`Mathematical Problems in EngineeringVolume 2012, Article ID 287195, 31 pageshttp://dx.doi.org/10.1155/2012/287195`
Research Article

## Vehicle Sliding Mode Control with Adaptive Upper Bounds: Static versus Dynamic Allocation to Saturated Tire Forces

Department of Mechanical Engineering, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran

Received 20 December 2011; Accepted 13 February 2012

Copyright © 2012 Ali Tavasoli and Mahyar Naraghi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Nonlinear vehicle control allocation is achieved through distributing the task of vehicle control among individual tire forces, which are constrained to nonlinear saturation conditions. A high-level sliding mode control with adaptive upper bounds is considered to assess the body yaw moment and lateral force for the vehicle motion. The proposed controller only requires the online adaptation of control gains without acquiring the knowledge of upper bounds on system uncertainties. Static and dynamic control allocation approaches have been formulated to distribute high-level control objectives among the system inputs. For static control allocation, the interior-point method is applied to solve the formulated nonlinear optimization problem. Based on the dynamic control allocation method, a dynamic update law is derived to allocate vehicle control to tire forces. The allocated tire forces are fed into a low-level control module, where the applied torque and active steering angle at each wheel are determined through a slip-ratio controller and an inverse tire model. Computer simulations are used to prove the significant effects of the proposed control allocation methods on improving the stability and handling performance. The advantages and limitations of each method have been discussed, and conclusions have been derived.

#### 1. Introduction

In recent years by rapid emergence of electronic control devices, employing all available actuators, or individual tire forces, for ground vehicle control has become possible [1]. The vehicle motion is governed by tire forces which are constrained based on friction circle notion. Thus, tire saturation constraints must be taken into account for a proper control design. The problem of optimal actuators selection to execute a control task, while minimizing effort and satisfying constraints, is known as Control Allocation (CA). To tackle actuator constraints in control design of a general over-actuated nonlinear system, two approaches of Static Control Allocation (SCA) and Dynamic Control Allocation (DCA) are proposed. In SCA, the total body forces/moments of a high-level controller are allocated to available actuators by optimizing a suitable cost function at each sampling time, whereas DCA generates a dynamic update law for actuators. No optimization problem needs to be solved by DCA.

In the field of Integrated Vehicle Dynamics Control (IVDC), Optimal Distribution of tire Forces (ODF) was introduced to meet various objectives, such as maximizing longitudinal acceleration [2], minimizing total tire workload usage [3], or adaptive-optimal coordination of braking and steering [4]. Attempts to account for actuator constraints were rare in these works.

To allocate control objectives to the actuators with limited amplitude/rate, SCA methods have been subject of different studies. Algorithms using various optimization methods, such as quadratic programming [5], multiparametric nonlinear programming [6], and fixed point [1], were developed. Kou [7] proposed an SCA scheme based on the model predictive control and fixed point method. The main problem in static control allocation is its computational burden for practical applications, due to numerical solution of a constrained optimization problem at each sampling instant.

To deal with this difficulty, Johansen [8] developed a dynamic control allocation method for a particular class of nonlinear systems. In this regard, a dynamic update law leads the desired actuator efforts to converge to the solution of a definite optimization problem, without solving the optimization problem. DCA was extended for systems with unknown parameters [9] and applied to vehicle yaw control [10].

This paper addresses nonlinear vehicle CA constrained to tire saturation conditions. Referring to schematic view of IVDC structure in Figure 1, the required body lateral force and yaw moment for vehicle motion are determined by a high-level sliding mode enhanced adaptive controller. The adaptive control methodology is utilized to update the perturbation and sliding mode control gains, so that the upper bounds of uncertainties are not required to be known in advance. Then, the body force and moment, along with the driver’s desired braking force, are allocated to individual lateral and longitudinal forces of each tire. Considering tires saturation induces nonlinear constraints in CA problem. To tackle this problem, we look into SCA and DCA methods for vehicle control. For static allocation purpose, the interior point is formulated and employed to solve the nonlinear inequality constrained optimization problem. To formulate dynamic allocation control for the proposed problem, a dynamic update law is derived and utilized in the IVDC scheme. The desired lateral force of each tire by CA modules is mapped into the corresponding active steering angle through an inverse tire model. In addition, the desired longitudinal forces are tracked by a low-level slip-ratio control scheme. Simulation results are conducted to evaluate the effectiveness of each method.

