Advance Modeling of a Skid-Steering Mobile Robot for Remote ...

2.2 Generalized Modeling 31⎡v ix⎡v x − y i ω z + z i ω yv i = v iy = v − ˆp i ω = v y + x i ω z − z i ω x(2.55)⎢⎣v⎤⎥⎦⎢⎣iz v z − x i ω y + y i ω⎤⎥⎦xwhere ˆp i is the skew-symmetric matrix of the vector p i .Finally, we can write the generalized dynamic model expressed in the inertial frame by combining(2.49) and (2.51), adding the reaction forces and torques vector (2.54) and incorporatingthe non-holonomic constraint (2.45) by using the vector of Lagrange multipliers ζ:M g ¨q + R(q) + G = B(q)τ + A T (q)ζ. (2.56)To eliminate the unknown ζ, we multiply equation (2.56) by the matrix S T (q) defined in(2.40) and then we use the constraint (2.16), obtaining the following dynamic model in thegeneralized coordinates q:¯M ¨q + ¯R + Ḡ = ¯Bτ (2.57)where ¯M = S T M g , ¯R = S T R, Ḡ = S T G and ¯B = S T B.For control purpose, it is convenient to express the dynamic model of SSMR in terms ofη, ˙η. Thus, by including the kinematic equation (2.40) and its time derivative in (2.57), weobtained the dynamic model:¯M ˙η + ¯Cη + ¯R + Ḡ = ¯Bτ (2.58)where ¯M = S T M g S and ¯C = S T MṠ .It is worth to notice that the dynamic model in (2.58) depends on q, and more preciselyon the value of Θ, only for the matrix Ḡ. In fact, we can rewrite the matrices in (2.58) byemploying relations (2.46),(2.51),(2.54) as follows:¯M = S T M g S = T T J T (J −1 ) T MJ −1 JT = T T (J −1 J) T MJ −1 JT = T T MT¯C = S T M g Ṡ = T T J T (J −1 ) T MJ −1 (JṪ + ˙ JT) = T T MṪ + T T MJ −1 ˙ JT¯R = S T R = T T J T (J −1 ) T F r = T T F r(2.59)Ḡ = S T G = T T J T G = T T [−m( l R g g) T0 T 3 ] T¯B = S T B = T T J T (J −1 ) T T f τ = T T T f τ.