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Research Papers

# A Quasi-Static Model for Planar Compliant Parallel Mechanisms

[+] Author and Article Information
Cyril Quennouelle

Laboratoire de Robotique, Département de Génie Mécanique, Université Laval-Québec, QC, G1V 0A6, Canadacyril.quennouelle.1@ulaval.ca

Clément Gosselin

Laboratoire de Robotique, Département de Génie Mécanique, Université Laval-Québec, QC, G1V 0A6, Canadagosselin@gmc.ulaval.ca

Often in literature these quantities are referred to as vectors. However, since in a purely mathematical sense they are not vectors, they are referred to here as arrays. In this paper, the term vector will be reserved to quantities that are mathematical vectors, such as the position vector of a point for instance.

$d=D2+(L+l)2−D(L+l)cos(π/2)=8.5 cm$.

J. Mechanisms Robotics 1(2), 021012 (Jan 20, 2009) (9 pages) doi:10.1115/1.3046144 History: Received January 22, 2008; Revised November 01, 2008; Published January 20, 2009

## Abstract

In this paper, the mobility, the kinematic constraints, the pose of the end-effector, and the static constraints that lead to the kinematostatic model of a compliant parallel mechanism are introduced. A formulation is then provided for its instantaneous variation—the quasi-static model. This new model allows the calculation of the variation in the pose as a linear function of the motion of the actuators and the variation in the external loads through two new matrices: the compliant Jacobian matrix and the Cartesian compliance matrix that give a simple and meaningful formulation of the model of the mechanism. Finally, a simple application to a planar four-bar mechanism is presented to illustrate the use of this model and the new possibilities that it opens, notably the study of the kinematics for any range of applied load.

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## Figures

Figure 1

Schematic description of revolute and prismatic compliant actuators

Figure 2

Compliant parallel mechanism with a compliant actuated joint θ1a

Figure 3

Evolution of T for kρ=1 N cm−1

Figure 4

Evolution of T for kρ=1 N cm−1 and ρ0=28.5 cm

Figure 5

Evolution of the KSM: θ1a=F(ϕ)

Figure 6

Example of notations for joint coordinates ψ, ϕ, and λ

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