# Non-linear bifurcation stability analysis for articulated vehicles with active trailer differential braking systems.

ABSTRACTThis paper presents nonlinear bifurcation stability analysis of articulated vehicles with active trailer differential braking (ATDB) systems. ATDB systems have been proposed to improve stability of articulated vehicle systems to prevent unstable motion modes, e.g., jack-knifing, trailer sway and rollover. Generally, behaviors of a nonlinear dynamic system may change with varying parameters; a stable equilibrium can become unstable and a periodic oscillation may occur or a new equilibrium may appear making the previous equilibrium unstable once the parameters vary. The value of a parameter, at which these changes occur, is known as "bifurcation value" and the parameter is known as the "bifurcation parameter". Conventionally, nonlinear bifurcation analysis approach is applied to examine the nonlinear dynamic characteristics of single-unit vehicles, e.g., cars, trucks, etc. Little attention has been paid to investigate the feasibility and effectiveness of the bifurcation analysis method for nonlinear stability analysis of articulated vehicles under varied operating conditions, e.g., varied forward speed and trailer payload. This motivates the research to examine stability boundaries of equilibrium and limit cycles in the parameter space and to predict qualitative changes in system's behaviour (bifurcations) occurring at their equilibrium points. To this end, a nonlinear yaw-roll model with 6 degrees of freedom (DOF) is generated to simulate the nonlinear dynamics of articulated vehicles. The nonlinear yaw-roll model is also used to design an ATDB controller based on a fuzzy logic control technique. The bifurcation analysis based on the phase-plane method is conducted to evaluate the yaw and roll stability of the articulated vehicle. Built upon the conventional bifurcation analysis for single-unit vehicles, an innovative bifurcation analysis technique is developed in order to effectively assess the nonlinear stability of articulated vehicles. The applicability and effectiveness of the newly developed technique is examined and demonstrated.

KEYWORDS: nonlinear bifurcation analysis method, phase-plane analysis method, fuzzy logic control, active trailer differential braking, articulated vehicles, lateral stability, nonlinear yaw-roll car-trailer model

CITATION: Sun, T., Lee, E., and He, Y., "Non-Linear Bifurcation Stability Analysis for Articulated Vehicles with Active Trailer Differential Braking Systems," SAE Int. J. Mater. Manf. 9(3):2016, doi:10.4271/2016-01-0433.

INTRODUCTION

A car-trailer (CT) system consists of a towing unit, e.g., a car, and a towed unit, i.e., a trailer. The car and trailer are connected at an articulated point by a hitch [1]. With respect to single-unit cars, car-trailer systems may exhibit poor yaw and roll stability at high speeds because of their double-unit structure. For these vehicles, three unique unstable motion modes have been identified, including the trailer swing (oscillatory instability), jack-knifing, and rollover. These unstable motion modes are the common causes for fatal accidents occurring on highways [2]. In order to evaluate the yaw and roll stability of articulated vehicles, a stability evaluation method based on eigenvalue analysis of a linear vehicle model was reported [1, 2]. However, typical fatal accidents of articulated vehicles are mainly resulted from the nonlinear dynamics of these vehicles, e.g., saturation of lateral tire forces, large articulation angles between adjacent vehicle units, etc. [3]. In general, the behaviors of a nonlinear dynamic system may change with varying parameters. Equilibrium can become unstable and a periodic solution may appear or a new stable equilibrium could appear making the previous equilibrium unstable. The value of a parameter, at which these changes occur, is known as "bifurcation value", and the parameter, which is varied, is called the "bifurcation parameter" [4]. Therefore, the bifurcation analysis technique may be applied as an effective dynamic analysis approach to the evaluation of the yaw and roll stability of articulated vehicles considering the unique nonlinear dynamic features of these vehicles with varying parameters.

The past two decades have witnessed the application of the phaseplane analysis approach and its theoretical base, bifurcation theory, to the stability analysis of road vehicles [5]. The side-slip angle phraseogram stable zone boundary was examined, which has been applied to the design of stability control systems for road vehicles [6]. The phase-plane analysis method was used to evaluate the performance of a stability controller based on results of simulations and tests [7]. Bifurcation analysis was used to evaluate the robustness of a vehicle stability controller and to examine the nonlinear dynamic characteristics of the vehicle system [8]. To date, the phase-plane analysis method and the bifurcation analysis technique are mainly applied to the assessment of handling characteristics and lateral stability of single-unit vehicles, such as cars, trucks, and sport utility vehicles (SUVs), etc. Little attention has been paid to investigating the feasibility and effectiveness of the phase-plane analysis method and the bifurcation analysis technique for yaw and roll stability analysis of articulated vehicles. Articulated vehicles, such as car-trailer systems, have multiple vehicle units. Multi-unit vehicles have the unique dynamic features compared against single-unit vehicles [9]. Considering the unique configurations and dynamic features of articulated vehicles, it is questionable whether the bifurcation analysis based on single-unit vehicles is feasible, applicable and effective for analyzing of the yaw and roll stability of articulated vehicles.

Under variable operating conditions, such as forward velocity and trailer payload, the behaviors of articulated vehicle systems may change. Therefore, the main objectives of the paper are: (1) examining equilibrium and periodic orbits of articulated vehicles with variable parameters, (2) determining stability boundaries of equilibrium and limit cycles in the parameter space, and (3) predicting qualitative changes of system's behaviors (bifurcations) occurring at these equilibrium points.

