0 ⎥
⎥
⎢
0 1 ⎦
⎣0
− s3
c3
0
0
− s6
c6
0
0
0 l 2 c3 ⎤
0 l 2 s 3 ⎥⎥
1 0 ⎥
⎥
0 1 ⎦
0 l 5 c6 ⎤
0 l5 s6 ⎥⎥
1 0 ⎥
⎥
0 1 ⎦
(8)
Finally we obtain the product of all six link transforms:
0
T6 = 0T1 1T2 2T3 3T4 4T5 5T6
(9)
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3. Dynamics of the Manipulator
In this section, the torque of each joint is calculated. To show the effectiveness of the proposed
manipulator, the joint torques are calculated using the proposed manipulator (using three motors only)
and the conventional manipulators (a motor for each joint).
Let us assume for concreteness that the center of mass of each link is at its geometric center. For the
manipulator used in our experiments, the mass of links without the motors are as follow: ml1 = 760 gm,
ml2 = 720 gm, ml3 = 680 gm, ml4 = 640 gm, and finally ml5 = 600 gm. These masses are calculated for the
manipulator with l1 = 19 cm, l2 = 18 cm, l3 = 17 cm, l4 = 16 cm, l5 = 15 cm and d2 = 21 cm.
The mass of each motor is 1,500 gm; for the manipulator of the proposed design, the first motor and
the second motor are located on the base and not on the links themselves. Therefore, for our manipulator,
the mass of the first link will be equal to the mass of this link (760 gm) plus the mass of the motor
(1,500 gm) that controls the next links. Because there are no more motors, the mass of the links will be:
m1 = 2,260 gm, m2 = 720 gm, m3 = 680 gm, m4 = 640 gm, and m5 = 600 gm. Figure 9(a) shows the mass
of each link with its motor for the manipulator of the proposed design.
Figure 9. The position of mass for (a) the proposed manipulator; (b) the conventional manipulator.
z-axis
ml4
ml5
z-axis
ml4+mmotor6
y-axis
ml3
origin
ml2
mmotor2
ml1+mmotor3
target
point
target
point
ml3+mmotor5
x-axis
ml5
y-axis
origin
ml2+mmotor4
x-axis
mmotor2
ml1+mmotor3
For the conventional three dimensional planar manipulator (one motor for each link), the mass of the
first link will equal to the mass of link itself plus the mass of the motor which controls the second link
position, i.e., 760 + 1,500 gm. The mass of the second link will equal to the mass of link itself plus the
mass of the motor which controls the third link position, i.e., 720 + 1,500 gm. The mass of the third link
will equal to the mass of third link plus the mass of the motor which controls the fourth link position, i.e.,
680 + 1,500 gm. The mass of the fourth link will equal to the mass of fourth link plus the mass of the
motor which controls the fifth link position, i.e., 640 + 1,500 gm, while the mass of the last link will
equal to the mass of the link itself because there are no more motors, i.e., 600 gm. Figure 9(b) shows the
mass of each link using the manipulator with five motors, while Table 2 shows the values of mass of the
links using both the manipulator with two motors and the manipulator with five links.
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Table 2. The mass of links for both the proposed and conventional manipulators.
mn(gm)
m1(gm)
m2(gm)
m3(gm)
m4(gm)
m5(gm)
m6(gm)
Proposed manipulator
1,500
2,260
720
680
640
600
Conventional manipulator
1,500
2,260
2,220
2,180
2,140
600
It is clearly noted how the proposed method could be used to decrease the weight of manipulator.
Decreasing the weight leads to a decrease of the torques of each link. The next section shows the results
of the torques of each joint when the end-effector is following a desired path. Using the Lagrangian
formulation, the dynamical equations of motion of the manipulator is:
6
∑M
j =1
ij
q j + Vi + Gi = Qi
(10)
for i = 1,2,….,6.
The first term in this equation is the inertia forces, the second term represents the Coriolis and
centrifugal forces, and the third term gives the gravitational effects [1,20,21]. Dynamics equations of the
manipulator are discussed in details in the Appendix.
As shown by dynamics equations, increasing the weight of motors will increase the torques needed to
control the manipulator. In order to decrease the effect of the motors weight on the inertia of
manipulators, parallel manipulators are used, as we mentioned earlier. For example in reference [22], the
parallel manipulator is actuated by three servo-motors located at the base which contributes to reducing
the inertia of manipulators. Reference [23] shows another way to decrease the effect of the motors
weight on the inertia of manipulators. This reference shows a simple configuration design, which
comprises of only three joints: two at the shoulder and one at the hand. In this design, the moment of
inertia of the arm is constant and independent from the joint angles. In contrast for our manipulator, we
see from Equations A21–A25 that the moment of inertia value is dependent on the joint angles.
