MAGMA: Multiplier-Augmented Geometric Motion Algorithm

César E. Ramos Chuquiure; Vansh Thakur; Challen Enninful Adu; Ram Vasudevan

cs.ROOct 15, 2026

MAGMA: Multiplier-Augmented Geometric Motion Algorithm

Authors: César E. Ramos Chuquiure, Vansh Thakur, Challen Enninful Adu, Ram Vasudevan

Organizations: Robotics, University of Michigan, Ann Arbor, MI

Abstract

Solving optimal control problems for high-dimensional mechanical systems subject to constraints remains challenging due to nonlinear dynamics and non-convex feasible sets. This paper introduces MAGMA(Multiplier-Augmented Geometric Motion Algorithm), an Affine Geometric Heat Flow (AGHF)-based trajectory optimizer that incorporates an augmented Lagrangian framework solved via a primal-dual flow. A theoretical characterization of the primal-dual AGHF is provided, establishing that MAGMA is a gradient flow on an appropriately defined energy. MAGMA is validated across a variety of challenging trajectory generation problems with obstacles, state limits, and actuation limits. Comparative evaluations against state-of-the-art solvers demonstrate that MAGMA achieves high quality solutions with significantly reduced tuning effort on the hardest problems.

Figures & tables

Panel of Figure 1 — Fig. 1: Qualitative comparison of the solution trajectories on Hard Scenario 2 (see Table I ). The Initial Guess (left column) is a straight line trajectory in configuration space that collides with the obstacles. MAGMA pushes this initial guess out of collision and converges to a feasible solution in 17.8 s, BLAZE does it in 6.6 s and Aligator in 72.2 s. The top row shows an isometric view; the bottom row shows a side view of the same trajectories, chosen to make the interaction with the obstacle visible.

Panel of Figure 2 — Fig. 2: This figure shows a visualization of one of the task-based scenarios (Scenario 7). The initial trajectory for the scenario is shown in the left image in red and collides with the obstacles along the path. MAGMA is able to push this initial guess out of collision and generate the optimal collision-free solution shown in green in 11.9s for this scenario.

Trajectory Optimization

Trajectory optimization computes motion or control sequences that minimize costs while satisfying system dynamics, safety limits, and task constraints.

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Oct 14, 2026J/K move · Enter open · S save

LEAP: Gradient Flow Augmented Lagrangian Trajectory Optimization for Contact Constrained Robots

César E. Ramos Chuquiure, Ryan Tsai, Yutong Zhou, Zihang Huang+7

Department of Robotics, University of Michigan, Ann Arbor, MI, USA. · Computer Science and Engineering, University of Michigan, Ann Arbor, MI, USA. · Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA.

Hypergradient Optimization Trajectory Optimization Robotic Optimization Augmented Lagrangian Method Robot Locomotion

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Generating dynamically feasible motions for legged and manipulating robots requires solving a large, nonlinear, non-convex optimal control problem (OCP) in which contact wrenches must satisfy friction-cone constraints throughout the motion, not only at isolated instants. Existing solvers are sensitive to the initial guess and scale poorly with the number of joints. This paper introduces LEAP, which transcribes the OCP with pseudospectral collocation and, instead of a Newton or SQP solve, integrates a primal--dual gradient flow of a Powell--Hestenes--Rockafellar augmented Lagrangian to compute a solution. Because the flow is stiff, this paper designs a linearly implicit integrator with a Gauss--Newton metric, momentum, and a merit-based step controller that keeps the augmented Lagrangian monotone. Across locomotion benchmarks on five robot platforms, from planar hoppers to 32-joint humanoids, LEAP is the only method to solve every tested problem, achieves substantial speedups over Crocoddyl and Aligator on several benchmarks, and reaches trackable references at smaller compute budgets. The resulting trajectories are validated by tracking them in a high-fidelity simulator and on hardware. We will make the code public upon final acceptance of this paper.

Abstract

Trajectory Optimization

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