Guaranteed Reachable Set Computation for Unknown Nonlinear Systems on Riemannian Manifolds

Core Concepts

This paper presents a method to underapproximate the guaranteed reachable set for unknown nonlinear control-affine systems operating on Riemannian manifolds, without requiring full knowledge of the system dynamics.

Abstract

The key highlights and insights of this content are: The authors consider nonlinear control-affine systems operating on complete Riemannian manifolds, where the system dynamics are unknown. They introduce the concept of a "guaranteed reachable set" (GRS), which is the set of states that are provably reachable by the unknown system within a finite time horizon. To compute the GRS, the authors first define the "guaranteed velocity set" (GVS) - the set of velocities that the unknown system can provably achieve from any state on the manifold. The GVS is underapproximated using only local information about the dynamics at a single point, and Lipschitz bounds on the rate of change of the dynamics. The underapproximation of the GVS is then used to construct a control system whose trajectories provide a guaranteed lower bound on the GRS. The results are general enough to apply to any complete Riemannian manifold, and the authors illustrate the approach on examples of a pendulum on a sphere and a 3D rotational system. The key advantage of this approach is that it can provide provable guarantees on reachability without requiring full knowledge of the system dynamics, which is important for scenarios where the dynamics may change abruptly.

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Guaranteed Reachability on Riemannian Manifolds for Unknown Nonlinear Systems

by Taha Shafa,M... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09850.pdf

Deeper Inquiries

How can the proposed method be extended to handle more complex system dynamics, such as those with disturbances or parametric uncertainties

The proposed method can be extended to handle more complex system dynamics by incorporating techniques to account for disturbances or parametric uncertainties. One approach could involve integrating robust control strategies to address disturbances that may affect the system dynamics. Robust control techniques, such as H-infinity control or sliding mode control, can help mitigate the impact of disturbances on the system's behavior. By incorporating robust control methods into the trajectory planning algorithm, the system can adapt to uncertainties and disturbances more effectively. For parametric uncertainties, methods like adaptive control or model predictive control (MPC) can be employed. Adaptive control algorithms can adjust the controller parameters in real-time based on the system's response, allowing the system to adapt to varying parameters. MPC, on the other hand, can optimize control inputs over a finite time horizon, taking into account uncertainties in the system model. By integrating these advanced control strategies into the trajectory planning algorithm, the system can handle more complex dynamics with disturbances and parametric uncertainties, ensuring robust and reliable performance in real-world scenarios.

What are the limitations of the Riemannian Lipschitz bounds assumed in this work, and how could they be relaxed or estimated in practice

The Riemannian Lipschitz bounds assumed in this work have certain limitations that need to be considered. One limitation is the reliance on Lipschitz constants, which may not always accurately capture the true dynamics of the system. In practice, estimating these Lipschitz constants can be challenging, especially for highly nonlinear systems or systems with complex dynamics. To relax these limitations, one approach could be to incorporate adaptive estimation techniques to update the Lipschitz bounds online based on real-time data. Adaptive estimation algorithms, such as recursive least squares or Kalman filters, can continuously update the Lipschitz constants based on the system's behavior, providing more accurate and dynamic bounds. Another approach could involve using data-driven methods to learn the system dynamics and estimate the Lipschitz bounds from experimental data. Machine learning algorithms, such as neural networks or Gaussian processes, can be trained on historical data to predict the Lipschitz constants, allowing for more flexible and data-driven bounds. By relaxing the assumptions on the Lipschitz bounds and incorporating adaptive estimation or data-driven techniques, the method can be enhanced to handle uncertainties more effectively and provide more accurate underapproximations of the guaranteed reachable set.

Can the ideas in this paper be combined with data-driven techniques to further improve the underapproximation of the guaranteed reachable set

The ideas presented in this paper can be combined with data-driven techniques to further improve the underapproximation of the guaranteed reachable set. By integrating data-driven methods, such as neural networks or reinforcement learning, the algorithm can learn from real-world data to refine the estimation of the reachable set and adapt to changing dynamics. One approach could be to use neural networks to approximate the system dynamics and predict the evolution of the reachable set over time. By training the neural network on historical data, the algorithm can learn the system's behavior and make more accurate predictions of the reachable set under uncertainties. Additionally, reinforcement learning techniques can be employed to optimize the control inputs and trajectories to maximize the reachable set while considering uncertainties and disturbances. By combining reinforcement learning with the proposed method, the system can adapt and optimize its trajectories in real-time based on the evolving dynamics. Overall, integrating data-driven techniques with the proposed method can enhance the accuracy and robustness of the reachable set estimation, making the system more adaptive and capable of handling complex scenarios.