# Potential Based Diffusion Motion Planning

Yunhao Luo  Chen Sun  Joshua B. Tenenbaum  Yilun Du

###### Abstract

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. Project website at https://energy-
based-model.github.io/potential-motion-plan.

Machine Learning, ICML, Diffusion Model, Motion Planning, Compositionality,
Robotics

##  1 Introduction

Motion planning is a fundamental problem in robotics and aims to find a
smooth, collision free path between a start and goal state given a specified
configuration space, and is heavily used across a variety of different
robotics tasks such as manipulation or navigation (Laumond et al., 1998). A
variety of approaches exist for motion planning, ranging from classical
sampling based approaches (Kavraki et al., 1996; Kuffner & LaValle, 2000;
Karaman & Frazzoli, 2011; Gammell et al., 2015) and optimization based methods
(Ratliff et al., 2009; Mukadam et al., 2018; Kalakrishnan et al., 2011). A
recent body of works have further explored how learned neural networks can be
integrated with motion planning for accelerated performance (Ichter & Pavone,
2019; Qureshi et al., 2019; Fishman et al., 2023; Yamada et al., 2023; Le et
al., 2023).

Figure 1:  Illustrative Example of Composing Diffusion Energy Potentials. 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.

A classical approach towards motion planning is potential based motion
planning (Khatib, 1986; Koren et al., 1991; Ratliff et al., 2009, 2018; Xie et
al., 2020), where both obstacles and goals define energy potentials through
which trajectories are optimized to reach. An advantage of potential based
motion planning is that different constraints to motion planning can be
converted into equivalent energy potentials and directly combined to optimize
for motion plans. However, potential based methods rely on gradient
optimization with respect to local geometry, resulting in the long-standing
local minima issues (LaValle, 2006). In addition, such motion planning
techniques typically require implicit obstacle representations, which is hard
to obtain in real-world settings.

We present a potential based motion planning approach leveraging diffusion
models (Sohl-Dickstein et al., 2015; Ho et al., 2020) where diffusion models
are used to parameterize and learn potential landscapes across configuration
space trajectories between start and goal states. These potential functions
can be directly inferred from raw perceptual inputs, removing the need for
implicit object representations. Furthermore, the annealed optimization
procedure across a sequence of potential energy landscapes in diffusion models
(Du et al., 2023) can help avoid local minima in optimization. This
optimization procedure is further stochastic, enabling the planner to generate
a multitude of motion plans with diverse morphology for a specific problem,
offering various motion plan candidates for selection in test time. Finally,
as our potential is defined globally over an environment, it is guided by both
local and global environment geometry, and thus our method provides more
accurate plans that require significantly less collision checking, compared
with problem-independent sampling-based planners.

One major hurdle of learning-based motion planners is their generalizability
to environments with unseen, more complex constraints. For example, a learned
model trained on sparse obstacles will perform poorly in scenarios with
cluttered obstacles, as such a setting is out of distribution. Our approach
addresses this through compositionality – our learned potentials can be
additively composed together to jointly solve motion planning problems with
larger sets of constraints than seen at training time. As illustrated in
Figure 1, combining two potentials from different diffusion models enables us
to optimize for trajectories that satisfy both constraints, one to avoid
obstacles in a cross, and a second to avoid obstacles in a square. Such
flexibility to ad-hoc composition of constraints is especially useful in
robotics where agents will often experience new sets of motion constraints in
its environment over the course of execution.

Overall, in this paper, our contributions are three-fold. (1) We present an
approach to learned potential based motion planning using diffusion models.
(2) We illustrate the effectiveness of our approach, outperforming existing
classical and learned motion planning algorithms in accuracy and collision
checks. (3) We illustrate the compositionality of motion planner, enabling it
to generalize to multiple sets of motion constraints as well as an increased
number of objects.

##  2 Related Work

Motion Planning. Classic sampling-based motion planners (Kavraki et al., 1996;
Kuffner & LaValle, 2000; Elbanhawi & Simic, 2014; Gammell et al., 2014; Janson
et al., 2015; Choudhury et al., 2016; Strub & Gammell, 2020) have gained wide
adoption due to their efficiency and generalizability. However, problem-
independent nature of these methods can result in inefficiency particularly
when planning for similar problems repetitively. Reactive methods, such as
potential-based approaches (Khatib, 1986; Ratliff et al., 2018; Xie et al.,
2020), velocity obstacles (Fiorini & Shiller, 1998; Van den Berg et al.,
2008), and safety barrier certificates (Wang et al., 2017) can provide fast
updates and have the guarantee for obstacle avoidance. However, their
performance is typically constrained by local minima or numerical instability
(LaValle, 2006), and they usually need to construct obstacle representations
in the robot configuration space, which is hard to obtain from raw perception.
To address these issues, recent works have proposed many deep-learning based
motion planners (Ichter & Pavone, 2019; Bency et al., 2019; Fishman et al.,
2023). Other works combine neural network with sampling-based methods (Qureshi
et al., 2019; Johnson et al., 2021; Yu & Gao, 2021; Lawson & Qureshi, 2022),
or combine trajectory level generative models with specifically designed cost
functions (Saha et al., 2023; Carvalho et al., 2023) that requires privileged
information. These methods can increase planning speed, expand the planning
horizon, or reduce the access queries to the environment by leveraging learned
knowledge. However, the performance of learned motion-planning approaches on
out-of-distribution environments often sharply degenerates. In addition, many
existing methods are only constrained to simple 2D environments (Yonetani et
al., 2021; Chaplot et al., 2021; Toma et al., 2021; Chang et al., 2023;
Carvalho et al., 2022). In contrast, we present a learning-based potential
motion planning approach, requiring no ground truth knowledge of the optimized
cost objective, which we illustrate can effectively generalize to new
environments through composability of potentials.

Diffusion Models for Robotics. Many recent works have explored the application
of diffusion model for robotics (Janner et al., 2022; Chen et al., 2022;
Kapelyukh et al., 2023; Ha et al., 2023). Current research spans a variety of
robotics problems, including action sequence generation (Liang et al., 2023;
Fang et al., 2023; Li et al., 2023), policy (Wang et al., 2023; Kang et al.,
2023), grasping (Urain et al., 2023; Huang et al., 2023), and visuomotor
planning or control (Dalal et al., 2023; Yang et al., 2023a; Chi et al.,
2023), with recent work also exploring their application in solving
manipulation constraints (Yang et al., 2023b). In contrast to these works, we
focus on how diffusion models can be used to explicitly parameterize and learn
potentials in potential-based motion planning. We illustrate the efficacy of
such an approach across motion-planning settings and its ability to generalize
to new environments through composition with other learned potentials.

Figure 2:  Trajectory Denoising Process. The trajectory is randomly
initialized from Gaussian in timestep $S=100$. Noise is iteratively removed
via the gradient of the energy function as given in equation 5. A feasible
trajectory can be obtained at timestep $S=0$.

##  3 Method

We first introduce potential based motion planning in Section 3.1. We then
discuss how potential based motion planning can be implemented with diffusion
models in Section 3.2. We further discuss how such an approach enables us to
combine multiple different potentials together in Section 3.3. Following this,
we discuss how we can refine motion plans generated by diffusion models in
cases of collision in Section 3.4. Finally, we show the probabilistic
completeness of the proposed method in Section 3.5.

###  3.1 Potential Based Motion Planning

Given a specified start state $q_{\text{st}}$ and end state $q_{\text{e}}$ in
a configuration space $\mathbb{R}^{n}$, motion planning is formulated as
finding a collision-free trajectory $q_{1:N}$ which starts from
$q_{\text{st}}$ and ends at $q_{\text{e}}$. To solve for such a collision-free
trajectory $q_{1:N}$ in potential based motion planning (Khatib, 1986; Koren
et al., 1991), a potential function $U(q):\mathbb{R}^{n}\rightarrow\mathbb{R}$
on the configuration space composed of

$U(q)=U_{\text{att}}(q)+U_{\text{repel}}(q)$ (1)

is defined, where $U(q)$ assigns low potential value to the goal state
$q_{\text{e}}$ and high potential to all states which are in collision. In
Equation 1, $U_{\text{att}}(q)$ represents a attraction potential that has low
values at the the end state $q_{\text{end}}$ and high values away from it, and
$U_{\text{repel}}(q)$ represents a repulsion potential that has high values
near obstacles and low values away them. The functional form of the potential
function $U(q)$ provides an easy approach to integrate additional obstacles in
motion planning by adding a new potential $U_{\text{new}}(q)$ representing
obstacles to the existing potential in Equation 1.

To obtain a motion plan from a potential function $U(q)$, a collision-free
trajectory $q_{1:N}$ from $q_{\text{st}}$ to $q_{\text{e}}$ is obtained by
iteratively following the scaled gradient of the potential function

$q_{t}=q_{t-1}-\gamma\nabla_{q}U(q)$ (2)

with a successful motion plan constructed when the optimization procedure
reaches the minimum of the potential function $U(q)$. A major limitation of
above approach in Equation 2 is local minima – if the optimization procedure
falls in such a minima, the motion plan will no longer successfully construct
paths from $q_{\text{st}}$ to $q_{\text{e}}$ (Yun & Tan, 1997; Teli & Wani,
2021; LaValle, 2006).

###  3.2 Potential Based Diffusion Motion Planning

We next discuss how to learn potentials for potential motion planning that
enables us to effectively optimize samples. Given a motion plan $q_{1:T}$ from
start state $q_{\text{st}}$ to end state $q_{\text{e}}$ and a configuration
space characterization $C$ (i.e. the set of obstacles in the environment), we
propose to learn a trajectory-level potential function $U_{\theta}$ so that

$q_{1:T}^{*}=\operatorname*{arg\,min}_{q_{1:T}}U_{\theta}(q_{1:T},q_{\text{st}},q_{\text{e}},C)$
(3)

where $q_{1:T}^{*}$ is a successful motion plan from $q_{\text{st}}$ to
$q_{\text{e}}$. To learn the potential function in Equation 3, we propose to
learn an EBM (LeCun et al., 2006; Du & Mordatch, 2019) 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 Equation 3.

