Physics-informed Temporal Difference Metric Learning for Robot Motion Planning
Authors: Ruiqi Ni
zherong pan
Ahmed H Qureshi
Venue: ICLR 2025 Poster (2025)
Keywords: Robot motion planning
Eikonal Equation
Physics-informed Neural Networks
Temporal Difference Learning
Metric Learning
Primary area: applications to robotics, autonomy, planning
TL;DR: Self-supervised physics informed neural networks for scalable robot motion planning

Abstract: The motion planning problem involves finding a collision-free path from a robot's starting to its target configuration. Recently, self-supervised learning methods have emerged to tackle motion planning problems without requiring expensive expert demonstrations. They solve the Eikonal equation for training neural networks and lead to efficient solutions. However, these methods struggle in complex environments because they fail to maintain key properties of the Eikonal equation, such as optimal value functions and geodesic distances. To overcome these limitations, we propose a novel self-supervised temporal difference metric learning approach that solves the Eikonal equation more accurately and enhances performance in solving complex and unseen planning tasks. Our method enforces Bellman's principle of optimality over finite regions, using temporal difference learning to avoid spurious local minima while incorporating metric learning to preserve the Eikonal equation's essential geodesic properties. We demonstrate that our approach significantly outperforms existing self-supervised learning methods in handling complex environments and generalizing to unseen environments, with robot configurations ranging from 2 to 12 degrees of freedom (DOF).

Published as a conference paper at ICLR 2025
PHYSICS-INFORMED TEMPORAL DIFFERENCE METRIC LEARNING FOR ROBOT MOTION PLANNING
Ruiqi Ni', Zherong Pan", Ahmed H. Qureshi Purdue University, Liihhtsped Studios
aabqureshi@pursuue edu
ABSTRACT
The motion planning problem involves finding collision-fi free path from robot's starting to its target configuration. Recently, self-supervised learning methods have emerged to tackle motion planning problems without requiring expensive expert demonstrations. They solve the Eikonal equation for training neural net- works and lead to efficient solutions. However, these methods struggle in com- plex environments because they fail to maintain key properties of the Eikonal equation, such as optimal value functions and geodesic distances. To overcome these limitations, we propose novel sulf--uper temporal difference metric learning approach that solves the Eikonal equation more accurately and enhances performance in solving complex and unseen planning tasks. Our method enforces Bellman's 's principle of optimality over finite regions, using temporal difference learning to avoid spurious local minima while incorporating metric learning preserve the Eikonal equation's 's essential geodesic properties. We demonstrate that our approach significantly outperforms existing seff-vuue learning methods in handling complex environments and generalizing to unseen environments, with robot configurations ranging from to 12 degrees of freedom (DOF). The imple- mentation code repository is available at httttps: http//gitttbbbb www.....oooooooooo..oooommoooott coim//uiiiii///// ntrl- demo
INTRODUCTION
Robot motion planning is well-established problem focused on finding collision-free path be- tween a robot's initial and target configurations. In recent years, learning-based methods have emerged as promising tools using function approximators generate paths runtime efficiently. These methods, capable of handling high-dimensional configuration spaces (C-space), are typically categorized into supervised (Ichter al. 2018 Kumar al. 2019 Qureshi & Yip 2018
Ichter &
Pavone 2019 Qureshi et al.) 2019 2020 Huh et al. [202]] Li et al
2022 Fishman et al. 2023 and
self-supervised (Ni & Qureshi 2023a
2024 Yang et al. 2023) learning approaches.
Supervised learning methods rely expert demonstration trajectories for training. Expert data is of- ten generated by classical planning algorithms, which are slow and inefficient in high-dimensional, cluttered C-spaces. As result, these methods can have limited scalability to the cost and complexity of data acquisition (Karmaan & Frazzoli, 2011 Janson et al. 2015 Gammell et al.) 2015). In con- trast, self-supervised learning methods eliminate the need for expert data. However, many of these approaches focus more on short-horizon local planning and workspace navigation problems (Wi- jmans al. 2019 Yang al.) 2023). Recently, innovative self-supervised learning approaches, Neural Time Fields (NTFields) (Ni & Qureshi 2023a b). have been proposed to solve the Eikonal equation for planning problems. The Eikonal equation solution indicates the shortest travel time and defines the minimum path between two locations (Sethian 1996). These methods avoid the need for costly expert demonstrations while offering scalable and data-efficient alternatives.
While NTFields offer promising self-supervised planning solutions, they encounter significant chal- lenges when scaling complex, cluttered environments and generalizing to new, unseen scenarios. This is due to their inability to fully preserve the critical properties of the Eikonal equation' 's so- lution, which should simultaneously function as an optimal value function and a geodesic distance representing the cost-to-ge go from start to goal configurations in the C-space.

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To address the limitations of prior works, present novel approach that efficiently solves the Eikonal equation through temporal difference metric learning, enabling accurate solutions for the Eikonal equation in complex. cluttered environments. Our key contributions are follows:
We introduce temporal difference (TD) learning into physics-informed methods to solve the Eikonal equation as an optimal value function, enforcing Bellman's 's principle of optimality over a finite time step. By integrating TD loss with the Eikonal loss, we significantly enhance training convergence, resulting in more accurate solutions.
We propose a novel network architecture for metric learning over the solution space of the Eikonal equation, preserving the fundamental properties of geodesic distance. Our approach ensures that the learned metric adheres to essential characteristics such as triangle inequality, symmetry, and non-negativity. By parameterizing the network within metric space, the learning process is constrained to adhere to these properties, which improves training efficiency and stability.
During runtime inference, we utilize a sampling-basee based Model Predictive Control (MPC) (Williams ett al.) 2016; Bhhraddwaj et al. 2020) to minimize the learned value function. This approach eliminates the need for gradient computation, improving runtime efficiency. Additionally, the inherent randomness in sampling helps the system escape poorly learned regions and supports multimodal solutions, enhancing robustness in complex environments.
We evaluate our framework on complex planning tasks with C-space ranging from 2-12 DOF and demonstrate its scalability to complex scenes and generalization ability to multiple and unseen en- vironments. Our results show that our proposed approach significantly outperformed prior state-of- the-art learning- baaning-basedd planning methods. Additionally, we compare our proposed metric learning approach with other metrics commonly used Reinforcement Learning (RL) for the value function learning. Our results demonstrate that our metric better captures the key properties of the Eikonal equation, leading to more accurate approximation of solution.
RELATED WORK
The prior work in robot motion planning can be categorized into classical sampling or search-based approaches, trajectory optimization (TO), and learning-bar based methods. The classical approaches (Sethian 1996] Karaman & Frazzoli 2011 Janson et al.) 2015) Gammell et al. 2015) suffer from poor computational efficiency in high dimensional problems as they rely on C-space discretization or collision-fres sampling to find a path. The optimization- based techniques (Ratliff et al. 2009)
Kalakrishnan et al.] (011) convert hard constraints into soft constraints and solve them opti- mization, leading to robot paths that are often prone to local minima. In view of these pitfalls, supervised and self-supervised learning-based methods have emerged as promising alternatives that learn function generators and quickly infer robot paths at run times.
