controlling vehicle instability through stable handling ...

CONTROLLING VEHICLE INSTABILITY THROUGH STABLE HANDLING ENVELOPES

Craig E. Beal ∗ Dynamic Design Laboratory Dept. of Mechanical Engineering Stanford University Stanford, California 94305 Email: [email protected]

Carrie G. Bobier Dynamic Design Laboratory Dept. of Mechanical Engineering Stanford University Stanford, California 94305 Email: [email protected]

ABSTRACT Loss of control accidents result in thousands of fatalities in the United States each year. Production stability control systems are highly effective in preventing these accidents, despite their reliance on a hand-tuned response to data from a small set of sensors. However, improvements in sensing offer opportunities to determine stabilizing actions in a more systematic manner. This paper presents an approach that utilizes the yaw-sideslip phase plane to choose boundaries that eliminate unstable and undesirable driving regimes. These boundaries may be varied to obtain desirable performance and driver acceptance and form the basis for a driver assistance system that augments the driver input to maintain the vehicle within the bounds of a safe handling envelope. Experimental results from a model predictive controller used to enforce the envelope boundaries on a steer-by-wire vehicle are presented to demonstrate the viability of this framework for implementing stability boundaries.

INTRODUCTION Loss of control accidents account for more than 14,000 fatalities in the United States each year [1]. Many of these accidents occur in vehicles not equipped with Electronic Stability Control (ESC), a technology that utilizes measurements of vehicle yaw rate, speed, lateral acceleration, and steering wheel position to estimate the vehicle states and apply individual brakes to assist the driver in maintaining control. Despite the limited measurement suite available to the system, ESC can be attributed to more than a 30% reduction in single-vehicle crashes in equipped light vehicles [2]. Because of these benefits, all new light vehi∗ Address

all correspondence to this author.

J. Christian Gerdes Dynamic Design Laboratory Dept. of Mechanical Engineering Stanford University Stanford, California 94305 Email: [email protected]

cles manufactured for sale in the United States after 2011 will be equipped with ESC [1]. Yet the structure of current systems requires them to be hand-tuned to achieve the desired performance. Fortunately, there are emerging technologies such as electric power steering, steer-by-wire, and electric drive that offer new opportunities to sense the vehicle behavior [3, 4]. Furthermore, the cost and precision of GPS and inertial sensors continue to improve at a rapid rate. These advances in state sensing can be leveraged to improve the process of developing a system to assist the driver in stabilizing the vehicle. The operation of the production Bosch VDC system is described in several publications by van Zanten, et. al. that explain the criteria for VDC intervention and the dynamics of the actuators [5, 6]. From these papers, the dependence of the system on the choice of parameters and hand-tuned thresholds is clear. In contrast, Inagaki, et. al. [7] present a method of examining vehicle stability using the phase plane and demonstrate unstable regions reachable by driver steering input. A controller that seeks to keep the vehicle away from regions of instability is also presented, utilizing the phase plane as a tool to shape the controller response. Shibahata, et. al. also present a state-based analysis of the vehicle dynamics; their analysis demonstrates the importance of sideslip angle in stability [8]. Common to these previous publications is the underlying basis on readily available vehicle sensors. However, if a reliable estimate of the tire-road friction coefficient is assumed to be available, the critical measurements and even the process of designing the system change. These insights about the design process that are presented in this paper are gleaned primarily from analysis of the yaw-sideslip plane. In the context of the friction information, the yaw rate becomes significantly more useful as an early

equations of motion for this model can be written as:

a

b

Uy

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δ

αr

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Figure 1: BICYCLE MODEL

indicator of instability since the sideslip dynamics are distinctly unfavorable at excessive yaw rates. Drawing upon this analysis, a framework for control is presented which assumes an accurate estimate of the sideslip can be obtained and the tire-road friction coefficient can be roughly estimated. A set of boundaries can be derived from this information to form a handling envelope in which the vehicle dynamics are stable and predictable. This envelope also allows the driver significant freedom to access driving regimes that might otherwise be disallowed by conventional stability control systems. The yaw-sideslip phase plane and the development of the boundaries of the envelope are based on a nonlinear single-track model described in the following section. Using this model, phase plots are presented with discussion of the instabilities that appear as a result of changes in environmental conditions and driver input. The consistent location of the instabilities in the phase plane leads to a natural expression of state boundaries, described in detail in the handling envelope section of the paper. Together with the discussion of boundary selection for performance and driver acceptance, these sections form a framework for the synthesis and analysis of advanced stabilization systems.

