Vehicle System Dynamics 549
the double benefit of decreasing cost and reducing the estimator’s reliance on measurement of
the input forces. Another benefit of the proposed methodology is once the inertial parameters
have beenmeasuredan estimate of thenon-lineartyre forces is possible[42],further increasing
the accuracy of information provided to active vehicle controllers.
Theproposedmethodisbasedonmodalparameterestimationusingthefree-decayresponses
of the vehicle, estimation of the system characteristic matrix using state space methods, least
squares analysis, combined withasimplifiedvehicle model. Knowledge of the vehicle’s equiv-
alent linear spring stiffness is also required. This is a valid assumption in the opinion of the
authors; this information is well known by the vehicle manufacturers and can be measured,
is reasonably linear for vehicles such as passenger cars, SUVs and pickups and has much
less parameter variation than damping coefficients, aerodynamic drag, road friction forces or
varying gravity forces due to changes in road grade.
2.1. Measurement sensors
The number, location and type of sensors mounted onto the vehicle’s sprung chassis mass
will determine the maximum degrees of freedom, what parameters can be measured and the
mathematical basis for the estimation model. An overview of sensors and sensor location is
presented in this section.
The first step is to choose the type of sensors, linear or angular, to be mounted onto the
vehicle chassis. For simplicity and proof of concept, the examples presented in this paper
uses three accelerometers mounted to the vehicle chassis, allowing for estimates of the sprung
mass, pitch and roll mass moments of inertia as well as lateral and longitudinal centre of
gravity locations of the vehicle.With this configuration, measurement of the yaw inertia is not
strictly possible; however, it is well documented in the literature [24,25] that the yaw and pitch
inertia values are within a few percentage points of each other, and therefore a measurement
of the pitch inertia will yield a reasonable estimate of the current yaw mass moment of inertia.
Other sensors such as gyroscopes or Linear Variable Differential Transformers (LVDTs) can
also be used.
The location of the sensors is also very important. If only accelerometers are used then they
should be spaced as far apart as possible to maximise the signal-to-noise ratio of the phase and
amplitude information. Another important consideration for positioning the accelerometers
is related to the wheelbase filtering effect. Most frequency components that excite vehicle
vibrations come from the road surface, but the geometric properties of the vehicle can result in
an amplification or attenuation of certain frequencies due to the front and rear axles following
the same profile delayed in time. This phenomenon is known as wheelbase filtering and is
dependent on the speed of the vehicle and the wheelbase. Best [43] shows that accelerometers
placed near the centre of the wheelbase will display null points and false resonance peaks.
Accelerometers placed further away from the centre of gravity position, for example above the
wheelstations,will detectboththe bounceand pitchmodesand theeffect ofwheelbasefiltering
is minimised.Therefore, in the opinionofthe authors, the optimum locationforaccelerometers
to minimise wheelbase filtering effects and to maximise the signal-to-noise ratio for phase
information is at the outer edges of the vehicle. For simplicity, in this paper the sensors are
placed directly above the unsprung masses, but this does not preclude other locations.
2.2. Extraction of free-decay response
The method relies on the successful extraction of the dynamic system’s free-decay response.
The method for extracting the free-decay response is dependent on the type of input to which
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