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79
Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups
 IEEE Transactions on Automatic Control
, 2000
"... In this paper, we provide controllability tests and motion control algorithms for underactuated mechanical control systems on Lie groups with Lagrangian equal to kinetic energy. Examples include satellite and underwater vehicle control systems with the number of control inputs less than the dimensi ..."
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Cited by 94 (26 self)
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In this paper, we provide controllability tests and motion control algorithms for underactuated mechanical control systems on Lie groups with Lagrangian equal to kinetic energy. Examples include satellite and underwater vehicle control systems with the number of control inputs less than the dimension of the configuration space. Local controllability properties of these systems are characterized, and two algebraic tests are derived in terms of the symmetric product and the Lie bracket of the input vector fields. Perturbation theory is applied to compute approximate solutions for the system under smallamplitude forcing; inphase signals play a crucial role in achieving motion along symmetric product directions. Motion control algorithms are then designed to solve problems of pointtopoint reconfiguration, static interpolation and exponential stabilization. We illustrate the theoretical results and the algorithms with applications to models of planar rigid bodies, satellites and underwater vehicles.
Kinematic controllability for decoupled trajectory planning in underactuated mechanical systems
 IEEE Transactions on Robotics and Automation
, 2001
"... Abstract — We introduce the notion of kinematic controllability for secondorder underactuated mechanical systems. For systems satisfying this property, the problem of planning fast collisionfree trajectories between zero velocity states can be decoupled into the computationally simpler problems of ..."
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Cited by 83 (20 self)
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Abstract — We introduce the notion of kinematic controllability for secondorder underactuated mechanical systems. For systems satisfying this property, the problem of planning fast collisionfree trajectories between zero velocity states can be decoupled into the computationally simpler problems of path planning for a kinematic system followed by timeoptimal time scaling. While this approach is well known for fully actuated systems, until now there has been no way to apply it to underactuated dynamic systems. The results in this paper form the basis for efficient collisionfree trajectory planning for a class of underactuated mechanical systems including manipulators and vehicles in space and underwater environments.
Selfreconfiguration planning with compressible unit modules
 In Proc. IEEE ICRA
, 1999
"... We discuss a robotic system composed of Crystalline modules. Crystaline modules can aggregate together to form distributed robot systems. Crystalline modules can move relative to each other by expanding and contracting. This actuation mechanism permits automated shape metamorphosis. We describe the ..."
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Cited by 78 (5 self)
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We discuss a robotic system composed of Crystalline modules. Crystaline modules can aggregate together to form distributed robot systems. Crystalline modules can move relative to each other by expanding and contracting. This actuation mechanism permits automated shape metamorphosis. We describe the crystalline module concept and show the basic motions that enable a crystalline robot system to selfreconfigure. We present an algorithm for general selfreconfiguration and describe simulation experiments. 1
Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction
, 1996
"... In this paper we establish necessary conditions for optimal control using the ideas of Lagrangian reduction in the sense of reduction under a symmetry group. The techniques developed here are designed for Lagrangian mechanical control systems with symmetry. The benefit of such an approach is that it ..."
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Cited by 47 (5 self)
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In this paper we establish necessary conditions for optimal control using the ideas of Lagrangian reduction in the sense of reduction under a symmetry group. The techniques developed here are designed for Lagrangian mechanical control systems with symmetry. The benefit of such an approach is that it makes use of the special structure of the system, especially its symmetry structure and thus it leads rather directly to the desired conclusions for such systems. Lagrangian reduction can do in one step what one can alternatively do by applying the Pontryagin Maximum Principle followed by an application of Poisson reduction. The idea of using Lagrangian reduction in the sense of symmetry reduction was also obtained by Bloch and Crouch [1995a,b] in a somewhat different context and the general idea is closely related to those in Montgomery [1990] and Vershik and Gershkovich [1994]. Here we develop this idea further and apply it to some known examples, such as optimal control on Lie groups and principal bundles (such as the ball and plate problem) and reorientation examples with zero angular momentum (such as the satellite with moveable masses). However, one of our main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation in
Geometric mechanics, Lagrangian reduction and nonholonomic systems
 in Mathematics Unlimited2001 and Beyond
, 2001
"... This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. Reduction theory for mechanical systems with symmetry has ..."
