Original Article
Simultaneous vehicle-handling and path-tracking improvement using adaptive dynamic surface control via a steer-by-wire system
Proc IMechE Part D: J Automobile Engineering 227(3) 345–360 Ó IMechE 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954407012453240 pid.sagepub.com
Amir Ali Janbakhsh, Mohsen Bayani Khaknejad and Reza Kazemi
Abstract This paper deals with the simultaneous vehicle-handling and path-tracking improvement through a steer-by-wire system, using non-linear adaptive dynamic surface sliding control. The designed adaptive dynamic surface controller, which is insensitive to system uncertainties, offers an adaptive sliding gain to eliminate the precise determination of the bound of uncertainties. The sliding-gain value is obtained using a simple adaptation law that does not require an extensive computational load. Achieving the improved vehicle-handling and path-tracking characteristics requires both accurate state estimation and well-controlled steering inputs from the steer-by-wire system. A second-order sliding-mode observer provides accurate estimation of the lateral and longitudinal velocities while the yaw rate is available from the angular rate sensor. A driver control model is also presented according to the preview or look-ahead strategy to generate appropriate steering angles using the vehicle state feedback and future information about the path to be followed. Moreover, because of the inertia and viscous damping of the steering mechanism, and the effects of the Coulomb friction and selfaligning moment of the front tyres, the steering-system controller is designed based on the proposed adaptive dynamic surface scheme, to control the front steering angle. A complete stability analysis based on the Lyapunov theory is presented to guarantee closed-loop stability. The simulation results confirmed that the proposed adaptive robust controller not only improves the vehicle-handling and path-tracking performance but also reduces the chattering problem in the presence of uncertainties in the tyres’ cornering stiffnesses.
Keywords Vehicle handling, path tracking, adaptive dynamic surface control, steer-by-wire system, sliding-mode observer, driver model, steering control system
Date received: 30 December 2011; accepted: 7 June 2012
Introduction In recent years, significant efforts have been undertaken in new drive-by-wire (DBW) systems in which mechanical and hydraulic subsystems such as the steering, braking and suspension are being replaced by electronic actuators, controllers and sensors. The benefits of applying electronic technology such as DBW systems are obvious: improved overall performance and driving convenience, reduced power consumption and significantly enhanced passenger safety. In steerby-wire (SBW) systems, which are a part of the DBW systems, the conventional mechanical interface between the steering wheel and the front wheels is replaced with electronic actuators. The elimination of parts, such as the steering column, gearbox and hydraulic pump,
provides noteworthy advantages including saving energy, decreasing noise and vibration, reducing weight and removing environmentally hazardous hydraulic fluids. Moreover, in front-end collisions, the danger that a driver is crushed is reduced as there is no steering column. Undoubtedly the most dominant virtue of an SBW system is its active steering capability, i.e. the ability to
Research and Development Center of Advanced Vehicle Systems, KN Toosi University of Technology, Tehran, Iran Corresponding author: Amir Ali Janbakhsh, Research and Development Center of Advanced Vehicle Systems, KN Toosi University of Technology, Tehran, Iran. Email:
[email protected]
346 change the driver’s steering input to improve manoeuvrability or stability. Therefore, research institutes and the automotive industry pay considerable attention to the potential benefits of SBW systems, particularly for improving vehicle-handling behaviour. Over the last two decades, a number of studies have been carried out on robust control of vehicle handling and stability using the SBW architecture. Significant work was carried out by Ackermann,1,2 who used yaw rate feedback in combination with active steering to decouple robustly the yaw rate from the lateral acceleration. The effectiveness of his approach was demonstrated by experimental results; through this method, the generated yaw angle is cancelled out during braking on split-friction roads. Hebden et al.3 devised sliding-mode control and a sliding-mode observer (SMO) to heighten the vehicle stability in a split-m manoeuvre. They put the controller and a conventional anti-lock braking system (ABS) together to guarantee safe and effective braking through an SBW system. In another important study, Yih4 developed a steering control strategy and implemented it on a test vehicle which is equipped with the SBW system. He also utilized the active steering capability of the SBW system, using vehicle full-state feedback in order to improve the vehicle-handling performance. Moreover, the results of observation of the vehicle states are compared favourably with a baseline side-slip estimation method using a combination of Global Positioning System and Inertial Navigation System sensors. The dynamic surface control (DSC) method has been widely used for vehicle control systems during the current decade. Girard and Hedrick5 proposed a combination of dynamic surface sliding control and hybrid systems to control multiple ocean vehicles. Each vehicle performs several manoeuvres; therefore, a dynamic surface controller is designed for each manoeuvre and switches between manoeuvres, and communication protocols between vehicles are also represented using hybrid system formalisms. In another dominant study, Kazemi and Zaviyeh6 proposed a new ABS for passenger cars which is called a servo ABS. The presented servo ABS hydraulic pressure modulator is equipped with servo valves for improving the speed, precision and controllability of the system in comparison with the conventional ABS. The DSC method is also employed to control the longitudinal wheel slip, using multiple sliding surfaces and synthetic inputs. In this paper, a non-linear vehicle dynamic model, defined in the road-fixed frame, is chosen at the first step. As some states cannot be measured directly by sensors, a non-linear SMO, based on the broken supertwisting algorithm,7 is used to provide some states such as the lateral and longitudinal velocities for designing the controllers. Next, the non-linear dynamic surface sliding controller is designed as a baseline controller because of its high accuracy in final tracking and its ideal transient performance in the presence of both kinds of system uncertainty. Moreover, using the DSC
Proc IMechE Part D: J Automobile Engineering 227(3) method, the need for model differentiation was eliminated, and so the problem of too many terms (called ‘the explosion of terms’) and a resultant complex control law was prevented. In traditional sliding-mode schemes, an upper bound for the system uncertainties should be determined as precisely as possible to calculate the sliding gain. As the upper bound becomes higher, the sliding gain should be increased to obtain the same level of tracking error. Therefore, the control effort will increase, which is undesirable in practice. By assuming that the uncertainties in the cornering stiffnesses of the tyres are the only source of parametric uncertainties in the system, a new non-linear scheme is proposed for sliding gain adaptation in DSC. This eliminates the requirement for the bound of uncertainties. A simple adaptation law, which does not require an extensive computational load, is chosen to adapt the sliding gain. The chattering increment dramatically decreases the actuators’ working life in SBW systems and may also excite high-frequency unmodelled non-linear dynamics. The proposed adaptive robust controller not only improves the vehiclehandling and path-tracking performance but also considerably reduces the system’s chattering because of its robustness against parametric uncertainties. Thus, a much smoother control input is obtained.
