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<!DOCTYPE html>
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content="Potential Based Diffusion Motion Planning">
<meta name="keywords" content="Machine Learning, ICML, Diffusion Model, Motion Planning, Compositionality, Robotics">
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<title>Potential Based Diffusion Motion Planning</title>
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<h1 class="title is-1 publication-title">Potential Based Diffusion Motion Planning</h1>
<div class="is-size-5 publication-authors">
<span class="author-block">
<a href="https://devinluo27.github.io/">Yunhao Luo</a><sup>1</sup>,
</span>
<span class="author-block">
<a href="https://chensun.me/">Chen Sun</a><sup>1</sup>,
</span>
<span class="author-block">
<a href="http://web.mit.edu/cocosci/josh.html">Joshua B. Tenenbaum</a><sup>2</sup>,
</span>
<span class="author-block">
<a href="https://yilundu.github.io/">Yilun Du</a><sup>2</sup>
</span>
</div>
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<span class="author-block">
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<sup>1</sup>Brown University,
</span>
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<sup>2</sup>MIT
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<br>ICML 2024
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contact: yunhao_luo [at] gatech [dot] edu
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</video>
<p>
<!-- via Composing Potentials -->
<b>Potential Based Diffusion Motion Planning and Generalization.</b>
<!-- by iteratively denoising -->
Our method encodes motion planning constraints to latent vectors and uses the potential functions to generate motion plans.
<!-- The potentials can then be composed and generalize to unseen out-of-distribution environments. -->
<!-- Our method can generalize to unseen out-of-distribution environments by composing potentials. -->
<!-- By composing potentials, our method can generalize to unseen out-of-distribution environments. -->
<!-- We show that the potential functions are trained on sparse homogeneous environments, -->
<!-- while can directly generalize to cluttered heterogeneous environments via composing potentials. -->
We show that our method can directly generalize to cluttered heterogeneous environments via composing potentials,
while the potential functions are only trained on simple homogeneous environments.
</p>
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<h2 class="title is-3">Abstract</h2>
<div class="content has-text-justified">
<p>
Effective motion planning in high dimensional spaces is a long-standing open problem in robotics. One class of traditional motion planning algorithms corresponds to potential-based motion planning. An advantage of potential based motion planning is composability -- different motion constraints can be easily combined by adding corresponding potentials. However, constructing motion paths from potentials requires solving a global optimization across configuration space potential landscape, which is often prone to local minima. We propose a new approach towards learning potential based motion planning, where we train a neural network to capture and learn an easily optimizable potentials over motion planning trajectories. We illustrate the effectiveness of such approach, significantly outperforming both classical and recent learned motion planning approaches and avoiding issues with local minima. We further illustrate its inherent composability, enabling us to generalize to a multitude of different motion constraints.
</p>
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</section>
<section class="hero teaser">
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<figure>
<img width="70%" src="materials/0-teaser-icml-website-v2.png" style="min-width: 300px">
<p class="caption">
<b>Illustrative Example of Composing Diffusion Energy Potentials.</b>
Our approach learns different potential functions over motion planning trajectories $q_{1:N}$ (orange dashed lines). Different potentials can be combined and optimized to construct new motion plans that avoid obstacles encoded in both potential functions.
</p>
</figure>
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</section>
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We discover a set of compositional concepts given a dataset of unlabeled images. Score functions representing each concept $\{c^1, \dots, c^K\}$ are composed together to form a compositional score function that is trained to denoise images. The inferred concepts can be used to generate new images.
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<h2 class="title is-3">Method</h2>
<div class="content has-text-justified">
<!-- We next discuss how to learn potentials for potential motion planning that enable us to effectively optimize samples. -->
<b>Learning Potentials for Motion Planning. </b>
Given a motion plan $q_{1:T}$ from start state $q_{\text{st}}$ to end state $q_{\text{e}}$ and a characterization of the configuration space $C$ ($\textit{i.e.}$ the set of obstacles in the environment), we propose to learn a trajectory-level potential function $U_\theta$ so that
<div class="mjx-rendered">
\begin{equation}
q_{1:T}^* = \text{arg\,min}_{q_{1:T}} U_\theta(q_{1:T}, q_{\text{st}}, q_{\text{e}}, C),
\label{eqn:potential_traj}
\end{equation}
</div>
where $q_{1:T}^*$ is a successful motion plan from $q_{\text{st}}$ to $q_{\text{e}}$.
