In this paper, we explore the impulsive dynamics common to single-joint, two-link models of walking and brachiating gaits with respect to slope and switching time. In particular, we investigate how the stability of a gait and bifurcations encountered within a family of gaits change under time-based and state-based switching of the impulsive dynamics.
Figures & tables
Figure 1 : Example period-one walking and brachiating gaits of a two-link robot.
Figure 2 : (left plot) Curves representing two symmetric gait families projected onto the q1(0) - q2(0) (the configuration q(t) at t=0 ) plane under a TBS strategy. Red represents unstable gaits and blue stable gaits. The black arrows depict unstable gaits on one family converging to stable gaits on the mirrored family. Each curve has NS and FD bifurcation points. Finally, the top-left quadrant shows the model, and the bottom-right quadrant shows an example brachiating gait on a curve. Its mirrored gait (not shown) brachiates in the opposite direction. (right plot) The evolution of the example brachiating gait, which is unstable. Under SBS, it converges to a stable period-one fixed point. With TBS, it converges to a stable period-seven fixed point.
Figure 3 : (top) Example gait families with their stability properties (red unstable, blue stable) and bifurcation points under SBS. (below) A similar plot, but with gaits subject to TBS. To avoid overlapping curves, we plot joint angles beyond a range of 2π on the left axes of each plot. The corresponding values for q1(0)∈(−π,π] under modulo arithmetic are shown on the right axes.
Figure 4 : (top left) A brachiating branch of gaits (part of the top-most curve in Figure 3 ) undergoing a PD bifurcation under an SBS strategy. (top right) The same branch undergoing an NS bifurcation under a TBS strategy. (bottom left) The corresponding eigenvalues are displayed for each bifurcation. (bottom right) A state-space projection of the evolution of an unstable period-one gait (red) converging to a closed invariant curve (blue) during the NS bifurcation, with 1,000 impacts plotted (purple).
Figure 5 : Convergence of unstable points after an FD bifurcation for the brachiating branch under a TBS method.
Figure 6 : The evolution of unstable period-one walking gaits (UWGs) along a segment in red after 1,000 impacts (points in cyan) under SBS (gaits evolve along lines of constant slope in the plot). Stable period-one walking and brachiating gaits are in blue. The points in cyan are also stable periodic orbits. The insets show zoomed in regions of interest. For example, the inset of the bottom-left corner shows UWGs converging to stable walking gaits of period- 2k ( k>1∈Z ), and the inset in the bottom-right corner shows UWGs converging to stable period- 2k brachiating gaits. Most UWGs converge to stable period-one brachiating gaits.
This paper presents the kinematic and dynamic modeling, trajectory generation, and stability analysis of an 8-degree-of-freedom (DOF) biped robot walking on flat and inclined terrain. Denavit-Hartenberg (DH) parameters and homogeneous transformations are used to derive the forward kinematics, while closed-form inverse kinematics maps the desired hip and swing-foot Cartesian trajectories, generated with cubic splines, to joint angles. Joint torques are computed using the Newton-Euler iterative algorithm, and dynamic stability is evaluated using the zero moment point (ZMP) criterion. A genetic algorithm (GA) optimizes the hip height, maximum swing-foot lift, and frontal-plane tilt angle by minimizing the work done by the joints subject to a ZMP feasibility penalty. Simulation results in MATLAB show that the nominal 8-DOF model remains ZMP-stable for step completion times down to 0.5 s and for slope inclinations up to 22.5 degrees with the given foot geometry. Beyond these limits, the ZMP leaves the support polygon, and either the foot dimensions or the trajectory parameters must be modified. The results also show that ZMP stability is governed by the mass distribution among the links rather than the total mass of the robot.
Madhav Rijal
Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India
Brachiation enables primates to move across overhead supports when ground paths are blocked, suggesting a complementary locomotion mode for robots operating in cluttered or hazardous environments. Bringing this capability to high-DoF humanoid robots is difficult because the controller must discover a long-horizon release-swing-capture sequence, coordinate alternating contacts with whole-body momentum, and act without reliable measurements of segment-relative displacement or hook-contact state. We present SwingBot, a learning framework for continuous humanoid brachiation with passive wrist hooks. SwingBot makes the task trainable by organizing learning around the structure of brachiation: biomimetic keyframes make rare release-swing-capture transitions reachable during early exploration, and recurrent privileged-state estimation provides compact position and contact latents for deployment. Hardware experiments demonstrate continuous bar traversal and robustness to payload, external disturbances and different bar spacings, showing that this formulation offers a practical route to whole-body robotic brachiation.
Yujie Xiong, Peng Zhai, Taixian Hou +6
College of Intelligent Robotics and Advanced Manufacturing, Fudan University
In legged locomotion, divergent components of motion (DCMs) have emerged as characteristic states for balance control. They isolate the unstable mode of the dynamics but, in existing formulations, apply only to reduced models such as the linear inverted pendulum. In this study, we show how DCMs can be more generally formulated as Koopman eigenfunctions. Whereas Koopman analysis typically targets eigenvalues near zero, which capture conserved or slowly varying quantities, our investigation leads us to deliberately search for unstable eigenpairs with large eigenvalues. The resulting Koopman DCMs are data-driven observables trained using only real-robot data. On a real biped, DCMs learned from one hour of robot data improve tracking of reference walking patterns. We further show how learned DCMs provide state-based viability constraints when combined with model predictive control.
Stéphane Caron
Institute for Intelligent Systems and Robotics CNRS and Sorbonne University, Paris, France