Figure 1: Overall structure of the integrated vehicle dynamics control scheme.

The rest of the paper is organized as follows. The high-level control is described next. Section 3 presents the formulation of optimal distribution of tire forces in IVDC considering nonlinear tire saturation constraints. In Section 4, the SCA is formulated based on the interior-point method. In Section 5, the DCA approach is utilized to derive an optimized dynamic update law for individual tire forces. The low-level slip-ratio controller has been addressed in Section 6. Sections 7 and 8 are devoted to simulation results and concluding remarks.

#### 2. High-Level Sliding Mode Control with Adaptive Upper Bounds

In this section we first design a high-level controller for vehicle handling and stability based on the conventional Sliding Mode Control (SMC), then an SMC with updated upper bounds of uncertainties is considered.

##### 2.1. Conventional Sliding Mode Control

In general, vehicle handling and stability are achieved through the control of yaw rate and side-slip angle, respectively. The design procedure is based on the 2DoF vehicle model, where the basic equations are [11] where and denote the total mass and yaw moment of inertia, from which only the estimates of and are available, and is the vehicle velocity. and stand for the actual vehicle side-slip angle and the yaw rate, respectively. and are sum of external moments in the yaw direction and lateral forces acting on the vehicle, respectively. To account for the unmodelled dynamics and uncertainties in modelling the actual nonlinear vehicle dynamics, the unknown, but bounded, disturbance terms, and , are embedded into each channel to get To design the total lateral force (), for a zero desired side-slip angle, the sliding surface,, is selected as Differentiating this equation and considering (2.2), where the term is assumed to be bounded by a known value : To guarantee the sliding condition [12] the desired body lateral force is considered as in which is our estimate of and is to be designed. Insert (2.8) into (2.5) to get the left side of (2.7) as where the mass estimation error is assumed to satisfy whose combination with (2.9) and (2.6) results in To achieve (2.7), is considered to be where is the signum function, and By substituting (2.12) into (2.8), the desired body lateral force is attained. To mitigate the problem of chattering, the sign function is replaced by saturation function with a boundary layer thickness of . Thus, the final control law becomes In order to design the body yaw moment (), for tracking the desired yaw rate (), the sliding surface,, is adopted as where the integral term is used to mitigate the undesirable yaw angle offset and to ensure the desired vehicle heading. Differentiating (2.15) along with (2.3) leads to The design process of the desired body yaw moment is similar to that of (2.14): where is an estimate of and with and being the upper bounds for and , respectively.

##### 2.2. Sliding Mode Control with Adaptive Upper Bounds

From (2.13) and (2.18) it can be observed that the selection of the SMC gains and depends on upper bounds of uncertainties in vehicle dynamics and body mass and inertia, that is, ,, , and . In practice, uncertainties and disturbances depend primarily on the highly nonlinear dynamics of vehicle and tire which are not completely known, and one cannot determine their exact bounds too. Therefore, no universal method is available yet to tune the controller gains and these gains should be tuned by trial and error approach in practical implementations. In this regard, the controller tends to be overconservative, which may induce poor tracking performance as well as undesirable oscillations in control signal. To overcome this drawback, adaptive control methodology with control parameters updated online is a promising approach. In this section, we use the adaptive control technique to attain a sliding mode controller with adaptive upper bounds. To design sliding mode controls with variable gains, the following modified control laws are established: where and the varying controller gains are updated as follows: Assume that there are positive constants,, , and that satisfy It should be noted that we need only to assure that such constants exist without acquiring the knowledge of these upper bounds to use in control laws. Also, consider Then, the stability of the considered adaptive-sliding mode control laws can be shown through Lyapunov candidates: To prove the stability of side-slip angle under (2.19) with adaptation laws (2.22) and (2.23), first insert (2.19) into (2.5) so that Then, differentiate (2.30) to get Replacing (2.32) in (2.33) and considering (2.28), we have that Using adaptation laws (2.22) and (2.23) in (2.34) and considering (2.26) results in where we use . Accordingly, the convergence of to zero and also boundedness of and are resulted by Barbalat’s lemma [12].

