Example 6.4:: Transferring a Wrench (MATLAB)

This example transfers a wrench from the end-effector frame $O_{\mathrm E}{\mathrm -}xyz$ to the link 6 frame $O_{6}{\mathrm -}xyz$ .

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Given Wrench $^{\mathrm E}\overline{\mathbf W}_{\mathrm E}$
The homogeneous transformation from $O_{\mathrm E}{\mathrm -}xyz$ to $O_{6}{\mathrm -}xyz$ .
Extract the rotation matrix ${\mathbf R}_{\mathrm E}^{6}$ and the position vector $^{6}{\mathbf X}_{{\mathrm E}/6}$ from ${\mathbf T}_{\mathrm E}^{6}$ .
Create the cross product operator $[\boldmath{X}]$ .
Form the wrench coordinate transfer matrix ${\mathbf \Lambda}_{\mathrm E}^{6}$ .
Wrench transferred from $O_{\mathrm E}{\mathrm -}xyz$ to $O_{6}{\mathrm -}xyz$ .

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Given Wrench $^{\mathrm E}\overline{\mathbf W}_{\mathrm E}$

WEResE = [-10*sind(54); 0; 10*cosd(54); 0; -30; 0]; % [Force (N); Moment (N-cm)]

The homogeneous transformation from $O_{\mathrm E}{\mathrm -}xyz$ to $O_{6}{\mathrm -}xyz$ .

TETo6 = [1 0 0  0;
         0 1 0  0;
         0 0 1 22;
         0 0 0  1];

Extract the rotation matrix ${\mathbf R}_{\mathrm E}^{6}$ and the position vector $^{6}{\mathbf X}_{{\mathrm E}/6}$ from ${\mathbf T}_{\mathrm E}^{6}$ .

RETo6 = TETo6(1:3,1:3)
XERel6Res6 = TETo6(1:3,4)

RETo6 =

     1     0     0
     0     1     0
     0     0     1


XERel6Res6 =

     0
     0
    22

Create the cross product operator $[\boldmath{X}]$ .

XCross = [ 0             -XERel6Res6(3)  XERel6Res6(2);
           XERel6Res6(3)  0             -XERel6Res6(1);
          -XERel6Res6(2)  XERel6Res6(1)  0           ]

XCross =

     0   -22     0
    22     0     0
     0     0     0

Form the wrench coordinate transfer matrix ${\mathbf \Lambda}_{\mathrm E}^{6}$ .

LambdaETo6 = [RETo6         zeros(3);
              XCross*RETo6  RETo6 ]

LambdaETo6 =

     1     0     0     0     0     0
     0     1     0     0     0     0
     0     0     1     0     0     0
     0   -22     0     1     0     0
    22     0     0     0     1     0
     0     0     0     0     0     1

Wrench transferred from $O_{\mathrm E}{\mathrm -}xyz$ to $O_{6}{\mathrm -}xyz$ .

W6Res6 = LambdaETo6*WEResE

This MATLAB example illustrates a computation from the textbook Fundamentals of Robot Mechanics by G. L. Long, Quintus-Hyperion Press, 2015. See http://www.RobotMechanicsControl.info for other relevant files.

Example 6.4:: Transferring a Wrench (MATLAB)

Contents

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Given Wrench

The homogeneous transformation from to .

Extract the rotation matrix and the position vector from .

Create the cross product operator .