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Technical Briefs

# A New Method for Dynamic Modeling of Flexible-Link Flexible-Joint Manipulators

[+] Author and Article Information
M. Vakil, P. N. Nikiforuk

R. Fotouhi1

1

Corresponding author.

J. Vib. Acoust 134(1), 014503 (Dec 28, 2011) (11 pages) doi:10.1115/1.4004677 History: Received July 12, 2010; Revised March 25, 2011; Published December 28, 2011; Online December 28, 2011

## Abstract

In this article, by combining the assumed mode shape method and the Lagrange’s equations, a new and efficient method is introduced to obtain a closed-form finite dimensional dynamic model for planar Flexible-link Flexible-joint Manipulators (FFs). To derive the dynamic model, this new method separates (disassembles) a FF into two subsystems. The first subsystem is the counterpart of the FF but without joints’ flexibilities and rotors’ mass moment of inertias; this subsystem is referred to as a Flexible-link Rigid-joint manipulator (FR). The second subsystem has the joints’ flexibilities and rotors’ mass moment of inertias, which are excluded from the FR; this subsystem is called Flexible-Inertia entities (FI). While the method proposed here employs the Lagrange’s equations, it neither requires the derivation of the lengthy Lagrangian function nor its complex derivative calculations. This new method only requires the Lagrangain function evaluation and its derivative calculations for a Single Flexible link manipulator on a Moving base (SFM). By using the dynamic model of a SFM and the Lagrange multipliers, the dynamic model of the FR is first obtained in terms of the dependent generalized coordinates. This dynamic model is then projected into the tangent space of the constraint manifold by the use of the natural orthogonal complement of the Jacobian constraint matrix. Therefore, the dynamic model of the FR is obtained in terms of the independent generalized coordinates and without the Lagrange multipliers. Finally, the joints’ flexibilities and rotors’ mass moment of inertias, which are included in the FI, are added to the dynamic model of the FR and a closed-form dynamic model for the FF is derived. To verify this new method, the results of simulation examples, which are obtained from the proposed method, are compared with those of a full-nonlinear finite element analysis, where the comparisons indicate sound agreement

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## Figures

Figure 1

Schematic of a two-link FF with its links’ and joints’ coordinate frames

Figure 2

(a) FR for the two-link FF presented in Fig. 1. (b) FI for two-link FF presented in Fig. 1.

Figure 3

Schematic of a SFM (ith SFM of an n-link FR)

Figure 4

Decomposing of a FR in Fig. 2 into two SFMs

Figure 5

Schematic of the ith link of the FF with the applied actuator torques

Figure 6

Detail schematic of a two-link FF and the parameter “Dee” used in Eq. 43; Manipulator composed of dotted-line is the shadow rigid-link manipulator

Figure 7

Inverse dynamic torques applied by the actuators located at the shoulder and elbow joints to the two-link FF

Figure 8

(a) Shoulder joint’s rotation (θ1) J. (b) Shoulder link’s rotation (θ1) L. (c) Deviation of the shoulder joint’s rotation from the shoulder link’s rotations, Dsh=(θ1) J-(θ1) L. (d) Deviation of the shoulder joint’s rotation from the shoulder joint’s rotation of FR, DFRsh=(θ1) J-(θ1) JFR.

Figure 9

(a) Elbow joint’s rotation (θ2) J. (b) Elbow link’s rotation (θ2) L. (c) Deviation of the elbow joint’s rotation from the elbow link’s rotations, Del=(θ2) J-(θ2) L. (d) Deviation of the elbow joint’s rotation from the elbow’s joint rotation of FR, DFRel=(θ2) J-(θ2) JFR.

Figure 10

(a) End-effector path. (b) Component of the end-effector velocity in X direction. (c) Component of the end-effector velocity in Y direction.

Figure 11

(a) Lateral deflection ξ1(L1) in meters. (b) Lateral deflection ξ2(L2) in meters.

## Errata

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