## Abstract

A multi-objective optimization (MOO) approach is utilized to design a controller for a novel rear-steering technology named “Brake-Actuated Steering” (BAS). This system uses individually controlled brakes to generate differential longitudinal forces on each side of an axle, causing it to steer. Compared to other active rear-steering solutions utilizing path-following control, the BAS system is expected to provide comparable maneuverability performance, while offering approximately a 50% reduction in both mass and costs. Two objective criteria that define the performance and control effort of the BAS system are considered. Constraints are imposed limiting the feasible set of design variables to ensure stability of the controller, and sufficient centering capability of the steering system in emergency braking conditions. The optimization is performed for low-speed cornering of a tractor-semitrailer under various operating conditions, including low-friction surfaces, different axle loadings, and vehicle speeds. The optimization provides a set of Pareto-optimal fronts, minimizing the objectives. Simulations are used to compare the performance of a nonoptimal design for the BAS axle prototype to that of the optimized axle design. These validate the superior performance resulting from the optimization, with root mean square error of the steering angle and the energy consumed by the towing unit reduced by 48% and 21%, respectively. Model and controller validation and the performance of the system are verified by experiments on a prototype vehicle system.

## 1 Introduction

Rear-steering of heavy goods vehicles can provide significant benefits in vehicle maneuverability, increasing vehicle capacity, and reducing fuel consumption [1,2]. However, rear-steering systems are heavy and expensive, and their widespread adoption is therefore limited [3].

The concept of using differential longitudinal forces across a steered axle to generate a yawing moment about the kingpin, thereby causing the axle to steer, has been proposed in literature. These systems are mainly proposed to provide an additional back-up steering system for the tractor unit in emergency steering conditions [4,5]. It is important to note that while these systems enable some degree of directional control, their steering accuracy is limited.

Recently, a partnership between automotive manufacturers has developed a brake-to-steer function [6] aimed at enhancing safety for highly automated and autonomous vehicles by utilizing existing actuators. However, its implementation necessitates fully electrically controlled brakes (x-by-wire), which are not yet standard.

There is scarce literature on the use of differential braking for trailer directional control. A “Steer by Brake” system has been implemented by replacing self-steering axles with standard rigid axles [6]. During cornering, the trailer electronic brake system can selectively brake the inner rear wheel of the last axle, reducing the equivalent wheelbase and improving vehicle maneuverability. Note that this approach introduces tire wear due to skid steering with a locked axle.

A patent issued to Vlastuin Group B.V. details the application of parking brakes for controlling self-steering axles during reverse maneuvers [7]. The absence of a physical prototype hinders the verification of the concept's practical implementation. McAdam et al. [8] showed that a simple differential braking control applied to a dolly connecting an A-double vehicle was able to generate limited axle steer due to the axle's suspension compliance and the lateral forces were able to significantly improve yaw stability. This was found to reduce the rearward amplification at high speeds.

This paper introduces a novel concept named “Brake-Actuated Steering” (BAS), which utilizes existing brakes on the sides of the trailer axles to accurately control the differential longitudinal forces and the directional performance of the vehicle [9]. This eliminates the need for any dedicated electrohydraulic steering actuator and associated hardware on the axle. The proposed solution is expected to reduce the difference between state-of-the-art active steering solutions and standard fixed axles by approximately 50% in terms of both additional costs and mass. Furthermore, owing to its precision steering control, the BAS system can provide accurate path following.

A comparison of steering solutions is detailed in Table 1, where it is evident that the BAS system on two axles has similar cost but half the mass of a hydraulic command (passive) steering system. The BAS system is expected to offer comparable maneuverability performance to commercially available electrohydraulic steering systems, but with significantly reduced mass, cost, and complexity, making it an attractive commercial steering solution.

