Page 119 - Handbook of Biomechatronics

Model-Based Control of Biomechatronic Systems 115

The rack and its connection to the wheel spindle, as well as the interme-
diate steering shaft, are combined and represented as a single inertia at the
pinion. The dynamics of the steering pinion are described by

€ _
J p θ p ¼ K p θ p b p θ p + T tb + T a + T SAT (23)
where θ p , J p , and b p are, respectively, angular displacement, inertia, and
damping of the pinion, and K p is the stiffness induced by the inclined kingpin
axis on the rack displacement. T SAT and T a represent the self-aligning torque
(SAT) and the assist torque provided by the EPS system, respectively.
In the single track, the vehicle’s center of mass velocity (V) makes an
angle β with the longitudinal direction of the vehicle. Considering the side-
slip angle (β) and yaw rate (ω z ) of the vehicle as the state variables of the
single track model, the equations of motion are expressed as follows:

_

mv x β + ω z ¼ F yf cos δ ðÞ + F yr (24)
I zz _ ω z ¼ L f F yf cos δðÞ L r F yr (25)
where F yf and F yr are front and rear lateral force of the wheels and are
approximated by a linear tire model (in contrast to a nonlinear tire model
used in the high-fidelity model):

(26)
F yf ¼ C αf α f
F yr ¼ C αr α r (27)
Assuming small steer angles, the front and rear slip angles can be approxi-
mated as follows:

v y + L f ω z
α f ¼ δ (28)
v x
v y L r ω z
α r ¼ (29)
v x
where v x and v y , respectively, are the longitudinal [v x ¼Vcos(β)] and lateral
[v y ¼Vsin(β)] components of the vehicle mass center velocity, and v x is
assumed to be constant during the simulations. The steering angle of the
front wheel is represented by δ¼θ p /G steering , and G steering is the ratio of
the rotation of steering wheel angle to the average value of left and right
wheel steer angles. The SAT, which is created by the interaction between
the tire and the road, is a linear function of slip angle (α f ) for small slip angles
(T SAT ¼C Tα α f ), where C Tα is a SAT coefficient.