Optimal Reconfiguration Control of the Yaw Stability of the Tractor-Semitrailer Vehicle

Abstract

An optimal reconfiguration control scheme based on control allocation (CA) is proposed to stabilize the yaw dynamics of the tractor-semitrailer vehicle. The proposed control scheme is a two-level structure consisting of an upper level of sliding mode yaw moment controller (SMYC) and a lower optimal brake force distributor (BFD). The upper SMYC is designed to follow the tractor yaw rate and the combination of the hitch angle and trailer slip angle and outputs the corrective yaw moment, respectively, for the tractor and the trailer. The optimal brake force allocation and reconfigurable control problem is transformed to a problem of error minimization and control minimization combination formulated by constrained weighted least squares (CWLS) optimization and further solved with active set (AS) algorithm. Simulation results reveal that the CA technique-based optimal reconfigurable control is rather effective for the tractor-semitrailer vehicle to enhance the yaw stability performance and the reliability in case of actuator failure thanks to the multiple-axle structure enriching the alternatives of possible actuator combinations in CA optimization.

1. Introduction

Tractor-semitrailer vehicle which is the combination of a tractor and a semitrailer through the hitch point presents particular handling and stability properties compared with the passenger cars. In critical driving situations, a tractor-trailer combination may experience different forms of lateral instability such as jackknife, trailer swing, and trailer oscillation [1, 2]. In emergency situations, it is difficult for a general driver to handle the tractor-semitrailer vehicle as he recognizes the vehicle falling into instability. Fortunately, similar to the passenger cars, the stability of a tractor-trailer combination can also be obviously improved by active control approach. In recent years, there is increasing interest by both the industry and academia in having electronic stability control (ESC) systems developed for and implemented in the heavy vehicles [3–5]. The conventional ESC systems both for the passenger cars and the commercial vehicles are mainly based on the concept of direct yaw moment control (DYC) which is generally realized by single-wheel braking [5–9]. However, this method cannot ensure the reliability or the safety especially in the situations such as the actuator failure, and low friction adhesion of the brake actuator.

In fact, for a typical ESC system, controlled states are typically lateral velocity and yaw motion, while the actuation could potentially include individual wheel driving/braking, comprising a redundantly actuated system especially for the commercial heavy vehicles which are generally with multiple axles and wheels [10]. Coordinating all the available brake actuators to expand the operational envelope and thereby improve the reliability refers to the problem of control allocation (CA) [10], which has been widely used in the control of aircraft and marine vessels having become more and more popular in vehicle dynamics control recently [10–14]. A CA scheme with the quadratic programming algorithm and a linear quadratic regulator is proposed by [13] to distribute the control effort among the front and rear brake torque differences and steering angle to track a desired yaw rate and minimize the sideslip. Wang and Longoria [10] proposed a coordinated reconfigurable vehicle dynamics control system achieved by high-level control of generalized forces/moment, distributed to the slip and slip angle of each tyre based on the fixed-point (FP) CA method. Tagesson et al. [14] tried to apply the CA method to the yaw stability control of a type of heavy truck with one front axle and two rear axles by coordinating the brake actions of the six wheels. Though CA method has been used in the stability control of passenger cars for some time, very little reports on the heavy vehicle especially the tractor-trailer combination can be found. Different from the single-body vehicles such as passenger cars and three-axle heavy trucks, it is rather complex to implement CA technique to the tractor-semitrailer vehicle due to the particular double-body structure. In this work, optimal reconfiguration control of the yaw stability of the tractor-semitrailer vehicle based on CA technique is studied by optimizing and distributing the brake forces of the tractor and trailer wheels in real time.

The rest of the paper is organized as follows: in Section 2, a nonlinear tractor-semitrailer combination vehicle model is developed to perform the control scheme validation by simulations. The proposed control scheme is presented in Section 3 with the upper level sliding mode yaw moment controller (SMYC) and the lower level constrained weighted least squares (CWLS) based brake force distributor (BFD) designed in this section. Simulations are conducted in Section 4 to evaluate the proposed control scheme by a single lane change maneuver, followed by the conclusions given in Section 5.

2. Nonlinear Vehicle Model

A typical five-axle tractor-semitrailer vehicle is considered in this study with the nonlinear schematic model illustrated by Figure 1. A 14-degree-of-freedom (14-DOF) vehicle model is constructed in this section involving the body planar motion, body roll motion, suspension and tyre forces, and wheel rotational dynamics.

(a)

(b)

Figure 1 

Nonlinear tractor-semitrailer vehicle schematic model. (a) 𝑋-𝑌 plane; (b) 𝑍-𝑋 plane.