VEHICLE MODELS There are a wide variety of models that can be used for analysis of vehicle dynamics. These models range in complexity and common variants are formed by the inclusion or neglection of vehicle track width, longitudinal dynamics, suspension and body motion, and compliance effects. For the purposes of the analysis done in this paper, the bicycle model provides a sufficiently detailed representation of the vehicle dynamics while remaining simple enough for analytical calculation of critical dynamics.

Bicycle Model The bicycle model is a two-state model of the vehicle dynamics that yields a pair of linear equations that describe the rotational and lateral velocities of the vehicle. The model, illustrated in figure 1, utilizes small angle assumptions and the approximation that the tires on each axle can be lumped together. A constant vehicle longitudinal velocity Ux is also assumed. The

where β is the ratio of the lateral to the longitudinal velocity and the parameters m and Izz are the vehicle mass and yaw moment of inertia, respectively. The parameters a and b are the distances from the vehicle CG to the front and rear axles, respectively. The slip angles, α f and αr , are described in terms of the vehicle states and the steering angle input from the driver: αf = β+

ar − δ, Ux

αr = β −

br . Ux

(2)

These slip angle equations can be used with a parameterized tire model to determine the forces in equation 1. Lateral Brush Tire Model

Tire Force Magnitude

Fyr

Slip Angle

Figure 2: RESPONSE CURVE - BRUSH TIRE MODEL

A brush tire model, for which the response is illustrated in figure 2, provides the tire information in this investigation. The model uses a parabolic pressure distribution like Fiala [9] and others, and a single tire-road friction coefficient. The model is parameterized by the linear tire cornering stiffness (Cα ) and the peak friction force (µFz ). The relationship between the slip angle and the tire force is given by the following expression if the tire is operating at a slip angle equal to or less than the angle of full sliding: Fy = −Cα tan α +

Cα2 Cα3 tan α |tan α| − tan3 α 3µFz 27 (µFz )2

(3)

where the angle associated with peak force, and full sliding, can be found by differentiation of the model. Beyond this angle,  αslide = arctan the tire force is simply Fy = µFz .

 3µFz , Cα

(4)

INSTABILITY DYNAMICS Using the bicycle model, it is possible to develop phase plane representations of the vehicle dynamics in the linear and nonlinear handling regimes. These plots allow the designer of a stability control system to identify regions in which the dynamics are unfavorable and in which the vehicle would fail to respond in a manner that most drivers would expect. These are the regions in which production stability control systems activate to assist the driver; a stability controller based on handling envelopes would prevent the vehicle from entering these regions at all. Phase Plots The first set of phase plots shown in this section illustrates the dynamics of a vehicle being driven at 10 m/s on a surface with an average coefficient of friction of 0.6. The parameters for the vehicle are those of a test vehicle constructed by students at Stanford University, seen in figure 3. Despite the lack of a typical vehicle body, this vehicle has handling properties and mass similar to a sports sedan, with the exception of a low yaw moment of inertia due to the central placement of the batteries. The vehicle is mildly understeering in the linear region, with an understeer gradient of 0.02 rad/g. At the handling limits, the vehicle is close to neutral, but tends slightly to limit oversteering. The driver steering input in the set of plots progresses from 3◦ at the road wheels in figure 4a up to 12◦ in figure 4d. For each phase plot in figures 4a-4c, there are three equilibria. In these plots, the steering angles are less than 10◦ and there are stable equilibria at moderate yaw rate and sideslip angles. There are also saddle points that separate regions of stable handling from unstable regions. These saddle points are located along lines of constant yaw rate, at approximately 0.59 rad/s. As the steering angle of the vehicle is increased, the stable equilibria move to higher yaw rates and larger (negative) sideslip angles, while the regions of instability shift, as seen by the movement of the saddle points toward lower sideslip angles. Figure 4d illustrates the vehicle behavior when the steering angle is increased to 12◦ . At this steering angle, the vehicle is unstable, demonstrating the bifurcation from a stable to unstable equilibrium point with increased steering. Nearly all trajectories of the system result in the growth of the vehicle yaw rate developing a large sideslip angle. Since the vehicle is nearly neutral at the limits, the yaw rate remains reasonable as the vehicle slides. These plots yield interesting insights for the design of stability controllers. Even though figures 4a-4c show vehicle behaviors with stable dynamics immediately surrounding the equilibria, unstable regions exist for all of these conditions. Therefore, a stability control system should seek to prevent the vehicle from entering these regions in case of disturbances to the system. Fortunately, the unstable dynamics occur in the same location, regardless of the driver’s steering input. The dotted black line on each of the phase plots in figure 4 represents the maximum equilibrium yaw rate for the vehicle. This yaw rate may be found utilizing the maximum tire force at each axle. The yaw rate is in

