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Exercise 5.1 - Threelink effort control using Symbolic Matrices

In this exercise you will use the matrices computed in Exercise 4.1 to control the Threelink manipulator in simulation using Simulink.

Start the Simulation

StartTutorialApplication('Simulation','Controller', 'Effort', 'Model','threelink', 'Docker', false);
StartTutorialApplication('Trajectory', 'Docker', false);
StartTutorialApplication('Safety_nodes','docker',false, 'model','threelink'); %sends a 0 torque when no other command has been sent

Remember that you can slow down the simulation as:

SetSimulationSpeed( SpeedFactor, 'docker', false)

Convert Symbolic Matrices to Functions

To use symbolic variables inside Simulink we have to convert them to a function. 

syms symbolic1 symbolic2 real
my_sym_variable = [symbolic1^-1, symbolic2*4];
my_sym_vec = [symbolic1, symbolic2]
my_sym_vec =

$$ \displaystyle \left(\begin{array}{cc} {\textrm{symbolic}}_1 & {\textrm{symbolic}}_2 \end{array}\right) $$ 

matlabFunction(my_sym_variable, 'vars',{my_sym_vec}, 'File','my_test_subs_function');

To use this function: 

test_configuration = [2,2]

test_configuration = 1x2
     2     2

my_sym_variable_subsituted = my_test_subs_function(test_configuration)

my_sym_variable_subsituted = 1x2
    0.5000    8.0000

hint: This can be done using the MatlabFunction block in Simulink

Convert your Symbolic Matrices here

Unrecognized function or variable 'B'.

Parameters

Setup the tau limit as:

$$ {\textrm{tau}}_{\lim } =\left\lbrack \begin{array}{c} 120\newline 120\newline 60 \end{array}\right\rbrack \left\lbrack \textrm{Nm}\right\rbrack $$

and the desired configuration (both speed and position)

try the configurations

$$ q\in \left\lbrace \left\lbrack \begin{array}{c} -\frac{\pi }{3}\newline \frac{\pi }{3}\newline \frac{\pi }{10} \end{array}\right\rbrack ,\left\lbrack \begin{array}{c} -\pi \;\newline \frac{\pi }{5}\newline \frac{\pi }{6}\; \end{array}\right\rbrack ,\left\lbrack \begin{array}{c} \frac{\pi }{8}\newline -\frac{\textrm{pi}}{2}\newline \frac{\textrm{pi}}{3} \end{array}\right\rbrack \right\rbrace $$

store them as:

  • q_desired_1
  • q_desired_2
  • q_desired_3
  • qd_desired 
    taulim = [120,120,60]';
q_desired_1 = [-pi/3, pi/3, pi/10]'; 
q_desired_2 = [-pi,pi/5,pi/6]'; 
q_desired_3 = [pi/8,-pi/2,pi/3]'; 
qd_desired = [0,0,0]';

to visualize the target transforms in Rviz:

load("5.Control/Resources/targetTransform_threelink.mat");
StaticFrameBroadcaster(targetTransform_threelink_1, 'target1');

Published static transform: base_link → target1

StaticFrameBroadcaster(targetTransform_threelink_2, 'target2');

Published static transform: base_link → target2

StaticFrameBroadcaster(targetTransform_threelink_3, 'target3');

Published static transform: base_link → target3

Gains

Select the gains for your system, as this is an inverse dynamic scheme, follow the approach from Exercise 4.2. Don't scale the gains yet, you will be able to do so during simulation. 

w_i = 1x3
    3.8095    4.9689    7.1429

Dashboard

In the Simulink file you will find the dashboard section that allows you to switch between the configurations, see the current torque output and scale the Kp and Kd matrix during simulation.

Configuration Selector

Check one of these boxes to select the desired configuration.

image_0.png

this selection block is linked to:

image_1.png

Scale Kd and Kp

By using the sliders you can alter the gain value of their corresponding K_scale blocks:

image_2.png

View Torque Trajectory

The Dashboard scope allows you to see the current torques live during simulation (like a scope).

image_3.png

Task 1

Open the File Exercise_5_1_1.slx you will find a setup to be used for this exercise. From the outputs q and qd (left Subsystem) you will receive the current position and velocity of the joints as a column vector.

The input to the right subsystem accepts a column vector and sends the torques to the simulation.

To import the results of your simulation into matlab: 

q_data_1 = out.position; 
qd_data_1 = out.velocity; 
tau_data_1 = out.tau; 
t_data_1 = out.tout;

Plot your results in matlab.

figure_0.png

Task 2

Reduce the computational load by only considering the diagonal terms of the B and C matrix and analyze the behavior and compare them to the results of Task 1.

Task 2.1

Setup the new symbolic matrix and convert them to a function.

Task 2.2

Open the File Exercise 5_1_2.slx and setup the plant with the new matrices B' and C'.

To import the results of your simulation into matlab:

figure_1.png