2.1. Body Motion Models

Body planar motions for the tractor and the trailer are, respectively, described. In the modeling, the tandem axle is simplified to one single axle. For the tractor, the longitudinal, lateral, yaw, and roll dynamics are, respectively, given by (2.1)–(2.4) as 𝑚₁𝑣̇_𝑥1−𝛾₁𝑣_𝑦1=𝐹_𝑥1+𝐹_𝑥2cos𝛿_𝑓+𝐹_𝑥3+𝐹_𝑥4−𝐹_𝑦1+𝐹_𝑦2sin𝛿_𝑓+𝐹_ℎ𝑥𝑚₁,𝑣̇_𝑦1+𝛾₁𝑣_𝑥1+𝑚_1𝑠ℎ_𝑠1𝜙̈₁=𝐹_𝑦1+𝐹_𝑦2cos𝛿_𝑓+𝐹_𝑦3+𝐹_𝑦4+𝐹_𝑥1+𝐹_𝑥2sin𝛿_𝑓−𝐹_ℎ𝑦𝐼_𝑧1,𝛾̇₁=𝐹_𝑦1−𝐹_𝑦2sin𝛿_𝑓+𝐹_𝑥2−𝐹_𝑥1cos𝛿_𝑓𝑡_𝑤12−𝐹_𝑦3+𝐹_𝑦4𝑏₁+𝐹_𝑥4−𝐹_𝑥3𝑡_𝑤22+𝐹_𝑦1+𝐹_𝑦2cos𝛿_𝑓+𝐹_𝑥1+𝐹_𝑥2sin𝛿_𝑓𝑎₁+𝐹_ℎ𝑦𝑙_𝑝,𝐼_𝑥1𝜙̈₁+𝑚_1𝑠𝑣̇_𝑦1+𝛾₁𝑣_𝑥1ℎ_𝑠1=𝑚_1𝑠𝑔ℎ_𝑠1sin𝜙₁+𝐹_𝑧2−𝐹_𝑧1𝑡_𝑤12+𝐹_𝑧4−𝐹_𝑧3𝑡_𝑤22−𝐹_𝑦1+𝐹_𝑦2ℎ_𝑓1cos𝛿_𝑓−𝐹_𝑦3+𝐹_𝑦4ℎ_𝑟1−𝐹_ℎ𝑦ℎ_𝑝−ℎ_𝑟1+𝑀_ℎ𝑡, (2.1)(2.2)(2.3)(2.4) where𝑚₁ and 𝑚_1𝑠 denote the total mass and the sprung mass of the tractor; 𝐼_𝑧1 and 𝐼_𝑥1 the yaw moment of inertia and roll moment of inertia of the tractor; 𝑎₁, 𝑏₁, and 𝑙_𝑝, respectively, the distance from the centre of mass (CM) of the tractor to its front, rear axle, and the hitch point; 𝑡_𝑤1, 𝑡_𝑤2 the track width of tractor front axle and rear axle; ℎ_𝑓1 and ℎ_𝑟1 the height of front, rear roll centre of the tractor; ℎ_𝑠1 the distance from the tractor CM to its roll axis; ℎ_𝑝 the height of the hitch point; 𝛾₁, 𝜙₁, 𝑣_𝑥1, and 𝑣_𝑦1, respectively, the yaw rate, roll angle, longitudinal velocity and lateral velocity of the tractor; 𝐹_𝑥𝑖, 𝐹_𝑦𝑖, and 𝐹_𝑧𝑖 (𝑖=1∼4), respectively, the longitudinal, lateral, and normal force of the tractor wheels; 𝐹_ℎ𝑥, and 𝐹_ℎ𝑦 the longitudinal and lateral interaction force between the tractor and the trailer; 𝑀_ℎ𝑡 the coupling roll moment between the tractor and the trailer.

The equations of longitudinal, lateral, yaw, and roll motions of the trailer can, respectively, be expressed by (2.5)–(2.8) as 𝑚₂𝑣̇_𝑥2−𝛾₂𝑣_𝑦2=𝐹_𝑥5+𝐹_𝑥6−𝐹_ℎ𝑥cos𝜃+𝐹_ℎ𝑦𝑚sin𝜃,₂𝑣̇_𝑦2+𝛾₂𝑣_𝑥2+𝑚_2𝑠ℎ_𝑠2𝜙̈₂=𝐹_𝑦5+𝐹_𝑦6+𝐹_ℎ𝑦cos𝜃−𝐹_ℎ𝑥𝐼sin𝜃,_𝑧2𝛾̇₂=𝐹_ℎ𝑦cos𝜃−𝐹_ℎ𝑥sin𝜃𝑎₂−𝐹_𝑦5+𝐹_𝑦6𝑏₂+𝐹_𝑥6−𝐹_𝑥5𝑡_𝑤32,𝐼_𝑥2𝜙̈₂+𝑚_2𝑠𝑣̇_𝑦2+𝛾₂𝑣_𝑥2ℎ_𝑠2=𝐹_𝑧6−𝐹_𝑧5𝑡_𝑤32+𝑚_2𝑠𝑔ℎ_𝑠2sin𝜙₂+𝐹_ℎ𝑦ℎ_𝑝−ℎ_𝑓2−𝐹cos𝜃_𝑦5+𝐹_𝑦6ℎ_𝑟2−𝑀_ℎ𝑡, (2.5)(2.6)(2.7)(2.8) where 𝑚₂ and 𝑚_2𝑠 denote the total mass and sprung mass of the trailer; 𝐼_𝑧2 and 𝐼_𝑥2 the yaw moment of inertia and roll moment of inertia of the trailer; 𝑎₂, and 𝑏₂ the distance from the trailer CM to the hitch point and the rear axle of the trailer; 𝑡_𝑤3 the track width of the trailer axle; ℎ_𝑓2 and ℎ_𝑟2 the front and rear roll centre height of the trailer; ℎ_𝑠2 the distance from the trailer CM to its roll axis; 𝛾₂, 𝜙₂, 𝑣_𝑥2, and 𝑣_𝑦2, respectively, the yaw rate, roll angle, longitudinal velocity, and lateral velocity of the trailer; 𝜃 the hitch angle between the tractor and the trailer; 𝐹_𝑥𝑖, 𝐹_𝑦𝑖, and 𝐹_𝑧𝑖 (𝑖=5,6), respectively, the longitudinal, lateral, and normal force of the trailer wheels.

The couplings exist in the longitudinal, lateral, yaw, and roll motions between the tractor and the trailer which can, respectively, be described by (2.9) as 𝑣_𝑥2=𝑣_𝑥1𝑣cos𝜃,_𝑦2=𝑣_𝑦1−𝑙_𝑝𝛾₁cos𝜃−𝑎₂𝛾₂−𝑣_𝑥1𝛾sin𝜃,₂=𝛾₁+̇𝑀𝜃,_ℎ𝑡=𝐾_ℎ𝑡𝜙₂−𝜙₁cos𝜃, (2.9) where 𝐾_ℎ𝑡 is the coupling roll stiffness between the tractor and the trailer.

