This formula can be modified to obtain the velocity of P by operating on its trajectory P (t) measured in the fixed frame F. Substituting the inverse transform for p into the velocity equation yields: v P = [ T ˙ ˙ ( t ) ] [ T ( t ) ] − − 1 P ( t ) = [ v P 0 ] = [ A ˙ ˙ d ˙ ˙ 0 0 ] [ A d 0 1 ] − − 1 [ P ( t ) 1 ] = [ A ˙ ˙ d ˙ ˙ 0 0 ] A − − 1 [ 1 − − d 0 A ] [ P ( t ) 1 ] = [ A ˙ ˙ A − − 1 − − A ˙ ˙ A − − 1 d + d ˙ ˙ 0 0 ] [ P ( t ) 1 ] = [ A ˙ ˙ A T − − A ˙ ˙ A T d + d ˙ ˙ 0 0 ] [ P ( t ) 1 ] v P = [ S ] P . The matrix [ S ] is given by: [ S ] = [ Ω Ω − − Ω Ω d + d ˙ ˙ 0 0 ] where [ Ω Ω ] = A ˙ ˙ A T , is the angular velocity matrix.