In an identical way, the stability of the yaw motion can be demonstrated, first, by combining (2.16) and (2.20) so that Differentiate (2.31) and replace (2.36) for to get Using adaptation laws (2.24) and (2.25) as well as the inequalities in (2.27), we have that Thus, according to the Barbalat’s Lemma, the system state can be driven to the sliding surface and the controller gains and will be bounded. Furthermore, to tackle the chattering problem saturation function is used to derive the final adaptive control laws

#### 3. Control Allocation in Vehicle System

The total body lateral force and yaw moment, as well as the braking acceleration command by driver, are generated by longitudinal and lateral forces of each tire. In this paper, a 4-wheel vehicle system with each wheel being braked/derived and steered independently is considered. Such a full tire-actuated vehicle can be available through X-by-wire systems [1, 13]. Thus, the overall control system contains eight actuators and only three control objectives, raising an overactuated control system. A general approach to resolve redundancy is to optimize a cost function for specific performance. The well-accepted cost function in IVDC is the sum of work load of four wheels: where denotes wheel number, and stand for desired values of longitudinal force, , and lateral force, , is the vertical load, all defined in the vehicle body fixed coordinate system, as shown in Figure 2, is the weighting coefficient and is the friction coefficient, of the th tire. Defining the 8 × 1 actuator vector of as the cost function is written in matrix form in which is a diagonal weighting matrix. To get and , by assuming small steering angles in Figure 2, individual tire forces must satisfy the equality constraints: Also, the longitudinal acceleration, , by driver is generated by longitudinal forces Equations (3.4)–(3.6) can be expressed in linear matrix form as where is a constant matrix and the vector of generalized forces/moment, , is given by On the other hand, the resultant force of each tire is constrained to friction circle Using (3.2), (3.9) can be written as or where is a 4 × 1 vector with the th component The optimization problem is summarized as follows:

Figure 2: Schematic view of forces acting on the vehicle.

#### 4. Static Control Allocation: Application of Interior-Point Method

SCA solves the optimization problem (3.13) at each sampling instant. A powerful approach for nonlinear programming is the set of interior-point (IP) methods. A benefit of such methods is that the distance from the optimum is always known, so that one can terminate the optimization algorithm when the solution reaches within desired tolerance. Convergence is also uniform toward the optimal solution. The description of IP algorithm in this paper is an excerpt from [14, 15].

##### 4.1. Karush-Kuhn-Tucker Conditions

At the outset to avoid infeasible solutions, the equality constraints are embedded in cost function to get which is written in the quadratic form where IP methods use barrier logarithmic functions to satisfy the inequality constraints. In this regard, the optimization problem turns into Applying KKT formulation [16] to (4.4), the optimality conditions can be expressed as follows, where is the vector of Lagrange multipliers and is a 4 × 4 diagonal matrix whose diagonal elements are the components of the vector .

##### 4.2. Primal-Dual Path-Following Method Steps

The Primal-dual path-following IP method steps for IVDC are as follows:

(1) Newton’s Step Direction
In each iteration, the step direction is obtained by applying Newton’s method to KKT conditions (4.5): where the residuals , , and are obtained as Equation (4.6) is solved to achieve the step directions

(2) Step Length Calculation
To satisfy nonnegativity condition , the new iteration is calculated as where

(3) Updating the Barrier Parameter
The sequence of barrier parameters converges to zero by the update law where to update the centring parameter, first, the predictor (affine scaling) direction and the corresponding longest step lengths and are calculated by setting in (4.9) and (4.11). Then the value of complementarity along the affine scaling step is defined to be The centring parameter is updated as follows:

(4) Stopping Criteria
The algorithm is terminated, when the following error function with has converged sufficiently close to zero All of these steps have been summarized in Pseudocode 1. To reduce the time of control allocation, the last allocated is chosen as the initial value of the control input . Then, is modified (reduced) to satisfy tire saturation constraints.

