Climbing favours the tripod gait over alternative faster insect gaits

Climbing favours the tripod gait over alternative faster insect gaits
Supplementary Figure 1 | An insect model based on Drosophila melanogaster. (a)
Side and ventral images of adult female flies used to calculate the sizes of body and leg
segments. Scale bar is 0.3 mm. Green, yellow, and red lines illustrate examples of leg,
head, and thoracic measurements, respectively. (b) Corresponding side and ventral
views of the insect model. Scale bar is 0.3 mm. (c) Image of the model’s front right leg.
Leg segments and the degrees of freedom for each joint are labeled in black and grey,
respectively. (d, e) Sample high-speed video images of D. melanogaster walking (grey)
are overlaid by semi-transparent images of the insect model as seen from the side (d)
or from below (e).
Supplementary Figure 2 | Ground reaction forces for the insect model. Ground
reaction forces (GRF) for ideal (a) tripod-A and (b) bipod-B gaits. Shown are GRFs for
each leg along the anterioposterior axis (left; positive values indicate GRFs pointing in
the forward direction – propulsive forces), mediolateral axis (middle; positive values
indicate GRFs pointing medially), and normal axis (right; positive values indicate GRFs
pointing away from the surface). Gray boxes highlight stance epochs for each leg during
tripod-A and bipod-B locomotion. Gray arrowheads indicate an instance of ground
contact with minimal normal force.
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Supplementary Figure 3 | Convergence of fastest forward locomotor velocities
during gait optimization. Forward velocities of the fastest individuals for each iteration
during gait optimization for forward velocity while (a) climbing upward, (b) downward, (c)
or sideways on a vertical surface using leg adhesion, (d) walking on the ground with leg
adhesion, or (e) walking on the ground without leg adhesion. N = 15 experiments per
condition. Each trace represents a single experiment and is color-coded according to
the gait class of the experiment’s fastest individual.
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Supplementary Figure 4 | Gait classes and quantitative justification of gait
classification. (a) Representative and (b) idealized footfall diagrams showing stance
(black) and swing (white) phases for each of the six gait classes identified. Two walking
cycles are shown for each footfall diagram. The phase of motion for each leg is
indicated. (c-h) Sum of the difference between leg phases of motion for each optimized
gait (sorted by class) versus the idealized (c) tripod-A, (d) tripod-B, (e) tripod-C, (f)
bipod-A, (g) bipod-B, or (h) bipod-C gait. Optimized gaits are color-coded by class. Data
points are randomly scattered along the x-axis for clarity. Grey boxes highlight
optimized gaits within their own, assigned class.
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Supplementary Figure 5 | Footfall diagrams for each optimized gait. Footfall
diagrams showing stance (black) and swing (white) periods for each experiment. Shown
are results for gait optimization of forward velocity while (a) climbing upward, (b)
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climbing downward, (c) or climbing sideways on a vertical surface using leg adhesion,
(d) walking on the ground with leg adhesion, or (e) walking on the ground without leg
adhesion.
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Supplementary Figure 6 | Duty factors for each optimized gait. The duty factor or
fraction of time each leg is in contact with the substrate relative to the stride period for
all optimized gaits. Shown are duty factors of gaits optimized for (a) climbing upward,
(b) climbing downward, (c) or climbing sideways on a vertical surface using leg
adhesion, (d) walking on the ground with leg adhesion, or (e) walking on the ground
without leg adhesion. A dashed black line indicates 50% time in contact with the
substrate. Optimized gaits are color-coded by class. Data points are randomly scattered
along the x-axis for clarity. N = 15 for each condition.
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Supplementary Figure 7 | Cost of transport for optimized gaits. The cost of
transport (dimensionless) of gaits optimized for forward velocity while climbing upward
(left), climbing downward (center-left), or climbing sideways (center) on a vertical
surface using leg adhesion, walking on the ground with leg adhesion (center-right), or
walking on the ground without leg adhesion (right). Optimized gaits are color-coded by
class. Data points are randomly scattered along the x-axis for clarity. N = 15 for each
condition.
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Supplementary Figure 8 | Transferring bipod and tripod gaits to a hexapod robot.
(a) Image of the robot’s leg. Degrees of freedom for each joint are labeled in black text.
(b) Inverse kinematics approach for mapping the position of the robot’s pretarsus (x 1,y1)
to the model’s pretarsus despite a reduction from four to two flexion/extension joints.
