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

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]𝑚𝑉𝐼𝛽̇+𝑟=𝑌,_𝑧𝑟̇=𝑀,(2.1) where 𝑚 and 𝐼_𝑧 denote the total mass and yaw moment of inertia, from which only the estimates of 𝑚 and 𝐼_𝑧 are available, and 𝑉 is the vehicle velocity. 𝛽 and 𝑟 stand for the actual vehicle side-slip angle and the yaw rate, respectively. 𝑀 and 𝑌 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, 𝜔_𝛽 and 𝜔_𝑟, are embedded into each channel to get𝑚𝑉𝐼𝛽̇+𝑟=𝑌+𝜔_𝛽,(2.2)_𝑧𝑟̇=𝑀+𝜔_𝑟.(2.3) To design the total lateral force (𝑌), for a zero desired side-slip angle, the sliding surface,𝑠_𝛽, is selected as𝑠_𝛽=𝛽.(2.4) Differentiating this equation and considering (2.2),𝑠_𝛽̇=𝑌𝑚𝑉−𝑟+Δ𝑚^𝛽,(2.5) where the term Δ_𝛽=𝜔_𝛽/𝑉 is assumed to be bounded by a known value Δ_𝛽:||Δ_𝛽||<Δ_𝛽.(2.6) To guarantee the sliding condition [12]𝑠_𝛽𝑠_𝛽̇<0,(2.7) the desired body lateral force is considered as𝑌=𝑉𝑚𝑟+𝑣_𝛽(2.8) in which 𝑚 is our estimate of 𝑚 and 𝑣_𝛽 is to be designed. Insert (2.8) into (2.5) to get the left side of (2.7) as𝑠_𝛽𝑠_𝛽̇=𝑠_𝛽𝑚⁻¹Δ_𝛽+𝑣_𝛽+𝑚𝑟,(2.9) where the mass estimation error 𝑚=𝑚−𝑚 is assumed to satisfy||𝑚||=||𝑚−𝑚||<𝑚,𝑚>0,(2.10) whose combination with (2.9) and (2.6) results in𝑠_𝛽𝑠_𝛽̇≤𝑚⁻¹𝑠_𝛽𝑣_𝛽+||Δ_𝛽||||𝑠_𝛽||+||𝑚|||𝑟|||𝑠_𝛽||<𝑚⁻¹𝑠^𝛽𝑣_𝛽+Δ_𝛽||𝑠_𝛽||+𝑚|𝑟|||𝑠_𝛽||.(2.11) To achieve (2.7), 𝑣_𝛽 is considered to be𝑣_𝛽=−𝑘_𝛽sgn𝑠_𝛽,(2.12) where sgn(⋅) is the signum function, and𝑘_𝛽>Δ_𝛽+𝑚|𝑟|+𝜂_𝛽,𝜂_𝛽>0.(2.13) 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 Φ_𝛽>0. Thus, the final control law becomes𝑌=𝑉𝑚𝑟−𝑘_𝛽sat𝑠^𝛽Φ_𝛽.(2.14) In order to design the body yaw moment (𝑀), for tracking the desired yaw rate (𝑟_𝑑), the sliding surface,𝑠_𝑟, is adopted as𝑠_𝑟=𝑟−𝑟_𝑑+𝜆_𝑟^𝑡0𝑟−𝑟_𝑑𝑑𝜏,𝜆_𝑟>0,(2.15) 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 to𝑠_𝑟̇=𝑀𝐼_𝑧+𝜔^𝑟𝐼_𝑧−𝜏,𝜏=𝑟_𝑑̇−𝜆_𝑟𝑟−𝑟_𝑑.(2.16) The design process of the desired body yaw moment is similar to that of (2.14):𝑀=𝐼_𝑧𝑟_𝑑̇−𝜆_𝑟𝑟−𝑟_𝑑−𝑘_𝑟sat𝑠^𝑟Φ_𝑟,Φ^𝑟>0,(2.17) where 𝐼_𝑧 is an estimate of 𝐼_𝑧 and𝑘_𝑟>Δ_𝑟+𝐼_𝑧||𝑟_𝑑̇−𝜆_𝑟𝑟−𝑟_𝑑||+𝜂_𝑟,𝜂_𝑟>0,(2.18) with Δ_𝑟>0 and 𝐼_𝑧>0 being the upper bounds for |𝜔_𝑟| and |𝐼_𝑧|=|𝐼_𝑧−𝐼_𝑧|, 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 𝑘_𝛽 and 𝑘_𝑟 depends on upper bounds of uncertainties in vehicle dynamics and body mass and inertia, that is, Δ_𝑟,Δ_𝛽, 𝑚, and 𝐼_𝑧. 