Path and Trajectory Planning - Robot Mechanics and Control

1. Environment Used

• MATLAB R2018a
• Toolbox: Robotics Toolbox

2. Robot Model

• Proposed Mechanism: 3-link cylindrical manipulator (RPP)
• Workspace is cylindrical

2.1 Forward Kinematics:

• Axis Assignment

• DH parameters

• Transformations Matrices

• Defining Robot Model in MATLAB

%% Robot Model Definition
d1 = 5;
L1 = Revolute('a', 0, 'd', d1, 'qlim', [(1*pi/4),(3*pi/4)],'standard');
L2 = Prismatic('a', 0, 'alpha', -pi/2, 'qlim',[5,200],'standard');
L3 = Prismatic('a', 0, 'qlim',[2,abs(R)],'standard');

robot1 = SerialLink( [L1 L2 L3],'name', 'RPP/Cake Robot');
text(0,0,0,' RPP ROBOT')
robot.teach('rpy')

2.2 Inverse Kinematics

• Inverse Kinematics Equations

3. Workspace

• Proposed Workspace: Cake-shaped Workspace
• Cake-shaped Workspace is drawn by generating every possible values of EE Pose (x, y, z of T(0,3))
• When plotting each point using point3, check det(Jacobian) for any singularity)
• Singularity for RPP occurs when 3rd link variable becomes zero |Jacob| = d3
• Boundary: Drawn using meshgrid -- Radius is 282 units -- Width is 400 units -- Height is 200 units -- H/W ratio: 0.5 -- Theta = pi/2

4. Path and Trajectory Plotting in Restricted Workspace

Steps Involved:

• First Get start and goal EE position from user in terms of cartesian coordinates
• Using inverse kinematics to get joint configuration values (initial and start)
• Using mstraj (Multi-segment multi-axis trajectory) to generate values of - joint configuration for the trajectory
• Loop through each joint configuration and check for singularity/joint limits. Quit the program if required
• Draw the trajectory using plot3() function
• Animate the robot to follow the trajectory

4.1 Velocity and Acceleration Graphs

• Using jtaj command of robotics toolbox to get q, qd, qdd

5. Overall Program Flow

• Get Start and Goal cartesian coordinates from the user
• Map those coordinates to EE pose/axis
• Use inverse kinematics to get Start and Goal joint configuration
• Use mstraj to get various joint configuration between start and goal config
• If joint limit exceeds, quit the program
• Robot Animation: using robot.plot
• Drawing Trajectory: Use fwd kinematics to get x,y,z
• for each joint config
• Use plot3 to draw points
• Display the coordinates (start, goal)
• Calculating time to each goal using cputime
• Ask again for another goal position (Homing in start ONLY)

5.1 Sample End and Start Cartesian Coordinates in the Workspace

• [-200,50,120]
• [-255,70,140]
• [-150,-70,140]