arXiv subjects

Adeel Akhtar

Publications and source records attributed to Adeel Akhtar.

13 recordsLinked to original sources

arXiv2026

Dual Path Following for an Aerial Manipulator Under State and Input Constraints via Transverse Feedback Linearization

This paper studies dual path following for a quadrotor carrying a three-degree-of-freedom arm. The center of mass of the combined system must follow a vehicle path and the end effector must follow a second path, while the arm respects its joint limits and avoids its singular postures. The vehicle and the arm are mechanically coupled, so one controller must handle both paths. With the center of mass and the end-effector path coordinates as outputs, and a dynamic extension of the thrust, the system admits a normal form without internal dynamics. We characterize the states where this normal form fails. We formulate the transverse feedback linearizing controller as a quadratic program that relaxes only the speed of the end effector along its path and imposes safety and actuator limits as inequality constraints. The program reduces to a scalar problem with a closed-form solution. If it remains strictly feasible for all time, the feedback is locally Lipschitz, both paths are invariant and attractive, and every constraint holds. In the physics engine Drake, the controller resolves a conflict between the two speed references by slowing the end effector while both paths remain invariant. By contrast, a cascade with a safety filter leaves both paths. Code and animations are available at https://gradslab.github.io/am_dual_pf-/

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arXiv2026

Intrinsic Path Following on $\mathsf{SO}(3)$

This paper studies intrinsic path following for fully actuated rigid-body attitude dynamics over a broad class of attitude paths, namely arbitrary smooth embedded closed curves on $\mathsf{SO}(3)$. We construct a tangential--transverse state decomposition on a tubular neighborhood using normal and pullback bundle geometry. The resulting second-order dynamics admit a smooth feedback transformation that independently assigns tangential and transverse covariant accelerations. A transverse proportional--derivative law renders the path-following manifold invariant and locally uniformly exponentially stable on an explicit forward-invariant neighborhood, while leaving the tangential dynamics freely assignable. This allows independent regulation of progression along the path, stabilization of a selected attitude on the path, or tracking of a time-varying motion. For a representative fiber path, the tubular coordinates are obtained in closed form up to the natural cut-locus obstruction, and simulations illustrate point stabilization and trajectory tracking.

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arXiv2026

Path Invariance of a Quadrotor System under Cyber Attacks with Theoretical Guarantees

This paper presents a path-following controller for a quadrotor system to guarantee safe maneuvers, in terms of forward path invariance, in the presence of cyber-physical attacks. We assume that an adversarial agent can control any one of the rotors through a false data injection (FDI) type of attack. A feedback controller is designed using transverse feedback linearization which guarantees that the system follows a class of smooth curves under FDI attacks. Our proposed controller is computationally efficient, with a closed-form analytical expression, that not only mitigates the effect of bounded malicious signal but also ensures mission success. We provide theoretical guarantees of forward path invariance under FDI attacks with realistic assumptions and demonstrate the effectiveness of our approach through simulation.

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arXiv2026

Modeling and Control of a Unicycle Robot on Manifolds

This paper studies path-following control for a kinematic unicycle evolving on a smooth manifold embedded in $\mathbb{R}^3$. The objective is to stabilize a one-dimensional path on the manifold without imposing a temporal parameterization, thereby distinguishing the problem from trajectory tracking. Working in a geometric framework, we propose two feedback laws: a static controller for prescribed nonzero translational speed and a dynamic controller based on a dynamic extension of the translational speed. By lifting the path to the unit tangent bundle of the surface, we characterize the corresponding path-following manifold and establish its control invariance. We then show that the proposed controllers render the path-following manifold locally exponentially stable under suitable regularity conditions. A numerical simulation illustrates the proposed controller. Code and animations are publicly available at https://gradslab.github.io/UnicycleOnManifolds/.

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arXiv2026

Quasi-Static Fault-Tolerant Feedback Control of a Quadrotor under Rotor Failure with Provable Safety Guarantees

This paper presents a nonlinear control law for a quadrotor unmanned aerial vehicle (UAV) under single-rotor failure that guarantees set stabilization via quasi-static feedback (QSF). Given a geometric curve in three-dimensional space, we characterize and stabilize the zero-dynamics manifold, also known as the path-following manifold, which represents all feasible motions along the path. Stabilizing this manifold ensures path-invariance: a UAV with a failed rotor initialized on the path with an appropriate orientation remains on the path for all future time. Furthermore, local exponential convergence to the manifold is guaranteed under certain conditions, implying that, under the stated assumptions, rotor failure during flight does not cause transverse deviation from the path. The proposed controller thus provides theoretical safety guarantees, which are validated through numerical experiments in the Drake physics-based simulation engine. The code is publicly available at https://gradslab.github.io/quasistatic-ftc/.

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arXiv2026

Decentralized Safe Path Following for Multiple Quadrotors on Intersecting Paths with Theoretical Guarantees

This paper studies decentralized safe path following for multiple quadrotors on intersecting paths, where safety requires both collision avoidance and strict adherence to pre-assigned routes. The proposed controller reformulates transverse feedback linearization as a constrained quadratic program with four equality constraints: two enforce convergence to and strict adherence to the path, and two prescribe the desired speed and heading. Safety is achieved by relaxing only the along-path speed constraint, while the path and heading constraints remain hard. Under the stated assumptions, and for admissible initial conditions, the controller guarantees that all agents converge to and thereafter follow their assigned paths, avoid collisions, and avoid attitude singularities. We evaluate the controller in the Drake physics engine on non-planar intersecting paths and compare it against two nominal-plus-safety-filter cascades. Code and additional results are available at https://gradslab.github.io/safe_multiquad_pf/.

