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

2.2 Generalized Modeling 30⎡g R(J −1 ) T l 0 3(q) = ⎢⎣ ˆX g R g l R⎤⎥⎦ , T f =l⎡ ⎤1 10 00 00 0−h⎢⎣−c−h⎥⎦d. (2.52)In order to complete the generalized dynamic model of SSMRs, we must add the reactionforces and torques in equation (2.49) and include the non-holonomic constraint (2.45) byusing the Lagrange multipliers. As previously said, in this section we are only consideringthe reaction forces at the i th contact point as unknown functions f ix , f iy , f iz . Thereby, similarlyto case of the active forces and torques, we can write the reaction force and torque vector expressedin the inertial frame, by first calculating it in the local frame, F r = [ f x f y f z τ x τ y τ z ] T ,and then multiplying by the Plücker transform. By considering the vector of reaction forcesand torques at each contact point defined as F i = [ f ix f iy f iz 0 0 0] T , we can calculatethe reaction force and torque vector expressed in the local frame as:⎡∑ 4i=1 f ix⎤F r =⎡4∑ I 3 0 3⎢⎣ˆp i I⎤⎥⎦ F i =3i=1∑ 4i=1 f iy∑ 4i=1 f iz∑ 4i=1(y i f iz − z i f iy )∑ 4i=1(−x i f iz + z i f ix )⎢⎣ ∑ 4⎥⎦i=1(x i f iy − y i f ix )(2.53)Thereby, the reaction force and torque vector expressed in the inertial frame can be calculatedas follows:R(q) = (J −1 ) T (q)F r (2.54)As it will be described in Chapter 4, the reaction forces f ix , f iy , f iz exerted on the i th contactpoint depend respectively by the wheel center velocities v ix , v iy , v iz . The relation between thewheel center velocity v i and the robot linear and angular velocity can be written as follows: