Koopman Operator Based Modeling and Control of Rigid Body Motion Represented by Dual Quaternions

Abstract

In this paper, we systematically derive a finite set of Koopman based observables to construct a lifted linear state space model that describes the rigid body dynamics based on the dual quaternion representation. Methods such as the Extended Dynamic Mode Decomposition (EDMD) can compute finite approximations of the Koopman operator for different classes of problems but in general, they cannot offer guarantees that the computed approximation of the nonlinear dynamics is sufficiently accurate unless an appropriate set of observables is available. State-of-the-art methods in the field compute approximations of the observables by using neural networks, standard radial basis functions (RBFs), polynomials or heuristic approximations of these functions. However, these observables might not yield a sufficiently accurate approximation of the dynamics. In contrast, we first show the pointwise convergence of the derived observable functions to zero. Next, we use the derived observables in EDMD to compute the lifted linear state and input matrices for the rigid body dynamics. Finally, we show that an LQR type (linear) controller, which is designed based on the truncated linear state space model, can steer the rigid body to a desired state while its performance is commensurate with that of a nonlinear controller. The efficacy of our approach is demonstrated through numerical simulations.