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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>MAGMA: Multiplier-Augmented Geometric Motion Algorithm</title>
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<body>
<!-- ==================== HEADER ==================== -->
<header>
<h1>MAGMA</h1>
<p class="subtitle">
Multiplier-Augmented Geometric Motion Algorithm
</p>
<p class="authors">
César E. Ramos Chuquiure, Vansh Thakur, Challen Enninful Adu, and Ram Vasudevan
</p>
<p class="lab">
ROAHM Lab, University of Michigan
</p>
</header>
<!-- ==================== COVER VIDEO ==================== -->
<div class="cover-band">
<figure class="cover-figure">
<video autoplay loop muted playsinline>
<source src="videos/Sequence 01_3.mp4" type="video/mp4">
</video>
<figcaption>
MAGMA generates constraint-satisfying trajectories for a Kinova Gen3 manipulator across cluttered task-oriented scenes.
</figcaption>
</figure>
</div>
<div class="container">
<!-- ==================== ABSTRACT ==================== -->
<section id="abstract">
<h2>Abstract</h2>
<p>
Trajectory optimization for robots, from autonomous vehicles to manipulators and
legged systems, demands four things at once: <strong>speed</strong> for near real-time
planning, <strong>scalability</strong> to high-dimensional state spaces, support for
<strong>nonlinear dynamics and task constraints</strong>, and <strong>reliable convergence</strong>
from poor initial guesses. Existing methods struggle to satisfy all four simultaneously.
HJB-based dynamic programming is intractable beyond moderate dimensions. DDP variants are
fragile far from the optimum. Collocation-based nonlinear programs are slow and sensitive
to discretization.
</p>
<p>
The <strong>Affine Geometric Heat Flow (AGHF)</strong> reformulates trajectory optimization as
a PDE that evolves an infeasible trajectory toward dynamic feasibility, scaling gracefully to
high-dimensional systems. But prior AGHF solvers lean on penalty and barrier formulations to
enforce constraints, which forces hand-tuning of weights, brittle behavior when cost and
feasibility compete, and sensitivity to the initial guess.
</p>
<p>
We introduce <strong>MAGMA</strong> (Multiplier-Augmented Geometric Motion Algorithm), an
AGHF-based trajectory optimizer built on an <strong>augmented Lagrangian structure solved
via a primal-dual geometric flow</strong>. By co-evolving primal and dual variables,
MAGMA automatically balances optimality and constraint satisfaction. This eliminates
manual weight tuning, handles general nonlinear constraints and arbitrary cost functions,
and remains robust to poor initialization. We additionally develop an infinite-dimensional
primal-dual gradient flow model that provides a theoretical foundation for constrained
AGHF methods, and demonstrate MAGMA on high-dimensional manipulation tasks against
state-of-the-art solvers.
</p>
</section>
<!-- ==================== ADVANTAGES ==================== -->
<section id="advantages">
<h2>MAGMA features</h2>
<div class="features">
<div class="feature-card">
<h3>Dramatically Reduced Manual Tuning</h3>
<p>The dual flow automatically accomodates constraint violations, resulting in a fundamentally different optimization landscape.</p>
</div>
<div class="feature-card">
<h3>Robust Initialization</h3>
<p>Barrier methods can be give rise to numerically large values in the gradient for bad initial guesses, MAGMA circumvents that completely!</p>
</div>
<div class="feature-card">
<h3>High Accuracy</h3>
<p>MAGMA retains and betters the solution quality of previous AGHF formulations, while simultaneously being better than classical DDP methods</p>
</div>
<div class="feature-card">
<h3>General Constraints</h3>
<p>Handles general nonlinear constraints and arbitrary cost functions, going beyond previous AGHF formulations.</p>
</div>
</div>
</section>
<!-- ==================== METHOD ==================== -->
<section id="method">
<h2>Method</h2>
<p>
MAGMA extends the AGHF equation to handle constraints by coupling it with an
augmented Lagrangian structure, co-evolving primal trajectories alongside dual
multipliers. We characterize this flow theoretically, proving that MAGMA is a
gradient flow on an appropriately defined energy.
</p>
<p>
The numerical implementation utilizes a pseudospectral method-of-lines approach.
To handle constraints dynamically, it approximates constraint derivatives using
a smooth activation function.