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 denoising diffusion training objective (Sohl-Dickstein et al., 2015; Ho
et al., 2020). In this objective, we explicitly train the energy landscape so
gradient with respect to the energy function can denoise and recover a motion
plan $q_{1:T}$ across many differing levels of noise corruption
$\\{1,\ldots,S\\}$ ranging from mostly correct motion paths to fully corrupted
Gaussian noise trajectories. By shaping the gradient of the energy function to
generate motion plans $q_{1:T}$ from arbitrary initialized trajectories, our
learned energy landscape is able to effectively optimize for motion paths.

Formally, to train our potential, we use the energy based diffusion training
objective in (Du et al., 2023), where the gradient of energy function is
trained to denoise noise-corrupted motion plans $q_{1:T}^{i}$ from $D$,

$\mathcal{L}_{\text{MSE}}=\|\mathbf{\epsilon}-\nabla_{q_{1:T}}E_{\theta}(\sqrt{1-\beta_{s}}q_{1:T}^{i}+\sqrt{\beta_{s}}\mathbf{\epsilon},s,q_{\text{st}}^{i},q_{\text{e}}^{i},C^{i})\|^{2}$

(4)

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 diffusion
potential function.

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

$\displaystyle
q_{1:T}^{s-1}=q_{1:T}^{s}-\gamma\epsilon+\xi,\quad\xi\sim\mathcal{N}\bigl{(}0,\sigma^{2}_{s}I\bigl{)},$
$\displaystyle\epsilon=\nabla_{q_{1:T}}E_{\theta}(q_{1:T},s,q_{\text{st}},q_{\text{e}},C)$
(5)

To parameterize the energy function
$E_{\theta}(q_{1:T},s,q_{\text{st}},q_{\text{e}},C)$, we use classifier-free
guidance scale of 2.0 (Ho & Salimans, 2022) to form a peaker composite energy
function conditioned on $C$. $\gamma$ and $\sigma^{2}_{s}$ are diffusion
specific scaling constants111A 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.

Algorithm 1 Compositional Potential Based Planning

1: Models: compositional set of $N$ diffusion potential functions
$E^{i}_{\theta}(q_{1:T},t,q_{\text{st}},q_{\text{e}},C_{i})$

2: Hyperparameters: trajectory horizon $T$, number of denoising diffusion
steps $S$

3: Input: start position $q_{\text{st}}$, end position $q_{\text{e}}$, $N$
motion planning constraints $C_{1:N}$

4: Initialize $q_{1:T}^{S}\sim\mathcal{N}(0,I)$

5: for $s$ = $S\dots 1$ do

6: # Combining Different Energy Potentials Together

7:
$\epsilon_{\text{comb}}=\sum_{i=1}^{N}\nabla_{q_{1:T}}E_{\theta}^{i}(q_{1:T}^{s},s,q_{\text{st}},q_{\text{e}},C_{i})$

8: # Transit to Next Diffusion Time Step

9:
$q_{1:T}^{s-1}=q_{1:T}^{s}-\gamma\epsilon_{\text{comb}}+\xi,\quad\xi\sim\mathcal{N}\bigl{(}0,\sigma^{2}_{s}I\bigl{)}$

10: end for

11: return $q_{1:T}^{0}$

###  3.3 Composing Diffusion Potential Functions

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. 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. To sample from the new diffusion potential function
$E^{\text{comb}}$, we can follow

$\displaystyle
q_{1:T}^{s-1}=q_{1:T}^{s}-\gamma\epsilon^{\text{comb}}+\xi,\quad\xi\sim\mathcal{N}\bigl{(}0,\sigma^{2}_{s}I\bigl{)},$
(6)
$\displaystyle\epsilon^{\text{comb}}=\nabla_{q_{1:T}}(E_{\theta}^{1}(q_{1:T},s,q_{\text{st}},q_{\text{e}},C_{1})+E_{\theta}^{2}(q_{1:T},s,q_{\text{st}},q_{\text{e}},C_{2}))$

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 (see Appendix C), and
otherwise serves as approximate proxy to optimize multiple constraints.

Applications of Composing Potential Functions. The ability to combine multiple
separate potential functions for motion planning offers a variety of different
ways to generalize and extend existing motion planning systems. First, in many
motion planning problems, there is often a heterogenous set of different types
of constraints or collisions that limit possible configuration space paths.
For instance, in autonomous driving, constraints that can arise may include
moving pedestrians, traffic lanes, road work, or incoming cars. Oftentimes, we
cannot enumerate all potential combinations, but we wish motion planning
systems to be able to handle all possible combinations of constraints. Jointly
learning a single motion planning model for all constraints may be difficult,
as at test time, we may see novel combinations that we do not have training
data for. By learning separate diffusion potential fields for each constraint,
we can combine them in an ad-hoc manner at test time to deal with arbitrary
sets of constraints. We provide two concrete implementations of composing
potentials together below and a detailed procedural in Algorithm 1.

##### Generalization over More Obstacles

Suppose that the potential function $E_{\theta}$ is trained on environments
with 4 obstacles $|C|=4$. However, in test time, we want to generalize to a
more complex environment that has 6 obstacles
$C^{\prime}=\\{o_{1},o_{2},o_{3},o_{4},o_{5},o_{6}\\}$. This can be achieved
by adding the potentials evaluated on two sets of obstacles, where
$C_{1}=\\{o_{1},o_{2},o_{3},o_{4}\\}$ and
$C_{2}=\\{o_{3},o_{4},o_{5},o_{6}\\}$. This formulation can be further
extended to $N$ sets of obstacles $C_{1:N}$ and the composite diffusion
potential function is given by:

$E_{\theta}^{\text{comb}}(q_{1:T},s,q_{\text{st}},q_{\text{e}},C_{1:N})=\sum\limits^{N}_{i=1}E_{\theta}(q_{1:T},s,q_{\text{st}},q_{\text{e}},C_{i})$

(7)

Figure 3:  Visualization of the Motion Refining Scheme. A proposal plan is
first generated by denoising an initial Gaussian noise. If collision is
detected, a small noise is added to the proposal and the new plan is generated
by denoising the partially noisy trajectory.

Generalization over Static and Dynamic Obstacles. Many real-life scenarios
involve dynamic real-time interaction. For instance, to construct motion plans
for an autonomous vehicle, we must both avoid static lane obstacles as well as
moving cars. While static obstacles are often known a priori, the motion
patterns of dynamics obstacles often change with time, making it advantageous
to be able to combine different dynamic constraints with static ones. We can
directly implement this by adding diffusion potential function
$E_{\theta_{s}}^{i}$ that only trained on static obstacles $C^{s}_{i}$ and
diffusion potential function $E_{\theta_{d}}^{j}$ that only trained on dynamic
obstacles $C^{d}_{j}$. In a more general form, to condition on a set of
$N_{1}$ static obstacles $C^{s}_{1:N_{1}}$ with their diffusion potential
functions $E_{\theta_{s}}^{1:N_{1}}$ and a set of $N_{2}$ dynamic
$C^{d}_{1:N_{2}}$ obstacles with their diffusion potential functions
$E_{\theta_{d}}^{1:N_{2}}$, the composite diffusion potential function can be
written as:

$\begin{aligned}
&E_{\theta}^{\text{comb}}(q_{1:T},s,q_{\text{st}},q_{\text{e}},[C_{1:N_{1}}^{s},C_{1:N_{2}}^{d}])=\\\
&\sum^{N_{1}}_{i=1}E_{\theta_{s}}^{i}(q_{1:T},s,q_{\text{st}},q_{\text{e}},C_{i}^{s})+\sum^{N_{2}}_{j=1}E_{\theta_{d}}^{j}(q_{1:T},t,q_{\text{st}},q_{\text{e}},C_{j}^{d})\end{aligned}$

(8)

Algorithm 2 Refining Motion Plans

1: Model: compositional potential denoiser
$f_{\theta}(q_{1:T},s,q_{\text{st}},q_{\text{e}},C_{1:N})$

2: Hyperparameters: number of refine attempts $R$, noise scale $k$

3: Input: trajectory $q_{1:T}$, start position $q_{\text{st}}$, end position
$q_{\text{e}}$, $N$ constraints $C_{1:N}$

4: $Z$ = $\mathtt{Get\\_Collision\\_Sections(}$$q$) A Set of Indices of
Collision Sections in $q_{1:T}$

5: for $r$ = $1\dots R$ do

6:
$q_{1:T}^{k}=\sqrt{\bar{\alpha}_{k}}q_{1:T}+(1-\bar{\alpha}_{k})\xi,\quad\xi\sim\mathcal{N}\bigl{(}0,\sigma^{2}_{t}I\bigl{)}$
Add Noise to $q_{1:T}$

7: $q^{\prime}=f_{\theta}(q_{1:T}^{k},k,q_{\text{st}},q_{\text{e}},C_{1:N})$,
Get new Denoised Trajectory

8: for all $z_{i}\in Z$ do

9: if $\mathtt{is\\_section\\_good}$($q^{\prime}[z_{i}]$) then

10: $q[z_{i}]=q^{\prime}[z_{i}]$; $Z=Z\setminus z_{i}$ Refine $q_{1:T}$ and
Remove $s_{i}$ from set $S$

11: end if

12: end for

13: end for

14: return $q_{1:T}$

###  3.4 Refining Motion Plans

In practice, the predicted motion plan $q_{1:T}$ might occasionally contains
sections that violate the constraints of the environment (i.e., collide with
obstacles). To tackle this issue, both classical and learned motion planners
(Kuffner & LaValle, 2000; Qureshi et al., 2019) provide mechanisms to refine
trajectories subject to collisions in configuration space. With diffusion
potential fields, we can likewise refine a trajectory, $q_{1:T}$ with
collision, by locally perturbing it into a noisy trajectory $q_{1:T}^{k}$
defined by the $k$-th step of the diffusion forward process:

$q_{1:T}^{k}=\sqrt{\bar{\alpha}_{k}}q_{1:T}+(1-\bar{\alpha}_{k})\xi,\quad\xi\sim\mathcal{N}\bigl{(}0,\sigma^{2}_{t}I\bigl{)}\vspace{0pt}$
(9)

as in (Ho et al., 2020). A new motion plan $q_{1:T}^{\prime}$ can be obtained
by denoising the noisy trajectory following Equation 5, where
$q_{1:T}^{\prime}=f_{\theta}(q_{1:T}^{k},k,q_{\text{st}},q_{\text{e}},C_{1:N})$
and $f_{\theta}(\cdot)$ is an iterative diffusion potential denoiser that
outputs a clean trajectory. To fix the problematic trajectory $q_{1:T}$, the
collision sections in $q_{1:T}$ will be replaced by the corresponding sections
in $q_{1:T}^{\prime}$ if these new sections are coherent and collision-free.
This refining procedural can be repeated until a desired trajectory is found.
The warm-start denoising scheme can enable faster re-planning and maintain the
morphology of the original plan, supporting planning on energy-critical mobile
devices or when the plan has been executed. Algorithm 2 displays the complete
refining pipeline and Figure 3 provides a corresponding visualization.

