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Int. J. Vehicle Autonomous Systems, Vol. 8, Nos. 2/3/4, 2010

Vehicle stabilisation in response to exogenous impulsive disturbances to the vehicle body Jing Zhou Department of Mechanical Engineering, The University of Michigan, G041 Lay AutoLab, 1231 Beal Ave., Ann Arbor, MI 48109, USA E-mail: [email protected]

Jianbo Lu Active Safety Research and Advanced Engineering, Ford Motor Company, 2101 Village Rd., Dearborn, MI 48124-3993, USA E-mail: [email protected]

Huei Peng* Department of Mechanical Engineering, The University of Michigan, G036 Lay AutoLab, 1231 Beal Ave., Ann Arbor, MI 48109, USA E-mail: [email protected] *Corresponding author Abstract: Multi-event traffic crashes typically result in a higher toll than single-event crashes do. One type of multi-event crash includes the case where the initial harmful event leads to a subsequent loss of directional control. Vehicle stabilisation countermeasures in response to exogenous impulsive disturbances are addressed here. Based on a dedicated collision model, a sensing scheme is proposed to detect the crash events. The stabilisation controller, developed from the model predictive supervisory control approach and optimal control allocation, is then activated to attenuate vehicle motions. This stabilisation scenario can be deemed as a functional extension to Electronic Stability Control (ESC) systems. Keywords: vehicle stability control; MPC; model predictive control; optimal tyre force allocation; vehicle collision model; crash impulse detection. Reference to this paper should be made as follows: Zhou, J., Lu, J. and Peng, H. (2010) ‘Vehicle stabilisation in response to exogenous impulsive disturbances to the vehicle body’, Int. J. Vehicle Autonomous Systems, Vol. 8, Nos. 2/3/4, pp.242–262.

Copyright © 2010 Inderscience Enterprises Ltd.

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Biographical notes: Jing Zhou received his BS/MS from Tongji University, Shanghai, China. From 2002 to 2008, he was a graduate student in the Department of Mechanical Engineering at the University of Michigan, Ann Arbor, working in the area of vehicle active safety, including vehicle stability control, human driver modelling, and adaptive cruise control. He is currently a Control Engineer working on hybrid transmission systems at Chrysler Group LLC. Jianbo Lu received a PhD in Aerospace Engineering from Purdue University in 1997. He was with Delphi Corp. from 1997 to 2000 and joined Ford Motor Company in 2000, where he is a Technical Specialist at Active Safety Research and Advanced Engineering. He held more than 50 US patents and is a recipient of Henry Ford Technology Award. He currently serves as an Associate Editor for IEEE Trans. on Control Systems Tech. and IFAC J. Control Engineering Practice. Huei Peng received a PhD in Mechanical Engineering from the University of California, Berkeley, in 1992. He is currently a Professor with the Department of Mechanical Engineering, University of Michigan, Ann Arbor. His research interests include adaptive control and optimal control, with an emphasis on their application to vehicular and transportation systems. He has been an active member of SAE and the ASME Dynamic System and Control Division.

1

Introduction

Despite advances in vehicle safety technology, the death toll of road traffic accidents remains steady. According to traffic safety statistics, approximately 6 million motor vehicle crashes were reported to the police during 2006 in the USA (NHTSA, 2008). The accident data of National Automotive Sampling System – Crashworthiness Data System (NASS-CDS) from 1988 through 2004 shows that about 2.9 million light passenger vehicles are involved in tow-away crashes annually in USA. Approximately 31% of them have at least one additional harmful event following the initial collision. The NASS-CDS data analysis also showed that risks of both injury and fatality increased with the number of events involved in accidents. A separate accident study performed by the German Insurance Association confirms that even a vehicle involved in a light impact (e.g., the collisions with impact forces under certain thresholds) is likely to experience a severe secondary crash, and one third of all accidents with severe injuries consist of multiple events (Langwieder et al., 1999). A representative case example (#NASS-CDS 2003-079-057) can illustrate the severe outcome resulting from an initial minor collision (Figure 1). During this multi-event accident, as Vehicle 1 negotiated a lane change, its left front corner contacted the right rear corner of Vehicle 2. Vehicle 2 spun clockwise and rolled over about six quarter turns. Eventually Vehicle 2 was towed with rollover damage and had deployed side curtain air bags. However, the rear bumper of Vehicle 2 sustained only minor cosmetic damages, and photos in the police report showed no substantial structural deformation in the area struck by Vehicle 1. These facts are indications that despite the grave consequences for Vehicle 2, the initial collision with Vehicle 1 was minor.