Steering solution | No. of axles | Additional mass (kg) | Additional cost (£) | Swept path width (%) | Tail swing (%) |
---|---|---|---|---|---|

Self-steering | 1 | 200 | 2300 | −6 | +37.5 |

Command steering | 2 | 1145 | 6600 | −23 | +346 |

Brake-actuated steering | 2 | 500 | 5600 | −27 | −88 |

Active steering | 2 | 1050 | 12,000 | −27 | −88 |

Steering solution | No. of axles | Additional mass (kg) | Additional cost (£) | Swept path width (%) | Tail swing (%) |
---|---|---|---|---|---|

Self-steering | 1 | 200 | 2300 | −6 | +37.5 |

Command steering | 2 | 1145 | 6600 | −23 | +346 |

Brake-actuated steering | 2 | 500 | 5600 | −27 | −88 |

Active steering | 2 | 1050 | 12,000 | −27 | −88 |

Maneuverability performance is assessed in simulation by measuring the swept path width and tail swing delta during the maneuver.

The use of steered rear axles will also significantly reduce tire wear and enable low rolling resistance tires to be used under all applications, reducing fuel consumption by 7–9% [13].

Due to the novelty of the BAS concept and the strong coupling between mechanical and control parameters affecting the performance of the system, it is necessary to develop a methodology to provide optimal design parameters. In this paper, a multi-objective performance optimization is performed to identify optimal designs for the prototype BAS system and improve its low-speed cornering performance, while limiting the control effort required. A standard cornering maneuver is used, and various vehicle operating conditions are tested to provide a comprehensive evaluation of the BAS performance.

This paper is organized as follows: In Sec. 2, the vehicle model and controller developed for the BAS system are introduced. In Sec. 3, the performance evaluation criteria for the BAS system are presented. Section 4 provides details about the initial design and performance requirements of an experimental BAS prototype axle built for a test vehicle. In Sec. 5, the formulation of the multi-objective problem and the optimization strategy used in this study are presented. Section 6 presents the optimization results and experimental validation of the BAS model for low friction tire-road condition. The section also includes a sensitivity analysis of the optimization objectives to parameter variations. Finally in Sec. 7, conclusions are drawn.

## 2 Simulation Model and Brake-Actuated Steering Controller

### 2.1 Vehicle Model.

Figure 1 shows a 15 degrees-of-freedom (DOF) yaw-plane model of an articulated vehicle. The trailer was simulated using a double-track vehicle model that accounted for the differential brake forces acting on each axle and the resulting effects on the lateral and vertical loads on each tire. A single-track model was used to model the tractor propulsion unit because only its directional performance is needed for optimizing the trailer system. The bodies of both vehicle units were assumed to be rigid and to have 3DOF corresponding to longitudinal, lateral, and yawing motions. Roll motion was ignored. To simulate the coupling of the combination vehicle at the fifth wheel, kinematic constraints were incorporated, resulting in the elimination of 2DOF in the longitudinal and lateral directions. Consequently, the motion of the sprung masses could be modeled with 4DOF. The rotational speeds of the wheels contributed eight additional DOF, while the steering angles of the trailer axles (average of left and right wheels) introduced three more. Variables related to the tractor and semitrailer units are denoted with the subscripts “1” and “2”, respectively.

The model accounted for both the longitudinal and lateral load transfers to capture the effects of tire scrubbing (which is important in multi-axle suspensions) and variations in longitudinal accelerations of the vehicle bodies due to the brake forces acting on the trailer. It also accounted for the effects of lateral load transfer on the variation of the vertical tire forces during the steering maneuver. Fancher's nonlinear combined tire slip model (developed specifically for truck tires) was used to model the tires in the case of simultaneous longitudinal and lateral slip [14]. To reduce the computational costs of the optimization simulations in this study, the vehicle model was combined with a simplified braking system submodel, consisting of a first-order transfer function with time delay and time constant *τ*, equal to 1 s. The nonlinear model of the tractors-semitrailer with conventional (nonsteering) trailer axles was experimentally validated in Ref. [9]. The complete set of equations of motion and corresponding estimated vehicle parameters can be found in Ref. [9].