Vehicle dynamics The vehicle is assumed to be a mass moving in the yaw plane (Figure 1), for which the non-linear equations of motion are derived from the coordinates representing the deviation of the vehicle from the lane centre. The lateral deviation of the vehicle’s centre of gravity (CG) is symbolized by e and the angular heading error, which is the angular difference between the direction in which the vehicle is pointing and the direction of the road centre-line, is symbolized by c. Assuming that the yaw rate is small in comparison with the vehicle’s longitudinal velocity, the motions of the left tyres and the right tyres are the same and the lateral tyre forces from both sides are equal. Moreover,
Figure 1. Vehicle model in the road-fixed frame.
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the steering angles of the front left tyre and the front right tyre are the same. With these assumptions the non-linear equations of motion8 are described in roadfixed coordinates as m€ s = 2Fyf sin (df + c) + 2Fyr sin c m€ e = 2Fyf cos (df + c) + 2Fyr cos c € = 2aFyf cos df 2bFyr Iz c
ð1Þ
Tyre model The tyre characteristics have crucial effects on the dynamic behaviour of the vehicle. In a study of vehicle handling, it is of primary importance to describe the generation of lateral forces by the tyres. The wellknown non-linear ‘magic formula’ model9 is chosen as
Figure 2. Steering-system dynamics.
Fyf, r = Df, r sin(G arctanfBf, r af, r L½Bf, r af, r arctan(Bf, r af, r )g) R(Fa)f, r Bf, r = GDf, r Df, r = mFzf, r
Fzf, r R(Fa)f, r = C1f, r sin 2 arctan C2f, r
ð2Þ
where Fzf and Fzr are the normal loads on the front tyres and the rear tyres respectively and can be described as b 2(a + b) a Fzr = mg 2(a + b)
Figure 3. Tyre self-aligning torque at a side-slip angle.
Fzf = mg
ð3Þ
For a small slip angle a, the lateral tyre forces are linearly proportional to the slip angle according to Fyf, r = Caf, r af, r
ð4Þ
where Caf and Car are recognized as the cornering stiffnesses of the front tyres and the cornering stiffnesses of the rear tyres respectively, and the slip angle a is defined as the angle between the direction in which the tyre is pointing and the direction of its movement (see Figure 3 later) as given by _ 1 uy + ac af = tan df ux uy bc_ ar = tan1 ð5Þ ux
t aligning = 2Fyf (tp + tm )
The steering system shown in Figure 2 is described by the differential equation6 ð6Þ
where Jw is the total moment of inertia of the system, bw is the viscous damping and ta and tf represent the self-aligning moment and the Coulomb friction respectively of the tyre. Furthermore, rs is the steering ratio
ð7Þ
The resisting torque t f is treated as t f = 2 tp m Fzf sgn(d_ f )
Model of the steering system
Jw € df + bw d_ f + t f + t a = rs t M
and t M is the steering actuator torque which is known as the control input. The self-aligning moment of the tyre is a function of the steering geometry (in particular, the caster angle) and also the deformation of the tyres which generates lateral forces. According to Figure 3, tp is the pneumatic trail, which is the distance between the resultant point of application of the lateral force and the centre of the tyre, tm is the mechanical trail, which is the distance between the centre of the tyre and the point on the ground about which the tyre pivots as a result of the wheel’s caster angle, and v is the velocity of the centre of the tyre. The total aligning moment6 is given by
ð8Þ
where m is the coefficient of friction and the normal load Fzf on the front tyres is available according to equation (3).
Vehicle model for controller design The transformation between the vehicle-fixed frame and the road-fixed frame is given by8
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ux = s_ cos c + e_ sin c uy = e_ cos c s_ sin c
ð9Þ
By means of the transformation in equation (9), the slip angles of the front and rear tyres are redefined in the road-fixed frame, so that the linear tyre model can be transferred to the road-fixed frame. Assuming that the steering angle is small, and using the linear tyre model in equation (4), the set of equations (1) can be rewritten as e_ cos c s_ sin c sin c s_ cos c + e_ sin c c_ + 2(Car b + Caf a) sin c 2Caf df sin c s_ cos c + e_ sin c e_ cos c s_ sin c cos c m€ e = 2(Caf + Car ) s_ cos c + e_ sin c c_ cos c + 2Caf df cos c + 2(Car b Caf a) s_ cos c + e_ sin c € = 2(Car b Caf a) e_ cos c s_ sin c Iz c s_ cos c + e_ sin c c_ 2(Car b2 + Caf a2 ) + 2aCaf df _s cos c + e_ sin c m€ s = 2(Car Caf )
ð10Þ
The dynamic model expressed in equation (10) is used in the next section to design the non-linear SMO
Figure 4. Schematic diagram of the SBW control structure.
and the non-linear adaptive dynamic surface control (ADSC).