4. Simulation Results
This section shows the effectiveness of using the proposed manipulator to be used when it is desired
to make the end-effector follow a desired path. This section has two examples. The first example
calculates the torques using both manipulators (the proposed one and the conventional three dimensional
planar manipulator) and shows how effective the proposed manipulator is in decreasing the torque of
each joint required to move the manipulator. To verify the estimation results and compare between them
and the results measured from the manipulator itself, the second example has been shown. This example
shows the results if the torque using: (1) the conventional three dimensional planar manipulator with
defined desired joint angles path, (2) the proposed manipulator with the defined desired joint angles path
and finally (3) the proposed manipulator with the measured joint angles path when the joint angles
follow the desired joint angles path.
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Case One:
Torque of each joint for both the manipulators is calculated to show the effectiveness of using
the proposed manipulator to decrease the torque of each joint. Using the same manipulator with
l = [19,18,17,16,15]T, and d2 = 21 where all lengths are in cm, the joint angles path is defined as:
θ1 (t ) = −0.5 cos(4t )
(11)
θ 2 (t ) = − cos(2t ) + 1
(12)
θ 3 (t ) = −4 cos(t ) + 3
(13)
It should be remembered that when using the proposed manipulator, θ3, θ4, θ5, θ6 are equal. To show
the effectiveness of the proposed manipulator in decreasing the torque, Figure 10 shows the values of the
torques of the first joint using both the manipulators, the proposed manipulator (with three motors) and
the manipulator of six motors.
Figure 10. The values of the torques of the first joint using the both manipulators.
20
Conventional manipulator
T1 (N.m)
10
Proposed manipulator
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
-10
-20
-30
-40
time (sec)
Figure 11 shows the values of the torques of the second joint using the manipulators.
Figure 11. The values of the torques of the second joint using the both manipulators.
60
50
Conventional manipulator
T2 (N.m)
40
30
20
Proposed manipulator
10
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
time (sec)
Figure 12 shows the absolute values of the torques of the third joint while Figure 13 shows the
absolute values of the torques of the fourth joint using the both manipulators.
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Figure 12. The values of the torques of the third joint using the both manipulators.
35
30
Conventional manipulator
T3 (N.m)
25
20
15
10
5
Proposed manipulator
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
time (sec)
Figure 13. The values of the torques of the fourth joint using the both manipulators.
14
12
10
Conventional manipulator
T4 (N.m)
8
6
4
Proposed manipulator
2
0
-2
-4
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
time (sec)
Figure 14 shows the torques of the fifth joint and finally Figure 15 shows the torques of the sixth joint
angle using the both manipulators.
Figure 14. The values of the torques of the fifth joint using the both manipulators.
4
3
T5 (N.m)
2
Conventional manipulator
1
Proposed manipulator
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
-1
-2
-3
time (sec)
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Figure 15. The values of the torques of the sixth joint using the both manipulators.
0.8
0.6
T6 (N.m)
0.4
0.2
Proposed and
conventional manipulator
0
-0.2
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
-0.4
-0.6
-0.8
time (sec)
First of all, it is noted that the torque of the sixth joint has the same value using both the manipulators
because the sixth link has the same mass for both the manipulators, in other words the mass of the sixth
link is equal to the mass of the link itself only because it does not hold any motor.
Secondly, as mentioned earlier for the proposed manipulator, the third motor should balance the
torque of all the third, fourth, fifth and the sixth joint. In other words, the torque of the third motor should
equal to (T3 + T4 + T5 + T6) for the proposed manipulator. Figure 16 shows the power that the third motor
should balance for both the manipulators. It is noted from this example that using the proposed
manipulator not only decreases the number of motors used in the manipulator, but also decreases the
torques of the motors used to control it.
Figure 16. The values of the torques of the third motor using the both manipulators.
35
30
25
T (N.m)
20
Torque that the third motor
should balance in the
conventional manipulator
15
10
Torque that the third motor
should balance in the
proposed manipulator
5
0
-5
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45
time (sec)
Case Two:
The trajectory applied to robot in verification experiments in this case is:
θ 1 (t ) =
e 0.7 t
25
(13)
θ 2 (t ) =
e 0.5 t
3
(14)
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θ 3 (t ) =
e 0 .9 t
150
(15)
Figure 17 shows the estimated (white) and measured (red) angle, angular velocity and angular
acceleration of the first joint angle defined above. Figure 18 shows the estimated (white) and measured
(red) angle, angular velocity and angular acceleration of the second joint angle.