Figure 3: P1 STEER AND DRIVE BY-WIRE RESEARCH TESTBED equilibrium when

r˙ =

aFy f − bFyr =0 Izz b Fy f = Fyr . a

(5) (6)

If there is weight transfer or variation in the friction coefficient on the front and rear axles, one axle will limit the maximum yaw rate. If the driver has pushed the vehicle to the equilibrium associated with this limiting tire force, then β˙ = 0 and

r=

  Fyrmax (1+b/a)

Fy fmax ≥ ab Fyrmax

 Fy fmax (1+a/b)

Fy fmax < ba Fyrmax

mUx

mUx

.

(7)

Therefore, the maximum equilibrium yaw rate separates the convergent sideslip dynamics from the divergent dynamics. Furthermore, examining figure 4, when the driver steers far enough to cause instability, the yaw rate grows rapidly prior to exceeding the equilibrium yaw rate and developing large sideslip. The ability to determine this yaw boundary explains the main differences between the conclusions drawn in the work of Inagaki, et. al., Shibahata, et. al., and those presented here. In the absence of context from the tire grip limits, a yaw rate may be associated with either stable or unstable dynamics, leaving the sideslip angle as the best indicator of instability. However, when the tireroad friction properties are known, the easily measured yaw rate state is a better choice for control. This observation also makes clear the underlying reason for the efficacy of stability control through independent braking to produce a yaw moment, or what many authors term Direct Yaw Control. By generating a corrective yaw moment, the controller moves the vehicle directly into a region of stable dynamics. While the handling properties of the vehicle used for the phase plots in figure 4 are quite neutral at the limits of handling, the observations made about the regions of instability are also valid for understeering and oversteering vehicles. Not only is

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Figure 4: NEUTRAL VEHICLE DYNAMICS AT 10M/S WITH 0.6 SURFACE FRICTION COEFFICIENT this important for vehicles with different handling properties, it is also critical since the application of drive and brake torques can alter the handling behavior at the friction limits. Figure 5 shows the same vehicle as before, but with the maximum available force on the front axle reduced in figures 5a and 5b to produce limit understeer and with the maximum force reduced on the rear axle in figures 5c and 5d to produce limit oversteer. In each case, the axle with reduced force capability is modeled with a friction coefficient of 0.55, reduced from 0.6. When the vehicle understeers, the stability problems of the neutral vehicle disappear, but the oscillatory behavior associated with an understeering vehicle becomes apparent. Like the neutral vehicle, there is also convergence to the equilibria within the maximum yaw equilibrium bounds. The oversteering vehicle, as expected, demonstrates significant degradation of stability. For 7◦ of steering angle, the stable equilibrium is quite close to the saddle point, beyond which the dynamics are strongly divergent.

