Institute for Systems Research Technical Reports
Permanent URI for this collectionhttp://hdl.handle.net/1903/4376
This archive contains a collection of reports generated by the faculty and students of the Institute for Systems Research (ISR), a permanent, interdisciplinary research unit in the A. James Clark School of Engineering at the University of Maryland. ISR-based projects are conducted through partnerships with industry and government, bringing together faculty and students from multiple academic departments and colleges across the university.
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Item The Dynamics of a Forced Sphere-Plate Mechanical System(2000) Hristu, Dimitrios; ISR; CDCSSWe study the dynamics and explore thecontrollability of a family of sphere-plate mechanical systems. Theseare nonholonomic systems with a five-dimensional configuration spaceand three independent velocities. They consist of a sphere rollingin contact with two horizontal plates. Kinematic models ofsphere-plate systems have played an important role in the controlsystems literature addressing the kinematics of rolling bodies, aswell as in discussions of nonholonomic systems. However, kinematicanalysis falls short of allowing one to understand the dynamicbehavior of such systems. In this work we formulate and study adynamic model for a class of sphere-plate systems in order to answerthe question: "Is it possible to impart a net angular momentum to asphere which rolls without slipping between two plates, given thatthe position of the top plate is subject to exogenousforces?"The research and scientific content in this material will appearin IEEE Transactions on Automatic Control. Item Control and Stabilization of a Class of Nonlinear Systems with Symmetry(1998) Manikonda, Vikram; Krishnaprasad, P.S.; ISR; CDCSSThe focus of this dissertation is to study issues related to controllability and stabilization of a class of underactuated mechanical systems with symmetry. In particular we look at systems whose configuration can be identified with a Lie group and the reduced equations are of the Lie-Poisson type. Examples of such systems include hovercraft, spacecraft and autonomous underwater vehicles. We present sufficient conditions for the controllability of affine nonlinear control systems where the drift vector field is a Lie-Poisson reduced Hamiltonian vector field. In this setting we show that depending on the existence of a radially unbounded Lyapunov type function, the drift vector field of the reduced system is weakly positively Poisson stable. The weak positive Poisson stability along with the Lie algebra rank condition is used to show controllability. These controllability results are then extended to the unreduced dynamics. Sufficient conditions for controllability are presented in both cases where the symmetry group is compact and noncompact. We also present a constructive approach to design feedback laws to stabilize relative equilibria of these systems. The approach is based on the observation that, under certain hypotheses the fixed points of the Lie-Poisson dynamics belong to an immersed equilibrium submanifold. The existence of such equilibrium manifolds, along with the center manifold theory is used to design stabilizing feedback laws.Item Motion Control for Nonholonomic Systems on Matrix Lie Groups(1998) Struemper, Herbert Karl; Krishnaprasad, P.S.; ISR; CDCSSIn this dissertation we study the control of nonholonomic systems defined by invariant vector fields on matrix Lie groups. We make use of canonical constructions of coordinates and other mathematical tools provided by the Lie group setting. An approximate tracking control law is derived for so-called chained form systems which arise as local representations of systems on a certain nilpotent matrix group. After studying the technique of nilpotentization in the setting of systems on matrix Lie groups we show how motion control laws derived for nilpotent systems can be extended to nilpotentizable systems using feedback and state transformations. The proposed control laws exhibit highly oscillatory components both for tracking and feedback stabilization of local representations of nonholonomic systems on Lie groups. Applications to the control and analysis of the kinematics of mechanical systems are discussed and numerical simulations are presented.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 Planning for Nonholonomic Kinematic Chains(1995) Tsakiris, D.P.; Krishnaprasad, P.S.; ISRIn this dissertation we examine a class of systems where nonholonomic kinematic constraints are combined with periodic shape variations, giving rise to snake-like undulating motion of the system. Within this class, we distinguish two subclasses, one where the system possesses enough kinematic constraints to allow the control of its motion to be based entirely on kinematics and another which does not; in the latter case, the dynamics plays a crucial role in complementing the kinematics and in making motion control possible. An instance of these systems are the Nonholonomic Variable Geometry Truss (NVGT) assemblies, where shape changes are implemented by parallel manipulator modules, while the nonholonomic constraints are imposed by idler wheels attached to the assembly. We assume that the wheels roll without slipping on the ground, thus constraining the instantaneous motion of the assembly. These assemblies can be considered as land locomotion alternatives to systems based on legs or actuated wheels. Their propulsion combines features of both biological systems like skating humans and snakes, and of man-made systems like orbiting satellites with manipulator arms. The NVGT assemblies can be modeled in terms of the Special Euclidean group of rigid motions on the plane. Generalization to nonholonomic kinematic chains on other Lie groups (G) gives rise to the notion of G -Snakes.Moreover, we examine systems with parallel manipulator subsystems which can be used as sensor- carrying platforms, with potential applications in exploratory and active visual or haptic robotic tasks. We concentrate on specifying a class of configuration space path segments that are optimal in the sense of a curvature-squared cost functional, which can be specified analytically in terms of elliptic functions and can be used to synthesize a trajectory of the system.