Steer-by-wire system control structure The schematic diagram in Figure 4 represents the vehicle’s SBW controllers and the interface. As can be seen, the vehicle’s yaw rate is directly measured by the angular rate sensor while the other states, such as the lateral and longitudinal velocities, cannot be measured directly because of the high measuring cost. Therefore, the SMO was used to estimate the unmeasured states. The driver control model generates an appropriate steering angle dfd, according to the vehicle and path information, when the reference model, which specifies the desired vehicle-handling characteristics, computes the desired yaw rate based on the driver input and the states of the vehicle. The upper controller utilizes the desired yaw rate and also the ideal path to determine the desired steering angle dfdes, while the lower controller applies a suitable torque to the steering actuator so that the front tyres reach dfdes. The designed control structure improves the vehicle handling and path tracking simultaneously according to the driver-demanded steering angle and the desired path.
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Second-order sliding-mode observer Accurate vehicle state estimation is necessary for the control algorithm. Because of environmental disturbances and the high measuring cost of the production vehicles, some states, such as the lateral velocity, cannot be measured directly through automotive sensors. As such, an observer plays a crucial rule in controller design. A second-order SMO based on the so-called broken super-twisting algorithm proposed by Davila et al.7 was used to estimate the yaw angle and the lateral and longitudinal velocities, while the yaw rate was measured directly with a yaw rate sensor. The importance of this observer was evident when seeking a finite time convergence, and when ensuring the system’s robustness against parameter variations. Moreover, observation error reduction and a limited computational load are advantages of the SMO.10 The set of equations (10) can be expressed in terms of a non-linear state space formulation as x_ = fðx, uÞ ð11Þ
y = h(x)
_ T , u = ½df , y = ½c _ and x_ 3 = x4 . with x = (s_ e_ c c) A general method for determining the rank of nonlinear systems was proposed by Fridman et al.11 By this method, the rank of the system in equation(11) was 4; thus, the system is observable. The SMO equations based on the super-twisting algorithm12 can be written as ^_ + Z ^_ 1=2 sgn c_ c ^_ = fðx ^, uÞ + g c_ c x eq 1 ^ y = C^ x
^_ Z_ 1 = g1 sgneq c_ c
ð12Þ
Where g and g1 the SMO gain vectors in R431 (all terms in equation (12) are vectors). To avoid chattering effects,10 the sgn function is replaced with an equivalent function sgneq in the control system as sgneq (ex) =
2 arctan (Lex) p
ð13Þ
where L as a design parameter is used to adjust the slope of the arctan function, as shown in Figure 5. By increasing the coefficient L, the slope is increased and as a result the chattering increases. Therefore, the sign function with L = 1 can be an appropriate choice for the SMO in equation (12).
Figure 5. Choosing the equivalent sign function for an SMO.
c_ des =
^s_ cos c ^ + ^e_ sin c ^ 2 dfd ^ + ^e_ sin c ^ a + b + kus ^s_ cos c g
where Kus is the understeer gradient and dfd is the driver-demanded steering angle. As the lateral acceleration is limited by the friction coefficient between the tyres and the road surface, the desired yaw rate is also limited by the value13 of 0.85mg/ux.
Driver model In fact, the modelling of human drivers is a difficult task, mainly because there are no general equations describing the complex human mind and because the driver is willing to adapt to the different vehicles and traffic situations,14 thereby changing his or her strategy and tactics. Today, it would be nearly impossible to find a general model for the driver, but several control models exist which are suited to specific tasks such as the longitudinal controller and the lateral controller for keeping distance and for changing lanes respectively. The lateral controllers can be divided into compensation tracking and preview tracking models. Preview or look-ahead models are a group of models which, unlike the compensatory models, not only use the vehicle state feedback but also utilize future information about the path to be followed as the controller inputs (Figure 6). The multi-input preview tracking model15 used the lateral position error edes e and also the vehicle yaw angle c to generate the
Vehicle reference model The desired vehicle-handling performance is expressed as a reference model which gives the desired responses to a command signal. The desired yaw rate as an output of the reference model was computed on the basis of the driverdemanded steering angle and the vehicle states.13 Using the transformation in equation (9), the desired or reference yaw rate c_ des could be expressed in the road-fixed frame as
ð14Þ
Figure 6. Basic structure of the preview model.
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Figure 7. Multi-input preview tracking driver model.
driver-demanded steering angle as the controller output in Figure 7. This driver model is described by ð15Þ
Ke (TLe s + 1)
and Kc (TLc s + 1)
The controllers’ characteristics are crucial when combining the two control techniques in the ADSC approach. With the existence of unmodelled non-linear dynamic uncertainties and external disturbances in the system, an adaptive controller may become unstable and, moreover, the transient performance of this controller is unclear. Contrary to the adaptive controller, a robust controller can guarantee the final tracking accuracy and the transient performance in the presence of model uncertainties. In this paper, DSC was chosen as the baseline control in order to guarantee the transient performance and also to prevent the explosion-of-terms problem.
et s T1 s + 1
ð16Þ
where Ke and Kc are the gain parameters which represent the proportional action of the driver to the lateral error and yaw angle respectively. The two parameters t and T1 are known as the dead time and the delay due to the human muscular system respectively, which should be kept constant. The driver controls the vehicle by predicting future values using TLc sþ1 factors as the model of lead or predictive action. Moreover, the driver’s mental workload is evaluated by the sum of the lead time constants TLe+TLc, and so a larger value denotes a higher mental workload and vice versa. Another significant parameter is the preview distance or look-ahead La, which is multiplied by the vehicle yaw angle (Figure 7). Using this preview distance makes intuitive sense as drivers look forwards while driving, utilizing future information to help in the path-tracking task. Using no preview distance makes the tracking task extremely difficult.
Non-linear adaptive dynamic surface control For a vehicle model, and generally for dynamic systems, two types of model uncertainty exist: these are unmodelled non-linear dynamic uncertainties and parametric uncertainties.16 In designing a robust controller, such as a dynamic surface sliding controller, only a robust term such as a sign or saturation function overcomes these two model uncertainties in order to achieve robust stability. As the model uncertainties (especially the parametric uncertainties) increase, the sliding gain adopts a higher value in order to obtain the same tracking error. As a result, the performance of the DSC is degraded significantly. To address the high-gain nature of robust controllers, adaptive control and DSC techniques are used simultaneously. By combining DSC and adaptive control, the dynamic surface would not be the only controller that deals with the model uncertainties, and so the resultant adaptive robust controller has the advantages of both controller design techniques to provide robust stability and robust performance.
