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

Authors: Xiujian Yang
Journal: Mathematical Problems in Engineering
 (2012)

---

## 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.


---

## Body

## 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 Section2, 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 Figure1. 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.Nonlinear tractor-semitrailer vehicle schematic model. (a)X-Y plane; (b) Z-X plane.
(a)
(b)


### 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
(2.1)m1(v˙x1-γ1vy1)=(Fx1+Fx2)cosδf+Fx3+Fx4-(Fy1+Fy2)sinδf+Fhx,(2.2)m1(v˙y1+γ1vx1)+m1shs1ϕ¨1=(Fy1+Fy2)cosδf+Fy3+Fy4+(Fx1+Fx2)sinδf-Fhy,(2.3)Iz1γ˙1=((Fy1-Fy2)sinδf+(Fx2-Fx1)cosδf)tw12-(Fy3+Fy4)b1+(Fx4-Fx3)tw22+((Fy1+Fy2)cosδf+(Fx1+Fx2)sinδf)a1+Fhylp,(2.4)Ix1ϕ¨1+m1s(v˙y1+γ1vx1)hs1=m1sghs1sinϕ1+(Fz2-Fz1)tw12+(Fz4-Fz3)tw22-(Fy1+Fy2)hf1cosδf-(Fy3+Fy4)hr1-Fhy(hp-hr1)+Mht,
wherem1 and m1s denote the total mass and the sprung mass of the tractor; Iz1 and Ix1 the yaw moment of inertia and roll moment of inertia of the tractor; a1, b1, and lp, respectively, the distance from the centre of mass (CM) of the tractor to its front, rear axle, and the hitch point; tw1, tw2 the track width of tractor front axle and rear axle; hf1 and hr1 the height of front, rear roll centre of the tractor; hs1 the distance from the tractor CM to its roll axis; hp the height of the hitch point; γ1, ϕ1, vx1, and vy1, respectively, the yaw rate, roll angle, longitudinal velocity and lateral velocity of the tractor; Fxi, Fyi, and Fzi (i=1~4), respectively, the longitudinal, lateral, and normal force of the tractor wheels; Fhx, and Fhy the longitudinal and lateral interaction force between the tractor and the trailer; Mht 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
(2.5)m2(v˙x2-γ2vy2)=Fx5+Fx6-Fhxcosθ+Fhysinθ,(2.6)m2(v˙y2+γ2vx2)+m2shs2ϕ¨2=Fy5+Fy6+Fhycosθ-Fhxsinθ,(2.7)Iz2γ˙2=(Fhycosθ-Fhxsinθ)a2-(Fy5+Fy6)b2+(Fx6-Fx5)tw32,(2.8)Ix2ϕ¨2+m2s(v˙y2+γ2vx2)hs2=(Fz6-Fz5)tw32+m2sghs2sinϕ2+Fhy(hp-hf2)cosθ-(Fy5+Fy6)hr2-Mht,
where m2 and m2s denote the total mass and sprung mass of the trailer; Iz2 and Ix2 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; tw3 the track width of the trailer axle; hf2 and hr2 the front and rear roll centre height of the trailer; hs2 the distance from the trailer CM to its roll axis; γ2, ϕ2, vx2, and vy2, respectively, the yaw rate, roll angle, longitudinal velocity, and lateral velocity of the trailer; θ the hitch angle between the tractor and the trailer; Fxi, Fyi, and Fzi (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
(2.9)vx2=vx1cosθ,vy2=(vy1-lpγ1)cosθ-a2γ2-vx1sinθ,γ2=γ1+θ˙,Mht=Kht(ϕ2-ϕ1cosθ),
where Kht 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:
(2.10)ΔFzilat={Ks11ei+Ks12ei5+Ds1e˙i,i=1,2Ks21ei+Ks22ei5+Ds2e˙i,i=3,4Ks31ei+Ks32ei5+Ds3e˙i,i=5,6,
where K(·), and D(·) are, respectively, the stiffness coefficients and damping coefficients; ei is the deformation of the ith suspension which can be formulated as
(2.11)e1=tw12ϕ1,e2=-tw12ϕ1,e3=tw22ϕ1,e4=-tw22ϕ1,e5=(tw32ϕ2)cosθ+l2ϕ2sinθ,e6=-(tw32ϕ2)cosθ+l2ϕ2sinθ.
