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‎index.html‎

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<li><a href="publications/gjk-acceleration"> GJK++: Leveraging Acceleration Methods for Faster Collision Detection. (IEEE T-RO 2024)</a></li>
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<li><a href="publications/consensus-to-pl"> Enforcing the consensus between Trajectory Optimization and Policy Learning for precise robot control.</a></li>
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<li><a href="publications/raisim-revisited"> Reconciling RaiSim with the Maximum Dissipation Principle. (IEEE T-RO 2024)</a></li>
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<li><a href="publications/proxddp-tro-2025"> ProxDDP: Proximal Constrained Trajectory Optimization. (IEEE T-RO 2025)</a></li>
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‎publications/proxddp-tro-2025/index.html‎

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content="Parallelized proximal constrained LQR">
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<meta name="keywords" content="Numerical optimization, Numerical optimal control, LQR, Riccati">
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More Research
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<a class="navbar-item" href="https://ieeexplore.ieee.org/document/9981586">
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Constrained DDP: a primal-dual augmented Lagrangian approach
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</a>
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<a class="navbar-item" href="https://ieeexplore.ieee.org/document/9811647">
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Implicit differential dynamic programming
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<a class="navbar-item" href="https://ieeexplore.ieee.org/document/10521997/">
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Condensed Semi-Implicit Dynamics for Trajectory Optimization in Soft Robotics
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<h1 class="title is-1 publication-title">PROXDDP: Proximal Constrained Trajectory Optimization</h1>
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<h2 class="subtitle is-5 is-italic">Published in IEEE Transactions on Robotics, Volume 41, 2025</h2>
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<div class="is-size-5 publication-authors">
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<span class="author-block">
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<a href="https://manifoldfr.github.io">Wilson Jallet</a><sup>1,2</sup>,
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</span>
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<span class="author-block">
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<a href="https://github.com/edantec">Antoine Bambade</a><sup>1</sup>,
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<span class="author-block">
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<a href="https://github.com/etiennear">Etienne Arlaud</a><sup>1</sup>,
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<span class="author-block">
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<a href="https://github.com/sarah-quinones">Sarah El-Kazdadi</a><sup>1</sup>,
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<span class="author-block">
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<a href="https://gepettoweb.laas.fr/index.php/Members/NicolasMansard">Nicolas Mansard</a><sup>2</sup>,
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<span class="author-block">
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<a href="https://jcarpent.github.io">Justin Carpentier</a><sup>1</sup>,
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<sup>1</sup>Willow team, INRIA and Département d'Informatique de l'École normale supérieure, Paris, France<br>
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<sup>2</sup>Gepetto team, LAAS-CNRS, 7 av. du colonel Roche, 31400 Toulouse, France
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<h2 class="title is-3">Abstract</h2>
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<p>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.</p>
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<p>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.</p>
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<p>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.</p>
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<h2 class="title is-3">Results</h2>
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<h3 class="title is-4">Solving cyclic LQ problems</h3>
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<p>One of the side-effects of our formulation, is the ability to tackle linear-quadratic problems with cyclical constraints of the form \(G_0x_0 + G_Nx_N + g = 0\).</p>
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<p>
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The following is a two-dimensional LQ problem over \(N = 20\) steps with no initial condition but a cyclical condition \(x_0 = x_N\), and some transient costs that steer the generated trajectory close to two points \(\bar{x}_5\) and \(\bar{x}_{15}\).
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<div class="content has-text-centered">
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<img src="./static/images/gar-cyclic-lqr-2d.png" alt="2D cyclic LQ problem"
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<h2 class="title is-3">Related Links</h2>
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This work heavily relies on the
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<a href="https://github.com/Simple-Robotics/aligator">aligator</a>
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and
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<a href="https://github.com/stack-of-tasks/pinocchio">Pinocchio</a>
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libraries, as well as the
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<a href="https://github.com/inria-paris-robotics-lab/quadruped-reactive-walking">quadruped-reactive-walking</a> framework for whole-body NMPC on Solo.
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<section class="section" id="BibTeX">
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<h2 class="title">BibTeX</h2>
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<pre><code>@article{jalletPROXDDPProximalConstrained2025,
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title = {PROXDDP: Proximal Constrained Trajectory Optimization},
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shorttitle = {PROXDDP},
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author = {Jallet, Wilson and Bambade, Antoine and Arlaud, Etienne and {El-Kazdadi}, Sarah and Mansard, Nicolas and Carpentier, Justin},
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year = {2025},
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journal = {IEEE Transactions on Robotics},
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pages = {1--20},
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issn = {1941-0468},
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doi = {10.1109/TRO.2025.3554437},
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urldate = {2025-04-04},
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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}
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}</code></pre>
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