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Guaranteed Reachability on Riemannian Manifolds for Unknown Nonlinear Systems

Taha Shafa, Melkior Ornik

TL;DR

The paper addresses certifiable reachability for unknown nonlinear control-affine systems evolving on complete Riemannian manifolds. It develops a constructive framework based on a guaranteed velocity set (GVS) derived from local dynamics and Riemannian Lipschitz bounds, and then obtains a guaranteed reachable set (GRS) by solving an ordinary differential inclusion tied to the GVS and projecting tangent-space trajectories back onto the manifold via the exponential map. A key contribution is a computable underapproximation of the GVS, enabling real-time certifiable reachability analyses on manifolds, demonstrated through a detailed $SO(3)$ example where the underapproximation lies inside the true reachable set. The approach generalizes reachability under uncertainty from Euclidean spaces to arbitrary complete Riemannian manifolds and offers a practical path toward safe autonomous planning under unknown dynamics. Potential extensions include leveraging additional dynamical information, combining data from multiple trajectories, and handling stochastic or neural-network-based model uncertainties with provable guarantees.

Abstract

Determining the reachable set for a given nonlinear system is critically important for autonomous trajectory planning for reach-avoid applications and safety critical scenarios. Providing the reachable set is generally impossible when the dynamics are unknown, so we calculate underapproximations of such sets using local dynamics at a single point and bounds on the rate of change of the dynamics determined from known physical laws. Motivated by scenarios where an adverse event causes an abrupt change in the dynamics, we attempt to determine a provably reachable set of states without knowledge of the dynamics. This paper considers systems which are known to operate on a manifold. Underapproximations are calculated by utilizing the aforementioned knowledge to derive a guaranteed set of velocities on the tangent bundle of a complete Riemannian manifold that can be reached within a finite time horizon. We then interpret said set as a control system; the trajectories of this control system provide us with a guaranteed set of reachable states the unknown system can reach within a given time. The results are general enough to apply on systems that operate on any complete Riemannian manifold. To illustrate the practical implementation of our results, we apply our algorithm to a model of a pendulum operating on a sphere and a three-dimensional rotational system which lives on the abstract set of special orthogonal matrices.

Guaranteed Reachability on Riemannian Manifolds for Unknown Nonlinear Systems

TL;DR

The paper addresses certifiable reachability for unknown nonlinear control-affine systems evolving on complete Riemannian manifolds. It develops a constructive framework based on a guaranteed velocity set (GVS) derived from local dynamics and Riemannian Lipschitz bounds, and then obtains a guaranteed reachable set (GRS) by solving an ordinary differential inclusion tied to the GVS and projecting tangent-space trajectories back onto the manifold via the exponential map. A key contribution is a computable underapproximation of the GVS, enabling real-time certifiable reachability analyses on manifolds, demonstrated through a detailed example where the underapproximation lies inside the true reachable set. The approach generalizes reachability under uncertainty from Euclidean spaces to arbitrary complete Riemannian manifolds and offers a practical path toward safe autonomous planning under unknown dynamics. Potential extensions include leveraging additional dynamical information, combining data from multiple trajectories, and handling stochastic or neural-network-based model uncertainties with provable guarantees.

Abstract

Determining the reachable set for a given nonlinear system is critically important for autonomous trajectory planning for reach-avoid applications and safety critical scenarios. Providing the reachable set is generally impossible when the dynamics are unknown, so we calculate underapproximations of such sets using local dynamics at a single point and bounds on the rate of change of the dynamics determined from known physical laws. Motivated by scenarios where an adverse event causes an abrupt change in the dynamics, we attempt to determine a provably reachable set of states without knowledge of the dynamics. This paper considers systems which are known to operate on a manifold. Underapproximations are calculated by utilizing the aforementioned knowledge to derive a guaranteed set of velocities on the tangent bundle of a complete Riemannian manifold that can be reached within a finite time horizon. We then interpret said set as a control system; the trajectories of this control system provide us with a guaranteed set of reachable states the unknown system can reach within a given time. The results are general enough to apply on systems that operate on any complete Riemannian manifold. To illustrate the practical implementation of our results, we apply our algorithm to a model of a pendulum operating on a sphere and a three-dimensional rotational system which lives on the abstract set of special orthogonal matrices.
Paper Structure (15 sections, 7 theorems, 58 equations, 1 figure)

This paper contains 15 sections, 7 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Preliminaries
  4. Lipschitz Continuity on Riemannian Manifold
  5. Guaranteed Velocity Set
  6. Guaranteed Reachable Set
  7. Guaranteed Velocity Set
  8. Reachable Set
  9. Numerical Examples
  10. The Set of all 3 x 3 Real Orthogonal Matrices
  11. Conclusion and Future Work
  12. Acknowledgements
  13. Appendix
  14. Proofs of Supporting Lemmata
  15. Proof of Theorem \ref{['Theorem 1']}

Key Result

Lemma 1

Let $v,\,a_i,\,A,$ and $H_x$ be defined as above. Then and

Figures (1)

  • Figure 1: True reachable set (blue) with the underapproximation $\overline{\mathcal{R}}(T,x_0)$ (red) numerically calculated for $T \in \{0.1, 0.2, 0.3\}$ seconds. Reachable sets display the set of all reachable orthonormal triads with initial condition $X_0$ up to time $T$ using the fully known dynamics \ref{['controlSystem_RotationalMatrices']} (blue) and the underapproximated system \ref{['ControlSystem1_GRS']} (red).

Theorems & Definitions (22)

  • Definition 1: Riemannian inner product
  • Lemma 1
  • Definition 2: Riemannian length functional
  • Definition 3: Riemannian distance
  • Definition 4: Classical Lipschitz Constant
  • Lemma 2: Lemma II.A.2.4, canary2006fundamentals
  • Lemma 3
  • Definition 5: Guaranteed velocity set
  • Definition 6: Forward reachable set
  • Definition 7: Guaranteed reachable set
  • ...and 12 more