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Parallel and Proximal Constrained Linear-Quadratic Methods for Real-Time Nonlinear MPC

Wilson Jallet, Ewen Dantec, Etienne Arlaud, Justin Carpentier, Nicolas Mansard

TL;DR

The paper addresses real-time NMPC with equality constraints and implicit dynamics by developing a proximal, dual-regularized LQ solver whose backward pass is recast as a block-sparse Riccati-like recursion. It introduces parametric extensions and a parallelization scheme that splits the horizon into multiple legs and solves a condensed consensus system in parallel, with a block-tridiagonal reduction for efficiency. The approach is implemented in the ALIGATOR framework and validated on high-dimensional robotic benchmarks (e.g., Talos and Solo-12), demonstrating real-time capability and meaningful speedups, albeit with partial parallel efficiency still limited by overheads. The work advances scalable, structure-exploiting NMPC solvers for complex, constrained, and implicit-dynamics problems, enabling longer horizons and higher fidelity models in practice.

Abstract

Recent strides in nonlinear model predictive control (NMPC) underscore a dependence on numerical advancements to efficiently and accurately solve large-scale problems. Given the substantial number of variables characterizing typical whole-body optimal control (OC) problems - often numbering in the thousands - exploiting the sparse structure of the numerical problem becomes crucial to meet computational demands, typically in the range of a few milliseconds. Addressing the linear-quadratic regulator (LQR) problem is a fundamental building block for computing Newton or Sequential Quadratic Programming (SQP) steps in direct optimal control methods. This paper concentrates on equality-constrained problems featuring implicit system dynamics and dual regularization, a characteristic of advanced interiorpoint or augmented Lagrangian solvers. Here, we introduce a parallel algorithm for solving an LQR problem with dual regularization. Leveraging a rewriting of the LQR recursion through block elimination, we first enhanced the efficiency of the serial algorithm and then subsequently generalized it to handle parametric problems. This extension enables us to split decision variables and solve multiple subproblems concurrently. Our algorithm is implemented in our nonlinear numerical optimal control library ALIGATOR. It showcases improved performance over previous serial formulations and we validate its efficacy by deploying it in the model predictive control of a real quadruped robot.

Parallel and Proximal Constrained Linear-Quadratic Methods for Real-Time Nonlinear MPC

TL;DR

The paper addresses real-time NMPC with equality constraints and implicit dynamics by developing a proximal, dual-regularized LQ solver whose backward pass is recast as a block-sparse Riccati-like recursion. It introduces parametric extensions and a parallelization scheme that splits the horizon into multiple legs and solves a condensed consensus system in parallel, with a block-tridiagonal reduction for efficiency. The approach is implemented in the ALIGATOR framework and validated on high-dimensional robotic benchmarks (e.g., Talos and Solo-12), demonstrating real-time capability and meaningful speedups, albeit with partial parallel efficiency still limited by overheads. The work advances scalable, structure-exploiting NMPC solvers for complex, constrained, and implicit-dynamics problems, enabling longer horizons and higher fidelity models in practice.

Abstract

Recent strides in nonlinear model predictive control (NMPC) underscore a dependence on numerical advancements to efficiently and accurately solve large-scale problems. Given the substantial number of variables characterizing typical whole-body optimal control (OC) problems - often numbering in the thousands - exploiting the sparse structure of the numerical problem becomes crucial to meet computational demands, typically in the range of a few milliseconds. Addressing the linear-quadratic regulator (LQR) problem is a fundamental building block for computing Newton or Sequential Quadratic Programming (SQP) steps in direct optimal control methods. This paper concentrates on equality-constrained problems featuring implicit system dynamics and dual regularization, a characteristic of advanced interiorpoint or augmented Lagrangian solvers. Here, we introduce a parallel algorithm for solving an LQR problem with dual regularization. Leveraging a rewriting of the LQR recursion through block elimination, we first enhanced the efficiency of the serial algorithm and then subsequently generalized it to handle parametric problems. This extension enables us to split decision variables and solve multiple subproblems concurrently. Our algorithm is implemented in our nonlinear numerical optimal control library ALIGATOR. It showcases improved performance over previous serial formulations and we validate its efficacy by deploying it in the model predictive control of a real quadruped robot.
Paper Structure (40 sections, 3 theorems, 94 equations, 8 figures, 2 tables, 4 algorithms)

This paper contains 40 sections, 3 theorems, 94 equations, 8 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Equality-constrained linear-quadratic Problems
  3. Problem statement
  4. Lagrangian and KKT conditions
  5. Proximal regularization of the LQ problem
  6. Riccati Equations for the proximal LQ Problem
  7. Solving the proximal LQ by block substitution
  8. Terminal stage
  9. Middle blocks
  10. Initial stage
  11. Riccati-like algorithm
  12. Block-sparse factorization for the stage KKT system
  13. Extension to Parametric LQ problems
  14. Parametric Lagrangians
  15. Expression of the value function $\mathcal{E}_\mu(x_0,\theta)$
  16. ...and 25 more sections

Key Result

Proposition 3

$\mathcal{E}_\mu$ is a quadratic function in $(x_0, \theta)$. There exist $\sigma_0\in\mathbb{R}^{n_{\theta}}$, $\Lambda_0\in\mathbb{R}^{n_x\times n_\theta}$ and $\Sigma_0 \in \mathbb{R}^{n_\theta\times n_\theta}$ such that

Figures (8)

  • Figure 1: LQ problem with cyclical constraint $x_0=x_{30}$ ,in one dimension. No other initial condition was provided for $x_0$.
  • Figure 2: Cyclic LQ problem in the 2D plane. Cost function $\ell(x,u) = 10^{-3}\|x\|^2 + \|u\|^2$ except at for $t\in\{5,15\}$ where $\ell(x,u)=0.2\|x-\bar{x}_t\|^2 + \|u\|^2$.
  • Figure 3: Timings for a backward-forward sweep of the solver on a synthetic benchmark.
  • Figure 4: C++ benchmarks of a trajectory optimization problem involving two forward steps with the whole-body model of the Talos robot, with single support time $T_\mathrm{ss}$ set to 0.6s, 0.8s and 1s from left to right. Each instance is run 40 times on every solver to produce a mean and standard deviation. Here, the Turboboost feature on the CPU was disabled and clock speed fixed to 2200MHz.
  • Figure 5: Snapshot of a PyBullet coumansPyBulletPythonModule2016 simulation featuring Talos walking with pre-defined feet trajectories (blue rectangles). The Bullet simulation timestep is set to 1ms.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Proposition 3
  • proof
  • Proposition 4
  • Proposition 5