</p>
<h3>Two-Phase Solver</h3>
<div class="pipeline">
<div class="step">
<strong>Phase 1</strong>
Find a feasible trajectory<br>(no control effort minimization)
</div>
<span class="arrow">→</span>
<div class="step">
<strong>Phase 2</strong>
Refine with control effort<br>(u²) minimization
</div>
<span class="arrow">→</span>
<div class="step">
<strong>Result</strong>
Optimized, constraint-satisfying trajectory
</div>
</div>
</section>
<!-- ==================== CONSTRAINTS ====================
<section id="constraints">
<h2>MAGMA Supports These Constraints</h2>
<table>
<thead>
<tr>
<th>Type</th>
<th>Description</th>
<th>Key Parameters</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>obstacles</code></td>
<td>Sphere obstacle avoidance</td>
<td><code>obstacles_info</code>, <code>c_cons_obs</code>, <code>k_cons_obs</code></td>
</tr>
<tr>
<td><code>boxes</code></td>
<td>Box obstacle avoidance</td>
<td><code>boxes_data</code>, <code>joint_radii</code>, <code>c_cons_boxes</code>, <code>k_cons_boxes</code></td>
</tr>
<tr>
<td><code>state</code></td>
<td>Joint position / velocity limits</td>
<td><code>x_lower</code>, <code>x_upper</code>, <code>c_cons_state</code>, <code>k_cons_state</code></td>
</tr>
<tr>
<td><code>input</code></td>
<td>Torque limits</td>
<td><code>u_lower</code>, <code>u_upper</code>, <code>c_cons_input</code>, <code>k_cons_input</code></td>
</tr>
</tbody>
</table>
</section> -->
<!-- ==================== USAGE ====================
<section id="usage">
<h2>Quick Start</h2>
<h3>Installation</h3>
<pre><code># Clone and set up environment
conda env create -f ps_aghf/new_ps_aghf_v1_env.yml
conda activate ps_aghf_v1
# Build C++ backend
cd ps_aghf/cpp && make
# Set Python path
export PYTHONPATH="/path/to/AGHF/ps_aghf/src:/path/to/AGHF/ps_aghf/src/ps_aghf"</code></pre>
<h3>Running Experiments</h3>
<pre><code># Experiment 2 — sphere obstacles
python timing_scripts/constrained/augLag_exp2.py
# Experiment 3 — box constraints
python timing_scripts/constrained/augLag_testing.py</code></pre>
<h3>Python API</h3>
<pre><code>from ps_aghf.experiment import Experiment
from ps_aghf.parameter_set import ParameterSetActivatedCompositeALM
import numpy as np
# Define boundary states
X0 = np.array([q_start, qd_start]).flatten()
Xf = np.array([q_end, qd_end]).flatten()
# Configure solver
pset = ParameterSetActivatedCompositeALM(
p=7, N=7, X0=X0, Xf=Xf,
name="my_experiment",
s_max=500, k=1e5,
abs_tol=1e-4, rel_tol=1e-4,
method_name="cvode",
max_steps=int(1e8),
j_type=j_type,
fp_urdf=fp_urdf,
constraint_list=constraint_list
)
# Run two-phase optimization
experiment = Experiment(
parameter_sets_ph2=[pset],
timeout=60,
folder_store="./results"
)
result_states, result_multipliers = experiment.run_ALM_two_phase(
pset, pset_ph1=pset_ph1,
mode="ALM", skip_ph1=False, force_ph2=True
)</code></pre>
</section> -->
<!-- ==================== RESULTS ==================== -->
<section id="results">
<h2>Results</h2>
<p>
MAGMA was validated across a variety of challenging trajectory generation problems
featuring obstacles, state limits, and actuation limits, and evaluated against
state-of-the-art solvers.