Figure 17. The values of the estimated and measured angular position, velocity and
acceleration of the first joint angle (white: estimated, red: measured).
Figure 18. The values of the estimated and measured angular position, velocity and
acceleration of the second joint angle (white: estimated, red: measured).
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Figure 19 shows the estimated (white) and measured (red) angle, angular velocity and angular
acceleration of the third joint angle of the manipulator. It should be remembered again that using the
proposed manipulator, θ3, θ4, θ5, θ6 are equals.
Figure 19. The values of the estimated and measured angular position, velocity and
acceleration of the third joint angle (white: estimated, red: measured).
Figures 20–25 show the comparison between the torque of each joint angle for: (1) the conventional
three dimensional planar manipulator using the estimated joint angles path; (2) the proposed manipulator
using the estimated joint angles path; and finally (3) the proposed manipulator using the measured
angular position, velocity and acceleration of the manipulator joints.
Figure 20. The torque of the first joint angle.
1.2
1
0.8
Simulation results
T1 (N.m)
0.6
Proposed manipulator
(three motors)
0.4
0.2
conventional manipulator
(six motors)
0
-0.2
-0.4
-0.6
0
5
time (sec)
10
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Figure 21. The torque of the second joint angle.
40
35
Simulation results
30
T2 (N.m)
25
Proposed manipulator
(three motors)
20
15
conventional manipulator
(six motors)
10
5
0
-5
0
5
10
time (sec)
Figure 22. The torque of the third joint angle.
25
20
Simulation results
T3 (N.m)
15
10
Proposed manipulator
(three motors)
5
0
conventional manipulator
(six motors)
-5
-10
-15
0
10
5
time (sec)
Figure 23. The torque of the fourth joint angle.
12
10
8
Simulation results
6
T4 (N.m)
4
Proposed manipulator
(three motors)
2
0
-2
conventional manipulator
(six motors)
-4
-6
-8
-10
0
5
time (sec)
10
Sensors 2012, 12
6885
Figure 24. The torque of the fifth joint angle.
4
3
Simulation results
T5 (N.m)
2
Proposed manipulator
(three motors)
1
0
conventional manipulator
(six motors)
-1
-2
-3
0
10
5
time (sec)
Figure 25. The torque of the sixth joint angle.
0.5
0.4
0.3
Simulation results
0.2
T6 (N.m)
0.1
Proposed manipulator
(three motors)
0
-0.1
conventional manipulator
(six motors)
-0.2
-0.3
-0.4
-0.5
0
5
10
time (sec)
Figure 26. The torque of the third motor.
40
Torque of the third motor (N.m)
30
Simulation results
20
Proposed manipulator
(three motors)
10
0
conventional manipulator
(six motors)
-10
-20
-30
0
5
10
time (sec)
The results obtained from verification experiments indicate that there is a good agreement between
the torque of the joint angles for the proposed manipulator using the estimated joint angles path (green)
and the measured joint angles path (red). These figures show the effectiveness of the proposed manipulator
in decreasing the torque of the joint angles using the proposed manipulator.
Sensors 2012, 12
6886
As mentioned in the first example that for the proposed manipulator, the third motor should balance
the torque of all the third, fourth, fifth and the sixth joint, i.e., the torque of the third motor should equal
(T3 + T4 + T5 + T6) for the proposed manipulator, Figure 26 shows that even though that this motor (third
motor) should balance the torques of four links, this motor could be smaller in size (less power) in the
proposed manipulator than the third motor in the conventional three dimensional planar manipulator.
5. Conclusions
This paper presents a mechanical design for a three dimensional planar redundant manipulator.
Theoretically, for each degree of freedom there should be one motor. However, in this design only three
motors are needed to control any n degrees of freedom three dimensional planar redundant manipulator.
Therefore, this design can be used to decrease the weight of the manipulator significantly. The design
steps of this manipulator are explained in detail. The dynamical equations are calculated for both the
proposed and the conventional three dimensional planar manipulators (with n motors) and it is
concluded from the result, that even though the proposed manipulator has less motors, these motors
could be even smaller (as regard to power) than the motors used with conventional three dimensional
planar manipulators.
References
1.
Tsai, L.-W. Robot Analysis: The Mechanics of Serial and Parallel Manipulators; John Wiley &
Sons, Inc.: Hoboken, NJ, USA, 1999.
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and basic experiments. In Proceeding of IEEE International Conference on Robotics and
Automation, Nice, France, 12–14 May 1992; Volume 3, pp. 2094–2100.
