This paper proposes global accelerated nonconvex geometric (GANG) optimization algorithms for optimizing a class of nonconvex functions on the compact Lie group SO(3). Nonconvex optimization is a challenging problem because the objective function may have multiple critical points, including saddle points. We propose two accelerated geometric algorithms to escape maxima and saddle points using random perturbations. The first algorithm uses the value of the Hessian of the objective function and random perturbations to escape the undesired critical points. In contrast, the second algorithm uses only the gradient information and random perturbations to escape maxima and saddle points. The efficacy of these geometric algorithms is verified in simulations.
@inproceedings{AkhSan2023,author={Akhtar, Adeel and Sanfelice, Ricardo G.},booktitle={2023 American Control Conference (ACC),},title={Global Accelerated Nonconvex Geometric Optimization Methods on SO(3)},html={https://ieeexplore.ieee.org/abstract/document/10155919},year={2023},volume={},number={},pages={3554-3559},publisher={IEEE},abbr={IEEE ACC},bibtex_show={true},selected={true},doi={10.23919/ACC55779.2023.10155919}}
IEEE TIE
Path Following of a Quadrotor With a Cable-Suspended Payload
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.
@article{AkhSalSha2023,author={Akhtar, Adeel and Saleem, Sajid and Shan, Jinjun},journal={IEEE Transactions on Industrial Electronics,},title={Path Following of a Quadrotor With a Cable-Suspended Payload},year={2023},volume={70},number={2},pages={1646-1654},publisher={IEEE},abbr={IEEE TIE},html={https://ieeexplore.ieee.org/abstract/document/9992659},bibtex_show={true},doi={10.1109/TIE.2022.3167130}}
2022
IEEE CDC
Hybrid Geometric Controllers for Fully-Actuated Left-invariant Systems on Matrix Lie Groups
Akhtar, Adeel,
and Sanfelice, Ricardo G.
In 2022 IEEE 61st Conference on Decision and Control (CDC)
2022
@inproceedings{AkhSan2022,author={Akhtar, Adeel and Sanfelice, Ricardo G.},booktitle={2022 IEEE 61st Conference on Decision and Control (CDC)},title={Hybrid Geometric Controllers for Fully-Actuated Left-invariant Systems on Matrix Lie Groups},year={2022},volume={},number={},pages={1855-1860},abbr={IEEE CDC},bibtex_show={true},selected={true},doi={10.1109/CDC51059.2022.9992659}}
IEEE ACC
A Class of Hybrid Geometric Controllers for Robust Global Asymptotic
Stabilization on S^1
This paper proposes a hybrid geometric control scheme for the classical problem of globally stabilizing a pointmass 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.
@article{AkhRic2022,title={A Class of Hybrid Geometric Controllers for Robust Global Asymptotic
Stabilization on S^1},author={Akhtar, A. and Sanfelice, R.},journal={IEEE American Control Conference (ACC),},year={2022},doi={10.23919/ACC53348.2022.9867666},publisher={IEEE},selected={true},abbr={IEEE ACC},bibtex_show={true}}
IEEE ACC
Coordinated Path Following for a Class of Underactuated Multi-agent System using Nested Set Stabilization
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.
@article{Akh2022,title={Coordinated Path Following for a Class of Underactuated Multi-agent System using Nested Set Stabilization},author={Akhtar, A.},journal={IEEE American Control Conference (ACC),},year={2022},publisher={IEEE},selected={true},doi={10.23919/ACC53348.2022.9867249},abbr={IEEE ACC},bibtex_show={true}}
2021
IEEE TCST
Path Invariant Controllers for a UAV with a Cable-suspended Payload using a Global Parameterization
A. Akhtar, S. Saleem,
and Shan, J.
IEEE Transactions on Control Systems Technology,
2021
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 the 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.