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
(2.12)Fz1=m1gb12l1+m2gb2(b1-lp)2l1l2+12F1lon+ΔFz1lat,Fz2=m1gb12l1+m2gb2(b1-lp)2l1l2+12F1lon+ΔFz2lat,Fz3=m1ga12l1+m2gb2(a1+lp)2l1l2+12F2lon+ΔFz3lat,Fz4=m1ga12l1+m2gb2(a1+lp)2l1l2+12F2lon+ΔFz4lat,Fz5=m2ga22l2+m2ax2hc22l2+ΔFz5lat,Fz6=m2ga22l2+m2ax2hc22l2+ΔFz6lat,
with
(2.13)F1lon=-m1ax1hc1l1-m2ax2hc2(b1-lp)l1l2,F2lon=m1ax1hc1l1-m2ax2hc2(b1-lp)l1l2.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
(2.14)Fxi=Cxiλi1-λif(Si),i=1,2…6,Fyi=Cyitanαi1-λif(Si),i=1,2…6,
with
(2.15)Si=μFzi(1-εrvxwiλi2+tan2αi)2Cxi2λi2+Cyi2tan2αi(1-λi),f(Si)={1,Si>1Si(2-Si),Si<1,
where Cxi and Cyi are, respectively, the tyre longitudinal and lateral stiffness and it is assumed that the tyres on the same axle are with the same stiffness. αi and λi 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 tyrei with the active brake torque Tbi, the wheel rotational dynamics can be formulated by
(2.16)Iwiω˙wi=-Tbi-RwiFxi,i=1,2…6,
where Rwi is the wheel rolling radius, Iwi the rotary moment of inertia, ωwi the wheel angular velocity, and Tbi the wheel brake torque of the ith wheel.


## 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
(2.1)m1(v˙x1-γ1vy1)=(Fx1+Fx2)cosδf+Fx3+Fx4-(Fy1+Fy2)sinδf+Fhx,(2.2)m1(v˙y1+γ1vx1)+m1shs1ϕ¨1=(Fy1+Fy2)cosδf+Fy3+Fy4+(Fx1+Fx2)sinδf-Fhy,(2.3)Iz1γ˙1=((Fy1-Fy2)sinδf+(Fx2-Fx1)cosδf)tw12-(Fy3+Fy4)b1+(Fx4-Fx3)tw22+((Fy1+Fy2)cosδf+(Fx1+Fx2)sinδf)a1+Fhylp,(2.4)Ix1ϕ¨1+m1s(v˙y1+γ1vx1)hs1=m1sghs1sinϕ1+(Fz2-Fz1)tw12+(Fz4-Fz3)tw22-(Fy1+Fy2)hf1cosδf-(Fy3+Fy4)hr1-Fhy(hp-hr1)+Mht,
wherem1 and m1s denote the total mass and the sprung mass of the tractor; Iz1 and Ix1 the yaw moment of inertia and roll moment of inertia of the tractor; a1, b1, and lp, respectively, the distance from the centre of mass (CM) of the tractor to its front, rear axle, and the hitch point; tw1, tw2 the track width of tractor front axle and rear axle; hf1 and hr1 the height of front, rear roll centre of the tractor; hs1 the distance from the tractor CM to its roll axis; hp the height of the hitch point; γ1, ϕ1, vx1, and vy1, respectively, the yaw rate, roll angle, longitudinal velocity and lateral velocity of the tractor; Fxi, Fyi, and Fzi (i=1~4), respectively, the longitudinal, lateral, and normal force of the tractor wheels; Fhx, and Fhy the longitudinal and lateral interaction force between the tractor and the trailer; Mht 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
(2.5)m2(v˙x2-γ2vy2)=Fx5+Fx6-Fhxcosθ+Fhysinθ,(2.6)m2(v˙y2+γ2vx2)+m2shs2ϕ¨2=Fy5+Fy6+Fhycosθ-Fhxsinθ,(2.7)Iz2γ˙2=(Fhycosθ-Fhxsinθ)a2-(Fy5+Fy6)b2+(Fx6-Fx5)tw32,(2.8)Ix2ϕ¨2+m2s(v˙y2+γ2vx2)hs2=(Fz6-Fz5)tw32+m2sghs2sinϕ2+Fhy(hp-hf2)cosθ-(Fy5+Fy6)hr2-Mht,
where m2 and m2s denote the total mass and sprung mass of the trailer; Iz2 and Ix2 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; tw3 the track width of the trailer axle; hf2 and hr2 the front and rear roll centre height of the trailer; hs2 the distance from the trailer CM to its roll axis; γ2, ϕ2, vx2, and vy2, respectively, the yaw rate, roll angle, longitudinal velocity, and lateral velocity of the trailer; θ the hitch angle between the tractor and the trailer; Fxi, Fyi, and Fzi (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
(2.9)vx2=vx1cosθ,vy2=(vy1-lpγ1)cosθ-a2γ2-vx1sinθ,γ2=γ1+θ˙,Mht=Kht(ϕ2-ϕ1cosθ),
where Kht 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:
(2.10)ΔFzilat={Ks11ei+Ks12ei5+Ds1e˙i,i=1,2Ks21ei+Ks22ei5+Ds2e˙i,i=3,4Ks31ei+Ks32ei5+Ds3e˙i,i=5,6,
where K(·), and D(·) are, respectively, the stiffness coefficients and damping coefficients; ei is the deformation of the ith suspension which can be formulated as
(2.11)e1=tw12ϕ1,e2=-tw12ϕ1,e3=tw22ϕ1,e4=-tw22ϕ1,e5=(tw32ϕ2)cosθ+l2ϕ2sinθ,e6=-(tw32ϕ2)cosθ+l2ϕ2sinθ.
