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The user can change simulation_time and n_pts. The rest of the scripts initialize the arrays required by the model to execute the simulations. All the values are initialized to 0

For a 5-DOF (Degrees of Freedom) SCU with pitch, roll, yaw, x, and y, the descriptions of Cartesian space and joint space are as follows:

Cartesian Space

In this context, Cartesian space refers to the position and orientation of the quadrupole in a five-dimensional coordinate system:

  • x: Linear displacement along the x-axis.
  • y: Linear displacement along the y-axis.
  • Pitch: Rotation about the x-axis.
  • Roll: Rotation about the y-axis.
  • Yaw: Rotation about the z-axis.

The Cartesian space describes the desired position (x, y) and orientation (pitch, roll, yaw) of the quadrupole within the working environment.

Joint Space

For a 5-DOF SCU, the joint space describes the states of its individual actuators. Each actuator's position or angle contributes to the overall configuration of the strongback. The joint space is represented by a set of five coordinates, one for each actuator, which could include:

  • A1: Position or angle of the first actuator.
  • A2: Position or angle of the second actuator.
  • A3: Position or angle of the third actuator.
  • A4: Position or angle of the fourth actuator.
  • A5: Position or angle of the fifth actuator.

In summary:

  • Cartesian Space: The quadrupole's position (x, y) and orientation (pitch, roll, yaw).
  • Joint Space: The specific positions of the SCU's five linear actuators.

Definition of a simple Point To Point motion in the cartesian space

Code Block
titleConfiguration
linenumberstrue
collapsetrue
%% Geometric parameters
quadrupole_offset = [925.92, 0, -95.25]; % Quadrupole offset
left_mounting_line_offset = -695.4; % Offset for left mounting line
right_mounting_line_offset = 983.17; % Offset for right mounting line
d_4q = 633.175 + 925.92; % Distance for 4th quadrupole
d_5q = 1003.65 - 925.92; % Distance for 5th quadrupole

% Distance calculations
d_aq = quadrupole_offset(1) - left_mounting_line_offset;
d_ab = right_mounting_line_offset - left_mounting_line_offset;
d_bc = 400; % Distance between B and C

% Trajectory parameters
start_x = -1; end_x = 1;
start_y = 0; end_y = 0;
start_pitch = -0.00127; end_pitch = 0.00127;
start_roll = 0; end_roll = 0;
start_yaw = 0; end_yaw = 0;

% Generate trajectory
x = start_x + time / time(end) * (end_x - start_x);
y = start_y + time / time(end) * (end_y - start_y);
pitch = start_pitch + time / time(end) * (end_pitch - start_pitch);
roll = start_roll + time / time(end) * (end_roll - start_roll);
yaw = start_yaw + time / time(end) * (end_yaw - start_yaw);

% Plot command positions
figure();
plot(time, y);
hold on;
plot(time, x);
title('Command positions');
xlabel('Time [s]');
ylabel('Position [mm]');
legend('y', 'x');
grid on;

% Plot command orientations
figure();
plot(time, pitch);
hold on;
plot(time, roll);
hold on;
plot(time, yaw);
title('Command orientations');
xlabel('Time [s]');
ylabel('Orientation [rad]');
legend('pitch', 'roll', 'yaw');
grid on;

% Calculate joint positions using inverse kinematics
tic;
res = arrayfun(@(a, b, c, d, e) inverse_kin_5(a, b, c, d, e, d_aq, d_ab, d_bc, d_4q, d_5q)', x, y, pitch, roll, yaw, 'UniformOutput', false);
toc;
joint_pos = cell2mat(res);

% Plot joint positions
figure();
plot(time, joint_pos(1, :));
hold on;
plot(time, joint_pos(2, :));
hold on;
plot(time, joint_pos(3, :));
hold on;
plot(time, joint_pos(4, :));
hold on;
plot(time, joint_pos(5, :));

% Apply compensations
x_compensation = 0;
y_compensation = 0;
A1_position = [time' joint_pos(1, :)' - y_compensation'];
A2_position = [time' joint_pos(2, :)' - y_compensation'];
A3_position = [time' joint_pos(3, :)' - y_compensation'];
A4_position = [time' joint_pos(4, :)' + x_compensation'];
A5_position = [time' joint_pos(5, :)' + x_compensation'];

% Update initial states
A1_initial_state = A1_position(1, 2);
A2_initial_state = A2_position(1, 2);
A3_initial_state = A3_position(1, 2);
A4_initial_state = A4_position(1, 2);
A5_initial_state = A5_position(1, 2);

...

This is located at project_root/Functions/inverse_kin_5.m

If we consider y the vector that represents the 5 actuator positions and x the vector that represents the quadrupole's 5 degree of freedom, we can write the relation between the two as:

Latex
\( \vec{y} = K^{-1}(\vec{x})\)

Where K^-1 is the inverse kinematics function.

Now that the y vector is computed for all the time samples of the desired motion, all the required blocks to run our simulation are ready.

Running the simulation

Code Block
titleExecute the code
linenumberstrue
collapsetrue
% comment the next line if you want to see the output on the Mechanics
% Explorer
set_param('SCU_FULL', 'SimMechanicsOpenEditorOnUpdate', 'off');
simOut = sim(fullPath, 'StopTime', string(simulation_time));  

Running the simulation is quite straightforward, the only command to execute is "sim()", and it will return a Simulink.SimulationOutput object. This is where all the simulation output signals are stored

Data analysis