@article{AkhSalSha2021,title={Path Invariant Controllers for a UAV with a Cable-suspended Payload using a Global Parameterization},author={A. Akhtar, S. Saleem and Shan, J.},volume={30},number={5},pages={2002-2017},journal={IEEE Transactions on Control Systems Technology,},year={2021},publisher={IEEE},doi={10.1109/TCST.2021.3133693},abbr={IEEE TCST},bibtex_show={true},selected={true}}
Tech. Report
Technical Report: Path Following of a Quadrotor with a Cable-Suspended Payload
Aerial robots and drone-based payload delivery is in the spotlight and is considered revolutionary for the logistics and transportation sectors. In most use cases, the payload is required to precisely follow a given path for safe and secure operation in a cluttered environment. This work addresses the problem of designing a path-following controller for a point-mass payload tethered to a quadrotor. Specifically, we design a smooth
dynamic feedback controller that forces the suspended load to converge and follow a
large class of both closed and non-closed embedded curves. Using a local representation
of the dynamical system consisting of a quadrotor and the cable-suspended payload, we
show that the system has a well-defined vector relative degree. We treat the given curve
as a smooth manifold and then use set stabilization to find the maximal control invariant
manifold. The resulting controller guarantees that once the system reaches the path,
it stays on the path indefinitely. We demonstrate the performance of the proposed
controller through extensive simulations with practical sensor noise and parametric
uncertainties. Moreover, successful real-world experimental validation of the proposed
controller is demonstrated on a Quanser QDrone UAV platform with a cable-suspended
payload
@techreport{AkhSalSha2021_TR,author={Akhtar, A. and Saleem, S. and Shan, J.},title={Technical Report: Path Following of a Quadrotor with a Cable-Suspended Payload},group={SDCN},abbr={Tech. Report},year={2021},institution={York University, Toronto},month={},date-added={2021},pdf={Technical_report_quadload_2021.pdf},date-modified={2021},bibtex_show={true}}
In this article, we propose a novel family of Lie algebra valued functions F_R on 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 FR form a controller class CR . This novel controller class CR contains both local and almost-global asymptotically stable controllers, and this article presents geometric stability results of the whole controller class C_R.
@article{AkhWas2020,title={Controller Class for Rigid Body Tracking on SO(3)},author={Akhtar, A. and Waslander, S.},journal={IEEE Transactions on Automatic Control,},%volume={34},%number = {4},doi = {doi: 10.1109/TAC.2020.3008295.},year = {2020},publisher = {IEEE},abbr = {IEEE TAC},html = {https://ieeexplore.ieee.org/document/9137655},bibtex_show = {true},selected = {true}}
IEEE Access
Path Following for a Class of Underactuated Systems Using Global Parameterization
A large number of both aerial and underwater mobile robots fall in the category of underactuated systems that are defined on a manifold, which is not isomorphic to Euclidean space. Traditional approaches to designing controllers for such systems include geometric approaches and local coordinate-based representations. In this paper, we propose a global parameterization of the special orthogonal group, denoted by SO(3), to design path-following controllers for underactuated systems. In particular, we present a nine-dimensional representation of SO(3) that leads to controllers achieving path-invariance for a large class of both closed and non-closed embedded curves. On the one hand, this over-parameterization leads to a simple set of differential equations and provides a global non-ambiguous representation of systems as compared to other local or minimal parametric approaches. On the other hand, this over-parameterization also leads to uncontrolled internal dynamics, which we prove to be bounded and stable. The proposed controller, when applied to a quadrotor system, is capable of recovering the system from challenging situations such as initial upside-down orientation and also capable of performing multiple flips.
@article{AkhSalWas2020,abbr={IEEE Access},author={{Akhtar}, A. and {Saleem}, S. and {Waslander}, S. L.},journal={IEEE Access,},title={Path Following for a Class of Underactuated Systems Using Global Parameterization},year={2020},volume={8},number={},pages={34737-34749},keywords={Manifolds;Aerospace electronics;Vehicle dynamics;Control design;Tools;Robots;Quaternions;Feedback linearization;nonlinear control;path following;quadrotor;underactuated system},doi={10.1109/ACCESS.2020.2974153},issn={2169-3536},month={},html={https://ieeexplore.ieee.org/document/8999577},bibtex_show={true}}
2019
IEEE ACC
Feedback Linearizing Controllers on SO(3) using a Global Parametrization
Akhtar, Adeel,
Saleem, Sajid,
and Waslander, Steven L.
