Adeel Akhtar

Establishing stability guarantees for dynamical systems on Lie groups is a fundamental challenge, as classical Lyapunov methods developed for Euclidean spaces do not directly transfer to curved geometries. In this paper, we propose a framework for learning maximal Lyapunov functions for systems evolving on the special orthogonal group SO(n). Theoretically, we introduce a neural Lyapunov architecture based on the logarithmic map with proven approximation capabilities, and we formulate the learning problem via a Zubov-type characterization of the maximal region of attraction. A key technical contribution is the derivation of explicit, numerically tractable formulas for the derivative of the logarithmic map, enabling training through a two-phase algorithm that balances computational efficiency and accuracy. Empirically, we validate the approach on a low-dimensional nonlinear system.

This work studies the Schrödinger bridge problem for the kinematic equation on a compact connected Lie group. The objective is to steer a controlled diffusion between given initial and terminal densities supported over the Lie group while minimizing the control effort. We develop a coordinate-free formulation of this stochastic optimal control problem that respects the underlying geometric structure of the Lie group, thereby avoiding limitations associated with local parameterizations or embeddings in Euclidean spaces. We establish the existence and uniqueness of solution to the corresponding Schrödinger system. Our results are constructive in that they derive a geometric controller that optimally interpolates probability densities supported over the Lie group. To illustrate the results, we provide numerical examples on SO(2) and SO(3). The codes and animations are publicly available at https://gradslab.github.io/SbpLieGroups.

This article addresses the design of a path following controller for a payload tethered to a quadrotor. Specifically, the goal is to design a smooth dynamic feedback controller that forces the suspended payload to converge and follow a large class of embedded curves. The given curve is treated as a smooth manifold, and set stabilization is used to find the maximal control invariant manifold. The controller design guarantees that once the aerial robot approaches the path, the payload never leaves the path. The performance of the proposed controller is verified through simulations with practical sensor noise and parametric uncertainties. Moreover, experimental implementation and verification of the proposed control scheme are performed on a Quanser UAV platform in an indoor flying arena.

Payload delivery using unmanned aerial vehicles (UAVs) has attained central importance for smart logistics and transportation systems in the context of the fourth industrial revolution. This work considers the problem of designing a smooth dynamic feedback control law for a point mass payload suspended to a quadrotor and making the load to follow a large class of curves that includes both closed and non-closed curves. Typically, the load path following problem is solved using either a coordinate-free or a local-coordinate-based approach. However, in this article, we adopt an alternative methodology. First, we express the system dynamics in an extended Euclidean space using a global coordinate system. Second, we propose two families of functions that lead to the design of almost-global and local controllers in terms of region of convergence. We cast the load path following problem in the framework of set stabilization, and as a result, the proposed controllers make the given path an invariant manifold. The resulting controllers guarantee that once the suspended load converges to the path, it stays on the path indefinitely while satisfying other application-specific constraints. Finally, to complement the theoretical results, we provide a successful real-world experimental validation of the proposed controller on a Quanser QDrone UAV platform with a cable-suspended payload.

In this article, we propose a novel family of Lie algebra valued functions on the special orthogonal group SO(3). Each function belonging to this family induces a novel controller that stabilizes a rigid body attitude. All controllers induced by the family form a controller class. This novel controller class contains both local and almost-global asymptotically stable controllers, and this article presents geometric stability results of the whole controller class.

This paper presents an approach for designing path-following controllers for the kinematic model of car-like mobile robots using transverse feedback linearization with dynamic extension. This approach is applicable to a large class of paths and its effectiveness is experimentally demonstrated on a Chameleon R100 Ackermann steering robot. Transverse feedback linearization makes the desired path attractive and invariant, while the dynamic extension allows the closed-loop system to achieve the desired motion along the path.

This paper addresses the multi-agent coordinated path following problem consisting of a class of underactuated vehicles capable of following a class of closed embedded curves in the three-dimensional space and proposes a solution using a nested set stabilization approach. The stabilization of the first set restricts each agent to the assigned path following manifold and ensures that each system stays on the path indefinitely when initialized in the neighborhood of the path. Stabilizing the second set, contained inside the first, solves the position, velocity, acceleration, or jerk-related coordination constraints. The third set, contained in the second set, satisfies the yaw, or yaw-rate-related coordination constraints. The reduction theorem establishes the asymptotic stability of each nested set, and as a result, the multi-agent coordinated path following problem is solved.

This paper proposes a hybrid geometric control scheme for the classical problem of globally stabilizing a point-mass system on a unit circle, as it is impossible to design a smooth globally asymptotically stable controller for this problem. Unlike most existing solutions that rely on coordinates and rely on a particular controller construction, our proposed solution is coordinate free, or geometric, and belongs to a class of controllers that we also characterize. Specifically, we propose a geometric hybrid controller that uses a local geometric controller from the said class and an open-loop geometric controller. The system achieves global asymptotic stability when each controller from the local geometric class is combined with the geometric open-loop controller using a hybrid systems framework. Moreover, the hybrid geometric controller guarantees robust asymptotic stability. Simulations validate the stability properties of the proposed hybrid geometric controller.