244 Figure 1

J. Zhou et al. Accident sketch of case #NASS-CDS 2003-079-057 (see online version for colours)

Few studies have directly addressed vehicle dynamics control in response to exogenous impulsive disturbances applied to the vehicle body. One prominent example is the study reported by Chan and Tan (2001), in which a steering control system was developed to demonstrate the feasibility of post-impact manoeuvres to mitigate accident consequences. A number of collision scenarios were simulated to demonstrate the effectiveness. However, its controller relied on the information about vehicle position in lane and heading angle, which are challenging to retrieve unless a computer vision or a magnetic marker sensing system is installed. Furthermore, the targeted collision scenarios were for intelligent transportation system where impacts occur mainly in longitudinal direction without causing large loss of control of the involved vehicles (peak post-impact heading angle <10°). For collisions with higher severity, the loss of control could happen with changes in vehicle heading much larger than 10 degree. Also for severe collisions, the steering control alone usually cannot stabilise the post-impact vehicle motion. In 2007 Bosch released a prototype feature called Secondary Collision Mitigation (SCM) (Robert Bosch, 2007), which networks between restraint control module and brake control module. It triggers automatic ABS braking on four wheels as soon as at least one airbag-firing criterion is met, so that the vehicle speed can be maximally reduced. Since the total kinetic energy decays fast, the potential of secondary collisions can be reduced or at least their severity is mitigated. Electronic Stability Control (ESC) systems have been widely equipped on modern vehicles for stability enhancement and handling transparency. A simulation assessment of a research ESC system, which mimics the performance of the existing ESC, for the

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post-impact stabilising purposes was conducted by Thor (2007). Since an ESC is designed to stabilise a vehicle in the event of a mismatch between the vehicle and the driver request due to tyre force variations (i.e., the disturbance and the control mechanisms are from the same source – the tyre force variations), the vehicle motion induced by an exogenous impact is likely beyond the operation range of the ESC. It is of interest to counteract the post-impact loss of control using the similar control mechanism as ESC systems upon the detection of an impact. Such a strategy is likely to request braking forces larger than those normally used in ESC to quickly attenuate undesired vehicle motions (spin-out, skid, and roll), so that subsequent crashes can be avoided or mitigated. The proposed stabilisation method constitutes an incremental step towards a comprehensive vehicle safety system that consists of conventional active safety systems, post-crash active safety measures, and passive safety systems, along with their interactions (Figure 2). Such a total safety system expands the operational horizon of active safety systems from preventive measures to post-event mitigation measures, which have previously been the sole responsibility of passive safety devices, such as airbags. Figure 2

Concept of a comprehensive vehicle safety system (see online version for colours)

The main contribution of this paper includes designing a post-impact control system for vehicle handling so that multiple loss-of-control conditions (spinning, skidding, rollover, etc.) as well as unnecessary vehicle speed may be attenuated. The multiple control objectives can be coordinated and prioritised. MPC-based strategy is used to generate virtual commands and optimal tyre force allocation. It explicitly accommodates constraints on actuator control variables while seeking to achieve better handling performance right after the impact. The rest of this paper is organised as follows. Using the collision model of characterising vehicle motions immediately after light impacts, as developed in Zhou et al. (2008) and Zhou (2009), a real-time crash sensing and validation scheme is proposed in Section 2. The design of a post-impact stabilisation system built on model predictive control and optimal Control Allocation (CA) is conducted in Section 3. The control effectiveness is demonstrated in angled rear-end collisions simulated with CarSim vehicle dynamics simulation package in Section 4. Section 5 concludes the paper.

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Detection and validation of impulsive disturbances

For the proposed vehicle stabilisation system to operate effectively, it has to be activated at the right moment without false activation due to faulty measurements, sensor noises or other non-impact events. While many severe impact events can be detected by the restraint control module of airbag systems, those less severe collisions with impact strengths below airbag thresholds but triggering abnormal vehicle motions need to be addressed. Namely, a real-time sensing and validation scheme of potential events using motion sensors (e.g., those equipped for vehicle stability control systems such as ESC) is needed. In this section, we consider the detection of an impulsive disturbance and the time instant at which subsequent abnormal and undesired vehicle motions follow. As an example, we consider using yaw rate and lateral acceleration sensors to do so, while the other sensor configurations are also possible. Suppose the sampling time is 0.01 s, same as the one typically used in the brake control module. If there are three consecutive but large inter-sample changes in properly filtered yaw rate and lateral acceleration, where the absolute magnitude of the changes is beyond the normal ranges corresponding to drivers’ manual operations, then it is assumed that the vehicle has experienced certain impact. Upon the detection of such impacts, the vehicle stabilisation system is activated. As an example, the sensing thresholds for the gradients of yaw rate and lateral acceleration used in this paper are set at ∆ωz = 3 deg/s and ∆Ay = 0.1 g between consecutive samples respectively. The choice of the values is based on calibrations with computer simulations which can be tuned for different vehicle types or platforms. In practice, an abnormal change in consecutive samples does not necessarily lead to the conclusion that a crash is on-going, since it may also be due to sensor faults or other accidental noises. In order to eliminate the possibility of improper characterisation of an impact event (false positive), the crash event is also validated through continuously monitoring key kinematic variables. Figure 3 shows the conceptual validation procedure of an impulsive disturbance. The actual duration of the crash is from the time instant O to D. This crash is presumably detected at B by meeting the aforementioned detection criteria, as a consequence the stabilisation system is then activated. The estimated crash onset is positioned at the time instant A since three sampling intervals are used. Then the crash severity and location as well as the predicted vehicle responses at a future time instant C are computed, which is five samples downstream from the current time step. When the actual time course reaches C, vehicle yaw rate measurement and lateral velocity estimate are compared with their predicted values. If they agree with each other, the crash event detected at B is validated. Otherwise, the impact event is invalidated and the stabilisation control will be de-activated. One critical step in the procedure is to estimate the magnitude and the location of the impulse, given accessible vehicle states and a limited set of nominal vehicle parameters. A previously developed model for light impacts is applied for this purpose. The modelling approach for light vehicle-to-vehicle impacts suitable for this study has been detailed in our previous work (Zhou et al., 2008), and will not be repeated here. The difference between this collision model and the models normally used for crashworthiness and occupant protection is that the model used here focuses on the change in vehicle kinematic states, such as longitudinal/lateral velocities and yaw rate

