James Richard Forbes

Nonlinear Estimator Design on the Special Orthogonal Group Using Vector Measurements Directly

The convergence properties of popular nonlinear attitude estimators can be traced to the choice of an attitude error function. This paper considers a nonlinear deterministic direction cosine matrix estimator whose form is derived from an alternate attitude error function. While the resulting estimator shares several properties with those previously presented in the literature, the careful selection of an attitude error function results in an estimator with superior convergence properties. The attitude estimate is propagated using a rate gyroscope measurement and corrected using two or more vector measurements. Simulation and experimental results are presented that highlight the desirable properties of the proposed estimator.

Magnetic Attitude Control of a Flexible Satellite

1 M.A.Sc. Candidate, University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, Ontario, Canada, M3H 5T6, and Student Member AIAA. 2 Assistant Professor, Ryerson University, Department of Aerospace Engineering, 350 Victoria Street, Toronto, ON, Canada, M5B 2K3, and Senior Member AIAA. 3 Assistant Professor, McGill University, Department of Mechanical Engineering and Centre for Intelligent Machines, 817 Sherbrooke Street West, Montreal, QC, Canada, H3A 0C3, and Member AIAA. 4 Associate Professor, University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, Ontario, Canada, M3H 5T6, and Senior Member AIAA. 5 Professor, University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, Ontario, Canada, M3H 5T6, and Associate Fellow AIAA. 6 Research Scientist, Canadian Space Agency, Space Science and Technology, 6767 route de l’Aeroport, St.Hubert,QC, J3Y 8Y9, and AIAA member.

Dynamic Modeling and Passivity-Based Control of a Single Degree of Freedom Cable-Actuated System

In this paper, a lumped-mass dynamic model of a single degree of freedom cable-actuated system is derived, and passivity-based control is considered. The dynamic model developed takes into consideration the changing cable stiffness and mass as the cable is wrapped around a winch. In addition, the change in the winch inertia as the cable is wrapped around the winch is modeled. It is assumed that the mass of the payload is much greater than the mass of the cables and the equivalent mass of the winches, which allows for an approximation where the rigid dynamics can be decoupled from the elastic dynamics of the system. This approximation enables the definition of a modified input torque and modified output rate, allowing the establishment of passive input-output mappings. Passivity-based controllers are investigated, shown to render the closed-loop system input-output stable, and tested in simulation.

Dynamic Modeling and Noncollocated Control of a Flexible Planar Cable-Driven Manipulator

This paper investigates the dynamic modeling and passivity-based control of a planar cable-actuated system. This system is modeled using a lumped-mass method that explicitly considers the change in cable stiffness and winch inertia that occurs when the cables are wound around their respective winches. In order to simplify the modeling process, each cable is modeled individually and then constrained to the other cables. Exploiting the fact that the payload is much more massive than the cables allows the definition of a modified output called the μ -tip rate. Coupling the μ-tip rate with a modified input realizes the definition of a passive input-output map. The two degrees of freedom of the system are controlled by four winches. This overactuation is simplified by employing a set of load-sharing parameters that effectively reduce four inputs to two. The performance and robustness of the controllers are evaluated in the simulation.

Passivity-Based Attitude Control on the Special Orthogonal Group of Rigid-Body Rotations

Set-point regulation and tracking control of a rigid body is considered. The rigid body, which could be a spacecraft, underwater vehicle, or unmanned aerial vehicle, is equipped with various sensors, including those that provide unit-length vector measurements. At no point is the rotation matrix associated with the rigid-body’s orientation parameterized. The control algorithms developed in this paper are posed directly on the special orthogonal group of rigid-body rotations SO(3). The set-point controller presented is composed of a proportional control term and an angular velocity control term. The proportional control term is a function of the vector measurements and a set of desired vector measurements that are used to compute the orientation error of the rigid body. Passive systems theory is used to motivate the use of a strictly positive real angular velocity controller. The set-point regulator is robust to modeling errors associated with the mass distribution of the body. Tracking control is also con...