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
(2.12)Fz1=m1gb12l1+m2gb2(b1-lp)2l1l2+12F1lon+ΔFz1lat,Fz2=m1gb12l1+m2gb2(b1-lp)2l1l2+12F1lon+ΔFz2lat,Fz3=m1ga12l1+m2gb2(a1+lp)2l1l2+12F2lon+ΔFz3lat,Fz4=m1ga12l1+m2gb2(a1+lp)2l1l2+12F2lon+ΔFz4lat,Fz5=m2ga22l2+m2ax2hc22l2+ΔFz5lat,Fz6=m2ga22l2+m2ax2hc22l2+ΔFz6lat,
with
(2.13)F1lon=-m1ax1hc1l1-m2ax2hc2(b1-lp)l1l2,F2lon=m1ax1hc1l1-m2ax2hc2(b1-lp)l1l2.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
(2.14)Fxi=Cxiλi1-λif(Si),i=1,2…6,Fyi=Cyitanαi1-λif(Si),i=1,2…6,
with
(2.15)Si=μFzi(1-εrvxwiλi2+tan2αi)2Cxi2λi2+Cyi2tan2αi(1-λi),f(Si)={1,Si>1Si(2-Si),Si<1,
where Cxi and Cyi are, respectively, the tyre longitudinal and lateral stiffness and it is assumed that the tyres on the same axle are with the same stiffness. αi and λi 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 tyrei with the active brake torque Tbi, the wheel rotational dynamics can be formulated by
(2.16)Iwiω˙wi=-Tbi-RwiFxi,i=1,2…6,
where Rwi is the wheel rolling radius, Iwi the rotary moment of inertia, ωwi the wheel angular velocity, and Tbi 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 Figure2. 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 Tb=[Tb1Tb2⋯Tb6]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 inputδf at the speed of vx is given by
(3.1)γ-1d=vx/l11+Ksvx2δf,
with
(3.2)Ks=b1l2m1+(b1-lp)b2m2l12l2Cyf-a1l2m1+(a1+lp)b2m2l12l2Cyr,
where Cyf, and Cyr 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,
(3.3)|γ-1d|vx≤μg.
So the dynamic reference response of the tractor yaw rate may be reasonably formulated as
(3.4)γ1d={γ-1d,|γ-1d|vx<μg,μgvx,|γ-1d|vx≥μg.
Similarly, the reference response of the hitch angle can be obtained as well from the linear 4-DOF model as
(3.5)θd=p1+(p2+p3)vx21+Ksvx2δf,
with
(3.6)p1=-lp-b1+l2l1,p2=a2m2l1l2Cyt,p3=-a1l2m1+(a1+lp)b2m2l12l2Cyr,
here Cyt 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:
(3.7)θ˙d=∂θd∂vxdvxdt=2(Ku1Δl-Ku2l1)vxv˙x(l1+Ku1vx2)2δf,
with
(3.8)Ku1=b1l2m1+(b1-lp)b2m2l1l2Cyf-a1l2m1+(a1+lp)b2m2l1l2Cyr,Ku2=a1l2m1+(a1+lp)b2m2l1l2Cyr-a2m2l2Cyt,Δl=lp-b1+l2.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, β2d=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 statesx=[vy1γ1θθ˙]T=[x1x2x3x4]T and the 4-DOF single-track model equations with the active corrective yaw moment ΔM=[ΔM1ΔM2]T can be rewritten as
(3.9)x˙=Ax+B1u1+B2u2,
with A∈R4×4, A=M-1V, B1∈R4×1, B1=M-1W, B2∈R4×2, B2=M-1Y, u1=δf, and u2=ΔM, where
(3.10)M=(m1lpIz100m1+m2-m2(lp+a2)0-m2a2m1a2Iz20Iz20010),V=1vx(v11v1200v21v22Cytvxl2Cytv31v32-b2Cytvx-b2l2Cyt000vx),v11=-(a1+lp)Cyf-(lp-b1)Cyr,v21=-(Cyf+Cyr+Cyt),v31=-Cyfa2-Cyra2+Cytb2,v12=-a1(a1+lp)Cyf-b1(b1-lp)Cyr-vx2lpm1,v22=-a1Cyf+b1Cyr+(l2+lp)Cyt-(m1+m2)vx2,v32=-a1a2Cyf+a2b1Cyr-b2(l2+lp)Cyt-m1a2vx2,W=((a1+lp)CyfCyfa2Cyf0),Y=(10000100).