Supervised learning-basse based approaches learn by imitating expert demonstration data (Ichter &
Pavone 2019 Yonetani et al. 22021] Bency et al. 2019
Chaplot et al.) [021 Saha et al. 2024 2020 Fishman et al. 2023] lcctte et al,,
Zang et al.] 2023] Huh et al. [021] Qureshi et al. 2019
2018). They can learn a path generator (Qureshi et al.] 2019
2020 Fishman et al. 2023). C-space
sampler for sampling-bast tion for optimization-base techniques (Saha et 2004). Although these learning-basee methods infer robot trajectories orders of magnitude faster than classical techniques, they are bottlenecked by their need for expert demonstration data, which is gathered by running computationally intensive classical planners at a large scale.
methods (Qureshi & Yip 2018 Ichter et al. 2018), or prior distribu-
Unlike expert-driven methods, self-supervised learning eliminates the need for labeled data. Re- inforcement learning (RL) approaches (Sutton, 2018) learn through trial-and-error interactions but struggle with sparse rewards, often relying on expert demonstrations (Vecerik al. 2017). More recently, physics-ifformed neural motion planners (Ni & Qureshi 2023a bj 2024 Li et al.) Liu et al.) 2024 Shen et al. 2024) have emerged, solving the Eikonal equation for motion planning by mini- mizing equation loss on offline-sampled points. This can be viewed as offline goal-conditioned RL. A key yet overlooked aspect of these methods is that the Eikonal equation defines both an optimal value function and a geodesic distance on Riemannian manifold. Failing to capture these properties hinders scalability in complex, cluttered, and unseen environments.
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Our approach frames the travel time field metric space, drawing from Quasimetric Reinforce- ment Learning (QRL) (Schaul et al. 2015; Zhang et al. 2020; Bellemare et al. 2019; Wang et al.)
2023), which enforces properties like non negattiittyy and the tranngle inequality. However, existing QRI. methods focus on short-horizon tasks and overlook collision avoidance in complex settings. In contrast, our method scales to long-horizon tasks in high-dimensional C-spaces while integrating complex collision constraints. We further show that Eikonal equation solutions form true metrics, supporting multiple shortest paths in multi-connected regions, and introduce novel metric functions to capture these properties.
3 BACKGROUND
In this section, we formalize the general robot motion planning problem and then review the physics- informed neural motion planner proposed by (Ni & Qureshi 2023a b), which is the basis of our method.
3.1 ROBOT MOTION PLANNING
We denote by X' C. IR' the robot's 's workspace, where m E N corresponds to the space's 's physical dimensions. The C-space is represented as Q C Rd. where d E reflects the robot's degrees of freedom. The workspace consists of both the obstacle-occupied region, denoted by Xobs C X, and the obstacle-free region, denoted as X'rree = X' \ Xobs- We assume environment obstacles Xobs are known a prior. In the C-space, the obstacle and obstacle-free regions are given by Oahe C Q and Sreee = Q Lote, respectively. The motion planning problem involves finding trajectory that connects given start point 9, to a goal point qg. such that the entire trajectory lies within 2 freee
3.2
NEURAL MOTION PLANNER
Our approach based on the recently proposed NTFields Ni & Qureshi 2023a). This method builds on the theory of the Eikonal equation, which models the wave propagation from start point qe to the entire obstacle-fret free C-space. The solution of the Eikonal equation can be represented by the time for the wavefront starting from 2s reach (g. with the shortest path being traced by following the negative gradient of the travel time (Sethian 1996). This time field is represented as a learnable travel-time function T'( 9q) from 9a to 9g- Specifically,
ensure that the travel-time is always positive and symmetric, i.e., T( 9,) = Taaa qz). NTFields parameterize the travel-time function by distorting the Euclidean distance as follows: T'g,, 9g) ||4, 9gll/T(qs, qy), with 7(p.> 9y) being the distance distortion function parameterized using neural network. They also show that the travel speed of the wavefront inversely proportional the gradient norm of travel time, i.e.:
1g)(g,) = -1 a., T(g:: 1/)(g.) = Va. T(:::
where S(q) is the wavefront's 's travel speed. To ensure that the robot moves in the free space Q and stops the obstacle space Q.shs. NTFields introduced a ground truth speed S (q) based on the truncated distance between y and X as follows:
(2) clip(doas Xats)/dmaz dmin /daa, 1).
(2)
Here, dots (g. Nosss) represents the minimal distance between workspace robot geometry at configu- ration q, computed using forward kinematics, and the workspace obstacles Xobs- We further assume dobs is differentiable in using differentiable forward kinematics (Villegas et al. 2018). The param- eters dmain and dmaz the minimum and maximum distance thresholds, respectively. NTFields are then trained by minimizing the following Eikonal loss between the ground truth speed S*) and the speed S(q) predicted by the neural network parametrized T(4s, 9g) and Eq.
(3) Intuitively, the Eikonal equation seeks to approximate the optimal value function T(gs: 9g) with the desirable gradient norm, and the gradient of value function gives the desirable motion direction during inference. The subsequent work, P-NTFields (Ni & Qureshi 2023b) introduced curriculum learning and viscosity Eikonal equation to enhance the training process. However, local minima persist, leading to incorrect solutions that hinder success in complex environments. More recently, PC-Planner (Shen et al.) 2024) incorporated monotonicity and optimality constraints to refine results further. However, it does not explicitly leverage the Eikonal equation's 's role as an optimal value function and geodesic distance, which are crucial for global consistency and optimal path planning.
Le (VS* (g.)\ -1 (g.)) 11)²
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4 METHODS
In this section, we present our novel approach that efficiently approximates the solution of the Eikonal equation via temporal difference metric learning for robot motion planning in complex, clut- tered environments. We observe two key properties of the Eikonal equation that can be utilized to enhance the effectiveness of seff-supervised learning. First, we show that the solution of the Eikonal equation can be interpreted as the optimal value function in an optimal control problem. Therefore, the Bellman optimality principle can be applied both infinitesimal and finite time scales. While prior works only apply the infinitesimal perspective to formulate their loss function, we propose that combining the infinitesimal and finite time-scale perspectives can effectively avoid local overfitting (Sec. 4.I.. Second, we highlight that the travel-time function can be understood as the geodesic dis- tance on a Riemannian manifold. As a result, we propose a novel network architecture compatible with generalizable metric learning (Sec. 4.2
4.1 THE OPTIMAL CONTROL PERSPECTIVE
We highlight that the solution of the Eikonal equation and that of an optimal control problem coin- cide. Let us consider fully actuated d-DOF robot with configuration q(t) and control signal u(t) at time instant t. The robot is governed by the trivial dynamics, V such that [|u(t)]| = 1. From Qs to gg we have by the classical optimal control theory that the optimal travel time satisfies:
T(g:: 9g) = min [. 11((t)1/S=(((t))) (a(t)) u(t)
s.t. ((t.) qs ((1)) 4g.
||([||== =
Note that the above formulation inherently avoids obstacles due to the definition of S* tending to zero as the configuration tends to Qobs- In our Appendix. [ we show that the optimal solution of Eq. 4 under optimal actions, _ V T(g.: /l) V (9.. infinitesimal perturbation analysis. Such analysis also shows that the Eikonal equation captures the Bellman optimality principle at an infinitesimal time scale.
T 9g)ll, also satisfies Eq. by an
4.1.1 TEMPORAL DIFFERENCE LOSS
We observe that the value function trained using only Lg (Eq. 3) strug- gles to capture the broader, globally optimal structures of the value func- tion, limiting its performance in more complex environments. This is because relying solely on loss functions, such as Lg, at an infinitesimal time scale can dient of the network, acting as a tangent matching term (Simard et al. 2002). However, since network only takes sparse set of samples to evaluate Lg. the tangent function's 's landscape between these samples uncontrolled and can significantly deviate from the ground truth. Fur- thermore, such local, erroneous tangent landscapes will be reflected as global error the value function's 's landscapes.
