Efficient Riccati recursion for optimal control problems with pure-state equality constraints

TL;DR

This paper introduces an efficient Riccati recursion method for solving optimal control problems with pure-state equality constraints by transforming them into mixed constraints, significantly reducing computational complexity.

Contribution

The paper proposes a novel transformation of pure-state constraints into mixed constraints and derives a Riccati recursion algorithm with linear time complexity for such problems.

Findings

The method achieves linear time complexity in the horizon length.

Numerical experiments show improved efficiency over existing methods.

Effective in whole-body quadrupedal gait control involving contact switches.

Abstract

A novel approach to efficiently treat pure-state equality constraints in optimal control problems (OCPs) using a Riccati recursion algorithm is proposed. The proposed method transforms a pure-state equality constraint into a mixed state-control constraint such that the constraint is expressed by variables at a certain previous time stage. It is showed that if the solution satisfies the second-order sufficient conditions of the OCP with the transformed mixed state-control constraints, it is a local minimum of the OCP with the original pure-state constraints. A Riccati recursion algorithm is derived to solve the OCP using the transformed constraints with linear time complexity in the grid number of the horizon, in contrast to a previous approach that scales cubically with respect to the total dimension of the pure-state equality constraints. Numerical experiments on the whole-body optimal control of quadrupedal gaits that involve pure-state equality constraints owing to contact switches demonstrate the effectiveness of the proposed method over existing approaches.