We present a methodology for studying the stabilization problem of a fully-actuated rotating rigid body. Since a rigid body attitude is represented by a rotation matrix in three dimensions, we exploit this fact and use each element of the rotation matrix as a parameter. This nine-parameter representation is global as well as unique, and results in a simplified set of nonlinear differential equations. We apply feedback linearization to design both local and almost global controllers. We also propose two novel definitions of feedback linearization functions, and prove that they lead to a well-defined vector relative degree and, as a result, almost-globally and locally stable controllers with bounded internal states. Using the proposed methodology, we present detailed examples of two such functions, demonstrating stabilization performance for each resulting controller on a rigid body system.
@inproceedings{AkhSalWas2021,author={Akhtar, Adeel and Saleem, Sajid and Waslander, Steven L.},booktitle={American Control Conference (ACC),},title={Feedback Linearizing Controllers on SO(3) using a Global Parametrization},year={2019},volume={},number={},pages={1441-1446},doi={10.23919/ACC45564.2020.9147963},bibtex_show={true},abbr={IEEE ACC},html={https://ieeexplore.ieee.org/document/9147963}}
2018
Ph.D Thesis
Nonlinear and Geometric Controllers for Rigid Body Vehicles
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.
@article{AkhWasNie2013Journal,abbr={IEEE T-RO},author={Akhtar, A. and Nielsen, C. and Waslander, S. L.},journal={IEEE Transactions on Robotics,},title={Path Following Using Dynamic Transverse Feedback Linearization for Car-Like Robots},year={2015},volume={31},number={2},pages={269-279},keywords={closed loop systems;control system synthesis;feedback;mobile robots;path planning;robot kinematics;Chameleon R100 Ackermann steering robot;car-like mobile robots;closed-loop system;dynamic extension;dynamic transverse feedback linearization;kinematic model;path following;path-following controller design;Approximation methods;Kinematics;Manifolds;Polynomials;Robot kinematics;Vectors;Control design;mobile robots;motion control;nonlinear control systems;nonlinear dynamical systems;nonlinear systems;robot control},doi={10.1109/TRO.2015.2395711},issn={1552-3098},month=apr,html={https://ieeexplore.ieee.org/document/7042843},bibtex_show={true},video={https://www.youtube.com/watch?v=6FEgU0kYqlM}}
2013
IEEE CDC
Fault tolerant path following for a quadrotor
Akhtar, A.,
Waslander, S.L.,
and Nielsen, C.
In 52nd IEEE Conference on Decision and Control (CDC),
2013
This paper presents a path following controller for a quadrotor UAV experiencing a single rotor failure. A smooth, dynamic feedback control law is proposed that allows the quadrotor to follow both closed and non-closed embedded curves while maintaining a desired velocity profile along the path when one out of four motors is completely disabled due to failure. The nonlinear model of the quadrotor is transformed into a partially linear model by a coordinate and feedback transformation. A path following controller is designed for transformed system that guarantees invariance of the path. The uncontrolled nonlinear portion of the dynamics (internal dynamics) are shown to be bounded.
@inproceedings{AkhWasNie2013,abbr={IEEE CDC},author={Akhtar, A. and Waslander, S.L. and Nielsen, C.},booktitle={52nd IEEE Conference on Decision and Control (CDC),},title={Fault tolerant path following for a quadrotor},year={2013},month=dec,pages={847-852},keywords={autonomous aerial vehicles;control system synthesis;fault tolerant control;feedback;mobile robots;nonlinear control systems;path planning;autonomous aerial vehicles;closed embedded curve;fault tolerant path following controller design;feedback transformation;nonclosed embedded curve;nonlinear model;partially linear model;quadrotor UAV;single rotor failure;smooth dynamic feedback control law;uncontrolled nonlinear portion;velocity profile;Fault tolerance;Fault tolerant systems;Rotors;Trajectory;Vectors;Vehicle dynamics;Vehicles},doi={10.1109/CDC.2013.6759988},issn={0743-1546},bibtex_show={true},html={https://ieeexplore.ieee.org/document/6759988}}
2012
AIAA
Relative Position-Based Visual Servoing Control for Quadrotors
Tribou, M.,
Akhtar, A.,
and Waslander, S. L.