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in real-time, rather than the structural deformations of the vehicle body in response to the collision impulses. Figure 3

Temporal representation of the crash validation steps (see online version for colours)

The planar components of the collision impulse (Px, Py) and its location (xA, yA) are inferred on the basis of the following equations (1)–(3) accounting for vehicle longitudinal, lateral and yaw motions. A list of symbols can be found at the end of this paper. Equations (1)–(3) apply to the striking vehicle (subscript ‘1’), and similar equations can be derived for the struck vehicle. The empirical coefficients of restitution and tangential friction (Brach, 1991) account for the interaction of the colliding vehicles. By proper rearrangement, the unknown post-impact states for both vehicles can be computed by numerically solving the involved algebraic equations. Px = M 1 ⋅ (V1x − v1x ) − M 1

∆t (V1 y Ω1z + v 1 y ω1z ) 2

∆t (V1x Ω1z + v 1x ω1z ) − mR1h1 ⋅ ( Ω1x − ω1x ) 2  V1 y + a1Ω1z v1 y + a1ω1z  ∆t  V1 y − b1Ω1z v1 y − b1ω1z  ∆t + Cf1  − −  + Cr 1   2 V1x v1x V1x v1x   2  

(1)

Py = M 1 ⋅ (V1 y − v1 y ) + M 1

(2)

Py x A − Px y A = I zz1 ( Ω1z − ω1z ) + I xz1 ( Ω1x − ω1x )  V1 y + a1Ω1z v1 y + a1ω1z  ∆t a1C f 1  −  V1x v1x 2   V b Ω v b ω − −   ∆t 1y 1 1z 1y 1 1z − b1Cr1  − . V v 2 1x 1x  

+

(3)

Since the envisioned scenario is light collisions without substantial vehicle deformation, the location of the impact is assumed to fall on the vehicle periphery instead of an arbitrary position inside the vehicle. Consequently, when solving for xA and yA from the above equations, two cases (side and rear-end impact assumptions) will need to be dealt with simultaneously. Only the geometrically feasible answers will be accepted. A linear extrapolation is used to predict (Px, Py) at a future time step, because during the brief interval of an on-going crash, impulses are monotonically increasing values. A short prediction horizon (e.g., 50 ms) will be employed. Based on the difference between the estimated current impulses and the predicted future impulses, the magnitude

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of collision forces within the short prediction horizon can be derived. Then a four-DOF vehicle dynamics model (Gillespie, 1992) can be used to make projections on vehicle kinematic states at the end of the prediction window, which will be compared with the measurements afterwards to check their agreements and to further verify the occurrence of an impact event. Further details about the crash detection method can be found in Zhou et al. (2009)

3

Development of the stabilisation control system

The post-impact stabilisation control is approached by treating the struck vehicle with substantial initial conditions induced by the impact and the subsequent control efforts to bring the vehicle back to its desired stable states. The control objective is to attenuate a vehicle’s sideslip angle, yaw rate, and roll rate as soon as an impulsive crash disturbance is detected such that the vehicle can regain its stability as quickly as possible. The following general assumptions are made. First of all, for the proposed control system to operate effectively, the braking and steering systems are assumed to function normally despite the collision. That is, the impact is the one where the impact level is low and the impact does not cause severe damage to the vehicle structure, its components, subsystems, and electronic system. Furthermore, the controller is assumed to have access to all the necessary vehicle states, for instance, measured yaw rate, longitudinal velocity, lateral acceleration, as well as estimations such as lateral velocity, tyre forces, and so on. Concerns over practical constraints as well as system robustness against varying road surface conditions and tyre force estimation inaccuracy will be explored in the later phase of this research.

