Another early paper is Cheng and Orin, вЂњEfficient Algorithms for Optimal Force Distribution, the Compact Dual LP MethodвЂќ, IEEE Trans.

Kerr and Roth, supra., was an early work in this area. The basic force optimization problem has been studied for several years. Each point on the trajectory has an associated external wrench and the computation finds the maximum grasping force over the set of external wrenches. For example, this arises when finding the maximum grasping force that is required to move an object along a given trajectory. A FOP is solved for each grasp configuration and the overall computation finds the configuration with the minimum grasping force.ДЄnother related problem is finding the maximum over a set of related force optimization problems. For example, this arises when choosing a desirable grasp configuration on an object. One is finding the minimum over a set of related force optimization problems. There are several other related problems. вЂњQuantitative Steinitz's theorems with applications to multifingered graspingвЂќ, Proceedings of the 20 th ACM Symposium on Theory of Computing, pages 341-351, 1990 and in Ferrari and Canny, вЂњPlanning optimal grasps, Proceedings of the 1992 IEEE International Conference on Robotics and Automation, pages 2290-2295, Nice, France, May 1992. The problem of finding a set of optimal forces is referred to as the вЂњforce optimization problemвЂќ or вЂњFOPвЂќ.ДЄ related problem is the ability to resist an arbitrary external wrench. Various formulations of the problem use various specific functions, as described below. A good choice is one with minimum force, as measured by some function of the contact forces. There may be many possible choices that achieve equilibrium. An early paper discussing this issue is Kerr and Roth, вЂњAnalysis of multifingered handsвЂќ, The International Journal of Robotic Research, 4(4): 3-17, 1986. The most basic requirement is the ability to restrain an object against a specified external wrench, for example that due to gravity. A survey of robotic grasping may be found in Bicchi and Kumar, вЂњRobotic grasping and contact: A reviewвЂќ, IEEE International Conference on Robotics and Automation ( ICRA), pp 348-353, 2000. The issues may be illustrated by considering grasping by a robotic hand, where the problem is choosing a set of contact forces used to grasp an object. A third example is multi-legged vehicles, where the kinematic-chains are closed through the terrain. Another example is cooperating manipulators, where the kinematic chains are closed through the load.

One example is grasping by a robotic hand, where the kinematic chains are closed through the object grasped. In such situations, an important problem is choosing an optimal distribution of contact forces to achieve some objective. The present invention is concerned with mechanisms having closed kinematic chains involving multiple frictional contacts between an actively controlled structure and an object. The present invention relates generally to the computation of contact forces and, more particularly, to the computation of optimal contact forces with friction.

L force inward h force outward