In order to achieve these goals, a nonlinear yaw-roll model with 6 degrees of freedom (DOF) is generated to simulate the nonlinear dynamics of articulated vehicles. An ATDB controller is designed to improve the lateral stability of a CT system using a fuzzy logic control technique. The bifurcation analysis based on the phase-plane method is conducted to evaluate the yaw and roll stability of the articulated vehicle. Built upon the conventional bifurcation analysis for single-unit vehicles, an innovative bifurcation analysis technique is developed in order to effectively assess the nonlinear stability of articulated vehicles. The applicability and effectiveness of the newly developed technique is examined and demonstrated.

The remainder of this paper is organized as follows. The second section introduces the nonlinear 6-DOF yaw-roll car-trailer model. The third section presents nonlinear stability analysis based on the bifurcation analysis method. The ATDB controller based on the fuzzy logic control technique for the 6-DOF model is presented in the fourth section. Finally, conclusions are drawn in the fifth section.

NONLINEAR MODEL FOR CAR-TRAILER COMBINATIONS

In this paper, the CT system with a car-trailer combination is represented by the nonlinear 6-DOF yaw-roll model to identify the unstable motion modes and predict the critical speed(s) of the nonlinear CT system. As shown in Figure 1, each axle is represented by a single wheel. Based on the body fixed coordinate systems, [X.sub.c] - [Y.sub.c] - [Z.sub.c] and [X.sub.t] - [Y.sub.t] - [Z.sub.t], for the car and trailer, respectively, the governing equations of motion can be derived.

In the vehicle modeling, it is assumed that the car front wheel steering angle [delta] is given. The pitch and bounce motions of the car and trailer as well as the aerodynamic forces are ignored. The tire dynamics is simulated using the Magic Formula tire model, which specifies the nonlinear relationship between the tire force and the tire side-slip angle. The roll stiffness and damping coefficients of the vehicle suspension systems are constant when roll motions are involved. The motions considered in the 6-DOF yaw-roll model are: (1) car forward speed [U.sub.c], (2) car lateral speed [V.sub.c], (3) car yaw rate [r.sub.c], (4) car roll angle [[phi].sub.c], (5) trailer yaw rate [r.sub.t], and (6) trailer roll angle [[phi].sub.t]. Based on Newton's law of dynamics, the equations of motion for the car and trailer can be derived and are shown as:

The equations of motion for the car are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The equations of motion for the trailer are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

In summary, the nonlinear equations describing the CT model can be expressed in a state-space representation as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The state variable vector is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

Table 1 lists the notations and the values of the primary parameters of the CT system. All parameters of the Magic Formula tire model are listed in Table 2 and the lateral tire force is determined using the equation (11) as follows:

F(x) = D * sin[C * arctan{B * x - E * (B * x - arctan(B))}] (11)

where x is the tire side-slip angle.

NON-LINEAR BIFURCATION STABILITY ANALYSIS

In this section, a systematic nonlinear analysis method is presented to analyze the yaw and roll stability of the nonlinear yaw-roll CT model. The nonlinear vehicle stability analysis is based on the phase-plane method and Lyapunov stability analysis theory. The phase-plane state trajectories are used to predict qualitative changes of system's behaviors (bifurcations) occurring with variable parameters. The phase-plane state trajectories' envelopes are used to provide an approximate stability range. The Lyapunov stability analysis can be used to judge whether the equilibrium point of the nonlinear system is stable.

Phase-Plane Stability Analysis

For articulated vehicle systems, varied steering input, forward velocity and trailer payload are regarded as external input variables and important factors on the yaw and roll stability of the CT system. Therefore, to conduct the bifurcation analysis and to investigate the effect of variable driving conditions on the yaw and roll stability of the nonlinear CT model, the numerical simulations of the nonlinear 6-DOF yaw-roll model with varying forward velocity are conducted. Under the simulated maneuver, the car front wheel steer input maintains constant at an angle of 0.0175rad, the vehicle forward speed will take different values (for different cases) from the speed series of 60, 65, 70,..., 130km/h (with a constant increment step of 5 km/h), and the road friction coefficient [mu] takes the value of 0.55.

According to the kinematic analysis based on the yaw-roll model shown in Figure 1, the relationship among the side-slip angles of the car and the trailer, the yaw rate of the car and the trailer, the roll angles of the car and the trailer, and the articulation angle between the car and the trailer can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

For a zero initial condition, the articulation angle can be derived as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where the notations and the primary parameters of the CT system are shown in Table 1. The [[beta].sub.c] and [[beta].sub.t] are the side-slip angle at the respective center of gravity (CG) of the car and the trailer, [h.sub.c]cos([[phi].sub.c]) and [h.sub.t]cos([[phi].sub.t]) are the vertical distance between the body CG to the roll center of the car and the trailer, respectively. Note that the relationship shown in Equation (12) will be used to aid the phase-plane analysis to evaluate the yaw and roll stability of the nonlinear model.

