Semitrailer Steering Control for Improved Articulated Vehicle Manoeuvrability and Stability

2.1 Equations of motion

The vehicle model used in the study is illustrated in Figure 1.

Fig. 1

AHV Linear Vehicle Model

The equations of motion are given as:

Equations (1) to (3) correspond to the tractor’s lateral, yaw, and roll motions, respectively.

Equations (4) to (6) correspond to the semitrailer’s lateral, yaw, and roll motions, respectively.

The kinematic constraint equation at the coupling point is derived by equating the velocities at the fifth wheel with respect to both vehicle units:

Note that for small articulation angles, we may assume equal forward velocities for vehicle units (u₁ = u₂).

One can construct a state-space representation of the linear system by defining the following state variables; details can be found in [19]:

In equation (10), δ₁ is the reference input imposed by the driver’s steering action and are the control inputs corresponding to semitrailer steering and roll control torque.

The assumptions in developing the model are as listed below.

• Effect of longitudinal forces are neglected (Constant forward speed),

• Pitch motions are neglected,

• Tyre and unsprung mass dynamics are neglected,

• Articulation, tyre slip, roll and steering angles are assumed to be small (linear equations),

• Vehicle parameters are time invariant (constant),

• Fifth wheel is assumed to allow free yawing without any stiffness or damping,

• Rigid sprung masses are assumed for vehicle units,

• Effect of lateral load transfer is neglected (single track/bicycle model),

• Tyres on semitrailer axles are assumed to be lumped in an equivalent single tyre.

In order to be able to simulate low-speed manoeuvres, a nonlinear version of the proposed model is also generated by relaxing the small angle assumption (keeping the trigonometric relations) due to large articulation and steering angles generated in low-speed turning manoeuvres.

2.2 Model validation

In order to validate the vehicle model, a standard single lane change manoeuvre defined in SAE J2179 [20] is performed with vehicle data presented in the Appendix. Simulation results from the linear model and the commercial software TruckSim model for the vehicle trajectory, lateral acceleration response, yaw rate response, and roll angle response are shown in Figure 2. It can be seen that there is, in general, good agreement between the results. The tractor’s roll motion has been accurately predicted by the linear model, but the roll response of the semitrailer experiences less oscillations in the linear approach which is possibly due to larger levels of simplification in the roll-plane model by ignoring the unsprung mass dynamics. But the linear model still gives acceptable overall values within the scope of this study which enables us to partially investigate the effect of active semitrailer steering on the roll motion.

Fig. 2

Model and TruckSim Responses for SAE J2179

As a result, the linear model is shown to provide acceptable dynamic behaviour.

3.1 control algorithms

LQR technique is applied with the objective of RA ratio regulation at high-speed for two purposes, namely the reduction of the semitrailer roll angle and high speed off-tracking. In the literature, there is no agreement on the desired value of the RA ratio. Islam [1] suggests that any deviation of the RA ratio from unity will correspond to a tracking error at high-speeds. This conclusion is supported by the physical explanation of the RA phenomenon given in the introduction section. With this understanding, the designer must try to keep the RA ratio as close to 1.0 as possible. Otherwise, tracking errors come to existence in the form of outwards semitrailer motion for RA ratio values larger than unity and inwards semitrailer motion for RA ratio values smaller than unity. On the other hand, it is desirable to reduce RA ratio as much as possible in order to decrease roll motion of the semitrailer unit.

Since the inwards motion of the semitrailer is possibly less destructive and more expectable than its outwards motion, it might be desirable to reduce RA ratio to values below unity to some extent, as a trade-off, in order to provide better roll stability as an important factor. This approach has been preferred in some previous works [13, 14].

In order to avoid such a compromise of the tracking ability, it is suggested to introduce another control input to the system to take care of the roll motion. In other words, it is possible to combine the active semitrailer steering with active roll control torque to achieve both targets. This is likely to come at a higher cost from both the hardware and energy requirement perspectives. Explicit discussion on the evaluation of relative merits of the two approaches is yet to appear in the literature. This study makes an attempt to fulfil the gap and suggests a general design approach which is based on the operational task and the vehicle of interest.

As a result of the aforementioned considerations, three different control algorithms are developed and suggested in this study and the results from simulations are compared to each other in order to provide a metric to be used in each individual case.

Control strategy 1: The algorithm used for this controller is developed such that it minimises the amount of High-Speed Transient Off-Tracking (HSTO) by providing the RA ratio of 1.0 without letting the ratio to go down further.

Control strategy 2: This controller is based on the algorithm which reduces the RA ratio below unity resulting in further reduction in the semitrailer’s roll motion, at the expense of larger HSTO values.

Control strategy 3: Making use of an additional roll control torque input, this controller keeps the RA ratio as close to 1.0 as possible, while simultaneously constraining semitrailer roll motions.