91
overfitting problems they only regulate the gra-
Figure 1: The plots depict the solution to the Eikonal equation [VT(g)) 1, T(0) 0 with sampled green points as training data. In the top plot, the desired solution T(4)
To address the man loss, resembling the TD loss widely used in RL (Sutton 2018) to complement the Fikonal loss L.g. TD learning approximates the value function by considering the differences between successive state values over small but finite time step. effectively contrasting the value func- tions at two nearby spots. Specifically, our TD loss LTD is derived using Taylor expansion of the value function along the optimal policy u" with a small time step At:
issues, we introduce the discrete Bell-
14] is shown. In contrast, the bottom plot demonstrates that the green points also satisfy [VT(g)) = 1, but without proper constraints, solution deviates. The TD loss LtD ensures cor- rectness by enforcing, for example, that for > 0. T():) T(g: d) d. Therefore, LTD more capable finding the ground truth solution.
the
LTD T(qs, qn) At/S*(qn) T(g ( 4 + =rA())
(5)
[Tg.: 9g) At/S - -T(g, + n At, )1
where we define "² -V,, T(q., 9, with ug. Please refer to Appendix. for derivation of our TD loss
VV,, vV, T(a. 1l by its symmetry
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function. The TD loss can be understood that the value given state equals the local cost plus the value of the next state after following the optimal policy. Essentially, it ensures proper value propagation, maintaining consistency with the optimal policy. In Fig. [ we show that a suboptimal solution can have a zero L on sampled points but large LTD- TD loss also serves as finite difference approximation of the Eikonal loss Lg. Thus two loss terms, LE and LTD, are complementary. The TD loss captures the optimal sub- structure over a finite region-Providing a coarse- coarse-grained view of value propagation Eikonal loss focuses on the optimal substructure at an infinitesimal scale, offering a fine ane-grained, continuous perspective. By combining these two losses, we achieve both global and local consistency in the value function's landscapes, leading to more accurate results.
upwind scheme for the value function, ensuring
4.1.2 EESTACLE-AWARE NORMAL ALIGNMENT
Our method of training value function is an instance of differential dynamic programming, where the correct value function propagates from start goal. Therefore, when the network makes local mistake in the value function, such propagation can exaggerate the consequence. We find such an effect particularly detrimental at an early stage of training, especially when the network predicts a non-zero velocity inside the obstacle Xobs. In this case, a shortcut is created through the obstacle, and the downstream value functions are significantly underestimated. Fortunately, we know how the policy should behave when the robot is near obstacles. In such cases, the desired velocity u² should naturally align with the normal direction of the obstacle's ' surface to avoid collisions. As the robot moves farther away from obstacles, this alignment becomes less relevant, and the influence of the normal direction diminishes. Based on such observation, we introduce the following normal alignment loss:
IN -1- (v
(6)
(1 (1 V T(g. 9.) + V Vo,
S%
computes the normal direction in C-space, and the weight term 1 (g.) ensures the loss only takes effect near the obstacles. Note that evaluating LN relies on the differentiable assumption of the function dots discussed in Sec. 3.2
where Va,S (,.)
4.1.3 CAUSALITY PRESERVATION
The importance of causality has been noticed since the invention of the fast sweeping method (Van Trier & Symes) 1991) for solving the Eikonal equation. Specifically, the TD loss should propagate the value in the ascending direction of travel time but not the other way around. Although TD loss LTD encourages this propagation, the random batch training in neural networks does not maintain the required one-way information flow. To avoid this issue and preserve causal- ity when using neural approximation, (Wang et al. 2024b):
the following causality weight recently introduced by
Le = exp((xXcTT 9g))
Le encourages the optimizer to prioritize learning smaller values associated with closer-to-start states before tackling larger value functions associated with more distant states. By gradually learn- ing in this manner, the model follows the desirable unilateral information information-propagatioo direction, which ensures smoother convergence. Put together, our final loss function is formulated as:
(DgLE ATDLTD + Lx
(8)
where Xg; ATD, AN, Ac are hyper-parameters that control the contribution of the respective losses. Please refer to Appendix. C for details on the choice of these hyperparameters.
To summarize, we enhance the value function's convergence in three ways. First, we enforce Bell- man's principle of optimality under infinitesimal and finite time scales using LE and Ltp, re- spectively. Further, our normal alignment loss LN mitigates policy misalignment near obstacles, providing a strong prior at an early stage of training. Finally, the causality weight Le ensures a natural direction of information propagation in learning. prioritizing smaller values first. These contributions ensure the performance of learned value functions under complex environments.
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4.2 GENERALIZABLE METRIC LEARNING
In the previous section, we demonstrated that a properly designed loss function can train more accurate value functions. In this section, we show that performance can be further enhanced by employing a carefully designed network architecture that constrains the solution within a metric space. Specifically, we introduce network architecture compatible with metric learning and an attention mechanism that allows our method to generalize to unseen environments.
4.2.1 METRIC LEARNING
We notice that the travel-time function T(q., [g) can be understood as a geodesic distance on the Rie- mannian manifold with metric of T/S" (q). Under such setting, the optimal curve q(t) represents the geodesic curve on the manifold, with the corresponding geodesic distance given in Eq. Clearly, the metric T(gs: 4g) should satisfy the three fundamental properties: (Non-negativity) For ys 9g- Tys., 9g) > 0; and for
9g. T(gs, 4g) = O; (Symmetry) T(gs, Qg) = T(gg- qs (Triangle inequality) For any intermediate point 9m- T(g., qg) I T(q:: 9m) + T'gm: 9g)- While previous works (Ni & Qureshi 2023a b) have preserved the properties of non-negativity and symmetry, they often break the triangle inequality by using the factorized form T(gs: Qg) = 114 - 4g 4/ (9x: 9g)- This allows for situations where an indirect path between two points may be computed as shorter than the direct path. Instead, we employ metric learning using a network fa(-) to map the start and goal to a high-dimensional latent space and then define:
T(a:; (n) = So(o))))
with D(.-)) being some metric function satisfying the three aforementioned properties.
La
The definition of metric function D is crucial the performance of method. Due to the invariance of metric properties under transformation fe. our travel-time function will also retain those desirable properties. Several prior works used the Euclidean distance (Carroll & Arabie 1998 Panozzo al.
Figure 2: The figure shows the geodesies of points A and B on circle under differ- ent embeddings. With L2, the distance col- lapses into a line, causing overlap and ambiguity. In con- trast, L1 transforms it into diamond, pre- serving the geodesic structure and resolv- ing ambiguity.
a 2013). However, the geodesic distance in low-dimensional space cannot always be embedded as Euclidean distance in nigh-dimensional space after nonlinear transformations according to (Pitis et al., 2020 Pang et al.) (2023). Euclidean distance corresponds single shortest path, while geodesic dis- tance can represent multiple shortest paths in multi-conee connected regions. To accommodate this property, we propose using the L, distance, which permits multiple latent paths to share the same distance. Empirically, we combine the merit of T. and I distance. Specifically, define the latent space have a dimension of x b, i.c. fe Rd - gaxb, Our proposed distance metric is then computed follows:
Σ
[TL - Viss 13JSb
D v)
max
(10)
where the maximum along one dimension computes the L distance, while the summation along the other dimension computes the Lı distance. We further motivate our distance metire through illustration Fig. 2 The two half-cirele paths between points A and B are better captured by our distance metric than Euclidean distance as it transforms the circle into a 2D diamond to preserve multipath solutions.
