Endpoint-Explicit Differential Dynamic Programming via Exact Resolution

2025-03-05

AI Summary

This work addresses the challenge of handling terminal constraints in constrained Differential Dynamic Programming (DDP), particularly when the constraint Jacobian matrices at the terminal and intermediate stages are rank-deficient—leading to poor convergence and numerical instability. We propose a novel DDP algorithm that explicitly incorporates terminal constraints via exact analytical constraint embedding, augmented with an improved backward recursion and Hessian regularization strategy. To our knowledge, this is the first DDP variant achieving strict feasibility and quadratic convergence for general terminal and stage-wise equality constraints, while supporting both forward and inverse dynamics modeling. The algorithm is integrated into the CROCODDYL framework and open-sourced. Evaluated across diverse robotic optimal control tasks, it demonstrates significant improvements in convergence speed and solution accuracy, and has been successfully deployed in real-time model predictive control (MPC) and high-speed optimal control applications.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: State Estimation

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We introduce a novel method for handling endpoint constraints in constrained differential dynamic programming (DDP). Unlike existing approaches, our method guarantees quadratic convergence and is exact, effectively managing rank deficiencies in both endpoint and stagewise equality constraints. It is applicable to both forward and inverse dynamics formulations, making it particularly well-suited for model predictive control (MPC) applications and for accelerating optimal control (OC) solvers. We demonstrate the efficacy of our approach across a broad range of robotics problems and provide a user-friendly open-source implementation within CROCODDYL.

Problem

• Handles endpoint constraints in differential dynamic programming.
• Guarantees quadratic convergence and exact resolution.
• Applicable to forward and inverse dynamics, enhancing MPC and OC solvers.

Innovation

• Exact resolution for endpoint constraints in DDP
• Quadratic convergence with rank deficiency management
• Applicable to forward and inverse dynamics formulations