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A fuzzy inference system for understeer/oversteer detection towards model-free stability control.

ABSTRACT

In this paper, a soft computing approach to a model-free vehicle stability control (VSC) algorithm is presented. The objective is to create a fuzzy inference system (FIS) that is robust enough to operate in a multitude of vehicle conditions (load, tire wear, alignment), and road conditions while at the same time providing optimal vehicle stability by detecting and minimizing loss of traction. In this approach, an adaptive neuro-fuzzy inference system (ANFIS) is generated using previously collected data to train and optimize the performance of the fuzzy logic VSC algorithm. This paper outlines the FIS detection algorithm and its benefits over a model-based approach. The performance of the FIS-based VSC is evaluated via a co-simulation of MATLAB/Simulink and CarSim model of the vehicle under various road and load conditions. The results showed that the proposed algorithm is capable of accurately indicating unstable vehicle behavior for two different types of vehicles (SUV and Sedan). The algorithm can do this without any significant parameter adjustment, illustrating its robustness against the considered uncertainty.

CITATION: Hirche, B. and Ayalew, B., "A Fuzzy Inference System for Understeer/Oversteer Detection Towards Model-Free Stability Control," SAE Int. J. Passeng. Cars - Mech. Syst. 9(2):2016.

INTRODUCTION

Vehicle stability control systems are a key factor in reducing the number of road fatalities. The National Highway Traffic Safety Administration (NHTSA) estimates a total of 1,144 lives saved in 2012 by VSC systems [1]. It is therefore not surprising that extensive research has been conducted to improve the workings of these systems [2]. However, current systems heavily rely on linearized planar vehicle models for sideslip and direct yaw moment control (DYC) using brake intervention [3]. These models use predefined vehicle parameters such as vehicle weight or tire lateral stiffness. In reality these parameters are subject to change, thus reducing the optimal operation point of these model-based VSC systems to a narrow window of road and load conditions. This also causes a considerable amount of tuning effort during the development process of VSC algorithms for production vehicles. Every new prototype stage or configuration will have a new weight distribution, tire parameters, or suspension alignment which makes it necessary to adapt the VSC algorithm. Hence, more robust algorithms are necessary [4]. Early approaches to compensate for nonlinearities and uncertainties used adaptive controllers [5] or H[infinity] control theory [6]. Arat et al [7] treated the degradation of performance from model-based systems due to estimation errors during evasive maneuvers. They proposed a sensor fusion approach that can estimate tire slip angle under varying surface friction conditions. Dengo et al [8] used model predictive control to calculate the amount of steering angle and stabilizing yaw moment (DYC). An optimal allocation algorithm is then designed to generate this yaw moment. Their algorithm showed promising results in terms of performance but lacked in robustness considerations.

In recent times, fuzzy logic-based approaches have proven to outperform advanced model-based control algorithms in industrial applications where models are inaccurate or not available. Fuzzy logic uses human linguistic reasoning and possibility distributions instead of traditional binary logic and allows for model uncertainty by imposing generalized rules [9]. Junwei et al and Uzunsoy combine a conventional yaw and side-slip angle tracking algorithm with fuzzy logic instead of a PID controller. By doing so they reduce the amount of tuning that is necessary to adjust PID characteristics, such as overshoot and settling times, while at the same time making the controller more robust against parameter changes such as load conditions or estimation errors from side-slip estimators or road friction estimators [10] [11]. Kaldas et al [12] and Rao et al [13] used fuzzy logic to control the vertical dynamics of suspension systems. Since the input to suspension systems is difficult to measure model-based controls are difficult to implement. A comparison with passive and model-based LQR controlled suspensions showed vast improvements in ride and dynamic tire load over a wide range of road inputs [13] [12].

Even though previous approaches have shown good results in dealing with model uncertainty and external inputs, there are a few drawbacks to be addressed. The key component of every fuzzy-inference system (FIS) is a set of if-then rules that mimic human knowledge and reasoning for the problem at hand. The FIS, unlike a human operator, cannot learn and adapt to changing external conditions without extensive training or adaptation [14]. Previous approaches have tried to generate arbitrary if-then rule sets and use adaptation algorithms to give the FIS system a means to optimize itself without supervision [12].

