iLQR for Piecewise-Smooth Hybrid Dynamical Systems

Trajectory optimization is a popular strategy for planning trajectories for robotic systems. However, many robotic tasks require changing contact conditions, which is difficult due to the hybrid nature of the dynamics. The optimal sequence and timing of these modes are typically not known ahead of time. In this work, we extend the Iterative Linear Quadratic Regulator (iLQR) method to a class of piecewise-smooth hybrid dynamical systems with state jumps by allowing for changing hybrid modes in the forward pass, using the saltation matrix to update the gradient information in the backwards pass, and using a reference extension to account for mode mismatch. We demonstrate these changes on a variety of hybrid systems and compare the different strategies for computing the gradients.

I. Introduction

For robots to be useful in real world settings, they need to be able to interact efficiently and effectively with their environments. However, systems like the quadcopter perching example shown in Fig. 1 often have highly nonlinear dynamics and complex, time-varying environmental interactions that make trajectory planning computationally challenging. These systems are often modeled as mechanical systems with impacts, a type of hybrid dynamical system (Def. 1), [1]–[3]. Hybrid dynamical systems differ from smooth dynamical systems in many ways which make planning and control more difficult, including: 1) they contain a discrete component of state (the "hybrid mode") over which the continuous dynamics may differ. 2) These modes are connected by a reset function that applies a discrete (and potentially discontinuous) change to the state. 3) There may be different control authority available in each mode.