Combined yaw stability and path-tracking controller design. Lateral, longitudinal and yaw dynamics were used to design the SBW dynamic surface controller. The first step in designing the controller is to choose a suitable sliding or switching surface. Then the control input is obtained to drive the system trajectories to the switching surface in the presence of model uncertainties and external disturbances. As the system trajectories reach the sliding surface, the closed-loop dynamics are completely governed by the equations that define the surface. The independence of these closed-loop dynamics from the system uncertainties guarantees the system’s robustness against model uncertainties. This task was split between the robust and adaptive controllers so that DSC only dealt with unmodelled dynamic uncertainties while ADSC dealt with parameter perturbations.16 The tracking error E is defined as the weighted combination of the difference between the actual and desired values of the yaw rate and the lateral displacement according to E = (c_ c_ des ) + z(^ e edes )
ð17Þ
where z is a positive gain; by setting z = 0, the combined controller changes to the yaw stability controller and will track only the desired yaw rate. Anyhow, the switching or sliding surface can be defined as S=E
ð18Þ
where the tracking error is chosen as a sliding surface. A common motion under sliding-mode control consists of two phases: a reaching phase in which all trajectories start away from the switching surface S = 0, must move towards the surface and reach it in finite time,17 and the next phase which is the sliding mode during which the trajectory motions will be confined to this surface.16 The system dynamics while in the sliding mode can be written as S_ + lS = 0
ð19Þ
where l is a strictly positive constant; thereby the tracking error E(t) converges to zero exponentially. The computation of the derivative of S could lead to too many terms, called the explosion-of-terms problem, and as a
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result a very complex control law would be obtained. The DSC method18 eliminates the need for model differentiation. That is the basic idea of DSC. First, we passed c_ des through a first-order low-pass filter y c_ z_ + z = c_ des
ð20Þ
where y c_ is a filter time constant and z serves as an estimate of c_ des , with a derivative that is easily computed as z_ =
1 _ (c z) y c_ des
ð21Þ
Using z in place of c_ des , the sliding surface in equation (18) can be rewritten as S = (c_ z) + z(^ e edes )
ð22Þ
Figure 8. Schematic diagram of the saturation function.
Hence, equation (19) can be written in terms of z and z_ according to € z) _ + z(^e_ e_des ) + l ½(c_ z) + z(^ (c e edes ) = 0 ð23Þ
approximate the control around the sliding surface within a boundary layer and is defined as S S if jSj 4 F ð26Þ sat = F S if jSj . F sgn F F
Equation (23) expresses the condition in which the trajectories will remain on the sliding surface. By substituting the yaw acceleration from equation (10) into equation (23), the continuous non-linear control law, which is called the equivalent control or Ueq, is obtained as ( ^ ^s_ sin c ^ Iz Car b Caf a ^e_ cos c Ueq = 2 ^s_ cos c ^ + ^e_ sin c ^ IZ 2aCaf c_ Car b2 + Caf a2 +2 ^s_ cos c ^ + ^e_ sin c ^ IZ ) ^ _ e edes ) + z_ z (e_ e_des ) l ½(c z) + z(^
Using equation (26), the continuous control law can be written as ( ^ ^s_ sin c ^ Iz Car b Caf a ^e_ cos c dfdes = 2 ^s_ cos c ^ + ^e_ sin c ^ IZ 2aCaf
ð24Þ
ð27Þ
To achieve robust stability in the presence of model uncertainties and external disturbances in the system, a term that is discontinuous across the surface S = 0 is added to Ueq according to U = Ueq K sgn(S)
ð25Þ
where K, as the controller design parameter or the sliding gain, indicates the system speed in approaching the switching surface. This constant parameter is replaced with a time-variant parameter in the adaptive controller. Chattering occurred in the control progress because of the discontinuity term sgn(S) in the non-linear equivalent control. This chattering will increase the control effort and may also excite the high-frequency unmodelled non-linear dynamics. To overcome this disadvantage, the boundary layer concept, as proposed by Slotine and Li,16 was used in the dynamic surface controller. The sign function was replaced by the saturation function sat(S/F) to eliminate the discontinuity and chattering of the control input. The saturation function sat(S/F), as shown in Figure 8, is used to
c_ Car b2 + Caf a2 ^s_ cos c ^ + ^e_ sin c ^ Iz z (^e_ e_des ) l ½(c_ z) + z(^ e edes ) S + z_ K sat F
+2
where F . 0 is the boundary layer thickness. The control input in equation (27) is the desired steering angle dfdes of the front tyres. Therefore, the corrective steering angle dfc is the difference between the non-linear DSC output and the driver steering input,13 which is given as dfc = dfdes dfd
ð28Þ
As was defined before, dfd is the front steering angle that is demanded by the driver. The undesirable effects of model uncertainties in non-linear control systems can be severe because nonlinear systems are inherently sensitive to model uncertainties. Considering equation (10), we can assume that the uncertainties in the tyres’ cornering stiffnesses Caf and Car are the only source of parametric uncertainties that significantly degrade the DSC performance. By this approach, the uncertainties DCaf and DCar in the tyres’ cornering stiffnesses are added to the cornering stiffness
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terms. If we assume that the uncertainties in the cornering stiffnesses of the front and rear tyres are equal to DCa, after modification of the yaw acceleration, equation (10) can be rewritten as ^ ^ ^ ^ € = 2(Car b Caf a) e_ cos c s_ sin c c ^s_ cos c ^ + ^e_ sin c ^ Iz 2 2 _ c 2(Car b + Caf a ) ^s_ cos c ^ + ^e_ sin c ^ Iz
is set to zero. The following assumptions should be considered to provide the trajectory tracking. Assumption 1. An unknown finite non-negative sliding gain K exists such that K . dmax + h ð33Þ
dmax 5jd(t)j8t ð29Þ
where h is a positive constant. Note that this assumption implies that the system’s parametric uncertainties are bounded magnitudes.