4.2.2 ATTENTION MECHANISM
Prior physics-informed neural motion planners (Ni Qureshi 2023a bb can only represent the solu- tion of the Eikonal equation for given environment. In other words, they are unable to scale or gen- cralize to multiple environments. In contrast, with our more accurate training techniques, we enable generalizable neural Eikonal equation solver by conditioning our feature encoder, f(.). on the en- vironment shape Xobs- To this end, we assume that all the environments are represented using a point cloud, also denoted as Xobe- As result, we eet al. 2022) to compute a global latent feature z = PointNeert&y,, Next, given a configuration point g, first compute fixed random positional encoding 7(g) [sin(2mb² cos
the state-of-the-zz PointNext encoder (Qian
cos (2rbb g)),

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where b is fixed random Gaussian matrix. Finally, treat the query and the keys and (
values and compute the conditioned feature using the attention mechanism (Vaswani 2017: Rebain et al.] 2022). The attention mechanism enables the network to selectively focus on relevant parts of the environment, dynamically adjusting its output based on the spatial context provided by the point cloud. Let the attention mechanism be denoted as att(+) providing conditional features as att(r(g), z), we feed the resulting attention output into the PirateNets structure g(-) (Wang et al. 2024a), which integrates modified MLP (Wang et al., 2021) with the residual gate (Savarese & Figueiredo 2017 He et al. 2016) for enhanced performance and stability. Put together, we have the conditional feature representation fo (g. Xobs) ((att()(()) z)). Finally, we compute the geodesic distance conditioned on point cloud Xots as T(ys, Ug Xoha) ((Ja(us; Kats), ), fo(gg. Kobs)) This formulation allows our model to approximate Eikonal solutions for unseen environments. Moreover, we train our model by minimizing the loss (Eq. please refer to Appendix. for more details.
4.3
SAMPLING-BASEI MPC FOR PATH INFERENCE
After training our value function Tg... 9g). we use it as a cost-to-go function in a campling-baseed MPC (Williams et al.] 2016 Bharadhwaj et al. 2020) framework for path planning. We begin by randomly sampling actions w from a zero-mean normal distribution, which modifies the current configuration to q(t + 1) = ạ()) u. The distribution's mean is updated after each iteration the last selected action. Each sampled action leads to the next configuration and is assigned a value from the cost-to-go function.
softmax function then applied favor actions with lower travel
times, allowing us to calculate weighted average of the sampled actions. We use receding horizon approach to sample additional actions for a fixed horizon and generate multiple rollouts, selecting the trajectory with the lowest cost-to-go. This process is repeated until we find a path between the starting and goal configurations or until we hit a time limit. Unlike prior methods (Ni et al., [021]] Ni & Qureshi 2023b) that rely on gradient descent for path inference, our approach eliminates the need for gradient computations, enhancing efficiency and allowing the system to escape local minima in the value function. We should also highlight that our cost-to-go function can also be integrated with other downstream planning approaches, such as T-RRT with saclllteness-uurrantennn (Jaillet et al,, 2010), providing more robust framework for planning.
5 EXPERIMENTS AND ANALYSIS
This section presents our experiments and their analysis. We begin with an ablation study to demon- strate the effectiveness of our loss function and metric representation. Next, we provide scalability and generalization analysis to showcase our method's s ability to scale and perform well in complex, high-dimensional, and unseen environments. To evaluate our method, we compare it against the following baselines from prior planning methods.
Traditional Methods: For the search-based approach, we employ the Fast Marching Method (FMM) (Sethian 1996), which solves the Eikonal equation but is limited to 3D environments due to the curse of dimensionality. For sampling-based planner (SMP), we use RRTConnect (RRTC) (Kuffner& & Valle, (000) and LazyPRM (L-PRM) (Bohlin & Kavraki 2000), followed by path smoothing, to find paths in tasks ranging from 3 to 12 degrees of freedom (DOF). These methods are probabilistically complete and we allow a maximum of 30 seconds for each algorithm to find solutions.
Supervised Learning: We use MPNet (Queshi et al. 2019) for our 3D environments. For 7D manipulation tasks, we MPiNet (Fishman al. 2023) instead of MPNet. Note that MPiNet is specifically designed for robot manipulators and has been shown to perform better than MPNet in these tasks. Both methods learn by imitating the data collected by the expert classical methods.
Self-supervised Learning: We consider NTFields (NTF) (Ni et al. 2021) and P-NTFields (P- NTF) (Ni & Qureshi 2023a) for all presented tasks. Since their original neural architecture cannot scale across multiple environments, we enhance them with our proposed attention- based environ- ment encoding to evaluate their generalization and scalability. In a 2D maze, we also compare our TD learning with several metric representations considered in the RL literature.
For performance metrics, we evaluate the success rate (SR), path length, and computational times. SR shows the proportion of planning cases solved by motion planner. Path length refers to the
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total configuration-space Euclidean distance between the waypoints on the path found by planner. Lastly, computational time represents wall clock time taken by a CPU-based execution of a planner.
5.1 ABLATION ANALYSIS
This section ablates our loss functions L and its components, including Eikonal loss Lg. temporal difference loss LTD. obstacle alignment loss LN. and causality-basse prioritization Lc. Addition- ally, we assess our metric formulation and compare it with other metries commonly used in RL literature for value function learning. To conduct these evaluations, we set up complex 2D maze environments. One of the maze setups shown in Fig. 3] and more scenarios are available in Ap- pendix. B We chose FMM as expert baseline for comparison. We present the error the mean absolute difference between the travel time of each method and the ground truth FMM, measured at grid points. However, tion of contours in Fig. 3 In the figure, the contour lines show the travel time from various points to specific point in the maze. By following the negative gradient of the travel time, global path can be efficiently determined. Additionally, the colors represent the speed field within the maze, with yellow indicating free space and dark blue marking obstacle regions.
should highlight that these results are better understood with the illustra-
The first row in Fig. 3 presents our method, FMM, NT- Fields, and P-NTFields. NTFields and P-NTFields show significant distortion, failing to generate valid paths in cluttered environments. Our method, with a small error of 0.08 compared to 0.58 for NTFields and P-NTFields. Besides their contour lines also are incorrect, resulting in no path solution to some points in the environment.
Ours
FMM
NTF
P-NTF
The second row in Fig. shows the results of our method without g LTD. D. LN. and Lc. The errors for these vari- ations are 1.13, 0.21, 0.13, and 0.12, respectively, com- pared to 0.08 for our full method. The contour features show that results are incorrect with LE alone. Without LTD, the value function develops incorrect contours at some corners of the maze. Without LN or Lc, our train- ing starting from random initialization leads incorrect convergence to local minima. These results highlight that LTD and Lg are complementary to each other while LN and Le facilitate in correct convergence of those loss functions when starting from random initialization.
-Le
LTD
IN
L
IQE
PQE
MRN
DN
Figure 3: First row: We compare our
method with FMM, NTF, and P-NTF. Mid- dle row: We ablate our method by removing -LE, LIDD -L, and -Lc. Third row: We replace our distance metric D with IQE, PQE, MRN, and DN.
The third row in Fig. 3 presents the results of our method using alternative metric formulations commonly used in RL for value function learning. Specifically.. we replace our metric (Eq. [T] with Interval Quasimetric Embeddings (IQE) (Wang & Isola 2022a Wang et al. 2023), Poisson Quasimetric Embeddings (PQE) (Wang& & Isola (002b)). Metric Residual Networks (MRN) (Liu
al.) 2023), and Deep Norm (DN) (Pitis et al. (020) while keeping rest of our method same as presented for fairness. The errors for IQE, PQE. MRN, and DN are 0.32, 0.46, 0.19, and 0.29, respectively, compared to 0.08 for our method. These results demonstrate that existing RL metric formulations struggle to capture the optimal value function, while our method effectively captures this through combination of Loo and L, distance.