Two sliding surfaces are, respectively, designed for the control of the tractor and the trailer as
(3.11)s1=x2-x2d,s2=ξ1(β2-β2d)+ξ2(x4-x4d),0≤ξ1,ξ2≤1,
with(3.12)β2=x1vx-lpvxx2-a2vx(x2+x4),β2d=0.
Then the attractive equations can be formulated as
(3.13)s˙1=x˙2-x˙2d=-ε11sgn(s1)-ε12s1,(3.14)s˙2=ξ1(x˙1vx-lpvxx˙2-a2vx(x˙2+x˙4))+ξ2(x˙4-x˙4d)=-ε21sgn(s2)-ε22s2,
where εij (i,j=1,2) are positive, and sgn(s1,2) 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:
(3.15)x˙2d=∂x2d∂vxdvxdt=(1-Ksvx2)v˙x(1+Ksvx2)l1δf,x˙4d=∂x4d∂vxdvxdt=2(Ku1Δl-Ku2l1)(l1-3Ku1vx2)(l1+Ku1vx2)3v˙x2δf.
Attractive equations given by (3.13) and (3.14) can guarantee the deviations of state response asymptotically converge to zero. However, the positive constants εij 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
(3.16)ΔM1=1ξ2vx(m1+m2)-ξ1a2m1×{2[-ξ2vx(m2a1+m2lp+a1m1)+ξ1a1a2m1]Fyf+2[b1ξ2vxm1+ξ2vx(b1-lp)m2-ξ1a2b1m1]Fyr+[2m1lp(ξ2vx-ξ1a2+ξ1x4)]Fyt+ξ2vxlpm1m2(-vxx4+a2x˙4d)+ξ1vxlpm1m2a2(x2+x4)+m1Iz1ξ1a2(sgn(s1)ε11+s1ε12-x˙2d)-(m1+m2)Iz1ξ2vx(sgn(s1)ε11+s1ε12-x˙2d)-[ξ2vxm1m2lp(a2+lp)](sgn(s1)ε11+s1ε12-x˙2d)+a2lpm1m2vx(sgn(s2)ε21-s2ε22)},ΔM2=1ξ2vx(m1+m2)-ξ1a2m1×{[-2(ξ1Iz2+ξ2vxa2m2)](Fyf+Fyr)+[2(-ξ1a2l2m1+ξ2vx(m1l2+m2b2-2ξ1m1Iz2))]Fyt+[-ξ2vxm2m1(a22+a2lp)](sgn(s1)ε11+s1ε12-x˙1d)-Iz2ξ2vx(m1+m2)(sgn(s1)ε11+s1ε12-x˙2d)-Iz2ξ1lpm1(sgn(s1)ε11+s1ε12-x˙2d)-m1m2vxa22(sgn(s2)ε21+s2ε22-ξ1x4-ξ2x˙4d)-vxIz2(m1+m2)(sgn(s2)ε21+s2ε22-ξ1x4-ξ2x˙4d)+m1vx(m2a2ξ2vx+ξ1Iz2)x2+m2vx[ξ1(Iz2+m1a22)-ξ2a2vxm1](x2+x4)},
where the tyre force in the control law are calculated from a linear tyre model with nominal tyre cornering stiffness; those are Fyf=Cyfαf, Fyr=Cyrαr, Fyt=Cytαt. αf, αr, and αt, 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 Cyf, Cyr, and Cyt 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 momentΔM 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
(3.17)v=Bu,umin≤u≤umax,
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=[Fx1,Fx2,…,Fx6]T; umin and umax 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
(3.18)B=B-κ,
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
(3.19)κ=diag(k1,k2,…,k6),0≤ki≤1,
and for the ith wheel, ki=1 means this wheel brake actuator works in the normal control situation without any failure, ki=0 means the actuator is completely invalid due to the actuator failure and cannot produce any brake force while 0<ki<1 means the actuator partly falls in failure and can only provide part brake force compared to the normal situation of ki=1.With small steer angle and hitch angle assumptions, (2.3) and (2.7) can be rewritten as
(3.20)Iz1γ˙1=(Fy1+Fy2)a1-(Fy3+Fy4)b1+ΔM1+Fhylp,Iz2γ˙2=Fhya2-(Fy5+Fy6)b2+ΔM2,
with
(3.21)ΔM1=(Fx2-Fx1)tw12+(Fx4-Fx3)tw22,ΔM2=(Fx6-Fx5)tw32.
