Keywords

1 Introduction

Recent legislation has released further development of Steer-by-Wire systems [7]. Published standards describe supplementary redundancies in steering systems with limited performance [2] as optional design method. Implementation of differential braking has therefore been topic of several research in the past. Jonasson [6], Gauger [3], Sharma [10] and our recent work investigated limits regarding driving dynamics in steady-state conditions [9]. Maximum curvature depends on maximum feasible brake forces due to road wheel friction.

The constraint of differential braking in terms of Steer-by-Wire can be equivalent to understeer since desired yaw rate would not be achieved. We are demonstrating theoretically and practically how understeer can be reduced. Therefore overlaying drive torque and increased longitudinal slip was used to change the vehicle’s driving behavior.

A stability controller within a model following control scheme was implemented. As a result we were able to show the impact of longitudinal slip and drive torque on the vehicle’s driving behavior in practical application. To achieve greater magnitudes of yaw rate, increasing side slip was necessary. This can theoretically be considered as oversteer, but might be compensated due to vehicle free steer behavior and active countermeasures.

The results show increases in potential for lateral control using effects of combined slip. Variations in driveline configurations and steering kinematics give an outlook towards further vehicle applications. These findings extend previous research.

2 Vehicle Dynamics

Differential braking differs from conventional front axle steering regarding control input and front wheel steering system behavior. Literature [4, 6, 9] has described single track models where the front wheel steering angle is treated as an additional degree of freedom. Figure 1 shows the model considered within this paper. (a) is an extended single track model with differential brake forces [6]. (b) shows the steering model to describe mechanics of the vehicle’s free steer behavior. The model was parametrized by test bench measurements with the test vehicles steering system. Phenomena of combined tire slip are modeled through a coupled brush tire model according to Pacejka [8] by Eq. (1). Its output vector \(\boldsymbol{F}\) effectively describes the resulting forces \(F_x\) and \(F_y\) depending on the combination of theoretical slip quantities \(\sigma _x\) and \(\sigma _y\).

Fig. 1.
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(a) Vehicle Model [6] (b) Steering Model

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$$\begin{aligned} \boldsymbol{F}=\boldsymbol{\sigma }\frac{F}{\sigma } \end{aligned}$$
(1)

Yielded vehicle model behavior is nonlinear due to steering system friction and tire behavior. Therefore the phase plane method can describe system dynamics as practiced by Bobier-Tiu [1] to describe differential braking control. Figure 2 shows side slip and yaw rate \((\beta -\dot{\psi })\) phase portraits with free steering as a result of approximate differential brake forces \(F_{b,fl/rl}\) on front and rear axle.

Fig. 2.
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Nonlinear behavior represented by phase portraits for differential brake forces at \(v_0 = 20\) m/s and \(\mu =0.3\)

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The system reaches a steady state for all three variants. It has fewer yaw rate but is more stable at higher velocities than the same system would with steering angle fixed to zero. Greater dynamics can be achieved using two different methods. On the one hand, differential brake forces \(F_{b,fl/rl}\) can be increased by overlaying drive torque. Rear axle understeer interventions can be utilized to increase yaw rate on the other hand. They cause the vehicle to turn into the corner by transiently increasing side slip angles on front and rear axle. Both options yield potential oversteer and the necessity of equivalent countermeasures.

3 Controller Design

We implemented a model following controller in a vehicle with rapid prototyping platform to investigate performance improvement. Figure 3 shows the scheme. It comprises a reference model, feedforward block, a slip controller and stability controller as well. The stability controller in this scheme can bypass the slip controller with an additional brake input disregarding the separate anti-lock functionality of the slip controller. The controller prevents under- and oversteer depending on the driving state.

Fig. 3.
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Control concept for differential braking as steering backup with stability control. Inputs \(u_{C}\): Driver steering wheel angle \(\delta _{SW}\), longitudinal velocity \(v_x\), yaw rate \(\dot{\psi }\). Outputs \(u_{V}\): Brake pressures \(p_{b1...4}\)

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The feedforward block computes necessary brake inputs by the reference model’s desired yaw rate \(\dot{\psi }\) and takes into account vehicle longitudinal speed \(v_x\) and steering wheel angle \(\delta _{SW}\). The steering angle resulting from the rack position \(x_{Rack}\) is not considered as an input in the control algorithm. Sensor data for the rack position might not be available in the considered use case of differential braking.

The control scheme is used to determine the possible performance increase by understeer interventions yielding greater magnitudes of tire slip. Triggering drive torque ovelay is not shown.

4 Vehicle Testing and Performance Evaluation

Differential braking’s limit is primarily understeer, therefore the main goal was to determine the performance increase with understeer countermeasures on the rear axle. Since it is also able to impact stability and maximum yaw moment, a variant with drive torque overlay was tested as well. The maneuver “slowly increasing steering-wheel angle” from straight line driving according to ISO 4138 [5] was chosen for performance evaluation regarding circular driving behavior. To determine the effectiveness of understeer countermeasures, the tests were carried out in a modified prototype vehicle on a snow-covered driving dynamics surface with low friction. Vehicle behavior for conventional steering was measured as a reference and compared to the results of differential braking with positive scrub radius in three variants. The results with feedforward differential braking were compared to adding the stability controller and drive torque. To eliminate the impact of the positive scrub radius, the test sequence was repeated only applying brake and drive forces on the rear axle to analyze an equivalent vehicle with negative scrub radius. Initial vehicle speed was 20 m/s.

Figure 4 shows the results. Thereby steering-wheel and side slip angle are shared over lateral acceleration for both sequences and corresponding measurements. For each case, three measurement series have been examined to ensure statistical validity. Steering by feedforward differential braking shows to lead to expected understeer behavior for the front and rear axle braking as well as rear axle only. For both test sequences, understeer interventions through the stability controller and drive torque overlay increased feasible lateral acceleration compromising slip angle \(\beta \). Shown driving behavior can further be adjusted by controller design.

Fig. 4.
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Slowly increasing steering-wheel angle according to [5] at \(v_0 = 20\) m/s

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Results again show strong dependency of differential braking on road wheel friction and scrub radius. Analyzing effects of combined slip indicates novel improvements towards feasible lateral acceleration during low friction tests. Drive torque overlay has also shown promising results towards feasible vehicle dynamics. The downside of drive torque overlay was the negative effect on driving stability and that vehicle speed was reduced less to decrease accident severities regarding the steering failure. Another challenge is the impact of the front axle’s free steering behavior.

5 Outlook and Summary

This paper shows earlier derived limits of lateral control for differential braking to be extendable by effects of combined slip. We focussed on maximum lateral vehicle dynamics regarding circular driving behavior. Measurements for two vehicle variants have shown expectable different results in the linear region. Achieved vehicle reactions using greater magnitudes of longitudinal slip were comparable for both setups.

Feasible yaw moment was increased overlaying drive torque at both axles and analyzed in the same driving maneuver. The result in real vehicle tests was a performance increase as well as a tendency to oversteer, which yields the necessity for a stability controller as well.

Considering potential increase in vehicle dynamics as a chance and the risk of oversteer as a fact of differential braking, a stability controller seems essential for full use of differential braking’s capabilities as backup steering. Further testing has to be carried out in real world and virtual development for implementation.