2.2. Suspension and Tyre Models

Suspension model is built to predict the lateral load transfer when cornering which affects the tyre normal force. In heavy vehicles, leaf spring is usually used and the suspension force which is nonlinear with the deformation of the leaf spring can be calculated as [15] follows: Δ𝐹^lat_𝑧𝑖=𝐾_𝑠11𝑒_𝑖+𝐾_𝑠12𝑒⁵_𝑖+𝐷_𝑠1𝑒_𝑖̇𝐾,𝑖=1,2_𝑠21𝑒_𝑖+𝐾_𝑠22𝑒⁵_𝑖+𝐷_𝑠2𝑒_𝑖̇𝐾,𝑖=3,4_𝑠31𝑒_𝑖+𝐾_𝑠32𝑒⁵_𝑖+𝐷_𝑠3𝑒_𝑖̇,𝑖=5,6, (2.10) where 𝐾_(⋅), and 𝐷_(⋅) are, respectively, the stiffness coefficients and damping coefficients; 𝑒_𝑖 is the deformation of the 𝑖th suspension which can be formulated as 𝑒₁=𝑡_𝑤12𝜙₁,𝑒₂𝑡=−^𝑤12𝜙₁,𝑒₃=𝑡_𝑤22𝜙₁,𝑒₄𝑡=−^𝑤22𝜙₁,𝑒₅=𝑡_𝑤32𝜙₂cos𝜃+𝑙₂𝜙₂sin𝜃,𝑒₆𝑡=−^𝑤32𝜙₂cos𝜃+𝑙₂𝜙₂sin𝜃. (2.11) In order to capture the actual property of the tyre force in critical situations, nonlinear trye mode is introduced. The tyre normal load which relates the body motions to the tyre forces should be specially considered in the tyre force modeling. The tyre normal load for each wheel is, respectively, formulated as 𝐹_𝑧1=𝑚₁𝑔𝑏₁2𝑙₁+𝑚₂𝑔𝑏₂𝑏₁−𝑙_𝑝2𝑙₁𝑙₂+12𝐹^lon₁+Δ𝐹^lat_𝑧1,𝐹_𝑧2=𝑚₁𝑔𝑏₁2𝑙₁+𝑚₂𝑔𝑏₂𝑏₁−𝑙_𝑝2𝑙₁𝑙₂+12𝐹^lon₁+Δ𝐹^lat_𝑧2,𝐹_𝑧3=𝑚₁𝑔𝑎₁2𝑙₁+𝑚₂𝑔𝑏₂𝑎₁+𝑙_𝑝2𝑙₁𝑙₂+12𝐹^lon₂+Δ𝐹^lat_𝑧3,𝐹_𝑧4=𝑚₁𝑔𝑎₁2𝑙₁+𝑚₂𝑔𝑏₂𝑎₁+𝑙_𝑝2𝑙₁𝑙₂+12𝐹^lon₂+Δ𝐹^lat_𝑧4,𝐹_𝑧5=𝑚₂𝑔𝑎₂2𝑙₂+𝑚₂𝑎_𝑥2ℎ_𝑐22𝑙₂+Δ𝐹^lat_𝑧5,𝐹_𝑧6=𝑚₂𝑔𝑎₂2𝑙₂+𝑚₂𝑎_𝑥2ℎ_𝑐22𝑙₂+Δ𝐹^lat_𝑧6, (2.12) with 𝐹^lon₁𝑚=−¹𝑎_𝑥1ℎ_𝑐1𝑙₁−𝑚₂𝑎_𝑥2ℎ_𝑐2𝑏₁−𝑙_𝑝𝑙₁𝑙₂,𝐹^lon₂=𝑚₁𝑎_𝑥1ℎ_𝑐1𝑙₁−𝑚₂𝑎_𝑥2ℎ_𝑐2𝑏₁−𝑙_𝑝𝑙₁𝑙₂. (2.13)

Nonlinear Dugoff model [16] is used here which can well characterize the actual tyre force in critical situations. For the 𝑖th wheel, the longitudinal (driving/braking) tyre force can be formulated as 𝐹_𝑥𝑖=𝐶_𝑥𝑖𝜆_𝑖1−𝜆_𝑖𝑓𝑆_𝑖𝐹,𝑖=1,2…6,_𝑦𝑖=𝐶_𝑦𝑖tan𝛼_𝑖1−𝜆_𝑖𝑓𝑆_𝑖,𝑖=1,2…6, (2.14) with 𝑆_𝑖=𝜇𝐹_𝑧𝑖1−𝜀_𝑟𝑣_𝑥𝑤𝑖𝜆²_𝑖+tan²𝛼_𝑖2𝐶²_𝑥𝑖𝜆²_𝑖+𝐶²_𝑦𝑖tan²𝛼_𝑖1−𝜆_𝑖,𝑓𝑆_𝑖=1,𝑆_𝑖𝑆>1_𝑖2−𝑆_𝑖,𝑆_𝑖<1, (2.15) where 𝐶_𝑥𝑖 and 𝐶_𝑦𝑖 are, respectively, the tyre longitudinal and lateral stiffness and it is assumed that the tyres on the same axle are with the same stiffness. 𝛼_𝑖 and 𝜆_𝑖 are the tyre side slip angle and longitudinal slip ratio which can be derived from the vehicle physical geometry and vehicle states and will not be given here in detail any more for the space limit.

For the tyre 𝑖 with the active brake torque 𝑇_𝑏𝑖, the wheel rotational dynamics can be formulated by 𝐼_𝑤𝑖𝜔̇_𝑤𝑖=−𝑇_𝑏𝑖−𝑅_𝑤𝑖𝐹_𝑥𝑖,𝑖=1,2…6, (2.16) where 𝑅_𝑤𝑖 is the wheel rolling radius, 𝐼_𝑤𝑖 the rotary moment of inertia, 𝜔_𝑤𝑖 the wheel angular velocity, and 𝑇_𝑏𝑖 the wheel brake torque of the 𝑖th wheel.