3. Ma, S.; Hirose, S.; Yoshinada, H. Design and experiments for a coupled tendon-driven manipulator.
IEEE Control Syst. Mag. 1993, 13, 30–36.
4. Feliu, V.; Ramos, F. Strain gauge based control of single-link flexible very lightweight robots
robust to payload changes. Mechatronics 2005, 15, 547–571.
5. Albu-Schaffer, A.; Haddadin, S.; Ott, Ch.; Stemmer, A.; Wimböck, T.; Hirzinger, G. The DLR
lightweight robot: Design and control concepts for robots in human environments. Ind. Robot Int. J.
2007, 34, 376–385.
6. Hagn, U.; Nickl, M.; Jorg, S.; Passig, G.; Bahls, T.; Nothhelfer, A.; Hacker, F.; Le-Tien, L.;
Albu-Schaffer, A.; Konietschke, R.; et al. The DLR MIRO: A versatile lightweight robot for
surgical applications. Ind. Robot: Int. J. 2008, 35, 324–336.
7. Hirose, S.; Ma, S. Moray drive for multijoint. In Proceeding of 5th International Conference of
Advanced Robotics, Pisa, Italy, 19–22 June 1991; Volume 1, pp. 521–526.
8. Merlet, J.P. Parallel Robots, 2nd ed.; Springer: Dordrecht, The Netherlands, 2006.
9. Londi, F.; Pennestri, E.; Valentini, P.P.; Vita, L. Control and virtual reality simulation of tendon
driven mechanisms. Multibody Syst. Dyn. 2004, 12, 133–145.
10. Rodriguez-Donate, C.; Osornio-Rios, R.A.; Rivera-Guillen, J.R.; de Jesus Romero-Troncoso, R.
Fused smart sensor network for multi-axis forward kinematics estimation in industrial robots.
Sensors 2011, 11, 4335–4357.
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11. Lee, J.K.; Kim, K.; Lee, Y.; Jeong, T. Simultaneous intrinsic and extrinsic parameter identification
of a hand-mounted laser-vision sensor. Sensors 2011, 11, 8751–8768.
12. Yahya, S.; Moghavvemi, M.; Mohamed, H.A.F. Geometrical approach of planar hyper-redundant
manipulators: Inverse kinematics, path planning and workspace. Simul. Model. Pract. Theory 2011,
19, 406–422.
13. Mohamed, H.A.F.; Yahya, S.; Moghavvemi, M.; Yang, S.S. A new inverse kinematics method for
three dimensional redundant manipulators. In ICROS-SICE; Fukuoka International Congress
Center, Fukuoka, Japan, 18–21 August 2009; pp. 1557–1562.
14. Yahya, S.; Moghavvemi, M.; Mohamed, H.A.F. Singularity avoidance of a six degrees of freedom
three dimensional redundant planar manipulator. Comput. Math. Appl. 2012, doi:10.1016/j.camwa.
2011.12.073.
15. Yahya, S.; Moghavvemi, M.; Mohamed, H.A.F. A review of Singularity avoidance in the inverse
kinematics of redundant robot manipulators. Int. Rev. Autom. Control 2011, 4, 807–814.
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London, UK, 2002.
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& Sons, Inc.: Hoboken, NJ, USA, 2004.
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Appendix
This section explains the Dynamics of the manipulator used in experiments. In Equation (10):
6
(
T
M = ∑ J υTi mi J υi + J wi
I i J wi
i =1
)
(A1)
where Jυi is the Jacobian submatrix assosiated with the linear velocity of the center of mass of link i and
Jwi is the Jacobian submatrix assosiated with the angular velocity vector of link i.
J υi = [J υ1i , J υ2i ,...., J υi i ,0,0,...., 0 ]
(A2)
[
1
J wi = J wi
, J wi2 ,...., J wii ,0,0,...., 0
]
(A3)
Sensors 2012, 12
6888
where J υji and J wij are the j th column vectors of Jυi and Jwi, respectively. Applying the theory of
instantaneous screw motion [1] for j ≤ i, and because all the joints are revolute joints, we obtain:
J υji = z j −1 × j −1 pci∗
(A4)
J wij = z j −1
(A5)
The link Jacobian submatrices, Jυi for the proposed manipulator can be obtained as:
J υ1
⎡0 0 0 0 0 0 ⎤
= ⎢⎢0 0 0 0 0 0 ⎥⎥
⎢⎣0 0 0 0 0 0 ⎥⎦
(A6)
Jυ1 is a zero matrix because from the link parameters of the manipulator of Table 1, it is noted that a1
and d1 are equal to zero because the first motor is set on the base.