Pseudocode 1: Pseudocode for interior-point algorithm.

#### 5. Dynamic Control Allocation

DCA uses an optimizing update law for system inputs. Since it is not required to solve the optimization problem at each sampling time, the main advantage of DCA is its computational efficiency. In what follows, the procedure explained in [8] is applied to the proposed integrated vehicle control scheme. However, the interested reader might refer to [8, 9] for convergence study and detailed analysis.

##### 5.1. Dynamic Control Allocation Applied to IVDC

DCA is formulated by introducing the following Lagrangian based on (3.13): with and being a 3-component vector of Lagrange multipliers. DCA updates and in the form of the Newton-like update law where and , and for our problem is written as In (5.2) the feedforward-like terms and are chosen so that the following scalar algebraic equation holds: with

Assume that the stability of the high-level control could be shown through a Lyapunov function, then, by the Lyapunov function global exponential convergence to optimality conditions is achieved.

##### 5.2. Discussion and Modification

Consider the equality-constrained optimization problem with the Lagrangian (5.1). Then the optimizing conditions are Applying Newton’s conditions to (5.7) results in optimizing Newton’s steps and or By setting, ignoring the terms and , and considering the symmetry of , it can be demonstrated that the update law (5.2) represents the solution of and in (5.9), that is, one Newton’s step towards the optimum point. Therefore, (5.9) can be interpreted as one Newton’s step from the solution of the control allocation problem at the current sampling time for the solution at the next instant, for which the terms and are to compensate time-varying optimum solution.

Despite the log-barrier term in (5.1), depending on the values of and, there is no guarantee that the Newton-like step (5.2) will satisfy the inequality constraints (3.9). As the th inequality constraint is infringed, becomes negative and) and correspondingly the Lagrangian (5.1) turn meaningless. To tackle this problem, in this paper, a line search is adopted for the coefficient , so that the resulting and at the next sampling time satisfy the inequality constraints and. The idea of the line search approach is taken from the second step of the interior-point stages, described in the previous section.

Another problem with DCA is that (5.2) could induce a Newton’s step from an infeasible point, violating the inequality constraints (3.9), because of time-varying nature of these constraints. This arises because the term in (3.9) is time varying. Consequently, it is possible for the feasible solution at the current time to violate the inequality constraints of the next sample time. In such conditions, (5.2) needs to be modified for Newton’s steps from infeasible points [15].

After desired longitudinal and lateral forces of each tire are computed by either of the proposed control allocation algorithms, the active steering angle, , at wheel can be determined as follows: where is the longitudinal velocity of the th tire and is the side-slip angle of the th tire and is obtained using the inverse of a simple tire model as in which denotes the cornering stiffness of the th tire and s are chosen to approximate the saturation conditions. Longitudinal forces are fed into the low-level control unit.

#### 6. Low-Level Slip-Ratio Control Design

The longitudinal force of each tire is related to the corresponding longitudinal slip ratio and is adjusted through slip-ratio control (SRC). The slip ratio of the th tire,, is defined as where denotes the radius and is the angular velocity of the th wheel. In the case where longitudinal slip ratio is small, the longitudinal tire force is found to be proportional to the slip ratio. Then, it gains its maximum value at a typical value of , after which it starts to lessen. Experimental studies have established that the tire lateral force decreases with increasing slip ratios greater than as well [17].

##### 6.1. Description of the SRC Scheme

In this paper, when the tire slip ratio is smaller than , by neglecting the wheel rotational inertia [3], the applied braking/traction torque, , at wheel is obtained as In this case, the SRC works for Desired Longitudinal Force Generation (DLFG). However, when the demanded is too high, applying (6.3) would increase the slip ratio beyond , inevitably leading to wheel lock and lateral tire force drop. In such conditions, the idea of Antilock Braking System (ABS) is employed to keep the slip ratio of tires at . This idea is utilized during both braking and traction. When traction torque applies, the proposed slip-ratio control is in the Traction Control System (TCS) mode. The SRC scheme is shown in Figure 3.