To learn the potential function above, we propose to learn an
<a href="https://openai.com/index/energy-based-models/" class="citation-link">EBM</a>
<!-- <a href="#ref1" class="citation-link">[1,</a> -->
<!-- <a href="#ref2" class="citation-link">2]</a> -->
<!-- ~\citep{lecun2006tutorial,du2019implicit} -->
across a dataset of solved motion planning $D = \{q_{\text{st}}^i, q_{\text{e}}^i, q_{1:T}^i, C^i \}$, where $e^{-E_\theta(q_{1:T}|q_{\text{st}}, q_{\text{e}}, C)} \propto p(q_{1:T}|q_{\text{st}}, q_{\text{e}}, C)$. Since the dataset $D$ is of solved motion planning problems, the learned energy function $E_\theta$ will have minimal energy at successful motion plans $q_{1:T}^*$ and thus satisfy our potential function $U_\theta$.
<!-- in \Eqref{eqn:potential_traj}. -->
<br/>
<br/>
<br/>
<b>Learning EBM for Motion Plan Generation.</b>
To learn the EBM landscape that enables us to effectively optimize and generate motion plans $q_{1:T}^*$, we propose to shape the energy landscape using <a href="https://arxiv.org/abs/1503.03585" class="citation-link">denoising diffusion</a> training objective.
<!-- <a href="#ref3" class="citation-link">[1]</a> -->
<!-- <a href="#ref3" class="citation-link">[3,</a> -->
<!-- <a href="#ref4" class="citation-link">4]</a>. -->
<!-- ~\citep{sohl2015deep,ho2020denoising}. -->
Formally, to train our potential, we use the
<a href="https://energy-based-model.github.io/reduce-reuse-recycle/" class="citation-link">energy based diffusion</a>
training objective,
<!-- in <a href="#ref5" class="citation-link">[5]</a>, -->
<!-- where the gradient of energy function is trained to denoise noise corrupted motion plans $q_{1:T}^*$, -->
where the gradient of energy function is trained to denoise noise-corrupted motion plans $q_{1:T}^i$ from dataset $D$,
<div class="mjx-rendered">
\begin{equation}
\mathcal{L}_{\text{MSE}}=\|\mathbf{\epsilon} - \nabla_{q_{1:T}} E_\theta(\sqrt{1-\beta_s} q_{1:T}^i + \sqrt{\beta_s} \mathbf{\epsilon}, s, q_{\text{st}}^i, q_{\text{e}}^i, C^i)\|^2
\label{eqn:train_obj}
\end{equation}
</div>
where $\epsilon$ is sampled from Gaussian noise $\mathcal{N}(0, 1)$, $s \in\{1,2,...,S\}$ is the denoising diffusion step (we set $S = 100$), and $\beta_s$ is the corresponding Gaussian noise corruption on a motion planning path $q_{1:T}^i$. We refer to $E_\theta$ as the <i>diffusion potential function</i>.
<br/>
<br/>
<br/>
<b>Optimize and Sample from Diffusion Potential Function.</b>
To optimize and sample from our diffusion potential function, we initialize a motion path $q_{1:T}^S$ at diffusion step $S$ from Gaussian noise $\mathcal{N}(0, 1)$ and optimize for motion path following the gradient of the energy function $E_{\theta}$. We iteratively refine the motion path $q_{1:T}^s$ across each diffusion step following
<div class="mjx-rendered">
\begin{align}
& q_{1:T}^{s-1}=q_{1:T}^{s}-\gamma \epsilon + \xi, \quad \xi \sim \mathcal{N} \bigl(0, \sigma^2_s I \bigl), \notag \\
& \text{where} \: \: \epsilon = \nabla_{q_{1:T}} E_\theta(q_{1:T}, s, q_{\text{st}}, q_{\text{e}}, C)
\label{eqn:diffusion_opt}
\end{align}
</div>
To parameterize the energy function $E_\theta(q_{1:T}, s, q_{\text{st}}, q_{\text{e}}, C)$,
we use <a href="https://arxiv.org/abs/2207.12598" class="citation-link">classifier-free guidance</a>
<!-- <a href="#ref6" class="citation-link">[6]</a> -->
<!-- \cite{ho2022classifier} -->
to form a peaker composite energy function conditioned on $C$.