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:𝑌𝑀=𝑉=𝐼𝑚_𝑧𝑟−𝑘̂_𝛽1+𝑘̂_𝛽2|𝑟|sgn𝑠_𝛽,(2.19)𝜏−𝑘̂_𝑟1+𝑘̂_𝑟2|𝜏|sgn𝑠_𝑟,(2.20) where𝜏=𝑟_𝑑̇−𝜆_𝑟𝑟−𝑟_𝑑(2.21) and the varying controller gains are updated as follows:𝑘̂̇𝑘̂̇_𝛽1=𝛾^−1𝛽1||𝑠_𝛽||,𝛾_𝛽1>0,(2.22)_𝛽2𝑘̂̇=𝛾^−1𝛽2|𝑟|||𝑠_𝛽||,𝛾_𝛽2>0,(2.23)_𝑟1𝑘̂̇=𝛾^−1𝑟1||𝑠_𝑟||,𝛾_𝑟1>0,(2.24)_𝑟2=𝛾^−1𝑟2|𝜏|||𝑠_𝑟||,𝛾_𝑟2>0.(2.25) Assume that there are positive constants𝑘^𝑑𝛽1,𝑘^𝑑𝛽2, 𝑘^𝑑𝑟1, and 𝑘^𝑑𝑟2 that satisfy𝑘𝑘^𝑑𝛽1>||Δ_𝛽||,𝑘^𝑑𝛽2>||𝑚||,(2.26)_𝑑𝑟1>||Δ_𝑟||,𝑘^𝑑𝑟2>||𝐼_𝑧||.(2.27) 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, consider𝑘𝑘̃̃_𝛽1=𝑘̂_𝛽1−𝑘^𝑑𝛽1,𝑘̃_𝛽2=𝑘̂_𝛽2−𝑘^𝑑𝛽2,(2.28)_𝑟1=𝑘̂_𝑟1−𝑘^𝑑𝑟1,𝑘̃_𝑟2=𝑘̂_𝑟2−𝑘^𝑑𝑟2.(2.29) Then, the stability of the considered adaptive-sliding mode control laws can be shown through Lyapunov candidates:𝑉_𝛽=𝑚2𝑠^2𝛽+12𝛾^𝛽1𝑘̃^2𝛽1+12𝛾^𝛽2𝑉𝑘̃^2𝛽2,(2.30)_𝑟=𝐼^𝑧2𝑠^2𝑟+12𝛾^𝑟1𝑘̃^2𝑟1+12𝛾^𝑟2𝑘̃^2𝑟2.(2.31) 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 that𝑚𝑠_𝛽̇=𝑚𝑟+Δ_𝛽−𝑘̂_𝛽1+𝑘̂_𝛽2|𝑟|sgn𝑠_𝛽.(2.32) Then, differentiate (2.30) to get𝑉_𝛽̇=𝑚𝑠_𝛽𝑠_𝛽̇+𝛾_𝛽1𝑘̃_𝛽1𝑘̃̇_𝛽1+𝛾_𝛽2𝑘̃_𝛽2𝑘̃̇_𝛽2.(2.33) Replacing (2.32) in (2.33) and considering (2.28), we have that𝑉_𝛽̇=𝑚𝑟+Δ_𝛽−𝑘̂_𝛽1+𝑘̂_𝛽2|𝑟|sgn𝑠_𝛽𝑠_𝛽+𝛾_𝛽1𝑘̂̇_𝛽1𝑘̂_𝛽1+𝛾_𝛽2𝑘̂̇_𝛽2𝑘̂_𝛽2−𝛾_𝛽1𝑘̂̇_𝛽1𝑘^𝑑𝛽1+𝛾_𝛽2𝑘̂̇_𝛽2𝑘^𝑑𝛽2.(2.34) Using adaptation laws (2.22) and (2.23) in (2.34) and considering (2.26) results in𝑉_𝛽̇==𝑚||𝑚𝑟||+−Δ𝑘_{𝛽𝑑𝛽2}𝑠_𝛽−𝑘^𝑑𝛽1||𝑠_𝛽||−𝑘^𝑑𝛽2|𝑟|||𝑠_𝛽||≤||𝑚|||𝑟|+||Δ_𝛽||||𝑠_𝛽||−𝑘^𝑑𝛽1||𝑠_𝛽||−𝑘^𝑑𝛽2|𝑟|||𝑠_𝛽|||𝑟|||𝑠^𝛽||+||Δ_𝛽||−𝑘^𝑑𝛽1||𝑠^𝛽||<0,(2.35) where we use 𝑠_𝛽sgn(𝑠_𝛽)=|𝑠_𝛽|. Accordingly, the convergence of 𝑠_𝛽 to zero and also boundedness of 𝑘̃_𝛽1 and 𝑘̃_𝛽2 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 that𝐼_𝑧𝑠_𝑟̇=𝐼_𝑧𝜏+Δ_𝑟−𝑘̂_𝑟1+𝑘̂_𝑟2|𝜏|sgn𝑠_𝑟.(2.36) Differentiate (2.31) and replace (2.36) for 𝐼_𝑧𝑠_𝑟̇ to get𝑉_𝑟̇=𝐼_𝑧𝜏+Δ_𝑟−𝑘̂_𝑟1+𝑘̂_𝑟2|𝜏|sgn𝑠_𝑟𝑠_𝑟+𝛾_𝑟1𝑘̂̇_𝑟1𝑘̂_𝑟1+𝛾_𝑟2𝑘̂̇_𝑟2𝑘̂_𝑟2−𝛾^𝑟1𝑘̂̇_𝑟1𝑘^𝑑𝑟1+𝛾_𝑟2𝑘̂̇_𝑟2𝑘^𝑑𝑟2.(2.37) Using adaptation laws (2.24) and (2.25) as well as the inequalities in (2.27), we have that𝑉_𝑟̇=𝐼_𝑧𝜏+Δ_𝑟𝑠^𝑟−𝑘^𝑑𝑟1=||𝐼_𝑧||𝑠_𝑟||−𝑘^𝑑𝛽2|𝜏|||𝑠_𝑟||≤||𝐼_𝑧|||𝜏|+||Δ_𝑟||||𝑠_𝑟||−𝑘^𝑑𝑟1||𝑠_𝑟||−𝑘^𝑑𝑟2|𝜏|||𝑠_𝑟||||−𝑘^𝑑𝑟2|𝜏|||𝑠^𝑟||+||Δ_𝑟||−𝑘^𝑑𝑟1||𝑠_𝑟||<0.(2.38) Thus, according to the Barbalat’s Lemma, the system state can be driven to the sliding surface 𝑠_𝑟 and the controller gains 𝑘̃_𝑟1 and 𝑘̃_𝑟2 will be bounded. Furthermore, to tackle the chattering problem saturation function is used to derive the final adaptive control laws𝑌=𝑉𝑚𝑟−𝑘̂_𝛽1+𝑘̂_𝛽2|𝑟|sat𝑠^𝛽𝑀=𝐼_𝑧Φ_𝛽,𝜏−𝑘̂_𝑟1+𝑘̂_𝑟2|𝜏|sat𝑠^𝑟Φ_𝑟.(2.39)