The nonlinear stability analysis based on the phase-plane method examines the set of relation between: (1) the leading unit's side-slip angle and side-slip angular velocity ([[beta].sub.c] - d[[beta].sub.c]/dt), (2) the trailing unit's side-slip angle and side-slip angular velocity ([[beta].sub.t] - d[[beta].sub.t] /dt), (3) the side-slip angle of the leading unit, the yaw rate of the leading unit, and the articulation angle between the leading and trailing units ([[beta].sub.c] - [r.sub.c] - [psi]), (4) the side-slip angle of the trailing unit, the yaw rate of the trailing unit, and the articulation angle between the leading and trailing units ([[beta].sub.t] - [r.sub.t] - [psi]), and (5) the articulation angle and the articulation angular velocity ([psi] - d[psi]dt). Results of the simulation are analysed to assess the lateral stability of the articulated vehicle. Similarly, the leading unit's roll angle and roll angular velocity ([[phi].sub.c] - d[[phi].sub.c]/dt) and the trailing unit's roll angle and roll angular velocity ([[phi].sub.t] - d[[phi].sub.t]/dt), are examined to evaluate the roll stability of this combination.

Figures 2 and 3 show the phase-plane trajectories of the side-slip angle and side-slip angular velocity of the car and the trailer, i.e., [[beta].sub.c] - d[[beta].sub.c]/dt and [[beta].sub.t] - d[[beta].sub.t]/dt, respectively. Each of the state trajectories is associated with a specified value of the car forward speed. The X axis denotes the variable quantity of the car and trailer's side slip angle. The Y axis represents the variable rate of car and trailer's side slip angle rate. According to the phase-plane analysis method, if a state trajectory may return to the origin of the coordinate system or the state trajectory could converge to a final steady-state, this means that the system is stable. On the other hand, if the state trajectory can't return to the origin of the coordination system or the state trajectory can't converge a steady-state, this means that the system will loss stability. The effect of the vehicle forward speed on the yaw stability of the CT system is clearly illustrated by the trajectories of [[beta].sub.c] - d[[beta].sub.c]/dt and [[beta].sub.t] - d[[beta].sub.t]/dt.

As shown in Figures 2 and 3, all the state trajectories may reside in 3 regions:

1. Yaw Motion Stable Region. If the vehicle forward speed is less than 115km/h (i.e., within the range of 60km/h to 110km/h), all the state trajectories will converge to the corresponding points on the horizontal axis of the coordination system. All those points are the respective final steady-states of the vehicle units. Thus, the CT system remains yaw stability. This region swept by the state trajectories (in blue) is defined as the yaw motion stable region of the CT system. As shown in Figure 2, the stable region of the leading unit is enclosed in the phase-plane range of -0.003rad < [[beta].sub.c] < 0.015rad and -0.095rad/s < d[[beta].sub.c]/dt < 0.045rad/s, while the stable region of the trailing unit shown in Figure 3 is expanded to the phase-plane range of 0 < [[beta].sub.t] < 0.1rad and -0.1rad/s < d[[beta].sub.t] /dt < 0.15rad/s. This means that during the constant steering angle input maneuver, compared with the car, the trailer has amplified lateral motions. This phenomenon is usually called "Rearward Amplification", which is one of the unique dynamic features of articulated vehicles [3].

2. Yaw Motion Stability Boundary Region. When the vehicle forward speed reaches 115km/h, the state trajectories will also converge to the points located on the horizontal axis of the coordination system, but more time is needed for the cycles. When the vehicle forward speed is larger than 115km/h, all the state trajectories (in red) will diverge, thereby causing the unstable yaw motions of the CT system. Hence, the region around the envelope of the state trajectories associated with the forward speeds around 115 km/h is defined as the yaw motion stability boundary region, and the forward speed of 115 km/h may be simply considered as the critical speed of the nonlinear CT system, above which the CT system will lose the lateral stability.

3. Yaw Motion Unstable Region. If the vehicle forward speed is between 120km/h and 130km/h, all the state trajectories will diverge with increasing of time, thereby leading to the loss of yaw stability. The region that is swept by those state trajectories (in black) is defined as the yaw motion unstable region.

Figures 4 show the phase-plane trajectories of the articulation angle and articulation angular velocity, [psi] - d[psi]/dt. Each of the state trajectories is associated with a specified value of the vehicle forward speed. All the state trajectories may also reside in 3 regions corresponding to those shown in Figures 2 and 3. The stable region swept by the state trajectories (in blue) is enclosed in the phase-plane range of -0.03rad < [psi] < 0.05rad and -0.11rad/s < d[psi]/dt < 0.15rad/s. Similarly, the boundary region is around the envelope of the state trajectories (in red) when the vehicle speed is around 115 km/h. If the vehicle forward speed is above this critical speed, 115 km/h, all the state trajectories (in black) will diverge, and an unstable yaw motion mode will occur.

As shown in Figures 2 and 3, the phase-plane analysis based on [[beta].sub.c] - d[[beta].sub.c]/dt and [[beta].sub.t] - d[[beta].sub.t] /dt may be effective for assessing the yaw stability of the car and the trailer seperately. Actually, this phase-plane analysis has been widely applied to the yaw stability evaluation for single-unit vehicles [6, 7]. For the CT system, the relationship between the yaw motions of the car and the trailer is governed by Equation (12), implying that the side-slip angle of the car and trailer, the yaw rate of the car and trailer, and the articulation angle between the two vehicle units are interrelated. In order to use the phase-plane analysis to evaluate the lateral stability of the car and the trailer simultaneously, the 3D state trajectories of [[beta].sub.c] - [r.sub.c] - [psi] and [[beta].sub.t] - [r.sub.t] - [psi] are examined.

