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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<meta name="description"
content="Global Contact-rich Planning">
<meta name="keywords" content="Contact-riched Control and Planning, Sparse Polynomial Optimization">
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<title>SPOT: Sparse POP for Robotics</title>
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<h1 class="title is-1 publication-title">Global Contact-Rich Planning with Sparsity-Rich Semidefinite Relaxations</h1>
<!-- <h2 class="title is-2 publication-title">
<img src="static/images/rss-logo.png" width="3">
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Robotics: Science and Systems (RSS) 2025
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Robotics: Science and Systems (RSS) 2025
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<div class="is-size-5 publication-authors">
<span class="author-block"></span>
<a href="https://shuchengkang.github.io">Shucheng Kang</a><sup>*1</sup>,
</span>
<span class="author-block">
<a href="">Guorui Liu</a><sup>*2</sup>,
</span>
<span class="author-block">
<a href="https://hankyang.seas.harvard.edu">Heng Yang</a><sup>1</sup>
</span>
</div>
<div class="is-size-5 publication-authors">
<span class="author-block"><sup>1</sup>School of Engineering and Applied Sciences, Harvard University</span>
<span class="author-block"><sup>2</sup>School of Industrial and Systems Engineering, Georgia Institute of Technology</span>
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class="external-link button is-normal is-rounded is-dark">
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<h2 class="subtitle has-text-centered">
<span class="nameshort">SPOT</span> is a toolbox for exploiting sparsity in polynomial optimization for robotics and beyond.
</h2>
</div>
</div>
</section>
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</video>
<h2 class="subtitle has-text-centered">
<span class="nameshort">SPOT</span> is a toolbox for exploiting sparsity in polynomial optimization for robotics and beyond.
</h2>
</div>
</div>
</section> -->
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<video poster="" id="chair-tp" autoplay controls muted loop playsinline height="100%">
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<h2 class="title is-3">Abstract</h2>
<div class="content has-text-justified">
<p>
We show that contact-rich motion planning is also <b>sparsity-rich</b> when viewed as polynomial optimization (POP). We can exploit not only the <b>correlative</b> and <b>term</b> sparsity patterns that are general to all POPs, but also specialized sparsity patterns from the robot kinematic structure and the separability of contact modes. Such sparsity enables the design of high-order but sparse semidefinite programming (SDPs) relaxations---building upon Lasserre's moment and sums of squares hierarchy---that (<b>i</b>) can be solved in seconds by off-the-shelf SDP solvers, and (<b>ii</b>) compute near globally optimal solutions to the nonconvex contact-rich planning problems with small certified suboptimality. Through extensive experiments both in simulation (Push Bot, Push Box, Push Box with Obstacles, and Planar Hand) and real world (Push T), we demonstrate the power of using convex SDP relaxations to generate global contact-rich motion plans.
As a contribution of independent interest, we release the Sparse Polynomial Optimization Toolbox (<span class="nameshort">SPOT</span>)---implemented in C++ with interfaces to both Python and Matlab---that automates sparsity exploitation for robotics and beyond.
</p>
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<h2 class="title">BibTeX</h2>
<pre><code>@article{park2021nerfies,
author = {Park, Keunhong and Sinha, Utkarsh and Barron, Jonathan T. and Bouaziz, Sofien and Goldman, Dan B and Seitz, Steven M. and Martin-Brualla, Ricardo},
title = {Nerfies: Deformable Neural Radiance Fields},
journal = {ICCV},
year = {2021},
}</code></pre>
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</section> -->
<section class="hero teaser" style="margin-bottom: 4rem;">
<div class="container is-max-desktop">
<div
class="hero-body"
style="
display: grid;
grid-template-columns: repeat(2, 1fr);
grid-gap: 1rem;
"
>
<!-- Repeat the same video 9 times for a 3x3 grid -->
<video id="teaser1" autoplay muted loop playsinline style="width:100%;">
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</video>
<video id="teaser2" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/longT-3.mp4" type="video/mp4">
</video>
</div>
<h2 class="subtitle has-text-centered" style="margin-top: 0rem;">
Real-world push-T: whole procedure. (Left) “Clean” push T: no mismatches and disturbances; (Right) “Dirty” push T: severe mismatches and disturbances.