###  3.5 Probabilistic Completeness

In this section, we will show the probabilistic completeness of our method.
Let $f_{\theta}(q_{1:T})$ denote the probability density function of the
output distribution $\mathcal{D}_{o}$ of our diffusion potential model. In
such learned neural distribution, all data points $q_{1:T}$ are assigned
positive density, that is,

$\forall q_{1:T},\ f_{\theta}(q_{1:T})>0$ (10)

Define $\mathcal{J}_{c}$ as the a set of all valid trajectories subject to
constraint $C$. There always exists a small interval in the vicinity of a
random trajectory $q_{1:T}^{c}\in\mathcal{J}_{c}$, such that

$[q_{1:T}^{c}-\tau,q_{1:T}^{c}+\tau]\subseteq\mathcal{J}_{c},\quad\tau>0$ (11)

i.e., all trajectories in the interval satisfy the given constraint $C$. Let
$\mathbb{P}_{\tau}$ denote the probability for our model to sample a
trajectory from the interval, and according to equation 10 we have,

$\mathbb{P}_{\tau}=\int_{q_{1:T}^{c}-\tau}^{q_{1:T}^{c}+\tau}f(x)dx>0$ (12)

Let $A_{n}$ denote the event that there is at least one trajectory
$q_{1:T}\in\mathcal{J}_{c}$ among $n$ sampled trajectories. Clearly, as the
number of samples approaches infinity, event $A$ will happen almost surely,
i.e.,

$\lim_{n\to\infty}\mathbb{P}\ \big{(}A_{n}\big{)}=1\vspace{-2pt}$ (13)

Hence, our method is probabilistically complete.

Figure 4:  Environment Demonstration. Maze2D: a point robot moving in 2D
workspace with the highlighted blocks as obstacles. KUKA: robotic arm with 7
DoF operating on a tabletop. The grey cuboids are obstacles. Dual KUKA 14D:
Two side by side KUKA arms operate simultaneously. Figure 5:  Quantitative
Comparisons in Motion Planning Environments. Three metrics of three
environments from 2D to 14D are reported. From left to right: a) number of
collision checks, b) success rate, c) planning time.

##  4 Experiments

In this section, we first describe our environments and baselines in Section
4.1. Next, in Section 4.2, we discuss our experiments on the base environments
and the motion refining algorithm. Following, in Section 4.3, we present the
compositionality results by evaluating our motion planner on composite
environments. Then, we describe the real world motion planning performance in
Section 4.4.

###  4.1 Environments and Baselines

We first classify the environments that we evaluated on to 3 categories by the
level of generalization capability:

  * •

Base Environment: same number of constraints as in training phase, constraints
sampled from the same distribution as in training.

  * •

Composite Same Environment: more constraints than training phase, constraints
sampled from the same distribution as in training.

  * •

Composite Different Environment: more constraints than training phase,
constraints sampled from different distributions.

Our simulated motion planning environments are listed below and shown in
Figure 4. See Appendix A.1 for more details. In all environments, the motion
planning task is to generate a feasible trajectory from the start state to the
goal state. The agent trajectory is represented in C-space, while obstacles
are represented in workspace.

  * •

Maze2D. A point-robot moving in a 2D workspace. The configuration space is the
x-y coordinate of the agent. We offer two variants: Static Maze2D where
obstacles stay in the same locations and Dynamic Maze2D where obstacles are
moving in randomly generated linear trajectories.

  * •

KUKA. A KUKA arm of 7 DoF operating on a tabletop in a 3D workspace. The
start/goal is given as the 7D joint state of the KUKA arm.

  * •

Dual KUKA. Two KUKA arms are placed side by side on a tabletop and operating
simultaneously. The start/goal is given as the joint states of two KUKA arms
(14 DoF).

Figure 6: Qualitative Motion Plans in KUKA Environment. Results of two motion
planning problem are shown in two rows. The large green/pink ball indicates
the start/goal state. Our method in the left column generates smooth and near-
optimal trajectories, while in row (a), the trajectory of M$\pi$Net chooses a
longer route from behind and gets stuck in some local regions, and in row (b),
M$\pi$Net cannot pass through the narrow passage and keeps hovering near the
start state.

##### Baselines

For non-learning motion planning methods, our baselines include classic
sampling-based planning baseline RRT* (Karaman & Frazzoli, 2011), sampling-
based method with potential functions based heuristic P-RRT* (Qureshi & Ayaz,
2016), advanced sampling-based method BIT* (Gammell et al., 2015), sampling-
based method for dynamic environments SIPP (Phillips & Likhachev, 2011), and
traditional potential-based method RMP (Ratliff et al., 2018). For learned
motion planners, we compare our method with: MPNet (Qureshi et al., 2019),
M$\pi$Net (Fishman et al., 2023), and AMP-LS (Yamada et al., 2023). MPNet is
trained on trajectories with sparse waypoints and use MLPs to encode
environment configuration and predict the next robot state. M$\pi$Net is
trained on dense trajectory waypoints and predicts the movement vector instead
of the next robot state. AMP-LS encodes the robot state into a latent feature
and approaches the goal state by iteratively using the gradient of several
planning losses to update the latent. A sequence of latents are then decoded
and form a trajectory. In evaluation, all the start/goal states and
environment configurations are unseen to the models. For each experiment, we
evaluate on 100 different environments with 20 randomly sampled starts and
goals in each environment. No re-training is performed in test time.

###  4.2 Performance on Base Environments

We first evaluate motion planning performance in each base environment:
randomly generated environments that follow the same procedural generation
pipeline as the training environments. Quantitative results are shown in
Figure 5 and Appendix B.1. We include the full details of evaluation setup in
Appendix A.2.3.

##### Comparison to Sampling-based Planners

We set the planning time limit to 5 seconds for all sampling-based methods.
Both RRT*, P-RRT*, and BIT* can succeed in the base Maze2D environment.
However, their success rates suffer from a significant degradation when the
dimension of the configuration space increases. The planning time of the
sampling-based planners also rises dramatically as the dimension of the space
increases. By contrast, the success rates of our method surpass all baselines
and are consistent across different environments, achieving this in less than
11% of BIT*’s planning time and requiring up to two orders of magnitude fewer
collision checks.

Figure 7:  Qualitative Performance on Environments with Concave Obstacles. Potential-based planning method RMP is prone to local minima. BIT* is able to find a feasible solution, but the trajectories are rough and require much longer planning time, especially for the harder problem in the second row. Trajectories generated by our planner are both smooth and short in length.  | $R=3$ | $R=5$ | $R=10$
---|---|---|---
Env | Before | After | Before | After | Before | After
Maze2D | 96.3 | 99.8 | 95.3 | 99.0 | 95.8 | 100.0
KUKA | 71.3 | 90.0 | 69.5 | 94.3 | 69.8 | 94.8
Dual KUKA | 45.5 | 69.8 | 47.3 | 77.3 | 47.0 | 80.8
Table 1: Quantitative Results of Motion Plan Refining. Success rates before
and after motion refining are shown. $R$ denotes the number of refining
attempts. The proposed refining method can consistently boost success rates on
three base environments.

##### Comparison to Learning-based Planners

We also compare to three learning-based motion planning baselines: MPNet,
M$\pi$Net, and AMP-LS. Our method outperforms all baselines in both success
rate and number of collision check. In Dual KUKA, our method leads the state-
of-the-art learning-based planner M$\pi$Net by nearly 35% in success rate
while with less than 40% of its planning time and 7 times less of its
collision checks. We also observe that the performance of other planners drop
quickly as the difficulty increases, while our planner performs steadily
across all environments. Though the planning time of M$\pi$Net in Maze2D is
slightly shorter than ours, the gap is closing as the dimension of the
configuration space increases, and in Dual KUKA, our planner requires less
planning time than all baselines, probably due to the expensive cost of
collision checks and higher task complexity. Qualitative comparisons are shown
in Figure 6 and 18.

Figure 8:  Qualitative Compositional Generalization over More Obstacles. Two models that trained on only 6 obstacles are composed and tested on out-of-distribution environments with 7, 8, 9, 10, 11 obstacles, respectively. | Maze2D – Convex | Maze2D – Concave
---|---|---
Method | Success | Time | Success | Time
RRT* | 98.8 | 0.95 | 92.1 | 2.14
P-RRT* | 98.5 | 1.17 | 85.9 | 2.69
BIT* | 100.0 | 0.21 | 100.0 | 0.45
MPNet | 88.4 | 0.21 | 84.3 | 0.38
M$\pi$Net | 97.9 | 0.07 | 96.9 | 0.10
MPD | 77.9 | 2.99 | 44.4 | 3.93
RMP | 64.9 | 0.13 | 28.0 | 0.34
Ours | 100.0 | 0.12 | 100.0 | 0.15
Table 2:  Quantitative Performance on Convex and Concave Obstacles. Motion
planning performance on Maze2D environments with 6 convex obstacles (left) and
7 concave obstacles (right). The proposed diffusion potential planner
outperforms the traditional potential-based method RMP and diffusion motion
planning counterpart MPD by a margin. While both our method and the advanced
sampling-based method BIT* can solve all the problems, ours requires
significantly less planning time than BIT*.  Figure 9:  Compositional
Generalization. Quantitative comparisons of different planner on composite
environments. The shaded areas represent standard errors. Each graph’s
leftmost column displays results for environments that contain the same number
of obstacles as encountered during training. By composing potentials at test
time, our method (red line) can generalize to environments with much more
obstacles than training time.