In both cases, a setup of the problem which involves tools from differential geometry and the theory of Lie groups appears to be natural. In the case of G -Snakes, when the number of nonholonomic constraints equals the dimension of the group G, the constraints determine a principal fiber bundle connection. The geometric phase associated to this connection allows us to derive (kinematic) motion control strategies based on periodic shape variations of the system. When the G -Snakes assembly has one constraint less than the dimension of the group G, we are still able to synthesize a principal fiber bundle connection by taking into account the Lagrangian dynamics of the system through the so-called nonholonomic momentum. The symmetries of the system are captured by actions of non-abelian Lie groups that leave invariant both the constraints and the Lagrangian and play a significant role in the definition of the momentum and the specification of its evolution. The (dynamic) motion control is now based on periodic shape variations that build up momentum and allow propulsion and steering, as described by the geometric and dynamic phases of the system.
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 A Hybrid Control Strategy for Path Planning and Obstacle Avoidance with Nonholonomic robots(1994) Manikonda, Vikram; Krishnaprasad, P.S.; ISRThe primary focus is on providing a formal basis for behavior- based robotics using techniques that have been successful in control-based approaches for steering and stabilizing robots that are subject to nonholonomic constraints. In particular, behaviors for robots are formalized in terms of kinetic state machines, a motion description language and the interaction of the kinetic state machine with information coming in from (limited range) sensors. This allows us to create a mathematical basis for discussing these systems, including techniques for integrating sets of behaviors. In addition we suggest optimality criteria for comparing both atomic and compound behaviors in various environments. A hybrid architecture for the implementation of path planners that use the motion description language is presented. The design and implementation of a planner for path planning and examples of obstacle avoidance with nonholonomic robots are discussed.Item Averaging and Motion Control of Systems on Lie Groups(1994) Leonard, Naomi E.; Krishnaprasad, P.S.; ISRIn this dissertation, we study motion control problems in the framework of systems on finite-dimentional Lie groups. Nonholonomic motion control problems are challenging because nonlinear controllability theory does not provide an explicit procedure for constructing controls and linearization techniques, typically effective for nonlinear system analysis, fail to be useful. Our approach, distinguished from previous motion control research, is to exploit the Lie group framework since it provides a natural and mathematically rich setting for studying nonholonomic systems. In particular, we use the framework to develop explicit, structured formulas that describe system behavior and from these formulas we derive a systematic say of synthesizing controls to achieve desired motion.As our main tool we derive averaging theory for left-invariant systems on finite-dimensional Lie groups. This theory provides basis- independent formulas which approximate system behavior on the Lie group to arbitrarily high order in given small () amplitude, periodically time-varying control inputs. We interpret the average formulas geometrically and exploit this interpretation to prove a constructive controllability theorem for the average system. The proof of this theorem provides a constructive control synthesis methodology for drift-free systems which we use to derive algorithms which synthesize sinusoidal open-loop controls. We apply the algorithms to several under-actuated mechanical control problems including problems in spacecraft attitude control, unicycle motion control and autonomous underwater vehicle control. We illustrate the effectiveness of the synthesized controls by simulation and experimentation. We show further that as a consequence of the geometry inherited from the average formulas, our algorithms can be used to produce motion controls that adapt to changes in control authority such as loss of an actuator.
We also apply our theory to synthesize controls for bilinear control systems on Rn possibly with drift. Our approach is to control the system state by controlling the state transition matrix which evolves on a matrix Lie group. We design and demonstrate a controller for an example system with drift, a simple switched electrical network.
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.