$\gamma$ and $\sigma^2_s$ are diffusion specific scaling constants (A rescaling term at each diffusion step is omitted above for clarity). The final predicted motion path $q_{1:T}^*$ corresponds to the output $q_{1:T}^0$ after running $S$ steps of optimization from the diffusion potential function.
<br/>
<br/>
<br/>
<b>Composing Diffusion Potential Functions</b>
Given two separate diffusion potential functions $E^1_\theta(\cdot)$ and $E^2_\theta(\cdot)$, encoding separate constraints for motion planning, we can likewise form a composite potential function $E^{\text{comb}}(\cdot) = E^1(\cdot) + E^2(\cdot)$ by directly summing the corresponding potentials.
To sample from the new diffusion potential function $E^{\text{comb}}$, we can follow
<div class="mjx-rendered">
\begin{align}
&q_{1:T}^{s-1}=q_{1:T}^{s}-\gamma \epsilon^{\text{comb}}+\xi, \quad \xi \sim \mathcal{N} \bigl(0, \sigma^2_s I \bigl), \label{eqn:diffusion_opt_comb} \\
& \text{where} \: \: \epsilon^{\text{comb}} = \nabla_{q_{1:T}} (E_\theta^{1}(q_{1:T}, s, q_{\text{st}}, q_{\text{e}}, C_1) + E_\theta^{2}(q_{1:T}, s, q_{\text{st}}, q_{\text{e}}, C_2)). \notag
\end{align}
</div>
This potential function $E^{\text{comb}}$ will have low energy precisely at motion planning paths $q_{1:T}$ which satisfy both constraints, with sampling corresponding to optimizing this potential function. Hence, by iteratively sampling from the composite potential function, we can obtain motion plans that satisfy both constraints. Please see our
<a href="#comp-results-title" class="citation-link">compositional results</a> below.
<!-- (see Appendix \ref{sect:app_cond_indep}) -->
<!-- This composite potential is the ground truth potential combining constraints if the constraints are independent, which is satisfied if the set of trajectories given constraints are uniformly distributed, and otherwise serves as approximate proxy to optimize multiple constraints. -->
</div>
</div>
</div>
</div>
</section>
<hr>
<section class="section">
<div class="container is-max-desktop">
<!-- Animation. -->
<div class="columns is-centered">
<div class="column is-full-width">
<h2 class="title is-4">Trajectory Denoising Process using Diffusion Potential Function</h2>
<!-- <div class="content has-text-justified has-text-centered"> -->
<!-- <p> -->
<!-- Our method can decompose a set of unlabeled images from into objects without using any labels. -->
<!-- </p> -->
<!-- </div> -->
<!-- <img src="materials/decomposition/imagenet_decomposition.png" -->
<!-- alt="Object Decomposition."/> -->
<div class="video-container text-centered traj_denoise_row">
<!-- <center> -->
<video width="19%" playsinline="" autoplay="" preload="" muted="" loop>
<source src="materials/rm2d-per-step/luotest_env6.mp4" type="video/mp4">
</video>
<video width="19%" playsinline="" autoplay="" preload="" muted="" loop>
<source src="materials/rm2d-per-step/luotest_env10.mp4" type="video/mp4">
</video>
<video width="19%" playsinline="" autoplay="" preload="" muted="" loop>
<source src="materials/rm2d-per-step/luotest_env17.mp4" type="video/mp4">
</video>
<!-- </center> -->
<p>
<!-- \textbf{Trajectory Denoising Process.} -->
<b>Trajectory Denoising Process on Maze2D.</b>
Motion trajectories are randomly initialized from Gaussian distribution in timestep $S = 100$. Noises are iteratively removed via the gradient of the energy function. A feasible collision-free trajectory can be obtained at timestep $S = 0$.
