An apparatus for mastering a robot includes two balls, each of which has a spherical surface configured to simultaneously contact its respective set of mutually orthogonal locating surfaces of a fixture only when the robot is in a predetermined reference position and orientation. The fixture is mounted on a pedestal of the robot base in a fixed spatial position. One of the balls is mounted for movement with a free end of an outer arm of an arm assembly of the robot. The robot also includes a wrist mechanism. A first end of the wrist mechanism is connected to the free end of the outer arm and a second end of the wrist mechanism is adapted to support a tool. The second ball is mounted for movement with the second end of the wrist mechanism. First and second clamping mechanisms are associated with a fixture and their respective balls for clamping their respective balls so that the generally spherical surfce of each of the balls simultaneously contacts each of the surfaces of its set of locating surfaces to thereby secure the robot in the predetermined spatial position and orientation. Each of the clamping mechanisms includes a circular clamping surface having a center. Each clamping mechanism is supported for pivotal movement on the fixture about a pivot axis offset from its ball's center so that its clamping surface defines an eccentric cam. Preferably, the clamping surface is conical.