Partitioning and Control for Dynamical Systems Evolving on Manifolds

With the development and integration of cyber-physical and safety-critical systems, control systems are expected to achieve tasks that include logic rules, receptive decision-making, safety constraints, and so forth. For example, in a persistent surveillance application, an unmanned aerial vehicle might be required to "take photos of areas A and B infinitely often, always avoid unsafe region C, and return to the charging point when the battery level goes low." One possible design approach to achieve such complex specifications is automata-based planning using formal verification algorithms. Central to the existing formal verification of continuous-time systems is the notion of abstraction, which consists of partitioning the state space into cells, and then formulating a certain control problem on each cell. The control problem is characterized as finding a state feedback to make all the closed-loop trajectories starting from one cell reach and enter a consecutive cell in finite time without intruding any other cells. This essentially abstracts the continuous system into a finite-state transition graph. The complex specifications can thus be checked against the simple transition model using formal verification tools, which yields a sequence of cells to visit consecutively. While control algorithms have been developed in the literature for linear systems associated with a polytopic partitioning of the state space, the partitioning and control problem for systems on a curved space is a relatively unexplored research area. In this thesis, we consider $ SO (3) $ and $ \ mathbb {S} ^ 2 $, the two most commonly encountered manifolds in mechanical systems, and propose several approaches to address the partitioning and control problem that in principle could be generalized to other manifolds. Chapter 2 proposes a discretization scheme that consists of sampling point generation and cell construction. Each cell is constructed as a ball region around a sampling point with an identical radius. Uniformity measures for the sampling points are proposed. As a result, the $SO(3)$ manifold is discretized into interconnected cells whose union covers the whole space. A graph model is naturally built up based on the cell adjacency relations. This discretization method, in general, can be extended to any Riemannian manifold. To enable the cell transitions, two reference trajectories are constructed corresponding to the cell-level plan. We demonstrate the results by solving a constrained attitude maneuvering problem with arbitrary obstacle shapes. It is shown that the algorithm finds a feasible trajectory as long as it exists at that discretization level. In Chapter 3, the 2-sphere manifold is considered and discretized into spherical polytopes, an analog of convex polytopes in the Euclidean space. Moreover, with the gnomonic projection, we show that the spherical polytopes can be naturally mapped into Euclidean polytopes and the dynamics on the manifold locally transform to a simple linear system via feedback linearization. Based on this transformation, the control problems then can be solved in the Euclidean space, where many control schemes exist with safe cell transition guarantee. This method serves as a special case that solves the partition-and-control problem by transforming the states and dynamics on manifold to Euclidean space in local charts. In Chapter 4, we propose a notion of high-order barrier functions for general control affine systems to guarantee set forward invariance by checking their higher order derivatives. This notion provides a unified framework to constrain the transient behavior of the closed-loop trajectories, which is essential in the cell-transition control design. The asymptotic stability of the forward invariant set is also proved, which is highly favorable for robustness with respect to model perturbations. We revisit the cell transition problem in Chapter 2 and show that even with a simple stabilizing nominal controller, the proposed high-order barrier function framework provides satisfactory transient performance.QC 20201012