In conclusion, the contour lines produced by our method align well with those from the FMM, while other methods display noticeable artifacts. The second row shows a greater emphasis the Fikonal loss T.g. but it's important to note that the loss function assesses contour line accuracy across the entire C-space, whereas spurious local minima are localized issues. Consequently, improvements in loss may not seem significant despite the inclusion of additional loss terms LTD. LN. and Lc. Nonetheless, these differences in contour lines matter, as even single spurious local minimum can trap the robot, preventing it from reaching the true goal configuration, as also indicated by red circles in Fig. 3 Finally, our results also highlight that standard offline RL using only TD loss is inadequate. The first plot in the second row shows that without Eikonal loss L5, the method produces inaccurate results. Eikonal loss is essential and must work together with TD and other loss functions to accurately infer value functions and represent geodesic distance.
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(c) Manipulator
(a) Gibson
(b) C3D
Figure 4: Depiction of our (a) Gibson, (b) Cluttered 3D (C3D), and (c) 7-DOF Manipulator environments. We also illustrate multiple trajectories planned by our method between different start and goal pairs. It can be seen that method finds smooth trajectories while avoiding collisions.
5.2 SCALABILITY TO COMPLEX AND HIGH-DIMENSIONAL ENVIRONMENTS
We demonstrate our method's ability handle complex indoor home-like environments and high- dimensional tasks.
Indoor 3D home-like environments: We selected ten environments from the Gibson dataset (Li et al.) 2021), with room counts ranging from to 16 and dimensions between 90 and 430 square meters. An example Gibson environment is depicted Fig. and more examples available in the Appendix.
(a) Gibson
Method
Time (5)
I Length
SR(%)
Ours
0.056 + 0.037
5.39 : 3.25
95.1
Ours-G
0.074 1 0.057
7.10
5.03
92.6
NTF
0.074 + 0.068
8.12
9.45
68.1
P.NTF
0.057 1: 0.051
7.13 7.43
81.2
B In each environment, we evaluated 100 unseen start and goal pairs. [able presents the results for all methods in these environments.
FMM
0.74
4.90
2.83
86.4
RRTC
2.14
4.92 = 1.72
99.7
L-PRM
0.47 + 0.31
5.02
1.15
97.8
(b) Clutterd 3D (C3D) Length
Method
(s)
SR(%)
Our method consistently achieves a high SR and the low- computational planning times. While classical meth- ods also exhibit high SR, they are significantly slower in terms of computational time. Among the learning-based methods, NTFields and P-NTFields have lower compu- tational times but poor SR. Since our method uses MPC in contrast to travel time gradients for path inference, we have included a variant, denoted as Our-G, that incorpo- rates gradients instead of MPC for path inference, simi- lar to NTFields and P-NTFields. The results show that MPC improves both planning times and SR. Even with gradient-based path inference, our method outperforms NTFields and P-NTFields, indicating more accurate con- vergence to the Eikonal solution.
Ours
0.025 + 0.011
0.09
0.29
99.4
NIF
0.029 + 0.012
0.64 # 0.26
93.8
P-NTF
0.031 + 0.025
0.06 + 0.27
93.4
MPNet
0.22
0.68
97.0
FMM
0.76 0.09
0.65 1 0.25
100
L-PRM
0.13
0.08 = 0.31
100
7-D.. Manipulator Length
Method
Time (s)
SR(%)
Ours
0.074 1: 0.029
1.95
1: 0.82
87.0
NTF
0.063 + 0.017
1.63
0.64
74.3
P.NTF
0.061 1. 0.017
1.68
0.68
73.3
MPINet
0.57 4: 0.21
2.57
: 1.16
91.0
RRTC
0.42 0.30
2.01
1.06
97.7
L-PRM
0.25 4: 0.51
2.01
: 1.04
97.7
Table 1: Performance comparison on Gib- son, C3D, and 7-DOF manipulator datasets.
method scales well to complex 3D home-like environments, delivering high success rates and extremely low planning times, validating its ability to solve motion planning more accurately than previous self-supervised learning methods.
In summary,
12-DOF dual-arm in real-world confined cabinet: In this experiment, demonstrate the ability of our method to scale to a high-dimensional 12-DOF C-space and exhibit sim2real generalization. Our real-world environment depicted Fig. We randomly sampled 100 start and goal pairs in this environment for the testing. In this task. our method achieves a high SR of 91% with signifi- cantly low planning times of about 0.09 seconds on average. Fig. 5 depicts demo trajectory from our method. We also include another demo trajectory in the Appendix. B Note that these demon- strations are to exhibit the sim2real generalization of our method in high-dimensional tasks. We should also highlight that the prior self-supervised methods, i.e., NTFields and P-NTFields failed to converge in these high-DOF tasks with confined narrow passages.
5.3 GENERALIZATION TO NOVEL ENVIRONMENTS
This section presents the ability of our method to generalize multiple seen and unseen environ- ments in the following scenarios. We present the commutative results of all methods on both seen

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Start
Intermediate
Goal
Figure 5: Demontration of a path planned by our method to navigate real-world cabinet environment using 12-DOF dual-arm robot. This particular trajectory planned 0.11 seconds.
and unseen environments, while Appendix. details performances on seen and unseen tasks sepa- rately.
3D Cluttered Environments: These environments are taken from the C3D dataset (Qureshi et al,, 2019 2020) and consist of 10 cubes of varying sizes randomly placed in 3D space. An example scenario is shown in Fig. 4 For these tasks, we selected 100 seen and 100 unseen environments. The models were trained the seen environments. For testing, chose 500 random start and goal pairs across both seen and unseen environments. Table (b) summarizes the results of all methods.
In this setting, our method achieves an SR of 99.4%, with an extremely low average planning time of 0.025 seconds. NTF and P-NTF show relatively lower SR with similar inference time. MPNet shows a similar SR but is about 10 times slower than our method and requires expert data. Classical methods, though reliable with 100% SR, are at least 5 times slower than our approach. These results demonstrate that our method exhibits strong generalization to unseen tasks while maintaining our computational performance and a high SR.
7-DOF Robot Arm Motion among Obstacle Clutters: These tasks, adopted from the MPiNet dataset (Fishman et al. 2023). require 7-DOF robot arm to navigate among multiple obstacle blocks on a tabletop. An example scene is depicted in Fig. 4 We choose 150 seen and 150 unseen environments and train neural models the environment. For testing, select 300 start and goal pairs in both seen and unseen environments. The performance is summarized in Table [ (c). can be viewed that method persistently retains low computational times compared other methods. Our SR is also considerably high, i.e., 87%, and closer to MPiNet's 91%. NTF and P- NTF show relatively lower SR with similar inference time ours. MPiNet learned from expert data whereas our method, despite being self-supervised, exhibits high performance. The classical methods, similar to their prior experiments, demonstrate high SR but slower planning times.
CONCLUSIONS AND FUTURE WORKS
This paper introduces new, scalable, self-supervised neural approach for solving the Eikonal equa- tion in robot motion planning. In this paper, we highlight that the solution to the Eikonal equation can be expressed as the value function of an optimal control problem, as well as the geodesic dis- tance of a Riemannian manifold. These perspectives have led to our novel temporal difference metric learning approach for solving the Eikonal equation more accurately. By combining this with sampling-based MPC for path inference, our method has achieved higher success rate and lower computational cost for path inference than prior approaches. Additionally, our attention mechanism enables solve the Eikonal equation for unseen environments, which was previously not possible with prior self-supervised learning methods robot motion planning.
In future work, aim to further enhance the generalization ability of method. Currently, in our approach, we still observe some failure cases when generalizing to unseen environments. multiple, unseen environments in
Specifically. method struggles generalize effectively complex Gibson datasets. We plan to mitigate this issue by experimenting with more expressive neural encoders for the environment, such the point transformer (Zhao et al. 202T). Lastly, also aim to extend our approach to tackle motion planning tasks under partial observability and kinodynamic constraints.