The uncertainty terms for the yaw acceleration are collected together as d(t), which is given by " ^ ^s_ sin c ^ 2(b a) ^e_ cos c d(t) = DCa ^ + ^e_ sin c ^ Iz ^s_ cos c c_ 2(b2 + a2 ) 2a df + ^s_ cos c ð30Þ ^ + ^e_ sin c ^ Iz Iz
Assumption 2. The positive constant b must be chosen so that b5F and b51. The study of the system stability will be carried out using the Lyapunov stability theory. The candidate Lyapunov function19 is defined as
2aCaf + df + d(t) Iz
The adaptive dynamic surface controller was proposed to compensate for the above parametric uncertainties in the tyres’ cornering stiffnesses. The sliding gain K plays an important role in satisfying the sliding condition.16 To address the uncertainties in the system and to meet the sliding condition, an appropriate value for the sliding gain should be chosen. The upper bounds for the parameters and unmodelled dynamic uncertainties should be determined as precisely as possible in order to select the sliding gain accurately. In practice, it is very difficult to evaluate the uncertainty bounds. One solution is to set the sliding gain very high, but this approach will increase the control effort and, as a result, the controller performance will degrade. In order to overcome these difficulties, we propose to determine the sliding gain adaptively; therefore the constant K is replaced with a time-variant parameter, which is updated by an adaptation law. In this approach, the control effort in equation (27) is written as ( ^ ^s_ sin c ^ Iz Car b Caf a ^e_ cos c dfdes = 2 ^s_ cos c ^ + ^e_ sin c ^ IZ 2aCaf c_ Car b2 + Caf a2 ^s_ cos c ^ + ^e_ sin c ^ Iz ^ z (e_ e_des ) l ½(c_ z) + z(^ e edes ) S ^ + z_ K(t)b sat F +2
~ K(t) ~ V(t) = 12S(t)S(t) + 12K(t)
~ =K(t) ^ K. where S(t) is the sliding surface and K(t) The time derivative of the scalar V along the system trajectory is calculated as _ + K(t) _ = S(t)S(t) ~ K(t) ~_ V(t) ~ K(t) ^_ = S(E_ + lE) + K(t) € z_ + z(^e_ e_des ) + l ½(c_ z) + z(^ = Sfc e edes )g S ~ sat + KSb F S ^ sat S + (K^ K)Sb sat = S d(t) Kb F F ^ sat S ^ sat S + KSb = ½d(t)S KSb F F S KSb sat F S 4jd(t)jjSj KSb sat F S 4jd(t)jjSj (dmax + h)Sb sat F ( 2 jSj if jSj 4 F = jd(t)jjSj (dmax + h)b F jSj if jSj . F
jSj 4 hb ð31Þ
^ is the where l is the constant defined previously, K(t) estimated time-variant sliding gain and b is a positive constant. The adaptation law for updating the sliding gain K^ is defined as _K^ = Sb sat S ð32Þ F where b, as the positive constant, determines the adaption speed for the sliding gain and the initial condition
ð34Þ
F
jSj
ð35Þ if jSj 4 F if jSj . F
ð36Þ
Therefore _ 40 V(t)
ð37Þ
To obtain the inequality (36) as the sliding condition, the two above-mentioned assumptions as well as equations (18), (19), (23), (31), (32) and (26) were used. _ The function V(t)is positive definite and V(t)is negative semidefinite. Moreover, V(t) tends to infinity as ~ tend to infinity; therefore, according to S(t) and K(t) the Lyapunov direct method, the equilibrium at the
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~ = ½0, 0 is globally stable and the variorigin ½S(t), K(t) ~ ables S(t), K(t)and E(t) are bounded. It can be shown that each term in equation (35) is _ bounded; henceS(t)is bounded. One method for deter_ mining the uniform continuity of V(t)is to examine its derivative. From equation (34) it is deduced that d S _ € S(t) sat V(t) = dS Kb ð38Þ dt F _ is bounded. which is a bounded quantity because S(t) € _ V is bounded and V is a uniformly continuous function; therefore, from the Barbalat lemma,16 we conclude that V_ ! 0 as t ! ‘, which implies that, as t ! ‘, then S(t) ! 0 and E(t) ! 0 (i.e. the tracking error tends to zero as the time t tends to infinity). In addition, by satisfying the sliding condition, all trajectories starting off the sliding surface S = 0 will reach this surface in finite time and after that will remain on it. Steering control system design. Because of the inertia and viscous damping of the steering mechanism and the effects of the Coulomb friction and self-aligning moment of the front tyres, another adaptive dynamic surface controller, as the lower-level controller, was designed to determine the steering actuator torque tM, and so the steering system tracks the desired steering angle dfdes as the upper controller output (Figure 4). The tracking error E1 is defined as the difference between the actual and desired steering angles according to E1 = df dfdes
ð45Þ
The system dynamics while in the sliding mode S_ 1 = 0 can be defined as _ =0 S_ 1 = €df q€ + l1 (d_ f q)
ð46Þ
By substituting €df from equation (6) into equation (46), the adaptive control input can be obtained as Jw bw _ 1 1 _ tM = tf + t a + q€ l1 (d_ f q) df + Jw Jw rs Jw S1 ^ K1 (t)b1 sat ð47Þ F1 where b1 is a positive constant and K^1 (t), as the adaptive sliding gain, can be updated through the adaptation law in equation (32). Moreover, the aligning moment and the resisting torque of the front tyres can be obtained from equation(7) and equation (8) respectively. The study of the stability of the steering control system can be carried out using the same Lyapunov function (34) as was used for the upper controller.
