Robust Autonomy with Lie Groups

A large class of nonlinear systems in engineering and robotics evolves on geometric manifolds, such as fixed-wing aircraft, multi-rotors, etc., whose kinematic motion can be described using Lie groups, which provide a consistent representation of rigid-body motion. By leveraging the inherent symmetry, the nonlinear system behaviors can be represented and analyzed more efficiently, as the structure of Lie groups describes rotations and translations in a consistent framework, which can transform a complex complex dynamics into a simple and predictable structure. These features make Lie group-based methods highly suitable for real-time applications, including multi-rotor inertial navigation or trajectory tracking. In this dissertation, Lie group theory is leveraged for enhancing safety assurance in autonomous systems in an numerically efficient and stable way. %, where conventional numerical methods often lead to increased computational complexity or reduced stability. We investigate three problems related to robust autonomy: (i) we develop a log-linear dynamic inversion-based control law that compensates for nonlinearities arising from spatial curvature in Lie groups to enhance robustness in trajectory tracking, (ii) we extend the proposed control law to a multi-rotor dynamics and validate the effectiveness with real-world experimental testing, and (iii) we propose a geometric closed-form approach to solve the initial value problem for a class of nonlinear systems that evolves on the $SE_n(3)$ Lie group. The first problem proposes a robust control law for nonlinear systems evolving on Lie groups. We start with utilizing the derivative of the exponential map to derive the exact evolution of the logarithm of the tracking error for mixed-invariant systems, a class of systems capable of describing rigid body tracking problems in Lie groups. We then design a log-linear dynamic inversion-based control law to remove the nonlinearities due to spatial curvature and enhance the robustness of the controller. Furthermore, we employ Linear Matrix Inequalities (LMIs) to bound the tracking error given a bounded disturbance amplified by the distortion matrix and leverage the tracking error bound to create flow pipes. Our demonstrations show the effectiveness of our approach with Urban Air Mobility (UAM) scenarios using a 2D simplified kinematic aircraft model and ground vehicle scenarios using a 2D non-holonomic rover model. Extending our previous work to incorporate the dynamics of the UAV, a novel controller is proposed for a multi-rotor model and validate through real-world experiments.We present an approach that employs an exact logarithm map in Lie group theory and the Newton-Euler equations to derive exact linear error dynamics for a multi-rotor model, and apply these linear error dynamics with a novel log-linear dynamic inversion-based control law to simplify the nonlinear distortion and enhance the robustness of the system. In addition, we also leverage LMIs to bound the tracking error for a multi-rotor model in the presence of bounded disturbances. Our demonstrations in a Robot Operating System 2 (ROS 2)-based simulation environment and real-world experiments validate the performance of the proposed method and illustrate its applicability to autonomous aerial systems. In our last problem, we consider the numerical integration in the context of robust autonomy. In robust autonomy, an accurate integration scheme is required for both state estimation and control, since errors in numerical integration can accumulate over time, reducing the accuracy of localization, navigation, and system reliability. The existing techniques involve numerical integration methods such as Runge-Kutta integration to propagate their motion forward in time. However, these methods are approximate solutions and do not respect the geometric constraints of the nonlinear differential equation, leading to higher computational costs and numerical integration errors. To this end, we propose a geometric closed-form approach to solve the initial value problem for a class of nonlinear systems that evolves on the $SE_n(3)$ Lie group. Through numerical simulations, we show that our closed-form solution is more efficient than the Runge-Kutta $4^{th}$-order (RK-4) integrator and reduces floating-point operations by $45\%$ in a strapdown-inertial-navigation (SINS) case study.