Jυ 4
Jυ 5
⎤
0 0 0 0⎥
⎥
0 0 0 0⎥
⎥
0 0 0 0⎥⎥
⎦
Jυ 2
l1
⎡
⎢ d1c1 − 2 s1c 2
⎢
l
= ⎢ d1 s1 + 1 c1c 2
2
⎢
⎢
0
⎢⎣
Jυ 3
⎡
⎤
l2
l2
⎛
⎞
⎛
⎞
⎛ l2
⎞
⎢ d 1c1 − s1 ⎜ l1c 2 + 2 c 23 ⎟ − c1 ⎜ l1 s 2 + 2 s 23 ⎟ − c1 ⎜ 2 s 23 ⎟ 0 0 0 ⎥
⎝
⎠
⎝
⎠
⎝
⎠
⎢
⎥
l2
l2
⎛
⎞
⎛ l2
⎞
⎛
⎞
⎢
= d 1 s1 + c1 ⎜ l1c 2 + c 23 ⎟ − s1 ⎜ l1 s 2 + s 23 ⎟ − s1 ⎜ s 23 ⎟ 0 0 0 ⎥
⎢
⎥
2
2
⎝
⎠
⎝2
⎠
⎝
⎠
⎢
⎥
l
l2
⎢
0
0 0 0⎥
l1c 2 + 2 c 23
c 23
2
2
⎣⎢
⎦⎥
l1
c1 s 2
2
l
− 1 s1 s 2
2
l1
c2
2
−
⎡
⎤
l3
l3
l3
⎛
⎞
⎛
⎞
⎛
⎞
⎛ l3
⎞
⎢ d 1c1 − s1 ⎜ l1c 2 + l 2 c 23 + 2 c 234 ⎟ − c1 ⎜ l1 s 2 + l 2 s 23 + 2 s 234 ⎟ − c1 ⎜ l 2 s 23 + 2 s 234 ⎟ − c1 ⎜ 2 s 234 ⎟ 0 0 ⎥
⎝
⎠
⎝
⎠
⎝
⎠
⎝
⎠
⎢
⎥
l3
l3
l3
⎞
⎛
⎞
⎛
⎞
⎛
⎞
⎛ l3
⎢
= d 1 s1 + c1 ⎜ l1c 2 + l 2 c 23 + c 234 ⎟ − s1 ⎜ l1 s 2 + l 2 s 23 + s 234 ⎟ − s1 ⎜ l 2 s 23 + s 234 ⎟ − s1 ⎜ s 234 ⎟ 0 0 ⎥
⎢
⎥
2
2
2
⎝
⎠
⎝
⎠
⎝
⎠
⎝2
⎠
⎢
⎥
l
l
l3
⎢
0
0 0⎥
l1c 2 + l 2 c 23 + 3 c 234
l 2 c 23 + 3 c 234
c 234
2
2
2
⎢⎣
⎥⎦
(A7)
(A8)
(A9)
⎡
l4
l4
l4
⎛
⎞
⎛
⎞
⎛
⎞
⎢ d1c1 − s1 ⎜ l1c 2 + l 2 c 23 + l3 c 234 + 2 c 2345 ⎟ − c1 ⎜ l1 s 2 + l 2 s 23 + l3 s 234 + 2 s 2345 ⎟ − c1 ⎜ l 2 s 23 + l3 s 234 + 2 s 2345 ⎟
⎠
⎝
⎠
⎝
⎠
⎝
⎢
l4
l4
l4
⎛
⎞
⎛
⎞
⎛
⎞
⎢
= d1 s1 + c1 ⎜ l1c 2 + l 2 c 23 + l3 c 234 + c 2345 ⎟ − s1 ⎜ l1 s 2 + l 2 s 23 + l3 s 234 + s 2345 ⎟ − s1 ⎜ l 2 s 23 + l3 s 234 + s 2345 ⎟
⎢
2
2
2
⎝
⎠
⎝
⎠
⎝
⎠
⎢
l4
l4
⎢
0
l1c 2 + l 2 c 23 + l3 c 234 + c 2345
l 2 c 23 + l3 c 234 + c 2345
(A10)
2
2
⎢⎣
l
⎞
⎛
⎛l
⎞ ⎤
− c1 ⎜ l3 s 234 + 4 s 2345 ⎟ − c1 ⎜ 4 s 2345 ⎟ 0⎥
2
⎝
⎠
⎝2
⎠ ⎥
l4
⎛
⎞
⎛ l4
⎞ ⎥
− s1 ⎜ l3 s 234 + s 2345 ⎟ − s1 ⎜ s 2345 ⎟ 0
2
⎝
⎠
⎝2
⎠ ⎥
⎥
l
l4
0⎥
l3 c 234 + 4 c 2345
c 2345
2
2
⎥⎦
Sensors 2012, 12
Jυ 6
6889
⎡
l5
l5
⎛
⎞
⎛
⎞