3.1 Control architecture Active steering system, selective braking system, as well as active differentials are being implemented in many vehicle platforms from various automotive manufacturers. Those actuators greatly increase the flexibility and capability of vehicle dynamics controls. However, they also present potentials for overlapping or conflict since multiple actuators might influence the tyre forces in different ways. Given the interdependencies among these actuators and the tyre force generations, the challenge is to find a method to distribute the tyre forces in a unified and optimal way such that the available actuation resources can be better utilised and the overall control objectives can be achieved. The concept of Direct Yaw-Moment Control (DYC) was proposed by Shibahata et al. (1993), where the vehicle motion is controlled by yaw moment generated from the differential distribution of the tyre longitudinal forces. It demonstrated significant benefits in improving vehicle control performance in near-limit operating regions. Peng and Hu (1996) concentrated on the optimal distribution of tyre forces to deliver maximum longitudinal acceleration or deceleration while cornering successfully. The longitudinal and lateral forces of the tyres were solved by constrained nonlinear programming and assumed available when requested. Hattori et al. (2002) and Ono et al. (2006) proposed a Vehicle Dynamics Management (VDM) concept to control the forces of individual wheels. They used a feedforward control to compute the desired longitudinal and lateral forces and yaw moment of the vehicle. A nonlinear optimal

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method was used to distribute the desired forces and the moment of the vehicle to longitudinal and lateral forces on each wheel. A coordinated vehicle dynamics control system was presented in Wang et al. (2007). Sliding mode control theory was first applied to derive a higher-level controller for total desired tyre forces. Then an accelerated fixed-point method is proposed to numerically solve the constrained control allocation problem for individual tyre forces. Individual tyre slip ratios and sideslip angles are selected as the control variables to resolve the inherent tyre force nonlinear constraints. To control vehicle motions with coordinated actuations, it is crucial to determine proper tyre forces on each wheel to achieve a specific optimal control objective. The challenge here is that the longitudinal and lateral tyre forces cannot be arbitrarily assigned due to the coupling and the physical constraints on tyre forces. The proposed control design procedure is illustrated in Figure 4. The desired values of vehicle states are first compared with their measurements or estimations. Then, in response to the state errors, a supervisory controller at the first stage determines the desired control efforts, namely, the desired total longitudinal/lateral forces and yaw moment. These forces and moment are requested to achieve handling control objectives without violating vehicle dynamic constraints (mainly from the tyres). The adopted control scheme is based on the Model Predictive Control (MPC) methodology, because it handles multivariable and constrained nonlinear problems naturally. In the second stage, an optimal control allocator maps the total control demand onto individual actuators, in other words, slip ratios and sideslip angles on individual wheels. In the last stage, controllers at the actuator level manipulate physical variables to achieve the desired values dictated at the second stage, e.g., wheel cylinder braking pressures, traction torque, and steering angles. Finally, tyre forces generated affect vehicle motion and the resulting vehicle states are fed back to the supervisory controller to close the loop. Figure 4

Hierarchical framework for vehicle handling control (see online version for colours)

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3.2 Model predictive supervisory control Model Predictive Control is chosen as the design approach for the supervisory controller here. The applications of MPC in automotive systems can be found in lateral stabilisation (Falcone et al., 2008), emergency lane change (Chang and Gordon, 2007), and so on. A comprehensive discussion of MPC can be found in the literature (such as Camacho and Bordons, 2004), and the rest of this section outlines the major derivation steps. Applying Newton’s second law of motion to the 4-DOF model in Figure 5 and assuming ω x ≈ φ, the equations of motion (Segel, 1956) for the vehicle’s lateral, yaw, and roll dynamics can be written as M (v y + vxωz ) − mR hω x = Fy

(4)

I zz ω z = M z

(5)

I xxsω x − mR (v y + vxω z )h = mR ghφ − Dsωx − K sφ

(6)

where Fy and Mz are the total tyre lateral force and yaw moment applied to the vehicle, respectively. For instance, Fy = FxFL sin δ FL + FyFL cos δ FL + FxFR sin δ FR + FyFR cos δ FR + FxRL sin δ RL + FyRL cos δ RL + FxRR sin δ RR + FyRR cos δ RR .