Figures 5 and 6 show the 3D phase-plane trajectories interrelating the side-slip angle of the car and trailer, the yaw rate of the car and trailer, and the articulation angle between the car and the trailer. Each of the state trajectories of [[beta].sub.c] - [r.sub.c] - [psi] and [[beta].sub.t] - [r.sub.t] - [psi] is associated with specified the vehicle forward speed. The results shown in Figures 5 and 6 are consistent with those illustrated in Figures 2 and 3, indicating that the CT system will lose its lateral stability if the forward speed is above 115km/h. It is proposed that the 3D state trajectories of [[beta].sub.c] - [r.sub.c] - [psi] and [[beta].sub.t] - [r.sub.t] - [psi] are used to conduct the lateral stability analysis of the CT system.

Corresponding to the phase-plane trajectories shown in Figures 2 to 6 for yaw stability analysis of the CT system, Figures 7 and 8 show the phase-plane trajectories of the roll angle and roll angular velocity of the leading and trailing units, i.e., [[phi].sub.c] - d[[phi].sub.c]/dt and [[phi].sub.c] - d[[phi].sub.c]/dt, respectively, of the CT system. The phase-plane trajectories shown in Figures 7 and 8 may be used for roll stability analysis of the CT system. Similarly, each of the state trajectories illustrated in Figures 7 and 8 is associated with a specified vehicle forward speed from the speed series of 60, 65, 70,..., 130 km/h. A closed observation of the state trajectories shown in Figures 7 and 8 discloses an interesting phenomenon with the following features: (1) all state trajectories and even those associated with the high forward speeds, i.e., 120, 125 and 130 km/h, do not exhibit an obvious tendency of divergence; (2) each of the state trajectories converges at an individual point located on the horizontal axis; and (3) for those convergent points on the horizontal axis, the higher the forward speed with which a convergent point is associated, the longer the distance the point is located from the organ point.

Based on the above features of the observed phenomenon, we may conduct roll stability analysis for the CT system. First, since no state trajectory shows a tendency of divergence and all trajectories are enclosed in the limited area on the phase plane, it could be judged that the CT system is roll stable under the emulated maneuver once the vehicle forward speed takes different values in the speed series. With the second and third features aforementioned, it seems that the judgement of the roll stability of the CT system is questionable, since all state trajectories converge at different points on the horizontal axis instead of the original point. These features of the observed phenomenon may be explained considering the characteristics of the simulated maneuver, under which at a given forward speed, the front wheel steering input maintains constant at the angle of 0.0175rad. This maneuver is a typical steady-state maneuver, under which the vehicle units of the CT system may exhibit one of the following three handling characteristics: (1) oversteer, (2) neutral steer, and (3) understeer. If a vehicle unit has a certain extent of oversteer, under the maneuver at a constant forward speed, as time goes, the radius of curvature of the vehicle units' trajectories will become smaller and smaller. Eventually, the CT system will rollover. Since no state trajectory shown in Figures 7 and 8 exhibits a tendency of divergence, the vehicle units of the CT system may have some degree of understeer. If this is the case, under the maneuver at a constant forward speed, as time goes, the radius of curvature of the vehicle units' trajectories will become larger and larger. By the time of the termination of the simulated maneuver, the radius of a vehicle unit's trajectory may be large enough, and the rate of increment of the radius may be very low. At this instant, the lateral acceleration at the center of gravity (CG) of a vehicle unit can be determined as [U.sup.2]/R, where U is vehicle forward speed and R is the radius. Accordingly, the corresponding lateral force at the vehicle unit's CG will cause the sprung mass rotates at an angle, which is mainly determined by the vehicle forward speed. It can be predicted that the larger the forward speed, the bigger the sprung mass roll angle. The above dynamic analysis of the CT system well explains the second and third features of the observed features.

According to the phase-plane stability analysis, the vehicle forward speed of 115 km/h may be simply considered as the critical speed, above which the CT system will lose stability. The corresponding area enclosed by the envelope of the state trajectory associated with the critical speed could be approximately treated as the stable domain of the nonlinear CT system. Hence, the stabile domain may be quantitatively described using the data offered in Table 3. The stabile domain derived from the phase-plane stability analysis will be used to design the ATDB controller. Note that the data derived from each of Figures 2 to 8 can only be used to define a half of the stable range of the symmetric shape. The half stable range corresponds to the given turning direction of the steering angle of the car front steer angle. When the steer angle turns in opposite direction with the same magnitude (0.0175 rad), the similar simulation may be conducted, and the respective results can determine the other half of the stable range. This half and the previous half are symmetric with respect to the vertical axis of the coordinate system.

Bifurcation Analysis

For a nonlinear dynamic system, with given parameter values, an equilibrium is a constant solution of the system. For the lateral stability of the CT system, equilibrium points could contain a lot of important information. Firstly, the stable equilibrium can be used to dominate the stability of the system in a steady-state motion and to estimate the dynamic characteristics in the neighborhood of the equilibrium. Secondly, the unstable equilibrium can be used to indicate a rough boundary of the system's stable region [12].

As required by the bifurcation analysis approach, a state-space equation of the CT model can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where f(X, p) is generic and sufficiently smooth, in this case, f(X, p) = [M.sub.-1] * F(X, [delta]), and p denotes the parameters of the system, representing experimental central setting or variable inputs, such as the car front wheel steering input, forward velocity and trailer payload. Then, to solve the equation (14) when [??] = 0, and to find the solution [X.sub.p], which is considered as the equilibrium of the nonlinear dynamic system. So Equation (14) can be rewritten as:

f([X.sub.p],p) = 0 (15)

Now let's set :

Z = X - [X.sub.p] (16)

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

where O([|Z|.sup.2]) is an infinitesimal of higher order in the neighbourhood of the equilibrium, and J(p) is the Jacobian matrix of f(X, p) at the equilibrium X = [X.sub.p] as follows:

J(p) = [[partial derivative]f(X,p)]/[partial derivative]X (18)

According to the Lyapunov stability analysis theory, an equilibrium point [X.sub.p] of the differential equation is stable, if all the eigenvalues of the Jacobian matrix, J(p), evaluated at [X.sub.p] have negative real parts. The equilibrium point is unstable if at least one of the eigenvalues has a positive real part [13].