⎢ d1c1 − s1 ⎜ l1c 2 + l 2 c 23 + l3 c 234 + l 4 c 2345 + 2 c 23456 ⎟ − c1 ⎜ l1 s 2 + l 2 s 23 + l3 s 234 + l 4 s 2345 + 2 s 23456 ⎟
⎝
⎠
⎝
⎠
⎢
l5
l5
⎛
⎞
⎛
⎞
⎢
= ⎢d 1 s1 + c1 ⎜ l1c 2 + l 2 c 23 + l 3 c 234 + l 4 c 2345 + c 23456 ⎟ − s1 ⎜ l1 s 2 + l 2 s 23 + l 3 s 234 + l 4 s 2345 + s 23456 ⎟
2
2
⎝
⎠
⎝
⎠
⎢
l
5
⎢
l1c 2 + l 2 c 23 + l3 c 234 + l 4 c 2345 + c 23456
0
2
⎢⎣
l
l
l
⎛
⎞
⎛
⎞
⎛
⎞
− c1 ⎜ l 2 s 23 + l3 s 234 + l 4 s 2345 + 5 s 23456 ⎟ − c1 ⎜ l3 s 234 + l 4 s 2345 + 5 s 23456 ⎟ − c1 ⎜ l 4 s 2345 + 5 s 23456 ⎟
2
2
2
⎝
⎠
⎝
⎠
⎝
⎠
l5
l5
l5
⎛
⎞
⎛
⎞
⎛
⎞
− s1 ⎜ l 2 s 23 + l 3 s 234 + l 4 s 2345 + s 23456 ⎟ − s1 ⎜ l 3 s 234 + l 4 s 2345 + s 23456 ⎟ − s1 ⎜ l 4 s 2345 + s 23456 ⎟
2
2
2
⎝
⎠
⎝
⎠
⎝
⎠
l5
l5
l5
l 2 c 23 + l 3 c 234 + l 4 c 2345 + c 23456
l3 c 234 + l 4 c 2345 + c 23456
l 4 c 2345 + c 23456
2
2
2
(A11)
⎛l
⎞⎤
− c1 ⎜ 5 s 23456 ⎟⎥
⎝2
⎠⎥
⎛ l5
⎞⎥
− s1 ⎜ s 23456 ⎟
⎝2
⎠⎥
⎥
l5
c 23456 ⎥
2
⎥⎦
To formulate the submatrices of Jwi, following equations are used:
J w1
⎡0 0 0 0 0 0⎤
= ⎢⎢0 0 0 0 0 0 ⎥⎥
⎢⎣1 0 0 0 0 0 ⎥⎦
⎡0 s1
J w 2 = ⎢⎢0 − c1
⎢⎣1
0
(A12)
0 0 0 0⎤
0 0 0 0⎥⎥
0 0 0 0⎥⎦
(A13)
⎡0 s1
= ⎢⎢0 − c1
⎢⎣1
0
s1
− c1
J w4
⎡0 s1
= ⎢⎢0 − c1
0
⎣⎢1
s1
− c1
0
s1
− c1
0
0 0⎤
0 0 ⎥⎥
0 0 ⎥⎦
J w5
⎡0 s1
= ⎢⎢0 − c1
⎢⎣1
0
s1
− c1
0
s1
− c1
0
s1
− c1
0
0⎤
0 ⎥⎥
0 ⎥⎦
J w6
⎡0 s1
= ⎢⎢0 − c1
⎢⎣1
0
s1
− c1
s1
− c1
s1
− c1
0
0
0
s1 ⎤
− c1 ⎥⎥
0 ⎥⎦
J w3
0
0 0 0⎤
0 0 0⎥⎥
0 0 0⎥⎦
(A44)
(A15)
(A16)
(A17)
Now to formulate the inertia matrices of the manipulator, assuming that the moving links are
homogeneous with relatively small cross section, the inertia matrix iIi of link i about its center of mass
and expressed in the link link frame i th link fram is:
i
⎡0 0 0 ⎤
1
2⎢
I i = mi a i ⎢0 1 0 ⎥⎥
12
⎢⎣0 0 1 ⎥⎦
(A18)
for I = 1,2,…,6.