### 2.2 Axle Model.

Figure 2 shows a schematic of the BAS system. The steering hardware consists of (i) a freely steerable king-pin axle with modified conventional steering linkages; (ii) brake actuators that are individually controlled to provide the required asymmetric brake torques at the wheels; (iii) a yawing steering damper connecting the axle beam to the track rod to ensure stability of the steering yaw mode; and (iv) a pneumatic centering and locking mechanism (not shown) that is used to drive the axle to straight-ahead position when is not active (i.e., at high vehicle speed) or in the event of hardware and controller faults.

where $Jeq$ is the equivalent yaw inertia of the steering assembly about the kingpins,$\u2009c\delta $ is the yaw damping of the axle, $k\delta $ is the yaw stiffness, $Mf$ is axle friction torque, and $\u2211Mt$ is the sum of the moments applied by the tyre forces [15]. Each term contributing to $\u2211Mt$ is dependent on the steering linkage and suspension geometries. These will be discussed later in the paper.

### 2.3 Control Structure.

The objective of the BAS controller is to improve the low-speed maneuverability of articulated vehicles by using rear steering on the trailer unit. The BAS uses precise actuation of the brake torques at each wheel station to control the steering angles of the axles.

A “path-following steering” strategy was adopted [1]. This controls the “follow point” at the rear end of the trailer to accurately follow the path of the “lead point” (fifth-wheel coupling between the two units). The strategy is illustrated in Fig. 3.

In the initial analysis of control of the BAS system, it is assumed that the trailer has just one equivalent steering axle with steering angle $\delta r$. Later, this will be converted into steering angles for all three trailer axles.

Figure 4 shows a schematic of the BAS control structure consisting of: (i) a “Global Controller” (physically located in the tractor unit) containing a vehicle path-following reference model, calculating the rear steering angle demand ($\delta r,d$) based on the trailer vehicle speed ($u2$), yaw rate ($\Psi \u02d92$), and articulation angle ($\Gamma 1,2$); (ii) a “Local Controller” (physically located in the trailer) with two nested feedback loops. The outer loop takes the error in steering angle and outputs the differential brake torque demand $\Delta Tbd$ and provides a feedback compensating action for the steering angle ($\delta r$) to track the desired steering angle. The inner loop controller contains an allocation algorithm for distributing the differential brake torque demand between the left and right axle wheels ensuring that only one brake of the BAS axle is actuated at each time, depending on the sign of $\Delta Tbd$, and ensuring that the actuator limits are respected. It then regulates the differential brake torque ($\Delta Tb$) using a simple proportional-integral-derivative (PID) approach to track $\Delta Tbd$. The design of the controller, including all equations and control parameters, can be found in Ref. [9].

### 2.4 Pendulum Model.

In some of the control system studies, a simplified “pendulum model” was used representing the trailer. This model assumed the trailer to pivot about the fifth wheel and travel forward with a fixed longitudinal velocity $u2$, as shown in Fig. 5. Additionally, it was assumed that the trailer has one equivalent steering axle with steering angle $\delta r$. The simplified trailer model was used to calculate the response of the closed-loop steering system to a steering demand $\delta r,d$. Readers can refer to Ref. [9] for details of this model.

where $L(s)=C(s)G(s)$ is the open-loop transfer function of the system, $\omega o$ is the filtering frequency of the low pass filter used to limit the sensitivity of $C(s)$ to high-frequency measurement noise [16], and $s$ is the Laplace operator.

## 3 Evaluation Criteria of Controller Performance

The performance of the BAS controller can be characterized by various metrics and test methods. Since the aim of the controller is to improve the low-speed maneuverability of heavy vehicles in tight corners, the “UK standard roundabout” maneuver was selected to measure the vehicle performance. This roundabout sets the maneuverability requirements for all heavy vehicles using UK roads. It has an outer radius of 12.5 m and inner radius of 5.3 m [17]. In this study, numerical simulations were used to assess the dynamic performance of a tractor-semitrailer equipped with multiple BAS axles acting on the trailer unit.