##### Environments with Concave Obstacles

We construct additional Maze2D environments where obstacles are concave to
demonstrate the capability of our learned potential motion planner to avoid
local minima. As shown in Figure 7, potential-based method RMP tends to get
trapped in local minima in environments filled with concave obstacles. In
addition, we observe that BIT* is able to find a feasible motion plan, but the
morphology of the proposed trajectories are sharp and irregular. We further
present the quantitative results on base Maze2D environments in Table 2, where
we include an additional diffusion-based motion planning approach (Carvalho et
al., 2023). Notably, our diffusion potential planner is the only learning-
based approach that successfully solves every motion planning problems in our
evaluation set. Although the advanced sampling-based method BIT* can also find
all the solutions, its average planning time is significantly longer, taking
three times longer than our method in Maze2D – Concave. More results are
provided in Appendix B.4.

##### Motion Refining

We present quantitative and qualitative results of refining motion plans in
Table 1 and Figure 3. The gain of refining motion plans increases as the
dimension of the environment increases. As shown in Table 1, the success rate
generally increases as we increase the number of refining attempts $R$, but
the gain gradually saturates in 10 attempts. In this case, the proposed
trajectory probably suffers from a catastrophic collision, and to obtain a
successful motion plan, the diffusion potential model might need to resample a
trajectory from pure noise.

###  4.3 Performance on Composite Environments

##### Composing Obstacles

We first evaluate the compositionality by adding obstacles to the
environments. Qualitative results of composite Maze2D environments are given
in Figure 8, where we train our model on 6 obstacles and evaluate on
environments with up to 11 obstacles. Each orange block represents an extra
obstacle added to the test time environments. As we can see, the composed
model effectively proposes different trajectories according to the presented
obstacles by sampling from the composite potential. We report the quantitative
performance on three environments with different configuration space
dimensions in Figure 9. More results are shown in Appendix B.2. In addition,
we further include a baseline which directly learns demonstrations of
different numbers and types of obstacles in a single diffusion potential
function in Appendix B.6.

Figure 10:  Qualitative Real World Motion Plans, Hotel Scene.  The composed
model provides long-horizon motion plan that avoid 10 pedestrians, while only
trained on 5 pedestrians. In column (a) and (b), the composed plan is aware of
P1 (cyan) and P6 (pink) and overtakes them from above, while the baseline
model runs into them. In column (c), the composed motion plan chooses to move
faster so as to pass through the intersection with P7 (brown) before P7
arrives, but the baseline motion plan results in a collision due to its slower
speed. In column (d), the composed plan choose to go upward to avoid the
oncoming P8 (black).

| Base Dynamic | Static 1 + Dynamic | Static 2 + Dynamic
---|---|---|---
M | Suc | Time | Col | Suc | Time | Col | Suc | Time | Col
SIPP | 69.9 | 32.2 | 1M+ | 70.4 | 185.5 | 1.7M+ | 74.0 | 98.7 | 1.3M+
Ours | 99.5 | 0.13 | 168.8 | 96.6 | 4.31 | 828.6 | 97.5 | 4.23 | 646.4

Table 3: Quantitative Results on Base Dynamic and Static + Dynamic on Maze2D.
Static 1 and Static 2 refer to two different static Maze2D environments. Our
planner can generalize to environments with both static and dynamic obstacles
while only separately trained on static environments or dynamic environments.

##### Composing Multiple Constraints

We then investigate the compositionality to combine two different diffusion
potential functions together, i.e., models trained on completely different
environments. Specifically, we separately train a model on 6 small obstacles
(Static 1) and a model on 3 large obstacles (Static 2) and evaluate the
composed model on environments where both small and large obstacles are
presented. Quantitative and qualitative results are shown in Table 13 and
Figure 12. Moreover, we compose the models trained on static environments with
another model trained on dynamic environments, by which the composed model can
generalize to environments where both static and dynamic obstacles are
presented, namely Static 1 + Dynamic and Static 2 + Dynamic. The corresponding
quantitative results are shown in Table 3. The planning time limit for SIPP is
set to 60s/300s for base/composite dynamic environments. Please refer to
Figure 14 and Figure 15 in Appendix B.3 for more qualitative results.

###  4.4 Performance on Real World Datasets

Finally, we evaluate the effectiveness of our method on the real world ETH/UCY
(Pellegrini et al., 2010; Lerner et al., 2007) dataset. The dataset we used
consists of 6 scenes (ETH, Hotel, Zara01, Zara02, Students01, Students03),
where each scene contains human trajectories in world-coordinates collected by
manual annotation from a bird-eye-view camera. Our focus is to investigate if
our model can propose successful trajectories given the start and goal
locations of an agent in a random, cluttered street-level real-world
interaction. Specifically, the planner is trained to predict the trajectory of
the agent (highlighted in red), conditioned on the trajectories of 5 other
pedestrians. The training data contains 5 scenes and the held-out scene is
used for evaluation. In Figure 17, we present the qualitative results where 5
other pedestrians are presented. We also evaluate on 10 presented pedestrians
by composing two potential functions each constrained by 5 pedestrians, as
illustrated in Figure 10. Details settings and qualitative results are
presented in Appendix B.5.

##  5 Discussion

Limitations. Our existing formulation of potential based diffusion motion
planner has several limitations. First, although our motion trajectory is
accurate, it is often suboptimal, e.g., there exists a shorter path from start
to goal. This may be addressed by adding an additional potential to reach the
goal as soon as possible. Second, our approach to composing potentials scales
linearly with the number of composed models, requiring significantly more
computation power with additional models. This can be remedied by having
different potential operate on shared features in a network.

Conclusion. In this work, we have introduced a potential based diffusion
motion planner. We first formulate our diffusion potential motion planner and
describe its connections and advantages over traditional potential based
planner. We illustrate the motion planning performance of our approach in
terms of success rate, planning time, and the number of collision checks over
motion planning problems with dimension of 2D, 7D, 14D. We further illustrate
the compositionality of the approach, enabling generalization to both new
objects and new combinations of motion constraints. Finally, we illustrate the
potential of our work on real world scenes with multi-agent interaction.

## Acknowledgements

We acknowledge support from NSF grant 2214177; from AFOSR grant
FA9550-22-1-0249; from ONR MURI grant N00014-22-1-2740; from ARO grant
W911NF-23-1-0034; and from the Samsung Global Research Outreach Program. Yilun
Du is supported by a NSF Graduate Fellowship. We thank the anonymous reviewers
for their careful review. Our research was conducted using computational
resources at the Center for Computation and Visualization at Brown University.

## Impact Statement

This paper presents work whose goal is to advance the field of Machine
Learning. There are many potential societal consequences of our work, which we
believe should be similar to a generic machine learning paper. For example, if
the motion pattern in the dataset is biased, the motion plans generated by our
model might have similar biases as the training dataset. No other issues we
feel must be specifically highlighted here.

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##  Appendix A Appendix

In this appendix, we first present our dataset details in Section A.1. Next,
we provide implementation details in Section A.2, such as model architecture,
training and evaluation setups, hyperparameters. Further, We show additional
quantitative and qualitative results in Section B, including base
environments, composite environments, and real-world datasets. Lastly, in
Section C, we give a proof on the optimality of composing potentials of sets
of constraints.

###  A.1 Dataset Details

In this section, we present details of the three base environments. Each
dataset consists of feasible trajectories of randomly sampled start and goal
states. All datasets are collected by BIT*(Gammell et al., 2015). All
environments except Maze2D are simulated via PyBullet (Coumans & Bai,
2016–2021).

##### Maze2D

The workspace is a 5 $\times$ 5 square and the agent is a point-robot whose
state is its x-y coordinate in the workspace. In the base Maze2D environment
(Static 1), the obstacles are 6 square blocks of size 1 $\times$ 1\. We
construct another Maze2D environment (Static 2) where obstacles are 3 square
blocks of size 1.4 $\times$ 1.4. In addition, we construct a Maze2D
environment with 7 concave obstacles. For simplicity, the volume of the robot
agent is ignored – collision happens when the location is inside the region of
an obstacle. The training data contains 3,000 different environment
configurations and 25,000 states for each environments (approximately 500
trajectories).

##### KUKA

One KUKA LBR iiwa robotic arm is placed at the center $(0,0,0)$ in world
coordinate and obstacles are given as cubic of length 0.40 meter and are
randomly placed in the surrounding of the robot. The training data contains
2,000 different environment configurations and 25,000 states for each
environments.

##### Dual KUKA

Two KUKA LBR iiwa robotic arms are placed side by side on a tabletop. One is
at world coordinate $(-0.5,0,0)$ and the other is at $(0.5,0,0)$. Obstacles
are given as cubic of length 0.44 meters and are randomly placed in the
surrounding of the robots. The training data contains 2,500 different
environment configurations and 25,000 states for each environment.

##### ETH/UCY

The dataset we used contains 6 scenes, where in each scene there are hundreds
of street-level pedestrians trajectories. In each scene, a video is recorded
using a fixed bird-eye-view camera facing the street, and the coordinates of
each pedestrian are manually annotated according to the recorded video. We use
real-world pedestrian trajectories of 5 scenes to train our model and evaluate
on the one held-out scene.

###  A.2 Implementation details

##### Software:

The computation platform is installed with Red Hat 7.9, Python 3.8, PyTorch
1.10.1, and Cuda 11.1

##### Hardware:

For each of our experiments, we used 1 RTX 3090 GPU.

