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Research Papers

# A Constrained Lagrange Formulation of Multilink Planar Flexible Manipulator

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

R. Fotouhi1

To solve the governing equations in the AMM model, the fourth-order Runge–Kutta with a sampling time of $0.0008s$ was used. The solver of the FEA was the Newmark method with the full Newton–Raphson technique for updating matrices with the same sampling time of $0.0008s$.

For simplicity in the rest of this article, the term “coordinate” refers to the “generalized coordinate.” (The generalized coordinates of a system are the geometrical quantities that by knowing them, it is possible to draw a diagram of the system. The minimum number of generalized coordinates required to specify the position of a system is the number of degree of freedom (DOF) of that system (27).)

Lagrange’s equation, Eq. 2, must be satisfied for every single component of the vector $q$.

Hereafter, the terms “mass matrix” and “matrix representing the Coriolis and centrifugal forces” refer to the coefficients of the multiplier of the second and the first derivative of the generalized coordinate with respect to time, which is obtained after the differentiation of the kinetic energy, respectively.

If a velocity constraint is holonomic, then there exists an integration factor for which the Pfaffian form of the constraint equation becomes a perfect differential (27).

The difference between the end-effector location of a FLM and the end-effector location of a rigid, shadow, link manipulator with the same lengths and joint rotation is called end-effector deviation of a FLM with respect to the shadow manipulator (see Fig. 5).

From a simple analysis for a rotating single FLM, the first five nonzero natural frequencies were compared against the analytical values when increasing the number of elements from 2 to 10. The finding indicated that ten elements is reasonably capable of capturing up to five natural frequencies. Details can be found in Appendix .

1

Corresponding author.

J. Vib. Acoust 130(3), 031007 (Apr 03, 2008) (16 pages) doi:10.1115/1.2827455 History: Received April 11, 2007; Revised September 24, 2007; Published April 03, 2008

## Abstract

In this article, the closed-form dynamic equations of planar flexible link manipulators (FLMs), with revolute joints and constant cross sections, are derived combining Lagrange’s equations and the assumed mode shape method. To overcome the lengthy and complicated derivative calculation of the Lagrangian function of a FLM, these computations are done only once for a single flexible link manipulator with a moving base (SFLMB). Employing the Lagrange multipliers and the dynamic equations of the SFLMB, the equations of motion of the FLM are derived in terms of the dependent generalized coordinates. To obtain the closed-form dynamic equations of the FLM in terms of the independent generalized coordinates, the natural orthogonal complement of the Jacobian constraint matrix, which is associated with the velocity constraints in the linear homogeneous form, is used. To verify the proposed closed-form dynamic model, the simulation results obtained from the model were compared with the results of the full nonlinear finite element analysis. These comparisons showed sound agreement. One of the main advantages of this approach is that the derived dynamic model can be used for the model based end-effector control and the vibration suppression of planar FLMs.

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## Figures

Figure 1

Schematic of a SFLMB with a moving base

Figure 2

Schematic of the first link of a FLM

Figure 3

Schematic of a FLM

Figure 4

The ith link of a FLM with the applied actuator torques

Figure 5

Schematic of a flexible two-link manipulator

Figure 6

Example 1, applied bang-bang torque to the shoulder and elbow joints

Figure 7

Example 1, rotation of the shoulder joint, one element per link in FEA

Figure 8

Example 1, rotation of the elbow joint, one element per link in FEA

Figure 9

Example 1, end-effector paths, one element per link in FEA

Figure 10

Example 1, end-effector velocity, XI direction, one element per link in FEA

Figure 11

Example 1, end-effector velocity, YI direction, one element per link in FEA

Figure 12

Example 1, absolute value of D shown in Fig. 5, one element per link in FEA

Figure 13

Example 2, rotation of the shoulder joint, ten elements per link in FEA

Figure 14

Example 2, rotation of the elbow joint, ten elements per link in FEA

Figure 15

Example 2, end-effector paths, ten elements per link in FEA

Figure 16

Example 2, end-effector velocity, XI direction, ten elements per link in FEA

Figure 17

Example 2, end-effector velocity, YI direction, ten elements per link in FEA

Figure 18

Example 2, absolute value of D shown in Fig. 5, ten elements per link in FEA

Figure 19

Example 3, applied rigid torque to the shoulder motor and elbow joints

Figure 20

Example 3, rotation of the shoulder joint, ten elements per link in FEA

Figure 21

Example 3, rotation of the elbow joint, ten elements per link in FEA

Figure 22

Example 3, end-effector paths, ten elements per link in FEA

Figure 23

Example 3, end-effector velocity, XI direction, ten elements per link in FEA

Figure 24

Example 3, end-effector velocity, YI direction, ten elements per link in FEA

Figure 25

Example 3, absolute value of D shown in Fig. 5, ten elements per link in FEA

Figure 26

Schematic of a rotating single FLM

Figure 27

Comparison of FEA and analytical natural frequency errors as a function of number of elements for the first five modes of a rotating single flexible manipulator

## Errata

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