Then the input matrix can be defined as
(3.22)B-=(-tw12tw12-tw22tw22000000-tw32tw32).
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 umin and umax 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
(3.23)Fxmi≤Fxi≤0,
with
(3.24)Fxmi=σiFx0i,
where σi relates the longitudinal force limit to the normal force variation. According to the actual tyre force property [16, 22], define
(3.25)σi=τisin[c1arctan(c2τi)+c3arctan(c4τi)],
where c1~c4 are experiential constants, and τi is the ratio of the dynamic tyre normal force to the static value, that is,
(3.26)τi=FziFz0i.
In real implementation, the dynamic tyre normal force Fzi 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 Fxm0=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 umin≤u≤umax 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:
(3.27)minς∥Wu(u-ud)∥22+(1-ς)∥Wv(Bu-v)∥22s.t.umin≤u≤umax,
where ud is the preferred control effort vector and take ud=0 here to minimize the brake force effort; Wu and Wv are nonsingular weighting matrices with
(3.28)Wu=diag(Wu1,Wu2,…,Wu6),Wv=diag(Wv1,Wv2),
where ς is the weighting factor to weight the actuator effort and control error. The brake force lower bound vector umin is derived from brake force limit estimation in real time by (3.24)–(3.26), that is, umin=[Fxm1,Fxm2,…,Fxm6]T. The upper bound umax is set zero.Since(3.29)ς∥Wu(u-ud)∥22+(1-ς)∥Wv(Bu-v)∥22=∥((1-ς)WvBςWu)u-((1-ς)WvvςWuud)∥22=∥B~u-v~∥22,
the CA problem (3.27) is equivalent to the optimization problem
(3.30)u=argminumin≤u≤umax∥B~u-v~∥22,
with
(3.31)B~=((1-ς)WvBςWu),v~=((1-ς)WvvςWuud),
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.


## 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 inputδf at the speed of vx is given by
(3.1)γ-1d=vx/l11+Ksvx2δf,
with
(3.2)Ks=b1l2m1+(b1-lp)b2m2l12l2Cyf-a1l2m1+(a1+lp)b2m2l12l2Cyr,
where Cyf, and Cyr 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,
(3.3)|γ-1d|vx≤μg.
So the dynamic reference response of the tractor yaw rate may be reasonably formulated as
(3.4)γ1d={γ-1d,|γ-1d|vx<μg,μgvx,|γ-1d|vx≥μg.
Similarly, the reference response of the hitch angle can be obtained as well from the linear 4-DOF model as
(3.5)θd=p1+(p2+p3)vx21+Ksvx2δf,
with
(3.6)p1=-lp-b1+l2l1,p2=a2m2l1l2Cyt,p3=-a1l2m1+(a1+lp)b2m2l12l2Cyr,
here Cyt 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:
(3.7)θ˙d=∂θd∂vxdvxdt=2(Ku1Δl-Ku2l1)vxv˙x(l1+Ku1vx2)2δf,
with
(3.8)Ku1=b1l2m1+(b1-lp)b2m2l1l2Cyf-a1l2m1+(a1+lp)b2m2l1l2Cyr,Ku2=a1l2m1+(a1+lp)b2m2l1l2Cyr-a2m2l2Cyt,Δl=lp-b1+l2.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, β2d=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 statesx=[vy1γ1θθ˙]T=[x1x2x3x4]T and the 4-DOF single-track model equations with the active corrective yaw moment ΔM=[ΔM1ΔM2]T can be rewritten as
(3.9)x˙=Ax+B1u1+B2u2,
with A∈R4×4, A=M-1V, B1∈R4×1, B1=M-1W, B2∈R4×2, B2=M-1Y, u1=δf, and u2=ΔM, where
(3.10)M=(m1lpIz100m1+m2-m2(lp+a2)0-m2a2m1a2Iz20Iz20010),V=1vx(v11v1200v21v22Cytvxl2Cytv31v32-b2Cytvx-b2l2Cyt000vx),v11=-(a1+lp)Cyf-(lp-b1)Cyr,v21=-(Cyf+Cyr+Cyt),v31=-Cyfa2-Cyra2+Cytb2,v12=-a1(a1+lp)Cyf-b1(b1-lp)Cyr-vx2lpm1,v22=-a1Cyf+b1Cyr+(l2+lp)Cyt-(m1+m2)vx2,v32=-a1a2Cyf+a2b1Cyr-b2(l2+lp)Cyt-m1a2vx2,W=((a1+lp)CyfCyfa2Cyf0),Y=(10000100).