3. Reconfigurable Control System Design

The reconfigurable control system as a two-level structure consists of two primary parts which are the upper level SMYC and the lower level BFD. In addition, state observer used to estimate vehicle states such as slip angle, hitch angle, and hitch angular rate and the brake force limit estimator supplying the upper bound of brake forces to the CWLS optimization as the supplementary modules are also included. The overall structure of the configurable control system is presented in Figure 2. As shown by the figure, Δ𝐌=[Δ𝑀₁Δ𝑀₂]^𝑇 is the corrective yaw moment from the upper level SMYC for the tractor and the trailer and 𝐓_𝑏=[𝑇_𝑏1𝑇_𝑏2⋯𝑇_𝑏6]^𝑇 is the reconfigured brake torque from the lower level BFD for all wheels.

Figure 2 

Overall structure of the reconfigurable control system.

3.1. Upper Level SMYC Design

3.1.1. Reference Response

Model-following technique which is often used in vehicle stability control system, that is, the actual vehicle responses follow the reference or the desired responses produced by an on board model [9, 17, 18], is employed here. The on board model used to produce the reference response is derived from a linear single-track 4-DOF tractor-trailer model as shown in Figure 3 and the corresponding motion equations can be referred in detail to [2, 19].

Figure 3 

4DOF single-track tractor-trailer schematic model.

The reference response of the tractor yaw rate in steady-state with the front wheel steer input 𝛿_𝑓 at the speed of 𝑣_𝑥 is given by 𝛾_1𝑑=𝑣_𝑥/𝑙₁1+𝐾_𝑠𝑣²_𝑥𝛿_𝑓, (3.1) with 𝐾_𝑠=𝑏₁𝑙₂𝑚₁+𝑏₁−𝑙_𝑝𝑏₂𝑚₂𝑙²₁𝑙₂𝐶_𝑦𝑓−𝑎₁𝑙₂𝑚₁+𝑎₁+𝑙_𝑝𝑏₂𝑚₂𝑙²₁𝑙₂𝐶_𝑦𝑟, (3.2) where 𝐶_𝑦𝑓, and 𝐶_𝑦𝑟 are, respectively, the linear cornering stiffness of the tractor front and rear axle in the 4-DOF single track model. In fact, the yaw rate reference response (3.1) is restricted by the tyre adhesion limit, that is, 𝛾_1𝑑𝑣_𝑥≤𝜇𝑔. (3.3) So the dynamic reference response of the tractor yaw rate may be reasonably formulated as 𝛾_1𝑑=𝛾_1𝑑,𝛾_1𝑑𝑣_𝑥<𝜇𝑔,𝜇𝑔𝑣_𝑥,𝛾_1𝑑𝑣_𝑥≥𝜇𝑔. (3.4) Similarly, the reference response of the hitch angle can be obtained as well from the linear 4-DOF model as 𝜃_𝑑=𝑝₁+𝑝₂+𝑝₃𝑣²_𝑥1+𝐾_𝑠𝑣²_𝑥𝛿_𝑓, (3.5) with 𝑝₁𝑙=−^𝑝−𝑏₁+𝑙₂𝑙₁,𝑝₂=𝑎₂𝑚₂𝑙₁𝑙₂𝐶_𝑦𝑡,𝑝₃𝑎=−¹𝑙₂𝑚₁+𝑎₁+𝑙_𝑝𝑏₂𝑚₂𝑙²₁𝑙₂𝐶_𝑦𝑟, (3.6) here 𝐶_𝑦𝑡 is the linear cornering stiffness of the trailer axle in the 4-DOF single track model. Then the reference response of the hitch angular rate can thereby be derived as follows: 𝜃̇_𝑑=𝜕𝜃_𝑑𝜕𝑣_𝑥𝑑𝑣_𝑥=2𝐾𝑑𝑡^𝑢1Δ𝑙−𝐾_𝑢2𝑙₁𝑣_𝑥𝑣̇_𝑥𝑙₁+𝐾_𝑢1𝑣²_𝑥²𝛿_𝑓, (3.7) with 𝐾_𝑢1=𝑏₁𝑙₂𝑚₁+𝑏₁−𝑙_𝑝𝑏₂𝑚₂𝑙₁𝑙₂𝐶_𝑦𝑓−𝑎₁𝑙₂𝑚₁+𝑎₁+𝑙_𝑝𝑏₂𝑚₂𝑙₁𝑙₂𝐶_𝑦𝑟,𝐾_𝑢2=𝑎₁𝑙₂𝑚₁+𝑎₁+𝑙_𝑝𝑏₂𝑚₂𝑙₁𝑙₂𝐶_𝑦𝑟−𝑎₂𝑚₂𝑙₂𝐶_𝑦𝑡,Δ𝑙=𝑙_𝑝−𝑏₁+𝑙₂. (3.8)

In vehicle stability control system design based on model following method, the reference slip angle may be derived from the linear single-track model or directly set zero by simplicity which is also viable and can even obtain satisfying control performance [8, 9, 17, 18]. A zero reference slip angle for the trailer is employed here to be followed in the controller design, that is, 𝛽_2𝑑=0.