The approach proposed in this paper uses fuzzy-inference systems towards a model-free design of vehicle stability control systems. It builds on Anderson and Law's [15] approach of a fuzzy logic stability control that uses measurable vehicle signals instead of estimation algorithms and parameterized models. In this paper, we propose soft computing techniques to generate an adaptive neuro-fuzzy inference system (ANFIS), a rule set that can be trained to characterize a desired vehicle behavior. Deviations from the desired vehicle behavior are categorized as understeer or oversteer on a scale of +10 to -10. An optimized ANFIS system has the potential to simplify and unify the vehicle stability control design process for different vehicle classes by removing reliance on vehicle specific and potentially changing parameters such as tire properties, center of gravity location, and dimensions.

The paper will focus on the discussion of the design and illustration of the robustness of this upper level detection of understeer or overseer. A simple lower level brake force allocation algorithm will be used to compare the control performance against conventional systems. The model-free VSC is implemented in Matlab/Simulink[c] and co-simulated with two CarSim[c] vehicle models. The vehicle models represent two different types of vehicles, 1) a SUV, and 2) a Sedan. The performance of the model-free VSC is evaluated in simulations of a standardized ECE R13H Sine with Dwell test [16].

MODEL-FREE VSC ARCHITECTURE

The structure of the model-free VSC is shown in Figure 1. Conceptually, the implementation is organized hierarchically as upper and lower level control. The fuzzy detection stage in the upper level control uses vehicle signal inputs, like longitudinal vehicle speed (Vx), lateral acceleration (Ay), yaw rate (Avz), and road wheel angle (RWA), to detect and categorize unstable vehicle oversteer (OS) or understeer (US) events. This is done in a single FIS. The output is one final indicator that categorizes the unstable situation on a scale of -10 for heavy oversteer to +10 for heavy understeer. Logically, 0 stands for a neural stable vehicle. In the lower level control, responsible for allocation of the stabilizing yaw moments, the activation module uses the stability indicator from the detection stage and combines it with the yaw rate signals to decide which wheel to brake. The activation signal, which is proportional to the OS/US indicator is then fed through a proportional gain to convert the [+ or -]10 OS/US indicator signal to brake caliper pressures.

A companion paper by the authors [17] focuses on an active steering-based implementation of the lower-level of the VSC using the FIS for upper-level OS/US detection detailed in the following sections of this paper.

FUZZY OS/US DETECTION

An important task in modern VSC control systems is to reliably detect unstable events. The reliability here is one of the key factors towards customer satisfaction. VSC systems with misdetections or overly sensitive systems limiting the vehicles cornering capacity too early lead to customer complaints. The same applies to systems that are missing to detect unstable events, where the latter is not just a customer complaint but also a safety issue. Therefore, a VSC system has to detect unstable events early enough to be able to stabilize the vehicle, with the least amount of misdetections to satisfy the customer. It is believed that model-based approaches are prone to such misdetections because possible mismatches between reference model used for control and the actual vehicle. The goal for the model-free structure is to eliminate reliance on models for US/OS detection by using fuzzy logic modeling.

The FIS of the model-free VSC utilizes various vehicle signals, evaluates them and returns one combined oversteer/understeer number (OS/US Indicator) between -10 and +10 to indicate unstable events of various severity. The FIS module we propose uses five vehicle signals steering wheel angle (SWA), steering wheel rate (SWRATE), yaw rate (AVZ), yaw acceleration (AAZ), and lateral jerk (AAY). The first four signals are commonly used in conventional VSC approaches as well. The lateral jerk signal was included to identify the OS pattern (see Figure 3). It is the derivative of the lateral acceleration and goes to zero when the lateral acceleration peaks out (saturates).

