Static balancing of spatial six-degree-of-freedom parallel mechanisms with revolute actuators

The static balancing of spatial six-degree-of-freedom parallel mechanisms or manipulators with revolute actuators is studied in this paper. Two static balancing methods, namely, using counterweights and using springs, are used. The first method leads to mechanisms with a stationary global center of mass while the second approach leads to mechanisms whose total potential energy (including the elastic potential energy stored in the springs as well as the gravitational potential energy) is constant. The position vector of the global center of mass and the total potential energy of the manipulator are first expressed as functions of the position and orientation of the platform. Then, conditions for static balancing are derived from the resulting expressions. Finally, examples are given in order to illustrate the design methodologies. Introduction The balancing of mechanisms has been an important research topic for several decades (see for instance Lowen, 1983 for a literature review). A balanced mechanism leads to better dynamic characteristics and less vibrations caused by motion. Static and dynamic balancing of planar linkages has been studied extensively in the literature (see for instance Stevenson, 1973; Smith, 1975; Bagci, 1979; Gao, 1991; Ye, 1994). In the context of manipulators and motion simulation mechanisms, static balancing is defined as the set of conditions under corresponding author, Tel:(418)-656-3474, Fax:(418)-656-7415 1 which the weight of the links of the mechanism does not produce any torque (or force) at the actuators under static conditions, for any configuration of the manipulator or mechanism. This condition is also referred to as gravity compensation. Gravitycompensated serial manipulators have been designed in (Nathan, 1985; Hervé, 1986; Streit, 1989; Ulrich, 1991; Walsh, 1991) using counterweights, springs and sometimes cams and/or pulleys. A hybrid direct-drive gravity-compensated manipulator has also been developed in (Kazerooni, 1988). Moreover, a general approach for the static balancing of planar linkages using springs has been presented in (Streit, 1990). The balancing of spatial mechanisms has also been studied, for instance in (Bagci, 1983 and Walsh, 1991). However, to the knowledge of the authors, statically balanced spatial six-degree-of-freedom parallel manipulators or mechanisms cannot be found in the literature. Since spatial parallel mechanisms find more and more applications in robotics and flight simulation, their static balancing becomes an important issue. As mentioned above, a statically balanced parallel mechanism is one in which the actuators do not contribute to supporting the weight of the moving links, for any configuration. Hence, the actuators are used only to impart accelerations to the moving links, which leads to a reduction of the size and power of the actuators and results in the improvement of the accuracy of the control. In flight simulation, for instance, since the payload is very large (usually in the order of tons) and the motion of the platform of the mechanism is rather slow, the forces or torques exerted at the actuated joints are mainly due to the weight of the platform and links. Hence, if the mechanism is statically balCopyright  1998 by ASME anced, the actuating forces or torques will be greatly reduced, which will result in significant improvements of the control and energy efficiency. Finally, from a more general perspective, the design of a ' floating' gravity-compensated platform with six degrees of freedom may have several applications, including the simulation of space systems. In this paper, the static balancing of spatial six-degree-offreedom parallel mechanisms or manipulators with revolute actuators is addressed. Two approaches of static balancing are presented, namely, i) static balancing using counterweights and ii) using springs. When the mechanism is balanced using counterweights, a mechanism with a fixed global center of mass is obtained. In other words, the static balancing is achieved in any direction of the Cartesian space of the mechanism. This property is useful for applications in which the mechanism is needed to be statically balanced in all directions as for instance, when a system can be installed in different orientations with respect to the gravity vector. However, for some parallel mechanisms, static balancing with counterweights is difficult to realize. For example, in flight simulators, since the mass of the platform is very large, the counterweights required would be too large to be practical. Springs can be used in such instances. When springs are used, the total potential energy of the manipulator — gravitational and elastic — is set to be constant and the weight of the whole manipulator can be balanced with a much smaller total mass than when using counterweights, as pointed out in (Streit, 1990). However, a mechanism which is statically balanced using springs will be statically balanced for only one direction and magnitude of the gravity vector, which may be unsuitable for some applications. Both methodologies are discussed in this paper. Six-degree-of-freedom parallel mechanism with revolute actuators A spatial six-degree-of-freedom parallel mechanism or manipulator with revolute actuators is illustrated in Figs. 1 and 2. It consists of six identical legs connecting the base to the platform. Each of these legs consists of an actuated revolute joint attached to the base, a first moving link, a passive Hooke joint, a second moving link and a passive spherical joint attached to the platform. A parallel manipulator of this type was described in (Benea, 1996). The coordinate frame of the base, designated as the O x;y;z frame is fixed to the base with its Z-axis pointing vertically upward. Similarly, the moving coordinate frame O 0 x 0 ;y 0 ;z 0 is attached to the platform. The Cartesian coordinates of the platform are given by the position of point O with respect to the fixed frame, noted p = [x; y; z]T and the orientation of the platform (orientation of frame O xyz with respect to the fixed frame), represented by matrix 2 Figure 1. CAD model of a spatial six-degree-of-freedom parallel mechanism with revolute actuators.