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Cited by 38 (5 self)
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This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in mechanics ranges from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and utilizing their associated conservation laws. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many
Oscillations, SE(2)snakes and motion control: a study of the Roller Racer
 Dynamical Systems
, 2001
"... This report is concerned with the problem of motion generation via cyclic variations in selected degrees of freedom (usually referred to as shape variables) in mechanical systems subject to nonholonomic constraints (here the classical one of a disk rolling without sliding on a at surface). In earlie ..."
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Cited by 35 (15 self)
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This report is concerned with the problem of motion generation via cyclic variations in selected degrees of freedom (usually referred to as shape variables) in mechanical systems subject to nonholonomic constraints (here the classical one of a disk rolling without sliding on a at surface). In earlier work, we identi ed an interesting class of such problems arising in the setting of Lie groups, and investigated these under a hypothesis on constraints, that naturally led to a purely kinematic approach. In the present work, the hypothesis on constraints does not hold, and as a consequence, it is necessary to take into account certain dynamical phenomena. Speci cally we concern ourselves with the group SE(2) of rigid motions in the plane and a concrete mechanical realization dubbed the 2{node, 1{module SE(2){snake. In a restricted version, it is also known as the Roller Racer (a patented ride/toy). Based on the work of Bloch, Krishnaprasad, Marsden and Murray, one recognizes in the example of this report a balance law called the momentum equation, which is a direct consequence of the interaction of the SE(2){symmetry of the problem with the
Mechanical control systems on Lie algebroids
, 2003
"... This paper considers control systems defined on Lie algebroids. After deriving basic controllability tests for general control systems, we specialize our discussion to the class of mechanical control systems on Lie algebroids. This class of systems includes mechanical systems subject to holonomic ..."
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Cited by 33 (3 self)
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This paper considers control systems defined on Lie algebroids. After deriving basic controllability tests for general control systems, we specialize our discussion to the class of mechanical control systems on Lie algebroids. This class of systems includes mechanical systems subject to holonomic and nonholonomic constraints, mechanical systems with symmetry and mechanical systems evolving on semidirect products.
Nonholonomic integrators
, 2001
"... Abstract. We introduce a discretization of the Lagranged’Alembert principle for Lagrangian systems with nonholonomic constraints, which allows us to construct numerical integrators that approximate the continuous flow. We study the geometric invariance properties of the discrete flow which provide ..."
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Cited by 31 (0 self)
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Abstract. We introduce a discretization of the Lagranged’Alembert principle for Lagrangian systems with nonholonomic constraints, which allows us to construct numerical integrators that approximate the continuous flow. We study the geometric invariance properties of the discrete flow which provide an explanation for the good performance of the proposed method. This is tested on two examples: a nonholonomic particle with a quadratic potential and a mobile robot with fixed orientation.
Nonlinear Control of Mechanical Systems: A Riemannian Geometry Approach
, 1998
"... Nonlinear control of mechanical systems is a challenging discipline that lies at the intersection between control theory and geometric mechanics. This thesis sheds new light on this interplay while investigating motion control problems for Lagrangian systems. Both stability and motion planning aspec ..."
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Cited by 29 (0 self)
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Nonlinear control of mechanical systems is a challenging discipline that lies at the intersection between control theory and geometric mechanics. This thesis sheds new light on this interplay while investigating motion control problems for Lagrangian systems. Both stability and motion planning aspects are treated within a unified framework that accounts for a large class of devices such as robotic manipulators, autonomous vehicles and locomotion systems. One distinguishing feature of mechanical systems is the number of control forces. For systems with as many input forces as degrees of freedom, many control problems are tractable. One contribution of this thesis is a set of trajectory tracking controllers designed via the notions of configuration and velocity error. The proposed approach includes as special cases a variety of results on joint and workspace control of manipulators as well as on attitude and position control of vehicles. Whenever fewer input forces are available than deg...