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
LIPM is everywhere in legged-locomotion control, but almost always as a modeling choice rather than as something the controller's cost actually prefers. This note tries to make that link more explicit. Working from a small centroidal OCP that penalizes the rate of angular momentum, we look at what its optimum tends to look like. Three things come out. With full-rank stance, the optimum drifts toward a pendular force pattern at a rate determined by the SVD of the moment Jacobian; the constant is set by foot-span geometry and matches the experiments to within 16%. With N=2 stance, as in trot, the friction cone introduces a lower bound on ∥H˙G∥ that no amount of weight tuning fixes; we also see a non-smooth feasibility kink at a critical horizontal acceleration that we can write in closed form. Adding a task term that asks for a nonzero H˙G moves the optimum off the pendular set in a predictable way. None of this is far from the classical ZMP/DCM picture. We test these claims on a point-mass quadruped and on the Unitree Go1 in MuJoCo (open-loop QP and a torque-level closed-loop controller), and we note where the asymptotic story stops being a good description of what the closed loop actually does.
Lingxue Lyu, Zihui Liu
School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA 19104, USA · Department of Aeronautics & Astronautics, Stanford University, Stanford, CA 94305, USA
Physical joint limits are common in legged robots and can restrict workspace, constrain gait design, and increase the risk of hardware damage. This paper introduces MiNI-Q^2, a miniature, wire-free quadruped robot with independently actuated, mechanically unbounded 2-DOF leg joints. We present the mechanical design, kinematic analysis, and experimental validation of the proposed robot. The leg mechanism enables both oscillatory gaits and rotary locomotion while allowing the robot to fold to a minimum height of 2.5 cm. Experimentally, MiNI-Q achieves speeds up to 0.46 m/s and demonstrates low-clearance crawling, stair climbing, inverted locomotion, jumping, and backflipping. The wire-free architecture extends our previous Q8bot design, improving assembly reliability at miniature scale. All mechanical and electrical design files are released open source to support reproducibility and further research.
Daniel Koh, Suraj Shah, Yufeng Wu +1
Department of Mechanical and Aerospace Engineering, UCLA, Los Angeles, CA 90095, USA.
Robotics would gain by replicating the remarkable agility of arthropods in navigating complex environments. Here we consider the control of multi-legged systems which have 6 or more legs. Current multi-legged control strategies in robots include large black-box machine learning models, Central Pattern Generator (CPG) networks, and open-loop feed-forward control with stability arising from mechanics. Here we present a multi-legged control architecture for rough terrain using a segmental robot with 3 actuators for every 2 legs, which we validated in simulation for robots with 6 to 16 legs. Segments have identical state machines, and each segment also receives input from the segment in front of it. Our design bridges the gap between WalkNet-like event cascade controllers and CPG-based controllers: it tightly couples to the ground when contact is present, but produces fictive locomotion when ground contact is missing. The approach may be useful as an adaptive and computationally lightweight controller for multi-legged robots, and as a baseline capability for scaffolding the learning of machine learning controllers.
Zhuoyang Chen, Xinyuan Wang, Shai Revzen
The authors are with the University of Michigan, Ann Arbor, MI 48109, USA.