Boundedness issues in planning of locomotion trajectories for biped robots

Abstract

It is in general complex to consider the complete robot dynamics when planning trajectories for bipedal locomotion. We present an approach to trajectory planning, with the classical Linear Inverted Pendulum Model (LIPM), that takes explicit consideration of the unstable dynamics. We derive a relationship between initial state and the control input that ensures the overall system dynamics will converge to a stable steady state solution. This allows us to exploit the unstable dynamics to achieve system goals, while imposing constraints on certain degrees of freedom of the input and initial conditions. Based on this, we propose an approach to trajectory planning, and derive solutions for several typical applications. Experimental simulations using the REEM-C biped robot platform of Pal Robotics validate our approach.

Capturability-Based Pattern Generation for Walking With Variable Height

“…All of these works hinted at key features of 3D capture trajectories, but applied only to two-dimensional CoM motions in vertical planes. The key to lift this restriction is the 3D boundedness condition, which was first formulated in the case of the LIP by Lanari et al [21] and applied to model predictive control of the LIP in [44]. This condition can be more generally applied to different asymptotic behaviors, including but not restricted to stopping.…”

“…4 However, a careful match between future capture inputs and the initial condition ξ i can guarantee that ξ(t) converges as well. This choice is known as the boundedness condition [21]: Property 1 (Boundedness condition). Consider an input function λ(t), r(t) such that lim t→∞ λ(t) = λ f and lim t→∞ r(t) = r f .…”

“…The general solution (21) of the divergent component of motion is based on the antiderivative Ω(t) = t 0 ω(t)dt.…”

Real-time planning and execution of evasive motions for a humanoid robot

“…Use of the LIP model for realtime CoM trajectory generation is effective in spite of its simplicity [10]. In particular, we follow the Zero Moment Point (ZMP)-based approach of [15], [21] to achieve closedform generation of bounded CoM trajectories, a feature which will prove fundamental in the replanning context of Sect. VI.…”

“…This is achieved by making use of closed-form expressions throughout the method, and results in an algorithm suitable for real-time implementation. In particular, differently from recently proposed algorithms for on-line generation of humanoid motions [13], [14], we rely on the existence of analytical expressions relating a desired Zero Moment Point trajectory to the associated bounded Center of Mass trajectory, as illustrated in [15].…”