Wednesday, April 23, 2014

Skew symmetry property

The three types of derivatives: vectors-by-matrices, matrices-by-vectors, and matrices-by-matrices. These are not as widely considered and a notation is not widely agreed upon. However the second type (matrices-by-vectors) is widely used in Robotics.
If we consider the following state form for a robot:
 M(\theta )\ddot \theta  + C(\theta ,\dot \theta )\dot \theta  + g(\theta ) = \tau    

The matrix N(\theta ,\dot \theta ) = \dot M(\theta ) - 2C(\theta ,\dot \theta ) is skew-symmetric.
Here the matrix-by-vector derivative is:                                                              

{\dot m_{kj}} = \sum\limits_i^n {\frac{{\partial {m_{kj}}}}{{\partial {\theta _i}}}} {\dot \theta _i}                   

Note that this is not true if the robot dynamics is expressed in any other forms (for exampleM(\theta )\ddot \theta  + R(\theta ,\dot \theta ) + g(\theta ) = \tau ).