Institute for Systems Research
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Item Control Problems of Hydrodynamic Type(1998) Krishnaprasad, Perinkulam S.; Manikonda, Vikram; ISR; CDCSSIt has been known for some time that the classical work of Kirchhoff, Love,and Birkhoff on rigid bodies in incompressible, irrotational flows provideseffective models for treating control problems for underwater vehicles.This has also led to a better appreciation of the dynamics of suchsystems. In this paper, we develop results based on geometric mechanics andcenter manifold theory to solve controllability and stabilization questionsfor a class of under-actuated left invariant mechanical systems on Liegroups that include approximate models of underwater vehicles and surfacevehicles. We also provide numerical evidence to capture the globalproperties of certain interesting feedback laws.(This work appears as an invited paper in the Proc. IFAC Sympo. on NonlinearControl Systems Design (NOLCOS'98), (1998), 1:139-144)
Item Motion Control and Coupled Oscillators(1995) Krishnaprasad, Perinkulam S.; ISRIt is remarkable that despite the presence of large numbers of degrees of freedom, motion control problems are effectively solved in biological systems. While feedback, regulation and tracking have served us well in engineering, as useful solution paradigms for a wide variety of control problems including motion control, it appears that nature gives prominent roles to planning and co-ordination as well. There is also complex interplay between sensory feedback and motion planning to achieve effective operation in uncertain environments, for example, in movement on uneven terrain cluttered with obstacles.Recent investigations by neurophysiologists have brought to increasing prominence the idea of central pattern generators -- a class of coupled oscillators -- as sources of motion scripts as well as a means for coordinating multiple degrees of freedom. The role of coupled oscillators in motion control systems is currently under intense investigation.
In this paper we examine some unifying themes relating movements in biological systems and machines. An important insight in this direction comes from the natural grouping of degrees of freedom and time scales in biological and engineering systems. Such grouping and separation can be treated from a geometric viewpoint using the formalisms and methods of differential geometry, Lie groups, and fiber bundles. Coupled oscillators provide the means to bind degrees of freedom either directly through phase locking or indirectly through geometric phases. This point of view leads to fresh ways of organizing the control structures of complex technological systems.
Item Motion Control of Drift-Free, Left-Invariant Systems on Lie Groups, Part II: A General Constructive Control Algorithm(1994) Leonard, Naomi E.; Krishnaprasad, Perinkulam S.; ISRIn this paper we present a general algorithm for constructing open-loop controls to solve the complete constructive controllability problem for drift-free invariant systems on Lie groups that satisfy the Lie algebra controllability rank condition with up to ( p - 1) iterations of Lie brackets, p = 1,2,3. Specifically, given only the structure constants of the given system, an initial condition Xi, a final condition Xf and a final time tf, the algorithm specifies open-loop, small (e) amplitude sinusoidal controls such that the system starting from Xi, reaches Xf at t = tf, with O (ep) accuracy. The algorithm is based on the formulas and geometric interpretation of the average approximations to the solution given in Part I to this paper. To illustrate the effectiveness of the algorithms, we apply it to three problems: the spacecraft attitude control problem with only two controls available, the unicycle motion planning problem and the autonomous underwater vehicle motion control problem with only three controls available.Item Motion Control of Drift-Free, Left-Invariant Systems on Lie Groups(1994) Leonard, Naomi E.; Krishnaprasad, Perinkulam S.; ISRIn this paper we address the constructive controllability problem for drift free, left-invariant systems on finite-dimensional Lie groups with fewer controls than state dimension. We consider small (e) amplitude, low-frequency, periodically time-varying controls and derive average solutions for system behavior. We show how the pth-order average formula can be used to construct open-loop controls for point-to-point maneuvering of systems that require up to ( p - 1) iterations of Lie brackets to satisfy the Lie algebra controllability rank condition. In the cases p =2,3, we give algorithms for constructing these controls as a function of structure constants that define the control authority, i.e., the actuator capability, of the system. The algorithms are based on a geometric interpretation of the average formulas and produce sinusoidal controls that solve the constructive controllability problem with O (ep) accuracy in general (exactly if the Lie algebra is nipotent). The methodology is applicable to a variety of control problems and is illustrated for the motion control problem of an autonomous underwater vehicle with as few as three control inputs.Item Optimal Control and Poisson Reduction(1993) Krishnaprasad, Perinkulam S.; ISRIn this paper we make explicit a reduction of G-invariant optimal control problems on a Lie group G.Item High-Order Averaging on Lie Groups and Control of an Autonomous Underwater Vehicle(1993) Leonard, Naomi E.; Krishnaprasad, Perinkulam S.; ISRIn this paper we extend our earlier results on the use of periodic forcing and averaging to solve the constructive controllability problem for drift-free left-invariant systems on Lie groups with fewer