Introduction to Dynamics Analysis of Robots: Velocity Tensor Analysis

1. INTRODUCTION TO DYNAMICS ANALYSIS OF ROBOTS (Part 1)

2. Introduction to Dynamics Analysis of Robots (1) • 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: • Analyze the instantaneous motion of a rigid body • Derive the velocity tensor and defines its vector • Derive the instantaneous direction of sliding and rotation • Define the angular velocity vector and the instantaneous screw axis (ISA)

3. Z Q P Y Time t1 O X P Q Time t0 Velocity Tensor Consider the rotation of the object shown below. “P” and “Q’ are two points on the rigid body. Given the rotational tensor ‘R’ and with positional vectors of “P” and “Q” fixed at time t0, Differentiating the above w.r.t. time: But Using R-1=RT, we get

4. Velocity Tensor Let (t) is called the velocity tensor. This can be used to find the linear velocity at “P” and “Q” as a result of the rotation: A property of (t) is that it is skew-symmetric, T(t) = - (t)

5. Example: Velocity Tensor Given that =t/6, where  is a rotation about the X-axis. Find the velocity tensor. Solution: The rotation matrix about the X-axis is

6. Example: Velocity Tensor Given =t/6

7. Example: Application of Velocity Tensor Given at t=0, This vector rotates about the X-axis. The angle of rotation is governed by =t/6. What is the velocity of the vector at time t=1? Solution: The velocity tensor was found in the previous example as:

8. o Example: Application of Velocity Tensor

9. Example: Application of Velocity Tensor At t=1:

10. Vector of the Velocity Tensor From linear algebra, we can replace the (3x3) matrix (t) with a velocity tensor such that: where The linear velocity at “P” and “Q” as a result of the pure rotation can alternatively be expressed as

11. Example: Vector of the Velocity Tensor Given that =t/6, where  is a rotation about the X-axis. Find the vector of the velocity tensor and find the velocity for the following vector at time t=1, where Solution: The velocity tensor had been found in the previous example as:

12. Example: Vector of the Velocity Tensor At t=1:

13. V=PQ Q  P Q P is called the angular velocity vector of the body o Interpretation: Vector of the Velocity Tensor Consider a rotation in 2-D: Point “P” is fixed Extend the rotation concept to 3-D: Point “P” is fixed

14. 90º • The equation reveals the following: • is normal to the and If is parallel to then , i.e. all points “Q’ such that has zero relative velocity to P Instantaneous direction of sliding

15. since is called the sliding velocity The dot product indicates that the sliding velocity is the projection of the velocity of “Q” (or “P’) onto The direction of sliding is obvious along Instantaneous direction of sliding If is not parallel to then using Every point of the rigid body has the same sliding velocity

16. The sliding velocity is the projection of the velocity of “Q” onto Example: Instantaneous direction of sliding Find the sliding velocity given Solution: The direction of sliding is obvious along

17. Axis of rotation passes through the point Axis of rotation The angle of rotation  The displacement component parallel to the direction of rotation It is again obvious that the vectors and are parallel. It can be shown that Comparison with screw parameters for general rigid body motion The direction of sliding is obvious along Compare with the motion of a screw:

18. Z-axis Y-axis 1 X-axis Solution: 1 1 Example: Angular velocity vector Find the angular velocity vector for the following rigid body motion:

19. ISA parallel to The sliding velocity Axis of rotation passes through the point The rate of rotation Given a point on the rigid body along with Instantaneous screw axis (ISA) We can define the screw motion using

20. Z-axis Y-axis 1 X-axis Example: ISA Find a point where the ISA passes for the following rigid body motion: 1 Solution:

21. ISA parallel to The sliding velocity Axis of rotation passes through the point Axis of rotation Axis of rotation passes through the point The angle of rotation  The rate of rotation The displacement component parallel to the direction of rotation

22. Theorems In general, the instantaneous motion of a rigid body is described by a sliding and a rotation about a particular axis. This axis called the instantaneous screw axis (ISA) is parallel to the angular velocity vector and passes through the point A1 The difference of the velocities of any two points of a rigid body undergoing an arbitrary motion is normal to the ISA If the velocities of three non-collinear points of a rigid body are identical, the body undergoes a pure translation

23. 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: • The instantaneous motion of a rigid body • The velocity tensor and its vector • The instantaneous direction of sliding and rotation • The angular velocity vector and the instantaneous screw axis (ISA)