# DIRECT: A Differential Dynamic Programming Based Framework for Trajectory Generation

arXiv: https://arxiv.org/abs/2109.04686 · HTML: https://mlsanity.com/paper/2109.04686/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2021.

## Built on by (most cited first, 4 of 4)

- [UAVScenes: A Multi-Modal Dataset for UAVs](https://mlsanity.com/paper/2507.22412/) (2025)
- [Unsupervised UAV 3D Trajectories Estimation with Sparse Point Clouds](https://mlsanity.com/paper/2412.12716/) (2024)
- [TAME: Temporal Audio-based Mamba for Enhanced Drone Trajectory Estimation and Classification](https://mlsanity.com/paper/2412.13037/) (2024)
- [NVP-HRI: Zero Shot Natural Voice and Posture-based Human-Robot Interaction via Large Language Model](https://mlsanity.com/paper/2503.09335/) (2025)

## Abstract (arXiv)

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,