Static balancing of spatial four-degree-of-freedom parallel mechanisms

The static balancing of four types of spatial four-degree-of-freedom parallel mechanisms or manipulators is discussed in this paper. Two static balancing methods, namely, using counterweights and using springs, are used. The first method leads to a mechanism with a stationary global center of mass and the second one leads to a mechanism with constant total potential energy (including the elastic potential energy stored in the springs). In both cases, the resulting mechanisms can be brought to static equilibrium in any configuration of their workspace with zero actuator forces or torques. In order to obtain the balancing conditions, the position vector of the global center of mass and the total potential energy of the mechanisms are first expressed as functions of the position and orientation of the platform and of the special fifth leg. Then, the conditions for static balancing are derived from the expressions obtained. Examples are given to illustrate the application of the results to the design of statically balanced spatial four-dof mechanisms or manipulators.