3.1.2. SMYC Design

The yaw dynamics of the tractor and the trailer are individually stabilized by a corrective yaw moment to, respectively, follow the reference responses of the tractor yaw rate and the combination of the hitch angular rate and trailer slip angle. Sliding mode control method which is robust against parametric uncertainties is employed to complete the control objective. Define the system states 𝐱=[𝑣_𝑦1𝛾₁𝜃𝜃̇]^𝑇=[𝑥₁𝑥₂𝑥₃𝑥₄]^𝑇 and the 4-DOF single-track model equations with the active corrective yaw moment Δ𝐌=[Δ𝑀₁Δ𝑀₂]^𝑇 can be rewritten as 𝐱̇=𝐀𝐱+𝐁₁𝑢₁+𝐁₂𝐮₂, (3.9) with 𝐀∈𝑅^4×4, 𝐀=𝐌⁻¹𝐕, 𝐁₁∈𝑅^4×1, 𝐁₁=𝐌⁻¹𝐖, 𝐁₂∈𝑅^4×2, 𝐁₂=𝐌⁻¹𝐘, 𝑢₁=𝛿_𝑓, and 𝐮₂=Δ𝐌, where 𝑚𝐌=¹𝑙_𝑝𝐼_𝑧1𝑚00₁+𝑚₂−𝑚₂𝑙_𝑝+𝑎₂0−𝑚₂𝑎₂𝑚₁𝑎₂𝐼_𝑧20𝐼_𝑧2,10010𝐕=𝑣_𝑥𝑣₁₁𝑣₁₂𝑣00₂₁𝑣₂₂𝐶_𝑦𝑡𝑣_𝑥𝑙₂𝐶_𝑦𝑡𝑣₃₁𝑣₃₂−𝑏₂𝐶_𝑦𝑡𝑣_𝑥−𝑏₂𝑙₂𝐶_𝑦𝑡000𝑣_𝑥,𝑣₁₁=−𝑎₁+𝑙_𝑝𝐶_𝑦𝑓−𝑙_𝑝−𝑏₁𝐶_𝑦𝑟,𝑣₂₁=−𝐶_𝑦𝑓+𝐶_𝑦𝑟+𝐶_𝑦𝑡,𝑣₃₁=−𝐶_𝑦𝑓𝑎₂−𝐶_𝑦𝑟𝑎₂+𝐶_𝑦𝑡𝑏₂,𝑣₁₂=−𝑎₁𝑎₁+𝑙_𝑝𝐶_𝑦𝑓−𝑏₁𝑏₁−𝑙_𝑝𝐶_𝑦𝑟−𝑣²_𝑥𝑙_𝑝𝑚₁,𝑣₂₂=−𝑎₁𝐶_𝑦𝑓+𝑏₁𝐶_𝑦𝑟+𝑙₂+𝑙_𝑝𝐶_𝑦𝑡−𝑚₁+𝑚₂𝑣²_𝑥,𝑣₃₂=−𝑎₁𝑎₂𝐶_𝑦𝑓+𝑎₂𝑏₁𝐶_𝑦𝑟−𝑏₂𝑙₂+𝑙_𝑝𝐶_𝑦𝑡−𝑚₁𝑎₂𝑣²_𝑥,𝑎𝐖=¹+𝑙_𝑝𝐶_𝑦𝑓𝐶_𝑦𝑓𝑎₂𝐶_𝑦𝑓0,𝐘=.10000100 (3.10) Two sliding surfaces are, respectively, designed for the control of the tractor and the trailer as 𝑠₁=𝑥₂−𝑥_2𝑑,𝑠₂=𝜉₁𝛽₂−𝛽_2𝑑+𝜉₂𝑥₄−𝑥_4𝑑,0≤𝜉₁,𝜉₂≤1, (3.11) with𝛽₂=𝑥₁𝑣_𝑥−𝑙_𝑝𝑣_𝑥𝑥₂−𝑎₂𝑣_𝑥𝑥₂+𝑥₄,𝛽_2𝑑=0. (3.12) Then the attractive equations can be formulated as 𝑠₁̇=𝑥̇₂−𝑥̇_2𝑑=−𝜀₁₁sgn𝑠₁−𝜀₁₂𝑠₁,𝑠₂̇=𝜉₁𝑥̇₁𝑣_𝑥−𝑙_𝑝𝑣_𝑥𝑥̇₂−𝑎₂𝑣_𝑥𝑥̇₂+𝑥̇₄+𝜉₂𝑥̇₄−𝑥̇_4𝑑=−𝜀₂₁sgn𝑠₂−𝜀₂₂𝑠₂, (3.13)(3.14) where 𝜀_𝑖𝑗 (𝑖,𝑗=1,2) are positive, and sgn(𝑠_1,2) is the sign function. Parameters 𝜉₁, 𝜉₂ in the sliding surface (3.14) are designed to regulate the control weight between the yaw dynamics and the lateral stability of the trailer. The derivatives of the reference responses of the tractor yaw rate, the hitch angular rate, and trailer slip angle are, respectively, given as follows: 𝑥̇_2𝑑=𝜕𝑥_2𝑑𝜕𝑣_𝑥𝑑𝑣_𝑥=𝑑𝑡1−𝐾_𝑠𝑣²_𝑥𝑣̇_𝑥1+𝐾_𝑠𝑣²_𝑥𝑙₁𝛿_𝑓,𝑥̇_4𝑑=𝜕𝑥_4𝑑𝜕𝑣_𝑥𝑑𝑣_𝑥=2𝐾𝑑𝑡^𝑢1Δ𝑙−𝐾_𝑢2𝑙₁𝑙₁−3𝐾_𝑢1𝑣²_𝑥𝑙₁+𝐾_𝑢1𝑣²_𝑥³𝑣̇²_𝑥𝛿_𝑓. (3.15) Attractive equations given by (3.13) and (3.14) can guarantee the deviations of state response asymptotically converge to zero. However, the positive constants 𝜀_𝑖𝑗 still should be selected properly to improve the control and response quality.

