A geometric approach to deriving position/Force trajectroy in fine motion

The synthesis of position/force trajectories for compliant motion is an important task for the control of industrial robots in assembly tasks. The position and orientation of the manipulator can be specified by a set of six linear independent constraints. Some are caused by contact with other objects (natural position constraints) and the others are applied artificially (artificial position constraints). Since the uncertainties and the modeling errors make the exact natural position constraints incomputable, a set of contact forces (artificial reaction-force constraints) is servoed to implicitly reflect their effects. Two mappings, T-C and C-PF, are defined which derive the nominal position trajectory P(t) and freedom decomposition function S(t) from the attributes of a given task, and then the desired artificial position and reaction-force constraints. In the T-C mapping, for an insertion task, the convex operand shrinks to a line and the three 2-D cut-diagrams of the concave operand are derived and expanded correspondingly. The nominal position trajectory is determined from these expanded cut-diagrams. Three 2-D trapezoids are defined to model the uncertainties. The intersections between the trapezoids and the cut-diagrams are checked to determine the values of freedom decomposition function S(t). Consequently, in the C-PF mapping the time sequences of the artificial position constraints and artificial reaction-force constraints are derived from the nominal position trajectory and the freedom decomposition function.