The performance metrics measured from the simulation model are detailed below.

### 3.1 Stability Analysis.

The BAS mechanism presents some inherent instability associated with an underdamped yaw mode of the axle, which can cause sustained shimmy oscillations of the wheels about the kingpins during vehicle travel. Therefore, it is essential to carefully choose the axle design parameters to mitigate this phenomenon and ensure stable and responsive steering behavior. As shown in a previous study [9], both controller gains and physical components' values can be adjusted to improve the stability of the closed-loop steering system in response to a desired steering angle demand.

The damping ratio ζ is a measure of how quickly the oscillatory motion of the system yaw mode decays in response to a disturbance in the steering input. The damping ratio is a dimensionless parameter that is calculated from the real and imaginary parts of the dominant closed-loop poles, presenting the lowest damping ratio and slowest decay rate.

where $\lambda i=\rho i\xb1i\sigma i$ are the eigenvalues of the closed-loop response matrix $ACL$. For the steering system yaw mode to be considered sufficiently well dampened, the required damping ratio was set to be greater than 0.7.

### 3.2 Friction Utilization.

Brake-actuated steering axles are actuated by applying differential brake forces at the tire–road interface. These result from individually controlled brake torques at each wheel station. The longitudinal brake forces, combined with the lateral forces generated by the tires during cornering, utilize some of the maximum available friction (μ) for a given type of road. It was here assumed that the tires behave similarly in the longitudinal and lateral directions (i.e., $\mu x=\mu y$) [19].

where $i=fl$ (front-left), $fr$ (front-right), $ml$ (middle-left), $mr$ (middle-right),$\u2009rl$ (rear-left), and $rr$ (rear-right).

### 3.3 Low-Speed Cornering Performance.

The low-speed cornering performance was assessed by evaluating the tracking performance of the BAS steering controller during the test maneuver. This metric is a measure of how well the axle steering controller tracks the desired steering angles prescribed by the path-following controller for the three BAS axles.

where $\delta 2d,j$ is the demand steer angle at each axle *j*, $ti$ and $to$ are the times corresponding to the entry to and exit from the maneuver, and n is the number of BAS-steered axles.

### 3.4 Tractor Unit Propulsion Energy.

The engine propulsion energy $Ee$ can be used to quantify the impact of the trailer brakes and associated drag on the propulsion energy required. Note that this metric does not include the energy needed by the air compressor to provide the air used by the trailer braking system. This simplification should not impact the metric as minimal replenishment of the compressor is required for the low magnitude of brake pressures needed during cornering with the BAS system.

## 4 Hardware Design

### 4.1 Performance Requirements.

The steering axles on the test vehicle were designed to meet the following performance requirements:

To enable sufficient steering capabilities for the articulated vehicle to make prescribed turns, such as “the UK standard roundabout” maneuver.

Capacity to handle the maximum forces induced at the steering system track rod during the maneuvers or by external unforeseen disturbances. It was important to design the axle to ensure its structural integrity and to prevent the wheels from losing their controlled positions in any realistic circumstances.

To enable “full-lock” to “full-lock” movements of the axle at a frequency of at least 0.5 Hz. The steering system need to perform fast enough movements to avoid obstacles or during other unexpected maneuvers.

To quickly return the wheel to the straight-ahead position using an automated emergency centering and locking system. A response time to center and lock of 2 s was considered acceptable. The worst-case loading was taken as one brake being locked on.

The ability to operate during both forward and reverse travel.

### 4.2 Overview of the Initial Design.

A computer-aided design model of a prototype steering axle is shown in Fig. 6. The center-and-lock mechanism consists of an air spring actuator, which center the axle through a pair of levers attached to pivots on the axle. When the air spring is inflated these beams press the track rod to its central position, where it is locked in place with a pneumatic locking pin. Underneath the axle is a steering damper connected between the track rod and the axle.