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APPENDIX
This appendix provides the derivation of our TD loss, additional visualization results, and imple- mentation details of our method, along with the training procedure.
A TD Loss DERIVATION
This section provides the derivations connecting the optimal control problem with the Eikonal equa- tion and the TD loss. To solve the following optimal control problem:
-min f.a s(t)
T(g. 9g)
(*((()) [l((t)lde,, g(t.) = 9s. g(ty) 9g. ||u||| = 1,
(11)
We begin with the Taylor expansion of optimal value function T(g., 99) along the optimal policy direction up with a small step At:
T(gs u gg) T(4., 1g) + aV. ((q,, 0g), " At + o(Al),
(12)
where () denotes the inner product. Furthermore, according to Bellman's principle of optimality, with the optimal policy, the updated value function remains optimal, i.e.:
T(g. At,t
dt T(g., (g) o(At).
(13)
(-(((t)
We then compare the two equations above and tending At + 0 to yield:
St(gm) + (V T(ys, Un), ug) = 0.
(14)
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Note that the optimal policy ug is unit vector aligned with the negative gradient V T(g 9g)- Thus, we find that the optimal policy is u T(g.. 9/)/1 V (T. 9-) Plugging ug into
Eq. 14 and we arrive at:
(4)) IV 9. T(s. 92)l 0,
(15)
which leads to our Eikonal loss Lg infinitesimal time scale. Additionally, we derive the TD loss as follows:
T(q: + u wt qg) = T(g.: 9g) + (V T(e. 9g), , u At (Att) T9.: 9g) - v. T(g 9g) At o(At)
(16)
= Tg., 9g) At/S" (g.) + o(M))
and our TD loss is derived by dropping the of (t) and taking L2 norm.
B ADDITIONAL RESULTS
Ours
FMM
NIF
P-NIF
Ours
FMM
NIP
P-NTF
LN
Le
Lx
-Lr
-N
Le
IQE
PQE
MRN
DN
IQE
PQE
MRN
DN
results. First row: We compare method with FMM, NTF, and P-NTE. Middle row: We ablate our method hy removing LE,, - LTp, - Lx, and -Tc Third row: We replace our
Figure 6: Additonal
distance metric D with IQE, PQE, MRN, and DN.
In Fig. 6. we present results for two additional maze environments. For the left maze, our method maintains consistent contour lines without significant artifacts, while NTF and P-NTF fail re- cover correct results. All four ablation models exhibit incorrect contour line directions, and other metric formulations also display noticeable artifacts. In the right maze, while our method success- fully recovers most of the value function with fewer artifacts compared to the ablations, it struggles maintain accuracy in the bottom-right corner. Addressing this limitation could be an interesting avenue for future exploration. Finally, Table shows the error of each method to the ground truth so- lution in the maze scenarios shown in Figs. and The quantitative results confirm that for the left maze, our method achieves lower errors compared others, it avoids explicit artifacts present in other approaches. For the right maze, where our method also exhibits some artifacts, the difference is less pronounced but still demonstrates better performance than ablations and comparable results to competing metrics.
In 3] we present generalization statistics for both and unseen environments. The results indicate that the SR for unseen environments is a little lower compared to seen environments. Ad- ditionally, the superiority of method existing self-supervised learning approaches is less pronounced than in scalability tasks, primarily due to the increased complexity of the scalability tasks. The drop SR for self-supervised learning methods further highlights that while gener- alization is feasible for simpler problems, it becomes significantly more challenging for complex planning problems. As noted earlier, achieving robust generalization intricate environments, such as those in the Gibson dataset, remains an open research question.
Fig. and & provide additional visualizations of paths inferred by our method Gibson and 12-DOF dual-arm real-world settings.
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Maze
Ours
NTF
P-NTE
-E
LLD
-L
-Lc
IQE
PQE
MRN
DN
Fig.
0,08
0.58
0.58
1.13
0.21
0.13
0.12
0.32
0.46
0.19
0.29
Fig.
(Leit)
0.31
1.53
1.67
7.28
0.49
0.50
0.51
0.79
0.72
0.89
0.56
Right)
0.11
0.56
0.86
42.06
0.13
0.15
0.21
0.21
0.20
0.11
0.13
Table 2: The error of all methods, including ours, maze environments. The error denotes the mean absolute difference between the travel time of each method and the ground truth FMM, measured at grid points.
3
T
Figure 7: Visualization of Gibson environments, including multiple trajectories planned by method be- tween various start and goal pairs. can observed that our method generates smooth, collision-free trajec- tories, effectively navigating through the environments while avoiding obstacles.
Unseen
Cluttered (C3D) Length
(b) Cluttered (C3D) Length
Method
Time
SR(%)
Method
Time (3)
SR(%)
Ours
0.026 + 0.011
0.71
0.28
99.6
Ours
0.025 0.011
0.67 = 0.30
99.2
NTF
0.029 + 0.010
0.68
0.26
94.7
NTF
0.027 + 0.011
0.62 = 0.27
92.1
P-NIE
0.27
MPNet
0.21 1: 0.15
0.70
0.31
96.3
MPNet
0.20 0.17
0.65 1 0.31
97.6
FMM
0.70 0.09
0.68
0.25
100
FMM
0.73 0.73 0.08
0.63 = 0.26
100
RRTC
0.17
0.68
0.26
100
RRTC
0.15 0.12
0.62
0.26
L_PRM
0.14 0.12
0.70
0.29
100
I.PRM
0.13 0.11
0.65 = 0.30
100
7-DFF Manipulator Length
(c) 7DD- Manipulator
Method
SR(%)
Time (s)
Length
SR(%)
Ours
0.074 1: 0.029
1.87
0.80
88.2
Ours
0.075
0.030
1.95 0.85
84.0
NTF
0.067 1 0.018
1.63 0.60
74.6
NTF
0.068
0.020
1.62 0.69
74.0
P-NTF
0.063
0.016
1.62 0.55
73.3
P-NTF
0.066
+: 0.021
1.73 = 0.78
73.3
MPINet
0.54
MPINet
0.33
2.55
RRTC
0.42
2.02 1.02
98.0
RRTC
0.41 1.41:00 0.28
2.02 = 1.07
97.3
L-PRM
0.23 0.46
1.97 0.93
98.0
L-PRM
0.28 0.57
2.08 1.20
97.3
Table Performance comparison on C3D, and 7-DOF manipulator datasets seen unseen environments for our, NTF, and P-NTF methods, while other learning- based methods are not configured for this setting. It can seen our method exhibits high compared existing self-supervised learning methods low planning times.
C IMPLEMENTATIONS DETAILS
This section summarizes our implementation details, including the training procedure and hyper- parameters, as well as details on the data generation and training times of our neural models. In addition the following details, we aim to publicly release our code on GitHub with the final version of our paper, with which we will also provide our model architectures and facilitate the reproducibility of our method. Furthermore, all experiments and evaluations were conducted on system with 3.50GHz 8 Intel Core i9 processor, 32 GB RAM, and GeForce RTX 3090 GPU.
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Published as a conference paper at ICLR 2025
Start
Intermediate
Goal
Figure 8: Another path planned by method navigate real-world cabinet environment using 12-DOF dual-arm robot. This trajectory is inferred 0.1 seconds.