Simulation A 1997 Jeep Cherokee (Figure 9) was chosen for the simulation. Its parameters, given in Table 1, were published by the Vehicle Research and Test Center (VRTC) of the US National Highway Traffic Safety Administration.20
Verification ð40Þ
where l1 is a strictly positive constant. Using the DSC scheme, dfdes was passed through a first-order filter so that an explosion of terms was avoided to give y df q_ + q = dfdes
1 1 (dfdes q) 2 q_ 2 y df y df
ð39Þ
and afterwards the sliding surface can be written as S1 = E_ 1 + l1 E1
q€ =
ð41Þ
The non-linear three-degree-of-freedom model (1), with the non-linear tyre model (2), was verified by the 1997 Jeep Cherokee, which was provided by the VRTC.21. The manoeuvre used for the model verification was the J-turn test. According to the test conditions,21 a 43.2 km/h initial speed and a step steering input with a maximum handwheel angle of 160°, as shown in Figure 10,
where ydf is a filter time constant and q is an estimate of dfdes and its derivative is easily obtained as Table 1. Vehicle parameters.
1 (dfdes q) q_ = y df
ð42Þ
The sliding surface S1 can be redefined as S1 = d_ f q_ + l1 (df q)
ð43Þ
The derivative of S1 could not be conveniently obtained; therefore, dfdes was passed through a secondorder filter y 2df q€ + 2y df q_ + q = dfdes
and then we have
ð44Þ
Parameter
Symbol Value
Units
Total mass of the vehicle Yaw moment of inertia of the vehicle Distance from the CG to the front axle Distance from the CG to the rear axle Distance from the CG to the ground Track width Cornering stiffness of the front tyres Cornering stiffness of the rear tyres
m IZ a
1988 4513.4 1.15
kg kg m2 m
b
1.43
m
hcg T Caf Car
0.67 1.47 59,496 109,400
m m N/rad N/rad
CG: centre of gravity.
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Figure 9. 1997 Jeep Cherokee test vehicle. Figure 12. Comparison between the yaw rate in the simulation and the yaw rate in the field test.21
Table 2. Tyre parameters. Symbol
Value
Units
G L C2f C2r C1f C1r
1.3 23 5405 4346 59,496 109,400
— — N N N/rad N/rad
Table 3. Steering-system parameters. Figure 10. Handwheel input in the simulation and in the field test.21
Figure 11. Comparison between the lateral acceleration in the simulation and the lateral acceleration in the field test.21
Parameter
Symbol
Value
Units
Total moment of inertia Viscous damping Steering ratio Pneumatic trail Mechanical trail
Jw bw rs tp tm
10 200 30 0.012 0.010
N m s2/rad N m s/rad — m m
Moreover, as can be demonstrated from Figure 11 and Figure 12, the timing and peak levels of the lateral acceleration and the yaw rate for the simulation and field-test data are close. Since the lateral acceleration and the yaw rate of the simulation model were verified by field-test data, this model can be viewed as a full non-linear model which accurately represents the real Jeep Cherokee vehicle.
Results and analysis were applied to the simulation model. It is noteworthy that, although the vehicle speed is not high in comparison with severe handling manoeuvres, the large value of the steering input (Figure 10) compensates for the low vehicle speed. This can be evidently perceived from the high values of the lateral acceleration (Figure 11) and the yaw rate (Figure 12) when the vehicle response is in the non-linear region.
In order to confirm the effectiveness of the proposed ADSC for improving the vehicle-handling and pathtracking characteristics and the DSC performance, a computer simulation of the vehicle responses to the J-turn and double lane-change (DLC) manoeuvres was carried out. The verified vehicle and tyre models with parameters as given in Table 1 and Table 2 were used for the simulation, as also were the steering-system
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Table 4. Driver model parameters. Parameter
Symbol
Value
Units
Driver’s proportional action to the lateral error Driver’s proportional action to the yaw angle Dead time due to the muscular system Delay due to the muscular system Mental workload Look-ahead
Ke Kc t T1 TLe + TLc La
6.25 15 0.2 0.2 1 14
— — s s s m
Figure 13. Steering input for a J-turn manoeuvre.
Figure 14. Comparison of the actual lateral velocity with the estimated lateral velocity.
parameters and driver model parameters, which are shown in Table 3 and Table 4 respectively. To analyse the performance of the yaw stability controller individually, the gain z was set to zero, and it was assumed that the driver control model was omitted; then a severe J-turn manoeuvre on a dry road (m = 0.9) was chosen.22 According to the test procedure, the 80 km/h initial speed and the same steering angle (Figure 13) were exactly as considered for the simulation model. As Tomizuka and Tai23 proposed, we can
Figure 15. Comparison of the actual longitudinal velocity with the estimated longitudinal velocity.
assume that the uncertainties in the tyres’ cornering stiffnesses is approximately 30%. Finally, the gains of the SMO were chosen as g = [15, 15, 25, 0]T and g1 = [15, 15, 15, 0]T, and the ADSC gains were selected as l=25, b = 3, F = 0.1, and l1=20, b1 = 3.5, F1 = 0.15, for the upper controller and the lower controller respectively. The evaluation of the SMO was initially considered because of the importance of accurate state estimation for the controllers. In Figure 14 and Figure 15, the fast convergence and high accuracy of the SMO for the estimated velocities are shown. This highlights the ideal performance of the proposed observer. The constant sliding gain in the DSC was chosen as K = 10, whereas in the ADSC the estimated sliding ^ (Figure 16) was less than 3.5 while obtaining gainK(t) nearly the same level of tracking error (Figure 17). The estimated sliding gain starts from zero and then increases to just less than 3, at which point the ADSC can overcome the system uncertainties. Figure 18 shows that using an adapted sliding gain in ADSC reduced the chattering and smoothed the control input. Note that the chattering in the control input causes the steering actuator working life to decrease. The ADSC is insensitive to parametric uncertainties using an adapted sliding gain, and it is able to conquer the uncertainties in the tyres’ cornering stiffnesses with a much smaller gain. Therefore, system chattering due to a high sliding gain in the DSC was eliminated and
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Figure 19. ISO/TR 3888 DLC track.