(7)

For a non-steering axle, the corresponding steering angle δ is set to zero. Mz can be derived from the eight individual tyre force components in the same way and omitted here. After substitutions and reductions, it can be shown that v y = −vxω z +

ω x =

mR h ( mR gh − K s ) me I xxs

φ−

F mR hDs ωx + y me I xxs me

mR gh − K s D m h φ − s ω x + R Fy Ie Ie MI e

(8) (9)

where the equivalent mass and inertia are me = M − mR2 h 2 / I xxs and I e = I xxs − mR2 h 2 / M . Now the vehicle dynamics can be cast into a four-state system as in the following   0 − vx  v y       ωz  =  0 0  φ   0 0  ω    x  0 0  

mR h ( mR gh − K s ) me I xxs

−mR hDs me I xxs

0 0 mR gh − K s Ie

0 1 − Ds Ie

 1   m  e  v    y    ωz   0   +  φ   0   ω     x   mR h    MI  e

 0    vy  1      Fy   ωz  I zz    , x =  φ  . (10) M 0  z      ωx   0  

It can be treated as a linear parameter-varying system x = Ac (vx )x + Bc v,

with virtual control efforts v = ( Fy matrix in continuous time.

(11) M z ) , and Ac and Bc are state matrix and input T

Vehicle stabilisation in response to exogenous impulsive disturbances Figure 5

251

Schematic diagram of a 4-DOF vehicle model (see online version for colours)

Following the standard derivation of MPC procedures (Maciejowski, 2002), after the discretisation of equation (11), one obtains x k +1 = Ax k + Bv k . The sequence of the future states up to n-step ahead can be expressed as:  x k +1   A   B  x   A2   AB  k +2      x k + 3  =  A3  x k +  A2 B       #   #   #  x k + n   An   An −1 B

0   vk  B 0 " 0   v k +1  AB B 0 0   vk +2  .   # # % # #  An − 2 B " AB B   v k + n −1  0

"

0

(12)

It can be rewritten in a compact form by assigning symbols to the state and input matrices x = Ps x k + Pv v ,

→k

(13)

→ k −1

where  x k +1  x ≡  x k + 2  , →k  # 

 vk  v ≡  v k +1  , → k −1  # 

and Ps and Pv are their corresponding coefficient matrices. The cost function for optimisation is defined as the summation of weighted state deviation sequence e from the reference r and weighted control input sequence v. nx

nu

i =1

i =0

J = ∑ eTk + i Qe k + i + ∑ v Tk + i Rv k + i

(14)

where e k + i = x k + i − rk + i . For regulation problems, the desired states are zero (r = 0). The control objective is to minimise this cost function J with respect to future control sequence,

( )

min J = x v

→

→k

T

Q 0 " 0  0 Q " 0   x+  # # % #  →k    0 " 0 Q 

(v)

T

→ k −1

R 0 " 0  0 R " 0   v .  # # % #  → k −1    0 " 0 R 

(15)

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After substitutions and collecting similar terms, the objective function is formulated in a standard quadratic form

(

min J = Ps x k + Pv v v

→

) Q(P x T

→ k −1

T

s

k

+ Pv v

→ k −1

) + ( v ) R( v ) T

→ k −1

T

⇒ min J = v S v + 2 v f, →

v

→

→

→ k −1

(16)

→

where S and f are composed of system state transition and weighting matrices. In order to generate realistic total forces and yaw moment, constraints on the magnitudes and rates of change of control actions should be imposed. Suppose the control inputs are subject to upper and lower saturation limits v ≤ v ≤ v, and the lower and upper limits of the control increments sequence are ∆V ≤ ∆v ≤ ∆V . It can be shown eventually the MPC → problem can be formulated in a standard quadratic programming form, T

T

min J = v S v + 2 v f , subject to Lh v ≤ Rh v

→

→

→

→

→

(17)

where the matrices Lh and Rh consist of the linear constraints on control inputs. The routine ‘quadprog’ in the MATLAB Optimisation Toolbox can be called to solve this standard problem. By adjusting the weights on states, trade-offs among lateral, yaw or roll responses and control efforts can be favoured. The optimal solution of MPC itself does not ensure the stability of the closed-loop system. Presently an analytical proof of the stability of MPC schemes is still an active research area in control theory. Sometime the MPC formulation is augmented with a terminal cost and a terminal constraint set, carefully chosen to maintain closed-loop stability. A treatment of sufficient stability conditions goes beyond the scope of this work and can be found in the discussions in Mayne et al. (2000). After the optimal virtual control sequence v is found, only its first-step is used, → k −1 namely vk* . Given road friction, tyre vertical loads, and the vehicle states, the virtual controls can be attained through changing planar tyre forces Fxi and Fyi, which are in turn affected by tyre longitudinal slip ratios and sideslip angles. If independent braking is the only actuation mechanism, the actual control inputs are reduced to u = ( λFL

λFR

λRL

λRR ) . T

3.3 Control allocation scheme Mathematically, a Control Allocation (CA) algorithm solves an under-determined, typically constrained, set of equations (Hac et al., 2006). Given the total virtual control effort v(t) ∈ Rn requested by the supervisory stage, the allocator determines the true control input u(t) ∈ Rm, where m > n. For the linearised case, this mapping becomes B ⋅ u (t ) = v (t ) ,

(18)

where the control effectiveness matrix B is an n × m matrix with rank n. Typically the true control is also subject to upper and lower bounds u ≤ uk ≤ u .

