Differentiable Wheel-Terrain Interaction Model for Stable Trajectory Planning on Uneven Terrains

Core Concepts

A differentiable wheel-terrain interaction model is proposed that can accurately predict the 6-DOF pose of a wheeled vehicle on uneven terrain. This model is then leveraged to formulate a bi-level trajectory optimization problem that generates smooth and stable trajectories while avoiding risky terrain features.

Abstract

The paper presents a model-based approach for trajectory planning of wheeled vehicles on uneven terrains. The key contributions are: Fitting a functional form to the digital elevation data of the terrain using Fourier basis functions. This allows for a differentiable representation of the terrain. Formulating the wheel-terrain interaction as a set of coupled non-linear equations, which can be solved using a non-linear least squares (NLS) optimization. This provides a differentiable pose prediction model. Leveraging the differentiability of the pose prediction to formulate a bi-level trajectory optimization problem. The inner layer predicts the vehicle's pose along a given trajectory, while the outer layer optimizes the trajectory itself to minimize kinematic and stability costs. Extensive experiments and comparisons with a baseline show that the proposed approach can generate smooth and stable trajectories that successfully navigate uneven terrains, such as avoiding ditches and valleys. The paper first discusses the related works on learning-based and model-based approaches for wheel-terrain interaction and motion planning on uneven terrains. It then presents the preliminaries on trajectory parametrization and tip-over stability criteria. The main algorithmic contributions are then discussed. First, the functional form for the terrain model using Fourier basis is presented. Next, the differentiable wheel-terrain interaction model is derived by viewing the wheeled vehicle as a parallel manipulator and solving a non-linear least squares problem. The implicit differentiation of this NLS problem is also provided. Finally, the bi-level trajectory optimization problem is formulated, which leverages the differentiable pose prediction to efficiently compute the optimal trajectory using a projected gradient descent approach. The results demonstrate the accuracy of the pose prediction compared to a high-fidelity physics simulator, as well as the effectiveness of the trajectory optimizer in generating stable trajectories on uneven terrains.

Stats

The vehicle's position (xk, yk) and heading angle (αk) can be directly controlled. The vehicle's z-coordinate (zk), roll angle (βk), and pitch angle (γk) are functions of the yaw plane configuration and the terrain geometry.

Quotes

"Navigation of wheeled vehicles on uneven terrain necessitates going beyond the 2D approaches for trajectory planning. Specifically, it is essential to incorporate the full 6dof variation of vehicle pose and its associated stability cost in the planning process." "We improve the state-of-the-art in the following respects. First, we show that our NLS based pose prediction closely matches the output from a high-fidelity physics engine. This result coupled with the fact that we can query gradients of the NLS solver, makes our pose predictor, a differentiable wheel-terrain interaction model."

Key Insights Distilled From

Bi-level Trajectory Optimization on Uneven Terrains with Differentiable Wheel-Terrain Interaction Model

by Amith Manoha... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03307.pdf

Deeper Inquiries

How can the proposed approach be extended to handle dynamic effects and wheel-terrain interaction forces for more accurate prediction of vehicle behavior on uneven terrains?

The proposed approach can be extended to handle dynamic effects and wheel-terrain interaction forces by incorporating more sophisticated models for the vehicle dynamics and terrain interaction. One way to achieve this is by integrating a dynamic model of the vehicle that considers factors such as mass distribution, inertia, and wheel dynamics. By incorporating these dynamic effects into the pose prediction model, the accuracy of predicting the vehicle's behavior on uneven terrains can be significantly improved. Furthermore, the wheel-terrain interaction forces can be modeled using advanced contact mechanics principles to capture the complex interactions between the wheels and the terrain. This would involve considering factors such as wheel slip, traction, and terrain deformability in the prediction model. By accurately modeling these interaction forces, the proposed approach can provide more realistic predictions of the vehicle's motion on various types of terrains.

What are the potential limitations of the Fourier-based terrain representation, and how could alternative terrain modeling techniques be incorporated to handle more complex or discontinuous terrain features?

While the Fourier-based terrain representation offers a computationally efficient way to approximate terrain surfaces, it may have limitations in capturing highly complex or discontinuous terrain features. One potential limitation is the assumption of smoothness inherent in Fourier series, which may not accurately represent terrains with sharp edges, cliffs, or sudden changes in elevation. To address these limitations, alternative terrain modeling techniques can be incorporated to handle more complex terrain features. One approach is to use mesh-based representations that can capture detailed geometric information of the terrain surface. By representing the terrain as a mesh of interconnected vertices, edges, and faces, the model can better capture irregularities and discontinuities in the terrain. Another technique is to employ voxel-based representations, where the terrain is discretized into volumetric elements. This approach allows for a more detailed representation of the terrain's geometry and can handle complex features such as caves, overhangs, and rough surfaces more effectively. By integrating these alternative terrain modeling techniques into the proposed approach, the system can better handle a wider range of terrains with varying complexities and provide more accurate predictions of the vehicle's behavior in challenging environments.

Can the bi-level optimization framework be adapted to handle other types of wheeled or legged robots navigating on uneven terrain, and what modifications would be required to the underlying models and algorithms?

The bi-level optimization framework can be adapted to handle other types of wheeled or legged robots navigating on uneven terrain by making specific modifications to the underlying models and algorithms. For wheeled robots, the framework can be extended by incorporating additional constraints and dynamics specific to the robot's configuration and motion capabilities. In the case of legged robots, the optimization framework would need to account for the unique kinematics and stability criteria associated with legged locomotion. This would involve modifying the pose prediction model to consider leg configurations, contact points, and stability metrics relevant to legged robots. Additionally, the trajectory planning algorithm would need to be tailored to generate feasible trajectories that adhere to the legged robot's locomotion constraints. Furthermore, for both wheeled and legged robots, the stability cost function in the optimization framework may need to be adapted to account for the specific stability metrics and constraints relevant to each robot type. By customizing the models and algorithms to the characteristics of different robot types, the bi-level optimization framework can be effectively applied to a diverse range of robotic systems navigating on uneven terrains.