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:𝑓=⁴_𝑖=1𝐴_𝑖𝑋^2𝑖+𝑌^2𝑖𝜇_𝑖𝑍_𝑖²,(3.1) where 𝑖 denotes wheel number, 𝑋_𝑖 and 𝑌_𝑖 stand for desired values of longitudinal force, 𝐹_𝑥𝑖, and lateral force, 𝐹_𝑦𝑖, 𝑍_𝑖 is the vertical load, all defined in the vehicle body fixed coordinate system, as shown in Figure 2, 𝐴_𝑖 is the weighting coefficient and 𝜇_𝑖 is the friction coefficient, of the 𝑖th tire. Defining the 8 × 1 actuator vector of 𝐮 as𝐮=𝑋₁𝑋₂𝑋₃𝑋₄𝑌₁𝑌₂𝑌₃𝑌₄^𝑇,(3.2) the cost function is written in matrix form𝑓(𝐮)=𝐮^𝐓𝐖𝐮(3.3) in which 𝐖_𝟖×𝟖 is a diagonal weighting matrix. To get 𝑌 and 𝑀, by assuming small steering angles in Figure 2, individual tire forces must satisfy the equality constraints:𝑌=⁴_𝑖=1𝑀=₂𝑌_𝑖,(3.4)_𝑖=1𝐿_𝑓𝑌_𝑖−𝐿_𝑟𝑌_(𝑖+2)+𝑑2²_𝑖=1𝑋_(2𝑖)−𝑋_(2𝑖−1).(3.5) Also, the longitudinal acceleration, 𝑎_𝑥, by driver is generated by longitudinal forces𝑋=𝑚𝑎_𝑥=⁴_𝑖=1𝑋_𝑖.(3.6) Equations (3.4)–(3.6) can be expressed in linear matrix form as𝐀𝐮=𝐯,(3.7) where 𝐀∈ℜ^3×8 is a constant matrix and the vector of generalized forces/moment, 𝐯, is given by𝐯=[𝑋𝑌𝑀]^𝑇.(3.8) On the other hand, the resultant force of each tire is constrained to friction circle𝜇_𝑖𝑍_𝑖²−𝑋^2𝑖+𝑌^2𝑖≥0(𝑖=1,…,4).(3.9) Using (3.2), (3.9) can be written as𝜇_𝑖𝑍_𝑖²−𝑢^2𝑖+𝑢^2𝑖+4≥0(𝑖=1,…,4).(3.10) or𝐂_𝐈(𝐮)≥𝟎,(3.11) where 𝐂_𝐈 is a 4 × 1 vector with the 𝑖th component𝐶_𝐼𝑖𝑢_𝑖=𝜇_𝑖𝑍_𝑖²−𝑢^2𝑖+𝑢^2𝑖+4(𝑖=1,…,4).(3.12) The optimization problem is summarized as follows:smubinjeimctiztoe𝑓(ln𝐮ion)ne=alirn𝐮ee^𝐓aqr𝐖uain𝐮lie,tyqucaolintsytrcaoinntsst,raints,𝐂𝐀_𝐈𝐮=𝐯,(𝐮)≥𝟎.(3.13)