We adopt the Sugeno-based FIS, which uses 5 membership functions to fuzzify each of these input signals. The membership functions were not generated arbitrarily as in previous approaches. A combination of subtractive clustering and optimization lead to their shape and orientation. As a result of the optimization, 7 rules are retained that combine the 5 fuzzified inputs and generate a single OS/US Indicator between -10 and +10. Figure 2 shows an example of the detection process. Here, the two rules "If yaw rate (AVZ) is high and steering wheel angle (SWA) is medium then Indicator is high negative" and the rule "If yaw rate (AVZ) is medium then Indicator is 0" are combined. For the inputs of 30 deg/s yaw rate and 40 deg steering wheel angle, the two rules result in a final OS/US indicator of -5, indicating considerable oversteer.

The next section will explain in more detail how the membership functions and rules were chosen and optimized.

DESIGN OF OS/US DETECTION

Initial research into vehicle behavior during oversteer (OS) maneuvers indicated that there is a pattern in yaw rate and lateral acceleration displayed during most OS situations. Figure 3 illustrates such a behavior during a constant speed Sine with Dwell (SwD) maneuver. This vehicle shows heavy oversteering behavior after 2 seconds into the maneuver. Comparing the signals of lateral acceleration and yaw rate right before and right after 2 s one can see a clear pattern. Right before, the yaw rate follows the lateral acceleration; even though the absolute values are different, the pattern is similar. Right after two seconds, the lateral acceleration is constant, its jerk close to zero, indicating tire saturation. At the same time, the yaw rate changes rapidly. This behavior is displayed right after oversteer begins. However, it is desired to detect oversteer as early as possible. The optimal scheme could detect oversteer already in the buildup phase. This way the actuators have enough time to stabilize the vehicle.

To generate the optimal set of rules that approach such an ideal detection, an adaptive neuro fuzzy inference system (ANFIS) is adopted that can automatically train the fuzzy logic (membership functions & rules) that detects understeer and oversteer. This is detailed below.

GENERATING TRAINING DATA

The first step in training the fuzzy logic is to generate a set of input/output data. The input data consists of vehicle signals that can be used to determine and evaluate unstable situations. Recall that Figure 3 shows the pattern for OS detection. The first four signals are yaw rate, yaw acceleration, steering wheel angle, and lateral jerk. Excessive yaw rates indicate oversteer while small yaw rates combined with high steering angles indicate understeer. The combination of non-zero yaw acceleration and close to zero lateral jerk represent the identified OS pattern in Figure 3. Furthermore, the steering rate is used to indicate counter steer intentions of the driver. The output data consists of the desired OS/US Indicator for the set of input signals at hand. This is a tuning parameter for the control mechanism. Using different methods to determine the desired output will result in different fuzzy logic schemes. Changing the desired output is a way to set up the vehicle behavior, e.g. more understeering for rear wheel driven vehicles or to reduce the inherent understeer tendency of front heavy vehicles (sport mode).

The importance of the training data cannot be over emphasized. Not only is the content of the data itself important for the calculation of the desired output, but also the amount of training data is important. Too much training data can make it impossible for the ANFIS algorithm to minimize the error function to an acceptable level, resulting in spiky outputs and overly sensitive OS/US indicators or overtraining. On the other hand, not enough training data, in terms of not enough maneuvers, can mean that the training is insufficient. This means that the resulting fuzzy logic cannot deal with maneuvers that were not part of the training data and is unable to correctly detect OS or US. An insufficient amount of data points during one training maneuver, due to a too small sampling rate in the simulation, causes the signal to become spiky again. Summing up, the amount of training data and the kind of data as well as the sampling rate, have to be adjusted during the training process to generate an acceptable performance form the OS/US detection scheme.

In the present work, the training data was collected from simulations of Sine with Dwell maneuvers under different conditions. Figure 4 shows the signals of one of these maneuvers that was used to determine the desired OS/US Indicator. The upper two plots show the actual vehicle signals in green. Just from looking at the yaw rate plot, we can already see that, for the Sine with Dwell test, the vehicle is heavily oversteering. In order to create the desired fuzzy output one must compare the actual vehicle signals to a desired value. It must also be mentioned that arbitrary assignments of desired OS/US indicators for the data sets are not compatible with the mathematical ANFIS algorithm. This would create unreasonably high RMSE values resulting in poor OS/US detection. So, some reasonable desired behavior has to be specified at the training phase.

