This paper provides global formulations of Lagrangian and Hamiltonian variational dynamics evolvi... more This paper provides global formulations of Lagrangian and Hamiltonian variational dynamics evolving on the product of an arbitrary number of two-spheres. Four types of Euler-Lagrange equations and Hamilton's equations are developed in a coordinate-free fashion on two-spheres, without relying on local parameterizations that may lead to singularities and cumbersome equations of motion. The proposed intrinsic formulations of Lagrangian and Hamiltonian dynamics are novel in that they incorporate the geometry of two-spheres, resulting in equations of motion that are expressed compactly, and they are useful in analysis and computation of the global dynamics.
International Journal of Robust and Nonlinear Control, 2007
The 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degre... more The 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom; it is acted on by gravity and it is fully actuated by control forces. The 3D pendulum has two disjoint equilibrium manifolds, namely a hanging equilibrium manifold and an inverted equilibrium manifold. This paper shows that a controller based on angular velocity feedback can be used to asymptotically stabilize the hanging equilibrium manifold of the 3D pendulum. Lyapunov analysis and nonlinear geometric methods are used to assess the global closed-loop properties. We explicitly construct compact sets that lie in the domain of attraction of the hanging equilibrium of the closed-loop. Finally, this controller is shown to achieve almost global asymptotic stability of the hanging equilibrium manifold. An invariant manifold of the closed-loop that converges to the inverted equilibrium manifold is identified.
A high fidelity model is developed for an elastic string pendulum, one end of which is attached t... more A high fidelity model is developed for an elastic string pendulum, one end of which is attached to a rigid body while the other end is attached to an inertially fixed reel mechanism which allows the unstretched length of the string to be dynamically varied. The string is assumed to have distributed mass and elasticity that permits axial deformations. The rigid body is attached to the string at an arbitrary point, and the resulting string pendulum system exhibits nontrivial coupling between the elastic wave propagation in the string and the rigid body dynamics. Variational methods are used to develop coupled ordinary and partial differential equations of motion. Computational methods, referred to as Lie group variational integrators, are then developed, based on a finite element approximation and the use of variational methods in a discrete-time setting to obtain discrete-time equations T. Lee ( )
An efficient and accurate computational approach is proposed for optimal attitude control of a ri... more An efficient and accurate computational approach is proposed for optimal attitude control of a rigid body. The problem is formulated directly as a discrete time optimization problem using a Lie group variational integrator. Discrete necessary conditions for optimality are derived, and an efficient computational approach is proposed to solve the resulting two point boundary value problem. The use of geometrically exact computations on SO(3) guarantees that this optimal control approach has excellent convergence properties even for highly nonlinear large angle attitude maneuvers. Numerical results are presented for attitude maneuvers of a 3D pendulum and a spacecraft in a circular orbit.
International Journal of Non-linear Mechanics, Jul 1, 2003
We consider two linearly coupled masses, where one mass can have inelastic impacts with a fixed, ... more We consider two linearly coupled masses, where one mass can have inelastic impacts with a fixed, rigid stop. This leads to the study of a two degree of freedom, piecewise linear, frictionless, unforced, constrained mechanical system. The system is governed by three types of dynamics: coupled harmonic oscillation, simple harmonic motion and discrete rebounds. Energy is dissipated discontinuously in discrete amounts, through impacts with the stop. We prove the existence of a nonzero measure set of orbits that lead to infinite impacts with the stop in a finite time. We show how to modify the mathematical model so that forward existence and uniqueness of solutions for all time is guaranteed. Existence of hybrid periodic orbits is shown. A geometrical interpretation of the dynamics based on action coordinates is used to visualize numerical simulation results for the asymptotic dynamics.
Proceedings of the ... American Control Conference, Jul 1, 2007
A 3D pendulum implemented using a triaxial air bearing system is proposed for Earth-based testing... more A 3D pendulum implemented using a triaxial air bearing system is proposed for Earth-based testing of orbiting spacecraft attitude dynamics and closed loop systems. This proposal is based on prior research on attitude dynamics and control of orbiting spacecraft, attitude dynamics and control of the 3D pendulum, and an experimental implementation of the 3D pendulum in our laboratory, referred to as the Triaxial Attitude Control Testbed (TACT). These research themes are integrated to assess the strengths and weaknesses of such an Earth-based testbed for spacecraft attitude dynamics and control hardware and software components. Several different cases, based on the importance of orbital effects and gravitational effects, are analyzed.