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 get𝐽=𝜌𝐮^𝐓𝐖𝐮+(𝐀𝐮−𝐯)^𝐓(𝐀𝐮−𝐯),𝜌>0,(4.1) which is written in the quadratic form𝐽=12𝐮^𝐓𝐆𝐮+𝐜^𝐓𝐮+𝐡,(4.2) where𝐆=2𝜌𝐖+𝐀^𝐓𝐀,𝐜=−2𝐀^𝐓𝐯,𝐡=𝐯^𝐓𝐯.(4.3) IP methods use barrier logarithmic functions to satisfy the inequality constraints. In this regard, the optimization problem turns intominimize𝐿=12𝐮^𝐓𝐆𝐮+𝐜^𝐓𝐮+𝐡−𝜂⁴_𝑖=1subjectto𝐂_𝐈log𝐶_𝐼𝑖,𝜂>0,𝐩≥(𝐮𝟎).−𝐩=𝟎,(4.4) Applying KKT formulation [16] to (4.4), the optimality conditions can be expressed as follows,𝐆𝐮+𝐜+𝐏𝝀𝐀^𝐓𝐈−𝝀𝜂𝐞==𝟎,𝟎,𝐀𝐞_𝐈=(𝐮[)1=1−1∇1𝐂]_𝐈𝑇(𝐮),𝐂_𝐈(𝐮)−𝐩=𝟎,,𝐩>0,𝝀>0,(4.5) where 𝝀 is the vector of Lagrange multipliers and 𝐏 is a 4 × 4 diagonal matrix whose diagonal elements are the components of the vector 𝐩.