We use <a href="https://arxiv.org/abs/2010.02502" class="citation-link">DDIM</a> to accelerate sampling speed.
<!-- \eqref{eqn:diffusion_opt} -->
</p>
</div>
</div>
</div>
</div>
</section>
<hr>
<!-- ================================= -->
<section class="section">
<div class="container is-max-desktop">
<div class="columns is-centered">
<div class="column is-full-width">
<h2 class="title is-4">Motion Plans on Concave Obstacles</h2>
<section class="hero is-light is-small" style="margin: 0px; padding: 0px">
<div class="container is-max-desktop" style="margin: 0px; padding: 0px">
<div class="hero-body" style="margin: 0px; padding-bottom: 10px; padding-top: 10px">
<div class="container is-max-desktop" style="overflow: hidden; margin: 0px">
<div id="splide-rm2d-cc-base" class="splide">
<div class="splide__track">
<ul class="splide__list">
<li class="splide__slide splide_rm2d_cc">
<div style="text-align: center; margin-top:0px">
<p class="splide_kuka_prob_id">(a)</p>
<div class="splide_rm2d_cc_row">
<div>
<img src="materials/rm2d-concave/env4/rmp.png" alt="Image 1">
<div class="splide_img_caption_18">
RMP
</div>
</div>
<div>
<img src="materials/rm2d-concave/env4/rrt.png" alt="Image 1">
<div class="splide_img_caption_18">
RRT*
</div>
</div>
<div>
<img src="materials/rm2d-concave/env4/bit.png" alt="Image 1">
<div class="splide_img_caption_18">
BIT*
</div>
</div>
<div>
<img src="materials/rm2d-concave/env4/mpnet.png" alt="Image 1">
<div class="splide_img_caption_18">
MPNet
</div>
</div>
<div>
<img src="materials/rm2d-concave/env4/ours.png" alt="Image 1">
<div class="splide_img_caption_18">
Ours
</div>
</div>
</div>
</div>
</li>
<li class="splide__slide splide_kuka">
<div style="text-align: center; margin-top:0px">
<p class="splide_kuka_prob_id">(b)</p>
<div class="splide_rm2d_cc_row">
<div>
<img src="materials/rm2d-concave/env17/rmp.png" alt="Image 1">
<div class="splide_img_caption_18">
RMP
</div>
</div>
<div>
<img src="materials/rm2d-concave/env17/rrt.png" alt="Image 1">
<div class="splide_img_caption_18">
RRT*
</div>
</div>
<div>
<img src="materials/rm2d-concave/env17/bit.png" alt="Image 1">
<div class="splide_img_caption_18">
BIT*
</div>
</div>
<div>
<img src="materials/rm2d-concave/env17/mpnet.png" alt="Image 1">
<div class="splide_img_caption_18">
MPNet
</div>
</div>
<div>
<img src="materials/rm2d-concave/env17/ours.png" alt="Image 1">
<div class="splide_img_caption_18">
Ours
</div>
</div>
</div>
</div>
</li>
<li class="splide__slide splide_kuka">
<div style="text-align: center; margin-top:0px">
<p class="splide_kuka_prob_id">(c)</p>
<div class="splide_rm2d_cc_row">
<div>
<img src="materials/rm2d-concave/env66/rmp.png" alt="Image 1">
<div class="splide_img_caption_18">
RMP
</div>
</div>
<div>
<img src="materials/rm2d-concave/env66/rrt.png" alt="Image 1">
<div class="splide_img_caption_18">
RRT*
</div>
</div>
<div>
<img src="materials/rm2d-concave/env66/bit.png" alt="Image 1">
<div class="splide_img_caption_18">
BIT*
</div>
</div>
<div>
<img src="materials/rm2d-concave/env66/mpnet.png" alt="Image 1">
