A hybrid system is a dynamical system with both discrete and continuous state changes. For analysis purposes, it is often useful to abstract a system in a way that preserves the properties being analysed while hiding the details that are of no interest. We show that interesting classes of hybrid systems can be abstracted to purely discrete systems while preserving all properties that are definable in temporal logic. The classes that permit discrete abstractions fall into two categories. Either the continuous dynamics must be restricted, as is the case for timed and rectangular hybrid systems, or the discrete dynamics must be restricted, as is the case for o-minimal hybrid systems. In this paper, we survey and unify results from both areas.

A general strategy for solving the motion planning problem for real analytic, controllable systems without drift is proposed. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the system vector fields are added. Using formal calculations with a product expansion relative to P. Hall basis, another control is produced that achieves the desired result on the formal level. This provides an exact solution of the original problem if the given system is nilpotent. For a general system, an iterative algorithm is derived that converges very quickly to a solution. For nonnilpotent systems which are feedback nilpotentizable, the algorithm, in cascade with a precompensator, produces an exact solution. Results of simulations which illustrate the effectiveness of the procedure are presented.>

The existence of two- and three-finger grasps in the presence of arbitrarily small friction is shown for 2D and 3D smooth objects using a simple technique. No convexity of the objects is assumed. The existence of finger gaits for rotating a planar object using three and four fingers is proved. Additional results for smooth convex objects which describe two different finger gaits using four fingers are presented. The purpose is to move one or more fingers at any one time and at the same time to maintain a grip with the others, and thereby to move the object further than or in other ways than it could be moved with a fixed grip.>

The authors address the problem of optimal force distribution among three point fingers holding a planar object. A scheme that reduces the nonlinear optimization problem to an easily solved generalized eigenvalue problem is proposed. This scheme generalizes and simplifies results of Z. Ji and B. Roth (1988). The generalizations include all possible geometric arrangements and extensions to three dimensions and to the case of variable coefficients of friction. For the two-dimensional case with constant coefficients of friction, it is proved that except for some special cases, the optimal grasping forces, in the sense of minimizing the dependence on friction, are those for which the angles with the corresponding normals are all equal (in absolute value).>

An approach to manipulation tasks involving dextrous hands is presented. This approach treats the gripped object as a virtual finger and hence reduces the description of the task objectives to target forces and torques at reference points on the object. Computations are carried out for the case of three fingers holding a planar object. The authors implemented these ideas on the Four Finger Manipulator at New York University. Experimental results are presented for door opening and wall following with a gripped tool.>