1. Matrix Methods in Kinematics
Vector Methods with Matrix Notation
Rigid Body Rotation Matrices
Rigid Body – points have same relative position
Displacement = Rotation and Translation
Angular rotations described about
1. Right hand Cartesian axes (x,y,z)
2. Arbitrary Axis
3. Euler Angles

2. Matrix Methods in Kinematics
Holonomic Systems Holonomic Constraints
Relations between co-ordinates and possibly time)
Rigid body
xyz displacement is not order dependent

3. Matrix Methods in Kinematics
Rigid body rotations
Rotations not commutative (order dependent)

4. Matrix Methods in Kinematics
1. Rotate about Z (α)
Components in the fixed system
x-y
Rotation matrix

5. Matrix Methods in Kinematics
Rotation about three Cartesian axes y x z
X,Y,Z axes fixed in space

6. Matrix Methods in Kinematics
Plane rotation (2D) (rotation about z)
Spatial Rotation – sequence z,y,x

7. Matrix Methods in Kinematics

8. Matrix Methods in Kinematicsy u
2. Rotation about axis u (φ)
Decompose to rotations about x, y, z
Rotate u about z and then back again
+/- show directions x z

9. Matrix Methods in Kinematicsy x z u

10. Matrix Methods in Kinematics

11. Matrix Methods in Kinematics
3. Euler Angles
Displacement of a rigid body in terms
of three relative displacement angles
Each rotation about an axis whose
location depends on preceding
rotation 1 2 3
4 sets of axes initially coincident
xyz fixed in the body

12. Matrix Methods in Kinematics

13. Matrix Methods in Kinematics

14. Matrix Methods in Kinematics
12 different possible rotation sets
z-x-z
x-y-x
y-z-y
z-y-z
x-z-x
y-x-y
x-y-z
y-z-x
z-x-y
x-z-y
z-y-x
y-x-z

15. Matrix Methods in Kinematics

16. Matrix Methods in Kinematics

17. Matrix Methods in Kinematics
Euler Angles
Angular Velocity Vectors 3 1
Rigid
body 3 1 2 2

18. Matrix Methods in Kinematics
Rotation of Rigid Body in 2d Cartesian Space p1 q1
Fixed x-y
Vector form

19. Matrix Methods in Kinematics2 1

20. Matrix Methods in Kinematics
3x3 D = plane displacement matrix

21. Matrix Methods in Kinematics
Spatial (3D) Rigid Body Displacement
Replace
Cartesian
u Axis
Euler
Using Cartesian
new
original
4x4 matrix (3D)

22. Matrix Methods in Kinematics
Screw displacement matrix y q u q1
p=p1+su s p1 z x
Screw displacement matrix

23. Matrix Methods in Kinematics
12 non constant elements
3D space= 6 DOF, 6 elements are dependent
To Define D: euler angles, dir cos, points on body, etc…

24. Matrix Methods in Kinematics
Example Displacement of a point
Moving with a rigid body

25. Matrix Methods in Kinematics
3x3 Displacement matrix

26. Matrix Methods in Kinematics
Finite Rotation Pole – plane rotation about p0 po
With new position vectors
p1=p2=p0

27. Original displacement matrix
Matrix Methods in Kinematics
Previously
p1 and p2 in
Displacement matrix D now written as:
q2 is the same point
Original displacement matrix

28. Matrix Methods in Kinematics

29. Matrix Methods in Kinematics
HW #5 Salute Y X q
shoulder
30◦ p Z
p1=elbow
q1=tip of finger
Use Cartesian angles to find [D]
Treat as 3 independent rotations

30. Matrix Methods in Kinematics
HW #5 Salute Y Y
Changed axes
notation X X
shoulder 2 Z p 1 Z q
p1=elbow
move origins for
2,3 rotations Y
q1=tip of finger X
30◦
z-y-z rotation 3 Z

31. Matrix Methods in Kinematics
Finding the Displacement Matrix by Inversion y B1
Known points C1 C2 A1 A2 B2 x

32. Matrix Methods in Kinematics
Displacement Matrix by Inversion

33. Matrix Methods in Kinematicsq q1 d a x y
2D Planar motion z
D=d*a-1

34. Matrix Methods in Kinematics
Finding the inverse of A (by hand – the long way)
adj - adjoint
ith row=ith column
α=co factor
Mij= minor of A

35. Matrix Methods in Kinematics
Using MATLAB
inv(a)
ans =
d =
a =
D=d*a-1
Displacement matrix

36. Matrix Methods in Kinematics
HW 5 by Inversion
Using My arm

37. Matrix Methods in Kinematics
By Inverse method – position
D =

38. Matrix Methods in Kinematics
By Inverse method – position

39. Matrix Methods in Kinematics