<div class="splide_img_caption_18">
MPNet
</div>
</div>
<div>
<img src="materials/rm2d-concave/env66/ours.png" alt="Image 1">
<div class="splide_img_caption_18">
Ours
</div>
</div>
</div>
</div>
</li>
<li class="splide__slide splide_kuka">
<div style="text-align: center; margin-top:0px">
<p class="splide_kuka_prob_id">(d)</p>
<div class="splide_rm2d_cc_row">
<div>
<img src="materials/rm2d-concave/env97/rmp.png" alt="Image 1">
<div class="splide_img_caption_18">
RMP
</div>
</div>
<div>
<img src="materials/rm2d-concave/env97/rrt.png" alt="Image 1">
<div class="splide_img_caption_18">
RRT*
</div>
</div>
<div>
<img src="materials/rm2d-concave/env97/bit.png" alt="Image 1">
<div class="splide_img_caption_18">
BIT*
</div>
</div>
<div>
<img src="materials/rm2d-concave/env97/mpnet.png" alt="Image 1">
<div class="splide_img_caption_18">
MPNet
</div>
</div>
<div>
<img src="materials/rm2d-concave/env97/ours.png" alt="Image 1">
<div class="splide_img_caption_18">
Ours
</div>
</div>
</div>
</div>
</li>
<!-- <li class="splide__slide splide_kuka">
<div>
<video playsinline="" autoplay="" preload="" muted="" loop controls>
<source src="materials/kuka7d/pbd-2-600.mp4" type="video/mp4">
</video>
<p class="splide_kuka_caption">Ours</p>
</div>
<div class="splide_kuka_space"> </div>
<div>
<video playsinline="" autoplay="" preload="" muted="" loop controls>
<source src="materials/kuka7d/mpo-2-600.mp4" type="video/mp4">
</video>
<p class="splide_kuka_caption">MπNet</p>
</div>
</li> -->
</ul>
</div>
</div>
</div>
</div>
</div>
</section>
<br/>
<b> Qualitative Comparison of Motion Plans on Maze2D with Concave Obstacles.</b>
The green star indicates the start pose and the red star indicates the goal pose.
RMP tends to stuck in local minima and fails to reach the goals;
RRT* is slow in speed and may reach the planning time limit in some difficult problems;
trajectories by MPNet may occasionally go through obstacles in the environments;
our method is able to generate smooth and low-cost trajectories without collision while being up to 10 times faster than BIT*.
</div>
</div>
</div>
</section>
<!-- ================================================== -->
<section class="section">
<div class="container is-max-desktop">
<div class="columns is-centered">
<div class="column is-full-width">
<h2 class="title is-4">Motion Plans on KUKA</h2>
<section class="hero is-light is-small" style="margin: 0px; padding: 0px">
<div class="container is-max-desktop" style="margin: 0px; padding: 0px">
<div class="hero-body" style="margin: 0px; padding-bottom: 10px; padding-top: 10px">
<div class="container is-max-desktop" style="overflow: hidden; margin: 0px">
<div id="splide-kuka-base" class="splide">
<div class="splide__track">
<ul class="splide__list">
<li class="splide__slide splide_kuka">
<div style="text-align: center; margin-top:0px">
<p class="splide_kuka_prob_id">(a)</p>
<div class="kuka_base_row">
<div>
<video playsinline="" autoplay="" muted="" loop controls>
<source src="materials/kuka7d/pbd-1-600.mp4" type="video/mp4">
</video>
<p class="splide_kuka_caption">Ours</p>
