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Future active safety systems need more accurate information about the state of vehicles. This article proposes a method to evaluate the lateral state of a vehicle based on measured tyre forces. The tyre forces of two tyres are estimated from optically measured tyre carcass deflections and transmitted wirelessly to the vehicle body. The two remaining tyres are so-called virtual tyre sensors, the forces of which are calculated from the real tyre sensor estimates. The Kalman filter estimator for lateral vehicle state based on measured tyre forces is presented, together with a simple method to define adaptive measurement error covariance depending on the driving condition of the vehicle. The estimated yaw rate and lateral velocity are compared with the validation sensor measurements.

The vehicle state estimation has been a subject for numerous papers, especially because of the large scale market penetration of Electronic Stability Control (ESC) systems. Even then, more accurate and reliable vehicle state information is needed for upcoming active control applications such as active steering (front and rear), lane keeping and torque vectoring.

The main disadvantage of most of the vehicle state estimation approaches is the requirement for prior knowledge of tyre parameters such as cornering stiffness and friction coefficient [

The advantage of tyre force measurement can be seen in _{wind} and roll angle _{y}. If the state estimation is based on lateral acceleration the following errors exist:

roll angle θ and road inclination λ introduce offset for lateral acceleration a_{y} sensor due to gravity component (5,7° roll angle results 0.1g error for lateral acceleration)

Influence of side wind is not properly captured from the measured acceleration, but has to be carried by the tyres (and influence for v_{y} is missing)

v_{y} is not even parallel to compensated lateral acceleration

The roll angle and road inclination influence the yaw rate measurement as well, but contrary to the lateral acceleration, gravity does not participate and thus the overall impact is minor (the same applies for pitch angle).

Consequently, the direct measurement of tyre forces seems beneficial in contrast to body accelerations and rotational velocities. Possible technologies for the tyre force measurement could be:

strain measurement of suspension components [

measurement of tyre carcass displacement [

tyre tread displacement [

force sensing bearing [

Even though there have been many attempts to measure tyre forces to aid control systems, there are very few articles which study how to exploit them [_{y} based on measured tyre forces.

The optical tyre sensor was developed for the first time in the EC-funded APOLLO-project [

The OTS measuring principle is shown in

A test car setup can be seen in

The problem is to estimate lateral velocity and yaw rate at the centre of gravity of the vehicle with a minimal set of parameters. The estimator consists of virtual tyre sensors, a driving state estimator (linearity) and a Kalman filter. The overall structure of the estimator is shown in

The Kalman filter is an effective and recursive solution for the discrete data filtering problem from noisy measurements [_{k}:
_{k} and v_{k} represent process and measurement noise. The _{k}) estimate error is:

The

The Kalman gain weights the

The Kalman filter expects the process and measurements noise to be with normal probability distribution:

The test car was equipped with two optical tyre sensors. The sensor positions were at the left front and right rear wheels. However, the vehicle state estimator developed here requires individual tyre forces of all tyres or axle forces. Thus, the lateral tyre forces of right front wheel and left rear wheel have to be estimated. The natural way to estimate is to exploit the information from the tyre sensor at the same axle (only lateral dynamics considered here). Hence, the “similarity method” [e.g.,

The obvious starting point is to solve slip angle α of the tyre from tyre sensor forces F_{y} and F_{z}. The simple four parameter Magic Formula reads [

Another required variable for a virtual tyre sensor is vertical load. The wheel load deviation of a tyre sensor wheel is available:
_{z,1} or by recording F_{z,1} values as a static wheel load when F_{y,1} ∼ 0. The vertical load of the virtual tyre sensor (neglecting longitudinal load transfer and mass of vehicle is constant):

The lateral force of the virtual tyre sensor can be then calculated with the same

When considering the inverse magic formula method to solve the lateral force of the second tyre, it is clear that there has to be simpler method to obtain it. The normalized lateral force reads:

The vertical load F_{z,2} of the virtual tyre sensor is calculated as in

The Kalman filter requires a state transition matrix (without linearity assumption for tyre behaviour), which is here based on the lateral and yaw equations of motion:
_{z} is yaw moment of inertia, l_{f} and l_{r} are centre of gravity distances from the front and rear axles. The lateral equation can be written:
_{y,f}_{y,r}

Lateral tyre forces are assumed to act on the centre of the contact patch (neglecting pneumatic trail), thus the yaw acceleration can be expressed:

The _{x}, which is assumed to be slowly changing. T_{s} is time step. The tyre force state transition is modelled as an identity function. The measurement matrix reads:

A simple vehicle model is needed to provide vehicle yaw rate, lateral velocity and tyre forces for the other subsystems of the estimator. The single track (or bicycle model) [e.g.,

The axle forces F_{f} and F_{r} are assumed to be linear to the slip angle:
_{f} and C_{r} are the front and rear axle cornering stiffness. The slip angles for the front and rear axle read:

