An electromechanical system operates in part through physical interaction with an operator, and includes a multi-axis robot, a controller, and a counterbalance mechanism connected to the robot. The counterbalance mechanism includes a base structure connected to a set of linkages, a pneumatic cylinder, a spring-loaded cam assembly, and an optional constant force spring. The linkages form a four-bar parallelogram assembly connectable to a load. The cylinder and cam assembly, and optional constant force spring, each impart respective vertical forces to the parallelogram assembly. The forces combine to provide gravity compensation and self-centering functions or behaviors to the load, enabling the load to move with a vertical degree of freedom when manually acted upon by the operator, and to return the load to a nominal center position.