Two sliding surfaces are, respectively, designed for the control of the tractor and the trailer as
(3.11)s1=x2-x2d,s2=ξ1(β2-β2d)+ξ2(x4-x4d),0≤ξ1,ξ2≤1,
with(3.12)β2=x1vx-lpvxx2-a2vx(x2+x4),β2d=0.
Then the attractive equations can be formulated as
(3.13)s˙1=x˙2-x˙2d=-ε11sgn(s1)-ε12s1,(3.14)s˙2=ξ1(x˙1vx-lpvxx˙2-a2vx(x˙2+x˙4))+ξ2(x˙4-x˙4d)=-ε21sgn(s2)-ε22s2,
where εij (i,j=1,2) are positive, and sgn(s1,2) 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:
(3.15)x˙2d=∂x2d∂vxdvxdt=(1-Ksvx2)v˙x(1+Ksvx2)l1δf,x˙4d=∂x4d∂vxdvxdt=2(Ku1Δl-Ku2l1)(l1-3Ku1vx2)(l1+Ku1vx2)3v˙x2δf.
Attractive equations given by (3.13) and (3.14) can guarantee the deviations of state response asymptotically converge to zero. However, the positive constants εij 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
(3.16)ΔM1=1ξ2vx(m1+m2)-ξ1a2m1×{2[-ξ2vx(m2a1+m2lp+a1m1)+ξ1a1a2m1]Fyf+2[b1ξ2vxm1+ξ2vx(b1-lp)m2-ξ1a2b1m1]Fyr+[2m1lp(ξ2vx-ξ1a2+ξ1x4)]Fyt+ξ2vxlpm1m2(-vxx4+a2x˙4d)+ξ1vxlpm1m2a2(x2+x4)+m1Iz1ξ1a2(sgn(s1)ε11+s1ε12-x˙2d)-(m1+m2)Iz1ξ2vx(sgn(s1)ε11+s1ε12-x˙2d)-[ξ2vxm1m2lp(a2+lp)](sgn(s1)ε11+s1ε12-x˙2d)+a2lpm1m2vx(sgn(s2)ε21-s2ε22)},ΔM2=1ξ2vx(m1+m2)-ξ1a2m1×{[-2(ξ1Iz2+ξ2vxa2m2)](Fyf+Fyr)+[2(-ξ1a2l2m1+ξ2vx(m1l2+m2b2-2ξ1m1Iz2))]Fyt+[-ξ2vxm2m1(a22+a2lp)](sgn(s1)ε11+s1ε12-x˙1d)-Iz2ξ2vx(m1+m2)(sgn(s1)ε11+s1ε12-x˙2d)-Iz2ξ1lpm1(sgn(s1)ε11+s1ε12-x˙2d)-m1m2vxa22(sgn(s2)ε21+s2ε22-ξ1x4-ξ2x˙4d)-vxIz2(m1+m2)(sgn(s2)ε21+s2ε22-ξ1x4-ξ2x˙4d)+m1vx(m2a2ξ2vx+ξ1Iz2)x2+m2vx[ξ1(Iz2+m1a22)-ξ2a2vxm1](x2+x4)},
where the tyre force in the control law are calculated from a linear tyre model with nominal tyre cornering stiffness; those are Fyf=Cyfαf, Fyr=Cyrαr, Fyt=Cytαt. αf, αr, and αt, 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 Cyf, Cyr, and Cyt 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.


## 3.1.1. 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 inputδf at the speed of vx is given by
(3.1)γ-1d=vx/l11+Ksvx2δf,
with
(3.2)Ks=b1l2m1+(b1-lp)b2m2l12l2Cyf-a1l2m1+(a1+lp)b2m2l12l2Cyr,
where Cyf, and Cyr 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,
(3.3)|γ-1d|vx≤μg.
So the dynamic reference response of the tractor yaw rate may be reasonably formulated as
(3.4)γ1d={γ-1d,|γ-1d|vx<μg,μgvx,|γ-1d|vx≥μg.