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The control allocation problem can be posed as a constrained optimisation problem. In some studies, solutions to the CA problem have been found with the pseudo-inverse approach (Schiebahn et al., 2008), which can be easily implemented and computationally efficient. However, pseudo-inverse based control allocation fails to factor in actuator limitations directly and could lead to unrealistic solutions violating the physical bounds. In this work the control allocation is formulated as a minimal Least Squares (LS) problem. By first-order Taylor series expansion, the nonlinear function is locally approximated with an affine function at each sampling instant vk* = G (ζ , uk ) ≈ G (ζ , uk −1 ) +

∂G ∂u

k −1

⋅ ( uk − uk −1 ) ,

(19)

where ζ refers to accessible parameters and variables, such as tyre vertical loads, tyre model fitting parameters, and so on. Let vk = vk* − G (ζ , uk −1 ) +

∂G ∂u

k −1

⋅ uk −1 and T (ζ , uk −1 ) =

∂G (ζ , u ) ∂u

(20)

, k −1

then equation (19) is reduced to solving for uk so that the following linear mapping is satisfied in the least-square sense vk = T (ζ , uk −1 ) ⋅ uk .

(21)

The Jacobian matrix is denoted by ∂G (θ , u ) / ∂u = ( ∂Fy / ∂u ∂M z / ∂u ) . A tyre model has to be chosen and parameterised to serve the purpose. In this study, the Magic Formula tyre model for combined longitudinal and lateral slips is adopted (Pacejka, 2002). A typical resultant tyre behaviour is shown in Figure 6. In the top subplot, the slip ratio varies between 0 and 1, and the sideslip angles are fixed at discrete levels α = 0°, 2°, 5°, 10°, 20°. In the bottom subplot, the sideslip angle is varied and the slip ratios are fixed at five levels λ = 0%, 2%, 5%, 10%, 20%. In addition, two groups of local tangents, whose values are derived directly with the combined-slip MF model (Gordon and Best, 2006), are superimposed on the tyre force curves as a result of the computation of the partial derivatives. The fact that the gradients can be obtained analytically in this modelling approach provides great benefits to solve the control allocation problem. With the entries in the Jacobian matrix populated, one can solve the optimal allocation problem as a standard constrained linear least-square problem T

min T ⋅ uk − vk uk

2 2

, subject to u ≤ uk ≤ u .

(22)

Because the optimisation problem being solved is convex, there exists a global, not necessarily unique, solution. The optimal solution with the least l2-norm will be chosen, since excessive wheel slip ratios are not desired. Saturation and rate limit constraints on the optimisation variables can readily incorporated into the problem formulation. In the end, the optimal allocation problem becomes uk = arg min u u∈Ω

2 2

2

and Ω = arg min Wλ ( T ⋅ u − vk ) 2 . u≤u≤u

(23)

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Figure 6

Tyre forces in combined-slip Magic Formula model with superimposed local gradients (see online version for colours)

3.4 Control implementation At the actuator level, in order to achieve the slip ratio commands, a wheel-slip control module based on the sliding model control theory is used, which is adopted from the derivations in Bang et al. (2001). It tracks the slip ratio commands by generating the physically control inputs to feed through the controller/plant interface: wheel cylinder braking pressure commands. In order to generate realistic braking responses, a hydraulic braking module with practical performance is used. The pressure buildup behaviour of the nonlinear braking model can be approximated by a first-order time lag, a time-delay, and a rate limiter. Pdeliver e−Td s = , Pdeliver ≤ RateLim. Pcmd TL s + 1

(24)

For the front axle, the fitting parameters are Td = 0.06 s, TL = 0.12 s, RateLim = 230 bar/s; for the rear axle, the fitting parameters are Td = 0.02 s, TL = 0.05 s, RateLim = 550 bar/s. The complete collision and post-impact stability control process is simulated in the nonlinear multi-body vehicle dynamics simulation tool CarSim (Mechanical Simulation, 2006). CarSim simulates and analyses the dynamic behaviour of light vehicles on 3-D road surfaces and it is commercial available from Mechanical Simulation Co. CarSim is capable of predicting 3D forces and vehicle motions in response to driver inputs (steering, braking, etc) as well as other external sources (for instance, impacts and wind gusts). The control system detailed in previous subsections is implemented in Simulink/Matlab and interfaced with the vehicle dynamics model in the form of a CarSim S-function. The overall system structure is modularised. It consists of three major subsystems as illustrated in Figure 7: system activation and de-activation, impact estimation and prediction, as well as the generation of braking commands. The simulated scenario and results are described in the next section.