In the phase-plane stability analysis part, all the state trajectories reside in 3 regions: stable region, critical region (or stability boundary region) and unstable region. Therefore, in this bifurcation analysis part, three vehicle forward speeds, i.e., 80km/h, 115km/h and 120km, are chosed to represent a typical case in each of the 3 regions, respectively.

As discussed above, the equilibrium of the nonlinear 6-DOF yaw-roll CT model is the solution of Equation (9) when [??] = 0. In the paper, the state variables are define as, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, the equilibrium is the intersection of all phase-plane state trajectories with the line of [??] = 0. When the vehicle forward speed is 80km/h, the system has 4 equilibrium points, denoted as X_(s1-80), X_(s2-80), X_(s3-80), X_(s4-80), are as follows:

[X.sub.s1-80] = [[ -0.0021 0 0.0001 0 0.0464 - 0.0006 0.0010 0.0111 21.9282].sup.T],

[X.sub.s2-80] = [[ 0.0037 0 0.0160 0 - 0.0248 - 0.1378 0.0121 1.3458 21.1760].sup.T],

[X.sub.s3-80] = [[ 0.0006 0 - 0.0057 0 - 0.0080 0.0484 0.0224 - 0.4920 20.8997].sup.T], and

[X.sub.s4-80] = [[ -0.0002 0 0.0023 0 0.0031 - 0.0192 - 0.0087 0.1931 20.6339].sup.T].

All eigenvalues of the Jacobian matrices for the equilibriums at 80km/h are as follows

All eigenvalues of the Jacobian matrices for the equilibriums at 80km/h are as follows

[[lambda].sub.s1-80] = [[ -9.7675 - 16.7048i - 9.7675 + 16.7048i - 6.0059- 9.4582i - 6.0059 + 9.4582i -4.9826 - 1.8636i -4.9826 + 1.8636i -0.4813 - 2.9255i -0.4813 + 2.9255i -0.0014].sup.T],

[[lambda].sub.s2-80] = [[ -9.1548 - 17.1794i -9.1548 + 17.1794i -6.0142 - 9.4505i -6.0142 + 9.4505i -4.7096 - 1.9624i -4.7096 + 1.9624i -0.3284 - 2.5089i -0.3284 + 2.5089i - 0.0022].sup.T],

[[lambda].sub.s3-80] = [[ -9.7777 - 16.6878i -9.7777 + 16.6878i -6.0149 - 9.4405i - 6.0149 + 9.4405i - 5.0891 - 2.0210i -5.0891 + 2.0210i -0.5539 - 2.9031i -0.5539 + 2.9031i - 0.0004].sup.T], and

[[lambda].sub.s4-80] = [[ -9.8615 - 16.6090i -9.8615 + 16.6090i -6.0175 - 9.4368i -6.0175 + 9.4368i - 5.1350 - 2.1326i -5.1350 + 2.1326i - 0.6056 - 2.9422i - 0.6056 + 2.9422i - 0.0000].sup.T].

All eigenvalues of the Jacobian matrices for the equilibriums don't have a positive part. Hence, all equilibriums of the nonlinear 6-DOF yaw-roll CT model at 80 km/h are stable.

When the vehicle forward speed is 115km/h, the system has 5 equilibrium points, denoted as X_(s1-115), X_(s2-115), X_(s3-115), X_(s4-115), X_(s5-115), which are as follows:

[X.sub.s1-115] = [[ -0.0145 0 -0.0130 0 0.1113 0.2104 - 0.8663 - 1.4862 31.7007].sup.T]

[X.sub.s2-115] = [[ 0.0189 0 0.0205 0 - 0.1345 - 0.4674 1.4148 2.7163 31.2302].sup.T]

[X.sub.s3-115] = [[ -0.0105 0 0.0125 0 0.0637 0.7197 - 0.7027 0.0527 30.0644].sup.T]

[X.sub.s4-115] = [[ 0.0099 0 - 0.0186 0 - 0.0706 - 0.7273 0.6420 - 0.7877 28.7571].sup.T]

[X.sub.s5-115] = [[ -0.0086 0 0.0231 0 0.0732 0.1823 - 0.5114 9.5430 26.8522].sup.T]

All eigenvalues of the Jacobian matrices for these equilibriums at 115km/h are as follows:

[[lambda].sub.s1-115] = [[ -9.0012 - 17.3189i -9.0012 + 17.3189i - 5.8756 - 9.6537i -5.8756 + 9.6537i -2.5390 - 1.8293i -2.5390 + 1.8293i 0.2136 - 2.4276i 0.2136 + 2.4276i - 0.0123].sup.T],

[[lambda].sub.s2-115] = [[ -8.4718 - 17.6011i - 8.4718 + 17.6011i - 5.8613 - 9.6376i -5.8613 + 9.6376i -6.2143 -2.7632 0.0214 - 0.3298 - 1.6010i -0.3298 + 1.6010].sup.T],