A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees... more A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. Symmetry assumptions are shown to lead to the planar 1D pendulum and to the spherical 2D pendulum models as special cases. The case where the rigid body is asymmetric and the center of mass is distinct from the pivot location leads to the 3D pendulum. Full and reduced 3D pendulum models are introduced and used to study important features of the nonlinear dynamics: conserved quantities, equilibria, invariant manifolds, local dynamics near equilibria and invariant manifolds, and the presence of chaotic motions. These results demonstrate the rich and complex dynamics of the 3D pendulum.
Mathematical and Computer Modelling of Dynamical Systems, Jun 1, 2003
The Triaxial Attitude Control Testbed has been developed as part of a research program at the Uni... more The Triaxial Attitude Control Testbed has been developed as part of a research program at the University of Michigan on multibody rotational dynamics and control. In this paper, equations of motion are derived and presented in various forms. Actuation mechanisms are incorporated into the models; these include fan actuators, reaction wheel actuators and proof mass actuators that are £xed to the triaxial base body. The models also allow incorporation of unactuated auxiliary bodies that are constrained to move relative to the triaxial base body. The models expose the dynamic coupling between the rotational motion of the triaxial base body, the relative or shape motion of the unactuated auxiliary degrees of freedom, and dynamics associated with actuation mechanisms. Many different model simpli£cations and approximations are developed. Control models for the triaxial attitude control testbed are formulated that re¤ect speci£c assumptions.
A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees... more A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. 3D pendulum dynamics have been much studied in integrable cases that arise when certain physical symmetry assumptions are made. This paper treats the nonintegrable case of the 3D pendulum dynamics when the rigid body is asymmetric and the center of mass is distinct from the pivot location. 3D pendulum full and reduced models are introduced and used to study important features of the nonlinear dynam-Communicated by R. Sepulchre.
A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees... more A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. 3D pendulum dynamics have been much studied in integrable cases that arise when certain physical symmetry assumptions are made. This paper treats the nonintegrable case of the 3D pendulum dynamics when the rigid body is asymmetric and the center of mass is distinct from the pivot location. 3D pendulum full and reduced models are introduced and used to study important features of the nonlinear dynam-4 J Nonlinear Sci (2011) 21: 3-32 ics: conserved quantities, equilibria, relative equilibria, invariant manifolds, local dynamics, and presence of chaotic motions. The paper provides a unified treatment of the 3D pendulum dynamics that includes prior results and new results expressed in the framework of geometric mechanics. These results demonstrate the rich and complex dynamics of the 3D pendulum.
... He received a B.Sc (Eng.) degree in Aeronautical Engineering from Punjab Engineering College,... more ... He received a B.Sc (Eng.) degree in Aeronautical Engineering from Punjab Engineering College, Chandigarh, India, in 1982, and an MSE degree in Aerospace Engineering from the University of Michigan, Ann Arbor, in 1984, where he is currently at the Department of ...
Abstract. An efficient and accurate computational approach is pro-posed for a nonconvex optimal a... more Abstract. An efficient and accurate computational approach is pro-posed for a nonconvex optimal attitude control for a rigid body. The problem is formulated directly as a discrete time optimization prob-lem using a Lie group variational integrator. Discrete time necessary conditions for ...
International Journal of Robust and Nonlinear Control, 2000
We study a control problem for a special class of underactuated mechanical systems, namely for me... more We study a control problem for a special class of underactuated mechanical systems, namely for mechanical systems with several directly actuated degrees of freedom and a single unactuated degree of freedom that must be controlled through the system coupling. Speci"c assumptions are introduced that de"ne this class, which includes many important models of mechanical system examples. The main result of the paper is the construction of a discontinuous nonlinear feedback controller for which the closed loop equilibrium at the origin is made &globally attractive'. The control construction approach is introduced in detail, and a proof of attractiveness is presented. The results are applied to control of the planar motion of a rigid body with a single unactuated internal degree of freedom.