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 {Δ𝐩,Δ𝐮,Δ𝝀} is obtained by applying Newton’s method to KKT conditions (4.5): 𝐫_𝐜+𝐫𝐆+𝚲_𝟐𝚫𝐮+𝐀^𝐓𝐈𝚫𝝀=𝟎,_𝐛𝐫−𝐀_𝐈𝚫𝐮−𝚫𝐩=𝟎,_𝐬+𝐏𝚫𝝀+𝚲𝚫𝐩=𝟎,(4.6) where the residuals 𝐫_𝐜, 𝑟_𝐛, and 𝑟_𝐬 are obtained as 𝐏=diag𝐫𝑝_𝐜1=𝐆𝐮+𝐜+𝐀^𝑇𝐈𝝀,𝐫_𝐛=𝐂_𝐈(𝐮)−𝐩,𝐫_𝐬=𝐏𝝀−𝜂𝐞,(4.7),𝑝₂,𝑝₃,𝑝₄,𝚲=diag𝜆₁,𝜆₂,𝜆₃,𝜆₄,𝚲_𝟐=2diag(𝚲,𝚲).(4.8) Equation (4.6) is solved to achieve the step directions Δ𝐮=𝐆+Δ𝚲𝝀_𝟐=+−𝐀𝐏^𝐓𝐈𝐏^−𝟏₋₁𝚲𝐀_𝐈⁻¹𝐀^𝐓𝐈𝐏⁻¹𝐫_𝐬+𝚲𝐫_𝐛−𝐫_𝐜,Δ𝐩𝐫=_𝐬+𝐫_𝐛𝚲𝐫_𝐛+𝐏⁻¹𝚲𝐀_𝐈Δ𝐮,−𝐀_𝐈Δ𝐮.(4.9)

(2) Step Length Calculation
To satisfy nonnegativity condition (𝐩,𝝀)≥𝟎, the new iteration (𝐮⁺,𝐩⁺,𝝀⁺) is calculated as 𝐮⁺=𝐮+𝛼^𝑠maxΔ𝐮,𝐩⁺=𝐩+𝛼^𝑠max𝚫𝐩,𝝀⁺=𝝀+𝛼^𝜆max𝚫𝝀,(4.10) where 𝛼𝛼^𝑠max=max{𝛼𝜖(0,1]∶𝐩+𝛼Δ𝐩≥(1−𝜏)𝐩},𝜏𝜖(0,1),_𝜆max=max{𝛼𝜖(0,1]∶𝝀+𝛼Δ𝝀≥(1−𝜏)𝝀}.(4.11)

(3) Updating the Barrier Parameter 𝜂
The sequence of barrier parameters {𝜼_𝐤} converges to zero by the update law 𝜂_𝑘+1=𝜎_𝑘𝐩^𝐓𝐤𝝀_𝐤𝟒,𝜎^𝑘𝜖[0,1],(4.12) where to update the centring parameter𝜎_𝑘, first, the predictor (affine scaling) direction {Δ𝐮^aff,Δ𝐩^aff,Δ𝝀^aff} and the corresponding longest step lengths 𝛼^𝑠aff and 𝛼^𝜆affare calculated by setting 𝜂=0 in (4.9) and (4.11). Then the value of complementarity along the affine scaling 𝜂_aff step is defined to be 𝜂_aff=𝐩_𝐤+𝛼^𝑠affΔ𝐩^aff^𝑇𝝀_𝐤+𝛼^𝜆affΔ𝝀^aff.(4.13) The centring parameter is updated as follows: 𝜎_𝑘=𝜂^aff𝐩^𝐓𝐤𝝀_𝐤/𝟒³.(4.14)

(4) Stopping Criteria
The algorithm is terminated, when the following error function with 𝜂=0 has converged sufficiently close to zero 𝐸(𝐮,𝐩,𝝀,𝜂)=max‖‖𝐆𝐮+𝐜+𝐀^𝑇𝐈𝝀‖‖,‖‖𝐂_𝐈(𝐮)−𝐩‖‖,‖𝐏𝝀−𝜂𝐞‖.(4.15) All of these steps have been summarized in Pseudocode 1. To reduce the time of control allocation, the last allocated 𝐮 is chosen as the initial value of the control input 𝐮_𝟎. Then, 𝐮_𝟎 is modified (reduced) to satisfy tire saturation constraints.

Given A, W , and v  Compute G, c, and h in (4.2).  Choose the allocated control input of the previous sample time as the initial value 𝐮𝟎  If each entry of 𝐮𝟎 violates the inequality (3.9), reduce it to satisfy(3.9)  Compute initial values for 𝐩0 and𝝀0.  Compute the initial value for barrier parameter𝜂.  Compute residuals 𝐫𝐜,𝐫𝐛, and 𝐫𝐬 given in(4.7)  Select error tolerance 𝜀 and set𝑘←0. repeat until 𝐸(𝐮,𝐩,𝝀,0)≤𝜀  Compute step direction from equation (4.9),  Compute step size to satisfy nonnegativity conditions using (4.11),  Update the variables 𝐮,𝐩, and 𝝀 using (4.10),  Compute affine scaling 𝜂aff step from (4.13) and 𝜎𝑘 according to (4.14),  Update the barrier parameter 𝜂𝑘+1 based on (4.12),  Compute residuals 𝐫𝐜,𝐫𝐛, and 𝐫𝐬 from (4.7),  Set 𝑘←𝑘+1 end

