ASME Journal of Mechanisms and Robotics, Vol. 1, No. 3, August 2009, Paper No. 031005, pp. 1-16. ©2009 ASME 1 Marco Carricato University of Bologna, mar- co.carricato@mail.ing.unibo.it Clément Gosselin Université Laval gosselin@gmc.ulaval.ca A Statically Balanced Gough/Stewart-type Platform: Conception, Design and Simulation Gravity compensation of spatial parallel manipulators is a relatively recent topic of investigation. Perfect balancing has been accomplished, so far, only for parallel mechanisms in which the weight of the moving platform is sustained by legs comprising purely rotational joints. Indeed, balancing of parallel mechanisms with translational actuators, which are among the most common ones, has been traditionally thought possible only by resorting to additional legs containing no prismatic joints between the base and the end-effector. This paper presents the conceptual and mechanical designs of a balanced Gough/Stewart-type manipulator, in which the weight of the platform is entirely sustained by the legs comprising the extensible jacks. By the integrated action of both elastic elements and counterweights, each leg is statically balanced and it generates, at its tip, a constant force contributing to maintaining the end-effector in equilibrium in any admissible configuration. If no elastic elements are used, the resulting manipulator is balanced with respect to the shaking force too. The performance of a study prototype is simulated via a model in both static and dynamic conditions, in order to prove the feasibility of the proposed design. The effects of imperfect balancing, due to the difference between the payload inertial characteristics and the theoretical/nominal ones, are investigated. Under a theoretical point of view, formal and novel derivations are provided of the necessary and sufficient conditions allowing (i) a body arbitrarily rotating in space to rest in neutral equilibrium under the action of general constant-force generators, (ii) a body pivoting about a universal joint and acted upon by a number of zero-free-length springs to exhibit constant potential energy, (iii) a leg of a Gough/Stewart-type manipulator to operate as a constant-force generator. Keywords: gravity compensation, static balancing, force balancing, Gough-Stewart platform, parallel manipulators. 1 Introduction A mechanism is said to be statically balanced (or gravity- compensated) if zero external actions are required to maintain it at rest in any assumable configuration. In such an instance, the actu- ators impart accelerations to the moving members, but they do not contribute to supporting their weight. A mechanism is said to be dynamically balanced if the inertia forces globally acting on it are equivalent to zero, namely if both their resultant vector (shaking force) and resultant moment about an arbitrary pole (shaking mo- ment) are nought. The first condition is satisfied if the overall cen- tre of mass (c.m.) remains stationary during movement, while the second one (for which the vanishing of the shaking force is a ne- cessary requisite) is met if the total angular momentum remains constant. A dynamically balanced mechanism transmits a resultant constant force (equal to its weight) to the fixed frame for any ad- mitted motion. If only the shaking force identically vanishes, the mechanism is simply addressed as force balanced (and it is, inhe- rently, gravity-compensated). There are several reasons calling for the incorporation of some balancing system in mechanism design. Potential advantages (de- pending on the type of balancing strategy adopted) may be, in- deed, the following (Rivin 1988): − a less onerous and more uniform loading of actuators due to the reduction in peak-torque requirements induced by gravity and/or inertia forces, resulting in enhanced energy efficiency and a need for lighter and less powerful motors; − improvement of safety and undersizing of brakes (a balanced system is less vulnerable to the action of gravity forces in the occurrence of power-supply failures), leading to further weight saving and better efficiency (disengagement of fail- safe brakes requires energy supply during the operation of the mechanism); − enhancement of dynamic response and control-system per- formances, due to the reduction in coupling and nonlinear terms in the equations of motion; − degrease in structural compliance and improved accuracy; − lower sensitivity of input torques and tracking peformances in respect to payload variations; − reduction in vibrations and wear induced by base-transmitted unbalanced forces. Benefits of balancing are more prominent in direct-drive ma- nipulators. In this case, no speed reduction is provided by me- chanical gearings, and motors (as well as brakes) are required to exert large amount of continuous torque for long times, making size and overheating a critical issue; complicated dynamics have also a more direct influence upon driving systems (Asada and Youcef-Toumi 1984). Balancing may mitigate such problems, indeed. In this paper, we are mainly interested in force and static ba- lancing. There are two principal design approaches by way of which these may be attained, namely mass compensation and elas- tic compensation. By the first method, force balancing is achieved by suitably distributing link masses within the mechanism, normally append- ing counterweights in convenient locations. By the second ap- proach, static balancing is attained by adding elastic elements (typically springs) to the system, so that the variations of gravita- tional energy caused by motion are compensated by opposite changes in the elastic energy stored by the springs. Each approach has its own merits and drawbacks (Mahalingam and Sharan 1986; Gopalswamy, Gupta and Vidyasagar 1992; Kolarski,