1. Mon 30 July 2007
Overview of the course
Forward Kinematics (given theta, find end-point)
Inverse Kinematics (given end-point find theta)
Dynamics (given input torques find theta as a function of time)
Control (design torque to follow a given end-point trajectory)

2. A Note on Control Robot dynamical model
Control Problem - Given q(t) find appropriate tau
Two solutions:
Feedforward control – calculate tau from the above equation (since LHS is known)
Feedback control – let tau = f(q desired – q measured) – control design means how to find the function f.
When to use feedforward control and when to use feedback control?
Stability, optimality, robustness, etc., several issues to be considered for practical control

3. Representation in frame 1
Rotation matrix from frame 1 to frame 0

4. Example

5. Exercise
Fill in

6. Rotation Matrix
Rotation matrix columns are of unit length and perpendicular to each other (proof needed).
Columns are unit vectors of new x, y, and z axes expressed in the old frame.
Determinant of the rotation matrix is plus or minus one (proof needed).
SO(3) – set of all 3 x 3 rotation matrices

7. Composition of Rotations
also
Phi deg about the current y-axis (i.e., y0)
Theta deg about the current z-axis (i.e., z1)
Theta deg about the current z-axis (i.e., z0)
Phi deg about the current y-axis (i.e., y1)

8. About Fixed Axes
For example, phi about y0 and then rotate theta about z0 (not about z1 as in the previous example).
Is the composition rotation matrix
R = R(y,phi) R(z,theta) ?

9. About fixed axes Red – frame 0 Blue – frame 1 Green – frame 2
phi
theta
Phi about y0 Theta about z0
We want a rotation matrix from frame 2 to frame 0.

10. Pre-multiply for rotation about fixed axes
dashed frame is frame 0 rotated by theta about z0
phi
theta
Rotation matrix from green frame to the dashed frame is the same as from blue frame to red frame which is R(y,phi).
Rotation matrix from dashed frame to red frame is R(z,theta)
Finally the rotation matrix from green frame to red frame is: (first green to dashed and then dashed to red)
R(z,theta) R(y,phi)
Pre-multiply for rotation about fixed axes
Post-multiply for rotation about current axes

11. See Similarity Transform (Section 2.3.1 p47)
vector p in frame i is written as frame k is frame 0 after theta rotation about k-axis
Rotation matrix from frame 2 to frame 0 is the same as from frame 3 to frame k.
See Similarity Transform (Section p47)
Frame 3 from frame 0 by
first R(z,alpha) to align x0 with x-y plane projection of k-axis – frame 1 then R(y,beta) to align z1 with the k-axis – frame 2 then R(z,theta) to obtain frame 3.

12. R(k,theta)
Nine elements but only three independent parameters – what are they?
Any R in SO(3) can be written as R(k,theta) where
theta = acos(trace(R) – 1)/2 and
k = (1/2 sin(theta))[r32-r23; r13-r31; r21-r12]
Representation is non-unique

13. S(k) is a skew symmetric matrix

14. Rigid body rotation
Euler angles – {theta, phi, psi}-Rotations about current axes R = R(z,phi) R(y,theta) R(z,psi)
Roll, Pitch, and Yaw Angles – Rotations about fixed-axes

15. Homogeneous Transformation
Can we do this operation using matrices?

17. Elementary Transformations

18. Skew Symmetric Matrix ss(3)

19. Example
Find
What is k?

20. Angular velocity and Acceleration
That vector which parameterises S(t), i.e., the vector of the three components in S(t), is known as the angular velocity.

21. Example

22. Velocity and Acceleration
Transverse acceleration
Centripetal acceleration
Coriolis acceleration

23. Addition of Angular Velocities

24. Forward Kinematics

25. DH Representation

26. DH Representation - Frames
(Convention) Frame i is attached to link i. The inertial frame is Frame 0 and Earth is link 0. Joint i joins links i-1 and i.
(Another convention) The joint i+1 rotates about axis zi
(DH1) The axis xi is perpendicular to the axis zi-1
(DH2) The axis xi intersects the axis zi-1
DH convention imposes two constrains thus enabling the use of only four parameters instead of six.

27. DH1

28. DH2

29. Assign frames at the two joints and at the end-point based on DH convention
Write A1, A2, and A3
Frame i is attached to link i.
The inertial frame is Frame 0 and Earth is link 0.
Joint i joins links i-1 and i.
The joint i+1 rotates about axis zi
Link ai αi di 1 2

30. Three link cylindrical robotai αi di i 1 2 3

32. Stanford Manipulator
Link ai αi di i 1 2 3 4 5 6

33. SCARA Manipulator
Link ai αi di i 1 2 3 4

34. Inverse Kinematics