Vehicle stabilisation in response to exogenous impulsive disturbances Figure 7

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Simulation architecture of the control system (see online version for colours)

Simulation results

To evaluate the effectiveness of the proposed control system, simulations are performed to compare vehicle motions with various post-impact vehicle dynamic response measures. The simulated collision has a layout illustrated in Figure 8, where Vehicle 1 is a target vehicle receiving impact force from the bullet Vehicle 2. The road is assumed to be straight and have homogeneous adhesion with coefficient µR = 0.70. Both vehicles are parameterised using the ‘baseline big SUV’ dataset in CarSim (M = 2450 g, a = 1.105 m, b = 1.745 m, hCG = 0.66 m, Izz = 4946 kg-m2, etc). Figure 8

A light-impact collision scenario that generates impulsive disturbances (see online version for colours)

It is assumed both the target (Vehicle 1) and the bullet (Vehicle 2) vehicles are travelling along their own longitudinal axes when the collision occurs, with v1x = 29 m/s (104 km/h, or 65 mph), v2x’ = 33.5 m/s (120 km/h, or 75 mph). The target vehicle is aligned with road tangent, whereas the bullet vehicle has an orientation angle θ2 = 18°. At the time instant of crash, the impact location on the bullet vehicle is at the centre of its front bumper, whereas the impact location on the target vehicle is 0.1 m to the left of its rear bumper centre. The coefficient of restitution (e) is assumed to be 0.20 for this angled

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rear-end crash. After the collision computation, the post-impact lateral velocity and yaw rate of the target vehicle change fast from zeros to Vy = 3.3 m/s and Ωz = –64 deg/s respectively. All the computational results below are for the target vehicle. The impact is detected by the sensing algorithm described in Section 2 and the stabilisation control module is activated in due time. For the post-impact vehicle motion regulation, a prediction horizon of 0.3 s (or 30 samples) is used for the MPC supervisory controller. The constraints on the virtual controls are found at saturation limits |Fy| ≤ 11500 N and |Mz| ≤ 7700 Nm, selected based on vehicle mass and road adhesion conditions. Figure 9 shows the comparison of the desired ground forces dictated by MPC, which are tracked by the actual ground forces generated by the nonlinear vehicle model in the CarSim simulation tool. After the vehicle states have been substantially mitigated, the driver takes over, steers the vehicle back to its original lane and resumes the previous driving direction. The traces starting at 4.2 s in Figure 9 is highlighted just to exemplify the optimal control sequence determined at that instant by the MPC scheme. Figure 9

Comparison of the desired ground force dictated by MPC and the actual ground force (see online version for colours)

Figures 10 and 11 present a comparison of the trajectories and time responses for three vehicles subject to the same type of collision impulses, but with different control approaches. In each case, the space between two horizontal dashed lines represents one typical traffic lane. Without applying proper braking or steering intervention (Scenario 3), the vehicle quickly develops substantial sideslip angle and heading angle in response to the impulse, and eventually departs from the designated lanes. With full ABS braking and driver’s steering inputs (Scenario 2), the vehicle velocity is significantly reduced. However, the high yaw rate cannot be effectively attenuated, and the vehicle keeps spinning and leads to a heading angle beyond recovery (~150°). With the proposed post-impact stability control scheme actuated through differential braking (Scenario 1), the disturbed target vehicle states are well regulated. Despite the limited lateral deviations, the vehicle maintains its directional stability and orientation. Eventually its yaw rate and sideslip angle are attenuated to insignificant levels and the driver can easily manoeuvre the vehicle and stop safely or proceed to a safe location afterwards.

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Figure 10 Comparison of vehicle trajectories in angled rear-end collision events (see online version for colours)

Figure 11 Comparison of vehicle responses in an angled rear-end collision event (see online version for colours)

The simulations and observations above assume that the driver does not intervene until the vehicle has been sufficiently stabilised, if at all. In the real world, after a collision, it is likely for the driver to initiate reactions after she or he has perceived the collision. During these short intervals without human intervention, the proposed control system can preemptively attenuate a portion of impact-induced vehicle motions, so that when the driver does start to respond, the vehicle is more manageable in terms of lower side-slip angle and yaw rate. It should be noted that the proposed safety system is designed to assist drivers, instead of overtaking drivers’ role.

