Paper Insights: NTFields: Neural Time Fields for Physics-Informed Robot Motion Planning

Personal Motivation:

Although this paper was written in 2023, I have several reasons for reading this paper. Firstly, Dr. Ahmed Qureshi recently gave a talk at the Maryland Robotics Center at the University of Maryland, and I enjoyed his talk and work. He discussed Motion Planning using physics PDE priors. I have recently also become interested in leveraging first-principles physical laws for machine learning problems, as well as implicit neural representations. Thus, this paper was a good introduction to learning about physics with path planning.

I believe in the continued potential of using physics as a prior for many data-limited problems, as well as the need to eliminate data in some problems.

Research Motivation:

Neural Motion Planners (NMPs) are neural network-based motion planners that require training data for learning. Fundamentally, motion planning is the problem of finding a path from a given start point (q_s) to a given end point (q_g) while satisfying all constraints. A fault with NMPs is that they require expert trajectories, usually generated by traditional motion planning algorithms, which isn’t ideal in some cases.

Alongside this, there is another research direction: physics-informed deep learning and the use of neural models to solve PDEs such as the Navier-Stokes, Burgers, and Schrödinger equations. The authors bridged neural motion planning and physics, specifically the Eikonal equation, to find the shortest path while eliminating the need for expert trajectories.

Method:

Eikonal Equation:

The Eikonal equation is a first-order non-linear PDE approximating wave propagation:

Eikonal Equation

where T(x) is the travel time for a wave to reach point x, c(x) is the wave speed, and ∇T(x) is how fast travel time changes in space. Eikonal creates a gradient that gives the local direction and rate of fastest increase in travel time. Thus, its solution is the shortest arrival time to a destination.

Let S(x) be the speed at point x and T(x,y) be the wave arrival time from point x to point y. Then we can represent the Eikonal equation as follows:

where S(y) is the speed at point y and ∇_y T(x,y) is the arrival time with respect to point y. The eikonal equation breaks down at the starting point because the distance is zero there. They solve this by proposing a factorization:

where τ(q_s, q_g) is the time vectorized field that will be predicted by the neural network(NTField), q_s is the start point, and q_g is the goal point. τ will always be between 0 and 1, where τ will become 0 if the robot hits an obstacle. There is a symmetric property that shows that going from point A to B is the same as going from B to A. This can use the derivative to find the speed, and the neural network can be trained to follow speed and obstacle constraints.

Nueral Time Fields(NTFields):

They begin by defining a ground-truth speed model for each point S(q).

where p represents the robot surface obtained via forward kinematics, and d is the distance between the robot surface p and obstacles. They clip the function to the min and max distances to an obstacle, but if the robot's distance to the obstacle is greater than a set d_max, the speed is set to a constant s_const. This ground truth speed function will later be compared with the loss function. They introduce 2 encoders: workspace g(q) and c-space f(q).

The Workspace encoder converts a point cloud of the environment(obstacles) into a voxel grid using a 3D convolutional neural network and uses robotics geometry to understand its position relative to the obstacles. The point cloud is first converted into a voxel grid. Then, a 3D Convolutional Neural Network (3D CNN) takes the voxel grid and produces a volume that represents occupancy at each voxel, along with a learned feature vector that captures the local geometry in that area. Using forward kinematics, the system can calculate the 3D coordinates of the robot's surface. But the robot's surface won’t align with the voxel center. Thus, they find the 8 closest corners of the voxel grid and perform trilinear interpolation.

The C-space encoder is a ResNet that takes the robot configuration and provides a direct latent representation. Then, to force a direct symmetric property, they a ⊗ b = [max(a,b), min(a,b)]. Therefore, our configuration feature vectors are combined as [f(q_s)⊗f(q_g), g(q_s)⊗g(q_g)].

Finally, a time field generator, ResNet-based style architecture, takes the encoded and symmetric features from the encoders and produces the τ(q_s, q_g). We can then use this time field and inversely calculate the predicted speed. The loss function compares the ground-truth speed with the predicted speed. By following the gradient, there is no need to train data, and the system can rely on the physics of the eikonal equation. Please refer to the paper to see the results.

I thoroughly enjoyed this research and believe that using physics in certain problem spaces is very prevalent. For some problems, throwing data isn’t necessarily the solution, and using physics as either a prior or as a constraint can help us come up with effective and efficient solutions.