The reachable workarea of a manipulator

Extremal reaches in polynomial time

“…It is known [9,11,16,19] that in all critical configurations, the endpoint axis meets all the hinges. When applied to polygonal chains, three cases are distinguished: the endpoint axis meets hinge i (line through pi and pi+1) either inside the segment pipi+1, outside it, or at one of the endpoints.…”

“…A necessary condition, satisfied by all critical points of the end-to-end (squared) distance function (points where the differential is zero, which include the extrema) was identified in the early 1980's [9,11,16,19]. However, the number of critical points increases exponentially with the number of joints and the absence of a criterion for distinguishing maxima and minima among them hampered computational advances: neither gradient-based optimization nor Monte Carlo sampling guarantee the correctness or accuracy of their "solutions".…”

How far can you reach?

“…A necessary condition for extremal non-zero 5 configurations was recognized and proven in several papers [8,13,19,16]. In the words of [16], this necessary condition says: "the line of sight from the base-point to the hand must intersect all turning axes" 6 , where the base-point may be chosen arbitrarily, the end-point is called "hand" and the "turning axes" are what we call hinges. However, all critical points with non-zero value for the squared distance function between the extreme points (not just the maxima) satisfy this condition.…”

“…By contrast, the "classical" necessary condition of [8,13,19,16] leads only to a verification procedure for being a critical point, not necessarily an maximum. The number of critical points of the endpoint distance function could be exponentially large, and -to the best of our knowledge -there is no known procedure that can isolate from them the maxima, based on this information alone.…”

“…Papers studying the maximum reach of revolute jointed robotic manipulators have appeared in the robotics literature since at least 1969 [10], and gained momentum in the early 1980's, when a general, necessary condition satisfied by all critical points of the endpoint distance function was identified [8,13,19,16]. An approximation method received in 1985 the ACM Distinguished Dissertation Award [12].…”