Figure 3: The SRC scheme applied to the vehicle system.
##### 6.2. The ABS/TCS Design

The th wheel rotational dynamics is stated as where is wheel rotational inertia. To achieve slip-ratio differential equation when braking, by differentiating (6.1) and replacing for from (6.4) During decelerating, the longitudinal force can be stated as [17] in which the term represents the linear part, with being the tire longitudinal stiffness, and is the deviation from the linear part and is bounded by : The goal of the ABS is to regulate around the constant value. Applying the sliding mode control design procedure to (6.5), an ABS is designed so that to get the sliding condition the applied braking torque at the th wheel is Furthermore, the applied torque during acceleration, by TCS, is computed as

#### 7. Simulation Results

The considered methods are tested in several critical maneuvers. A 9DoF nonlinear vehicle model is used for simulation purpose [4]. The vehicle behavior is probed during an open-loop maneuver, with no driver model, and two closed-loop maneuvers, including the driver model validated experimentally in [18]. The overall control scheme for simulation is shown in Figure 4. Comparison is made with a well-recognized case [3], where tire saturation conditions are ignored in ODF.

Figure 4: Overall closed loop scheme of the proposed IVDC.

The simulations are performed by MATLAB/Simulink. For the sake of practical implementations, the time for each control allocation was computed. Using a PC based on a 2.6 GHz Intel Core 2 Duo T9500 processor, the maximum time for the total IP algorithm is about 0.002 seconds. This time can be still improved using efficient C-Programming and effective numerical techniques.

##### 7.1. Open-Loop Cornering Maneuver on a Split-𝜇 Road

The vehicle behaviour is examined in an open-loop split- manoeuvre, where the tire-road friction coefficients on the vehicle left and right are equal to 0.3 and 1, respectively. The vehicle is assumed to move with a velocity of 110 km/hr and to turn according to the steering angle in Figure 5. The deceleration demand is −0.1 g. In Figures 6 and 7, both Saturation Constrained ODF (SCODF) approaches, that is, SCA and DCA, have better yaw rate tracking and side-slip angle reduction. Using ODF, the allocated work load to the front-right wheel is substantially beyond its actual capacity, other tires have still margin to saturation, as Figure 8 shows. On the other hand, from Figures 9 and 10 it is seen that SCODF methods have balanced the workload distribution and managed to use the total capacity of tire forces by saturating all tires in critical conditions. Accordingly, as shown in Figures 11 and 12, these methods have generated larger lateral and longitudinal forces, compared to ODF. To compare SCA and DCA, it is noted (Figure 7) that SCA has less side slip at some points relative to DCA. This arises because DCA has failed to completely saturate the tire on rear left in this case. Furthermore, Figures 11 and 12 reveal that lateral and longitudinal forces by DCA have properly converged to the optimal solution of SCA, with minor errors in some cases.

Figure 5: Driver steering command in split- scenario.
Figure 6: Yaw angle velocity in split- scenario.
Figure 7: Side-slip angle in split- scenario.
Figure 8: Tire work-load distribution by ODF in split- scenario.
Figure 9: Tire work-load distribution by SCA in split- scenario.
Figure 10: Tire work-load distribution by DCA in split- scenario.
Figure 11: Actually generated tire lateral forces in split- scenario.
Figure 12: Actually generated tire longitudinal forces in split- scenario.
##### 7.2. Close-Loop Single-Lane Change Maneuver