[[lambda].sub.s3-115] = [[ -9.3096 - 17.1156i - 9.3096 + 17.1156i - 5.8976 - 9.6354i - 5.8976 + 9.6354i -2.9224 - 2.3370i -2.9224 + 2.3370i 0.0565 - 2.8404i 0.0565 + 2.8404i - 0.0096].sup.T],

[[lambda].sub.s4-115] = [[ -9.2893 - 17.1218i -9.2893 + 17.1218i - 5.9165 - 9.5700i -5.9165 + 9.5700i -6.6897 - 0.4465 - 2.0698i - 0.4465 + 2.0698i -0.0078 - 0.2498i -0.0078 + 0.2498i].sup.T], and

[[lambda].sub.s5-115] = [[ -7.9969 - 17.8022i - 7.9969 + 17.8022i - 5.9256 - 9.6035i - 5.9256 + 9.6035i -2.2941 - 1.5426i -2.2941 + 1.5426i -0.7041 0.4417 0.1588].sup.T].

Among all eigenvalues of the Jacobian matrices for the above equilibriums at 115km/h, X_(s4-115) is only stable equilibrium. Other equilibriums of the nonlinear 6-DOF yaw-roll CT model at 115 km/h are unstable.

When the vehicle forward speed is 125km/h,, the system has 4 equilibrium points, denoted as X_(s1-125), X_(s2-125), X_(s3-125), X_(s4-125), which are as follows:

[X.sub.s1-125] = [[-0.0198 0 - 0.0223 0 0.1466 0.2340 - 3.3535 - 3.1775 34.1702].sup.T]

[X.sub.s2-125] = [[0.0209 0 0.0004 0 - 0.4786 - 0.4945 5.8920 0.7320 33.1054].sup.T],

[X.sub.s3-125] = [[0.0201 0 0.0253 0 - 0.3922 - 0.5229 12.0237 8.2784 31.2419].sup.T], and

[X.sub.s4-125] = [[-0.0227 0 - 0.0206 0 0.3501 - 0.5407 4.8206 27.7225 - 3.0401].sup.T]

All eigenvalues of the Jacobian matrices for the above equilibriums at 125km/h are as follows:

[[lambda].sub.s1_125] = [[-8.3690 - 17.6519i -8.3690 + 17.6519; -5.7816 - 9.7233; -5.7816 + 9.7233i -0.4923 - 1.1754i; -0.4923 + 1.1754i; 0.0871 - 1.0566i; 0.0871 + 1.0566i - 0.0078].sup.T],

[[lambda].sub.s2_125] = [[-9.1523 - 17.2359i - 9.1523 + 17.2359i; - 5.7734 - 9.7280i; -5.7734 + 9.7280i; -0.8421 - 2.4994i; -0.8421 + 2.4994i; 0.7346 0.0008 - 0.7710].sup.T],

[lambda].sub.s3_125] = [[ -5.7715 - 9.7291i; -5.7715 + 9.7291i; -7.9914 - 17.8031i; -7.9914 + 17.8031i; 0.0228 - 0.3941i; 0.0228 + 0.3941i; - 0.1446 0.0866 - 0.1773i; 0.0866 + 0.1773;].sup.T], and

[lambda].sub.s4_125] = [[ -5.7741 - 9.7275i; - 5.7741 + 9.7275i; - 8.0143 - 17.7946i; - 8.0143 + 17.7946i; - 0.0544 - 0.5440i; - 0.0544 + 0.5440i; - 0.0514 0.0809 0.0366].sup.T].

All eigenvalues of the Jacobian matrices for the above equilibriums at 125km/h are unstable.

The aforementioned bifurcation analysis validates the results based on the phase-plane stability analysis.

ATDB CONTROLLER DESIGN

The proposed active trailer differential braking (ATDB) controller is designed using a fuzzy logic technique. The essential concept of this control strategy is to use the trailer yaw moment resulting from the trailer differential braking to enhance the stability of the CT system.

With the ATDB control strategy, the vehicle state variables from the sensors are analyzed and the performance measures are to be calculated. If there exists a potential unstable motion mode, such as the jack-knifing, the ATDB controller will manipulate the trailer braking system. Through the trailer differential braking, the resulting yaw moment of the trailer will align the trailer with the towing unit. Thus, the articulation angle between the towing and trailing units can be reduced and the jack-knifing can be prevented. In this process, the external trailer yaw moment [M.sub.Z] is produced through intelligently adjusting differential braking forces on the left- and right-side wheels of the trailer, which can be defined as in Equation (18) as

[M.sub.z] = T/2 * [DELTA][F.sub.t] (18)

where [DELTA][F.sub.t] is the longitudinal force difference between the left and right wheels of the trailer and T is the track between the left and right wheels of the trailer.

With the external trailer yaw moment [M.sub.Z], the trailer's yaw motion Equation (7) can be rewritten as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

The major parts for the proposed fuzzy logic controller (FLC) design include inputs and outputs variables, fuzzification, fuzzy rules and deffuzifications. The FLC allows receiving variable inputs and mapping them into fuzzified signals by selected membership functions known as fuzzy sets. The fuzzy rules relate the input and output fuzzy variables by applying the knowledge-based experience and behaviors to control interface system, and the control decision is made by defuzzifying the mapped information based on the set of rules.

According to the mathematical CT model described in Section 2, six out of the nine state variables are selected to estimate the required trailer yaw moment. The six FLC input variables are car lateral speed [V.sub.c], car yaw rate [r.sub.c], trailer lateral speed [V.sub.t], and trailer yaw rate [r.sub.t], car roll angle [[phi].sub.c], and trailer roll angle [[phi].sub.t].

