Object manipulation is the art of moving things. For example, an object can be moved from one point to another by a robot hand that is in contact with the object. The contacts serve two functions: they transmit forces and impose motion constraints on the object. The motion of the
robot is constrained by the interactions at the contacts for control of the motion of the object. If the robot hand has multiple fingers, there is a coordinated manipulation of the
object by the multiple fingers. Furthermore, the motion of the object may be constrained in some directions because of contact with fingers. The multi-fingered robot hand in contact with the object forms a closed loop. Hence, the fingers' effort and the contact forces and moments required for support and for moving the object are coupled. Any external force that acts on the object to bring about motion will cause a consequential readjustment of forces at the finger tips of the multi-fingered robot hand. The tips of the multi-fingered robot hand may deform in order to accommodate the object and prevent motion. In the robotics literature, the motion and force are related by the equation v=Cf [1], where v is the velocity of the object, C is the compliance of the fingers and f is the applied force from the fingers. C is a diagonal matrix, which means that motion is allowed only in particular directions.For example, in the compliance matrix C ¼ diag Cx; Cy; Cz, when Cz is set to zero, then motion is allowed in the x–y direction only. This analogy can be applied to fixturing so that a workpiece held
in a machining fixture can be considered as a special case within the same understanding. In the case of machining fixtures, the workpiece cannot accelerate on its own but is displaced when an external force, such as the machining force, acts on the workpiece. Since displacement is the main concern here for workpieces being machined, a displacement equation will be most appropriate.This paper examines the effect of fixture compliance and cutting conditions on workpiece stability and discusses the coordination of multiple-point contact of the fixture elements with the workpiece. The coordination problem is divided into a number of phases: determining the force by multiple fixture elements, determining the contact and fixture stiffness and fixture compliance and determining the workpiece displacement. The force is used for maintaining equilibrium and for generating the restoring force.Frictional constraints are incorporated in the force equation.