Figure 16. Estimated sliding gain.
Figure 20. Vehicle path tracking: passive results versus ADSC results on a dry road surface. ADSC: adaptive dynamic surface control.
Figure 17. Tracking error: DSC and ADSC. ADSC: adaptive dynamic surface control; DSC: dynamic surface control.
Figure 21. Driver’s steering effort: passive results versus ADSC results on a dry road surface. ADSC: adaptive dynamic surface control.
Figure 18. Control input: DSC and ADSC. DSC: dynamic surface control; ADSC: adaptive dynamic surface control.
uncertainties did not certainly influence the controller performance.
In the next step, the vehicle-handling performance and the path-tracking performance are evaluated simultaneously, considering the combined control sytem, in which the driver control model is employed. For this purpose, the DLC manoeuvre on a dry road (m = 0.9) and the 80 km/h initial speed were selected for the simulation model. Thereby, the DLC track exactly according to the ISO/TR 3888 standard24 (Figure 19) was applied to the driver model as the reference or desired path. The path tracking in Figure 20 shows that the vehicle without the upper controller (passive) can pass the test with difficulty and it is nearly unstable at the end of the track. Meanwhile, the ADSC assists the driver (Figure 21)
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Figure 22. Lateral acceleration: passive results versus ADSC results on a dry road surface.
Figure 25. Vehicle path tracking: passive results versus ADSC results on a snowy road surface with a side wind.
ADSC: adaptive dynamic surface control.
ADSC: adaptive dynamic surface control.
Figure 23. Yaw rate: passive results versus ADSC results on a dry road surface.
Figure 26. Yaw rate: passive results versus ADSC results on a snowy road surface with a side wind.
ADSC: adaptive dynamic surface control.
ADSC: adaptive dynamic surface control.
Figure 24. Steering actuator torque: passive results versus ADSC results on a dry road surface. ADSC: adaptive dynamic surface control.
in tracking the road centre-line as the desired path and also improves the vehicle-handling performance (Figure 22 and 23). The lower controller supplies an appropriate actuator torque (Figures 24 and 29) for the steering system, and so the front wheels can reach the desired steering angle dfdes determined by the upper controller.
When a vehicle comes out of a tunnel or crosses over a valley, it may experience an unexpected side wind. To investigate the controller’s performance in a more critical situation, the side wind force is applied to the vehicle model as a step input disturbance which starts from zero and increases until it reaches a value of 600 N. The 80 km/h initial speed and the snowy road surface (m = 0.3) are also considered for this situation, and, as the aerodynamic centre is so close to the CG, the yaw moment effect of the side wind force can be neglected.25 It can be revealed from Figure 25 that in this situation the driver model cannot control the vehicle, which is going to leave the road. In the meantime, the ADSC shows its robustness against the side wind as an external disturbance and also guarantees the vehicle handling and stability (Figures 26 and 27) with less steering effort (Figure 28). Note that, according to equation (17), the first priority of the upper controller is to improve the vehicle handling; thus during this severe manoeuvre the controller keeps the vehicle on the slippery road and guarantees its stability. The vehicle equipped with an SBW system and controlled with ADSC showed a significantly improved
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Conclusion
Figure 27. Lateral acceleration: passive results versus ADSC results on a snowy road surface with a side wind. ADSC: adaptive dynamic surface control.
Figure 28. Driver’s steering effort: passive results versus ADSC results on a snowy road surface with a side wind. ADSC: adaptive dynamic surface control.
This paper proposed a new non-linear adaptive dynamic surface sliding control for simultaneous vehicle-handling and path-tracking improvement through an SBW system. The DSC is inherently sensitive to the parametric uncertainties caused by changes in the tyre cornering stiffness. The adaptive control scheme presented here makes the system robust against model uncertainties. In addition, the constant sliding gain in the traditional sliding-mode scheme was replaced with a time-variant value estimated through an adaptation law. Therefore, the necessity to precisely compute the upper bound of the system’s uncertainties was avoided. Furthermore, the adaptation of the sliding gain allowed a much smaller sliding gain value to be used while achieving the same tracking error. Thus, with the proposed design, the control input will be smaller, there will be less chattering and, consequently, the control performance will be improved. Moreover, by means of the DSC method, the explosion-of-terms problem and a resultant complex control law was prevented. Computational simulations demonstrated that the proposed control system can considerably improve vehicle handling and path tracking under uncertainties in the tyre cornering stiffness and against the side wind as an external disturbance. Funding This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. Conflicts of interest The authors declare that there are no conflicts of interest. References
Figure 29. Steering actuator torque: passive results versus ADSC results on a snowy road surface with a side wind. ADSC: adaptive dynamic surface control.
vehicle-handling characteristic as well as a better path-tracking performance than did the passive vehicle. The rise times of the lateral acceleration and the yaw rate were much better, which implies that the SBW system with ADSC has a faster handling response to the driver steering input. During these
1. Ackermann J. Robust control prevents car skidding. 1996 Bode Prize Lecture. IEEE Control Systems Mag 1997; 17: 23–31. 2. Ackermann J. Yaw disturbance attenuation by robust decoupling of car steering. In: 13th IFAC world congress, San Francisco, CA, USA, 30 June–5 July 1996, vol. Q, pp. 1–6. Oxford: Pergamon. 3. Hebden RG, Edwards C and Spurgeon SK. An application of sliding mode control to vehicle steering in a splitmu maneuver. In: American control conference, Denver, CO, USA, 4–6 June 2003, vol. 5, pp. 4359–4364. New York: IEEE. 4. Yih P. Steer-by-wire: implication for vehicle handling and safety. PhD Dissertation, Department of Mechanical Engineering, Stanford University, Stanford, CA, USA, January 2005.