As a result, minor disturbances at this condition could cause the vehicle to spin. When the steering angle is increased to 10◦ , the vehicle exhibits the unstable tendencies seen at larger steering angles with the neutral steering vehicle. However, in this case, the yaw rate continues to grow large once the maximum equilibrium boundaries are exceeded and the vehicle “spins out” rather than experiencing a lateral slide. This can be seen in an experiment performed with the P1 vehicle as well, where the vehicle was initially driven to equilibrium cornering. At 10 m/s and at an equilibrium near 0.4 rad/s yaw rate, the accelerator pedal was used to upset the vehicle, yielding dynamics similar to the oversteering dynamics illustrated in the previous plots. No controller was active on the vehicle during the test, and the steering angle was held constant. Figure 6 shows the state trajectory of the vehicle in the phase plane. Once the vehicle is upset, the yaw rate increases into the region of unfavorable dynamics and the sideslip angle grows rapidly as

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Figure 5: UNDERSTEER AND OVERSTEER VEHICLE DYNAMICS AT 10M/S the vehicle spins out. The data is truncated for clarity, but the vehicle eventually reaches a sideslip angle of over 180◦ . Aside from the steering angle, the vehicle speed is also a critical parameter. From the expression of the lateral acceleration, ay = β˙ + rUx , it is clear that since the forces needed to produce lateral acceleration are limited, as the vehicle speed increases, the maximum yaw rate must decrease. Figure 7 shows the phase plane for the vehicle with a 7◦ driver steering angle on the same surface as the previous figures, but at 15 m/s. While the lines of maximum steady state yaw rate are located at lower values and the steering angle that destabilizes the vehicle is smaller, the dynamics are remarkably similar to those illustrated in figure 4d. In both plots, the unstable dynamics occur in the same location relative to the lines of maximum yaw equilibrium. From the plots shown in this section, it is clear that regions of unstable dynamics can be identified and separated from the

stable regions of handling by a constant yaw rate boundary. All of the stable equilibria for the bicycle model lie within these bounds, and the work of Hindiyeh and Gerdes [10] demonstrates that even the equilibria for drifting lie within these bounds. Supporting the importance of measuring and bounding the yaw rate is the observation that over the range of steering angles, speeds, and handling properties, the large sideslip angles develop only after the yaw rate grows large. Since a measurement of the yaw rate can be made directly, accurately, and inexpensively, this analysis suggests excellent opportunities for control.

CONTROLLING VEHICLE INSTABILITY In designing a stability controller, the objective is to assist the driver in maintaining control over the orientation and path of the vehicle. In the best case, the system would provide the driver with full authority in all situations in which the dynamics

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Figure 8: VEHICLE DYNAMICS AT 10M/S ON SLIPPERY SURFACE WITH CONTROLLER ACTIVE

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Figure 6: VEHICLE DYNAMICS AT 10M/S ON SLIPPERY SURFACE WITH LARGE STEER ANGLE

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Figure 7: NEUTRAL VEHICLE DYNAMICS AT 15M/S WITH 7 DEGREES STEERING AND 0.6 SURFACE FRICTION COEFFICIENT are favorable and intervene only in case the vehicle enters a region of unstable dynamics. Therefore, the insights gained from the phase plane in the previous section can be used to develop an envelope that includes a large set of the stable dynamics but excludes regions of instability. However, the size and shape of the envelope depends on the vehicle speed, tire grip, and other parameters. Fortunately, many vehicles being manufactured today are incorporating electric power steering and future vehicles may be equipped with steerby-wire, both of which have been shown by Hsu and Gerdes [4] to allow for estimates of both sideslip angle and tire-road friction coefficient. It is this additional information that enables the development of a controller using stable handling envelopes.

A Stable Handling Envelope In the previous section, analysis of the unstable vehicle dynamics showed that below the maximum equilibrium yaw rate, the vehicle dynamics are stable and the states converge toward stable equilibria. Therefore, it makes sense when designing a stable handling region to include a yaw rate constraint. Assuming that the axle normal loads are known and the tire-road friction coefficient can be estimated with a technique such as that presented by Hsu and Gerdes, the yaw rate constraint can be expressed simply as a function of the estimated peak tire forces as in equation 7. This constraint is, however, not sufficient to completely ensure the stability of the vehicle. In certain low-friction conditions, it is possible for disturbances or transients to excite large sideslip angles at low yaw rates, as demonstrated by Rock, et. al. [11]. To prevent the vehicle from reaching this condition, a second set of bounds can be used to close the stable handling envelope. Choosing a rear axle slip angle constraint results in natural scaling of the envelope. This is in part due to the fact that the rear slip angle, unlike the vehicle sideslip angle, remains small and exhibits no non-minimum phase behavior at low speeds. The rear slip angle is also a linear combination of the vehicle sideslip angle and the yaw rate and is dependent on the vehicle speed, as seen by the rear slip expression, αr = β − Ubx r. As a result, the allowed sideslip scales naturally with both the speed and the yaw rate, allowing the driver to utilize a significant portion of the state space associated with stable driving. One natural choice for the value of this rear slip angle boundary is the angle at which the tire reaches full saturation. Thus, a controller that enforces these boundaries maintains the vehicle within the tire grip limits. The envelope boundaries form a parallelogram in the yawsideslip plane, as illustrated in figure 8, where the stable handling envelope is plotted over the phase plane with a model predictive envelope controller active. From this figure, it is clear that the interior of the envelope contains only regions of the state space

