Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Endpoint-Explicit Differential Dynamic Programming via Exact Resolution

Maria Parilli, Sergi Martinez, Carlos Mastalli

TL;DR

This work tackles the challenge of enforcing endpoint constraints in constrained differential dynamic programming (DDP) by introducing an endpoint-explicit formulation that achieves exact, quadratic convergence. It derives two exact factorization strategies for the resulting saddle-point KKT system: a Schur-complement approach and a rank-deficient-friendly nullspace decomposition, enabling efficient Riccati recursions and separation of endpoint-independent and endpoint-dependent directions. The method supports both forward and inverse dynamics and is demonstrated on a broad set of robotics problems, including gymnastic maneuvers and MPC-like trials, showing superior endpoint satisfaction and convergence speed compared to penalty-based methods. The approach, implemented openly in Crocoddyl, promises real-time applicability to complex robotic control by reusing unconstrained Riccati structures and providing robust handling of rank deficiencies in endpoints and stagewise constraints.

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.

Endpoint-Explicit Differential Dynamic Programming via Exact Resolution

TL;DR

This work tackles the challenge of enforcing endpoint constraints in constrained differential dynamic programming (DDP) by introducing an endpoint-explicit formulation that achieves exact, quadratic convergence. It derives two exact factorization strategies for the resulting saddle-point KKT system: a Schur-complement approach and a rank-deficient-friendly nullspace decomposition, enabling efficient Riccati recursions and separation of endpoint-independent and endpoint-dependent directions. The method supports both forward and inverse dynamics and is demonstrated on a broad set of robotics problems, including gymnastic maneuvers and MPC-like trials, showing superior endpoint satisfaction and convergence speed compared to penalty-based methods. The approach, implemented openly in Crocoddyl, promises real-time applicability to complex robotic control by reusing unconstrained Riccati structures and providing robust handling of rank deficiencies in endpoints and stagewise constraints.

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.

Paper Structure

This paper contains 22 sections, 39 equations, 4 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contribution
  3. Quadratic programs with endpoint constraints
  4. Schur-complement resolution
  5. Nullspace resolution
  6. Optimal control with endpoint constraints
  7. Exploiting the structure of the problem
  8. Endpoint-independent search direction
  9. Forward dynamics
  10. Inverse dynamics
  11. Endpoint-dependent search direction
  12. Forward dynamics
  13. Inverse dynamics
  14. Endpoint multiplier and update direction
  15. Expected improvement, merit function, and rollouts
  16. ...and 7 more sections

Figures (4)

  • Figure 1: Snapshots of Talos performing various gymnastic maneuvers computed using our endpoint-explicit algorithm. The images show Talos executing a (top-left) handstand, (top-right) backflip, (bottom-left) frontflip, and (bottom-right) monkey bar maneuver. For each maneuver, the feet or hand placements were specified as endpoint constraints. To watch the video, click the picture or see https://youtu.be/RBohdOhgbWw.
  • Figure 2: Cost and endpoint feasibility evolution for forward and inverse dynamics formulations: (left) normalized cost, and (right) $\ell_1$-norm feasibility.
  • Figure 3: Average computation time for solving different factorizations and robotics problems over 10 trials, minimum time at the top.
  • Figure 4: Experiment trials with our endpoint-constrained algorithm. (Left) Movements of the Z1 robot demonstrating. (Right) Tracking performance improvements in experimental trials with the B1-Z1 quadruped robot.