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In order to realise the virtual control efforts determined in Figure 9, the optimal control allocation scheme is employed to distribute the generalised forces to four wheels, through changing the slip ratio commands. The results of the optimal control allocation are shown in the left subplot of Figure 12. The delivered pressures along with wheel dynamic states result in the actual wheel slips shown in the right subplot of Figure 12. The instantaneous wheel slips, along with tyre slip ratios (not actively controlled in this example) and normal loads, generate tyre longitudinal and lateral forces, whose combined effects have been shown in Figure 9 (dashed lines). It should be noted that in the CarSim simulations, the generation of tyre forces follow the internal MSC tyre model (Appendix D of Mechanical Simulation, 2006), which is approximated by the combined-slip MF model for controller design. Figure 12 Comparison of the slip ratio commands and the actual slip ratios generated in CarSim in a post-impact stabilisation manoeuvre (see online version for colours)

Finally, wheel cylinder braking pressure commands are shown in the left subplot of Figure 13. The actually delivered pressures are produced by processing the braking requests through hydraulic systems composed of time lag, time delay, and rate limiters (right subplot of Figure 13). Figure 13 Comparison of the braking pressures commanded and delivered in a post-impact stabilisation manoeuvre (see online version for colours)

In summary, after a collision event has been detected and confirmed, the supervisory stage dictates the desired total Fy and Mz via MPC scheme, so that the regulation objectives are fulfilled. At the intermediate stage, through constrained optimal allocation, wheel slip ratio commands are generated so that the difference between the desired total

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forces and those actually developed on the tyres is minimised in the least-square sense. The lower stage seeks to track the wheel slips requested by the intermediate stage, by manipulating the wheel cylinder braking pressures. The developed control system provides the benefits of a modularised approach to integrate vehicle stabilisation objective fulfillment, the optimality of tyre force allocation, and robust tracking of slip ratio commands.

5

Conclusions

In this investigation, a vehicle stabilisation control system is proposed to assist the driver to recover vehicle stability after an initial impact, so that secondary accidents such as crashes or rollovers can be avoided or mitigated. A crash sensing and validation scheme is first devised to activate the control system. Then a model-based hierarchical control framework is developed and applied to the post-impact stabilisation scenario. The developed control system consists of an MPC-based supervisory stage, an intermediate stage for optimal control allocation, and wheel slip ratio tracking at the actuator stage. It provides the benefits of a modularised approach to accommodate control constraints and coordinate multiple control effectors. The overall system is evaluated on a multi-body vehicle model provided by CarSim simulation tool under limit handling conditions. Simulations are conducted on a case in which the vehicle subject to a sizeable impulsive disturbance loses directional stability with manual steering and traditional braking control. However, with the proposed control system applied, the post-event vehicle stability can be recovered effectively. The usefulness and practical performance of the proposed system can be further studied by addressing the following issues. We will further study the stability and the robustness of the proposed MPC and optimal allocation scheme. On the application side, the arbitration and prioritisation among drivers’ actions and post-impact stabilisation control commands, as well as original ESC commands need to be examined in further depth. A driver’s steering intent has to be followed by active safety systems, but not necessarily ‘on its face value’, because drivers may behave irrationally and over-compensate. An interface needs to be designed to arbitrate and prioritise the three parties so as to streamline their interactions. In addition, although differential braking is illustrated in this paper, the proposed control framework can potentially be applied to vehicles equipped with other actuators and their combinations, such as active steering and active differential, so that their influences can be fully leveraged, and the overall control objective can be attained in a more coordinated and optimal way.

Acknowledgement The authors would like to thank Bengt Jacobson at Volvo Car Corp. and Daniel Eisele at Ford North America for their constructive inputs. The work was fully supported by Ford Motor Company under the Innovation Research Alliance between Ford and the University of Michigan.

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List of symbols a, b A, B Ay Cf , Cr Ds ek Fx Fy Fz H Ixxs Ixz Izz J Ks mR M Mz Pb P s, P v Px, Py Q R rk uk vk vx Vx vy Vy xA, yA xk

Distance from axles to vehicle CG State matrix and input matrix Vehicle CG lateral acceleration Cornering stiffness per axle Total suspension roll damping Error at sampling step k Longitudinal forces Lateral forces Wheel vertical load Sprung mass CG to roll axis distance Sprung mass roll moment of inertia Sprung mass roll-yaw product of inertia Yaw moment of inertia about z axis Performance index Total suspension roll stiffness Rolling (sprung) mass Total vehicle mass Yaw moment Wheel cylinder braking pressure State/input matrix in MPC formulation Planar components of collision impulse Weights penalising states Weights penalising control inputs Reference at sampling step k Actual control input at sampling step k Virtual control input at sampling step k Longitudinal velocity Post-impact longitudinal velocity Lateral velocity Post-impact lateral velocity Collision location w.r.t. vehicle CG States at sampling step k

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→ k −1

x

States stacked from time-step k + 1 to k + n

α β δ

Wheel slip angle

∆t

Time step or time increment

φ λ ωx

Sprung mass roll angle

→k

Vehicle CG sideslip angle Road-wheel steering angle

Wheel longitudinal slip ratio Sprung mass roll rate

Ωx

Post-impact roll rate

ωz

Vehicle yaw rate

Ωz

Post-impact yaw rate