Now to find Ii inertia matrix of link i about its center of mass and expressed in the base link frame, the
following equation is used:
I i = 0Ri i I i ( 0Ri )T
(A19)
Sensors 2012, 12
6890
Using Equations (A18) and (A19), we obtain:
⎡0 0 0 ⎤
I 1 = ⎢⎢0 0 0⎥⎥
⎣⎢0 0 0⎥⎦
(A20)
I1 is a zero matrix because a1 is equal to zero.
(
⎡ s12 + c12 c 22
ml ⎢
I2 =
s1c1 s 22 − 1
12 ⎢
⎢ − c1 (s 2 c 2 )
⎣
2
2 1
(
)
− c1 (s 2 c 2 )⎤
⎥
− s1 (s 2 c 2 )⎥
⎥
c 22
⎦
s1c1 s 22 − 1
c12 + s12 s 22
)
− s1 (s 2 c 2 )
(
(A21)
)
2
2
⎡ s12 + c12 c 23
− 1 − c1 (s 23 c 23 )⎤
s1c1 s 23
m3l 22 ⎢
⎥
2
2
2 2
s1c1 s 23 − 1 c1 + s1 s 23 − s1 (s 23 c 23 )⎥
I3 =
12 ⎢
2
⎢− c1 (s 23 c 23 ) − s1 (s 23 c 23 )
⎥
c 23
⎣
⎦
(
)
(
(A22)
)
2
2
⎡ s12 + c12 c 234
− 1 − c1 (s 234 c 234 )⎤
s1c1 s 234
m 4 l 32 ⎢
⎥
2
2
2 2
− s1 (s 234 c 234 )⎥
s1c1 s 234 − 1
c1 + s1 s 234
I4 =
⎢
12
2
⎢ − c1 (s 234 c 234 ) − s1 (s 234 c 234 )
⎥
c 234
⎣
⎦
(
)
(
(A23)
)
2
2
⎡ s12 + c12 c 2345
− 1 − c1 (s 2345 c 2345 )⎤
s1c1 s 2345
m5 l 42 ⎢
⎥
2
2
2 2
− s1 (s 2345 c 2345 )⎥
s1c1 s 2345 − 1
c1 + s1 s 2345
I5 =
⎢
12
2
⎢− c1 (s 2345 c 2345 ) − s1 (s 2345 c 2345 )
⎥
c 2345
⎣
⎦
(
)
(
(A24)
)
2
2
⎡ s12 + c12 c 23456
−1
− c1 (s 23456 c 23456 )⎤
s1c1 s 23456
m6 l52 ⎢
⎥
2
2
2 2
− s1 (s 23456 c 23456 )⎥
s1c1 s 23456 − 1
c1 + s1 s 23456
I6 =
⎢
12
2
⎢ − c1 (s 23456 c 23456 ) − s1 (s 23456 c 23456 )
⎥
c 23456
⎣
⎦
(
)
(A25)
Now to find the inertia manipulator matrix M, Equation (A1) can be rewritten as:
M = J υT1 m1 J υ 1 + J wT1 I 1 J w1 + J υT2 m 2 J υ 2 + J wT 2 I 2 J w 2 + J υT3 m3 J υ 3 + J wT 3 I 3 J w3 + J υT4 m 4 J υ 4 + J wT 4 I 4 J w 4
+ J υT5 m5 J υ 5 + J wT 5 I 5 J w5 + J υT6 m6 J υ 6 + J wT 6 I 6 J w 6
(A26)
Therefore, Equation (10) can be rewritten as:
⎡τ 1 ⎤ ⎡ M 11
⎢τ ⎥ ⎢ M
⎢ 2 ⎥ ⎢ 21
⎢τ 3 ⎥ ⎢ M 31
⎢ ⎥=⎢
⎢τ 4 ⎥ ⎢ M 41
⎢τ 5 ⎥ ⎢ M 51
⎢ ⎥ ⎢
⎢⎣τ 6 ⎦⎥ ⎣⎢ M 61
M 12
M 22
M 32
M 42
M 13
M 23
M 33
M 43
M 14
M 24
M 34
M 44
M 15
M 25
M 35
M 45
M 52
M 62
M 53
M 63
M 54
M 64
M 55
M 65
M 16 ⎤ ⎡θ1 ⎤ ⎡V1 (θ ,θ) ⎤ ⎡ G1 (θ ) ⎤
⎥
⎢ ⎥ ⎢
M 26 ⎥⎥ ⎢θ2 ⎥ ⎢V2 (θ ,θ) ⎥ ⎢⎢G 2 (θ ) ⎥⎥
M 36 ⎥ ⎢θ3 ⎥ ⎢V3 (θ ,θ) ⎥ ⎢G3 (θ ) ⎥
⎥+⎢
⎥
⎥⎢ ⎥ + ⎢
M 46 ⎥ ⎢θ4 ⎥ ⎢V4 (θ ,θ) ⎥ ⎢G 4 (θ ) ⎥
M 56 ⎥ ⎢θ5 ⎥ ⎢V5 (θ ,θ) ⎥ ⎢G5 (θ ) ⎥
⎥ ⎢
⎥
⎥⎢ ⎥ ⎢
M 66 ⎦⎥ ⎢⎣θ6 ⎥⎦ ⎢⎣V6 (θ ,θ) ⎥⎦ ⎣⎢G6 (θ ) ⎦⎥
