Now that we know how to derive displacement vectors for different types of robotic arms, let’s take a look at how to write displacement vectors in code.

# Two Degree of Freedom Robotic Arm

Here are the two displacement vectors we found earlier:

We’ve already found the rotation matrices for the two degrees of freedom robotic arm in a previous tutorial. I’ll start by copying and pasting that code into the program.

I will then write out the two displacement vectors.

Make sure that:

- servo_0_angle = 15 # Joint 1 (Theta 1)
- servo_1_angle = 60 # Joint 2 (Theta 2)

Here is the full code:

import numpy as np # Scientific computing library # Project: Displacement Vectors for a 2 DOF Robotic Arm # Author: Addison Sears-Collins # Date created: August 10, 2020 # Servo (joint) angles in degrees servo_0_angle = 15 # Joint 1 (Theta 1) servo_1_angle = 60 # Joint 2 (Theta 2) # Link lengths in centimeters a1 = 1 # Length of link 1 a2 = 1 # Length of link 2 a3 = 1 # Length of link 3 a4 = 1 # Length of link 4 # Convert servo angles from degrees to radians servo_0_angle = np.deg2rad(servo_0_angle) servo_1_angle = np.deg2rad(servo_1_angle) # Define the first rotation matrix. # This matrix helps convert servo_1 frame to the servo_0 frame. # There is only rotation around the z axis of servo_0. rot_mat_0_1 = np.array([[np.cos(servo_0_angle), -np.sin(servo_0_angle), 0], [np.sin(servo_0_angle), np.cos(servo_0_angle), 0], [0, 0, 1]]) # Define the second rotation matrix. # This matrix helps convert the # end-effector frame to the servo_1 frame. # There is only rotation around the z axis of servo_1. rot_mat_1_2 = np.array([[np.cos(servo_1_angle), -np.sin(servo_1_angle), 0], [np.sin(servo_1_angle), np.cos(servo_1_angle), 0], [0, 0, 1]]) # Calculate the rotation matrix that converts the # end-effector frame to the servo_0 frame. rot_mat_0_2 = rot_mat_0_1 @ rot_mat_1_2 # Display the rotation matrix print(rot_mat_0_2) # Displacement vector from frame 0 to frame 1. This vector describes # how frame 1 is displaced relative to frame 0. disp_vec_0_1 = np.array([[a2 * np.cos(servo_0_angle)], [a2 * np.sin(servo_0_angle)], [a1]]) # Displacement vector from frame 1 to frame 2. This vector describes # how frame 2 is displaced relative to frame 1. disp_vec_1_2 = np.array([[a4 * np.cos(servo_1_angle)], [a4 * np.sin(servo_1_angle)], [a3]]) # Display the displacement vectors print() # Add a space print(disp_vec_0_1) print() # Add a space print(disp_vec_1_2)

Now run the code. Here is the output for both the rotation matrix and the displacement vectors:

# References

Credit to Professor Angela Sodemann from whom I learned these important robotics fundamentals. Dr. Sodemann teaches robotics over at her website, RoboGrok.com. While she uses the PSoC in her work, I use Arduino and Raspberry Pi since I’m more comfortable with these computing platforms. Angela is an excellent teacher and does a fantastic job explaining various robotics topics over on her YouTube channel.