A closed-loop high-speed Single-Lane Changing (SLC) on a slippery road with the braking acceleration of −0.5 g is considered. The vehicle initial speed is 120 km/h and the tire-road friction coefficient is set at 0.5. By Figures 13, 14, 15, 16, 17, and 18, enhanced vehicle control in terms of vehicle trajectory, yaw rate, and side-slip angle, as well as balanced work load distribution by SCODF approaches are evident. Figure 19 illustrates that the generated tire lateral forces by ODF are smaller than SCODF approaches. For the tires at the vehicle rear, this is primarily due to the smaller work load assigned to these tires by ODF. However, reduction of the lateral forces at the front tires can be described referring to longitudinal forces and slip-ratio plots in Figures 20 and 21. The ODF method has designated larger braking forces to front tires at some points, compared with SCODF approaches. According to friction circle concept, this reduces the corresponding generated tire lateral forces by ODF. Accordingly, as Figure 21 demonstrates, higher longitudinal forces by ODF have increased the slip ratio of these wheels by . Then, ABS is activated by SRC to regulate the front slip ratios at . On the other hand, SCA and DCA moderate the role of longitudinal forces in vehicle control and leave out the use of ABS.

Figure 13: Vehicle path in SLC.
Figure 14: Yaw rate in SLC.
Figure 15: Side-slip angle in SLC.
Figure 16: Work-load distribution by ODF in SLC.
Figure 17: Tire work-load distribution by SCA in SLC.
Figure 18: Tire work-load distribution by DCA in SLC.
Figure 19: Actually generated tire lateral forces in SLC.
Figure 20: Actually generated tire longitudinal forces in SLC.
Figure 21: Tire longitudinal slip-ratios in SLC.
##### 7.3. Close-Loop Double-Lane Change Maneuver

Another case to evaluate the system is high-speed Double-Lane Changing (DLC) on a slippery road with driver’s braking. The maneuver conditions remain unchanged with respect to SLC. In Figures 22, 23, and 24, SCODF approaches are still superior in controlling the vehicle path, yaw rate, and side-slip angle.

Figure 22: Vehicle path in DLC with .
Figure 23: Yaw rate in DLC with .
Figure 24: Side-slip angle in DLC with .

To increase the challenge for the control systems, DLC is repeated on a more slippery road, where the tire-road friction coefficient is 0.4. Figures 25, 26, and 27 show the results in this case. Although both SCA and DCA methods have almost the same performance of vehicle control, oscillatory yaw rate response as well as more side-slip angle by DCA are clear. These are mainly due to transient response of DCA, specifically to panic reactions of driver in both steering and braking in critical conditions. The transient response is an inherent characteristic of the dynamic update law in DCA. In this scenario, the transient response has been triggered when the DCA tries to compensate for sudden braking release demand by driver. Another reason for oscillatory response by DCA is its relatively low robustness against highly nonlinear characteristics of vehicle dynamics and tire forces which are more substantial in more critical conditions. The oscillations by DCA grow more considerable once sudden changes occur in tire/road friction coefficient, for example, when the vehicle suddenly enters an icy road. Oscillations can intensify also after an actuator fails in a specific manoeuvre, where DCA is used for actuator failure compensation, which is one of the main advantages of control allocation methods [1]. In general, oscillatory response by DCA increases as the manoeuvre conditions worsen.

Figure 25: Vehicle path in DLC with .
Figure 26: Yaw rate in DLC with .
Figure 27: Side-slip angle in DLC with .

#### 8. Conclusion

Control allocation techniques can be applied effectively to enhance vehicle stability. Nonlinear vehicle motion control is split into three tasks. First, the total body lateral force and yaw moment were computed through a high-level sliding mode controller with adaptive upper bounds, where the knowledge of the upper bounds of uncertainties is not required. To allocate the vehicle high-level control objectives to saturation-constrained tire forces, static and dynamic control allocation techniques were formulated. Interior-point method was used to handle the nonlinear constrained optimization problem in SCA; a dynamic update law was derived using DCA. Simulation results were conducted, and vehicle operation under each method was evaluated and compared with a well-recognized work in the literature. It was seen that both SCA and DCA approaches manage to utilize whole capacity of tire forces more properly by offering balanced work-load distribution and rational adjustment of longitudinal and lateral tire forces. Since no optimization problem is required in DCA, the main advantage of this method is its computational efficiency for real-time implementations. Furthermore, although by DCA the set of distributed tire forces properly approaches the optimal one, that is, that of SCA, there is yet an amount of error, depending on the severity of maneuver. This error of DCA is mainly due to its transient response to panic behavior of driver and its relatively low robustness to highly nonlinear characteristics of vehicle dynamics and tire forces. Therefore, in maneuvers with more critical conditions, the performance of the vehicle control task can be degraded using DCA compared to SCA.

