A spherical robot is a nonlinear, nonholonomic, and unstable system which increases the difficulty of the direction and trajectory tracking problem. In this study, we propose a new direction controller Hierarchical Terminal Sliding Mode Controller (HTSMC), an instruction planning controller called Model Prediction Control-based Planner (MPCBP), and a trajectory tracking framework named MPCBP-HTSMC-HSMC (MHH). The HTSMC is designed by integrating a fast terminal algorithm, a hierarchical method, the motion features of a spherical robot, and its dynamics. In addition, the new direction controller has an excellent control effect with a quick response speed and strong stability. MPCBP can obtain optimal commands that are then transmitted to the velocity and direction controller. Since the two torque controllers in MHH are all Lyapunov-based sliding mode controllers, the MHH framework may achieve optimal control performance while assuring stability. Finally, the two controllers eliminate the requirement for MPCBP's stability and dynamic constraints. Finally, hardware experiments demonstrate the efficacy of the HTSMC, MPCBP, and MHH.

One controller cannot work on multiple and unknown terrains in the velocity control of the spherical robot, because the dynamic models of the robot vary on different terrains, and unmodeled dynamics and uncertainties exist in estimated dynamic models. Based on the above problem, a new velocity controller for spherical robot is designed. This controller combines a hierarchical sliding mode controller (HSMC), an adaptive RBF neural network (RBFNN) and a variable step-size algorithm. The RBFNN is used to online estimate the uncertainties, and the Lyapunov function is utilized to design the adaptive law for the RBFNN. In order to learn the uncertainties faster, while minimizing overshoot and preventing velocity oscillations, a variable step-size algorithm is proposed. The practical experiments demonstrate that, this controller of the spherical robot achieves velocity tracking on multiple and complex terrains, while eliminating steady-state error, having a good control effect, and maintaining high stability.

Multi-terrain trajectory tracking control is a prerequisite for robots to execute tasks in the wild and unknown environments. However, due to the uncertainties in both kinematics and dynamics, the current trajectory tracking framework for mobile robots, such as spherical robots cannot function effectively on multiple terrains, especially uneven and unknown ones. Thus, an efficient online adaptive algorithm that can rapidly obtain uncertainties online and assist the robot in adapting to multiple terrains is required. In this article, we propose an adaptive model predictive control-based instruction planner (VAN-MPC) and a multi-terrain trajectory tracking framework (VANMHH). The VAN-MPC is composed of three components as follows. 1) An adaptive dual-RBF neural network that online estimates all uncertainties based on the adaptive law and composite error. 2) A model predictive controller with a control Lyapunov function constraint. 3) The variable step-size algorithm for the adaptive controller to accelerate convergence and enhance stability. Without modifying the bottom velocity and direction controllers, we finally develop the multi-terrain trajectory tracking framework VANMHH just by employing the new instruction planner VAN-MPC. The practical experiments on multiple terrains demonstrate its effectiveness and robustness.

Model Predictive Control (MPC)-based trajectory planning has been widely used in robotics, and incorporating Control Barrier Function (CBF) constraints into MPC can greatly improve its obstacle avoidance efficiency. Unfortunately, traditional optimizers are resource-consuming and slow to solve such non-convex constrained optimization problems (COPs) while learning-based methods struggle to satisfy the non-convex constraints. In this letter, we propose SOMTP algorithm, a self-supervised learning-based optimizer for CBF-MPC trajectory planning. Specifically, first, SOMTP employs problem transcription to satisfy most of the constraints. Then the differentiable SLPG correction is proposed to move the solution closer to the safe set and is then converted as the guide policy in the following training process. After that, inspired by the Augmented Lagrangian Method (ALM), our training algorithm integrated with guide policy constraints is proposed to enable the optimizer network to converge to a feasible solution. Finally, experiments show that the proposed algorithm has better feasibility than other learning-based methods and can provide solutions much faster than traditional optimizers with similar optimality.

Motion control is essential for all autonomous mobile robots, and even more so for spherical robots. Due to the uniqueness of the spherical robot, its motion control must not only ensure accurate tracking of the target commands, but also minimize fluctuations in the robot's attitude and motors' current while tracking. In this paper, model predictive control (MPC) is applied to the control of spherical robots and an MPC-based motion control framework is designed. There are two controllers in the framework, an optimal velocity controller ESO-MPC which combines extend states observers (ESO) and MPC, and an optimal orientation controller that uses multilayer perceptron (MLP) to generate accurate trajectories and MPC with changing weights to achieve optimal control. Finally, the performance of individual controllers and the whole control framework are verified by physical experiments. The experimental results show that the MPC-based motion control framework proposed in this work is much better than PID in terms of rapidity and accuracy, and has great advantages over sliding mode controller (SMC) for overshoot, attitude stability, current stability and energy consumption.