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arXiv2026

On the Equivalence of Stochastic Control and Path Space Formulations for Schrödinger Bridges over Compact Connected Lie Groups

We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group. Using the geometric concepts of horizontal lift and stochastic anti-development, we derive a Girsanov-type change-of-measure result, and show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. Thus, the SBP is equivalently a path space relative entropy minimization problem subject to prescribed endpoint marginals.Our result has three useful implications. From an analytic viewpoint, the shown equivalence helps prove the existence and uniqueness of the SB. From a probabilistic viewpoint, it helps interpret the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the endpoint constraints. From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem and compute the optimal path measure. We illustrate the equivalence numerically on the torus $\mathbb{T}^2$. The code is publicly available at: https://github.com/gradslab/LargeDeviationSBP

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arXiv2026

Path-following Control of a Quadrotor using Quasi-Static Transverse Feedback Linearization

We propose a quasi-static transverse feedback linearization (QSTFL) controller for a quadrotor to follow a prescribed geometric path, rather than a time-parameterized trajectory. In contrast to existing dynamic-feedback approaches, the controller does not introduce additional controller states. The thrust input is computed algebraically from the current state, eliminating the need for thrust-derivative measurements and numerical integration. The proposed design renders the path-following manifold invariant, ensuring that trajectories initialized on the path remain on it for all future time, while simultaneously regulating tangential velocity and yaw. We establish a diffeomorphic coordinate transformation and prove local exponential stability of the path-following manifold. In addition, closed-form expressions are derived for the thrust and torque inputs. Compared with dynamic-feedback constructions, the controller requires inversion of only a $3\times 3$ decoupling matrix rather than a $4\times 4$ one, leading to a simpler control law and reduced computational complexity. Numerical simulations demonstrate the effectiveness of the proposed method. Code and animations are publicly available at \footnotesize{\texttt{\href{https://gitlab.com/a5akhtar/quasistatic-tfl-uav/}{https://gitlab.com/a5akhtar/quasistatic-tfl-uav/}}}.

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arXiv2026

A Relaxed Quadratic-Program-based Framework for Trajectory Tracking of Unicycle Robots with Singularity Avoidance

Dynamic feedback linearization (DFL) is a classical technique for trajectory tracking of unicycle-type mobile robots, but the resulting DFL-based controller becomes singular when the linear velocity vanishes, rendering standard DFL-based controllers unsuitable for stop-and-reverse maneuvers. This paper proposes a quadratic-program (QP)-based optimal control framework that avoids this singularity, while establishing local Lipschitz continuity of the resulting feedback law. Our approach reformulates the DFL constraints as an equality-constrained QP with a slack variable, ensuring feasibility for all states and reference signals, including at points where the robot's velocity vanishes. By introducing slack variables and tunable parameters, we demonstrate that the singular configuration can be avoided for a large class of reference trajectories. The effectiveness of the proposed approach for trajectory tracking is demonstrated through ROS 2-Gazebo simulations on a TurtleBot3 Waffle robot. The code is available at https://gradslab.github.io/DFL_QP_Unicycle/

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arXiv2026

A Geometric Solution of the Schrödinger Bridge Problem on $\mathsf{SO}(2)$ via Stochastic Optimal Control

We present a geometric coordinate-free solution to the isotropic Schrödinger bridge problem (SBP) for the kinematic equation on the Lie group $\mathsf{SO}(2)$. We consider the angular velocity of the system as the control input and assume that the given initial and terminal state probability density functions defined on $\mathsf{SO}(2)$ in our SBP are continuous and strictly positive. We solve the SBP by proving the existence and uniqueness of a solution to the so-called Schrödinger system of equations on $\mathsf{SO}(2)$, by showing that a fixed-point recursion is contractive in a complete metric space with respect to the Hilbert's projective metric. The geometric controller thus designed only uses the intrinsic geometric structure of $\mathsf{SO}(2)$ and does not embed it in the Euclidean plane to achieve the optimal density control. The numerical simulation verifies the validity of the theoretical construction of the Schrödinger bridge. The code and animations are publicly available at \texttt{\href{https://gitlab.com/a5akhtar/sbp}{https://gitlab.com/a5akhtar/sbp}}.

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arXiv2026

Learning Neural Maximal Lyapunov Functions on $\mathsf{SO}(n)$

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 $\mathsf{SO}(n)$. 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.

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arXiv2026

Schrödinger Bridge Over A Compact Connected Lie Group

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 $\mathsf{SO}(2)$ and $\mathsf{SO}(3)$. The codes and animations are publicly available at https://github.com/gradslab/SbpLieGroups.git .

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arXiv2025

A Safe Hybrid Control Framework for Car-like Robot with Guaranteed Global Path-Invariance using a Control Barrier Function

This work proposes a hybrid framework for car-like robots with obstacle avoidance, global convergence, and safety, where safety is interpreted as path invariance, namely, once the robot converges to the path, it never leaves the path. Given a priori obstacle-free feasible path where obstacles can be around the path, the task is to avoid obstacles while reaching the path and then staying on the path without leaving it. The problem is solved in two stages. Firstly, we define a ``tight'' obstacle-free neighborhood along the path and design a local controller to ensure convergence to the path and path invariance. The control barrier function technology is involved in the control design to steer the system away from its singularity points, where the local path invariant controller is not defined. Secondly, we design a hybrid control framework that integrates this local path-invariant controller with any global tracking controller from the existing literature without path invariance guarantee, ensuring convergence from any position to the desired path, namely, global convergence. This framework guarantees path invariance and robustness to sensor noise. Detailed simulation results affirm the effectiveness of the proposed scheme.

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