</p>
<!-- <h3>Obstacle Avoidance (Kinova Gen3, Random Sphere Obstacles)</h3>
<p>
MAGMA shows 100% success rate across a series of scenarios with randomly placed sphere obstacles, consistently finding feasible trajectories
</p> -->
<!-- <table>
<thead>
<tr>
<th>Solver</th>
<th>Success Rate</th>
<th>Avg. Solve Time</th>
</tr>
</thead>
<tbody>
<tr>
<td><strong>MAGMA</strong></td>
<td><strong>100%</strong></td>
<td><strong>4.83 s</strong></td>
</tr>
<tr>
<td>Aligator</td>
<td>83%</td>
<td>29.89 s</td>
</tr>
<tr>
<td>BLAZE</td>
<td>—</td>
<td>6.85 s</td>
</tr>
<tr>
<td>PHLAME</td>
<td>0%</td>
<td>—</td>
</tr>
</tbody>
</table> -->
<h3>Task-Based Scenarios</h3>
<p>
MAGMA was tested in practical, task-oriented scenarios involving cuboid obstacles,
such as reaching into shelves and placing items in confined bins, reflecting real-world
industrial and domestic deployments.
</p>
<p>
Below are visualizations of MAGMA solving these scenarios.
<span style="color: #e67e22; font-weight: 600;">Orange</span> represents the initial guess,
<span style="font-weight: 600;">white</span> is the final optimized solution, and
<span style="color: #e74c3c; font-weight: 600;">red</span> denotes obstacles.
</p>
<div class="video-grid">
<div class="scene-cell">
<div class="scene-label">Scene 1</div>
<video autoplay loop muted playsinline>
<source src="videos/scene2/init.mp4" type="video/mp4">
</video>
<div class="vid-caption">Initial Guess</div>
<video autoplay loop muted playsinline>
<source src="videos/scene2/fin.mp4" type="video/mp4">
</video>
<div class="vid-caption">Final Solution</div>
</div>
<div class="scene-cell">
<div class="scene-label">Scene 2</div>
<video autoplay loop muted playsinline>
<source src="videos/scene4/init.mp4" type="video/mp4">
</video>
<div class="vid-caption">Initial Guess</div>
<video autoplay loop muted playsinline>
<source src="videos/scene4/fin.mp4" type="video/mp4">
</video>
<div class="vid-caption">Final Solution</div>
</div>
<div class="scene-cell">
<div class="scene-label">Scene 3</div>
<video autoplay loop muted playsinline>
<source src="videos/scene16/init.mp4" type="video/mp4">
</video>
<div class="vid-caption">Initial Guess</div>
<video autoplay loop muted playsinline>
<source src="videos/scene16/fin.mp4" type="video/mp4">
</video>
<div class="vid-caption">Final Solution</div>
</div>
<div class="scene-cell">
<div class="scene-label">Scene 4</div>
<video autoplay loop muted playsinline>
<source src="videos/scene18/init.mp4" type="video/mp4">
</video>
<div class="vid-caption">Initial Guess</div>
<video autoplay loop muted playsinline>
<source src="videos/scene18/fin.mp4" type="video/mp4">
</video>
<div class="vid-caption">Final Solution</div>
</div>
<div class="scene-cell">
<div class="scene-label">Scene 5</div>
<video autoplay loop muted playsinline>
<source src="videos/scene20/init.mp4" type="video/mp4">
</video>
<div class="vid-caption">Initial Guess</div>
<video autoplay loop muted playsinline>
<source src="videos/scene20/fin.mp4" type="video/mp4">
</video>
<div class="vid-caption">Final Solution</div>
</div>
<div class="scene-cell">
<div class="scene-label">Scene 6</div>
<video autoplay loop muted playsinline>
<source src="videos/scene21/init.mp4" type="video/mp4">
</video>
<div class="vid-caption">Initial Guess</div>
<video autoplay loop muted playsinline>
<source src="videos/scene21/fin.mp4" type="video/mp4">
</video>
<div class="vid-caption">Final Solution</div>
</div>
</div>
</section>
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<!-- ==================== CITATION ==================== -->
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<section id="citation">
<h2>Citation</h2>
<pre><code>@article{magma2026,
title = {MAGMA: Multiplier-Augmented Geometric Motion Algorithm},
author = {Ramos Chuquiure, C{\'e}sar E. and Thakur, Vansh
and Enninful Adu, Challen and Vasudevan, Ram},
year = {2026}
}</code></pre>
<p style="margin-top: 0.8rem; font-size: 0.9rem; color: var(--muted);">
This work was supported by AFOSR MURI FA9550-23-1-0400.
</p>
</section>
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<!-- ==================== FOOTER ==================== -->
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<p>MAGMA — Multiplier-Augmented Geometric Motion Algorithm — ROAHM Lab, University of Michigan</p>
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