PROXDDP: Proximal Constrained Trajectory Optimization

Published in IEEE Transactions on Robotics, Volume 41, 2025

Wilson Jallet1,2, Antoine Bambade1, Etienne Arlaud1, Sarah El-Kazdadi1, Nicolas Mansard2, Justin Carpentier1,
1Willow team, INRIA and Département d'Informatique de l'École normale supérieure, Paris, France
2Gepetto team, LAAS-CNRS, 7 av. du colonel Roche, 31400 Toulouse, France
IEEEXplore HAL Repository for aligator
slaloming quadcopter

Abstract

Trajectory optimization has been a popular choice for motion generation and control in robotics for at least a decade. Several numerical approaches have exhibited the required speed to enable online computation of trajectories for real-time of various systems, including complex robots. Many of these said are based on the differential dynamic programming (DDP) algorithm—initially designed for unconstrained trajectory optimization problems— and its variants, which are relatively easy to implement and provide good runtime performance.

However, several problems in robot control call for using constrained formulations (e.g., torque limits, obstacle avoidance), from which several difficulties arise when trying to adapt DDP-type methods: numerical stability, computational efficiency, and constraint satisfaction. In this article, we leverage proximal methods for constrained optimization and introduce a DDP-type method for fast, constrained trajectory optimization suited for model-predictive control (MPC) applications with easy warm-starting.

Compared to earlier solvers, our approach effectively manages hard constraints without warm-start limitations and exhibits good convergence behavior. We provide a complete implementation as part of an open-source and flexible C++ trajectory optimization library called ALIGATOR. These algorithmic contributions are validated through several trajectory planning scenarios from the robotics literature and the real-time whole-body MPC of a quadruped robot.

Results

Benchmarks

An additional contribution of our journal paper is the inclusion of benchmarks against two other numerical solvers: IPOPT, which is a generic NLP solver often used in the OC community, and ALTRO, a tailored constrained solver.

UR5 reach, iterations UR5 reach, perf profile
UR5 reaching task: iterations and performance profile
UR10 ballistic, iterations UR10 ballistic, perf profile
UR10 ballistic task: iterations and performance profile

More details can be found in Section VIII of the paper.

Related Links

This work heavily relies on the aligator and Pinocchio libraries, as well as the quadruped-reactive-walking framework for whole-body NMPC on Solo.

BibTeX

@article{jalletPROXDDPProximalConstrained2025,
  title = {PROXDDP: Proximal Constrained Trajectory Optimization},
  shorttitle = {PROXDDP},
  author = {Jallet, Wilson and Bambade, Antoine and Arlaud, Etienne and {El-Kazdadi}, Sarah and Mansard, Nicolas and Carpentier, Justin},
  year = {2025},
  journal = {IEEE Transactions on Robotics},
  pages = {1--20},
  issn = {1941-0468},
  doi = {10.1109/TRO.2025.3554437},
  urldate = {2025-04-04},
  keywords = {Convergence,Heuristic algorithms,Legged Robots,Libraries,Linear systems,Minimization,Model-Predictive Control,Newton method,Optimization,Optimization and Optimal Control,Predictive control,Robots,Trajectory optimization}
}