Similarly, the reference response of the hitch angle can be obtained as well from the linear 4-DOF model as
(3.5)θd=p1+(p2+p3)vx21+Ksvx2δf,
with
(3.6)p1=-lp-b1+l2l1,p2=a2m2l1l2Cyt,p3=-a1l2m1+(a1+lp)b2m2l12l2Cyr,
here Cyt 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:
(3.7)θ˙d=∂θd∂vxdvxdt=2(Ku1Δl-Ku2l1)vxv˙x(l1+Ku1vx2)2δf,
with
(3.8)Ku1=b1l2m1+(b1-lp)b2m2l1l2Cyf-a1l2m1+(a1+lp)b2m2l1l2Cyr,Ku2=a1l2m1+(a1+lp)b2m2l1l2Cyr-a2m2l2Cyt,Δl=lp-b1+l2.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, β2d=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 statesx=[vy1γ1θθ˙]T=[x1x2x3x4]T and the 4-DOF single-track model equations with the active corrective yaw moment ΔM=[ΔM1ΔM2]T can be rewritten as
(3.9)x˙=Ax+B1u1+B2u2,
with A∈R4×4, A=M-1V, B1∈R4×1, B1=M-1W, B2∈R4×2, B2=M-1Y, u1=δf, and u2=ΔM, where
(3.10)M=(m1lpIz100m1+m2-m2(lp+a2)0-m2a2m1a2Iz20Iz20010),V=1vx(v11v1200v21v22Cytvxl2Cytv31v32-b2Cytvx-b2l2Cyt000vx),v11=-(a1+lp)Cyf-(lp-b1)Cyr,v21=-(Cyf+Cyr+Cyt),v31=-Cyfa2-Cyra2+Cytb2,v12=-a1(a1+lp)Cyf-b1(b1-lp)Cyr-vx2lpm1,v22=-a1Cyf+b1Cyr+(l2+lp)Cyt-(m1+m2)vx2,v32=-a1a2Cyf+a2b1Cyr-b2(l2+lp)Cyt-m1a2vx2,W=((a1+lp)CyfCyfa2Cyf0),Y=(10000100).
Two sliding surfaces are, respectively, designed for the control of the tractor and the trailer as
(3.11)s1=x2-x2d,s2=ξ1(β2-β2d)+ξ2(x4-x4d),0≤ξ1,ξ2≤1,
with(3.12)β2=x1vx-lpvxx2-a2vx(x2+x4),β2d=0.
Then the attractive equations can be formulated as
(3.13)s˙1=x˙2-x˙2d=-ε11sgn(s1)-ε12s1,(3.14)s˙2=ξ1(x˙1vx-lpvxx˙2-a2vx(x˙2+x˙4))+ξ2(x˙4-x˙4d)=-ε21sgn(s2)-ε22s2,
where εij (i,j=1,2) are positive, and sgn(s1,2) 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:
(3.15)x˙2d=∂x2d∂vxdvxdt=(1-Ksvx2)v˙x(1+Ksvx2)l1δf,x˙4d=∂x4d∂vxdvxdt=2(Ku1Δl-Ku2l1)(l1-3Ku1vx2)(l1+Ku1vx2)3v˙x2δf.
Attractive equations given by (3.13) and (3.14) can guarantee the deviations of state response asymptotically converge to zero. However, the positive constants εij 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
(3.16)ΔM1=1ξ2vx(m1+m2)-ξ1a2m1×{2[-ξ2vx(m2a1+m2lp+a1m1)+ξ1a1a2m1]Fyf+2[b1ξ2vxm1+ξ2vx(b1-lp)m2-ξ1a2b1m1]Fyr+[2m1lp(ξ2vx-ξ1a2+ξ1x4)]Fyt+ξ2vxlpm1m2(-vxx4+a2x˙4d)+ξ1vxlpm1m2a2(x2+x4)+m1Iz1ξ1a2(sgn(s1)ε11+s1ε12-x˙2d)-(m1+m2)Iz1ξ2vx(sgn(s1)ε11+s1ε12-x˙2d)-[ξ2vxm1m2lp(a2+lp)](sgn(s1)ε11+s1ε12-x˙2d)+a2lpm1m2vx(sgn(s2)ε21-s2ε22)},ΔM2=1ξ2vx(m1+m2)-ξ1a2m1×{[-2(ξ1Iz2+ξ2vxa2m2)](Fyf+Fyr)+[2(-ξ1a2l2m1+ξ2vx(m1l2+m2b2-2ξ1m1Iz2))]Fyt+[-ξ2vxm2m1(a22+a2lp)](sgn(s1)ε11+s1ε12-x˙1d)-Iz2ξ2vx(m1+m2)(sgn(s1)ε11+s1ε12-x˙2d)-Iz2ξ1lpm1(sgn(s1)ε11+s1ε12-x˙2d)-m1m2vxa22(sgn(s2)ε21+s2ε22-ξ1x4-ξ2x˙4d)-vxIz2(m1+m2)(sgn(s2)ε21+s2ε22-ξ1x4-ξ2x˙4d)+m1vx(m2a2ξ2vx+ξ1Iz2)x2+m2vx[ξ1(Iz2+m1a22)-ξ2a2vxm1](x2+x4)},
where the tyre force in the control law are calculated from a linear tyre model with nominal tyre cornering stiffness; those are Fyf=Cyfαf, Fyr=Cyrαr, Fyt=Cytαt. αf, αr, and αt, 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 Cyf, Cyr, and Cyt 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 momentΔM 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
(3.17)v=Bu,umin≤u≤umax,
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=[Fx1,Fx2,…,Fx6]T; umin and umax 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
(3.18)B=B-κ,
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
(3.19)κ=diag(k1,k2,…,k6),0≤ki≤1,
and for the ith wheel, ki=1 means this wheel brake actuator works in the normal control situation without any failure, ki=0 means the actuator is completely invalid due to the actuator failure and cannot produce any brake force while 0<ki<1 means the actuator partly falls in failure and can only provide part brake force compared to the normal situation of ki=1.With small steer angle and hitch angle assumptions, (2.3) and (2.7) can be rewritten as
(3.20)Iz1γ˙1=(Fy1+Fy2)a1-(Fy3+Fy4)b1+ΔM1+Fhylp,Iz2γ˙2=Fhya2-(Fy5+Fy6)b2+ΔM2,
with
(3.21)ΔM1=(Fx2-Fx1)tw12+(Fx4-Fx3)tw22,ΔM2=(Fx6-Fx5)tw32.
