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Research Papers

# On the Modeling of Leg Constraints in the Dynamic Analysis of Gough/Stewart-Type Platforms

[+] Author and Article Information
Marco Carricato

Department of Mechanical Engineering (DIEM), University of Bologna, Viale Risorgimento 2, 40136 Bologna, 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

Ordinarily, the U-joint 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. According to this practice, a leg such that represented in Fig. 2 would unrealistically lie, in its central configuration, in a horizontal plane.

It is the case, for instance, of legs whose extension is provided by a screw jack actuated by a motor mounted sideways with respect to the leg longitudinal direction.

In an inverse dynamic problem, the trajectory of the end-effector is assigned, whereas the motion of the other members, including those connected to the actuators, has to be determined. Conversely, in a direct dynamic problem, the motion imposed by the motors is given and the general movement of the mechanism, including the end-effector, has to be calculated.

It suffices to orderly project the equation of the moment equilibrium about $O$ of the leg on $y$, $X$, and $z$.

The last term at the right-hand side of Eq. 26 is the projection on $z$ of the resultant moment about $O$ of the inertia forces acting on member 3.

J. Comput. Nonlinear Dynam 4(1), 011008 (Nov 12, 2008) (8 pages) doi:10.1115/1.3007974 History: Received August 31, 2007; Revised February 20, 2008; Published November 12, 2008

## Abstract

This paper investigates the dynamic model of Gough/Stewart-type platforms, focusing on aspects concerning the proper modeling of leg constraints that seem to have been overlooked in a nonnegligible part of the available literature related to the subject. It shows, in particular, how failure in correctly modeling the arrangement of the universal joints connecting the legs to the base may cause positional, kinematic, and dynamic miscalculations. The effects of these errors are investigated and assessed for relevant cases of leg geometries. Numerical simulations are provided to validate the theory.

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## Figures

Figure 1

Constraint force (FU), constraint torque (MU), and relative instantaneous rotation (ω10) between the members connected by a universal joint

Figure 2

Euler-angle parametrization of the leg-axis orientation

Figure 3

Schematic of a GSP leg (U: universal joint, P: prismatic joint, and S: spherical joint)

Figure 4

Pose of the revolute axes of U(X,y) and U′(X′,y′) with respect to the leg axis

Figure 5

GSP models assessed in the numerical simulations; symmetric legs in (a) and asymmetric legs in (b) are equipped with screw jacks

Figure 6

Top view of the simulated GSP models

Figure 7

Simulation of the GSP with asymmetric legs and actuation jacks: graphs of Fa and ‖FS‖ as functions of the dimensionless time ft over the interval [0,2π] for the leg identified by the index h=6

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