Design considerations:

The moving parts were designed to achieve sufficient strokes to allow steering maneuver up to the limits specified by the requirement (1), without causing any interference.

A suitable yawing damper was selected to provide adequate axle damping. A linear characteristic was chosen throughout the range of displacements

The yaw stiffness was provided by the on-center stiffness of the air spring.

The design and material specifications of each component were established according to requirement (2) through structural analysis.

The pneumatic circuit architecture was designed based on requirements (3–4), which were also used as selection criteria for the center-and-lock mechanism.

Further information on the design can be found in Ref. [9].

Three BAS prototype axles were manufactured following the design guidelines outlined above. These axles were retrofitted onto an experimental trailer that was previously fitted with electrohydraulic active-steering axles [1]. The steering linkages were designed with Ackermann geometry. Most of the axle design parameters were carried over from the previous axle design, with kingpin angle, caster angle, and mechanical trail all set to zero.

## 5 Optimization of Steering Axle Design

### 5.1 Design Parameters

#### 5.1.1 Mechanical Parameters.

Selecting optimal values for the mechanical parameters of the axle is essential for optimizing the BAS system performance. As discussed in Sec. 2.2, the sum of the moments generated by the tire forces ($\u2211Mt)$, which governs the equation of motion of the BAS axle (see Eq. (1)), is dependent on the detailed geometry of the steering axle.

The important kinematic parameters included the kingpin axle geometry parameters, shown in Fig. 7, comprising caster trail ($nc$), caster angle (*ν*), king-pin inclination (*ϕ*), and king-pin offset at wheel center ($rw$). All of these parameters affect stability, steering effort, and returnability of the steering system [21,22].

Additional design parameters were the damping of the yaw damper ($c\delta $), and the axle yaw stiffness ($k\delta $), which were defined based on the stability and centering components discussed in Sec. 4. Note that the remaining geometry parameters (i.e., steering arms, track-rod length etc.) were here not considered as design variables. However, these were assumed to be fixed by Ackermann geometry, so as to minimize sideslip angles during turning, and consequently significantly reduce lateral tire forces and wear [23].

#### 5.1.2 Control Parameters.

The control parameters included the gains $KP\delta ,KI\delta $, and $KD\delta $ of the outer-loop steering tracking controller described in Sec. 2, and the brake torque distribution logic used to distribute the differential brake torque demand to each wheel. For the preliminary study, the torque distribution logic was set to the simple allocation strategy described in Sec. 2.3. As the optimization study focuses on the steering axle design, it was assumed for simplicity that the influence of the inner loop controller gains on the overall BAS system performance could be neglected.

#### 5.1.3 Vehicle Operating Conditions.

Various vehicle operating conditions were considered to provide a comprehensive assessment of the performance of the BAS system. The loading of the trailer axles was varied between 3*t* and 8*t*, corresponding to the case of an unladen and laden vehicle, respectively. The tire–road friction coefficient was simulated in the range from 0.2 (slippery road) to 0.8 (dry asphalt), and the vehicle speed was tested up to 20 km/h, which was assumed to be the threshold speed before activation of the automated centering and locking of the steering system. The axle performance is affected by friction in the steering mechanism, but this was neglected here for simplicity because effects are eliminated by the feedback controller.

where $xm\u2009\u03f5\u2009R6$ and $\u2009xc\u2009\u03f5\u2009R3$ refer to the mechanical and control design parameters, respectively.

The selection of lower and upper bounds for $x$ ($xlb$ and $xub)$ aimed at reasonably representing the geometry of conventional truck steering systems, as well as allowing packaging of wheel-hub components, and fitting of the centering and locking system. These bounding values are listed in Tables 2 and 3. Note that large negative values for $\nu $ and $nc$ are undesirable, as these would cause unstable axle motion [24].