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):ℓ=𝐮^𝐓𝐖𝐮+(𝐯−𝐀𝐮)^𝐓𝝀_dyn−𝜔_dyn⁴_𝑖=1log𝐶_𝐼𝑖𝑢_𝑖,𝑢_𝑖+4(5.1) with 𝜔_dyn>0and𝝀_dyn being a 3-component vector of Lagrange multipliers. DCA updates 𝐮 and 𝝀_dyn in the form of the Newton-like update law⎡⎢⎢⎣𝝀̇𝐮̇_dyn⎤⎥⎥⎦=−𝛾_dynℍ^𝑇ℍ+𝜀_dyn𝐈⁻¹ℍ⎡⎢⎢⎢⎢⎣𝜕ℓ𝜕𝜕ℓ𝐮𝜕𝝀_dyn⎤⎥⎥⎥⎥⎦+⎡⎢⎢⎣𝝓𝜻⎤⎥⎥⎦,(5.2) where 𝛾_dyn>0 and 𝜀_dyn≥0, and for our problem ℍ is written asℍ=⎡⎢⎢⎢⎣𝜕²ℓ𝜕𝐮²−𝜕(𝐀𝐮)^𝐓−𝜕(𝐀𝐮)𝜕𝐮𝜕𝐮𝟎⎤⎥⎥⎥⎦.(5.3) In (5.2) the feedforward-like terms 𝜻 and 𝝓 are chosen so that the following scalar algebraic equation holds:𝜶^𝐓_dyn𝜻+𝜷^𝐓_dyn𝝓+𝛿_dyn=0(5.4) with⎡⎢⎢⎣𝜶_dyn𝜷_dyn⎤⎥⎥⎦=ℍ⎡⎢⎢⎢⎢⎣𝜕𝜕𝜕ℓℓ𝐮𝜕𝝀_dyn⎤⎥⎥⎥⎥⎦,𝛿_dyn=−(𝐀𝐮−𝐯)^𝑇𝐯.̇(5.5)

Assume that the stability of the high-level control could be shown through a Lyapunov function𝑉₀(𝑡,𝑟,𝛽), then, by the Lyapunov function𝑉_dyn𝑡,𝑟,𝛽,𝐮,𝝀_dyn=𝜎_dyn𝑉₀(𝑡,𝑟,𝛽)+12𝜕𝜕ℓ𝐮^𝑇𝜕𝜕ℓ𝐮+𝜕𝜕𝝀ℓ^𝑇_dyn𝜕ℓ𝜕𝝀_dyn,𝜎^dyn>0,(5.6) 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⎡⎢⎢⎢⎢⎣𝜕ℓ𝜕𝜕ℓ𝐮𝜕𝝀_dyn⎤⎥⎥⎥⎥⎦=𝟎.(5.7) Applying Newton’s conditions to (5.7) results in optimizing Newton’s steps Δ𝐮 and Δ𝝀_dyn⎡⎢⎢⎢⎢⎣𝜕²ℓ𝜕𝐮²𝜕𝜕𝝀_dyn𝜕𝜕ℓ𝐮^𝑇𝜕𝜕𝐮𝜕𝝀𝜕ℓ_dyn𝜕²ℓ𝜕𝝀^𝟐_dyn⎤⎥⎥⎥⎥⎦⎡⎢⎢⎣𝚫𝚫𝝀𝐮_dyn⎤⎥⎥⎦=−⎡⎢⎢⎢⎢⎣𝜕𝜕𝜕ℓℓ𝐮𝜕𝝀_dyn⎤⎥⎥⎥⎥⎦(5.8) or⎡⎢⎢⎣𝚫𝚫𝝀𝐮_dyn⎤⎥⎥⎦=⎡⎢⎢⎢⎣𝜕²ℓ𝜕𝐮^𝟐−𝜕(𝐀𝐮)^𝑇−𝜕(𝐀𝐮)𝜕𝐮𝜕𝐮𝟎⎤⎥⎥⎥⎦⁻¹⎡⎢⎢⎢⎢⎣𝜕ℓ𝜕𝜕ℓ𝐮𝜕𝝀_dyn⎤⎥⎥⎥⎥⎦=ℍ₋₁⎡⎢⎢⎢⎢⎣𝜕ℓ𝜕𝜕ℓ𝐮𝜕𝝀_dyn⎤⎥⎥⎥⎥⎦.(5.9) By setting𝜀_dyn=0, ignoring the terms 𝜻 and 𝝓, and considering the symmetry of ℍ, it can be demonstrated that the update law (5.2) represents the solution of Δ𝐮 and Δ𝝀_dyn 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 𝜻 and 𝝓 are to compensate time-varying optimum solution.