The process noise variance matrix Q is assumed to be constant with very high process variance for tyre forces because a new measurement for tyre forces is always more accurate than _{y} and

The measurement noise covariance matrix R has to be modified continuously. A particularly noticeable bias is introduced to lateral velocity and yaw rate measurements from the single track model etc. during hard cornering. Consequently, during non-linear vehicle behaviour, the measurement variances for the single cycle model should have very high values compared with the process noise values. This allows Kalman gain to weight more direct tyre force integration instead of obviously biased v_{y} and

One way to evaluate whether the vehicle behaves as a linear system is to compare nominal yaw rate and actual yaw rate, which has been exploited in ESC-systems [

Depending on driving conditions, the corresponding measurement noise variance (for single track v_{y} and

Another method to judge vehicle non-linearity would be to evaluate the relation of lateral and vertical forces (

A test manoeuvre was sequential and aggressive lane changes on dry and horizontally even tarmac road. The lateral and vertical tyre forces for the test manoeuvre are shown in

_{y} (due to the integration step needed for _{y} remains tolerable. The yaw rate overestimation results in cumulation of error to the negative direction of v_{y}.

This paper presented one application for tyre sensors. The proposed vehicle lateral state estimator can similarly be used with other tyre force measurements than tyre sensors, such as suspension part or wheel hub strain measurements.

The main advantage of tyre force sensing is definitely the information given about the operating state of each tyre. In addition it is possible to calculate the lateral acceleration of a vehicle from the sum of tyre forces without any bias from body roll angle and road inclination. Also, the influence of side wind is realistically captured. The results show that lateral acceleration was accurately calculated from the tyre forces when measured by an optical tyre sensor. The yaw rate sensor, however, is much more complex to replace than the acceleration sensor. The required integration step makes the estimate extremely sensitive to errors in parameters l_{f} and l_{r}, which may arise, for example, from the pneumatic trail in addition to the mechanical movement of the wheel hub. Thus, the yaw rate is a valuable measure of the differences in front and rear axle forces acting on a vehicle. However, if the tyre sensor can produce an accurate and reliable estimate for the vertical force, the vehicle centre of gravity position can be calculated in a steady state condition.

One of the objectives of the estimator was to minimize the number of parameters.

The vehicle mass is naturally available from the vertical tyre forces, but it requires real force measurements instead of the virtual tyre sensors implemented in this paper. However, reasonable accuracy would also be possible with two tyre sensors.

Improvement of the optical tyre sensor operation region would enable accurate estimation of lateral force during high vertical force and high slip angle. The single track model could be extended to adapt the parameters to ensure accurate operation during low lateral excitation.

The main benefits of this proposed Kalman filter approach could be seen on slippery road conditions, on side wind, and on inclined roads, where the problems for the model based estimation based on non-linear vehicle model are seen. In addition, the tyre force based estimator can be fitted to totally new types of vehicles without any major parameter modifications as long as the axle length is known.

The virtual tyre sensor concept might be feasible together with a more production oriented tyre sensor, or other type of tyre force measurement. The virtual tyre sensor concept would also lower the threshold for production tyre sensors as half of the sensor costs would be saved if only two tyres of a car needed be equipped with tyre sensors. It is possible to do this research and development mainly with simulation models, with no significant investments needed for the tyre sensor prototypes. The main problems are the combined slip case and the slightly non-linear influence of the wheel load on tyre forces.

Lateral forces acting on one axle of cornering vehicle.

Optical tyre sensor measurement principle [

Tyre sensor calibration cycle for vertical force and comparison with test rig measurement.

Tyre sensor calibration cycle for lateral force and comparison with test rig measurement (vertical force and lateral forces during cycle, 60 km/h).

Measurement car setup.

Block diagram of the estimator.

A relay with hysteresis to select proper measurement noise variance.

Lateral and vertical tyre forces for a driving manoeuvre (v_{x}∼60 km/h, dry tarmac).

Lateral force deviation from single track model and measurement error variance during test run.

Lateral state estimate based Kalman filter estimator and for sensor measurement.

Required estimator parameters.

m | Vehicle mass | available from vertical tyre forces | 1,603 kg |

l | axle length | vehicle parameter | 2,575 m |

l_{f} |
Centre of gravity distance from front axle | available from vertical tyre forces in steady state condition | 1.05m |

l_{r} |
Centre of gravity distance from rear axle | available from vertical tyre forces in steady state condition | 1.525m |

I_{z} |
Vehicle yaw moment of inertia | roughly _{r}·l_{r} |
3,156 kg m^{2} |

Q | Process noise covariance | constant | diag([0.01 0.01 1e4 1e4]) |

R | Measurement noise covariance | derived in section 3.5 | variable |

C_{f} & C_{r} |
Cornering stiffness of the linear model (or characteristic velocity | ESC-system (nominal behaviour of a vehicle) | 76,614 N/rad & 82,087N/rad |