10.2.3. Chain Multiplication

Try these first

1. 1.

   What is the purpose of chain multiplication in robotics?

   Hint

   Think about the relationship between joint movements and end-effector location.

2. 2.

   Define what a transformation matrix is.

   Hint

   Consider how different frames of reference are connected.

3. 3.

   What is chain multiplication used for in robotics?

   • Determining the weight of the robot
   • Calculating the position of the end-effector
   • Designing robots

   Hint

   Think about the impact of joint movement on the end result.

4. 4.

   True or False: Chain multiplication can yield the overall transformation of a robot's end-effector.

   • True
   • False

   Hint

   Consider what we just discussed about overall transformations.

5. 5.

   A robotic manipulator uses chain multiplication to compute the final position of its end-effector. If the transformation matrices due to three joints are given as T1, T2, and T3, and their values are: T1 = [[1, 0, 0, 2], [0, 1, 0, 3], [0, 0, 1, 1], [0, 0, 0, 1]], T2 = [[0, -1, 0, 1], [1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], T3 = [[1, 0, 0, 2], [0, 1, 0, 1], [0, 0, 1, 3], [0, 0, 0, 1]]. Calculate the overall transformation matrix T = T1 * T2 * T3. What is the resultant T matrix?

   Hint

   Take it step by step; multiply the matrices two at a time.

6. 6.

   If errors occur in the transformation matrices by ±0.1 in each element, discuss how this would affect the position of the end-effector. Would the overall transformation still be reliable?

   Hint

   Think about how small changes multiply through the chain.