</div>
<!-- <div class="splide_kuka_space"> </div> -->
<div>
<video playsinline="" autoplay="" muted="" loop controls>
<source src="materials/kuka7d/mpo-1-600.mp4" type="video/mp4">
</video>
<p class="splide_kuka_caption">MπNet</p>
</div>
</div>
</div>
</li>
<li class="splide__slide splide_kuka">
<div style="text-align: center; margin-top:0px">
<p class="splide_kuka_prob_id">(b)</p>
<div class="kuka_base_row">
<div>
<video playsinline="" autoplay="" muted="" loop controls>
<source src="materials/kuka7d/pbd-2-600.mp4" type="video/mp4">
</video>
<p class="splide_kuka_caption">Ours</p>
</div>
<!-- <div class="splide_kuka_space"> </div> -->
<div>
<video playsinline="" autoplay="" muted="" loop controls>
<source src="materials/kuka7d/mpo-2-600.mp4" type="video/mp4">
</video>
<p class="splide_kuka_caption">MπNet</p>
</div>
</div>
</div>
</li>
<li class="splide__slide splide_kuka">
<div style="text-align: center; margin-top:0px">
<p class="splide_kuka_prob_id">(c)</p>
<div class="kuka_base_row">
<div>
<video playsinline="" autoplay="" muted="" loop controls>
<source src="materials/kuka7d/pbd-3-600.mp4" type="video/mp4">
</video>
<p class="splide_kuka_caption">Ours</p>
</div>
<!-- <div class="splide_kuka_space"> </div> -->
<div>
<video playsinline="" autoplay="" muted="" loop controls>
<source src="materials/kuka7d/mpo-3-600.mp4" type="video/mp4">
</video>
<p class="splide_kuka_caption">MπNet</p>
</div>
</div>
</div>
</li>
<li class="splide__slide splide_kuka">
<div style="text-align: center; margin-top:0px">
<p class="splide_kuka_prob_id">(d)</p>
<div class="kuka_base_row">
<div>
<video playsinline="" autoplay="" muted="" loop controls>
<source src="materials/kuka7d/pbd-4-600.mp4" type="video/mp4">
</video>
<p class="splide_kuka_caption">Ours</p>
</div>
<!-- <div class="splide_kuka_space"> </div> -->
<div>
<video playsinline="" autoplay="" muted="" loop controls>
<source src="materials/kuka7d/mpo-4-600.mp4" type="video/mp4">
</video>
<p class="splide_kuka_caption">MπNet</p>
</div>
</div>
</div>
</li>
<!-- <li class="splide__slide splide_kuka">
<div>
<video playsinline="" autoplay="" preload="" muted="" loop controls>
<source src="materials/kuka7d/pbd-2-600.mp4" type="video/mp4">
</video>
<p class="splide_kuka_caption">Ours</p>
</div>
<div class="splide_kuka_space"> </div>
<div>
<video playsinline="" autoplay="" preload="" muted="" loop controls>
<source src="materials/kuka7d/mpo-2-600.mp4" type="video/mp4">
</video>
<p class="splide_kuka_caption">MπNet</p>
</div>
</li> -->
</ul>
</div>
</div>
</div>
</div>
</div>
</section>
<b> Qualitative Comparison of Motion Plans on KUKA.</b>
We present motion plans by our diffusion potential planner and <a href="https://mpinets.github.io/" class="citation-link">MπNet</a>.
The large green/pink ball indicates the start/goal state. Our method in the left column generates smooth and near-optimal trajectories, while the trajectories of MπNet might choose a longer route, get stuck in some local regions, fail to pass through some narrow passages, or keep hovering near the start state.