Then the input matrix can be defined as
(3.22)B-=(-tw12tw12-tw22tw22000000-tw32tw32).
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 umin and umax 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
(3.23)Fxmi≤Fxi≤0,
with
(3.24)Fxmi=σiFx0i,
where σi relates the longitudinal force limit to the normal force variation. According to the actual tyre force property [16, 22], define
(3.25)σi=τisin[c1arctan(c2τi)+c3arctan(c4τi)],
where c1~c4 are experiential constants, and τi is the ratio of the dynamic tyre normal force to the static value, that is,
(3.26)τi=FziFz0i.
In real implementation, the dynamic tyre normal force Fzi 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 Fxm0=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 umin≤u≤umax 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:
(3.27)minς∥Wu(u-ud)∥22+(1-ς)∥Wv(Bu-v)∥22s.t.umin≤u≤umax,
where ud is the preferred control effort vector and take ud=0 here to minimize the brake force effort; Wu and Wv are nonsingular weighting matrices with
(3.28)Wu=diag(Wu1,Wu2,…,Wu6),Wv=diag(Wv1,Wv2),
where ς is the weighting factor to weight the actuator effort and control error. The brake force lower bound vector umin is derived from brake force limit estimation in real time by (3.24)–(3.26), that is, umin=[Fxm1,Fxm2,…,Fxm6]T. The upper bound umax is set zero.Since(3.29)ς∥Wu(u-ud)∥22+(1-ς)∥Wv(Bu-v)∥22=∥((1-ς)WvBςWu)u-((1-ς)WvvςWuud)∥22=∥B~u-v~∥22,
the CA problem (3.27) is equivalent to the optimization problem
(3.30)u=argminumin≤u≤umax∥B~u-v~∥22,
with
(3.31)B~=((1-ς)WvBςWu),v~=((1-ς)WvvςWuud),
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 Section2. 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.
m
1
8444 kg
h
c
1
1.18 m
K
s
21
4.32
×
10
4 N/rad
m
2
25000 kg
h
f
1
0.75 m
K
s
2
2
7.20
×
10
10 N/rad
m
1
s
5820 kg
h
f
2
0.60 m
K
s
31
7.68
×
10
4 N/rad
m
2
s
21640 kg
h
r
2
0.80 m
K
s
32
9.60
×
10
1
1 N/rad
I
x
1
10347 kg · m2
t
w
1
1.93 m
C
x
1
6.00
×
10
4 N/unit slip
I
x
2
58270 kg · m2
t
w
3
1.84 m
C
y
1
3.00
×
10
4 N/rad
I
z
1
65734.6 kg · m2
h
s
2
1.40 m
C
x
2
1.30
×
10
5 N/unit slip
I
z
2
560000 kg · m2
h
c
2
2.03 m
C
y
2
9.00
×
10
4 N/rad
a
1
2.12 m
h
r
1
0.82 m
C
x
3
1.70
×
10
5 N/unit slip
l
1
4.81 m
h
p
1.27 m
C
y
3
1.20
×
10
5 N/rad
l
p
2.60 m
K
h
t
6.90
×
10
6 Nm/rad
R
w
1
~
3
0.52 m
l
2
11.82 m
K
s
11
1.60
×
10
4 N/rad
I
w
1
40.8 kg · m2
t
w
2
1.84 m
K
s
12
2.40
×
10
10 N/rad
I
w
2
,
3
130 kg · m2
h
s
1
0.40 m
D
s
1
~
3
8.50
×
10
3 N/rad · s−1
g
9.81 m/s2Figure 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 Wu=diag(1.01.01.51.51.01.0), Wv=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 Figure8.Figures9 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 Figures13 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.



---

*Source: 602502-2012-10-02.xml*