Motion control of drift-free, left-invariant systems on Lie groups, (1995)

by N E LEONARD, P S KRISHNAPRASAD
Venue:IEEE Trans. on Automatic Control,
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Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups

by Francesco Bullo, Naomi Ehrich Leonard, Andrew D. Lewis - IEEE Transactions on Automatic Control , 2000
"... In this paper, we provide controllability tests and motion control algorithms for under-actuated 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 ..."
Abstract - Cited by 92 (22 self) - Add to MetaCart
In this paper, we provide controllability tests and motion control algorithms for under-actuated 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 symmet-ric product and the Lie bracket of the input vector fields. Perturbation theory is applied to compute approximate solutions for the system under small-amplitude forcing; in-phase signals play a crucial role in achieving motion along symmetric product directions. Motion control algorithms are then designed to solve problems of point-to-point reconfiguration, static inter-polation and exponential stabilization. We illustrate the theoretical results and the algorithms with applications to models of planar rigid bodies, satellites and underwater vehicles.

Dynamic Nonprehensile Manipulation: Controllability, Planning, and Experiments

by Kevin M. Lynch, Matthew T. Mason - International Journal of Robotics Research , 1998
"... We are interested in using low degree-of-freedom robots to perform complex tasks by nonprehensile manipulation (manipulation without a form- or force-closure grasp). By not grasping, the robot can use gravitational, centrifugal, and Coriolis forces as virtual motors to control more degreesof -freedo ..."
Abstract - Cited by 46 (14 self) - Add to MetaCart
We are interested in using low degree-of-freedom robots to perform complex tasks by nonprehensile manipulation (manipulation without a form- or force-closure grasp). By not grasping, the robot can use gravitational, centrifugal, and Coriolis forces as virtual motors to control more degreesof -freedom of the part. The extra motion freedoms of the part are exhibited as rolling, slipping, and free flight.

Optimal Gait Selection for Nonholonomic Locomotion Systems

by James P. Ostrowski, Jaydev P. Desai, Vijay Kumar , 2000
"... This paper addresses the optimal control and selection of gaits in a class of nonholonomic locomotion systems that exhibit group symmetries. We study optimal gaits for the snakeboard, a representative example of this class of systems. We employ Lagrangian reduction techniques to simplify the optimal ..."
Abstract - Cited by 44 (8 self) - Add to MetaCart
This paper addresses the optimal control and selection of gaits in a class of nonholonomic locomotion systems that exhibit group symmetries. We study optimal gaits for the snakeboard, a representative example of this class of systems. We employ Lagrangian reduction techniques to simplify the optimal control problem and describe a general framework and an algorithm to obtain numerical solutions to this problem. This work employs optimal control techniques to study the optimality of gaits and issues involving gait transitions. The general framework provided in this paper can easily be applied to other examples of biological and robotic locomotion. KEY WORDS---optimal control, robotic locomotion, geometric mechanics, locomotive gaits 1.

Practical stabilization of driftless systems on Lie groups: the transverse function approach

by Pascal Morin, Claude Samson, L Et - IEEE Trans. on Automatic Control,48,1496 , 2003
"... Abstract—A general control design approach for the sta-bilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented. The approach is based on the concept of bounded transverse functions, the existence of which is equivalent to the system’s controllability. Its ..."
Abstract - Cited by 42 (12 self) - Add to MetaCart
Abstract—A general control design approach for the sta-bilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented. The approach is based on the concept of bounded transverse functions, the existence of which is equivalent to the system’s controllability. Its outcome is the practical stabilization of any trajectory, i.e., not necessarily a solution of the control system, in the state–space. The possibility of applying the approach to an arbitrary controllable smooth driftless system follows in turn from the fact that any controllable homogeneous approximation of this system can be lifted (via a dynamic extension) to a system on a Lie group. Illustrative examples are given. Index Terms—Feedback law, Lie groups, nonlinear systems, stabilization.
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Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry

by Naomi Ehrich Leonard, Jerrold E. Marsden - Physica D , 1997
"... This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with r ..."
Abstract - Cited by 37 (7 self) - Add to MetaCart
This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with rigid motion symmetry, one gets stability but possibly with drift in certain rotational as well as translational directions. Motivated by questions on stability of underwater vehicle dynamics, it is of particular interest that, in some cases, we can allow the relative equilibria to have nongeneric values of their momentum. The results are proved by combining theorems of Patrick with the technique of reduction by stages. This theory is then applied to underwater vehicle dynamics. The stability of specific relative equilibria for the underwater vehicle is studied. For example, we find conditions for Liapunov stability of the steadily rising and possibly spinning, bottom-heavy vehicle, which corresponds to a relative equilibrium with nongeneric momentum. The results of this paper should prove