This paper provides global formulations of Lagrangian and Hamiltonian variational dynamics evolvi... more This paper provides global formulations of Lagrangian and Hamiltonian variational dynamics evolving on the product of an arbitrary number of two-spheres. Four types of Euler-Lagrange equations and Hamilton's equations are developed in a coordinate-free fashion on two-spheres, without relying on local parameterizations that may lead to singularities and cumbersome equations of motion. The proposed intrinsic formulations of Lagrangian and Hamiltonian dynamics are novel in that they incorporate the geometry of two-spheres, resulting in equations of motion that are expressed compactly, and they are useful in analysis and computation of the global dynamics.
International Journal of Robust and Nonlinear Control, 2007
The 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degre... more The 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom; it is acted on by gravity and it is fully actuated by control forces. The 3D pendulum has two disjoint equilibrium manifolds, namely a hanging equilibrium manifold and an inverted equilibrium manifold. This paper shows that a controller based on angular velocity feedback can be used to asymptotically stabilize the hanging equilibrium manifold of the 3D pendulum. Lyapunov analysis and nonlinear geometric methods are used to assess the global closed-loop properties. We explicitly construct compact sets that lie in the domain of attraction of the hanging equilibrium of the closed-loop. Finally, this controller is shown to achieve almost global asymptotic stability of the hanging equilibrium manifold. An invariant manifold of the closed-loop that converges to the inverted equilibrium manifold is identified.
A high fidelity model is developed for an elastic string pendulum, one end of which is attached t... more A high fidelity model is developed for an elastic string pendulum, one end of which is attached to a rigid body while the other end is attached to an inertially fixed reel mechanism which allows the unstretched length of the string to be dynamically varied. The string is assumed to have distributed mass and elasticity that permits axial deformations. The rigid body is attached to the string at an arbitrary point, and the resulting string pendulum system exhibits nontrivial coupling between the elastic wave propagation in the string and the rigid body dynamics. Variational methods are used to develop coupled ordinary and partial differential equations of motion. Computational methods, referred to as Lie group variational integrators, are then developed, based on a finite element approximation and the use of variational methods in a discrete-time setting to obtain discrete-time equations T. Lee ( )
An efficient and accurate computational approach is proposed for optimal attitude control of a ri... more An efficient and accurate computational approach is proposed for optimal attitude control of a rigid body. The problem is formulated directly as a discrete time optimization problem using a Lie group variational integrator. Discrete necessary conditions for optimality are derived, and an efficient computational approach is proposed to solve the resulting two point boundary value problem. The use of geometrically exact computations on SO(3) guarantees that this optimal control approach has excellent convergence properties even for highly nonlinear large angle attitude maneuvers. Numerical results are presented for attitude maneuvers of a 3D pendulum and a spacecraft in a circular orbit.
International Journal of Non-linear Mechanics, Jul 1, 2003
We consider two linearly coupled masses, where one mass can have inelastic impacts with a fixed, ... more We consider two linearly coupled masses, where one mass can have inelastic impacts with a fixed, rigid stop. This leads to the study of a two degree of freedom, piecewise linear, frictionless, unforced, constrained mechanical system. The system is governed by three types of dynamics: coupled harmonic oscillation, simple harmonic motion and discrete rebounds. Energy is dissipated discontinuously in discrete amounts, through impacts with the stop. We prove the existence of a nonzero measure set of orbits that lead to infinite impacts with the stop in a finite time. We show how to modify the mathematical model so that forward existence and uniqueness of solutions for all time is guaranteed. Existence of hybrid periodic orbits is shown. A geometrical interpretation of the dynamics based on action coordinates is used to visualize numerical simulation results for the asymptotic dynamics.
Proceedings of the ... American Control Conference, Jul 1, 2007
A 3D pendulum implemented using a triaxial air bearing system is proposed for Earth-based testing... more A 3D pendulum implemented using a triaxial air bearing system is proposed for Earth-based testing of orbiting spacecraft attitude dynamics and closed loop systems. This proposal is based on prior research on attitude dynamics and control of orbiting spacecraft, attitude dynamics and control of the 3D pendulum, and an experimental implementation of the 3D pendulum in our laboratory, referred to as the Triaxial Attitude Control Testbed (TACT). These research themes are integrated to assess the strengths and weaknesses of such an Earth-based testbed for spacecraft attitude dynamics and control hardware and software components. Several different cases, based on the importance of orbital effects and gravitational effects, are analyzed.