$KP\delta $ | $KI\delta $ | $KD\delta $ | |
---|---|---|---|

$\u2009xc,lb$ | 0 | 0 | 0 |

$\u2009xc,ub$ | 5 | 5 | 5 |

$KP\delta $ | $KI\delta $ | $KD\delta $ | |
---|---|---|---|

$\u2009xc,lb$ | 0 | 0 | 0 |

$\u2009xc,ub$ | 5 | 5 | 5 |

$nc$ (m) | $\nu $ (deg) | $\varphi $ (deg) | $rw$ (m) | $c\delta \u2009$ (Nms $rad\u22121$) | $k\delta $ (Nm $rad\u22121$) | |
---|---|---|---|---|---|---|

$\u2009xm,lb$ | −0.05 | −1.15 | −6.3 | 0.15 | 1000 | 100 |

$\u2009xm,ub$ | 0.2 | 4.01 | 6.3 | 0.35 | 3100 | 1500 |

$nc$ (m) | $\nu $ (deg) | $\varphi $ (deg) | $rw$ (m) | $c\delta \u2009$ (Nms $rad\u22121$) | $k\delta $ (Nm $rad\u22121$) | |
---|---|---|---|---|---|---|

$\u2009xm,lb$ | −0.05 | −1.15 | −6.3 | 0.15 | 1000 | 100 |

$\u2009xm,ub$ | 0.2 | 4.01 | 6.3 | 0.35 | 3100 | 1500 |

where $Fc$ is the centering force from the steering system, and $lt$ is the length of steering arm, whose values were estimated from the initial axle design in Sec. 4. For the limiting design case, it was assumed that the vehicle is laden and travels on dry asphalt road with $\mu $ = 0.8.

### 5.2 Problem Formulation.

To setup the MOO problem, the design objectives and constraints were defined first. The damping ratio $\zeta \u2009$(Eq. (3)), which measures the damping of the closed-loop steering system, is only weakly dependent on the axle geometry parameters. As a result, the constraint on the damping ratio $\zeta $ was treated separately from the optimization procedure.

where the maximum road friction coefficient $\mu max$ was set to 0.8 or 0.2, for the vehicle traveling on dry asphalt or a slippery road, respectively.

### 5.3 Optimization Process.

A two-step optimization approach was devised to reduce the computational cost of the MOO. Figure 8 is a flowchart of the optimization strategy, which consists of the following steps:

Step 1. *Narrow range of control parameters*: Narrow down the vector of design variables to include only the control parameters, $\u2009xc$, and the yaw damping and stiffness. Perform the optimization of the control parameters for a fixed set of the remaining axle geometry parameters. This helped to identify optimized PID gains for any axle configuration, as well as suitable bounds on $c\delta $ and $k\delta $ that satisfy the stability requirement, as set by constraint on the damping ratio $\zeta $. The definition of stricter upper and lower boundaries for the axle parameters aims to achieve a fast and successful convergence of the subsequent optimization process in Step 2.

where $w1$ and $w2$ are the weights of the cost function, and $tr$ and $os$ are the rise time and overshoot computed from system response in time domain, with corresponding desired values of $tr,d$ and $osd$ set to 0.3 s and 15%, respectively.

Inequality constraints were set to limit the minimum values of phase margin (PM) and gain margin (GM) of the open-loop response L(s) to the recommended values for ensuring stability of the closed-loop system, with $PMmin$ = 2, and $GMmin$ = 30 deg [26]. Additionally, an equality constraint was set for the 0 dB gain crossover frequency $\omega p$ of the open-loop response to meet the desired target of $\omega p,d$=3.5 rad/s. The problem formulated in Eq. (13) was solved using the constrained optimization command *fmincon* in MATLAB [27]. Both interior point and sequential quadratic programming algorithms were trialed. It was found that both algorithms handled the constrained optimization problem well, yielding similar results in terms of optimal PID gains. However, better accuracy was achieved using the interior point algorithm.

The optimization was called repeatedly by *fmincon* to evaluate the optimal control parameters in $\u2009xc$ and the damping ratio of the closed-loop system for each trialed set of damping and axle yaw stiffness, as well as vehicle speed.

