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Research Papers

# A Statically Balanced Gough/Stewart-Type Platform: Conception, Design, and Simulation

[+] Author and Article Information
Marco Carricato

Department of Mechanical Engineering (DIEM), University of Bologna, Viale Risorgimento 2, 40136 Bologna, BO, Italymarco.carricato@mail.ing.unibo.it

Clément Gosselin

Département de Génie Mécanique, Université Laval Pavillon Pouliot, 1065 Avenue de la Médecine, Québec, QC, G1V 0A6, Canadagosselin@gmc.ulaval.ca

When no means of storing elastic energy are used, static balancing conditions require the c.m. of the mechanism not to move in the direction of the gravity vector. For most mechanisms (in which point paths are necessarily closed curves), this results in keeping the c.m. stationary (a trivial exception are planar mechanisms moving parallelly to a horizontal plane).

The energy that may be accumulated in a spring is proportional to its own mass (for a given type and material) and the energy to be stored may be approximated by the weight of the moving bodies times their average displacement in the direction of gravity. By using this rule, Laliberté et al. (24) observed that for small systems the mass of the springs may be negligible, but this is not necessarily true for large ones.

In any case, a passive compensation system is likely to work well only if light changes in payload are expected during operation. Otherwise, an active self-adjusting gravity-balancing system is recommendable (37).

The concept of constant-force generators was presented by Nathan (46) referring to a hinged lever that generates a constant-magnitude vertical force throughout its rotation.

For instance, by alternatively setting $φl=0$, $π/2$, $π$, and $3π/2$, for all possible combinations of $l=1,2,3$, a set of linear equations may be obtained that only admits the trivial solution.

In this paper, the orientation of the $z$-axis is determined by point $P$, whereas in Refs. 25-26 the orientation of this axis is determined by the position of $C0$. It follows that, in these studies, it is also set, without loss of generality, $rC0,x=0$.

Ordinarily, the U-pair of each leg is arranged so as to maximize the leg range of motion, typically, in such a way that the joint cross plane is perpendicular to the leg in a reference central position. That is why the U-joint fixed-axis is unlikely to be vertical.

It may be noted that, in direct-drive applications, a major contribution to inertia forces comes from the high-speed rotation of screw-jack rotors (45). Screws 5, counter-rotating with respect to the main shaft, could provide, if conveniently sized, a natural means to limit the overall inertia torque transmitted to the frame.

In $J5,O$, the offset between the screw axis and the leg-axis is disregarded.

Equation 31 easily derives from Eqs. 8,11, by observing that $Q$ is the centroid of the equilateral triangle formed by the attachment points of the legs on the moving platform.

The total constraint system in the U-pair is made up by a force $U$ through $O$ and a moment $MU$ perpendicular to the joint cross plane (45).

For instance, by alternatively setting $φl=0,π/2,π,3π/2$, for all possible combinations of $l=1,2$, a set of linear equations is obtained that only admits the trivial solution.

J. Mechanisms Robotics 1(3), 031005 (Jul 14, 2009) (16 pages) doi:10.1115/1.3147192 History: Received August 02, 2008; Revised December 09, 2008; Published July 14, 2009

## Abstract

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, and (iii) a leg of a Gough/Stewart-type manipulator to operate as a constant-force generator.

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## Figures

Figure 1

A rigid body acted upon by constant forces and torques

Figure 2

Schematic of a constant-force generator of type UPS

Figure 3

Euler-angle parametrization of the leg-axis orientation

Figure 4

Mechanical model of a counterweighted UPS leg actuated by an electric screw-jack

Figure 5

A statically balanced GSP with electric screw-jack-based actuation: (a) tridimensional model and (b) top view

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