Generalized Maximum Entropy Differential Dynamic Programming

Abstract

We present a sampling-based trajectory optimization method derived from the maximum entropy formulation of Differential Dynamic Programming with Tsallis entropy. This method is a generalization of the legacy work with Shannon entropy, which leads to a Gaussian optimal control policy for exploration during optimization. With the Tsallis entropy, the policy takes the form of $q$-Gaussian, which further encourages exploration with its heavy-tailed shape. Moreover, the sampling variance is scaled according to the value function of the trajectory. This scaling mechanism is the unique property of the algorithm with Tsallis entropy in contrast to the original formulation with Shannon entropy, which scales variance with a fixed temperature parameter. Due to this property, our proposed algorithms can promote exploration when necessary, that is, the cost of the trajectory is high. The simulation results with two robotic systems with multimodal cost demonstrate the properties of the proposed algorithm.