Step 2. *Refine full design space*: Solve the MOO problem of Eq. (12), with the full design vector $x$, containing $xc$ and the bound vectors $xlb,xub$ refined from Step 1. Repeat optimization process until the maximum number of iterations $Nmax$ is reached. The MOO problem was iteratively solved by exploring different step increments for increasing N until the Pareto fronts approximately converged onto the same curve, thereby determining $Nmax$. Other types of termination criteria were explored. For instance, in Ref. [28] the authors use indicators to track the largest movement in the variable or normalized objective space from one Pareto front solution to its neighbor across subsequent generations. The change in the indicators can then compared to specified thresholds for termination.

The determination of $Nmax$ above resulted in acceptable convergence for this study, with negligible difference in the Pareto front for running more generations. It was therefore deemed unnecessary to explore alternative termination criteria due the computationally expensive nature of the highly nonlinear problem and the additional complexity in determining the threshold values.

In this step, computer simulations were developed in Matlab/Simulink based on the nonlinear vehicle model with multiple trailer-steered axles presented in Sec. 2. The full design vector $x$, including the optimal control parameters updated from Step 1, was used to compute the objective functions. While the initial control parameters were computed for a single equivalent steering axle in the pendulum model, the corresponding PID gains for the three steering axles in the nonlinear vehicle model were set proportionally to that of the equivalent axle based on the distance of each axle from the fifth wheel, i.e., a proportionality factor greater than 1was used for the rear axle, as it is located furthest from the fifth wheel.

The optimization search for the MOO problem was then conducted by using a Nondominant Sorting Genetic Algorithm (NSGA-II). In contrast to common gradient-descent optimization algorithms, which can struggle to identify global solutions for multi-objective constrained and nonconvex optimization problems, NSGA-II is a global, gradient-free, and computationally efficient algorithm that enables approximate global solutions (named Pareto optimal fronts) to multi-objective problems [29,30]. Therefore, it was deemed suitable to solve the problem defined in this work. More details about the NSGA-II algorithm can be found in Refs. [31,32].

## 6 Optimization Results and Experiments

### 6.1 Results.

Figure 9 shows the results of the optimization process from Step 1. It plots contours of the damping ratio as a function of the yaw damping $c\delta $ and the vehicle speed $u2$ for the axle yaw stiffness set to $k\delta =500\u2009Nm\u2009rad\u22121.$ The damping ratio was computed from the closed-loop system response, including the optimized controller based on Eq. (13). For sufficiently large dampers, the tuning space is shown to be close to 1, which corresponds to the desirable condition of a stable and critically damped system with no oscillations.

While the stability performance generally deteriorates at higher vehicles speeds and is strongly dependent on the value of $c\delta $, the contours of the damping ratio did not vary significantly for wide range of $k\delta $: 100 Nm $rad\u22121\u2009<k\delta <1000$ Nm $rad\u22121$. Consequently, bounds on the design variables for the next optimization steps were only imposed on the yaw damping coefficient. A lower bound on $c\delta $ was set to 1 kNms $rad\u22121$ for the axle to retain a minimal target damping ratio of ζ = 0.6 at the top speed of operation of the steering system (approximately 20 km/h), while an upper bound of $c\delta =2.8$ kNms $rad\u22121$ was chosen to provide ζ = 0.8 at the same top speed.

Figure 10 shows the Pareto-optimal fronts resulting from Step 2 of the MOO of the BAS axle performance for a low friction road. The focus of this optimization was to improve the axle performance by enabling the vehicle to generate adequate brake forces for directional control under the most challenging operating conditions, including an unloaded vehicle (axle load = 3*t*) with low road friction, *μ* = 0.2. The figure highlights a tradeoff between the conflicting requirements of minimizing steering error and energy consumption. While friction utilization is a constraint in the MOO, it has also been represented in this plot for each design using shading colors. Is it important to ensure that the friction utilization remains well below the limit (0.2) to prevent tires from experiencing excessive slip. Minimizing the steering error leads to the darker circles on the top left of the plot, indicating designs with higher friction utilization.

