EE 495 Modern Navigation Systems Navigation Mathematics Angular Velocity Monday, Jan 26 EE 495 Modern Navigation Systems Slide 1 of 8.

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EE 495 Modern Navigation Systems Navigation Mathematics Angular Velocity Monday, Jan 26 EE 495 Modern Navigation Systems Slide 1 of 8

Navigation Mathematics: Kinematics – Angular Velocity Monday, Jan 26 EE 495 Modern Navigation Systems Since the Relative & Fixed axis rotations must be performed in a particular order, their derivatives are somewhat challenging The Angle-Axis format, however, is readily differentiable as we can encode the 3 parameters by Hence, Slide 2 of 9

Navigation Mathematics : Kinematics – Angular Velocity Monday, Jan 26 EE 495 Modern Navigation Systems Slide 3 of 9

Navigation Mathematics : Kinematics – Angular Velocity Monday, Jan 26 EE 495 Modern Navigation Systems This definition of the angular velocity can also be related back to the rotation matrix. Recalling that Hence, Slide 4 of 9

Navigation Mathematics : Kinematics – Angular Velocity Monday, Jan 26 EE 495 Modern Navigation Systems Notice that Therefore, or Slide 5 of 9

Navigation Mathematics : Kinematics – Angular Velocity Monday, Jan 26 EE 495 Modern Navigation Systems Properties of skew-symmetric matrices  Hence, we can think of the skew-symmetric matrix as  or, in the case of angular velocity Slide 6 of 9

Navigation Mathematics : Kinematics – Angular Velocity Monday, Jan 26 EE 495 Modern Navigation Systems More properties of skew-symmetric matrices Therefore, or Slide 7 of 9

Navigation Mathematics : Kinematics – Angular Velocity Monday, Jan 26 EE 495 Modern Navigation Systems Angular velocity can be describe as a vector  The angular velocity of the b-frame wrt the a-frame resolved (i.e. coordinatized) in the c-frame Angular velocity can be describe as a skew-symmetric matrix  The skew-symmetric matrix is equivalent to the vector cross product o When pre-multiplying another vector Angular velocity can be related to the differential of the rotation matrix Slide 8 of 9

Navigation Mathematics : Kinematics – Angular Velocity Monday, Jan 26 EE 495 Modern Navigation Systems Slide 9 of 9