Two state variables, car roll angular velocity d[[phi].sub.c]/dt and trailer roll angular velocity d[[phi].sub.t]/dt, are not included, since their change tendencies are complicated in the fuzzy analysis due to their generation coming from the numerical derivation of car roll angle [[phi].sub.c], and trailer roll angle [[phi].sub.t]. The neglect of car roll angular velocity d[[phi].sub.c]/dt and the trailer roll angular velocity d[[phi].sub.t]/dt will not affect the FLC accuracy because they are exactly the roll rate to the two vehicle units. The third state variable, the car forward speed [U.sub.c] is also not included, because it is the input of the CT system.

The fuzzy input and output variables are defined Table 4, and they

are used in the FLC design. The popular membership functions for general FLC applications include triangular, trapezoidal and gbellmf etc. For the ATDB application in this paper, the gbellmf type is selected, as this type of membership function is simple and gives good control performance as well as easy to operate.

The fuzzy logic rule is used to associate the membership functions and variables received from the fuzzy inputs. To determine the result of the rule, a fuzzy associative memory is formed by evaluating the natural of each variable according to the level of fuzzy resolution chosen for the ATDB design purpose. The fuzzy logic controller for the ATDB system has six input variables. Each of them is divided into 3 levels respectively; therefore there will be 3 fuzzy rules in total. The output of the fuzzy controller is the estimate of the external trailer yaw moment [M.sub.Z]. A defuzzification strategy is used to represent this variable in 3 levels chosen by the definition of the 3 fuzzy rules. It is very important to notice, the magnitude and polarity of the selected fuzzy input variables are different at various CT system operational speeds, therefore the establishment of their fuzzy rules need to be adjusted accordingly.

To make the FLC design work properly, the CT system operational velocities are set at high speed. The main task using ATDB control system is to maintain high lateral stability. Figure 9 shows the membership functions for the above selected six fuzzy input variables at high speed case. Figure 10 shows the fuzzy rules.

To examine the effects of the ATDB controller on the lateral stability of the CT system, one case study is conducted under the simulated maneuver that the car front wheel steer angle is 0.0175 rad, and the vehicle forward speed remains constant at 125km/h.

Figures 11 to 15 show the simulation results of the case study with the ATDB control design, that is, the relation between: (1) the leading unit's side-slip angle and side-slip angular velocity ([[beta].sub.c] - d[[beta].sub.c]/dt), (2) the trailing unit's side-slip angle and side-slip angular velocity ([[beta].sub.t] - d[[beta].sub.t]/dt), (3) the articulation angle and the articulation angular velocity ([psi] - d[psi]/dt), (4) the leading unit's roll angle and roll angular velocity ([[phi].sub.c] - d[[phi].sub.c]/dt), and (5) the trailing unit's roll angle and roll angular velocity ([[phi].sub.t] - d[[phi].sub.t]/dt). For the purpose of comparison, the corresponding simulation results of the baseline CT system without the ATDB controller are also provided in these figures.

In the case with the ATDB controller, under the simulated maneuver at the vehicle forward speed of 125km/h, each of the state trajectories can return to the corresponding stable equilibrium point of the final steady state after a certain time period. However, in the case without the ATDB controller, except for the state trajectory of [[phi].sub.c] - d[[phi].sub.c]/dt, all other state trajectories diverge. This means that the CT system with the ATDB controller successfully improve the yaw and roll stability and converges to its final steady state condition.

CONCLUSIONS

This paper presents a systematic non-linear stability analysis method for articulated vehicles with ATDB control. In order to implement the proposed non-linear stability analysis method, a nonlinear 6-DOF yaw-roll CT system model is developed. An ATDB controller is designed to improve the lateral stability of the CT system represented by the nonlinear yaw-roll model. The bifurcation analysis based on the phase-plane method and the Lyapunov stability analysis theory are used to evaluate the lateral stability of the CT system through examining the following set of relation between: (1) the leading unit's side-slip angle and side-slip angular velocity ([[beta].sub.c] - d[[beta].sub.c]/dt), (2) the trailing unit's side-slip angle and side-slip angular velocity ([[beta].sub.t] - d[[beta].sub.t] /dt), (3) the articulation angle and the articulation angular velocity ([psi] - d[psi]/dt), (4) the leading unit's roll angle and roll angular velocity ([[phi].sub.c] - d[[phi].sub.c]/dt), and (5) the trailing unit's roll angle and roll angular velocity ([[phi].sub.t] - d[[phi].sub.t]/dt).

The nonlinear stability analysis based on the simulation results derived in the research leads to the following insightful findings: (1) the ATDB controller can significantly improve the yaw- and roll-stability of the CT system under the selected operating condition; (2) the superior performance measures of the ATDB controller can be reasonably evaluated using the proposed systematic non-linear stability analysis; (3) the proposed non-linear stability analysis method may be used in the design synthesis of articulated vehicles in order to determine lateral stability margins and to perform parametric studies; and (4) the proposed non-linear stability analysis method may be used to identify the threshold values to activate ATDB control of articulated vehicles in the implementation of the active safety system.

REFERENCES

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ACKNOWLEDGEMENTS

Financial support of this research by the Natural Sciences and Engineering Research Council of Canada [grant no. EGP 483295-15] is gratefully.

