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 G-Snakes: Nonholonomic Kinematic Chains on Lie Groups(1994) Krishnaprasad, Perinkulam S.; Tsakiris, D.P.; ISRWe consider kinematic chains evolving on a finite-dimensional Lie group G under nonholonomic constraints, where snake-like global motion is induced by shape variations of the system. In particular, we consider the case when the evolution of the system is restricted to a subspace h of the corresponding Lie algebra g, where h is not a subalgebra of g and it can generate the whole algebra under Lie bracketing. Such systems are referred to as G- snakes. Away from certain singular configurations of the system, the constraints specify a (partial) connection on a principal fiber bundle, which in turn gives rise to a geometric phase under periodic shape variations. This geometric structure can be exploited in order to solve the nonholonomic motion planning problem for such systems.G-snakes generalize the concept of nonholonomic Variable Geometry Truss assemblies, which are kinematic chains evolving on the Special Euclidean group SE (2) under nonholonomic constraints imposed by idler wheels. We examine in detail the cases of 3-dimensional groups with real non-abelian Lie algebras such as the Heisenberg group H(3), the Special Orthogonal group SO (3) and the Special Linear group SL(2).
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 Nonholonomic Variable Geometry Truss Assemblies I: Motion Control(1993) Krishnaprasad, Perinkulam S.; Tsakiris, D.P.; ISRWe consider the nonholonomic motion planning problem for a novel class of snakelike modular mobile manipulators, where each module is implemented as a planar parallel manipulator with idler wheels. This assembly is actuated by shape changes of its modules, which, under the influence of the nonholonomic constraints on the wheels, induce a global motion of the assembly.We formulate the kinematics for a generic assembly of this type and specialize to the 2-module case in order to study the motion planning problem in greater detail.
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 Modeling of Impact on a Flexible Beam(1993) Wei, Q.F.; Krishnaprasad, Perinkulam S.; Dayawansa, Wijesuriya P.; ISRWe consider the problem of modeling dynamical effects of impact of an elastic body on a flexible beam. We derive a nonlinear integral equation by using the Hertz law of impact in conjunction with the beam equation. This equation does not admit a closed form solution. We demonstrate the existence of solutions, derive a reliable numerical method for computing solutions, and compare the numerical results with those obtained by others.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 An Improved Model for the Dynamics of Spur Gear Systems with Backlash Consideration(1993) Shing, T.K.; Tsai, L.W.; Krishnaprasad, Perinkulam S.; ISRAn improved model which accounts for backlash effects is proposed for the dynamics of spur gear systems. This dynamic model is mainly developed for the purpose of real time control. The complicated variation of the meshing stiffness as a function of contact point along the line of action is studied. Then the mean value is used as the stiffness constant in the improved model. Two simulations, free vibration and constant load operation, are performed to illustrate the effects of backlash on gear dynamics. Also given are comparisons of the simulation results with that of the Yang and Sun's model. This model is judged to be more realistic which can be used in real time control to achieve high precision.
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