Despite the log-barrier term in (5.1), depending on the values of𝛾_dyn and𝜔_dyn, there is no guarantee that the Newton-like step (5.2) will satisfy the inequality constraints (3.9). As the 𝑖th inequality constraint is infringed, 𝐶_𝐼𝑖 becomes negative andlog(𝐶_𝐼𝑖) and correspondingly the Lagrangian (5.1) turn meaningless. To tackle this problem, in this paper, a line search is adopted for the coefficient 𝛾_dyn, so that the resulting 𝐮 and 𝝀_dyn at the next sampling time satisfy the inequality constraints 𝐂_𝐈>𝟎 and𝝀_dyn>0. 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(𝜇_𝑖𝑍_𝑖)² 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, 𝛿_𝑖, at wheel 𝑖 can be determined as follows:𝛿_𝑖≈𝛽+𝐿^𝑓𝑟𝑣_𝑥𝑖𝛿−𝛼_𝑖,𝑖=1,2,_𝑖≈𝛽−𝐿^𝑟𝑟𝑣_𝑥𝑖−𝛼_𝑖,𝑖=3,4,(5.10) where 𝑣_𝑥𝑖 is the longitudinal velocity of the 𝑖th tire and 𝛼_𝑖 is the side-slip angle of the 𝑖th tire and is obtained using the inverse of a simple tire model as𝛼_𝑖=⎧⎪⎨⎪⎩−𝑌^𝑖𝐶_𝑖−𝜇_𝑖,𝑋^2𝑖+𝑌^2𝑖≤𝑛∗𝜇_𝑖𝑍_𝑖²,0<𝑛<1,𝑍_𝑖∗arctan𝐾^𝑖𝑌_𝑖𝜇_𝑖𝑍_𝑖𝐶_𝑖,𝑋^2𝑖+𝑌^2𝑖>𝑛∗𝜇_𝑖𝑍_𝑖²,𝑖=1,…,4,(5.11) in which 𝐶_𝑖 denotes the cornering stiffness of the 𝑖th tire and 𝐾_𝑖s 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 𝑖th tire,𝜎_𝑖, is defined as𝜎_𝑖=𝑅𝜔^𝑖−𝑣_𝑥𝑖𝑣_𝑥𝑖𝜎duringbraking,(6.1)_𝑖=𝑅𝜔^𝑖−𝑣_𝑥𝑖𝑅𝜔_𝑖duringacceleration,(6.2) where 𝑅 denotes the radius and 𝜔_𝑖 is the angular velocity of the 𝑖th 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 𝜎^∗, after which it starts to lessen. Experimental studies have established that the tire lateral force decreases with increasing slip ratios greater than |𝜎^∗| as well [17].