3.2 Linear quadratic regulator

The tractor-semitrailer system has a reference steering input from the driver δ₁ and two control inputs δ₂ and T_AR2 as the semitrailer steering angle and the roll control torque. Adding the roll control torque would have the advantage that the Active Steering Control (ASC) system would take the responsibility for HSTO minimisation while the Active Roll Control (ARC) torque takes care of further roll motion limitation. The LQR problem is defined as following:

Substituting (14) into (13), the new state-space is given by:

Selection of the weighting matrices [Q] and [R] in (16) is the designer’s task in order to provide a suitable performance index for the control purposes. However, for the problem in hand, it would be very difficult to define proper quadratic cost functions expressing the Path-Following Off-Tracking (PFOT) at low-speeds and High-Speed Transient Off-tracking (HSTO) or RA at high-speeds. Specifically at high-speed, there is a phase difference between generations of lateral accelerations for the vehicle units.

Defining a single mathematical relationship for calculation of RA ratio would require dependency of the cost function to parameters such as vehicle forward speed. An alternative approach is to define the lateral accelerations of the vehicle units separately or simultaneously as the cost function which also increases the complexity in mathematical derivations as well as decreasing the accuracy level.

In order to eliminate such complexities and also increase the efficiency level of the LQR design, it is suggested to make use of the QPSO optimisation method in determining the weighting factors which are, in fact, the elements of [Q] and [R] matrices. In this approach, the designer assumes the general form of the weighting matrices and tries to find applicable values for their elements based on an iterative optimisation method. During the optimisation, another cost function called the fitness function is going to be calculated repeatedly in each loop in order to evaluate the overall performance of the controller and the level of fitness for each control parameter set. Then, the designer is free to choose any type of cost function even by looking at the overall performance of the system during its operation and/or looking at some specific times.

3.3 Optimisation of weighting factors

A relatively quick and simple method of optimisation, for numerical problems presented in the previous section, is introduced by Eberhart and Kennedy [21] as the Particle Swarm Optimisation (PSO) which is based on movement of a swarm of particles obeying the Newtonian mechanics. It is concluded that the PSO method has advantages over other methods such as faster convergence and the need for fewer iterations, which makes it suitable for application in LQR weighting factor optimisation. There has been an extension to the PSO, presented more recently in [22], which assumes that the particles follow the rules of quantum mechanics. QPSO has the advantage over PSO that it is less likely to get stuck in the local optima. The QPSO-based LQR has shown more desirable results in terms of overshoot, steady-state error, etc. compared to LQR which is tuned by PSO, genetic algorithm, and manual adjustment techniques [23].

Formulation of the general QPSO algorithm in a simple computer algorithm can be found in [22]. For the application, the design objective needs to be converted into a mathematical form, i.e. the fitness function, to be used in the optimisation procedure. Optimisation parameters as described in the previous section are to be tuned as well. Tuning of the optimisation parameters is quite a direct, sensible, and simple task compared to tuning of the LQR weighting factors, so the selection of LQR weighting factors is left to the optimisation process and the problem is reduced to the tuning of a couple of unambiguous coefficients. QPSO method is applied to both low and high-speed cases according to the suitable vehicle models.

At low-speed, the only design criterion is to minimise the PFOT. Since the calculation of exact PFOT for the low-speed case in a computer code could complicate the programming, a new function is defined to be used as an approximation for the value of PFOT:

The value of E function is, in fact, the distance between current position of semitrailer’s axle denoted by X₂, Y₂ and that of the tractor’s front axle at the corresponding previous time denoted by X1′,Y1′calculated according to vehicle forward speed.

The value of error E is calculated at the points of interest on the trajectory of the vehicle during the manoeuvre in order to constitute the fitness function for low-speed optimisation. The value of the steering input may also be added to the description of fitness function to limit the control energy:

Here c_i’s and d_j’s are weighting factors to give proper importance to each term.

At high-speed, the design criteria are to regulate RA and yaw rate response as well as HSTO improvement, and roll motion reduction. For the high-speed case, the designer only needs to compare the peak lateral accelerations of the vehicle units during the lane-change manoeuvre. This observation then results in the following fitness function:

which is calculated based on the results of each simulation at every iteration. It must be noted that the term c4′maxϕ2is only used in Controller 2 in order to give more priority to semitrailer’s roll motion compared to RA regulation which consequently results is RA values below unity. Since the controller must be able to perform properly at any vehicle speed within the selected range, the same procedure is applied to several forward speeds. More details for implementation of QPSO on the AHV system can be found in [19]. Once the weighting coefficients are determined, the state feedback matrix is calculated from:

In which, [P] is the solution to algebraic Riccati equation:

In this study, MATLAB’s “lqr” function from Control System Toolbox is used to solve for [K] directly by knowing [Q] , [R]. The calculated state feedback matrices for different controllers are given in Table 1 to Table 3.

Based on iterative simulations, for the current vehicle parameters, it is suggested that at some speeds interval there is no need for steering control neither for PFOT enhancement nor RA reduction. Thus the high-speed active steering may be applied at speeds above a certain speed due to the presence of RA at these speeds. The low-speed active steering is suggested to be applied at speeds below a specific due to significance of PFOT. The margins may be identified by looking at elements of [K] for each controller.