controls than state variables. In particular, we prove a third-order averaging theorem applicable to systems evolving on general matrix Lie groups and show how to use the resulting approximations to construct open loop controls for complete controllability of systems that require up to depth- two Lie brackets to satisfy the Lie algebra controllability rank condition. The motion control problem for an autonomous underwater vehicle is modeled as a drift-free left-invariant system on the matrix Lie group SE (3). In the general case, when only one translational and two angular control inputs are available, this system satisfies the controllability rank condition using depth-two Lie brackets. We use the third-order averaging result and its geometric interpretation to construct open loop controls to arbitrarily translate and orient an autonomous underwater vehicle.Item Optimal Control of a Rigid Body with Two Oscillators(1993) Yang, R.; Krishnaprasad, Perinkulam S.; Dayawansa, Wijesuriya P.; ISRThis paper is concerned with the exploration of reduction and explicit solvability of optimal control problems on principal bundles with connections from a Hamiltonian point of view. The particular mechanical system we consider is a rigid body with two driven oscillators, for which the bundle structure is (SO (3) x 者, 者, SO (3)). The optimal control problem is posed by considering a special nonholonomic variational problem, in which the nonholonomic distribution is defined via a connection. The necessary conditions for the optimal control problem are determined intrinsically by a Hamiltonian formulation. The necessary conditions admit the structure group of the principal bundle as a symmetry group of the system. Thus the problem is amendable to Poisson reduction. Under suitable hypotheses and approximations, we find that the reduced system possesses additional symmetry which is isomorphic to S1. Applying Poisson reduction again, we obtain a further reduced system and corresponding first integral. These reductions imply explicit solvability for suitable values of parameters.Item Averaging and Motion Control On Lie Groups(1993) Leonard, Naomi E.; Krishnaprasad, Perinkulam S.; ISRThe deeper investigation of problems of feedback stabilization and constructive controllability has drawn increased attention to the question of structuring control systems. Thus, for instance, it is interesting to know how to combine periodic open loop controls with intermittent feedback corrections to achieve prescribed behavior in robotic motion planning systems. As a first step towards understanding this type of question, it would be useful to obtain some insight into the average behavior of a periodically forced system. In the present paper we are primarily interested in periodic forcing of left-invariant systems on Lie groups such as would arise in spacecraft attitude control. We prove averaging theorems applicable to systems evolving on general matrix Lie groups with particular focus on the attitude control problem. The results of this paper also yield useful formulae for motion planning of a variety of other systems such as an underwater vehicle which can be modeled as a control system evolving on the Lie group SE (3).Item Dissipation Induced Instabilities(1992) Bloch, Anthony M.; Krishnaprasad, Perinkulam S.; Marsden, Jerrold E.; Ratiu, Tudor S.; ISRThe main goal of this paper is to prove that if the energy- momentum (or energy-Casimir) method predicts formal instability of a relative equilibrium in a Hamiltonian system with symmetry, then with the addition of dissipation, the relative equilibrium becomes spectrally and hence linearly and nonlinearly unstable. The energy-momentum method assumes that one is in the context of a mechanical system with a given symmetry group. Our result assumes that the dissipation chosen does not destroy the conservation law associated with the given symmetry group -- thus, we consider internal dissipation. Our result also includes the special case of systems with no symmetry and ordinary equilibria. Our result is proved by combining the techniques of Chetaev, who proved instability theorems using a special Chetaev- Lyapunov function, those of Hahn, which enable one to strengthen the Chetaev results from Lyapunov instability to spectral instability. Our main achievement is to strengthen these results to the context of the block diagonalization version of the energy momentum method given by Lewis. Marsden, Posbergh, and Simo. However, we also give the eigenvalue movement formulae of Krein, MacKay and others both in general and adapted to the context of the normal form of the linearized equations given by the block diagoanl form as provided by the energy-momentum method. A number of specific examples, such as the rigid body with internal rotors, are provided to illustrate the results.Item On the Geometry and Dynamics of Floating Four-Bar Linkages(1992) Yang, R.; Krishnaprasad, Perinkulam S.; ISRIn this paper, we investigate the kinematics and dynamics of floating, planar four-bar linkages. The geometry of configuration space is analyzed through the classical theory of mechanisms due to Grashof. The techniques of symplectic and Poisson reduction are used to understand the dynamics of the system. Bifurcations of relative equilibria for linkages admitting symmetric shapes are studied using the techniques of singularity theory. The problem of reconstruction of the full dynamics and its relation to geometric phases is discussed through some examples. This research reveals that a coupled mechanical system with kinematic loops possesses richer and more complicated dynamical aspects in comparison with systems which have the same number of degrees of freedom, but no kinematic loops.