####  A.2.1 Energy-based Diffusion Model

##### Model Architecture

The diffusion potential model $f_{\theta}$ consists of a CNN trajectory
denoiser based on U-Net similar to (Ajay et al., 2023) and a constraint (i.e.,
environment configuration) encoder. The U-Net contains repeated residual
blocks where each block consists of two temporal convolutions followed by
GroupNorm and SiLU nonlinearity (Hendrycks & Gimpel, 2016). The constraint
encoder uses the Transformer encoder structure (Vaswani et al., 2017), whose
input is a set of obstacle locations for static environments or obstacle
trajectories for dynamic environments. We remove the positional embedding,
since the obstacles information should be permutation invariance with each
other. We concatenate the learned class token from transformer with the time
embedding and feed the concatenated tensor to temporal convolution blocks in
U-Net for denoising. More details of the models are shown in Table A.2.1 and
Table A.2.1. Note that we do not further explore the selection of the concrete
model architecture, but we believe that some more advanced architectures could
further improve our performance.

Hyperparameters | Value
---|---
Base Feature Channels | 64
Feature Dimension Scale | (64, 256, 512)
Groups in GroupNorm | 8
Nonlinearity | SiLU

Table 4: Hyperparameter of U-Net.

Hyperparameters | Value
---|---
Base Embedding Channel | 64
Transformer Layers | 3
Attention Heads | 1
Nonlinearity | SiLU

Table 5: Hyperparameter of Constraint Encoder.

##### Energy Parameterization

To encode the energy $E(\cdot)$ of as in equation 4, we use L2 energy-
parameterization as given in equation 14. For more details on Energy-based
Diffusion Model, please refer to (Du et al., 2023).

$E_{\theta}^{L2}(x,t)=\frac{1}{2}||f_{\theta}(x,t)||^{2}$ (14)

####  A.2.2 Training Details

##### Training Pipeline

In training, the dataloader randomly samples trajectories of length equal to
the training horizon from the whole dataset. We provide detailed
hyperparameters for training our model in Table 6. We do not apply any
hyperparameter search nor learning rate scheduler. The training time of our
model is approximately two days, but we observe that the performance is close
to saturation within one day.

Hyperparameters | Value
---|---
Horizon | 48
Diffusion Time Step | 100
Probability of Condition Dropout | 0.2
Iterations | 2M
Batch Size | 512
Optimizer | Adam
Learning Rate | 2e-4
Table 6: Hyperparameters of Diffusion Potential Motion Planner (Training)

####  A.2.3 Evaluation Details

##### Our Evaluation Pipeline

The input of the planner is the start state, goal state, and the environment
constraints (e.g., obstacle locations represented in workspace), and the
output is the proposed trajectory. In test time, the number of denoising
timestep is set to 10 by using DDIM (Song et al., 2020). The intermediate
noise scale eta in DDIM is set to 0 for base environments and 1 for composite
environments. The evaluation pipeline of our model consists of three phases:
Propose Motion Plan Candidates, Candidate Selection, and Motion Refining. The
planner first generates multiple candidate trajectories using Algorithm 1. It
then accesses the environment and performs collision check to select a
successful trajectory from the candidates. Finally, if no desired candidate is
found, it will execute the motion refining as in Algorithm 2. Hyperparameters
used in evaluation is detailed in Table 8 and Table 8. We observe that our
planner can directly solve all the problems in Maze2D, hence we do not use
replanning during the evaluation of Maze2D environments.

Hyperparameters | Value
---|---
Horizon | 48
DDIM Time Step | 8
DDIM eta | 0.0
Guidance Scale | 2.0
# of Trajectory Candidate | 20
# of Refine Attempts $R$ | 0
Refine Noise Scale $k$ | –
Table 7:  Hyperparameters of Diffusion Potential Motion Planner (Evaluation on
Maze2D)

Hyperparameters | Value
---|---
Horizon | 52
DDIM Time Step | 10
DDIM eta | 0.0
Guidance Scale | 2.0
# of Trajectory Candidate | 20
# of Refine Attempts $R$ | 5
Refine Noise Scale $k$ | 3
Table 8:  Hyperparameters of Diffusion Potential Motion Planner (Evaluation on
KUKA and Dual KUKA)

##### Baselines Evaluation Pipeline

We try our best to re-implement every baseline and follow their original
settings. For MPNet (Qureshi et al., 2019), we follow their implementation and
use bi-directional path generation in test time. As for AMP-LS, we implement a
Variational Auto-Encoder (VAE) (Kingma & Welling, 2013) to encode the robot
pose state and leverage GECO loss (Rezende & Viola, 2018) in path
optimization. M$\pi$Net does not provide a replan scheme in their design. For
fair comparison, we boost M$\pi$Net with replan by backtracing to previous
timestep and adding a small random noise for restart when any collisions are
detected.

##  Appendix B Additional Results

In this section, we provide more quantitative and qualitative motion planning
results. In Section B.1, we present additional planning results on the base
environments. In Section B.2, we show planning performance on composite same
environments. In Section B.3, we provide planning performance on composite
different environments, including composing two different static models and
composing a static model and a dynamic model. In Section B.4, we show motion
planning performance on a more challenging Maze2D environment with concave
obstacles. In Section B.5, we present more quantitative and qualitative
evaluation results on the real-world dataset.

###  B.1 Performance on Base Environment

We provide detailed numerical motion planning results on three base
environments in Table 9. The corresponding bar plot visualization is given in
Figure 5. Besides, we provide an additional GNN-based motion planning baseline
(Yu & Gao, 2021) in the table below. We report the mean and standard error on
three main motion planning metrics: success rate, planning time and number of
collision checks. Our method consistently outperforms all other baselines on
success rate. Especially, in the hardest Dual KUKA environment, our planner
outperforms other learning-based baseline by 40-50% on success rate, and even
outperforms the advance sampling-based method BIT* by 15% while with one order
of magnitude less planning time and collision checks.

| Maze2D | KUKA | Dual KUKA
---|---|---|---
Method | Success | Time | Check | Success | Time | Check | Success | Time | Check
RRT* |  98.8$\pm\mathtt{0.3}$ |  0.95$\pm\mathtt{0.02}$ |  10453.4$\pm\mathtt{156.8}$ |  59.4$\pm\mathtt{1.9}$ |  3.15$\pm\mathtt{0.06}$ |  19508.6$\pm\mathtt{278.6}$ |  39.5$\pm\mathtt{1.6}$ |  4.16$\pm\mathtt{0.04}$ |  18560.5$\pm\mathtt{112.6}$
P-RRT* |  98.5$\pm\mathtt{0.3}$ |  1.17$\pm\mathtt{0.02}$ |  14158.9$\pm\mathtt{225.8}$ |  58.0$\pm\mathtt{1.9}$ |  3.44$\pm\mathtt{0.06}$ |  21471.4$\pm\mathtt{269.1}$ |  35.0$\pm\mathtt{1.6}$ |  4.54$\pm\mathtt{0.02}$ |  18117.7$\pm\mathtt{78.8}$
BIT* |  100.0$\pm\mathtt{0.0}$ |  0.21$\pm\mathtt{0.00}$ |  2067.4$\pm\mathtt{24.4}$ |  96.5$\pm\mathtt{0.7}$ |  0.86$\pm\mathtt{0.04}$ |  6047.6$\pm\mathtt{364.7}$ |  82.1$\pm\mathtt{1.6}$ |  2.37$\pm\mathtt{0.08}$ |  8925.4$\pm\mathtt{365.0}$
AMP-LS |  86.3$\pm\mathtt{0.9}$ |  1.41$\pm\mathtt{0.11}$ |  1025.4$\pm\mathtt{30.2}$ |  27.2$\pm\mathtt{4.3}$ |  3.89$\pm\mathtt{0.35}$ |  4176.4$\pm\mathtt{131.0}$ |  40.9$\pm\mathtt{3.2}$ |  7.38$\pm\mathtt{0.72}$ |  12091.3$\pm\mathtt{230.0}$
MPNet |  85.6$\pm\mathtt{0.9}$ |  0.41$\pm\mathtt{0.01}$ |  6315.7$\pm\mathtt{203.0}$ |  76.0$\pm\mathtt{3.9}$ |  0.10$\pm\mathtt{0.02}$ |  885.4$\pm\mathtt{135.5}$ |  42.0$\pm\mathtt{4.1}$ |  0.35$\pm\mathtt{0.04}$ |  1750.2$\pm\mathtt{195.6}$
M$\pi$Net |  97.9$\pm\mathtt{0.3}$ |  0.07$\pm\mathtt{0.00}$ |  178.7$\pm\mathtt{7.8}$ |  88.7$\pm\mathtt{0.8}$ |  0.22$\pm\mathtt{0.01}$ |  232.5$\pm\mathtt{11.6}$ |  63.9$\pm\mathtt{1.4}$ |  0.72$\pm\mathtt{0.02}$ |  875.2$\pm\mathtt{26.7}$
GNN |  100.0$\pm\mathtt{0.0}$ |  0.11$\pm\mathtt{0.00}$ |  1043.3$\pm\mathtt{5.5}$ |  98.9$\pm\mathtt{0.4}$ |  0.38$\pm\mathtt{0.01}$ |  2207.4$\pm\mathtt{153.6}$ |  94.6$\pm\mathtt{0.6}$ |  1.22$\pm\mathtt{0.03}$ |  6269.1$\pm\mathtt{344.7}$
Ours |  100.0$\pm\mathtt{0.0}$ |  0.12$\pm\mathtt{0.00}$ |  97.0$\pm\mathtt{0.3}$ |  99.7$\pm\mathtt{0.0}$ |  0.13$\pm\mathtt{0.00}$ |  75.5$\pm\mathtt{3.1}$ |  97.7$\pm\mathtt{0.5}$ |  0.27$\pm\mathtt{0.01}$ |  122.4$\pm\mathtt{6.1}$
Table 9: Quantitative Motion Planning Performance. Evaluated on 100 unseen
environments with 20 motion planning problems in each environment. We report
the mean over all environments and the standard error across different
environments. The table is the numerical results corresponding to the bar plot
visualization in Figure 5.

