DIRECT: A Differential Dynamic Programming Based Framework for Trajectory Generation

TL;DR

This paper presents a novel DDP-based framework for polynomial trajectory generation in flat systems, simplifying the problem to a finite horizon control and enabling efficient, linear-complexity algorithms validated through simulations and experiments.

Contribution

It introduces a state-space perspective that reduces trajectory generation to a finite horizon control problem, enabling efficient algorithms with linear complexity.

Findings

Linear complexity with respect to the number of segments.

Effective trajectory generation validated by simulations and experiments.

Unified approach for constrained and unconstrained problems.

Abstract

This paper introduces a differential dynamic programming (DDP) based framework for polynomial trajectory generation for differentially flat systems. In particular, instead of using a linear equation with increasing size to represent multiple polynomial segments as in literature, we take a new perspective from state-space representation such that the linear equation reduces to a finite horizon control system with a fixed state dimension and the required continuity conditions for consecutive polynomials are automatically satisfied. Consequently, the constrained trajectory generation problem (both with and without time optimization) can be converted to a discrete-time finite-horizon optimal control problem with inequality constraints, which can be approached by a recently developed interior-point DDP (IPDDP) algorithm. Furthermore, for unconstrained trajectory generation with preallocated time, we show that this problem is indeed a linear-quadratic tracking (LQT) problem (DDP algorithm with exact one iteration). All these algorithms enjoy linear complexity with respect to the number of segments. Both numerical comparisons with state-of-the-art methods and physical experiments are presented to verify and validate the effectiveness of our theoretical findings. The implementation code will be open-sourced,