Advance Modeling of a Skid-Steering Mobile Robot for Remote ...
Advance Modeling of a Skid-Steering Mobile Robot for Remote ... Advance Modeling of a Skid-Steering Mobile Robot for Remote ...
2.2 Generalized Modeling 26V = [v T ω T ] T = [v x v y v z ω x ω y ω z ] T be a kinematic screw vector, namely thetwist, where v, ω represent respectively the vector of the linear and angular velocity of therobot expressed in the local frame. The rotation matrix projecting the local frame onto theinertial frame is defined as:⎡⎤cos φ cos ψ − sin φ cos θ + cos φ sin ψ sin θ sin φ sin θ + cos φ sin ψ cos θg R l (φ, ψ, θ) = sin φ cos ψ cos φ cos θ + sin φ sin ψ sin θ − cos φ sin θ + sin φ sin ψ cos θ⎢⎣− sin ψ cos ψ sin θ cos ψ cos θ⎥⎦(2.37)With abuse of notation, we denote with ˙q = [ g v T g ω T ] T = [ g v x g v y g v z g ω x g ω y g ω z ] Tthe twist projected onto the inertial frame, i.e. the vector of the linear and angular velocity ofthe robot expressed in the inertial frame. It is well know from the kinematics of a rigid bodythat ˙q dq . In particular, the relationship between the vector q and the time derivative of qdtcan be written as follows [18], [2], [19]:where Ω p , Ω r are defined as:dqdt⎡⎢⎣= Ω ˙q = Ω p 0 30 3 Ω⎤⎥⎦r˙q (2.38)Ω p = I 3 , Ω r (φ, ψ, θ) =⎢⎣⎡⎤cos φ tan ψ sin φ tan ψ 1− sin φ cos φ 0sin φ ⎥⎦cos φcos ψcos ψ0(2.39)By using the Plücker transform notation for spatial velocity vectors [18], [19], the kinematicequation in (2.1) can be replaced by the following equation, representing the kinematicmodel of a free rigid-body motion:⎡g R lˆX g R l˙q = J(q)V = ⎢⎣0 g 3 R⎤⎥⎦ V (2.40)lwhere J(q) represents the geometric Jacobian projecting the twist V to the inertial frame,and ˆX represents the skew-symmetric matrix of the position vector X and is defined as:⎡⎤0 −Z YˆX = Z 0 −X⎢⎣−Y X 0⎥⎦(2.41)
2.2 Generalized Modeling 27By combining equation (2.38) with (2.40), we can also wrote the following relation:dqdt = ΩJ(q)V = J a(q)V (2.42)where J a (q) represents the analytical Jacobian of the system, considering q and V as statevector and control input respectively.Furthermore, the kinematic relation given in (2.3) does not hold anymore in the general casewhere v z , ω x , ω y , d Cz 0. In this case, in fact, the first equation in (2.2), combined with theconstraint d C · v = 0, leads to the following relations:ω x |p ICR | 2 = −y ICR v z + z ICR v yω x |p ICR | 2 = x ICR v z − z ICR v xω x |p ICR | 2 = −x ICR v y + y ICR v x(2.43)x ICR v x + y ICR v y + z ICR v z = 0where |p ICR | 2 = (x 2 ICR + y2 ICR + z2 ICR ).We can rewrite (2.43) in matrix form, leading to the following non-holonomic constraintexpressed in the local frame:⎡⎤0 −z ICR y ICR |p ICR | 2 0 0z ICR 0 −x ICR 0 |p ICR | 2 0V = D(d C )V = 0 (2.44)−y ICR x ICR 0 0 0 |p ICR | 2⎢⎣⎥⎦x ICR y ICR z ICR 0 0 0By combining equation (2.40) and (2.44), we obtain the non-holonomic constraint expressedin the inertial frame:D(d C )V = D(d C )J −1 (q) ˙q = A(q) ˙q = 0 (2.45)Similarly to the planar case, we notice from (2.45) that ˙q belongs to the null space of A(q),therefore, by solving the equations in (2.44) with respect to v y , v z , ω x , ω y and substitutingthem in V, we can rewrite the kinematic model in (2.40) as follows:˙q = J(q)T(d C )η = S (q)η (2.46)
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2.2 Generalized <strong>Modeling</strong> 27By combining equation (2.38) with (2.40), we can also wrote the following relation:dqdt = ΩJ(q)V = J a(q)V (2.42)where J a (q) represents the analytical Jacobian <strong>of</strong> the system, considering q and V as statevector and control input respectively.Furthermore, the kinematic relation given in (2.3) does not hold anymore in the general casewhere v z , ω x , ω y , d Cz 0. In this case, in fact, the first equation in (2.2), combined with theconstraint d C · v = 0, leads to the following relations:ω x |p ICR | 2 = −y ICR v z + z ICR v yω x |p ICR | 2 = x ICR v z − z ICR v xω x |p ICR | 2 = −x ICR v y + y ICR v x(2.43)x ICR v x + y ICR v y + z ICR v z = 0where |p ICR | 2 = (x 2 ICR + y2 ICR + z2 ICR ).We can rewrite (2.43) in matrix <strong>for</strong>m, leading to the following non-holonomic constraintexpressed in the local frame:⎡⎤0 −z ICR y ICR |p ICR | 2 0 0z ICR 0 −x ICR 0 |p ICR | 2 0V = D(d C )V = 0 (2.44)−y ICR x ICR 0 0 0 |p ICR | 2⎢⎣⎥⎦x ICR y ICR z ICR 0 0 0By combining equation (2.40) and (2.44), we obtain the non-holonomic constraint expressedin the inertial frame:D(d C )V = D(d C )J −1 (q) ˙q = A(q) ˙q = 0 (2.45)Similarly to the planar case, we notice from (2.45) that ˙q belongs to the null space <strong>of</strong> A(q),there<strong>for</strong>e, by solving the equations in (2.44) with respect to v y , v z , ω x , ω y and substitutingthem in V, we can rewrite the kinematic model in (2.40) as follows:˙q = J(q)T(d C )η = S (q)η (2.46)
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