Tao Sun, Eungkil Lee, and Yuping He

University of Ontario Institute of Technology

Table 1. The notations and the primary parameters of the CT system Car total mass [m.sub.c] Car sprung mass [m.sub.cs] Trailer total mass [m.sub.t] Trailer sprung mass [m.sub.ts] Yaw moment of inertia of the total mass of [I.sub.z1] the car Yaw moment of inertia of the total mass of [I.sub.z2] the trailer Roll moment of inertia of the sprung mass of [I.sub.xx1] the car Roll moment of inertia of the sprung mass of [I.sub.xx2] the trailer Roll-yaw product of inertial of the sprung [I.sub.xx1] mass of the car Roll- yaw product of inertial of the sprung [I.sub.xz2] mass of the trailer Longitudinal distance between the CG of the a car and front axle of the car Longitudinal distance between the CG of the b car and rear axle of the car Longitudinal distance between the CG of the d car and hitch Longitudinal distance between the CG of the e trailer and hitch Longitudinal distance between the CG of the f trailer and axle of the trailer Height of the CG of car sprung mass above [h.sub.1] roll axis Height of the CG of trailer sprung mass above [h.sub.2] roll axis Vertical distance between car roll center and [Z.sub.1] hitch Vertical distance between trailer roll center [Z.sub.2] and hitch Roll damping coefficient of the car suspension [cr.sub.1] Roll damping coefficient of the trailer [cr.sub.2] suspension Roll stiffness of the car suspension [kr.sub.1] Roll stiffness of the trailer suspension [kr.sub.2] Car total mass 1521 kg Car sprung mass 1306 kg Trailer total mass 2000 kg Trailer sprung mass 1864 kg Yaw moment of inertia of the total mass of 1816 [kgm.sup.2] the car Yaw moment of inertia of the total mass of 3000 [kgm.sup.2] the trailer Roll moment of inertia of the sprung mass of 846.6 [kgm.sup.2] the car Roll moment of inertia of the sprung mass of 707.3 [kgm.sup.2] the trailer Roll-yaw product of inertial of the sprung 0 mass of the car Roll- yaw product of inertial of the sprung 0 mass of the trailer Longitudinal distance between the CG of the 1.063 m car and front axle of the car Longitudinal distance between the CG of the 1.716 m car and rear axle of the car Longitudinal distance between the CG of the 2.937 m car and hitch Longitudinal distance between the CG of the 6.0 m trailer and hitch Longitudinal distance between the CG of the 0 trailer and axle of the trailer Height of the CG of car sprung mass above 0.325 m roll axis Height of the CG of trailer sprung mass above 0.676 m roll axis Vertical distance between car roll center and 0.305 m hitch Vertical distance between trailer roll center 0.391 m and hitch Roll damping coefficient of the car suspension 19000 Nms/rad Roll damping coefficient of the trailer 30000 Nms/rad suspension Roll stiffness of the car suspension 110000 Nm/rad Roll stiffness of the trailer suspension 230000 Nm/rad Table 2. Tires parameters for using the Magic formula tire model The front tire coefficient of the car [B.sub.c1] 0.2583 The front tire coefficient of the car [C.sub.c1] 1.3 The front tire coefficient of the car [D.sub.c1] 2588.4 The front tire coefficient of the car [E.sub.c1] -0.2567 The rear tire coefficient of the car [B.sub.c2] 0.2793 The rear tire coefficient of the car [C.sub.c2] 1.3 The rear tire coefficient of the car [D.sub.c2] 2013.7 The rear tire coefficient of the car [E.sub.c2] -0.0318 The tire coefficient of the trailer [B.sub.t] 0.1787 The tire coefficient of the trailer [C.sub.t] 1.3 The tire coefficient of the trailer [D.sub.t] 4626.7 The tire coefficient of the trailer [E.sub.t] -1.1189 Table 3. The stability domain of the CT system Car yaw motion stability domain [[beta].sub.c] -0.015 to 0.015 (rad) d[[beta].sub.c]/dt -0.045 to 0.045(rad/s) roll motion stability domain [[phi].sub.c] -0.014 to 0.014 (rad) d[[phi].sub.c]/dt -0.025 to 0.025(rad/s) hitch motion stability domain [psi] -0.06 to 0.06 (rad) d[psi]/dt -0.15 to 0.15 (rad) Trailer yaw motion stability domain [[beta].sub.t] -0.1 to 0.1 (rad) d[[beta].sub.t]/dt -0.15 to 0.15 (rad/s) roll motion stability domain [[phi].sub.t] -0.025 to 0.025 (rad) d[[phi].sub.t]/dt -0.036 to 0.036 (rad/s) Table 4. Input and output variables used in the FLC design Fuzzy Variables Description Labels Input [r.sub.c] Negative high af1 Medium af3 Positive high af5 [r.sub.t] Negative high bf1 Medium bf3 Positive high bf5 [V.sub.c] Negative high cf1 Medium cf3 Positive high cf5 [V.sub.t] Negative high df1 Medium df3 Positive high df5 [[phi].sub.t] Negative high ef1 Medium ef3 Positive high ef5 [[phi].sub.t] Negative high ffl Medium ff3 Positive high ff5 Output [M.sub.z] Negative high nfl Medium nf3 Positive high nf5

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Author: | Sun, Tao; Lee, Eungkil; He, Yuping |
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Publication: | SAE International Journal of Materials and Manufacturing |

Article Type: | Report |

Date: | Aug 1, 2016 |

Words: | 8034 |

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