Within the 40 – 70 km/h range, all the weighting factors are set to zero. Interpolation is applied to determine weighting factors for mid-speeds, and speeds below 10 km/h, are treated the same as 10 km/h.

The QPSO method is applied to all LQR designs with different objectives as described earlier and the results from the simulations are represented and compared in the next section.

3.4 Stability analysis

Since a linear model is used for high-speed case, a simple eigenvalue analysis provides stability status of the closed-loop system at a specific reference input from the driver’s command. On the other hand, for low-speed case, the stability of the nonlinear closed-loop system may be assessed using Lyapunov’s linearization method [24]. Based on this method, if the linearised system around the equilibrium point is strictly stable, the actual nonlinear system is concluded to be asymptotically stable, which is desirable in case of the articulated vehicle. The linearised closed loop system for both high and low-speed cases is in the form of (15). Thus, the new system matrix is given by:

Since several state feedback matrices have been introduced for a set of forward speeds, the eigenvalue analysis of [A^*] must be done at every speed interval. Based on the adopted vehicle parameters, matrices [K] , [A] , [B_c] are calculated as functions of forward speed for every controller within a selected speed interval (v_L < u₁ < v_H):

Note that [A] , [B_c] are directly dependant on u₁. Finally, using (29) the total system matrix [A^*] is obtained as a function of forward speed:

Eigenvalues of [A^*] for the three proposed controllers are plotted in Figure 3 within the speed range (10 – 120 km/h). All eigenvalues have negative real parts justifying the stability of the closed-loop control system in all three cases.

Fig. 3

Eigenvalues of Closed-Loop System (A, B, C for Controllers 1, 2, 3 respectively)

Note that for speeds lower than around 10 km/h, due to very small tyre slip angles and vehicle speed, a kinematic vehicle model is needed for simulating vehicle motion and the dynamic vehicle model will lose accuracy due to division by small values of u₁, u₂. At those speeds, stability of the vehicle is automatically realised with a kinematic model.

4.1 High-speed

The high-speed simulation results for the SAE J2179 manoeuvre are presented in this section in order to make a comparison between the introduced controllers. A sinusoidal steering input is applied to the vehicle’s front axle to generate the single 1.46 mlane change manoeuvre at speed of 88 km/h during 2.5 sec according to SAE J2179.

4.1.1 Baseline (passive) vehicle

The baseline vehicle’s response is given in Figure 4. The vehicle experiences RA especially at the end of the manoeuvre when the driver steers back to the starting position. This is followed by a deviation of the semitrailer from its expected path and oscillations in both trajectory and lateral acceleration response of the semitrailer at that time.

Fig. 4

Baseline Vehicle Response for SAE J2179

4.1.2 Controller 1 (Ra=1.0)

The vehicle equipped with Controller 1 responds to the steering input as depicted in Figure 5. It can be seen that Controller 1 has resulted in accurate tracking of the semi-trailer while the peak lateral acceleration of the semitrailer has been almost equal to that of the tractor resulting in RA ratio of almost 1.0. The roll motion, however, has not shown any significant improvement with respect to the results from baseline vehicle.

Fig. 5

Controller 1 Performance for SAE J2179

4.1.3 Controller 2 (Ra<1.0)

Application of Controller 2 results in the vehicle response as shown in Figure 6. Controller 2 reduced the RA ratio to less than unity to the extent that the HSTO and the control input do not exceed a predefined threshold; here, the RA ratio is allowed to be reduced by maximum of 50% to demonstrate the effect of this control strategy and compare it against other strategies. The roll angle of the semi-trailer is reduced significantly, but at the expense of introducing tracking errors and increased semitrailer steering input values. The settling time is also increased as a result of this control strategy. It must be noted that the responses provided here are based on an extreme case with high concentration on semitrailer’s roll reduction, in order to show the effect of control strategy explicitly. More priority could be given to the tracking error minimisation, to achieve a compromise between tracking errors and roll motion response and mitigating the undesired motion.

Fig. 6

Controller 2 Performance for SAE J2179

4.1.4 Controller 3 (Ra=1 & Rc torque application)

Controller 3, as the most complicated strategy among others, results in dynamic behaviour as shown in Figure 7. It can be seen that controller 3 results in the most desirable responses both from tracking perspective and the roll management. Obviously, this comes at a higher level of complexity for the system and larger amount of energy consumption necessary for the roll control torque.

Fig. 7

Controller 3 Performance for SAE J2179

Further simulations and discussion on the effects of ASC on other vehicle responses can be found in [19].

4.2 Low-speed

For the low-speed case, the objective of LQR is to reduce the PFOT of the vehicle which is defined as the radial offset between the front axle path and that of the rear axle during a 360 degree steady-state turn.

The performance of the controller at low-speed is evaluated by simulating a 360-degree manoeuvre at 10 km/h as presented in Figure 8. It can be observed that the LQR active semitrailer steering have successfully reduced the PFOT significantly with the proposed method. It is noted, however, that a small outward swing of the semitrailer in the transient cornering phase has appeared.

Fig. 8

Low-Speed 360-degree Turn