In addition, we present 4 qualitative comparisons with the state-of-the-art
learning-based motion planner M$\pi$Net on the KUKA environments in Figure 18
at the end of this section. In Figure 19, we demonstrate the versatile
morphology of our motion trajectories on the same motion planning problem.

###  B.2 Performance on Composite Same Environment

We construct composite same environments with more obstacles of the same type
than training phase environments. Since more obstacles are presented, these
evaluation environments are out of training distribution and hence more
challenging. The locations of each obstacle in the evaluation environments are
random and unseen to the model. We further provide a baseline Diffusion in
which we do not compose potentials. The quantitative results are shown in
Table 10, 11, 12, and qualitative results are shown in Figure 11. Our composed
model demonstrates superior performance over all benchmarks, which is
consistent with the expectation that composition enables more effective
generalization to more obstacles.

As presented in Section 3.3, when generalizing to more obstacles, we might
split the obstacles into several groups before composition. In our
experiments, the splitting of obstacles is random, and we do not observe
correlations between performance and the ways of splitting.

Figure 11:  Compositional Generalization over Increasing Obstacles. The green/pink star indicates the start/goal state. Our planner is trained with a dataset with only 6 obstacles and can generalize to scenarios with more obstacles by directly composing potentials in test time (without any re-training). The generated trajectories demonstrate various morphology and reach the goals with near optimal paths.  |  | 6 + 1 | 6 + 2 | 6 + 3
---|---|---|---|---
| Method | Success | Check | Success | Check | Success | Check
Maze2D | RRT* |  98.5$\pm\mathtt{0.3}$ |  11467.3$\pm\mathtt{176.8}$ |  97.7$\pm\mathtt{0.4}$ |  12675.6$\pm\mathtt{238.9}$ |  96.7$\pm\mathtt{0.5}$ |  14247.8$\pm\mathtt{302.1}$
P-RRT* |  97.5$\pm\mathtt{0.3}$ |  15308.9$\pm\mathtt{257.3}$ |  96.8$\pm\mathtt{0.5}$ |  17176.1$\pm\mathtt{324.3}$ |  92.8$\pm\mathtt{0.7}$ |  19280.9$\pm\mathtt{429.9}$
BIT* |  100.0$\pm\mathtt{0.0}$ |  2340.0$\pm\mathtt{35.4}$ |  100.0$\pm\mathtt{0.0}$ |  2635.7$\pm\mathtt{40.6}$ |  100.0$\pm\mathtt{0.0}$ |  2942.3$\pm\mathtt{53.7}$
AMP-LS |  74.1$\pm\mathtt{6.4}$ |  2845.2$\pm\mathtt{247.0}$ |  66.7$\pm\mathtt{6.2}$ |  3357.6$\pm\mathtt{298.0}$ |  56.5$\pm\mathtt{7.0}$ |  3973.5$\pm\mathtt{290.0}$
M$\pi$Net |  92.6$\pm\mathtt{0.8}$ |  349.7$\pm\mathtt{23.3}$ |  88.0$\pm\mathtt{1.1}$ |  481.2$\pm\mathtt{30.1}$ |  81.2$\pm\mathtt{1.5}$ |  685.3$\pm\mathtt{45.5}$
Diffusion |  99.9$\pm\mathtt{0.1}$ |  126.1$\pm\mathtt{5.6}$ |  99.2$\pm\mathtt{0.2}$ |  229.7$\pm\mathtt{11.8}$ |  95.4$\pm\mathtt{0.9}$ |  454.8$\pm\mathtt{34.2}$
Composed |  100.0$\pm\mathtt{0.0}$ |  102.8$\pm\mathtt{2.6}$ |  99.9$\pm\mathtt{0.1}$ |  147.9$\pm\mathtt{6.4}$ |  99.5$\pm\mathtt{0.3}$ |  218.8$\pm\mathtt{15.1}$
|  | 6 + 4 | 6 + 5 | 6 + 6
| Method | Success | Check | Success | Check | Success | Check
Maze2D | RRT* |  93.3$\pm\mathtt{0.6}$ |  15729.8$\pm\mathtt{307.4}$ |  88.7$\pm\mathtt{1.1}$ |  18508.4$\pm\mathtt{449.0}$ |  84.7$\pm\mathtt{1.3}$ |  18598.5$\pm\mathtt{416.9}$
P-RRT* |  89.0$\pm\mathtt{0.9}$ |  21280.1$\pm\mathtt{398.9}$ |  78.2$\pm\mathtt{1.7}$ |  24471.1$\pm\mathtt{526.0}$ |  76.7$\pm\mathtt{1.5}$ |  24491.8$\pm\mathtt{501.8}$
BIT* |  100.0$\pm\mathtt{0.0}$ |  3220.2$\pm\mathtt{59.7}$ |  100.0$\pm\mathtt{0.1}$ |  3563.4$\pm\mathtt{79.2}$ |  100.0$\pm\mathtt{0.1}$ |  3580.7$\pm\mathtt{61.1}$
AMP-LS |  45.1$\pm\mathtt{7.2}$ |  4560.4$\pm\mathtt{310.0}$ |  37.8$\pm\mathtt{7.7}$ |  4928.6$\pm\mathtt{361.0}$ |  33.5$\pm\mathtt{7.0}$ |  5107.2$\pm\mathtt{341.0}$
M$\pi$Net |  75.6$\pm\mathtt{1.5}$ |  860.7$\pm\mathtt{47.3}$ |  65.0$\pm\mathtt{1.8}$ |  1154.3$\pm\mathtt{50.8}$ |  66.2$\pm\mathtt{1.6}$ |  1120.9$\pm\mathtt{48.0}$
Diffusion |  85.2$\pm\mathtt{1.4}$ |  769.7$\pm\mathtt{41.4}$ |  71.9$\pm\mathtt{1.8}$ |  997.0$\pm\mathtt{38.1}$ |  64.0$\pm\mathtt{1.8}$ |  1080.1$\pm\mathtt{31.9}$
Composed |  98.8$\pm\mathtt{0.3}$ |  308.2$\pm\mathtt{22.1}$ |  97.0$\pm\mathtt{0.5}$ |  393.9$\pm\mathtt{23.0}$ |  97.0$\pm\mathtt{0.5}$ |  392.7$\pm\mathtt{25.7}$
Table 10:  Compositional Generalization over Increased Obstacles in Maze2D. In the top row, the left digit represents the number of obstacles that the model is trained on; the right digit denotes the number of extra obstacles added to the evaluation environments. Unlike other learning-based planners, whose performance significantly declines as the number of obstacles increases, our method sustains a near-optimal success rate even when the quantity of obstacles doubles.  |  | 4 + 1 | 4 + 2 | 4 + 3
---|---|---|---|---
| Method | Success | Check | Success | Check | Success | Check
KUKA | RRT* |  58.1$\pm\mathtt{2.1}$ |  18920.7$\pm\mathtt{281.3}$ |  56.5$\pm\mathtt{2.0}$ |  17964.2$\pm\mathtt{230.7}$ |  59.5$\pm\mathtt{2.3}$ |  17339.7$\pm\mathtt{261.3}$
P-RRT* |  55.7$\pm\mathtt{2.1}$ |  20356.2$\pm\mathtt{254.2}$ |  55.0$\pm\mathtt{2.1}$ |  19420.8$\pm\mathtt{216.0}$ |  56.8$\pm\mathtt{2.3}$ |  18287.8$\pm\mathtt{203.3}$
BIT* |  93.4$\pm\mathtt{1.1}$ |  7045.6$\pm\mathtt{483.9}$ |  91.5$\pm\mathtt{1.2}$ |  8055.9$\pm\mathtt{505.9}$ |  92.0$\pm\mathtt{1.2}$ |  7176.8$\pm\mathtt{477.9}$
AMP-LS |  28.8$\pm\mathtt{2.0}$ |  6767.0$\pm\mathtt{154.0}$ |  35.0$\pm\mathtt{1.6}$ |  7461.5$\pm\mathtt{401.0}$ |  37.0$\pm\mathtt{1.7}$ |  7277.4$\pm\mathtt{150.0}$
M$\pi$Net |  85.7$\pm\mathtt{1.2}$ |  262.6$\pm\mathtt{16.0}$ |  82.2$\pm\mathtt{1.4}$ |  314.7$\pm\mathtt{19.3}$ |  84.0$\pm\mathtt{1.3}$ |  301.7$\pm\mathtt{18.6}$
Diffusion |  92.7$\pm\mathtt{1.5}$ |  273.3$\pm\mathtt{45.1}$ |  83.9$\pm\mathtt{6.1}$ |  425.5$\pm\mathtt{99.6}$ |  82.8$\pm\mathtt{5.1}$ |  430.8$\pm\mathtt{79.8}$
Composed |  97.0$\pm\mathtt{0.8}$ |  171.5$\pm\mathtt{35.3}$ |  92.4$\pm\mathtt{1.5}$ |  283.4$\pm\mathtt{43.2}$ |  93.4$\pm\mathtt{1.2}$ |  259.9$\pm\mathtt{39.4}$
Table 11:  Compositional Generalization over Increased Obstacles in KUKA. In the top row, the left digit represents the number of obstacles that the model is trained on; the right digit denotes the number of additional obstacles. By composing potentials, our planner surpasses all the baselines by a margin.  |  | 5 + 1 | 5 + 2 | 5 + 3 | 5 + 4
---|---|---|---|---|---
| Method | Success | Check | Success | Check | Success | Check | Success | Check
Dual KUKA | RRT* |  36.2$\pm\mathtt{1.9}$ |  17843.8$\pm\mathtt{110.8}$ |  33.2$\pm\mathtt{1.7}$ |  17311.8$\pm\mathtt{93.9}$ |  33.9$\pm\mathtt{1.8}$ |  16847.9$\pm\mathtt{81.1}$ |  30.4$\pm\mathtt{1.7}$ |  16355.9$\pm\mathtt{72.6}$
P-RRT* |  30.4$\pm\mathtt{1.8}$ |  16921.2$\pm\mathtt{75.2}$ |  24.6$\pm\mathtt{1.4}$ |  16029.4$\pm\mathtt{74.3}$ |  23.1$\pm\mathtt{1.5}$ |  15387.9$\pm\mathtt{69.0}$ |  17.3$\pm\mathtt{1.3}$ |  14667.8$\pm\mathtt{81.9}$
BIT* |  76.2$\pm\mathtt{1.9}$ |  9470.5$\pm\mathtt{376.1}$ |  69.0$\pm\mathtt{2.2}$ |  9631.7$\pm\mathtt{284.6}$ |  57.9$\pm\mathtt{2.7}$ |  9034.9$\pm\mathtt{256.0}$ |  44.0$\pm\mathtt{3.0}$ |  8290.1$\pm\mathtt{176.1}$
AMP-LS |  0.1$\pm\mathtt{0.0}$ |  17512.0$\pm\mathtt{180.0}$ |  0.3$\pm\mathtt{0.1}$ |  17107.4$\pm\mathtt{219.0}$ |  0.1$\pm\mathtt{0.1}$ |  16725.1$\pm\mathtt{233.0}$ |  0.3$\pm\mathtt{0.1}$ |  16419.1$\pm\mathtt{210.0}$
M$\pi$Net |  60.5$\pm\mathtt{1.4}$ |  919.4$\pm\mathtt{27.5}$ |  55.1$\pm\mathtt{1.5}$ |  1034.9$\pm\mathtt{27.6}$ |  51.0$\pm\mathtt{1.7}$ |  1106.4$\pm\mathtt{30.3}$ |  48.8$\pm\mathtt{1.5}$ |  1134.0$\pm\mathtt{27.2}$
Diffusion |  91.7$\pm\mathtt{1.5}$ |  305.6$\pm\mathtt{29.8}$ |  80.9$\pm\mathtt{2.2}$ |  497.1$\pm\mathtt{35.5}$ |  69.0$\pm\mathtt{2.9}$ |  624.0$\pm\mathtt{35.8}$ |  58.1$\pm\mathtt{3.0}$ |  761.0$\pm\mathtt{31.0}$
Composed |  97.3$\pm\mathtt{0.7}$ |  250.4$\pm\mathtt{26.2}$ |  95.2$\pm\mathtt{1.1}$ |  373.9$\pm\mathtt{37.1}$ |  90.8$\pm\mathtt{1.3}$ |  545.7$\pm\mathtt{43.7}$ |  87.8$\pm\mathtt{1.5}$ |  648.4$\pm\mathtt{47.2}$
Table 12:  Compositional Generalization over Increased Obstacles in Dual KUKA.
In the top row, the left digit represents the number of obstacles that the
model trained on; the right digit denotes the number of additional obstacles.
In this most difficult Dual KUKA environment, our planner outperforms other
baselines by a larger margin. From 5 + 1 to 5 + 4, the advanced sampling-based
method BIT* drops by 32%, with a success rate of 44%, while our method only
drops by less than 10%, achieving 87.8% in success rate.

