04 Steering And Turning Vehicles - Department of Mechanical ...

Y
Recall: position and velocity in inertial frame
We defined the vehicle’s kinematic state in the inertial frame by,
l2
y
B
l1
x
ψ
So, for our simple (single-axle) vehicle,
the velocities in the inertial frame in
terms of the wheel velocities are,
ME 360/390 – Prof. R.G. Longoria
Vehicle System Dynamics and Control
X
X
Velocities in the local (body-fixed)
reference frame are transformed
into the inertial frame by the
rotation matrix,
or, specifically, qɺ = R( ψ ) ⋅qɺ
I
q
I
⎡ X ⎤
=
⎢
Y
⎥
⎢ ⎥
⎢⎣ ψ ⎥⎦
⎡ cosψ sinψ 0⎤
R(
ψ ) =
⎢
−sinψ
cosψ 0
⎥
⎢ ⎥
⎢⎣ 0 0 1⎥⎦
Inverting, we arrive at the velocities in the global reference
frame,
⎡ Xɺ ⎤ ⎡U ⎤ ⎡cosψ ⎢ ⎥
qɺ I = Yɺ ⎢
V
⎥
( ψ )
⎢
⎢ ⎥ =
⎢ ⎥
= Ψ ⋅ qɺ
=
⎢
sinψ ⎢ψ ⎥
⎣
ɺ
⎦
⎢⎣ Ω⎥⎦
⎢⎣ 0
−sinψ
cosψ 0
0⎤
⎡ vx
⎤
0
⎥ ⎢
v
⎥
⎥ ⎢ y ⎥
1⎥⎦
⎢⎣ ω ⎥ z ⎦
qɺ
I
⎡ Rw l2Rw ⎤
⎢ ( ω1 + ω2)cos ψ − ( ω1 −ω2
)sinψ
2
B
⎥
⎡Xɺ ⎤ ⎢ ⎥
⎢ ⎥ Rw l2Rw = Yɺ
⎢ ( ω1 ω2)sin ψ ( ω1 ω2)cos ψ ⎥
⎢ ⎥ = + + −
⎢ 2
B
⎥
⎢ψ ⎥
⎣
ɺ
⎦ ⎢ ⎥
⎢
Rw
( ω1 −ω2
)
⎥
⎢⎣ B
⎥⎦
Department of Mechanical Engineering
The University of Texas at Austin