A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees... more A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. Symmetry assumptions are shown to lead to the planar 1D pendulum and to the spherical 2D pendulum models as special cases. The case where the rigid body is asymmetric and the center of mass is distinct from the pivot location leads to the 3D pendulum. Full and reduced 3D pendulum models are introduced and used to study important features of the nonlinear dynamics: conserved quantities, equilibria, invariant manifolds, local dynamics near equilibria and invariant manifolds, and the presence of chaotic motions. These results demonstrate the rich and complex dynamics of the 3D pendulum.
Mathematical and Computer Modelling of Dynamical Systems, Jun 1, 2003
The Triaxial Attitude Control Testbed has been developed as part of a research program at the Uni... more The Triaxial Attitude Control Testbed has been developed as part of a research program at the University of Michigan on multibody rotational dynamics and control. In this paper, equations of motion are derived and presented in various forms. Actuation mechanisms are incorporated into the models; these include fan actuators, reaction wheel actuators and proof mass actuators that are £xed to the triaxial base body. The models also allow incorporation of unactuated auxiliary bodies that are constrained to move relative to the triaxial base body. The models expose the dynamic coupling between the rotational motion of the triaxial base body, the relative or shape motion of the unactuated auxiliary degrees of freedom, and dynamics associated with actuation mechanisms. Many different model simpli£cations and approximations are developed. Control models for the triaxial attitude control testbed are formulated that re¤ect speci£c assumptions.
A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees... more A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. 3D pendulum dynamics have been much studied in integrable cases that arise when certain physical symmetry assumptions are made. This paper treats the nonintegrable case of the 3D pendulum dynamics when the rigid body is asymmetric and the center of mass is distinct from the pivot location. 3D pendulum full and reduced models are introduced and used to study important features of the nonlinear dynam-Communicated by R. Sepulchre.
A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees... more A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. 3D pendulum dynamics have been much studied in integrable cases that arise when certain physical symmetry assumptions are made. This paper treats the nonintegrable case of the 3D pendulum dynamics when the rigid body is asymmetric and the center of mass is distinct from the pivot location. 3D pendulum full and reduced models are introduced and used to study important features of the nonlinear dynam-4 J Nonlinear Sci (2011) 21: 3-32 ics: conserved quantities, equilibria, relative equilibria, invariant manifolds, local dynamics, and presence of chaotic motions. The paper provides a unified treatment of the 3D pendulum dynamics that includes prior results and new results expressed in the framework of geometric mechanics. These results demonstrate the rich and complex dynamics of the 3D pendulum.
... He received a B.Sc (Eng.) degree in Aeronautical Engineering from Punjab Engineering College,... more ... He received a B.Sc (Eng.) degree in Aeronautical Engineering from Punjab Engineering College, Chandigarh, India, in 1982, and an MSE degree in Aerospace Engineering from the University of Michigan, Ann Arbor, in 1984, where he is currently at the Department of ...
Abstract. An efficient and accurate computational approach is pro-posed for a nonconvex optimal a... more Abstract. An efficient and accurate computational approach is pro-posed for a nonconvex optimal attitude control for a rigid body. The problem is formulated directly as a discrete time optimization prob-lem using a Lie group variational integrator. Discrete time necessary conditions for ...
International Journal of Robust and Nonlinear Control, 2000
We study a control problem for a special class of underactuated mechanical systems, namely for me... more We study a control problem for a special class of underactuated mechanical systems, namely for mechanical systems with several directly actuated degrees of freedom and a single unactuated degree of freedom that must be controlled through the system coupling. Speci"c assumptions are introduced that de"ne this class, which includes many important models of mechanical system examples. The main result of the paper is the construction of a discontinuous nonlinear feedback controller for which the closed loop equilibrium at the origin is made &globally attractive'. The control construction approach is introduced in detail, and a proof of attractiveness is presented. The results are applied to control of the planar motion of a rigid body with a single unactuated internal degree of freedom.
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Papers by N. McClamroch