INTRODUCTION TO DYNAMICS ANALYSIS OF ROBOTS (Part 4)

Introduction to Dynamics Analysis of Robots (4) This lecture continues the discussion on the analysis of the instantaneous motion of a rigid body, i.e. the velocities and accelerations associated with a rigid body as it moves from one configuration to another. After this lecture, the student should be able to: Derive the principles of relative motion between bodies in terms of acceleration analysis Solve problems of robot instantaneous motion using joint variable interpolation Calculate the Jacobian of a given robot

Summary of previous lecture Acceleration tensor and angular acceleration vector

Summary of previous lecture Moving FORs

Example: Acceleration and moving FORs B=2 C=1 Y0, Y1 X0, X1 Z0, Z1 Z2 X2 Y2 Z3 X3 Y3 Example: The 3 DOF RRR Robot: P What is the acceleration of point “P” after 1 second if all the joints are rotating at

Example: Acceleration and moving FORs We did the following: To get

Example: Acceleration and moving FORs We should get the same answer if we use transformation matrix method. We know that For

Example: Acceleration and moving FORs The answer is the same as that obtained earlier:

Relative angular acceleration We can differentiate the relative angular velocity to get the relative angular acceleration: where

Example: Relative Angular Acceleration B=2 C=1 Y0, Y1 X0, X1 Z0, Z1 Z2 X2 Y2 Z3 X3 Y3 Example: The 3 DOF RRR Robot: P What is after 1 second if all the joints are rotating at

Example: Relative angular acceleration Solution: We re-used the following data obtained from the previous lecture

Example: Relative angular acceleration

Example: Relative angular acceleration You should get the same answer from the overall rotational matrix and its derivative, i.e.

Example: Relative angular acceleration Using the data from the previous example:

Example: Relative angular acceleration The answer is the same as that obtained earlier:

Instantaneous motion of robots So far, we have gone through the following exercises: Given the robot parameters, the joint angles and their rates of rotation, we can find the following: The linear (translation) velocities w.r.t. base frame of a point located at the end of the robot arm The angular velocities w.r.t. base frame of a point located at the end of the robot arm The linear (translation) acceleration w.r.t. base frame of a point located at the end of the robot arm The angular acceleration w.r.t. base frame of a point located at the end of the robot arm We will now use another approach to solve the linear velocities and linear acceleration problem.

Jacobian for Translational Velocities In general, the position and orientation of a point at the end of the arm can be specified using Note that the position of the point w.r.t. {0} is The velocities of the point w.r.t. frame {0} is

Jacobian for Translational Velocities

Example: Jacobian for Translational Velocities Y0, Y1 X0, X1 Z0, Z1 What is the Jacobian for translational velocities of point “P”? Z2 X2 Y2 Z3 X3 Y3 P Given:

Example: Jacobian for Translational Velocities The transformation matrix of point “P” w.r.t. frame {3} is

Example: Jacobian for Translational Velocities

Example: Jacobian for Translational Velocities What is the velocity of point “P” after 1 second if all the joints are rotating at

Example: Jacobian for Translational Velocities Given a=3, B=2, C=1. At t=1, The answer is similar to that obtained previously using another approach! (refer to velocity and moving FORs)

Getting the Translational Acceleration If the angular acceleration for 1, 2, …, n are 0s then

Example: Getting the Translational Acceleration B=2 C=1 Y0, Y1 X0, X1 Z0, Z1 Z2 X2 Y2 Z3 X3 Y3 Example: The 3 DOF RRR Robot: P What is the acceleration of point “P” after 1 second if all the joints are rotating at

Example: Getting the Translational Acceleration

Example: Getting the Translational Acceleration At t=1, Given a=3, B=2, C=1.

Example: Getting the Translational Acceleration All the angular acceleration for 1, 2, …, n are 0s: The answer is the same as that obtained earlier:

Summary This lecture continues the discussion on the analysis of the instantaneous motion of a rigid body, i.e. the velocities and accelerations associated with a rigid body as it moves from one configuration to another. The following were covered: Principles of relative motion between bodies in terms of acceleration analysis Robot instantaneous motion using joint variable interpolation Jacobian of a robot