Infinite-Horizon Model Predictive Control for Periodic Tasks with Contacts

Abstract

Abstract-We present a method that combines offline trajectory optimization and online Model Predictive Control (MPC), generating robust controllers for complex periodic behavior in domains with unilateral constraints (e.g., contact with the environment). MPC offers robust and adaptive control even in high-dimensional domains; however, the online optimization gets stuck in local minima when the domains has discontinuous dynamics. Some methods of trajectory optimization that are immune to such problems, but these are often too slow to be applied online.In this paper, we use offline optimization to find the limit-cycle solution of an infinite-horizon average-cost optimal-control task. We then compute a local quadratic approximation of the Value function around this limit cycle. Finally, we use this quadratic approximation as the terminal cost of an online MPC.This combination of an offline solution of the infinite-horizon problem with an online MPC controller is known as Infinite Horizon Model Predictive Control (IHMPC), and has previously been applied only to simple stabilization objectives. Here we extend IHMPC to tackle periodic tasks, and demonstrate the power of our approach by synthesizing hopping behavior in a simulated robot. IHMPC involves a limited computational load, and can be executed online on a standard laptop computer. The resulting behavior is extremely robust, allowing the hopper to recover from virtually any perturbation.In real robotic domains, modeling errors are inevitable. We show how IHMPC is robust to modeling errors by altering the morphology of the robot; the same controller remains effective, even when the underlying infinite-horizon solution is no longer accurate.

Infinite-horizon deterministic dynamic programming in discrete time: a monotone convergence principle and a penalty method

“…To see this, note that if x ∈ X, both inequalities directly follow from (19) and (20). If x ∈ S \ X, both inequalities are immediate since v(x) = −∞ by (26).…”

“…Lemma 6.2: Suppose that there exists v ∈ W satisfying (19) and (20 {u(x, y) + βv * (y)} = (Bv * )(x), ( 4 7 ) where (46) uses Lemma 6.1. Since x ∈ X was arbitrary, it follows that v * =Bv * ; i.e.…”

“…Assume (21). We have v * ≤ v * since v ≤ v * by (19),B is order-preserving, and v * is a fixed point ofB by Theorem 2.1. We also have v * ≥ v * by Lemma 6.3.…”

“…The result seems significant since, to our knowledge, there have been very few applications of penalty methods to dynamic programming in the literature; among the exceptions are [18,19]. Another related study is [20], where a dynamic programming approach was used to solve a problem with penalty functions.…”

“…To complete the proof of Theorem 3.1, let v ∈ W satisfy (19) and (20). Then, by Lemma 6.2, {B n v} n∈N converges increasingly to v * pointwise, and v * is a fixed point ofB.…”