6.1. Description of the SRC Scheme

In this paper, when the tire slip ratio is smaller than 𝜎^∗, by neglecting the wheel rotational inertia [3], the applied braking/traction torque, 𝑇_𝑖, at wheel 𝑖 is obtained as𝑇_𝑖=𝑅𝑋_𝑖.(6.3) In this case, the SRC works for Desired Longitudinal Force Generation (DLFG). However, when the demanded 𝑋_𝑖 is too high, applying (6.3) would increase the slip ratio beyond 𝜎^∗, 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 𝜎^∗. 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 𝑖th wheel rotational dynamics is stated as𝐼_𝜔𝜔̇_𝑖=𝑇_𝑖−𝑅𝐹_𝑥𝑖,(6.4) where 𝐼_𝜔 is wheel rotational inertia. To achieve slip-ratio differential equation when braking, by differentiating (6.1) and replacing for 𝜔̇_𝑖 from (6.4) 𝜎̇_𝑖=𝑅𝐼_𝜔𝑣_𝑥𝑖𝑇_𝑖−𝑟𝐹_𝑥𝑖−1+𝜎_𝑖𝑣̇^𝑥𝑖𝑣_𝑥𝑖.(6.5) During decelerating, the longitudinal force can be stated as [17]𝐹_𝑥𝑖=𝐶_𝜎𝜎_𝑖+Δ𝐹_𝑥𝑖(6.6) in which the term 𝐶_𝜎𝜎_𝑖 represents the linear part, with 𝐶_𝜎 being the tire longitudinal stiffness, and Δ𝐹_𝑥𝑖 is the deviation from the linear part and is bounded by 𝐹_𝑥𝑖:0<Δ𝐹_𝑥𝑖<𝐹_𝑥𝑖.(6.7) The goal of the ABS is to regulate 𝜎_𝑖 around the constant value𝜎^∗. Applying the sliding mode control design procedure to (6.5), an ABS is designed so that to get the sliding condition𝑑𝑑𝑡12𝑠^2𝜎≤−𝜂_𝜎||𝑠_𝜎||,𝑠_𝜎=𝜎_𝑖−𝜎^∗,𝜎^∗<0,𝜂_𝜎>0,(6.8) the applied braking torque at the 𝑖th wheel is𝑇_𝑖=𝑅𝐶_𝜎𝜎^∗+𝐼^𝜔𝑣̇_𝑥𝑖𝑅1+𝜎^𝑖−𝐼^𝜔𝑣_𝑥𝑖𝑅𝑘^𝜎sat𝑠^𝜎𝜙_𝜎,𝑘^𝜎>𝑅²𝐼_𝜔𝑣_𝑥𝑖𝐹_𝑥𝑖+𝜂_𝜎,𝜙_𝜎>0.(6.9) Furthermore, the applied torque during acceleration, by TCS, is computed as𝑇_𝑖=𝑅𝐶_𝜎||𝜎^∗||+𝐼^𝜔𝑣̇_𝑥𝑖𝑅1−𝜎_𝑖−𝐼^𝜔𝑣_𝑥𝑖𝑅1−𝜎_𝑖²𝑘_𝜎sat𝑠^𝜎𝜙_𝜎,𝑠^𝜎=𝜎_𝑖−||𝜎^∗||.(6.10)

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-𝜇 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-𝜇 scenario.

Figure 6 

Yaw angle velocity in split-𝜇 scenario.

Figure 7 

Side-slip angle in split-𝜇 scenario.

Figure 8 

Tire work-load distribution by ODF in split-𝜇 scenario.

Figure 9 

Tire work-load distribution by SCA in split-𝜇 scenario.

Figure 10 

Tire work-load distribution by DCA in split-𝜇 scenario.

(a)

(b)

(c)

(d)

Figure 11 

Actually generated tire lateral forces in split-𝜇 scenario.

(a)

(b)

(c)

(d)

Figure 12 

Actually generated tire longitudinal forces in split-𝜇 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 𝜎^∗=0.2. Then, ABS is activated by SRC to regulate the front slip ratios at 𝜎^∗=0.2. 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.

(a)

(b)

(c)

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.

(a)

(b)

(c)

(d)

Figure 19 

Actually generated tire lateral forces in SLC.

(a)

(b)

(c)

(d)

Figure 20 

Actually generated tire longitudinal forces in SLC.

(a)

(b)

(c)

(d)

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 𝜇=0.5.

(a)

(b)

(c)

Figure 23 

Yaw rate in DLC with 𝜇=0.5.

Figure 24 

Side-slip angle in DLC with 𝜇=0.5.

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 𝜇=0.4.

(a)

(b)

(c)

Figure 26 

Yaw rate in DLC with 𝜇=0.4.

Figure 27 

Side-slip angle in DLC with 𝜇=0.4.

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
𝐼𝑧:	Estimation of yaw moment of inertia
𝐼𝑧:	Estimation of yaw moment of inertia error
𝐼𝑧:	Upper bound for 𝐼𝑧
𝐽:	Objective function in quadratic form
𝑘𝑟, 𝑘𝛽, 𝑘𝜎:	Sliding mode control gain
𝑘̂𝛽1, 𝑘̂𝛽2, 𝑘̂𝑟1, 𝑘̂𝑟2:	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.