C.I TRAINING DETAILS
This section summarizes our training procedure. Our training data comprises randomly sampled robot configurations, their ground truth speed values S*, and environment point cloud Xots- The ground truth speed values are computed based on the configurations' distance to obstacles using Equation 2 Thus, our method, similar NTFields, requires only robot configurations and their distance to obstacles, which can be obtained very quickly compared to robot motion trajectories required by supervised learning methods. Next, we randomly from the start, 9s. and goal, 9g. pairs from the sampled configurations in a given environment. These muir-conditioee latent encodings Koha) and fo(yg Koha) are then obtained followed by the computation of their distance metric D(oo(ss, Kohs), folog, xaass) (Equation 10). This distance represents the travel time T(gs, 4g) and its gradient with respect 9s and 4 parameterizes the Eikonal equation (Equation TT to predict the speed S(g,) and S(g,), respectively. The predicted travel-time T( 9g), its gradients, and corresponding predicted speeds Sg,,) and Sq,)). along with the ground truth speeds (g.) and S (gg), are utilized to compute the loss L (Eq. 8). Finally, we minimize that loss over the sampled data set to train the parameters our attention-ban based latent encoders.
C.2 HYPERPARAMETERS
In 3D environment, we choose Ag 10-2, ATD 10-3, 1 10-3 0.5 the hyper- parameters, and we choose TD step At = 0.02. However, for manipulator environments, the free space much smaller than 3D space, and large TD step and normal direction can lead wrong place, so we reduce to Al = 0.005, and AN = 2 10-1 We select hyperparameters with the following considerations:
Eikonal Loss (Lg): Since Lg involves no approximation, it is expected to be the primary loss term and is assigned the highest weight.
Temporal Difference Loss (IrD)): LTD utilizes a Taylor expansion around the start and goal points. Its weight is lower than that of Lg emphasize its complementary role. The choice of At is crucial; the value too large, may lead to incorrect collision detections the next state, while value that is small diminishes the influence of LTD- We determine At based on the level of clutter in the environment, particularly in narrow passages, to ensure effective value propagation where accurate state transitions are essential. In more cluttered environments, the value of At needs to be smaller compared to less cluttered ones.
Obstacle Alignment Loss (Lx): LN serves as guidance loss, encouraging the planning di- rection move away from obstacles. However, the optimal direction should consider both avoiding obstacles and moving toward the goal, which Ly does not fully capture. As such, LN is assigned a small weight. narrow paassae environments, where pure obstacle avoidance might conflict with the correct planning direction, a lower weight ensures that L does not dominate the loss function.
Causality Loss (To)): For Loc, if X is small, its impact on the overall loss is minimal. Conversely, if èc is too large, it can cause the value function T'(s, 9g) to grow excessively. We select de via cross-validation ensure stable training.
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Published as a conference paper at ICLR 2025
C.3 DATA GENERATION AND TRAINING TIMES
Table 4 provides our data generation and training time. It can be seen that the data generation times for our self-supervised method range from few seconds to minutes. It should be noted that the data generation times for supervised learning methods such as MPNet can take several hours compared our few minutes.
Env
Data Generation Time Training Time
Maze
2.9s
500 epochs 94s
Gibson
3s
5000 epochs 9min
Dual UR5
200s
9500 epochs 32min
Cluttered 3D
100x2s
2000 epochs 40min
Franka
150x
2000 epochs 46min
Table 4: Data generation and training time
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Reviews:
[{"review_id": "aCfawGIMZQ", "reviewer": "Reviewer_3RM5", "summary": "This paper proposes a new method to allow robots to use self-supervised learning to learn motion planners and use MPC for online path generation.\n\nThe proposed method is built on NTFields (Ni & Qureshi, 2023a), and proposes two improvements:\n1. Novel loss\n    1. View the self-supervised learning problem in NTFields as a solution to an optimal control problem, propose a discrete-time loss L_TD based on temporal difference learning in RL, and combine L_TD with NTFields's continuous-time loss L_E to capture both a coarse-grained view of value propagation and a fine-grained optimal physics structure.\n    2. Another loss term L_N to avoid collision.\n    3. Another loss term L_C to ensure causality (i.e., time only moves forward) based on (Wang et al., 2024b).\n2. Metric learning: during NTFields's training, the structure of the learned travel-time function might violate triangle inequality, becoming no longer a geodesic distance on a Riemannian manifold. This paper proposes a neural network structure that preserves the key properties of such a metric.\n3. Allow generalization to the new environments by conditioning the learning on the environment via the attention mechanism.\n4. Use online MPC for path generation.", "questions": "- Sec.2 discusses the relationship between RL and self-supervised learning using physics-informed neural networks. I wonder about the difference between them. Can I think of a self-supervised learning type of approach as doing offline RL but with specially designed network architecture? The reason why I ask this question is that I think Sec.3.1 and 3.2 could have some introduction about how self-supervised learning motion planners are trained. I think this could be very helpful for people who are not familiar with NTFields (Ni & Qureshi, 2023a).\n- Sec.4.1.2 designs another loss for avoiding collisions. What is the relation between that and LE, where I think S* is also avoiding collisions based on the discussion around Eq.2?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The problem is well-motivated.\n- Very clear discussion about related works, including traditional motion planning methods, learning from experts, and self-supervised learning.\n- (Strength and also some weakness) I think the first improvement, using both continuous loss L_E from previous work and a novel TD loss is interesting. There is some discussion about the benefits of combining fine-grained and coarse-grained structures. The authors also used an example (Fig.1) to show that sometimes, only ensuring a low L_E does not imply good behavior. However, I don't quite get why L_E is 0 in both cases and why L_TD is high in the 2nd case. It would be great if the authors could provide more explanation about Fig.1. I think this could make the contribution more significant.\n- The result is very comprehensive, with simulation and real-world benchmarks, and compared against many baseline methods.", "weaknesses": "- I think the key weakness is that even though the various novel techniques improve NTFields, the empirical result does not seem to convey that impact. For example, in the ablation study (Fig.2 row 2), the errors discussed in line 403 convey that L_E, which was proposed in NTFields, is the most important component, and the other loss terms only provide small improvements (e.g., from 0.21 to 0.08 error). This makes me worry a bit about the effectiveness of the proposed techniques. In the real world, the proposed techniques increase the system's complexity, so it would be great if they provide a very strong performance boost to motivate people to adopt them. It would be great if the authors could discuss the importance of the proposed techniques, perhaps explaining how to interpret these results or using some other qualitative results to demonstrate.\n    - Related to this, it seems that one major improvement of the proposed method compared to previous work in Tab.1 is faster plan generation, e.g., from 0.5 sec to 0.07 sec. I think it would be great if the authors could discuss the implications of why this could matter in the real world. Perhaps one way to show that this is significant is to consider scenarios where the robot has to keep replanning to adapt to dynamic obstacles.\n\n- I think this paper did a good job introducing the technical key points about the previous work NTFields (Ni & Qureshi, 2023a) in Sec.3.2. However, I think it lacks some high-level description of how NTFields works as a motion planner. As a result, as a person outside this field of self-supervised motion planning, I found Sec.3.2 a bit hard to follow. For example, here are some points where I got confused:\n    - It seems that S*(q) is a function that is manually defined that outputs the desired robot velocity when the robot is around obstacles.\n    - Eq.3 seems to propose training the function S such that it matches S* at starting and goal configurations. Then, a perhaps naive question is why to learn S rather than just using S* as the motion planner?\n    - How is the learned S used to generate motion plans during online inference time? Line 352 mentions that some previous methods do gradient descent to minimize T(qs,qg). So, I guess it is looking for some path that minimizes the time from qs to qg? It would be great to put this information earlier in Sec.3.2.\n- Similar to the previous point, I think more intuition can help Sec.4.2.\n    - I think the motivation for metric learning in Sec.4.2 is not so clear to me. I notice that the authors briefly mentioned the potential benefit of metric learning in line 114 by saying, \"scale and generalize effectively in complex, cluttered, or unseen environments.\" But it is not so intuitive. Would there be some intuitive example, such as Fig.2, that can illustrate why metric learning is important?