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Figure 9: RANGES OF POSSIBLE VEHICLE TRAJECTORIES ON ENVELOPE BOUNDARIES for which the handling dynamics are stable. The remaining task is to verify that it is possible to design a controller to enforce the invariance of the envelope. This is done by finding a control input that causes the yaw rate and sideslip derivatives to cut into the envelope on every portion of the boundary. When active front steer is used to enforce the boundaries of the envelope, the control action on the boundary is to alter the front axle lateral force. Figure 9 shows the envelope with possible state trajectories plotted at points on each of the boundaries. The set of arrows at each point shows the complete range from the minimum to the maximum front lateral force. From the diagram, it is possible to see that at every point, there is a trajectory that stays within the envelope, though any controller that is developed to enforce this envelope must take into account the steering limitations to ensure that these stabilizing trajectories may be obtained at all points. At the top right and bottom left corners, only the extreme lateral force keeps the vehicle in the envelope. However, this is sufficient to establish the feasibility of creating an invariant envelope. More complete statements may be developed by establishing inequalities using the bicycle model and the available tire forces. However, the graphical analysis gives a more intuitive understanding and is thus presented here. Similar analyses may be performed for other actuation schemes, including independent braking such as that used by production stability control, rear wheel steering, or differential drive. With a guarantee that there is a viable control action to keep the vehicle within the boundaries of the envelope, a variety of controllers may be used to implement this control strategy. In previous work, the authors have presented a model predictive controller [12] that augments the driver’s front steering input based on this stable handling envelope that has been implemented and validated with the P1 experimental vehicle. Model predictive control is well-suited to this problem since it can con-

Steer Angle (deg)

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Figure 10: EXPERIMENTAL DATA FROM SLALOM WITH ENVELOPE CONTROL AT 10M/S

sider the range and slew rate limitations of the steering system in determining inputs that keep the vehicle in the envelope. The yaw rate and rear slip angle from a slalom maneuver are plotted in figure 10 with the limits as well as the steering of thec driver and the total steering angle after the controller augmentation. Aside from disturbances that result from the uneven friction properties of the driving surface, the controller is able to use the steering to restrict the vehicle to the interior of the safe handling envelope. ALTERNATIVE ENVELOPES The envelope presented in figure 8 is a straightforward choice with a constant yaw rate boundary and rear slip limit at the angle of peak tire force. However, this rear slip limit may be constraining to skilled drivers who may be capable of driving with the rear tires in saturation. Likewise, the vehicle is capable of stably attaining higher yaw rates in transient maneuvers than are allowed by the controller. Fortunately, there are alternative envelope choices that eliminate the unstable dynamics in the phase plane but offer the designer of the assistance system to adjust the boundaries to achieve the desired performance. One possible boundary change is to allow wider rear slip limits to allow skilled drivers to drift the vehicle. Achieving additional yaw rate at small sideslip angles is also possible. Keeping the point at intersection of the maximum equilibrium yaw rate and angle of peak rear tire force fixed, the yaw rate bound may be inclined so that additional yaw rate is allowed at smaller sideslip angles. This change in the boundary also results in additional restriction of yaw rates allowed at larger

the dynamics of vehicle instability, but also to form a basis for designers of future stability control systems to synthesize controllers that may utilize various vehicle actuators and consider rollover or environmental hazards and also to make guarantees about vehicle performance.

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ACKNOWLEDGEMENTS The authors would like to thank the NISSAN MOTOR Co., Ltd. as well as project team members Yoshitaka Deguchi, Shinichiro Joe and Takuro Matsuda for sponsoring this research.