Joint angles are indicated in red. Leg segment lengths are shown in black. (c)
Visualization of a matched leg trajectory (orange) for the right middle leg pretarsus of
the robot (red) and the model (blue). A yellow arrow indicates the direction of heading.
(d) To track the robot’s legs automatically, red tape was affixed to their tips. A black
arrow indicates the direction of heading. (e) The forward displacement of each of the
robot’s legs during tripod (top), or bipod-B (bottom) locomotion. Scale bar is 6 cm.
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Supplementary Figure 9 | Optimized gaits for models of different sizes. Gaits were
optimized for 25 mm, or 250 mm long models for forward velocity while climbing upward
(left and middle-left), or walking on the ground without leg adhesion (middle-right and
right). (a) Tripod Coordination Strength (TCS) values indicating the degree of similarity
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to the classic tripod gait footfall diagram (tripod-A). (b) The average number of legs in
stance phase over five walking cycles. A dashed black line indicates three legs in
stance phase as expected for the classic tripod-A gait. (c) The percentage of time that
the model’s center of mass (COM) lies within a polygon of support delineated by each
leg in stance phase when the gait is tested during ground walking. Optimized gaits are
color-coded by class. Data points are randomly scattered along the x-axis for clarity. N =
15 for each condition.
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Body part
Type
Diameter
Length
Thickness
Mass
(mm)
(mm)
(mm)
(mg)
Abdomen
Capsule
0.8925
0.595
-
0.0062
Thorax
Sphere
0.952
-
-
0.0124
Head
Capsule
0.595
0.1785
-
0.0124
Wing
Pill-
1.19
1.2495
0.0595
1.236
x 10-5
shaped
Eye
Sphere
0.4165
-
-
(part
of
head)
Supplementary Table 1 | Geometric dimensions of the model’s body. For
experiments with larger models (25 mm and 250 mm in length) all dimensions were
scaled up while keeping the density of each body part the same.
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Body part
Type
Coxa
Capsule
Trochanter/Femur Capsule
Tibia
Capsule
Tarsus
Capsule
Pretarsus
Diameter
Length
Mass
(mm)
(mm)
(mg)
0.1547;
0.1547;
0.0494
0.1547;
0.0952;
0.1547
0.2737
0.1309;
0.5653;
0.1309;
0.5177;
0.1309
0.4879
0.0952;
0.5534;
0.0952;
0.4879;
0.0952
0.4165
0.0714;
0.6069;
0.0714;
0.5415;
0.0714
0.5355
0.119;
-; -; -
Sphere
0.0247
0.0247
0.0247
0.0124
0.119; 0.119
Supplementary
Table
2
|
Geometric
dimensions
of
the
model’s
legs
(hind/metathoracic; middle/mesothoracic; front/prothoracic). For experiments with
larger models (25 mm and 250 mm in length) all dimensions were scaled up while
keeping the density of each body part the same.
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basicTimeStep
0.2 ms
maxVelocity
100 rad s-1
maxForce
(torque for a rotational
2.1 x 10-8 Nm
joint/motor)
control
50
acceleration
not limited
springConstant
0
dampingConstant
0
Supplementary Table 3 | General and joint parameters values.
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Leg
Body-
Body-
Body-
Coxa-
Femur-
Tibia-
Coxa
Coxa
Coxa
Femur
Tibia
Tarsus
promotion/
abduction/
rotation
flexion/
flexion/
flexion/
remotion
adduction
extension
extension
extension
Hind
[-75, -45]
[40, 59]
[-55, -20]
[40, 107.2]
[50, 135]
20.5
Middle
[-25, 25]
25.44
0
[80, 90]
[80, 90]
25.5
Front
[70, 80]
[-40, 10]
[0, 40]
[90, 160]
[55, 125]
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Type
Supplementary Table 4 | The ranges of motion for each of the model’s joints
(degrees). Intervals indicate ranges, single values indicate constant position without
oscillation.
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Leg
Body-
Body-
Body-
Coxa-
Femur-
Tibia-
Coxa
Coxa
Coxa
Femur
Tibia
Tarsus
promotion/
abduction/
rotation
flexion/
flexion/
flexion/
remotion
adduction
extension
extension
extension
Hind
180
0
180
200
180
0
Middle
180
0
0
270
90
0
Front
0
210
0
0
20
0
Type
Supplementary Table 5 | The relative phase of oscillation for each of the model’s
joints (degrees).
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