This paper addresses the general problem of controlling a large number of robots required to move as a group. We propose an abstraction based on the definition of a map from the configuration space Q of the robots to a lower dimensional manifold A, whose dimension is independent of the number of robots. In this paper, we focus on planar fully actuated robots. We require that the manifold A has a product structure A=G/spl times/S, where G is a Lie group, which captures the position and orientation of the ensemble in the chosen world coordinate frame, and S is a shape manifold, which is an intrinsic characterization of the team describing the "shape" as the area spanned by the robots. We design decoupled controllers for the group and shape variables. We derive controllers for individual robots that guarantee the desired behavior on A. These controllers can be realized by feedback that depends only on the current state of the robot and the state of the manifold A. This has the practical advantage of reducing the communication and sensing that is required and limiting the complexity of individual robot controllers, even for large numbers of robots.

This paper develops a method for generating smooth trajectories for a moving rigid body with specified boundary conditions. Our method involves two key steps: 1) the generation of optimal trajectories in GA/sup +/ (n), a subgroup of the affine group in R/sup n/ and 2) the projection of the trajectories onto SE(3), the Lie group of rigid body displacements. The overall procedure is invariant with respect to both the local coordinates on the manifold and the choice of the inertial frame. The benefits of the method are threefold. First, it is possible to apply any of the variety of well-known efficient techniques to generate optimal curves on GA/sup +/ (n). Second, the method yields approximations to optimal solutions for general choices of Riemannian metrics on SE(3). Third, from a computational point of view, the method we propose is less expensive than traditional methods.

An approach to the control of constrained dynamic systems such as multiple arm systems, multifingered grippers, and walking vehicles is described. The basic philosophy is to utilize a minimal set of inputs to control the trajectory and the surplus input to control the constraint or interaction forces and moments in the closed chain. A dynamic control model for the closed chain is derived that is suitable for designing a controller in which the trajectory and the interaction forces and moments are explicitly controlled. Nonlinear feedback techniques derived from differential geometry are then applied to linearize and decouple the nonlinear model. These ideas are illustrated through a planar example in which two arms are used for cooperative manipulation. Results from a simulation are used to illustrate the efficacy of the method.< >

The instantaneous kinematics of a hybrid manipulation system, which combines the traditional serial chain geometry with parallelism in actuation, and the problem of coordination is discussed. The indeterminacy and singularities in the inverse kinematics and statics equations and measures of kinematic performance are analyzed. Finally, coordination algorithms that maintain an optimal force distribution between the actuators while avoiding or exploiting singularities are presented.>

A method is presented which addresses the problem of determinating the appropriate distribution of forces among the fingers of a multifingered gripper grasping an object. The finger-object interactions are modeled as point contacts. The system is statically indeterminate, and an optimal solution for this problem is desired for force control. A fast and efficient suboptimal method for computing the grasping forces is presented. In addition to determining the equilibrating forces (the forces required to maintain equilibrium), it is essential to superpose interaction forces (the forces which squeeze the object) to ensure that the fingers do not slip. The decomposition of the equilibrating forces into forces perpendicular and parallel to the load wrench provides a useful simplification. If it is assumed that the normals to the object at the point of contract pass through the centroid of the contact points, the computations required to find the interaction forces are considerably simplified. Some simple grasps are used to evaluate the proposed algorithms.>

The problem of force distribution in systems involving multiple frictional contacts between actively coordinated mechanisms and passive objects is examined. The special case in which the contact interaction can be modeled by three components of forces (zero moments) is particularly interesting. The Moore-Penrose generalized inverse solution for such a model (point contact) is shown to yield a solution vector such that the difference between the forces at any two contact points projected along the line joining the two points vanishes. Such a system of forces is described by a helicoidal vector field which is geometrically similar to the velocity field in a rigid body twisting about an instantaneous screw axis. A method to determine this force system is presented. The possibility of superposing another force field which constitutes the null system is investigated.>

The problem of force distribution in systems involving multiple frictional contacts between actively coordinated mechanisms and passive objects is examined. The special case in which the contact interaction can be modeled by three components of forces (zero moments) is particularly interesting. The Moore-Penrose generalized inverse solution for such a model (point contact) is shown to yield a solution vector such that the difference between the forces at any two contact points projected along the line joining the two points vanishes. Such a system of contact forces is described by a helicoidal vector field which is geometrically similar to the velocity field in a rigid body twisting about an instantaneous screw axis. A method to determine this force system is presented. The possibility of superposing another force field which constitutes the null system is also investigated.>

The work described in this paper addresses the problem of determination of the appropriate distribution of forces between the fingers of a multifingered gripper grasping an object. The system is statically indeterminate and an optimal solution for this problem is desired for force control. A fast and efficient sub-optimal method for computing the grasping forces is presented. This method is based on the superposition of finger-interaction forces on equilibrating forces. An interaction force is defined as the component of the vector difference of the finger contact forces at any two fingers along the line joining the two contact points. They are computed based on the assumption that the normals at the point of contact pass through the centroid of the contact points and are therefore independent of the actual geometry of the object. The contact interaction is modelled as a point contact. The problems associated with making the algorithm independent of the object geometry are explored.