Robotics fundamentals

1. Differential Kinematics

6. Skew Symmetric Matrices
Complex Case:
When axis of rotation are not
fixed
Angular velocity is the result of
multiple rotations about distinct
axis.
For general representation of
angular velocities Skew
symmetric Matrices were

8. Definition
Skew matrix is a square matrix A whose
transpose is also its negative; that is, it satisfies
the condition -A = AT.
If the entry in the ith row and jth column is aij,i.e.
A = (aij)
then the skew symmetric condition is aij = −aji.
For example, the following matrix is skew-
symmetric:

10. Properties

12. Derivative of rotation matrix
Differentiating both
sides
1

13. Which shows that S is a skew symmetric
Now as
Multiplying both sides of equation-01
by R we get

16. Angular Velocity and Acceleration
Kinematics
 Suppose that rotation Matrix R is time varying
i.e.R=R(t)
 Time derivative of R is(as proved above)

18. Thanks