NTFIELDS: NEURAL TIME FIELDS FOR PHYSICS-INFORMED ROBOT MOTION PLANNING

Abstract

Neural Motion Planners (NMPs) have emerged as a promising tool for solving robot navigation tasks in complex environments. However, these methods often require expert data for learning, which limits their application to scenarios where data generation is time-consuming. Recent developments have also led to physicsinformed deep neural models capable of representing complex dynamical Partial Differential Equations (PDEs). Inspired by these developments, we propose Neural Time Fields (NTFields) for robot motion planning in cluttered scenarios. Our framework represents a wave propagation model generating continuous arrival time to find path solutions informed by a nonlinear first-order PDE called the Eikonal equation. We evaluate our method in various cluttered 3D environments, including the Gibson dataset, and demonstrate its ability to solve motion planning problems for 4-DOF and 6-DOF robot manipulators where the traditional grid-based Eikonal planners often face the curse of dimensionality. Furthermore, the results show that our method exhibits high success rates and significantly lower computational times than the state-of-the-art methods, including NMPs that require training data from classical planners. Our code is released:

1. INTRODUCTION

Motion Planning (MP) is one of the core components of an autonomous robot system that aims to interact physically with its surrounding environments. MP algorithms find path solutions from the robot's start state to the goal state while respecting all constraints, such as collision avoidance. The quest for fast, scalable MP methods has led from traditional approaches such as RRT* (LaValle et al., 2001) , Informed-RRT* (Gammell et al., 2014), and FMT* (Janson et al., 2015) to NMPs that exhibit promising performance in high-dimensional spaces. However, a significant bottleneck in state-of-the-art NMPs is their need for expert trajectories from traditional MP methods, limiting their application to high-dimensional scenarios where large-scale data generation is time-consuming. Recent developments have provided us with ways to have physics-informed deep learning models (Raissi et al., 2019; Li et al., 2020; Smith et al., 2020) that can directly solve complex PDEs such as Navier-Stokes, Burgers, and Schrodinger equations. Inspired by these neural PDE solvers and to overcome the expert data needs of NMPs, this paper introduces NTFields for robot motion planning in cluttered environments. Our NTFields are generated by a physics-informed neural model driven by a first-order, non-linear PDE called the Eikonal equation, whose solution represents the shortest arrival time from a source to a destination location under a pre-defined speed model, and leads to the continuous shortest-path between two locations (Sethian, 1996; Clawson et al., 2014) . Our model generates a continuous time field between the robot's given start and goal configurations while respecting collision-avoidance constraints, leading to a path solution in sub-seconds. Our method's salient features and contributions are summarized as follows: 1) A novel physicsinformed PDE solving the Eikonal equation formulation for robot motion planning under collisionavoidance constraints in high-dimensional spaces. 2) A novel neural architecture design that encapsulates various properties of our PDE, resulting in a scalable, fast NMP method. 3) A novel bidirectional algorithm that quickly finds path solutions by iteratively following the gradient of neural time fields, marching towards each other from the start and goal configurations. 4) Unlike prior, state-of-the-art NMP methods requiring extensive expert motion trajectories, NTFields only require randomly sampled robot start and goal configurations and learn robot motion time fields by directly solving the PDE formulation to find path solutions in sub-second times. Our data generation takes less than 3 minutes, even in complex high-dimensional scenarios, and yet, our method computes paths significantly faster than traditional planners while retaining high success rates. 5) We demonstrate our method in various 3D environments, including the Gibson dataset, and also solve motion planning problems for 4-DOF and 6-DOF robot manipulators where traditional Eikonal equation based planners such as the Fast Marching Method (FMM) (Sethian, 1996) struggle due to computational time complexity. 6) Finally, we compare our method against state-of-the-art NMP methods, which require expert motion trajectories, and against existing Neural PDE solvers for the Eikonal equation to highlight their limitations in solving complex motion planning problems.

2. RELATED WORK

Perhaps the most relevant work is the FMM (Sethian, 1996; Valero-Gomez et al., 2013; Treister & Haber, 2016) , which numerically solves the Eikonal equation to determine the time field for robot motion planning. However, numerical approaches require discretization of configuration spaces, thus failing to provide the continuous-time fields and suffering from the curse of dimensionality in terms of computational tractability. There also exist other traditional methods for solving MP problems without relying on the Eikonal equation. These methods range from sampling-based MP (Karaman & Frazzoli, 2011; Janson et al., 2015; Gammell et al., 2015) to gradient-based optimization techniques (Ratliff et al., 2009; Kalakrishnan et al., 2011) . The former techniques relatively scale better than FMM but still exhibit large computation times in high-dimensional space. In contrast, the latter methods do not provide global solutions and often lead to local minima. Recent developments have introduced NMPs, which learn from expert data via imitation and improve the performance of classical techniques in higher-dimensional problems. For instance, (Ichter et al., 2018; Kumar et al., 2019; Qureshi & Yip, 2018) generate informed samples in the regions containing the path solution to guide the underlying classical MP methods. (Qureshi et al., 2019; 2020) generate end-to-end path solutions and revert to classical planners to inherit their completeness guarantees. (Ichter & Pavone, 2019) perform MP in latent spaces to find path solutions but lacks interpretability. In general, NMPs, despite their computational gains, usually rely on classical MP methods to provide the training data, which hinders their application in scenarios where data generation is challenging. Recent approaches (Ortiz et al., 2022; Liu et al., 2022; Saulnier et al., 2020; Finean et al., 2021) use gradient information from Neural Distance Field (NDF) (Chibane et al., 2020; Xie et al., 2021) for collision avoidance resulting in real-time robot motion generation methods. In a similar vein, (Adamkiewicz et al., 2022) uses gradient information from Neural Radiance Field (NeRF) (Mildenhall et al., 2020) for visual navigation. Other relevant and recent methods to our NTFields are Implicit Environment Functions (IEF) (Li et al., 2022) and cost-to-go (c2g) function (Huh et al., 2021) , which also generate time fields or c2g function, then compute gradients for robot motion planning. However, IEF and c2g are not physics-driven methods; instead, they are supervised learning approach that requires training data from FMM or PRM. Note that, unlike prior NMP methods, our approach does not require motion path trajectories for imitation learning and instead directly solves the PDE formulated by the Eikonal equation to generate a continuous-time field for motion planning. Recent deep learning advances encapsulate various physics-based continuous functions solving PDEs without relying on explicit numerical derivatives. These methods exploit the neural network's backpropagation for gradient computation of output concerning related inputs. Similarly, EikoNet (Smith et al., 2020) solves the Eikonal equation for modeling the time field concerning seismology. However, EikoNet, as we show in our experiments, does not generalize to motion planning tasks requiring speed constraints in obstacle regions. Instead, we introduce novel formulation and architecture design enabling continuous time field modeling through solving the Eikonal equation for complex robot motion planning tasks.

3. NTFIELDS: PROPOSED FRAMEWORK

This section presents our physics-informed NTFields framework (Fig. 1 ) and discusses its formulation, architecture design, training details, and overall bidirectional planning execution.

availability

https://github.com/ruiqini