#### Nomenclature

 𝑎𝑥: Driver’s braking acceleration 𝐴𝑖: Weighting coefficients of 𝑖th tire 𝐶𝑖: Cornering stiffness of 𝑖th tire 𝐂𝐈: Vector of saturation constraints 𝐶𝜎: Tire longitudinal stiffness 𝑑: Vehicle tread 𝑓: Objective function 𝐹𝑥𝑖: Actual longitudinal force of 𝑖th tire 𝐅𝑥𝑖: Maximum deviation from linear part for 𝐹𝑥𝑖 𝐹𝑦𝑖: Actual lateral force of 𝑖th tire 𝐼𝑤: Moment of inertia of each wheel 𝐼𝑧: Yaw moment of inertia of vehicle 𝐼𝑧: Estimation of yaw moment of inertia 𝐼𝑧: Estimation of yaw moment of inertia error 𝐼𝑧: Upper bound for 𝐼𝑧 𝐽: Objective function in quadratic form 𝑘𝑟, 𝑘𝛽, 𝑘𝜎: Sliding mode control gain ̂𝑘𝛽1, ̂𝑘𝛽2, ̂𝑘𝑟1, ̂𝑘𝑟2: Adaptive gains 𝑘𝑑𝛽1,𝑘𝑑𝛽2, 𝑘𝑑𝑟1, 𝑘𝑑𝑟2: Uncertainty upper bound 𝐿: Lagrangian 𝐿𝑓,𝑟: Distance between mass center and axle 𝑚: Vehicle mass 𝑚: Vehicle mass estimation 𝑚: Mass estimation error 𝑚: Upper bound for 𝑚 𝑀: Body yaw moment 𝑅: Wheel radius 𝑟: Yaw angle velocity 𝑟𝑏,𝑟𝑐,𝑟𝑠: IP residuals 𝑠𝑟: Sliding surface for yaw rate 𝑠𝛽: Sliding surface for side-slip angle 𝑠𝜎: Sliding surface for SRC 𝑇𝑖: Applied torque at the 𝑖th wheel 𝐮: Vector of allocated tire forces 𝑉: Vehicle velocity 𝑣𝑥𝑖: Longitudinal velocity of the 𝑖th tire 𝑉𝛽,𝑉𝑟,𝑉dyn,𝑉0: Lyapunov candidate 𝐯: Vector of virtual control input 𝐖: Matrix of weighting coefficients 𝑋: Body longitudinal force 𝑌: Body lateral force 𝑋𝑖: Allocated longitudinal force to 𝑖th tire 𝑌𝑖: Allocated lateral force to 𝑖th tire 𝑍𝑖: Vertical load of 𝑖th tire.
Greek Letters
 𝛼𝑖: Side-slip angle of 𝑖th tire 𝛼𝑠max,𝛼𝜆max: Newton’s step size 𝛽: Side-slip angle 𝛿𝑖: Steering angle of 𝑖th tire Δ𝛽, Δ𝑟: Uncertainty upper bound 𝜂𝑘: Barrier parameter Φ𝛽, Φ𝑟, 𝜙𝜎: Boundary layer thickness 𝝀: Vector of Lagrange multipliers Λ: Matrix of Lagrange multipliers 𝜎: Centering parameter 𝜎𝑖: Slip ratio of the 𝑖th tire 𝜎∗: Reference value of slip ratio 𝜇𝑖: Friction coefficient of 𝑖th tire 𝜔𝑟, 𝜔𝛽: Disturbance terms 𝜔𝑖: Angular velocity of 𝑖th tire.
Subscripts/Superscripts
 aff: Affine scaling 𝑓: Front 𝑟: Rear 𝑑: Desired value +: N iteration dyn: Dynamic.
Coordinate System
 (𝑥,𝑦,𝑧): Moving coordinate attached to vehicle centre.

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