Design of optimal strictly positive real controllers using numerical optimization for the control of flexible robotic systems

Abstract The design of optimal strictly positive real (SPR) controllers using numerical optimization is considered. We focus on how to parameterize the SPR controllers being optimized and the effect of parameterization. Minimization of the closed-loop H 2 - norm is the optimization objective function. Various single-input single-output and multi-input multi-output controller parameterizations using transfer functions/matrices and state–space equations are considered. Depending on the controller form, constraints are enforced (i) using simple inequalities guaranteeing SPRness, (ii) in the frequency domain or, (iii) by implementing the Kalman–Yakubovich–Popov Lemma. None of the parameterizations we consider foster an observer-based controller structure. Simulated control of a single-link and a two-link flexible manipulators demonstrates the effectiveness of our proposed controller optimization formulations.

Conic-sector-based control to circumvent passivity violations

This paper explores the use of the Conic Sector Theorem for both stability analysis and controller design. Ensuring input-output stability of plants experiencing passivity violations is the motivation behind this work. Given a previously designed controller and plant that has experienced a (partially unknown) passivity violation, a novel sector bound selection procedure is presented. This procedure can be used to assess input-output stability of the violated plant and original controller via the Conic Sector Theorem. Should input-output stability not be ensured, two original controller synthesis methods are suggested: one is designed to mimic the H2 controller, and the other is inspired by strictly positive real controller synthesis. Both methods guarantee input-output stability by selecting controllers within appropriate conic sectors, and involve only the evaluation of readily solvable linear matrix inequalities and algebraic Riccati inequalities. A numerical simulation is provided as a proof of concept.

Spacecraft constrained attitude control using positively invariant constraint admissible sets on SO(3) × ℝ 3

This paper presents a constrained attitude control approach for performing spacecraft reorientation maneuvers that maintain specified body vectors within inclusion zones and out of exclusion zones, while respecting control authority limits. The controller uses a supervisory switching strategy with an inner-loop Lyapunov SO(3)-based controller and an outer-loop set-point guidance. A virtual net of orientation equilibria covering SO(3) is introduced, and positively invariant constraint admissible sets on SO(3) × ℝ3 of the inner loop controller are constructed to determine if equilibria are connected by a feasible trajectory. Optimization procedures to maximize the size of the positively-invariant sets are discussed. Graph search is used in the outer-loop to compute the set-point sequence leading from an initial orientation to a final orientation that rigorously enforces constraints. The proposed methodology reduces the search space of possible attitude maneuver solutions, and has computational and implementation simplicity. Numerical simulation results are reported to illustrate the performance of the proposed constrained attitude control methodology.

Continuous-time norm-constrained Kalman filtering

This paper considers continuous-time state estimation when part of the state estimate or the entire state estimate is norm-constrained. In the former case continuous-time state estimation is considered by posing a constrained optimization problem. The optimization problem can be broken up into two separate optimization problems, one which solves for the optimal observer gain associated with the unconstrained state estimates, while the other solves for the optimal observer gain associated with the constrained state estimates. The optimal constrained state estimate is found by projecting the time derivative of an unconstrained estimate onto the tangent space associated with the norm constraint. The special case where the entire state estimate is norm-constrained is briefly discussed. The utility of the filtering results developed are highlighted through a spacecraft attitude estimation example. Numerical simulation results are included.

Synthesis of Optimal Finite-Frequency Controllers Able to Accommodate Passivity Violations

In this paper, we explore the relationship between the hybrid passivity/finite-gain systems framework and the generalized Kalman-Yakubovich-Popov (GKYP) lemma. In particular, we investigate how to optimally design finite-frequency (FF) controllers that possess strictly positive real (SPR) properties over a low-frequency range and bounded real (BR) properties over a high-frequency range. Such FF SPR/BR controllers will be used to control systems that have experienced a passivity violation. We first review the hybrid passive/finite-gain systems framework and how linear time-invariant hybrid passive/finite-gain systems relate to systems with low-frequency FF positive real (PR) or SPR properties, and high-frequency FF BR properties as characterized by the GKYP lemma. Optimal design of FF SPR/BR controllers is considered next. A convex optimization problem constrained by a set of linear matrix inequalities is posed where constraints are imposed using various forms of the GKYP lemma, yielding optimal FF SPR/BR controllers. The FF SPR/BR controllers are optimal in that they approximate the traditional H2 control solution. Finally, FF SPR/BR controllers are used within a gain-scheduling architecture to control a two-link flexible manipulator. Experimental results successfully demonstrate closed-loop stability and good closed-loop performance.