Computation of the Output Torque, Power and Work of the Driving Motor for a Redundant Parallel Manipulator

Introduction

There are mainly two different types of redundancy for the parallel manipulators: a) kinematic redundancy and b) actuation redundancy. A parallel manipulator is said to be kinematically redundant manipulator when its mobility of the mechanism is greater than the required degrees of freedom of the moving platform. On the other hand, a parallel manipulator is called redundantly actuated manipulator when the number of actuators is greater than the mobility of the mechanism. It is believed that redundancy can improve the ability and performance of parallel manipulator (Kim, 2001; Merlet, 1996; Nokleby, 2005; Wang, 2004; Cheng, 2003; Müller, 2005; Mohamed, 2005; Ebrahimi, 2007; Zhao, 2009). There are some advantages for the redundant parallel manipulator such as avoiding kinematic singularities, increasing workspace, improving dexterity, enlarging load capability and so on. It is shown that the redundantly actuated parallel manipulator has a better dynamic characteristic than its non-redundant counterpart considering the presented dynamic index (Zhao, 2009). Redundant actuation in a parallel manipulator can be implemented by the following approaches. The first one is to actuate some of the passive joints within the branches of parallel manipulator. The second one is to add some additional branches beyond the minimum necessary to actuate the parallel manipulator. The last one can be the hybrid of the above two approaches.

The dynamics of the manipulator are of importance in the areas of simulation, control and dynamic optimum design. Many works have been done on the dynamics of parallel manipulators. Several approaches, including the Newton-Euler formulation (Carvalho, 2001; Dasgupta, 1998), the Lagrangian formulation (Lee, 1988; Miller, 1992), the Kane formulation (Ben-Horin, 1998; Liu, 2000) and the virtual work principle (Li, 2005; Sokolov, 2007; Wang, 1998; Tsai, 2000; Zhu, 2005; Staicu, 2009) have been applied to the dynamics analysis of parallel manipulators. In fact, the inverse dynamics of parallel manipulators involve almost all of the mechanics principles. Along with these mechanics principles, many mathematic methods such as screw theory (Gallardo, 2003), Lie algebra (Muller, 2003), natural orthogonal complement (Khan, 2005), motor algebra (Sugimoto, 1987), group theory (Geike, 2003), symbolic programming (Geike, 2003; McPhee, 2002), geometric approach (Selig, 1999), parallel computational algorithms (Gosselin, 1996) and system identification (Wiens, 2002) have also been adopted to the dynamics of parallel manipulators. However, the results computed by different methods have been shown to be equivalent.