Convergent iLQR for Safe Trajectory Planning and Control of Legged Robots

TL;DR

This work introduces convergent iLQR ($\chi$‑iLQR), an optimization framework for hybrid, underactuated legged robots that explicitly minimizes worst‑case perturbation growth along a trajectory. By incorporating the fundamental solution matrix $\Phi$ and its 2‑norm $\chi = ||\Phi||_2$ into the cost, and using a two‑pass backward strategy plus a convergent reference trajectory, the method yields trajectories with superior closed‑loop convergence under an LQR tracker. The approach is demonstrated on a Rocket Hopper and a Planar Quadruped, showing reduced $\chi$, lower feedback effort, and greater robustness to large disturbances, compared to vanilla iLQR. The results suggest practical benefits for safely executing dynamic maneuvers (e.g., leaps) in uncertain environments, with potential extensions to hardware and other hybrid systems.

Abstract

In order to perform highly dynamic and agile maneuvers, legged robots typically spend time in underactuated domains (e.g. with feet off the ground) where the system has limited command of its acceleration and a constrained amount of time before transitioning to a new domain (e.g. foot touchdown). Meanwhile, these transitions can instantaneously change the system's state, possibly causing perturbations to be mapped arbitrarily far away from the target trajectory. These properties make it difficult for local feedback controllers to effectively recover from disturbances as the system evolves through underactuated domains and hybrid impact events. To address this, we utilize the fundamental solution matrix that characterizes the evolution of perturbations through a hybrid trajectory and its 2-norm, which represents the worst-case growth of perturbations. In this paper, the worst-case perturbation analysis is used to explicitly reason about the tracking performance of a hybrid trajectory and is incorporated in an iLQR framework to optimize a trajectory while taking into account the closed-loop convergence of the trajectory under an LQR tracking controller. The generated convergent trajectories recover more effectively from perturbations, are more robust to large disturbances, and use less feedback control effort than trajectories generated with traditional methods.

Figures (3)

• Figure 1: Similar quadruped gaits tracked with equivalent LQR controllers display enormous differences in final tracking performance. Results for 50 paired trials of trajectories sampled from an initial error covariance of $10^{-2}$ in all directions are shown. The top blue trajectory was generated with a standard (vanilla) iLQR algorithm, while the bottom green trajectory was generated with our novel $\chi$-iLQR, which improves the average closed-loop convergence by 92%. The horizontal distance between the displayed frames is exaggerated for visual clarity. Only the convergence of the position states is represented, and does not indicate convergence of the velocity states.
• Figure 2: Histogram of error ratio for 100 paired simulated trials of the quadruped model with small initial perturbations. Error ratio is the 2-norm of final errors divided by the 2-norm of initial errors.
• Figure 3: Plots showing success of LQR controllers at tracking vanilla ($V$) and convergent ($\chi$) trajectories over various initial perturbation covariances. $V_{50}$ and $\chi_{50}$ indicate proportion of trials where error ratio was below 50, $V_{10}$ and $\chi_{10}$ below 10, and $V_{5}$ and $\chi_{5}$ below 5.