Combining the 4-DOF vehicle model (3.9) and the attractive equations (3.13) and (3.14), the corrective yaw moment control law, respectively, for the tractor and the trailer can be formulated as Δ𝑀₁=1𝜉₂𝑣_𝑥𝑚₁+𝑚₂−𝜉₁𝑎₂𝑚₁×2−𝜉₂𝑣_𝑥𝑚₂𝑎₁+𝑚₂𝑙_𝑝+𝑎₁𝑚₁+𝜉₁𝑎₁𝑎₂𝑚₁𝐹_𝑦𝑓+2𝑏₁𝜉₂𝑣_𝑥𝑚₁+𝜉₂𝑣_𝑥𝑏₁−𝑙_𝑝𝑚₂−𝜉₁𝑎₂𝑏₁𝑚₁𝐹_𝑦𝑟+2𝑚₁𝑙_𝑝𝜉₂𝑣_𝑥−𝜉₁𝑎₂+𝜉₁𝑥₄𝐹_𝑦𝑡+𝜉₂𝑣_𝑥𝑙_𝑝𝑚₁𝑚₂−𝑣_𝑥𝑥₄+𝑎₂𝑥̇_4𝑑+𝜉₁𝑣_𝑥𝑙_𝑝𝑚₁𝑚₂𝑎₂𝑥₂+𝑥₄+𝑚₁𝐼_𝑧1𝜉₁𝑎₂sgn𝑠₁𝜀₁₁+𝑠₁𝜀₁₂−𝑥̇_2𝑑−𝑚₁+𝑚₂𝐼_𝑧1𝜉₂𝑣_𝑥sgn𝑠₁𝜀₁₁+𝑠₁𝜀₁₂−𝑥̇_2𝑑−𝜉₂𝑣_𝑥𝑚₁𝑚₂𝑙_𝑝𝑎₂+𝑙_𝑝sgn𝑠₁𝜀₁₁+𝑠₁𝜀₁₂−𝑥̇_2𝑑+𝑎₂𝑙_𝑝𝑚₁𝑚₂𝑣_𝑥sgn𝑠₂𝜀₂₁−𝑠₂𝜀₂₂,Δ𝑀₂=1𝜉₂𝑣_𝑥𝑚₁+𝑚₂−𝜉₁𝑎₂𝑚₁×−2𝜉₁𝐼_𝑧2+𝜉₂𝑣_𝑥𝑎₂𝑚₂𝐹_𝑦𝑓+𝐹_𝑦𝑟+2−𝜉₁𝑎₂𝑙₂𝑚₁+𝜉₂𝑣_𝑥𝑚₁𝑙₂+𝑚₂𝑏₂−2𝜉₁𝑚₁𝐼_𝑧2𝐹_𝑦𝑡+−𝜉₂𝑣_𝑥𝑚₂𝑚₁𝑎²₂+𝑎₂𝑙_𝑝sgn𝑠₁𝜀₁₁+𝑠₁𝜀₁₂−𝑥̇_1𝑑−𝐼_𝑧2𝜉₂𝑣_𝑥𝑚₁+𝑚₂sgn𝑠₁𝜀₁₁+𝑠₁𝜀₁₂−𝑥̇_2𝑑−𝐼_𝑧2𝜉₁𝑙_𝑝𝑚₁sgn𝑠₁𝜀₁₁+𝑠₁𝜀₁₂−𝑥̇_2𝑑−𝑚₁𝑚₂𝑣_𝑥𝑎²₂sgn𝑠₂𝜀₂₁+𝑠₂𝜀₂₂−𝜉₁𝑥₄−𝜉₂𝑥̇_4𝑑−𝑣_𝑥𝐼_𝑧2𝑚₁+𝑚₂sgn𝑠₂𝜀₂₁+𝑠₂𝜀₂₂−𝜉₁𝑥₄−𝜉₂𝑥̇_4𝑑+𝑚₁𝑣_𝑥𝑚₂𝑎₂𝜉₂𝑣_𝑥+𝜉₁𝐼_𝑧2𝑥₂+𝑚₂𝑣_𝑥𝜉₁𝐼_𝑧2+𝑚₁𝑎²₂−𝜉₂𝑎₂𝑣_𝑥𝑚₁𝑥₂+𝑥₄, (3.16) where the tyre force in the control law are calculated from a linear tyre model with nominal tyre cornering stiffness; those are 𝐹_𝑦𝑓=𝐶_𝑦𝑓𝛼_𝑓, 𝐹_𝑦𝑟=𝐶_𝑦𝑟𝛼_𝑟, 𝐹_𝑦𝑡=𝐶_𝑦𝑡𝛼_𝑡. 𝛼_𝑓, 𝛼_𝑟, and 𝛼_𝑡, respectively, which are related to the side slip angle of the front, rear axle of the tractor and the trailer axle in the linear single-track model, and 𝐶_𝑦𝑓, 𝐶_𝑦𝑟, and 𝐶_𝑦𝑡 are, respectively, the corresponding cornering stiffness. Though the cornering stiffness is assumed constant, it is still feasible since sliding mode control is robust and can make the actual states track the reference responses.

In real implementation, some vehicle states such as the tractor/trailer slip angle and hitch angle/angular rate. unsuitable to be measured directly by sensors for the cost considerations can be obtained by an observer since numerous state estimation algorithms have been proposed in the literature [20, 21]. For the space limit, the state observer is not presented here in detail and assuming the states can be obtained directly in simulations.

3.2. Lower Level BFD Design

After the upper level SMYC is designed, the following step is to develop the lower level BFD to map the corrective yaw moment Δ𝐌 from the upper level to a set of physical vehicle actuator commands/brake forces. The key issue of which is how to select a set of brake forces from all possible combinations since the solution to achieve the desired control is not unique for redundantly actuated systems. The problem concerned with CA can generally be formulated as 𝐯=𝐁𝐮,𝐮_min≤𝐮≤𝐮_max, (3.17) where 𝐯 is the virtual control vector with 𝐯=Δ𝐌 in this study; 𝐮 is the actuator effort vector corresponding to the brake force vector, that is, 𝐮=[𝐹_𝑥1,𝐹_𝑥2,…,𝐹_𝑥6]^𝑇; 𝐮_min and 𝐮_max are the actuator effort limits corresponding to the brake force limits, here mainly affected by the adhesion coefficient; 𝐁 is the control effectiveness matrix and can be further expressed as 𝐁=𝐁𝜿, (3.18) with 𝐁 denoting the input matrix, 𝜿 the actuator effectiveness matrix which is used to express the effectiveness level of the six wheel brake force. Define 𝜿=diag𝑘₁,𝑘₂,…,𝑘₆,0≤𝑘_𝑖≤1, (3.19) and for the 𝑖th wheel, 𝑘_𝑖=1 means this wheel brake actuator works in the normal control situation without any failure, 𝑘_𝑖=0 means the actuator is completely invalid due to the actuator failure and cannot produce any brake force while 0<𝑘_𝑖<1 means the actuator partly falls in failure and can only provide part brake force compared to the normal situation of 𝑘_𝑖=1.