Gait Kinematics for a Serpentine Robot

by Jim Ostrowski, Joel Burdick - In Proc. IEEE Int. Conf. on Rob. and Autom , 1996
"... : This paper considers the problem of serpentine, or snake-like, locomotion from the perspective of geometric mechanics. A particular model, which is similar to Hirose's Active Cord Mechanism (ACM), is analyzed. Using the kinematic constraints, we develop a connection, which describes the net m ..."
Abstract - Cited by 30 (1 self) - Add to MetaCart
: This paper considers the problem of serpentine, or snake-like, locomotion from the perspective of geometric mechanics. A particular model, which is similar to Hirose's Active Cord Mechanism (ACM), is analyzed. Using the kinematic constraints, we develop a connection, which describes the net motion of the machine as a function of variations in the mechanism 's shape variables. We present simulation results demonstrating three types of locomotive gaits, one of which bears an obvious resemblance to the serpentine motion of a snake. We also show how these algorithms can be used to optimize certain inputs given the particular choice of physical parameters for a snake robot. 1. Introduction Most mobile robots are wheeled vehicles, since wheels provide the simplest means for robotic mobility. The assumption that these wheels do not slip provides nonholonomic kinematic constraints on a vehicle 's motion. These kinematic nonholonomic systems have been extensively studied in the literature. F...
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Proportional Derivative (PD) Control On The Euclidean Group

by F. Bullo, R. M. Murray - In European Control Conference , 1995
"... . In this paper we study the stabilization problem for control systems defined on SE(3) (the special Euclidean group of rigid-body motions) and its subgroups. Assuming one actuator is available for each degree of freedom, we exploit geometric properties of Lie groups (and corresponding Lie algebras) ..."
Abstract - Cited by 30 (3 self) - Add to MetaCart
. In this paper we study the stabilization problem for control systems defined on SE(3) (the special Euclidean group of rigid-body motions) and its subgroups. Assuming one actuator is available for each degree of freedom, we exploit geometric properties of Lie groups (and corresponding Lie algebras) to generalize the classical proportional derivative (PD) control in a coordinate-free way. For the SO(3) case, the compactness of the group gives rise to a natural metric structure and to a natural choice of preferred control direction: an optimal (in the sense of geodesic) solution is given to the attitude control problem. In the SE(3) case, no natural metric is uniquely defined, so that more freedom is left in the control design. Different formulations of PD feedback can be adopted by extending the SO(3) approach to the whole of SE(3) or by breaking the problem into a control problem on SO(3) \Theta R 3 . For the simple SE(2) case, simulations are reported to illustrate the behavior of...
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Nonprehensile Robotic Manipulation: Controllability and Planning

by Kevin M. Lynch , 1997
"... the author and should not be interpreted as representing the o cial policies, either expressed or A good model of the mechanics of a task is a resource for a robot, just as actuators and sensors are resources. The e ective use of frictional, gravitational, and dynamic forces can substitute for extra ..."
Abstract - Cited by 29 (5 self) - Add to MetaCart
the author and should not be interpreted as representing the o cial policies, either expressed or A good model of the mechanics of a task is a resource for a robot, just as actuators and sensors are resources. The e ective use of frictional, gravitational, and dynamic forces can substitute for extra actuators; the expectation derived from a good model can minimize sensing requirements. Despite this, most robot systems attempt to dominate or nullify task mechanics, rather than exploit them. There has been little e ort to understand the manipulation capabilities of even the simplest robots under more complete mechanics models. This thesis addresses that knowledge de cit by studying graspless or nonprehensile manipulation. Nonprehensile manipulation exploits task mechanics to achieve a goal state without grasping, allowing simple mechanisms to accomplish complex tasks. With nonprehensile manipulation, a robot can manipulate objects too large or heavy to be grasped and lifted, and a low-degree-of-freedom robot can control more degrees-of-freedom of an object by allowing relative motion between the object and the manipulator. Two key problems are determining controllability of and motion planning for
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Nonlinear Control of Mechanical Systems: A Lagrangian Perspective

by Richard M. Murray , 1997
"... . Recent advances in geometric mechanics, motivated in large part by applications in control theory, have introduced new tools for understanding and utilizing the structure present in mechanical systems. In particular, the use of geometric methods for analyzing Lagrangian systems with both symmetr ..."
Abstract - Cited by 28 (6 self) - Add to MetaCart
. Recent advances in geometric mechanics, motivated in large part by applications in control theory, have introduced new tools for understanding and utilizing the structure present in mechanical systems. In particular, the use of geometric methods for analyzing Lagrangian systems with both symmetries and non-integrable (or nonholonomic) constraints has led to a unified formulation of the dynamics that has important implications for a wide class of mechanical control systems. This paper presents a survey of recent results in this area, focusing on the relationships between geometric phases, controllability, and curvature, and the role of trajectory generation in nonlinear controller synthesis. Examples are drawn from robotics and flight control systems, with an emphasis on motion control problems. Key Words. Geometric mechanics, nonlinear control, Lagrangian dynamics, motion control. 1. INTRODUCTION Mechanical systems form an important class of nonlinear control systems that h...

Nonlinear Control of Mechanical Systems: A Riemannian Geometry Approach

by Francesco Bullo , 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 ..."
Abstract - Cited by 27 (0 self) - Add to MetaCart
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...