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Figure 11: HANDLING ENVELOPE VARIANT FOR INCREASED PERFORMANCE sideslip angles. A trade-off must therefore be made to obtain an envelope that drivers will find unobtrusive. Figure 11 illustrates an envelope that allows 115% of the maximum equilibrium yaw rate at zero sideslip angle and has rear slip angle bounds at ±15◦ . When these boundaries were enforced with the model predictive controller introduced in [12], experienced drivers found this envelope more natural feeling and less intrusive than the envelope shown in figure 8.

CONCLUSIONS The development of electronic stability control has led to more than a 30% reduction in the number of fatal loss of control accidents for light vehicles in the United States. Yet previously, little had been published about the dynamics of vehicles under control of these systems. By examining the dynamics in the yawsideslip phase plane, specific regions in which unstable behavior occurs emerge. The location of these regions remains similar over a wide range of driver input and vehicle parameters and can be delineated by the maximum equilibrium yaw rate. The insight gained from examining the phase plane suggests an alternative to the approach used in production ESC systems. This approach specifies a region of stable handling in which the driver has full authority to control the vehicle. The boundaries of this stable handling envelope are developed naturally from estimated tire-road friction coefficients, scale with vehicle speed and tire grip, and can be obtained using new sensing opportunites from electric power steering (EPS). Physically motivated variants of the boundaries can be also be used to yield different handling characteristics at the maximum extents of the envelope to satisfy the preferences of different drivers. Once the envelope is chosen, a control scheme such as model predictive control can be used to enforce the boundaries and prevent the vehicle from leaving the stable handling envelope. The results presented in this paper serve not only to elucidate

REFERENCES [1] NHTSA, 2007. Federal Motor Vehicle Safety Standards; Electronic Stability Control Systems. http: //www.nhtsa.gov/Laws+&+Regulations/ Electronic+Stability+Control+(ESC). [2] Dang, J. N., 2007. Statistical analysis of the effectiveness of electronic stability control systems. Tech. rep., NHTSA. [3] Rajamani, R., Piyabongkarn, D., Lew, J., and Grogg, J., 2006. “Algorithms for Real-Time Estimation of Individual Wheel Tire-Road Friction Coefficients”. In Proc. of the American Control Conference, Minneapolis, Minnesota. [4] Hsu, Y.-H. J., and Gerdes, J. C., 2008. “The predictive nature of pneumatic trail: Tire slip angle and peak force estimation using steering torque”. In AVEC08. [5] Zanten, A. T. V., Erhardt, R., Pfaff, G., Kost, F., Hartmann, U., and Ehret, T., 1996. “Control Aspects of the BoschVDC”. In Proc. of the International Symposium on Advanced Vehicle Control. [6] Zanten, A. T. V., 2002. “Evolution of Electronic Control Systems for Improving the Vehicle Dynamic Behavior”. In Proc. of the Int. Symposium on Advanced Vehicle Control. [7] Inagaki, S., Kshiro, I., and Yamamoto, M., 1994. “Analysis on Vehicle Stability in Critical Cornering Using PhasePlane Method”. In Proceedings of the International Symposium on Advanced Vehicle Control. [8] Shibahata, Y., Shimada, K., and Tomari, T., 1993. “Improvement of Vehicle Maneuverability by Direct Yaw Moment Control”. Vehicle System Dynamics, 22, pp. 465–481. [9] Fiala, E., 1954. “Lateral forces on rolling pneumatic tires”. In Zeitschrift V.D.I. 96. [10] Hindiyeh, R. Y., and Gerdes, J. C., 2009. “Equilibrium Analysis of Drifting Vehicles for Control Design”. In Proc. of the ASME Dynamic Systems and Controls Conference. [11] Rock, K., Beiker, S., Laws, S., and Gerdes, J., 2005. “Validating GPS Based Measurements for Vehicle Control”. In Proceedings of ASME International Mechanical Engineering Congress and Exposition, Orlando, Florida. [12] Beal, C. E., and Gerdes, J. C., 2010. “A Method for Incorporating Nonlinear Tire Behavior into Model Predictive Control for Vehicle Stability”. In Proceedings of the ASME Dynamic Systems and Controls Conference.