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 Geometric Phases, Anholonomy, and Optimal Movement(1991) Krishnaprasad, Perinkulam S.; Yang, R.; ISRIn the search for useful strategies for movement of robotic systems (e.g. manipulators, platforms) in constrained environments (e.g. in space, underwater), there appear to be new principles emerging from a deeper geometric understanding of optimal movements of nonholonomically constrained systems. In our work, we have exploited some new formulas for geometric phase shifts to derive effective control strategies. The theory of connections in principal bundles provides the proper framework for questions of the type addressed in this paper. we outline the essentials of this theory. A related optimal control problem and its localizations are also considered.