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Exercise 4.1 - Dynamic Term Computations

In this exercise you will setup functions to compute the dynamics terms of the lagrange formulation for a threelink manipulator.

image_0.svg
Link Mass [kg] Link width [m] Link height [m] Link length [m] Center of mass [m]
1 5 0.1 0.1 0.3 0.15
2 3 0.1 0.1 0.5 0.25
3 3 0.1 0.1 0.5 0.25

The manipulator can be modeled using these DH parameters:

Link a [m] alpha d [m] theta
1 0.3 \(\displaystyle \frac{\pi }{2}\) 0.2 \(\displaystyle q_1\)
2 0.5 0 0 \(\displaystyle q_2\)
3 0.5 0 0 \(\displaystyle q_3\)

For this tutorial only consider the three bar links.

You can compute their inertia as follows:

image_1.png

$$ I_{\textrm{xx}} =\frac{1}{12}\cdot \;m\cdot \left(w^2 +h^2 \right) $$

$$ I_{\textrm{yy}} =I =\frac{1}{12}\cdot \;m\cdot \left(w^2 +h^2 \right) $$

$$ I_{\textrm{zz}} =\frac{1}{12}\cdot \;m\cdot \left(w^2 +L^2 \right) $$

Task 1: Setup

Calculate the inertias for the links in their Center of Mass and store them in the variables:

  • I1
  • I2
  • I3

Setup a symbolic array q containing the following real symbolic variables for the joint angles

  • q1
  • q2
  • q3

Setup a symbolic array qd containing the following real symbolic variables for the joint velocities

  • qd1
  • qd2
  • qd3 
    I1 = [];
I2 = []; 
I3 = []; 
q  = []; 
qd = [];

You can check your work by clicking the Run: 

check_exercise('4-1-1')

Task 2: Setup the Robot

Load the urdf file for the threelink manipulator by using

  • importrobot("threelink_noInertia.urdf")

Store it in the variable

  • threelink
  • Set set the gravity to \(\left\lbrack \begin{array}{c} 0\newline 0\newline -9\ldotp 81 \end{array}\right\rbrack\)
  • Set the Inertias in the correct frame
  • Set the center of mass for each body

hint: remember to use the parallel axis shift

threelink = [];

You can check your work by clicking the Run: 

check_exercise('4-1-2')

Task 3: Inertia Matrix

Build the Inertia matrix for the three link manipulator using the symbolic toolbox.

Store the matrix in the variable

  • B ( B should depend on q1, q2, q3) 
    B = 0;

You can check your work by clicking the Run: 

check_exercise('4-1-3')

Task 3: Coriolis Matrix

Build the Coriolis matrix for the three link manipulator using the symbolic toolbox.

Store the matrix in the variable

  • C ( C should depend on q1, q2, q3, qd1, qd2, qd3) 
    C = 0;

hint: in Case you verify your matrice yourself, remember that the toolbox function velocityProduct returns C*qd

You can check your work by clicking the Run: 

check_exercise('4-1-4')

Task 5: Gravity Compensation

Build the Gravity compensation term for the three link manipulator using the symbolic toolbox.

Store the matrix in the variable

  • G (G should depend on q1, q2, q3) 
    G = 0;

You can check your work by clicking the Run: 

check_exercise('4-1-5')