I have a question about acceleration equation. [ f x f y f z ] = [ u ˙ v ˙ w ˙ ] + [ 0 w − v − w 0 u v − u 0 ] [ p q r ] + g [ sin ⁡ θ − cos ⁡ θ sin ⁡ ϕ − cos ⁡ θ cos ⁡ ϕ ] Here, why uvw should be skew-symmetric matrix and why skew-symmetric matrix of uvw should be multiplied with pqr?

Question

I have a question about acceleration equation.      [                                        f                          x                                                                        f                          y                                                                        f                          z                                            ]    =      [                                                      u              ˙                                                                                      v              ˙                                                                                      w              ˙                                            ]    +      [                            0                          w                          −          v                                      −          w                          0                          u                                      v                          −          u                          0                      ]        [                            p                                      q                                      r                      ]    +  g      [                            sin          ⁡          θ                                      −          cos          ⁡          θ          sin          ⁡          ϕ                                      −          cos          ⁡          θ          cos          ⁡          ϕ                      ]  Here, why uvw should be skew-symmetric matrix and why skew-symmetric matrix of uvw should be multiplied with pqr?

Aspen Beard · Accepted Answer

This is likely related to motion in a rotating frame. The antisymmetric matrix follows because this term is of the form       R          −      1        d  R, where R is a rotation matrix. In particular R is orthogonal so       R          −      1        =      R    T   and       R    T    R  =            1      ^      Take the differential of this:  0  =  (  d      R    T    )  R  +      R    T    d  R  =  (      R    T    d  R      )    T    +      R    T    d  Rshowing that   Ω  =      R    T    d  R plus its transpose is nil, i.e.       Ω    T    +  Ω  =  0, meaning   Ω is antisymmetric.This term is the rate of change of a rotating frame as seen from a lab frame.So why should we need to consider       R    T    d  R? Take any fixed rotation matrix       R    0   and consider       R    0    R  =  r. The matrix r is simply the compound rotation of       R    0   and the original R , i.e. we have done a rotational shift of       R    0   to the coordinate system. Note then that       r    T    d  r  =      R    T        R    0    T        R    0    d  R since       R    0   is constant. Since       R    0   is a rotation,       R    0    T    R  =            1      ^       so that       r    T    d  r  =      R    T    d  R, independent of       R    0  , and thus independent of the shift of origin in the rotational coordinates.

Why we need a Skew-symmetric matrix to define acceleration?

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