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INTRODUCTION TO DYNAMICS ANALYSIS OF ROBOTS (Part 5). Introduction to Dynamics Analysis of Robots (5).

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## INTRODUCTION TO DYNAMICS ANALYSIS OF ROBOTS (Part 5)

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**INTRODUCTION**TO DYNAMICS ANALYSIS OF ROBOTS (Part 5)**Introduction to Dynamics Analysis of Robots (5)**• 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: • Solve problems of robot instantaneous motion using joint variable interpolation • Calculate the Jacobian of a given robot • Investigate robot singularity and its relation to Jacobian**Summary of previous lecture**Jacobian for translational velocities**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 angular velocities problem.**Jacobian for Angular Velocities**In general, the position and orientation of a point at the end of the arm can be specified using**Jacobian for Angular Velocities**Similarly:**Jacobian for Angular Velocities**Similarly: Jacobian for angular velocities**Y2**Y3 X2 X3 Z0, Z1 Z2 Z3 Y0, Y1 X0, X1 Example: Jacobian for Angular Velocities A=3 B=2 C=1 What is the Jacobian for angular velocities of point “P”? P Given:**Example: Jacobian for Angular Velocities**What is after 1 second if all the joints are rotating at The answer is similar to that obtained previously using another approach! (refer to the example on relative angular velocity)**Clarification**Why r1 r2 Note: every point on the link will rotate at the same angular velocity! However, the linear velocities at different points on the link are not the same!**Getting the Angular Acceleration**If the joint angular acceleration for 1, 2, …, n are 0s then**Y2**Y3 X2 X3 Z0, Z1 Z2 Z3 Y0, Y1 X0, X1 Example: Getting the Angular Acceleration A=3 B=2 C=1 Example: The 3 DOF RRR Robot: P What is after 1 second if all the joints are rotating at**Getting the Angular Acceleration**All the joints angular acceleration for 1, 2, …, n are 0s: The answer is similar to that obtained previously using another approach! (refer to the example on relative angular acceleration)**Transformation between Joint variables and the general**motion of the last link We can combine the Jacobians for the linear and angular velocities to get:**Y2**Y3 X2 X3 Z0, Z1 Z2 Z3 Y0, Y1 X0, X1 Example: Transformation between Joint variables and the general motion of the last link A=3 B=2 C=1 Example: The 3 DOF RRR Robot: P What is the Jacobian for the 3 DOF RRR robot?**Example: Transformation between Joint variables and the**general motion of the last link**Jacobian and Singularities**We know that The above is true only if the Jacobian is invertible. From algebra, we now that a matrix cannot be inverted if its determinant is zero (i.e. the matrix is singular)**Y2**Y3 X2 X3 Z0, Z1 Z2 Z3 Y0, Y1 X0, X1 Example: Jacobian and Singularities A=3 B=2 C=1 Example: The 3 DOF RRR Robot: P Investigate the singularities of the 3 DOF RRR robot**Example: Jacobian and Singularities**Under these two conditions, we cannot determine the joint angular velocities using the Jacobian**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: • Robot instantaneous motion using joint variable interpolation • The Jacobian of a given robot • Robot singularity and its relation to Jacobian

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