1. Rotations and Translations 1

2. Mathematical terms The inner product of 2 vectors a,b is defined as: The cross product of 2 vectors is defined as: A unit vector will be marked as: 2

3. Representing a Point 3D A three-dimensional point A is a reference coordinate system here 3

4. Representing a Point 3D (cont.) Once a coordinate system is fixed, we can locate any point in the universe with a 3x1 position vector. The components of P in {A} have numerical values which indicate distances along the axes of {A}. To describe the orientation of a body we will attach a coordinate system to the body and then give a description of this coordinate system relative to the reference system. 4

5. Example 5

6. Description of Orientation is a unit vector in B is a coordinate of a unit vector of B in coordinates system A (i.e. the projection of onto the unit direction of its reference) 6

7. Example Rotating B relative to A around Z by 7

8. Example In general: 8

9. Using Rotation Matrices 9

10. Translation 10

11. Combining Rotation and Translation 11

12. What is a Frame ? A set of four vectors giving position and orientation information. The description of the frame can be thought as a position vector and a rotation matrix. Frame is a coordinate system, where in addition to the orientation we give a position vector which locates its origin relative to some other embedding frame. 12

13. Arrows Convention An Arrow - represents a vector drawn from one origin to another which shows the position of the origin at the head of the arrow in terms of the frame at the tail of the arrow. The direction of this locating arrow tells us that {B} is known relative to {A} and not vice versa. 13

14. R otating a frame B relative to a frame A about Z axis by degrees and moving it 10 units in direction of X and 5 units in the direction of Y. What will be the coordinates of a point in frame A if in frame B the point is : [3, 7, 0] T ? Example 14

15. Extension to 4x4 We can define a 4x4 matrix operator and use a 4x1 position vector 15

16. Example If we use the above example we can see that: 16

17. P in the coordinate system A 17

18. Formula 18

19. Compound Transformation 19

20. Several Combinations 20

21. Example Rotating a frame B relative to a frame A about Z axis by degrees and moving it 1 units in direction of X and 2 units in the direction of Y. What will be ? 21

22. Example cont - 2 We create a homogeneous transformation using the function se2: T1 = se2(1, 2, 30*pi/180) T1 = 0.8660 -0.5000 1.0000 0.5000 0.8660 2.0000 0 0 1.0000 Note that this is a 2D matrix, we eliminate the z axis 22

23. Example cont - 3 We can also plot this, relative to the world coordinate frame, by: >> axis([0 5 0 5]); >>trplot2(T1, 'frame', '1', 'color', 'b') 23

24. Example cont - 4 We can plot a point by: >> P = [3; 2]; >> plot_point(P, ‘*’); 24

25. Example cont - 5 Now we got something like this: The point P is known with respect to {0} We want to determine the coordination of point P with respect to {1} 25

26. Example cont - 6 What do we know ? But… >> P1 = inv(T1)* [P; 1] >> P1 = h2e( inv(T1) * e2h(P) ); h2e: homogenous to euclidean 26

27. Example cont - 7 This also can be achieved by: >> homtrans(inv(T1), P) 27

28. Example Rotating a frame B relative to a frame A about Z axis by degrees and moving it 10 units in direction of X and 5 units in the direction of Y. What will be the coordinates of a point in frame A if in frame B the point is : [3, 7, 0] T ? >> T = se2(10,5,30*pi/180) >> P = [3;7;1] >> P2 = T*P >> P = inv(T)*P2 28

29. Notes Homogeneous transforms are useful in writing compact equations; a computer program would not use them because of the time wasted multiplying ones and zeros. This representation is mainly for our convenience. For the details turn to chapter 2. 29