</div>
</div>
</div>
</section>
<hr>
<section class="section">
<div class="container is-max-desktop">
<div class="columns is-centered">
<div class="column is-full-width">
<!-- <h2 class="title is-4"> Flexible Trajectory Morphology</h2> -->
<h2 class="title is-4"> Diverse Motion Plan Morphology</h2>
<div class="container bg-3">
<div class="row centered-caption">
<!-- TODO: Add More, these two videos start sec = 1, while others = 2. -->
<div class="video-container" style="gap: 0px">
<div class="col-sm-3 centered-caption" style="min-width: 235px;">
<video playsinline="" autoplay="" preload="" muted="" loop controls>
<source src="materials/kuka7d/pbd-6-600-morp1.mp4" type="video/mp4">
</video>
Trajectory 1
</div>
<!-- -->
<div class="col-sm-3 centered-caption" style="min-width: 235px;">
<video playsinline="" autoplay="" preload="" muted="" loop controls>
<source src="materials/kuka7d/pbd-6-600-morp2.mp4" type="video/mp4">
</video>
Trajectory 2
</div>
</div>
<div class="left-caption">
<!-- <b>Qualitative Visualizations of Different Trajectory Morphology.</b>
Our method can generate trajectories with various morphological shapes. The large green/pink ball indicates the location of the end effector of the start/goal state. Trajectory 1 and Trajectory 2 are generated under the same constraints (e.g., start state, goal state, obstacles locations) but with different starting noise, resulting in different route selection. Trajectory 1 routes through the central narrow passage between the two purple blocks and arrives the pink ball from above. Trajectory 2 avoids the obstacles by going under the blocks and finally arrives the same goal state from below. -->
<b>Stochastic Denoising Process Enables Diverse Motion Path Morphology.</b>
Trajectory 1 and Trajectory 2 are generated under the same constraints (e.g., start state, goal state, obstacles locations) and planning horizon, but with different denoising random seeds.
</div>
</div>
</div>
<div class="container bg-3">
<div class="row centered-caption">
<div class="video-container" style="gap: 0px">
<!-- <div class="col-sm-3 centered-caption" > -->
<!-- Maze2D 6+6 Obstacles -->
<!-- <br/> -->
<!-- <img src="./materials/rm2d-multi-h/comp_env60_base.png" alt="" width="100%" style="margin-top: 5px;"> -->
<!-- <img src="./materials/rm2d-multi-h/comp_env60_base.png" alt="" width="100%"> -->
<div class="multi_hor_rm2d_item">
<p class="multi_hor_rm2d_caption">Maze2D 6+6 Obstacles</p>
<!-- margin-top: 5px; -->
<img src="./materials/rm2d-multi-h/comp_env60_base.png" alt="" style="max-width: 235px">
</div>
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<div class="multi_hor_rm2d_item">
<!-- Horizon -->
<div class="multi_hor_rm2d_caption">Horizon
<div class="select is-small">
<select id="rm2d-comp-multi-h-1" onchange="update_rm2d_comp_multiH()">
<option value="60">60</option>
<option value="64">64</option>
<option value="68" selected="selected">68</option>
</select>
</div>
</div>
<!-- <br/> -->
<!-- width:100%; -->
<video id="rm2d-comp-multi-h-1-video" muted autoplay muted preload="" style="max-width: 235px">
<source src="materials/rm2d-multi-h/env60_h68.mp4" type="video/mp4">
</video>
<!-- </div> -->
</div>
<div class="left-caption">
<b>Different Planning Horizons Enable Diverse Motion Path Morphology.</b>
Left: Motion planning problem setup. The green/pink star indicates the start/goal state.
Right: Various motion path selection according to the planning horizon.
</div>
</div>
</div>
</div>
</div>
</div>
</section>
<hr>
<section class="section">
<div class="container is-max-desktop">
<div class="columns is-centered">
<div class="column is-full-width">
<!-- <h2 class="title is-4">Compositional Generalization over Increasing Obstacles</h2> -->
<h2 class="title is-4" id="comp-results-title">Generalization via Composing Potentials</h2>
<div class="content has-text-justified has-text-centered">
<p>
Our method can generalize to more difficult out-of-distribution environments by sampling from the composite diffusion potential function. We demonstrate qualitative results where we compose potentials of the same obstacle type, of different obstacle types, and of static and dynamic obstacles. Please see our paper for more comprehensive results.
</p>
</div>
<!-- ================================= -->
<section class="hero is-light is-small" style="margin: 0px; padding: 0px">
<div class="container is-max-desktop" style="margin: 0px; padding: 0px">
<div class="hero-body" style="margin: 0px; padding-bottom: 10px">
<div class="container is-max-desktop" style="overflow: hidden; margin: 0px">
<div id="splide-rm2d-comp-same" class="splide">
<div class="splide__track">
<ul class="splide__list">
<li class="splide__slide splide_rm2d">