Here, the desired vehicle behavior signals are assumed taken from yaw rate and yaw acceleration generated from a linear vehicle model with neutral steering. From this model, the desired yaw rate is given by Equation 1. Note that this model is used in off-line calculations at the training stage only, as opposed to its use for on-line reference generation in model-based approaches. One advantage of doing this is that the desired signal instantaneously calculates the desired OS/US value, while the actual signal from the vehicle (CarSim simulation) has delays and lags from the vehicle dynamics. This way unstable behavior can be indicated in advance and the VSC can be used to minimize delays or rise times during understeer

Equation 1. Desired Yaw Rate

AV[Z.sub.desired] = [[RW[A.sub.front]*[V.sub.x]]/[WB]]

Comparing the desired yaw rate and yaw acceleration (obtained by filtered differentiation of Eq. 1) to the actual signals, one can make several observations. Firstly, the difference between the values in the yaw rate plot determines the current behavior of the vehicle. Up to 1.5 s into the maneuver, the actual yaw rate is smaller than the desired yaw rate, indicating understeer. After 2.2 s, the actual yaw rate is higher than the desired yaw rate indicating oversteer. The time inbetween is a transition phase when the driver changes the steering direction, but the vehicle takes time to react to the driver input. As a result, the yaw rate comparison gives us an indication of the current vehicle behavior.

Secondly, the difference between the desired and the actual yaw acceleration determines the change of the vehicle behavior. Looking at the yaw rate plot OS can be detected at 2.2 s. However, at 1.6 s the desired yaw acceleration (i.e., slope of the desired yaw rate) reduces while the actual yaw acceleration is still decreasing. This indicates that the vehicle behavior is about to change from understeer to oversteer. The two crossing lines in the yaw acceleration plot at 1.6 seconds serve as an indicator for this change from understeer to oversteer.

Based on these observations, the desired/target OS/US indicator is calculated by combining the two differences in the actual and the desired signals of yaw rate and yaw acceleration. Weighing the yaw rate difference with 1 and the yaw acceleration difference with 0.5 emphasizes the current vehicle behavior over the future prediction. Adapting these weights towards the yaw acceleration is a practical means toward a more predictive control with earlier and more subtle control interventions. However, this can affect the misdetection rate. The third plot in Figure 6 shows the resulting desired OS/US Indicator to be used as the target output in the training data.

Several of these maneuvers are used to create the input/output data for the ANFIS training. It is important to mention that the Sine with Dwell maneuver, by definition, starts with a steering input to the left. Therefore, in every maneuver oversteer appears after the one second mark while the driver inputs a negative steering angle to the right. In order to have sufficient and balanced data to be able to detect oversteer in right and left steer maneuvers, it is necessary to use Sine with Dwell maneuvers with initial steering input to the right as well. This will result in symmetrical fuzzy logic surfaces, as can be seen in the further discussions below.

FIS MODEL IDENTIFICATION

A first set of fuzzy rules and membership functions must be generated in the next step. The set of input data (SWA, SWRATE, AVZ, AAZ, AAY) from the training maneuvers and the corresponding output data (desired OS/US Indicator) is used to create a first Sugeno fuzzy model. This model will later be optimized in the ANFIS algorithm. The objective in the model identification process is to look at the I/O data and find common pairs or clusters. These lead to a first set of membership functions. In this case, subtractive clustering, as proposed by Chiu [18], is used. Equation 2 defines the cluster density for each potential cluster center, or in case of subtractive clustering for each I/O point [x.sub.i] from the training data.

Equation 2. Cluster Density

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The clustering parameters, in particular, the cluster radius [r.sub.a], defines the robustness vs. performance trade-off for the FIS system. We can use the system to detect oversteer and understeer very accurately for one specific vehicle or vehicle configuration by specifying a small cluster radius. This will create smaller clusters and result in a rule set that is more tuned towards the particular vehicle configuration that generated the training data. The second option is to specify a larger cluster radius. This will generate larger clusters, and subsequently less if-then rules, resulting in a more robust detection. Therefore, a tradeoff between performance and robustness has to be made here.