</h2>
</section>
<!-- Text block between the two sections -->
<div
style="
margin: 2rem auto;
text-align: center;
font-size: 1.2rem;
max-width: 800px;
"
>
<!-- Bold text -->
<strong>For each replanning, we reset the arm to a fixed position to make sure the Apriltag can give accurate state estimation. In the following videos, we only show the procedure that the pusher is pushing the T-block — for simplicity.</strong>
</div>
<section class="hero teaser" style="margin-bottom: 4rem;">
<div class="container is-max-desktop">
<div
class="hero-body"
style="
display: grid;
grid-template-columns: repeat(3, 1fr);
grid-gap: 1rem;
"
>
<!-- Repeat the same video 9 times for a 3x3 grid -->
<video id="teaser1" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/shortT-10.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<video id="teaser2" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/shortT-2.mp4" type="video/mp4">
</video>
<video id="teaser3" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/shortT-3.mp4" type="video/mp4">
</video>
<video id="teaser4" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/shortT-4.mp4" type="video/mp4">
</video>
<video id="teaser5" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/shortT-5.mp4" type="video/mp4">
</video>
<video id="teaser6" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/shortT-6.mp4" type="video/mp4">
</video>
<video id="teaser7" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/shortT-7.mp4" type="video/mp4">
</video>
<video id="teaser8" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/shortT-8.mp4" type="video/mp4">
</video>
<video id="teaser9" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/shortT-9.mp4" type="video/mp4">
</video>
</div>
<h2 class="subtitle has-text-centered" style="margin-top: 0rem;">
Real-world push-T: "clean” case
</h2>
</div>
</section>
<section class="hero teaser" style="margin-bottom: 4rem;">
<div class="container is-max-desktop">
<div
class="hero-body"
style="
display: grid;
grid-template-columns: repeat(3, 1fr);
grid-gap: 1rem;
"
>
<!-- Repeat the same video 9 times for a 3x3 grid -->
<video id="teaser1" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/dirtyT-1.mp4" type="video/mp4">
Your browser does not support the video tag.
</video>
<video id="teaser2" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/dirtyT-11.mp4" type="video/mp4">
</video>
<video id="teaser3" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/dirtyT-3.mp4" type="video/mp4">
</video>
<video id="teaser4" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/dirtyT-12.mp4" type="video/mp4">
</video>
<video id="teaser5" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/dirtyT-10.mp4" type="video/mp4">
</video>
<video id="teaser6" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/dirtyT-6.mp4" type="video/mp4">
</video>
<video id="teaser7" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/dirtyT-7.mp4" type="video/mp4">
</video>
<video id="teaser8" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/dirtyT-8.mp4" type="video/mp4">
</video>
<video id="teaser9" autoplay muted loop playsinline style="width:100%;">
<source src="./static/videos/2024-guorui-realworld/compressed/dirtyT-9.mp4" type="video/mp4">
</video>
</div>
<h2 class="subtitle has-text-centered">
Real-world push-T: “dirty” case
</h2>
</div>
</section>
<!-- Text block between the two sections -->
<div
style="
margin: 2rem auto;
text-align: center;
font-size: 1.2rem;
max-width: 800px;
"
>
<!-- Bold text -->
<strong>More contact-rich trajectories with certifiable optimality...</strong>
</div>
<section class="hero teaser" style="margin-bottom: 4rem;">
<div class="container is-max-desktop">
<div
class="hero-body"
style="
display: flex;
justify-content: center;
align-items: center;
"
>
<img
src="./static/videos/kang/pushbot.gif"
alt="Animated GIF"
style="width: 100%; max-width: 1000px;"
/>
</div>
<h2 class="subtitle has-text-centered" style="margin-top: 0rem;">
Push Bot in simulation
</h2>
</div>
</section>
<section class="hero teaser" style="margin-bottom: 4rem;">
<div class="container is-max-desktop">
<div
class="hero-body"
style="
display: flex;
justify-content: center;
align-items: center;
"
>
<img
src="./static/videos/kang/pushbox.gif"
alt="Animated GIF"
style="width: 100%; max-width: 1000px;"
/>
</div>
<h2 class="subtitle has-text-centered" style="margin-top: 0rem;">
Push Box in simulation
</h2>
</div>
</section>
<section class="hero teaser" style="margin-bottom: 4rem;">
<div class="container is-max-desktop">
<div
class="hero-body"
style="
display: flex;
justify-content: center;
align-items: center;
"
>
<img
src="./static/videos/kang/pushT.gif"
alt="Animated GIF"
style="width: 100%; max-width: 1000px;"
/>
</div>
<h2 class="subtitle has-text-centered" style="margin-top: 0rem;">
Push T-block in simulation
</h2>
</div>
</section>
<section class="hero teaser" style="margin-bottom: 4rem;">
<div class="container is-max-desktop">
<div
class="hero-body"
style="
display: flex;
justify-content: center;
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Push Box with a Tunnel in simulation
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Planar Hand in simulation
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