With small steer angle and hitch angle assumptions, (2.3) and (2.7) can be rewritten as 𝐼_𝑧1𝛾̇₁=𝐹_𝑦1+𝐹_𝑦2𝑎₁−𝐹_𝑦3+𝐹_𝑦4𝑏₁+Δ𝑀₁+𝐹_ℎ𝑦𝑙_𝑝,𝐼_𝑧2𝛾̇₂=𝐹_ℎ𝑦𝑎₂−𝐹_𝑦5+𝐹_𝑦6𝑏₂+Δ𝑀₂, (3.20) with Δ𝑀₁=𝐹_𝑥2−𝐹_𝑥1𝑡_𝑤12+𝐹_𝑥4−𝐹_𝑥3𝑡_𝑤22,Δ𝑀₂=𝐹_𝑥6−𝐹_𝑥5𝑡_𝑤32. (3.21) Then the input matrix can be defined as −𝑡𝐁=2^𝑤1𝑡_𝑤12−𝑡_𝑤22𝑡_𝑤22𝑡000000−^𝑤32𝑡_𝑤32. (3.22) As can be observed from (3.22), the improvements of vehicle attitude by active wheel braking for the tractor and the trailer are uncoupled with each other for the purpose of simplifying the CA problem which is not the case actually. However, it is not necessary to worry about that since the corrective yaw moments Δ𝑀₁ and Δ𝑀₂ from the upper level SMYC are coupled.

In order to solve the CA problem (3.17), the actuator limits 𝐮_min and 𝐮_max should be determined first. In vehicle dynamics, the brake force limit for a certain wheel is not constant but varies with the driving conditions since the longitudinal tyre force is generally affected by the road friction adhesion coefficient and tyre normal force which directly relates to the maneuver. Estimating the brake force limit accurately in real time plays a key role in enhancing the reconfigurable control performance. Obviously, the brake force is negative according to the definition given by Figure 1. Now define the brake force limit of the 𝑖th wheel as 𝐹_𝑥𝑚𝑖≤𝐹_𝑥𝑖≤0, (3.23) with 𝐹_𝑥𝑚𝑖=𝜎_𝑖𝐹_𝑥0𝑖, (3.24) where 𝜎_𝑖 relates the longitudinal force limit to the normal force variation. According to the actual tyre force property [16, 22], define 𝜎_𝑖=𝜏_𝑖sin𝑐₁arctan𝑐₂𝜏_𝑖+𝑐₃arctan𝑐₄𝜏_𝑖, (3.25) where 𝑐₁∼𝑐₄ are experiential constants, and 𝜏_𝑖 is the ratio of the dynamic tyre normal force to the static value, that is, 𝜏_𝑖=𝐹_𝑧𝑖𝐹_𝑧0𝑖. (3.26) In real implementation, the dynamic tyre normal force 𝐹_𝑧𝑖 can be estimated according to (2.12) by the information of longitudinal and lateral accelerations which can be measured in real time. With the specified values of 𝑐₁=1.3, 𝑐₂=20, 𝑐₃=−1.99, and 𝑐₄=0.3 in this study and assuming 𝐹_𝑥𝑚0=10000𝑁, the influence of the tyre normal force on the brake force limit is illustrated by Figure 4. As shown in the figure, the proposed estimated brake force limit which is nonlinear against the tyre normal force is consistent with the actual tyre force property.

Figure 4 

Influence of the tyre normal force to the brake force limit.

For the CA problem formulated by (3.17), it can be understood as follows. Given a virtual control command 𝐯, determine 𝐮 satisfying the actuator limit 𝐮_min≤𝐮≤𝐮_max such that 𝐯=𝐁𝐮. If there exist more than one solution, take the best one. Otherwise, if there is no accurate solution, determine 𝐮 such that 𝐁𝐮 approximates 𝐯 as possible. In order to follow the reference responses of the on board model as well as the consideration of the brake force limits, the brake force optimization allocation is equivalent to a problem of error minimization and control minimization combination which can be formulated by CWLS optimization problem as follows: min𝜍𝐖_𝐮𝐮−𝐮_𝐝²₂+(1−𝜍)𝐖_𝐯(𝐁𝐮−𝐯)²₂s.t.𝐮_min≤𝐮≤𝐮_max, (3.27) where 𝐮_𝐝 is the preferred control effort vector and take 𝐮_𝐝=0 here to minimize the brake force effort; 𝐖_𝐮 and 𝐖_𝐯 are nonsingular weighting matrices with 𝐖_𝐮=diag𝑊_𝑢1,𝑊_𝑢2,…,𝑊_𝑢6,𝐖_𝐯=diag𝑊_𝑣1,𝑊_𝑣2, (3.28) where 𝜍 is the weighting factor to weight the actuator effort and control error. The brake force lower bound vector 𝐮_min is derived from brake force limit estimation in real time by (3.24)–(3.26), that is, 𝐮_min=[𝐹_𝑥𝑚1,𝐹_𝑥𝑚2,…,𝐹_𝑥𝑚6]^𝑇. The upper bound 𝐮_max is set zero.

Since 𝜍𝐖_𝐮𝐮−𝐮_𝐝²₂+(1−𝜍)𝐖_𝐯(𝐁𝐮−𝐯)²₂=(1−𝜍)𝐖_𝐯𝐁√𝜍𝐖_𝐮𝐮−(1−𝜍)𝐖_𝐯𝐯√𝜍𝐖_𝐮𝐮_𝐝²₂=𝐁𝐮−𝐯²₂, (3.29) the CA problem (3.27) is equivalent to the optimization problem 𝐮=argmin_{𝐮min≤𝐮≤𝐮max}𝐁𝐮−𝐯²₂, (3.30) with 𝐁=(1−𝜍)𝐖_𝐯𝐁√𝜍𝐖_𝐮,𝐯=(1−𝜍)𝐖_𝐯𝐯√𝜍𝐖_𝐮𝐮_𝐝, (3.31) which can be solved with the active set (AS) algorithm [23]. This method starts at an initial guess of the solution. The elements of 𝐮 which are on the border of the feasible set are made fixated, and other elements are left completely free. For these free elements the optimal solution is calculated.

