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

2.1 State of the Art 17these expressions for v Lx , v Rx in equation (2.6) and adding/subtracting the first two scalarequations, we obtain the following relation between the wheel angular velocities and thelongitudinal and angular velocity of the robot in its matrix form:⎡ ⎡⎤ ⎡v xη = ⎢⎣ω⎤⎥⎦ = r 1 − λ L 1 − λ R ω Lz2⎢⎣⎥⎦ ⎢⎣ω⎤⎥⎦R− 1−λ Lc1−λ Rc(2.11)In order to complete kinematic model of a SSMR, additional velocity constraints shouldbe considered. We can easily obtain the velocity constraint on the local frame from equation(2.8) as:v y + sω z = 0 (2.12)which can be rewritten with respect to the inertial frame, by using equation (2.1), as:[] []sin θ cos(θ) 0 ˙q + s˙θ = sin θ cos(θ) s ˙q = A(q) ˙q = 0 (2.13)It is worth to notice that constraint (2.13) represents a non-holonomic constraint since itis not integrable, therefore our system can be defined as non-holonomic system similarly toall conventional wheel mobile robot.Furthermore, since ˙q belongs to the null space of A(q), we can also write:˙q = S (q)ηwhere the expression of the matrix S (q) can be obtained from equation (2.1) by includingthe constraint (2.12) as follows:⎡⎤ ⎡ ⎡⎤cos(θ) − sin(θ) 0 v x cos(θ) s sin(θ)˙q = sin(θ) cos(θ) 0 −sω z = sin(θ) −s cos(θ) η = S (q)η (2.14)⎢⎣0 0 1⎥⎦ ⎢⎣⎤⎥⎦ ⎢⎣0 1⎥⎦ω zEquation (2.14) provides the constraint on velocity between the generalized velocity vector˙q and the control input η and represents our kinematic model of SSMR.The constraint on acceleration between ˙q, ¨q and V, ˙V can be obtained by differentiating withrespect to time equation (2.14):¨q = Ṡ (q)V + S (q) ˙V. (2.15)