Janbakhsh et al. 5. Girard AR and Hedrick JK. Formation control of multiple vehicles using dynamic surface control and hybrid systems. Int J Control 2003; 76(9–10): 913–923. 6. Kazemi R and Zaviyeh KJ. Development of a new ABS for passenger cars using dynamic surface control method. In: American control conference, Arlington, VA, USA, 25–27 June 2001, vol. 2, pp. 677–683. New York: IEEE. 7. Davila J, Fridman L and Levant A. Second-order sliding mode observer for mechanical systems. IEEE Trans Autom Control 2005; 50: 1785–1789. 8. Rossetter EJ. A potential field framework for active vehicle lanekeeping assistance. PhD Dissertation, Department of Mechanical Enginering, Stanford University, Stanford, CA, USA, December 2003. 9. Pacejka HB. Tyre and vehicle dynamics, 2nd edition. Oxford: Butterworth–Heinemann, 2002. 10. Perruquetti W and Barbot JP. Sliding mode control in engineering. New York: Marcel Dekker, 2002. 11. Fridman L, Shtessel Y, Edwards C and Yan XG. Higherorder sliding-mode observer for state estimation and input reconstruction in nonlinear systems. Int J Robust Nonlinear Control 2008; 18: 399–412. 12. Levant A. Sliding order and sliding accuracy in sliding mode control. Int J Control 1993; 58: 1247–1263. 13. Rajamani R. Vehicle dynamics and control. New York: Springer, 2006. 14. Apel A and Mitschke M. Adjusting vehicle characteristics by means of driver models. Int J Veh Des 1997; 18(6): 583–596. 15. Horiuchi S and Yuhara N. An analytical approach to the prediction of handling qualities of vehicles with advanced steering control system using multi-input driver model. J Dynamic Systems, Measmt Control 2000; 122: 490–497. 16. Slotine JE and Li W. Applied nonlinear control. Englewood Cliffs, NJ, Prentice-Hall, 1991. 17. Hamzah N B. Antilock braking control using robust control approach. MS Thesis, Department of Mechanical Engineering, Universiti Teknologi Malaysia, Johor, Malaysia, November 2006. 18. Swaroop D, Hedrick JK, Yip PP and Gerdes JC. Dynamic surface control of nonlinear systems. In: American control conference, Albuquerque, NM, 4–6 June 1997, Vol 5, pp. 3028–3034. New York: IEEE. 19. Qiao F, Zhu Q, Winfield AFT and Melhuish C. Adaptive sliding mode control for MIMO nonlinear systems based on a fuzzy logic scheme. Int J Automot Comput 2004; 1: 51–62. 20. Chen BC and Peng H. Differential-braking-based rollover prevention for sport utility vehicles with humanin-the-loop evaluations. Veh System Dynamics 2001; 36(4–5): 359–389. 21. Kamel Salaani M and Heydinger GJ. Model validation of the 1997 Jeep Cherokee for the National Advanced Driving Simulator. SAE paper 2000-01-0700, 2000. 22. A comprehensive evaluation of test maneuver that may include on-road, untripped, light vehicle rollover. Technical Report DOT HS 809 513, US Department of Transportation, National Highway Traffic Safety Administration, Washington, DC, USA, October 2002. 23. Tomizuka M and Tai M. Robust lateral control of heavy duty vehicles: final report. California PATH Research Report, California Partners for Advanced Transit and Highways, Institute of Transportation Studies, University of California Berkeley, CA, USA, July 2003.
359 24. Klumparend, L. Variation von Fahrverhaltensparametern unterbesonderer Beru¨cksichigung des Lenkradmoments. Diplomarbeit, Fachbereich Maschinenbau, Fachhochschule Aalen, Aalen, Germany, August 1999. 25. Technical documentation/J411-CFD specification sheet: O-I-LP-1068. Confidential Technical Report, Computeraided Engineering Department, Advanced Vehicle Systems Research Center, 2008.
Appendix Notation a b bw Caf Car C1f, C1r, C2f, C2r e E Fy Fz g G hcg Iz Jw K Ke Kus K^ Kc L La m rs sat sgn s S t tm tp T TLe + TLc T1 ux uy U Ueq
distance from the centre of gravity to the front axle distance from the centre of gravity to the rear axle viscous damping cornering stiffness of the front tyre cornering stiffness of the rear tyre tyre parameters lateral deviation of the centre of gravity from the lane centre tracking error lateral tyre force normal tyre force acceleration due to gravity tyre parameter distance from the centre of gravity to the ground yaw moment of inertia of the vehicle total moment of inertia sliding gain driver’s proportional action to the lateral error understeer gradient adaptive sliding gain driver’s proportional action to the yaw angle tyre parameter look-ahead total mass of the vehicle steering ratio saturation function sign function longitudinal deviation of the centre of gravity from the lane centre sliding surface time mechanical trail pneumatic trail track width mental workload delay due to the muscular system longitudinal velocity in the vehicle-fixed frame lateral velocity in the vehicle-fixed frame control input equivalent control
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V
Lyapunov stability function
yc_
a b g df dfd z L m t ta tf tM y df
slip angle of the tyre sliding gain adaption speed parameter sliding-mode observer gain matrix front steering angle driver-demanded steering angle weighting factor design parameter of the arctan function road friction coefficient dead time due to the muscular system self-aligning torque Coulomb friction torque steering actuator torque filter time constant for the lower controller
F c
filter time constant for the upper controller boundary layer thickness angular difference between the direction of the vehicle and the direction of the lane centre
Abbreviations ADSC CG DBW DSC SBW SMO VRTC
adaptive dynamic surface control centre of gravity drive-by-wire dynamic surface control steer-by-wire sliding-mode observer Vehicle Research and Test Center
