This paper proposes a nonplanar model predictive control (MPC) framework for autonomous vehicles operating on nonplanar terrain. To approximate complex vehicle dynamics in such environments, we develop a geometry-aware modeling approach that learns a residual Gaussian Process (GP). By utilizing a recursive sparse GP, the framework enables real-time adaptation to varying terrain geometry. The effectiveness of the learned model is demonstrated in a reference-tracking task using a Model Predictive Path Integral (MPPI) controller. Validation within a custom Isaac Sim environment confirms the framework's capability to maintain high tracking accuracy on challenging 3D surfaces.
Although extensive research efforts have addressed autonomous driving tasks, prior research has predominantly focused on flat terrain conditions [1]. However, in many real-world applications, autonomous vehicles must be able to operate in unstructured and off-road environments. In particular, off-road driving requires safe navigation over nonplanar or uneven terrain, including slopes, hills, banks, and bumps. Navigating autonomous vehicles in such terrain environments requires a precise understanding of 3D vehicle dynamics, which can then be incorporated into model predictive control.
Recent work has addressed dynamic modeling and navigation by accounting for terrain topology of nonplanar surfaces. Some studies on nonplanar modeling considered the road as a ribbon, and developed the dynamics of the vehicle based on the assumption that the vehicle moves on Darboux tangent frame at the nearby spine point. The idea was first introduced by Limebeer & Perantoni [2]. Fork & Borrelli in [3], [4] used Tait-Bryan (TB) angles along the trajectory for surface parameterization, and derived a kinematic model for the vehicle on the nonplanar parametric surface. Yu et al. [5] presented an extended bicycle model that takes into account the topology of the terrain through a rotation transformation using the TB angles. Piccinini et al. [6] developed an artificial race driver that integrates a kineto-dynamical vehicle model to learn the vehicle dynamics, plan, and execute minimumtime maneuvers on a 3D track. Other studies investigated planning directly on point cloud maps. For example, Krüsi et al. [7] introduced a motion planning framework on generic 2D manifolds embedded in the 3D space without requiring explicit topology extraction. Han et al. [8] proposed a physics-based framework to derive traversability constraints, and a planning framework that generates sampled rollouts on a 2D kinematic plane, then projects the rollouts on a 3D elevation map. Datar et al. [9] proposed a learning framework to model the forward vehicle-terrain dynamics, in which the terrain’s elevation map is incorporated as part of the neural network input. Lee et al. [10] proposed a terrain-aware kinodynamic model, which combined an elevation map encoder, a dynamics predictive neural network, and an explicit kinematic layer.
Due to the difficulty in analytically modeling complex vehicle dynamics, and the effects of the terrain geometry and interaction, learning-based nonplanar vehicle modeling has been considered in recent work [9], [10], which have used neural networks. However, neural networks require large training data, and the training process must be performed offline. As a result, the model lacks the ability to adapt online to capture the uncertainty of the varying terrain. Therefore, in this work, we propose a new approach to approximate the dynamics of an autonomous vehicle on nonplanar terrain in real-time. In particular, our approach relies on a sparse Gaussian process (GP) model with online recursive updates. In the proposed geometry-aware model, we combine a nominal single-track dynamic model with a data-driven residual model using a sparse GP that takes into account the geometry of the terrain. The sparse GP model is updated recursively given real-time collected data. The proposed model is then used in a model predictive control (MPC) formulation, and a model predictive path integral (MPPI) framework [11] is used to solve the complex MPC problem to generate the optimal control inputs in a receding horizon manner. We validate the proposed framework in a simulation environment where we generate different nonplanar tracks with slopes, hills, banks, and bumps.
The rest of this paper is organized as follows. In Section II, we present the vehicle model on a nonplanar surface. In Section III, we formulate the MPC problem and how to solve the problem with MPPI. In Section IV, we show the simulation results in the Isaac Sim environment, and we provide some concluding remarks in Section V.
In this section, we first present the problem of modeling vehicle dynamics on a nonplanar terrain and then develop a learning-based model using sparse GPs with recursive updates [12]. We consider the motion of a ground vehicle traveling over a nonplanar surface. We denote the vectors of states and control inputs of the vehicle as x and u, respectively. We define the state vector as x = [p x , p y , ψ, δ, v, β, r] ⊤ ∈ R 7 , where [p x , p y ] ⊤ are the x and y positions, ψ is the yaw angle, δ is steering angle, v is the velocity in car frame, β is side-slip angle, and r = ψ is yaw rate. The vector of control inputs is u = [a, v δ ] ⊤ ∈ R 2 that includes the longitudinal acceleration a and steering-rate v δ .
We model the terrain as a smooth height field h : R 2 → R over a global frame in which the z-axis is vertical and the (x, y) plane is the nominal ground plane. At each planar location (p x , p y ), we associate a unit surface normal n(p x , p y ) = [n x
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