###  B.3 Performance on Composite Different Environment

We present extra experiment results on composing two different models. Each
model is separately trained on one single obstacle type and we evaluate them
on more complex environments with multiple obstacle types.

##### Static 1 + Static 2

In Table 13, we present quantitative generalization results of composing two
different static Maze2D models. Static 1 is a model only trained on obstacles
of size $1\times 1$ and Static 2 is a model only trained on obstacles of size
$1.4\times 1.4$. The composite environment Static 1 + Static 2 contains six
$1\times 1$ obstacles and three $1.4\times 1.4$. As the setting in the
evaluation of the base environments, we evaluate on 100 different environments
with 20 randomly generated start and goal in each environment. Obstacles in
all environments are randomly placed. No extra training is required. We also
present qualitative results over increasing obstacles in the composite
different environment (Static 1 + Static 2) in Figure 12.

| Static 1 + Static 2
---|---
Method | Success | Check
RRT* | 90.3 | 18495.1
P-RRT* | 82.8 | 24959.9
BIT* | 100.0 | 3486.9
Ours | 98.9 | 304.7
Table 13: Quantitative Results on Maze2D Composite Different Environments.
Static 1 is a model only trained on obstacles of size $1\times 1$ and Static 2
is a model only trained on obstacles of size $1.4\times 1.4$. The testing
environments contain six $1\times 1$ obstacles and three $1.4\times 1.4$
obstacles. Though trained on different environments separately, the composed
model reaches near optimal success rate while requiring one order of magnitude
less collision checks than BIT*. Figure 12:  Compositional Generalization over
Different Obstacles. The green/pink star indicates the start/goal state. Two
separately trained models are composed: a model only trained on large blocks
(blue) of size $1.4\times 1.4$ and a model only trained on smaller blocks
(orange) $1\times 1$. By composing two models, our planner can generalize to
more complex scenarios with various obstacles in test time (without any re-
training). The generated trajectories demonstrate various morphology and reach
the goal with smooth and short paths.

##### Static 1 + Concave

In Figure 13, we provide qualitative results on composite environments of
square obstacles and concave obstacles. In the leftmost column, we show a
generated motion trajectory on a base environment with 6 square obstacles
only. Following this, we gradually add concave obstacles to the base
environment, from 6 + 1 to 6 + 6. Our planner can plan smooth and coherent
trajectories accordingly by composing the corresponding potentials.

Figure 13:  Compositional Generalization over Square and Concave Obstacles.
The green/pink star indicates the start/goal state. Two separately trained
models are composed: a model only trained on square blocks (blue) of size
$1.0\times 1.0$ and a model only trained on concave blocks (orange). Our
planner can adaptively propose motion trajectories according to the
environments by composing the potentials of square obstacles and concave
obstacles.

##### Static 1 + Dynamic Environment

We evaluate our method on composite different environments which contain both
static and dynamic obstacles. Note that our method is only trained on
environments that purely consist of static obstacles or dynamic obstacles.
Static 1 is a model that is only trained on six $1\times 1$ static obstacles
and Dynamic is a model that only trained on one moving $2\times 2$ obstacle.
The composite environments contain six $1\times 1$ static obstacles together
with one moving $2\times 2$ obstacle. The quantitative results are shown in
Table 3 and two qualitative results are shown in Figure 14.

Figure 14:  Qualitative Compositionality Generalization over Static 1 +
Dynamic Obstacles. The current position of the agent is shown in the pink
asterisk. The grey line indicates the moving trajectory of the dynamic
obstacle (the large blue block). In the first row, the planned trajectory goes
further left in order to avoid the moving obstacle from the upper right. At T
= 19, the trajectory is traveling right while can still avoid the other orange
static obstacles. In the second row, the dynamic obstacle is first moving
toward the bottom right corner and then going back to the upper left. At
around T = 35, the trajectory takes several extra moves, waiting for the blue
block passing through the goal position.

##### Static 2 + Dynamic Environment

Static 2 is a model that is only trained on $1.4\times 1.4$ static obstacles
and Dynamic is a model that only trained on one large moving $2\times 2$
obstacle. The composite environments contain three $1.4\times 1.4$ static
obstacles and one moving $2\times 2$ obstacle. The quantitative results are
shown in Table 3 and two qualitative results are shown in Figure 15.

Figure 15:  Qualitative Compositionality Generalization over Static 2 +
Dynamic Obstacles. The current position of the agent is shown in the pink
asterisk. The grey line indicates the moving trajectory of the dynamic
obstacle (the large blue block). Two motion plans are shown. In the first row,
the planned trajectory veers further left to circumvent the approaching moving
obstacle while still being aware of the other orange static obstacles. In the
second row, the dynamic obstacle first moves upwards and then goes downwards.
Both phrases of the obstacle’s motion can potential block the agent’s path,
especially when it is going downwards. The planned trajectory first navigates
through the gap between the blue block and the orange block, and then reserves
enough space and travel near the bottom of the environment to prevent any
collisions.

###  B.4 Performance on Concave Obstacles

We construct Maze2D environments with 7 concave obstacles and have shown the
qualitative and quantitative results in Figure 7 and Table 2 in the main
paper.

Specficially, in Table 2, we show the motion planning performance on Convex
and Concave obstacles side by side, where RMP (Ratliff et al., 2018) is a
traditional potential-based motion planner and MPD is a recent diffusion-based
motion planner (Carvalho et al., 2023). Similar to other experiments, all
obstacles are randomly placed and no post-training/fine-tuning is required. We
can see that all methods subject to a certain decline in environments with
concave obstacles. Notably, our learned diffusion potential motion planner can
still solve all the problems, surpassing the pure potential-based method by a
margin while requiring significantly less planning time than the state-of-the-
art sampling-based planners BIT*. In Figure 16, we present more qualitative
results of RMP, RRT*, BIT*, MPNet, and our method on environments with concave
obstacles.

Figure 16:  Qualitative Performance on Environments with Concave Obstacles.
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*.

