In order for robots to become more useful they must be able to adapt and operate in foreign or unpredictable environments. The goal of this thesis is to present an algorithm that will enable a robot to autonomously explore its environment by touch and then estimate the shape of objects it encounters. To demonstrate the feasibility and functionality of such an algorithm, it was fully implemented on a MERLIN 6540 industrial robot. A unique compliant end-effector (consisting of a trackball mounted to a force/torque sensor on a sliding mechanism) and a fuzzy logic force controller were developed to overcome the difficulties inherent in force control on a stepper motor robot. A Kalman filter based quadric shape estimator was then used to describe the objects encountered in the MERLIN's workspace. The minimization of a cost function based on the shape estimator's uncertainty guided the robot along an exploration trajectory designed to produce the fastest converging shape estimate. Results of various exploration trials using autonomous and preprogrammed trajectories are presented. In addition to shape estimates, surface curvature measurements were also obtained. The unique end-effector that provided compliance for the force controller was also able to measure the arc length traversed on the object's surface. Arc length combined with surface orientation makes it possible to determine local surface curvature. / Master of Science