A fast, accurate and robust position controller is proposed for a planar three degree of freedom robot arm actuated by rotary Shape Memory Alloy actuators and servomotors. Servomotors are used to actuate the first link and the gripper, while the remaining two links are actuated by a combination of shape memory alloy wires and pulleys. Initially, a model of the robot arm is developed for theoretical controller development. The model combines robot kinematics and dynamics with the SMA wire heat convection, constitutive law, and phase transformation equations. The model is then used to develop several nonlinear position controllers based on the Variable Structure Control, also called sliding mode control. The controller development is of an evolutionary nature starting from a simple switching control based only on position feedback and then adding velocity and integral feedbacks, respectively. Several simulations of the proposed controllers are presented. Several experiments have been performed with a desktop prototype of the robot arm. The experimental results verify the effective and accurate performance of the controllers despite significant modeling inaccuracies and parameter uncertainties.