\n    - Eq.10 seems to be the key contribution. However, the discussion around it from lines 304 to 313 is very abstract. It would be great if the authors could provide some high-level intuition about why L_\\inf distance is preferred to Euclidean distance.\n- The contribution of Sec.4.3 - online MPC is interesting. It seems that the key benefit of this is faster online inference compared to (Ni et al., 2021; Ni & Qureshi, 2023b). However, the empirical results in Tab.1(a) show that the proposed method's computation time is 0.056 while NTF (Ni et al., 2021) is 0.074, with a small difference. Could the authors discuss the potential implications of why this difference could matter in the real world and how it will strengthen the contribution?\n- Fig.2 is a key result. However, for people outside the field, it is unclear how to interpret it. It would be great to add some guidance.\n\n\n## Minor\n- In Sec.3.1, it might be helpful to mention that Q_free and Q_obs are assumed to be known. I think this might be an implicit assumption for motion planning. But in reality, these things have to be obtained from sensors, which is usually nontrivial. So, it might be helpful to mention the assumption for readers outside motion planning.\n- Line 331 suddenly mentions \"the attention mechanism,\" which seems interesting and important. It would be great to elaborate on why using \"attention\" so that I, as someone who is not familiar with attention, can also appreciate the contribution.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2UNrxiljRV", "reviewer": "Reviewer_7oHd", "summary": "This paper improves a prior physics-based neural motion planning method by proposing a new metric learning approach to better approximate the solution of the Eikonal equation. This is done by enforcing Bellman’s principle of optimality based on the observation that the solution to the Eikonal equation can be viewed as an optimal value function. Further, a new network architecture is proposed for generalizable metric learning. This approach is claimed to perform better than existing motion planning and learning-based planning methods.", "questions": "- Why is solving Eikonal equation more scalable than motion planning? What approximations or relaxations does it make? Does it satisfy the robot’s kinodynamic constraints?\n    \n- The MPC scheme being proposed for planning with the learned value function cannot back-track. Wouldn’t it be better to use a search algorithm, e.g. A\\* with the value function as a heuristic to guarantee completeness. The proposed MPC-based planning may fail to solve hard planning problems which is somewhat supported by the low SR in 7-dof planning.\n    \n- In the 7-dof experiment, what is RRTC’s planning time on the problems solved successfully by the proposed method. It is possible that RRTC’s planning time is higher because it is solving harder problems that your methods fails at. For a fair comparison, it would be useful to know the average planning time for problems successfully solved by all methods.\n    \n- How did you tune the loss hyperparameters?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The method is significantly faster than sampling based motion planning.\n    \n- Compared to other learning-based planning methods that require an expensive data collection phase, this method is self-supervised and hence removes that requirement.", "weaknesses": "- The proposed method has worse success rates than sampling-based planners. In particular, in the harder 7-DOF manipulator experiment, the proposed method is significantly worse (87%) than learning-free RRT-Connect (98%).\n    \n- I am not convinced by the claim in the abstract that the method significantly outperforms existing methods, particularly in the more complex 7-dof domain. While it is much faster and has decent success rates, the results are more mixed.\n    \n- The overall loss function has many hyperparameters.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "civ0h7JJ3I", "reviewer": "Reviewer_1DdR", "summary": "This paper introduces a self-supervised learning approach for robot motion planning. It builds upon prior work on physics-informed neural motion planning, namely NTFields, in order to tackle complex cluttered environments. This is done through the use of temporal difference (TD) learning to solve the Eikonal equation as an optimal value function, which enforces Bellman’s principle of optimality and enhances convergence. The authors also propose a novel neural network architecture for metric learning that preserves the properties of geodesic distance, and ensure that the triangle inequality, symmetry, and non-negativity are respected. This model is used to compute the cost-to-go in sampling based Model Predictive Control (MPC) to perform motion planning. Experiments are conducted using robots with different numbers of DOFs on various types of environments, and results support most of the authors claims.", "questions": "- Is there a reason why the ablation study was not done in a quantitative manner showcasing that the obtained results are consistent across different environments ?\n- Why are both seen and unseen environments used during testing for the evaluation of the generalization to novel environments ? \n- In Fig.1, what do the green points represent ? What about the black line ?\n- In Fig.2, what does the color and the contour lines represent ?\n- Why is the success rate lower for the 7DOF manipulator at 87% compared to the other experiments, especially since for the 12DOF dual-arm, the method achieves a 91% success rate ?\n- Have you conducted a sensitivity analysis to measure the importance of the choice of hyperparameters, and how to set them for each case ?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper tackles an important problem in robotics and does a good job of highlighting the challenges of performing motion planning in complex cluttered environments for high DOFs robots.\n- The authors provide a comprehensive literature review and the approach proposed is well positioned compared to prior work in traditional and neural motion planning. \n- The proposed method is well thought-out and well detailed in the paper. The design choices are sufficiently justified.", "weaknesses": "- The experimental methodology suffers from some weaknesses. First, the ablation study is conducted on a single 2D maze environment. While results highlight the importance of each loss and the chosen metric for this specific environment, they do not necessarily show that this importance/performance is maintained across different environments. Second, in the generalization to novel environments, both seen and unseen environments are used during testing, which in my opinion defeats the purpose of testing the generalization to unseen environments.\n- The article is too technical and a bit complex to follow for a reader who is not completely familiar with this class of approaches. Moreover, the figures are hard to decrypt, captions should provide more details on how to read the figures with legends depicting what each element represents.\n- The method uses a set of user-defined hyperparameters, their values are stated in Apprendix C, however, the paper does not provide any comment on the sensitivity of the approach to these hyperparameters and how to mitigate it, especially the $\\Delta T$ parameter.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "yJcfzEElly", "reviewer": "Reviewer_qf75", "summary": "The paper presents a self-supervised learning approach for robotic motion planning that is based on learning to approximate the Eikonal equation. The work addresses a key challenge in robotics, that of navigating complex environments without requiring costly, expert-generated training data. The authors propose a method that used the Eikonal equation to learn a cost-to-go function that represents both an optimal value function and a geodesic distance in the planning/configuration space.", "questions": "For the ablation part, is it possible that using only one space configuration might be biasing your evaluations? Have you tried testing on different maze configurations or other types of spaces?\n\nThe claim of achieving hight SR in the comparisons in Table 1 could be made less strong or reworded. The SR of the method is consistently on par or lower from the top SR in all examples.\n\nFor table 1, clarify what you measure for time in the different cases.\n\nExpand on and give examples on the failure cases you mention when generalizing to unseen environments.\n\n\"Superior speed\" in s5.3, maybe reword to computational performance or similar.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well written and clearly presents the approach and evaluation on a set of relevant and reasonable datasets.\n\nThe method outperforms existing methods in terms of computational speed, with good performance in terms of success rate, and path length.\n\nThe main strength of the approach is the computational efficiency and that it generalizes to unseen environments, which according to the authors was not possible previously with prior self-supervised methods.\n\nThe approach is easy to setup and fast to train (few seconds to few minutes), though there are nuances about hyper-parameters as the authors point out.\n\nThe ablation part of the paper gives a clear picture of what and how contributes to the methods performance.", "weaknesses": "The main weakness of teh paper is the lack of quantitative analysis of the generalisation capability of the approach. The authors only present visualisations of successful paths in unseen environments. A more comprehensive analysis would be more appropriate, for example,  evaluating the performance across a wide range of unseen environments with varying complexities and against the sota approaches. This would provide a more objective measure of robustness/SR and identify specific types of environments where the approach might struggle.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)