###  B.5 Performance on Real-World Dataset

Current robotic path planning problem is usually limited to simulation
environments and there is no widely-used real-world benchmark. Since
pedestrian trajectory is naturally a kind of demonstrations of human planning,
we leverage the ETH/UCY Dataset to evaluate real-world motion planning
capability. Specifically, the entire dataset is comprised of 6 scenes: ETH,
Hotel, Zara01, Zara02, Students01, Students03, where the models are trained on
5 scenes and tested on the held-out scene. Similar to our simulation
environments, the model takes as input the position of start, goal, and other
pedestrians, and outputs a predicted motion plan. The predicted motion plan is
desired to be as close as possible to the real human trajectory and thus we
report the Average Displacement Error (ADE) as defined in equation 15:

$\text{ADE}=\frac{\sum\limits_{i\in N}\sum\limits_{t\in
T}||q_{t}^{i}-\hat{q}_{t}^{i}||_{2}}{N\times T}$ (15)

where $\hat{q}_{t}^{i}\in\mathbb{R}^{2}$ is the step $t$ in the $i$th
predicted path and $q_{t}^{i}$ is the corresponding ground truth. ADE measure
the similarity between the predicted trajectories and human trajectories. A
smaller ADE value indicates the predictions are closer to the real human
behavior.

|  ETH |  Hotel |  Zara01 |  Zara02 |  Students01 |  Students03
---|---|---|---|---|---|---
Method |  ADE $\downarrow$ | Time | ADE | Time | ADE | Time | ADE | Time | ADE | Time | ADE | Time
MPNet | 18.11 | 0.12 | 28.60 | 0.11 | 11.06 | 0.12 | 17.22 | 0.11 | 10.37 | 0.12 | 8.93 | 0.11
M$\pi$Net | 37.70 | 0.26 | 44.49 | 0.29 | 1.14 | 0.22 | 13.66 | 0.23 | 12.76 | 0.18 | 1.54 | 0.22
Ours | 0.94 | 0.17 | 5.20 | 0.17 | 0.35 | 0.17 | 0.38 | 0.17 | 0.52 | 0.17 | 0.89 | 0.17
Table 14:  Quantitative Results on real-world ETH/UCY Dataset. We adopt the
ADE metric to show the similarity between the predicted motion plans and real
human motion trajectories on unseen scenarios. All motion plans are in the
world coordinate as given in the dataset. Our method can mimic human motion
behaviors more precisely in most scenes, while both MPNet and M$\pi$Net cause
drastic error compared to real human trajectories.

The per-scene quantitative performance is shown in Table 14. Each scene is
captured at different time of a day or different location, resulting in
different data distribution. The ADE is relatively smaller in Zara and
Students because the model can see a similar counterpart scene at training,
e.g., train set includes Zara01 when tested on Zara02. By contrast, ETH or
Hotel are more unique to other scenes (e.g., contain different scene layout
and human behavior patterns) and thus causing higher evaluation error. As
shown in Table 14, MPNet consistently falls short of the ADE, though with
slightly faster speed. We observe that M$\pi$Net can produce reasonable motion
plans in many cases, but it occasionally predicts random values, which causes
significant deviation from the target and leads to the notbaly larger ADE. Our
method demonstrates better generalizability and stability by precisely
mimicing the human trajectories in most held-out scenes. We also notice that
in Hotel, all methods suffer from a severe surge in ADE. Our speculation is
that the Hotel has different walkable world-coordination and in addition, it
contains various unseen types of pedestrians motion pattern, such as slowly
pacing people that are chatting or waiting, people stepping on or off the
train.

##### Base Real World Motion Planning

We visualize the motion trajectory planned by our model in Zara02 scene where
five other pedestrians are presented in Figure 17. The planner is trained on
the other 5 scenes and evaluated on this held-out scene. In the given
scenario, our agent (highlighted in red) is heading towards the bottom right
corner where P2 (yellow) is on its way. Also, the agent is about to enter an
intersection with multiple oncoming pedestrians. In this complex dynamic
scene, the trajectory planned by our model first chooses to follow P2 (shown
in T = 10) and adjust its pace to let other pedestrians pass through first. As
shown in in T = 22, our agent successfully crosses the intersection without
interrupting any other pedestrians.

Figure 17:  Qualitative Real World Motion Plan, Zara02 Scene. Each star
represents a pedestrian on the street. Red indicates the trajectory planned by
our model, while other colors represent 5 pedestrians in the surrounding. Our
motion plan smoothly passes through the intersection without any collision or
discontinuity. Figure 18:  Comparison with M$\pi$Net on KUKA. Trajectories of
4 motion planning problems. The large green/pink ball indicates the location
of the end effector of the start/goal state. Our planner can generate smooth
long-horizon motion trajectories and avoid being stuck in local geometry. For
example, in (a), our planner is able to select a feasible and shorter path
passing through the center of the workspace to reach the goal state; and in
(c), the trajectory by our planner navigates through narrow passage ways
without collision, e.g. travel between the green and purple blocks.  Figure
19:  Flexible Trajectory Morphology. 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. View a and b represent two
different viewing angles of the same trajectory. In the figure, 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.6 Comparison to Training on Multiple Types and Numbers of Obstacles

In this section, we train an additional baseline which learns all data with
different numbers and types of obstacles in a single diffusion potential
function. Thus, training this baseline requires additional motion planning
demonstrations of various obstacle combinations. Note that for our proposed
compositional motion planning method, models are only trained on a limited
number or type of obstacles, while can effectively generalize to various
combinations through composition.

We use Maze2D environment as the testbed. Let $s$ denote obstacle of type 1
and $l$ denote obstacle of type 2, so that $s6$ indicates 6 obstacles of type
1. We keep all the training and evaluation setups the same as in previous
experiments in B.2. In Table 15, Direct Training refers to a model directly
trained on various numbers/types of obstacles, and Composed refers to the
model trained on a fixed number of obstacles and generalizes to novel obstacle
combinations via composing potentials in test-time. For further comparison, we
include another baseline Process All Obstacles in the table, which is also
trained on a fixed number of obstacles, but is directly passed in different
numbers of obstacles at test-time (This baseline is equivalent to Diffusion
shown in Table 10).

Shown in Table 15, we observe that composing models obtains a comparable
performance to the Direct Training and substantially outperforms Process All
Obstacles. Note that the training and evaluating distribution is similar for
Direct Training, while very different for Composed, for example, the training
data in Direct Training includes the scenarios with 12 obstacles, while
Composed is only trained on 6 obstacles, highlighting the effectiveness of
composition for generalization.

|  $s6$ |  $s7$ |  $s8$ |  $s9$ | $s10$
---|---|---|---|---|---
Method | Success | Check | Success | Check | Success | Check | Success | Check | Success | Check
Process All Obstacles | 100.0 | 97.0 | 99.9 | 126.1 | 99.2 | 229.7 | 95.4 | 454.8 | 85.2 | 769.7
Direct Training | 100.0 | 99.5 | 100.0 | 86.3 | 100.0 | 95.7 | 100.0 | 118.4 | 99.9 | 140.5
Composed | 100.0 | 97.0 | 100.0 | 102.8 | 99.9 | 147.9 | 99.5 | 218.8 | 98.8 | 308.2

|  $s11$ |  $s12$ |  $s6+l1$ |  $s6+l2$ |  $s6+l3$
---|---|---|---|---|---
Method | Success | Check | Success | Check | Success | Check | Success | Check | Success | Check
Process All Obstacles | 71.9 | 997.0 | 64.0 | 1080.1 | – | – | –
Direct Training | 98.8 | 227.4 | 98.2 | 264.1 | 100.0 | 90.4 | 99.9 | 133.8 | 99.8 | 172.4
Composed | 97.0 | 393.9 | 97.0 | 392.7 | 99.9 | 184.2 | 99.2 | 304.1 | 98.9 | 304.7

Table 15:  Quantitative Comparison with Model Directly Trained on various
Types/Number of Obstacles. Motion planning performance on the Maze2D
environments with combinations of different types and numbers of obstacles,
e.g., $s6+l2$ denotes an environment with six obstacles of type 1 and two
obstacles of type 2. In columns with –, Processing All Obstacles baseline is
not applicable as different models are composed. Note that in Direct Training,
the model is trained on all 10 obstacle combinations shown above; while for
Composed, we only train a model on environments with six obstacles of type 1
and a model on environments with three obstacles of type 2.

##  Appendix C Proof of Conditional Independence for Composing Potentials

In Section 3.3, we present a way to generalize to multiple unseen out-of-
training-distribution constraints by composing corresponding potentials.
Specifically, in Equation 6, we assume that constraint $C_{1}$ and $C_{2}$ are
conditional independent. In this section, we will show how our assumption
holds in a general case, and thus the compositionality of our planner can be
achieved as in (Liu et al., 2022).

Assume that two set of constraints are given,
$C_{1}=\\{o_{1},o_{2},o_{3},o_{4}\\}$ and
$C_{2}=\\{o_{3},o_{4},o_{5},o_{6}\\}$. Let $f_{C_{i}}(q_{1:T})$ denote a
probability density function over trajectories, where positive likelihood is
uniformly assigned to the trajectory $q_{1:T}$ if it satisfies the constraint
$C_{i}$; otherwise, the likelihood is set to 0. Let $\mathcal{J}_{c_{i}}$
denote the set of trajectories that satisfies constraint $C_{i}$. Then, we
have

$f_{C_{i}}(q_{1:T})=\begin{cases}\rho_{i}&\text{if
}q_{1:T}\in\mathcal{J}_{c_{i}}\\\ 0&\text{if
}q_{1:T}\notin\mathcal{J}_{c_{i}}\end{cases}$ (16)

where $\rho_{i}$ is a small constant. Similarly, we can define $f_{C_{1}\cup
C_{2}}$ as the probability density function of trajectories that satisfy both
$C_{1}$ and $C_{2}$. Clearly, given any trajectory $q_{1:T}$, the probability
density of $q_{1:T}$ is positive if and only if both $f_{C_{1}}(q_{1:T})$ and
$f_{C_{2}}(q_{1:T})$ are positive. More specifically, we have

$f_{C_{1}\cup C_{2}}(q_{1:T})=\gamma f_{C_{1}}({q_{1:T}})f_{C_{2}}({q_{1:T}})$
(17)

where $\gamma$ is a constant (note the proportionality constant in the
previous equation). We can see that the joint probability density function
equals to the scaled product of $f_{C_{1}}$ and $f_{C_{2}}$. While the above
equation doesn’t indicate independence, sampling using the score function for
the left side of the equation is the same as sampling from the summed score
function for the right side of the equation, because the score function is the
gradient of the log probability and is invariant to the constant multiplier.
Therefore, the constant $\gamma$ here will not affect the test-time sampling
process and the proposed procedure of composing potentials for generalization
can be achieved in theory.