In American Institute of Aeronautics and Astronautics (AIAA) Guidance, Navigation, and Control Conference,
2012
This paper presents a position-based visual servoing control system that allows a quadrotor aerial robot with a single onboard camera to track a four degree of freedom relative pose trajectory with respect to a free-moving unknown target object composed of visual point features. The proposed vision system is able to estimate both the six degree of freedom relative motion and the target object model structure using only the images captured from the camera and recover scale using a minimal set of inter-feature point distances. An inner-loop attitude controller is used with an outer-loop position trajectory tracking control law to follow a relative trajectory with respect to the target object, while both bodies move in the inertial frame. The performance of the proposed algorithms are evaluated in simulation.copy; 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
@inproceedings{TriAkhWas2012,abbr={AIAA},bibtex_show={true},author={Tribou, M. and Akhtar, A. and Waslander, S. L.},booktitle={American Institute of Aeronautics and Astronautics (AIAA) Guidance, Navigation, and Control Conference,},title={Relative Position-Based Visual Servoing Control for Quadrotors},year={2012},month=aug,doi={10.2514/6.2012-5049},issn={9781622763238},html={https://arc.aiaa.org/doi/10.2514/6.2012-5049}}
IEEE CDC
Path following for a quadrotor using dynamic extension and transverse feedback linearization
Akhtar, Adeel,
Waslander, Steven L.,
and Nielsen, Christopher
In 51st IEEE Conference on Decision and Control (CDC),
2012
This work presents a path following controller for a quadrotor vehicle. A smooth, dynamic, feedback controller is designed that allows the quadrotor to follow both closed and non-closed embedded curves while maintaining a desired speed, a desired acceleration or while stabilizing desired points along the curves. The nonlinear dynamic model of the quadrotor is transformed into a linear system via a coordinate and feedback linearization transformation. Once transformed, a path following controller is designed that guarantees invariance of the path while enforcing the desired motion along the path.
@inproceedings{AkhWasNie2012,abbr={IEEE CDC},author={Akhtar, Adeel and Waslander, Steven L. and Nielsen, Christopher},booktitle={51st IEEE Conference on Decision and Control (CDC),},title={Path following for a quadrotor using dynamic extension and transverse feedback linearization},year={2012},month=dec,pages={3551 -3556},keywords={Acceleration;Dynamics;Heuristic algorithms;Level set;Manifolds;Nonlinear dynamical systems;Vectors;},doi={10.1109/CDC.2012.6425945},issn={0743-1546},bibtex_show={true},html={https://ieeexplore.ieee.org/document/6425945}}
2011
IEEE CDC
Path following for a car-like robot using transverse feedback linearization
and tangential dynamic extension
Akhtar, A.,
and Nielsen, C.
In 50th IEEE Conference on Decision and Control and European Control Conference, (CDC)-ECC,
2011
@inproceedings{AkhNie2011,abbr={IEEE CDC},bibtex_show={true},author={Akhtar, A. and Nielsen, C.},booktitle={50th IEEE Conference on Decision and Control and European Control Conference, (CDC)-ECC,},title={Path following for a car-like robot using transverse feedback linearization
and tangential dynamic extension},year={2011},month=dec,volume={},number={},pages={7949 -7979},html={https://ieeexplore.ieee.org/document/6161413},pdf={AkhWasNie2013Journal.pdf},video={https://www.youtube.com/watch?v=6FEgU0kYqlM},keywords={finite time stabilization;path following controller;planar vertical take-off and landing aircraft;set stabilization approach;static feedback controller;transverse feedback linearization;virtual constraints;aircraft;feedback;path planning;stability;},issn={0191-2216}}
MS. Thesis
Dynamic path following controllers
for planar mobile robots