In our study, the potential of a FIS to detect unstable behavior over multiple vehicle configurations or platforms will be shown. This requires high robustness against model uncertainty and consequently a larger cluster radius. The algorithm detected three big clusters in the data, resulting in three if-then rules. This is not a surprise, since the detectable vehicle states are OS, US, and stable.

Each input signal must be fuzzified with membership functions (MF), as well as each output must be defuzzified with a MF. In order to get an optimal and smooth output value, it is important to choose the right shape, amount and overlap for each signal's MFs. Triangular MFs are computationally better for real time applications but their rapid changes in gradient can cause jumpy output signals. One of the main requirements for the OS/US detection is a smooth transition in the signal, leading to a smooth control intervention. In order to achieve this, three bell MFs for the main operation points of the vehicle and two asymmetric trapezoid functions for the extreme values were chosen to fuzzify the input values. Figure 5 shows the membership function of the yaw rate.

The three bell functions (see Equation 3) and their overlap make the responses smooth, while the two outer asymmetric trapezoid functions catch extreme signal inputs.

Equation 3. Bell Function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The preliminary parameters[a.sub.i], [b.sub.i], [c.sub.i], defining the shape of each bell function, were chosen according to the position of the clustering results. These parameters are also subject to optimization.

Four more rules were included by the designer to account for the differences in training data between the uncontrolled vehicle and the vehicle signals that are generated by a vehicle controlled by the VSC system.

ADAPTIVE NEURO FUZZY INFERENCE SYSTEM

The ANFIS algorithm, as discussed in Jang [19], is able to enhance the performance of the previously generated FIS. It is a neuro adaptive network that works in the same way as a Sugeno FIS. This combination of supervised learning from the neural network part and the robustness and human reasoning from the fuzzy part can enhance the control performance when choosing the right learning rules.

Research has shown that the learning success heavily depends on the learning rule. In this case, a combination of least squares method and back-propagation gradient descent method is used. This prevented the optimization algorithm from being caught up in local minima and has proven to perform best.

The error function to be minimized (for the optimization) is the difference between the desired output (desired OS/US Indicator) and the output at the current optimization epoch. After 600,000 optimization epochs the RMSE for the training data is at approximately 0.61. This is an acceptable value, considering that the indicator can take values from -10 to 10 and we accept minor deviations from the desired output, especially during the transition phase from OS to US or vice versa.

Figures 6 and 7 illustrate the combination of the final trained and optimized membership functions and rules in the fuzzy detection scheme as input/output surfaces. Recall that Figure 3 showed that oversteer can be detected when the yaw rate is changing, while the lateral acceleration is close to constant. After the optimization, this scheme is represented in the fuzzy logic as well. Looking at Figure 6 one can see that at 0 yaw acceleration and zero lateral jerk the indicator is also zero. If the lateral jerk remains zero while the lateral acceleration deviates from zero, representing the OS pattern, the indicator changes. This indicates an unstable situation. Moreover, the surface is symmetrical and indicates changing yaw acceleration in both directions. The reason for this is the previously mentioned symmetrical training data that uses data from SwD maneuvers starting with initial left as well as initial right turns.

Figure 7 represents the US detection. If the yaw rate remains constant or changes very slow (close to 0 yaw acceleration), while the steering wheel angle increases or decreases the indicator goes down. This represents the US pattern. On the other hand, a small steering angle and high yaw acceleration causes a change in the opposite direction, indicating oversteer.

It must be mentioned at this point that the data in Figure 6 and 7 shows the fuzzy logic I/O surfaces prior to normalizing the output to [+ or -]10.

SIMULATION RESULTS

In this section, the model-free VSC incorporating the proposed detection algorithm will be evaluated against a common model-based VSC. We refer to the latter as the DYC (direct yaw control) VSC and it uses (on-line) model generated yaw rate reference tracking control.

