Impact-Invariant Control: Maximizing Control Authority During Impacts

TL;DR

This work introduces impact-invariant control, projecting velocity errors onto the subspace invariant to contact impulses to maintain robust tracking through impacts. The method is derived from a robust-optimal-control perspective and realized by a fast, closed-form projection embedded in an operational-space control framework. It is validated across walking, jumping, and running scenarios on both a planar five-link biped and the 3D Cassie robot, with successful hardware demonstrations showing reduced control effort spikes and improved stability near impact. The approach preserves most control authority while mitigating impact-induced disturbances, enabling more agile and reliable legged locomotion in the presence of contact uncertainties.

Abstract

When legged robots impact their environment executing dynamic motions, they undergo large changes in their velocities in a short amount of time. Measuring and applying feedback to these velocities is challenging, further complicated by uncertainty in the impact model and impact timing. This work proposes a general framework for adapting feedback control during impact by projecting the control objectives to a subspace that is invariant to the impact event. The resultant controller is robust to uncertainties in the impact event while maintaining maximum control authority over the impact-invariant subspace. We demonstrate the improved performance of the projection over other commonly used heuristics on a walking controller for a planar five-link-biped. The projection is also applied to jumping, box jumping, and running controllers for the compliant 3D bipedal robot, Cassie. The modification is easily applied to these various controllers and is a critical component to deploying on the physical robot. Code and video of the experiments are available at https://impact-invariant-control.github.io/.

Figures (16)

• Figure 1: Cassie is able to execute agile motions with non-negligible impacts like jumping (top), box jumping (bottom-left), and running (bottom-right) using impact-invariant control.
• Figure 2: We use both the planar biped Rabbit (left) and the 3D compliant bipedal robot Cassie (right) as concrete examples to highlight the advantages of impact-invariant control.
• Figure 3: Illustration of a system that undergoes an impact event. The desired velocity plan correctly includes the discontinuity as predicted by rigid body impact laws and the measured velocity is being properly regulated to match the desired plan. However, due to the mismatch in impact time, the velocity error inevitably spikes during the impact event.
• Figure 4: The velocities are shown for a periodic symmetric walking trajectory (a) for a planar five-link biped. Because the trajectory is constructed to be symmetric, only half of the gait is shown without loss of information. The instantaneous jump in velocities is due to an impact event when the right foot makes contact with the ground. The darker shade indicates the nominal trajectory while the lighter shade is an example of actual velocities when tracking to the nominal trajectory. Despite the qualitatively "good" tracking of the velocities, the error near the impact event still exhibits a sharp jump (c) due to the discontinuity from impact. The same velocities projected to the impact-invariant subspace (b), do not have experience this discontinuity and therefore results in a much smaller tracking error, which is a better reflection of the actual system tracking.
• Figure 5: Demonstration of the impact-invariant projection on joint velocity data from 8 consecutive jumping experiments on the physical Cassie robot. Joint velocities (top) during the landing event change rapidly. By projecting the same joint velocities to the impact-invariant subspace (bottom), the values are more consistent and more amenable for feedback control. Note, the change in joint velocities primarily occurs within a time span of only 10 - 20 ms. The L and R subscripts indicate the left and right leg respectively.

Theorems & Definitions (1)

• Remark 1