This paper presents the kinematic modeling and singularity analysis of a three-degree-of-freedom, double four-bar closed-loop quadruped leg mechanism. This leg is essential for transitioning between the two operational modes; therefore, it is examined in both joint and task spaces, leading to the development of its Jacobian matrix, which is vital for designing the gaits required for terrestrial mode and mid-air aerial reconfiguration (for aerial mode). Various gaits are devised for the system's motion in the MATLAB simulation, and their practical implementation on the single-leg test rig is demonstrated.
Figures & tables
Figure 1: (a) Ground and aerial bio-inspired robot (b) Aerial and aquatic bio-inspired robot
Figure 2: (a) Wheel and rotor-based multimodal robot in terrestrial configuration (b) Aerial configuration
Figure 3: (a) leg with rotors in terrestrial mode (b) leg with rotors in aerial mode
Figure 4: (a) Shape-morphable robot in terrestrial mode (b) Morph-material working (c) Aerial mode
Figure 5: CAD model of the proposed design of the multimodal reconfigurable robot
Figure 6: Transformation stages of reconfiguration between quadruped mode and pentarotor aerial mode
Frame ID
αi−1 (rad)
ai−1 (m)
θi (rad)
di (m)
1
3π/2
0
3π/2
y
2
3π/2
0
3π/2
x
3
3π/2
0
3π/2
z
4
0
0
ψ
0
5
3π/2
0
π/2+θ
0
6
π/2
0
3π/2+ϕ
0
Table 1: DH parameter table of the quadruped for the front-right locomotor.
Figure 7: DH Table of quadruped for front right locomotor
Parameter
Magnitude
Description
l
0.19 m
Length of the robot
w
0.14 m
Width of the robot
h1
0.04 m
Length between frames {FR0} and {FR1}
h2
0.05 m
Length between frames {FR1} and {FR2}
s1
0.11 m
Length of the thigh link
s2
0.14 m
Length of the shank link
Table 2: Geometrical parameters of the system.
Figure 8: Front view of the leg to calculate first joint angle, θ1
Figure 9: Effect of the joint angle θ1 on the pose of the leg in front view
Figure 10: Side view of the leg to calculate the angles, θ2 and θ3
Figure 11: Effect of rotation of the floating base
Figure 12: Leg’s pose with roll offset implementation
Figure 13: Leg’s pose with pitch offset
Figure 14: Test rig for the analysis of a single locomotor module
Figure 15: Kinematic diagrams of different kinematic chains of the test rig.
Table 18
Figure 16: Kinematic diagram of the rig with two closed-loop kinematic chains
Figure 17: Diagrammatic representation of mapping between different spaces of the rig
Figure 18: Reachable workspace of the leg in task space
Figure 19: Distribution of joint angles in the task space.
Figure 20: Workspace of the leg in joint-space
Figure 21: Singularity in the workspace in Cartesian space
Figure 22: Joint space singularity
Figure 23: Schematic diagram of the communication between the computer and the experimental setup
Figure 24: Single-leg test rig for the analysis of workspace
Figure 25: Single-leg test rig with all three joints and EDF
Figure 26: Time lapse of the workspace boundary generation on the test rig of the leg
Figure 27: Planned crouch gait within the workspace (Left), with transition on test bench (Right)
Figure 28: Planned trot gait overlaid onto the workspace (left), with temporal frame-wise representation onto the test bench (right)
Figure 29: Planned squat gait, overlaid onto the workspace (left), with time lapse of squat gait on hardware (right)
Figure 30: Time lapse of reconfiguration of the locomotor
School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA 19104, USA · Department of Aeronautics & Astronautics, Stanford University, Stanford, CA 94305, USA