The main purpose of a vehicle stability control is to detect unstable vehicle behavior and bring the vehicle into a state that makes it controllable for the driver. FMVSS 126 [20] and ECE R13H [16] define a method of evaluating this capability. An adapted form of the procedure defined in these regulations, the Sine with Dwell (SwD) maneuver, will be used here. The vehicle will be exposed to the open loop steering wheel input of the SwD maneuver. Figure 8 shows the steering input as well as the evaluation parameters used.

Figure 9 and 10 show the simulation results of the SwD test at 80 km/h on a road surface with friction coefficient 0.9 for an SUV. As we can see, the fuzzy controller detects mild understeer during the first half of the maneuver and oversteer after that. Due to that the VSC activates the left rear brake first, to reduce understeer in the initial left turn, and the right rear brake to reduce understeer in the following right turn to increase the yaw rate and help the vehicle to turn. This understeer compensation generates high yaw rates and pushes the vehicle to go into oversteer. To prevent this, the left front brake is applied to generate a stabilizing yaw moment, as soon as oversteer is detected. Figure 9 shows that the understeer control could increase the yaw rate peaks of the vehicle using the model-free VSC. Even though the vehicle with fuzzy VSC can achieve a higher yaw rate than the conventional yaw tracking VSC, the fuzzy VSC still stabilizes the SUV sooner. The reason for this is the predictive aspect embedded during the ANFIS training. The fuzzy logic monitors the changing vehicle response and detects potential oversteer before it happens. This gives the lower level control enough time to apply brakes and stabilize the vehicle.

Figure 9 showed that the model-free approach can stabilize a vehicle and outperform conventional DYC VSC systems. The main advantage of the model-free approach however, is that the algorithm does not rely on changing model parameters, such as estimations of road surfaces or linear approximations of tire models.

In all of the above discussions, the fuzzy algorithm was created using training data of an SUV. As seen in the previous example it performs well, when controlling the vehicle it was trained for. Figure 11 and 12 show the simulation results for an overloaded Sedan, a vehicle and loading condition that the fuzzy algorithm has not been trained for. The model-based DYC VSC's model parameters were adapted to those of the overloaded Sedan in order to indicate the correct desired yaw moment, whereas the fuzzy algorithm is used in this vehicle without any change from the SUV to illustrate its robustness.

Since the fuzzy algorithm uses human reasoning and perception instead of fixed models, it can also detect oversteer and understeer in the unknown Sedan, as can be seen in Figure 12. Figure 11 shows the yaw rate plots. The Sedan without VSC is uncontrollable and spins out. The DYC VSC with the adapted parameters can stabilize the overloaded vehicle after approximately 7 seconds. The model free VSC does not only increase the peak yaw rate, but also stabilizes the vehicle within 4 seconds.

CONCLUSION

In this paper, an adaptive fuzzy inference system (FIS) is proposed for OS/US detection towards a model-free approach for the development of vehicle stability control systems. The proposed approach uses training data to tune the controlled vehicle response to the desired behavior. Patterns in this data were identified using subtractive clustering and are used to generate a FIS. The ANFIS algorithm with a hybrid learning rule was adopted to optimize the performance and robustness of the algorithm. The fuzzy VSC was compared to an uncontrolled and a model-based DYC controlled vehicle using brake intervention. The test results showed that the fuzzy VSC is robust enough to reliably detect understeer and oversteer under heavy nonlinear maneuvers and for different vehicles without any parameter changes.

This paper concentrated on the upper level fuzzy detection. An area of further research is the formalization of dedicated test maneuvers that can create an optimal set of training data. In a companion paper [21], we will focus on the design of a similar ANFIS system for the lower level application of stabilizing yaw moments via active four wheel steering actuator sets.

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ACKNOWLEDGMENTS

The authors acknowledge the support provided by Honda for conducting the research presented here, which was conducted at the Clemson University-International Center for Automotive Research.

Benjamin Hirche

Ford Motor Co.

Beshah Ayalew

Clemson Univ.
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Author:Hirche, Benjamin; Ayalew, Beshah
Publication:SAE International Journal of Passenger Cars - Mechanical Systems
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Date:Jun 1, 2016
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