3
3 cos
(A27)
where:
M
1
4
12
3
cos
4
1
12
2 cos
4
1
2 cos
2 cos
4
12
1
2 cos
cos
4
cos
2 cos
3
cos
cos
12
2 cos
cos
cos
cos
cos
cos
cos
cos
cos
Sensors 2012, 12
M
1
2
M
M
6891
sin
2 sin
2
1
2
M
2
sin
sin
2 sin
sin
sin
2
sin
sin
2 sin
2
sin
sin
2
sin
1
2
M
M
M
sin
2
2
sin
1
d l m sin θ
2
M
θ
1
192
192
θ
1
2
M
M
θ
cos
2 cos
2
cos 2
2
cos 4
1 cos 2
cos
96 2
cos
2 cos
48
M
M
M
M
M
1
2
192
2
1
96
2
M
cos 4
cos 2 2
cos
2
cos
192
96
32
cos
cos
4
sin 2 cos 2
cos 2
16
48
192
cos
sin 2
2
cos 4
2
16
cos
48
64
2
cos
cos 2
2
cos 2 2
2
cos 2
cos 4
2
cos 4
1 cos 2
96 2
cos
cos
2
cos
64
192
cos 2 2
cos
cos
192
96
96
cos
4
64
sin 2
2
cos 2
cos 2 2
192
cos
cos 2 2
2
32
cos 4
1 cos 2
cos
96 2
cos
2
cos
2
2
cos
192
64
cos
1 cos 2
48
cos
96
2 cos
1
48 cos
96
cos
θ
cos
cos 2
2
cos 2
48
θ
cos 2 2
64
192
1
192
cos
192
cos
2
cos 4
2
32
θ
cos
cos
4
2
2m sin θ
sin
2
M
l m
θ
2
M
sin
2
96
cos
cos
96
2
cos
cos
cos 4
cos
cos
cos
32
2
96
cos
cos 4
16
cos 4
cos 2
cos 4
32
cos
1 cos 2
cos
cos
cos
32
1 cos 2
cos
cos
cos
64
cos 4
Sensors 2012, 12
6892
1
96
96
M
2
32
cos 4
1 cos 2
2
cos
1
96
M
M
cos
2
96
M
cos
48
M
M
M
cos 4
96
cos
1 cos 2
96
2
32
1 cos 2
2 cos
96
cos 4
1 cos 2
48
cos
cos
cos
cos
1 cos 2
32
32
cos 4
48 cos
cos 4
cos
96
32
96
cos 4
1 cos 2
32
cos
cos 4
32
1
96
M
1 cos 2
32
48
32
cos
cos 4
1 cos 2
48 cos
1
96
cos 4
32
2 cos
1 cos 2
96
1 cos 2
cos 4
cos
96
48 cos
cos 4
96
1 cos 2
1 cos 2
48 cos
1
96
M
32
1 cos 2
1
96
M
M
cos 4
cos
cos
1
96
cos 4
1 cos 2
2
cos
cos 4
48 cos
32
cos
32
M
96
32
32
cos
32
cos 4
cos
16
96
cos 2
cos
48 cos
1
96
M
M
96
cos
cos
1
96
M
96
sin 2
1 cos 2
cos
cos
48
96
96
cos 4
2
2
1 cos 2
2
32
96
M
cos 4
1 cos 2
cos
cos
32
96
32
cos 4
cos 4
1 cos 2
1 cos 2
The second term (the Coriolis and centrifugal forces term) and third term (gravitational effects term)
of Equation (A27) can be calculated as follow:
6
Vi = ∑
j =1
6
⎛ ∂M ij
∑ ⎜⎜ ∂q
k =1
⎝
k
−
1 ∂M jk
2 ∂qi
⎞
⎟⎟q j q k
⎠
(A28)
6
Gi = −∑ m j g T J υi j
j =1
(A29)
© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article
distributed under the terms and conditions of the Creative Commons Attribution license
(http://creativecommons.org/licenses/by/3.0/).
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