4. Simulation Results

Single lane change maneuver is carried out in this section to evaluate the proposed reconfigurable control scheme based on CA technique by numerical simulations on the 14-DOF nonlinear vehicle model constructed in Section 2. Three cases which are normal control, actuator failure control, and uncontrolled are considered in the simulations. The normal control relates to the case where the wheel brake actuator works well without any failure; in the actuator failure control case the brake actuator partly falls in failure and can only provide part brake force compared to the normal control case while the uncontrolled case means without active control applied to the vehicle to correct the yaw dynamics. One cycle sinusoid signal with the amplitude of 0.08 rad is used for the front wheel steer angle for the single lane change maneuver illustrated by Figure 5. Vehicle parameters in the simulation are given by Table 1.

Table 1 

Vehicle parameters.

Figure 5 

Front wheel steer angle.

The normal case and the failure case are realized through setting the actuator effectiveness matrix equation (3.19). For the normal control case, set 𝜿=diag(111111) and set 𝜿=diag(0.010.010.0050.00500) for the failure case which means that the front and rear axle brake actuators are partly failed while the trailer axle brake actuators are completely failed. Setting the weighting matrices 𝐖_𝐮=diag(1.01.01.51.51.01.0), 𝐖_𝐯=diag(1.0,1.0), the weighting factor 𝜍=0.2, and the initial vehicle speed of 22 m/s, conduct the single lane change maneuver simulation, respectively, in the three cases and the state response results are shown in Figures 6, 7, and 8 including the tractor and trailer yaw rate, slip angle, hitch angle and vehicle speed.

Figure 6 

Yaw response comparison.

Figure 7 

Side slip response comparison.

Figure 8 

Vehicle speed response comparison.

As given by the figures, in the normal control case with the proposed reconfigurable control scheme, the vehicle represents satisfying response performance with respect to the steer input except for a bit tractor side slip. Further, in the failure control case where the trailer brake actuators are completely fails and the tractor actuators are heavy partly fails, owing to the reconfigurable coordination of the remaining functioning actuators/efforts, the vehicle can still keep stable and can generally follow the steer input except for a bit decrement of response performance compared to the normal control case. However, in the uncontrolled case, the vehicle loses control rapidly. Due to the brake actuator damage, the brake force/torque for the failure case decreases considerably and the vehicle speed shall be relatively lightly affected compared to the normal case as illustrated by Figure 8.

Figures 9 and 10 illustrated the estimations of the brake force limit of all wheels in the lane change maneuver. Brake force limit which is directly related to the vehicle handling attitude is estimated in real-time for the CWLS optimization based CA algorithm to determine the optimal brake force effort. Comparing the two cases, we can find that the difference of vehicle response leads to different brake force limit. With the estimated brake force limit, the optimized brake forces are obtained by solving the CA problem (3.27), and by multiplying the wheel radius the reconfigured optimal brake torques can be achieved and are illustrated in Figures 11 and 12, respectively, for the normal and failure case. It is obvious that the damaged but still functioning actuators 1–4 are reconfigured optimally to realize the corrective yaw moment from the upper level SMYC as possible. Note that no brake effort has been assigned to the trailer wheels since the actuators of which are completely failed.

Figure 9 

Brake force limit estimation in the normal control case.

Figure 10 

Brake force limit estimation in the failure control case.

Figure 11 

Brake torque effort for the normal control case.

Figure 12 

Brake torque effort for the failure control case.

Similarly, the difference in wheel longitudinal slip ratio can also be found as shown in Figures 13 and 14 which corresponds to the brake force. All slip ratios are below 0.2 without wheel locking for heavy brake. Figures 15 and 16, respectively, show the realization of the corrective yaw moment vector 𝐯=Δ𝐌 from the upper level SMYC for the normal and fail case. Since the reference corrective yaw moment Δ𝐌 is calculated by sliding mode control to follow the reference response of the on board model, it does not mean that the corrective yaw moment can be realized completely by active wheel brake due to the physical limit of the brake force. So the actual produced corrective yaw moment presents a remarkable gap to the desired value from SMYC and this is much more remarkable for the failure control case due to the fail of actuator.

Figure 13 

Longitudinal slip ratio in the normal control case.

Figure 14 

Longitudinal slip ratio in the failure control case.

Figure 15 

Corrective yaw moment in the normal control case.

Figure 16 

Brake force limit estimation in the failure control case.

The above simulations and analysis reveal that reconfigurable control method based on CA technique is feasible to be used in the stability control system design of tractor-semitrailer double-body vehicle. Especially, the reconfigurable control system can still keep the vehicle stable in the situation of brake actuator being damaged greatly by reconfiguring the remaining functioning actuators/efforts and thereby the reliability and safety are improved compared with the conventional single wheel brake method.

5. Conclusions

In order to evaluate the proposed reconfigurable control scheme, a 14-DOF nonlinear model is constructed for a typical five-axle tractor-semitrailer vehicle which can generally reflect the actual vehicle dynamics especially in critical driving situations. A two-level structure reconfigurable control scheme is proposed with the upper level of SMYC and the lower level of yaw moment allocator based on CWLS optimization algorithm. As a particular feature of this control scheme, the brake force limit as one of the required conditions for the CWLS optimization generally varying with the handling maneuver is estimated in real time to improve the CA performance. The reconfigurable control issue based on CA technique is transformed to a combination of error minimization and control minimization problem formulated by CWLS optimization and solved with AS algorithm. Simulations conducted on the 14-DOF nonlinear model reveal that the CA technique based reconfigurable control is rather significant for the tractor-semitrailer vehicle since its multiple-axle structure can enrich the alternatives of possible actuator combinations in CA optimization.

Acknowledgment

This project was supported by the National Natural Science Foundation of China (no. 51005109). The authors are greatly thankful for the financial support.