The initial axle design in Sec. 4 is denoted as “*I*,*”* for which the optimal control parameters were selected following STEP 1 of the MOO.

#### 6.1.1 Validation of Optimization.

To validate the optimization study, a particular Pareto-optimal design denoted as “*O”* was selected and compared with solution *I.* While the selection of point “*O”* along the Pareto front is arbitrary for this study, it is preferred to minimize steer error while remaining within the flatter region of energy use around 130 kJ. The selected point has a friction utilization of approximately 0.16, making it suitably far from the limit set by the coefficient of friction of the road.

Note that the energy associated with turning, even for a vehicle with fixed axles, is a secondary factor within a real driving cycle when compared to the primary sources of energy use. The latter include tire rolling resistance losses (proportional to static load), aerodynamic drag, and braking power when decelerating to the corner (approximately 4 MJ is dissipated by a 40*t* tractor-semitrailer for a stop from 48 km/h).

In the context of a typical route for a heavy vehicle, the reduced mass with BAS, compared to state-of-the art active steering systems, potentially has a greater impact on reducing the energy usage than optimizing the cornering losses.

The full set of design variables and computed optimization objectives for the two investigated axle designs is detailed in Tables 4–6. The important differences between these two designs are due to the optimized values of kinematic parameters, $\varphi $ and $rw$, resulting in a higher value of the scrub radius d for the design *O.* Both $nc$ and $\nu $ remained substantially small in both cases. Other key differences between the two designs are the values of $c\delta $ and $k\delta $. The control parameters were optimized for each design.

$KP\delta $ | $KI\delta $ | $KD\delta $ | |
---|---|---|---|

$\u2009xc(I)$ | 2.86 | 0.68 | 2.13 |

$\u2009xc(O)$ | 2.66 | 0.70 | 1.63 |

$KP\delta $ | $KI\delta $ | $KD\delta $ | |
---|---|---|---|

$\u2009xc(I)$ | 2.86 | 0.68 | 2.13 |

$\u2009xc(O)$ | 2.66 | 0.70 | 1.63 |

$nc$ (m) | $\nu $ (deg) | $\varphi $ (deg) | $rw$ (m) | $c\delta \u2009$ (Nms $rad\u22121$) | $k\delta \u2009\u2009(Nm\u2009rad\u22121)\u2009$ | |
---|---|---|---|---|---|---|

$\u2009xm(I)$ | 0 | 0 | 0 | 0.274 | 3000 | 100 |

$\u2009xm(O)$ | −4.1 × 10^{−3} | −3.44 | −4.01 | 0.254 | 2228 | 822 |

$nc$ (m) | $\nu $ (deg) | $\varphi $ (deg) | $rw$ (m) | $c\delta \u2009$ (Nms $rad\u22121$) | $k\delta \u2009\u2009(Nm\u2009rad\u22121)\u2009$ | |
---|---|---|---|---|---|---|

$\u2009xm(I)$ | 0 | 0 | 0 | 0.274 | 3000 | 100 |

$\u2009xm(O)$ | −4.1 × 10^{−3} | −3.44 | −4.01 | 0.254 | 2228 | 822 |

$Ee$ (kJ) | $\epsilon \delta $ (deg) | $\mu u$ (-) | $d$ (m) | $\zeta $^{a} (-) | |
---|---|---|---|---|---|

I | 162.4 | 0.242 | 0.22 | 0.274 | 0.90 |

O | 128.3 (−21%) | 0.126 (−48%) | 0.17 | 0.288 | 0.87 |

$Ee$ (kJ) | $\epsilon \delta $ (deg) | $\mu u$ (-) | $d$ (m) | $\zeta $^{a} (-) | |
---|---|---|---|---|---|

I | 162.4 | 0.242 | 0.22 | 0.274 | 0.90 |

O | 128.3 (−21%) | 0.126 (−48%) | 0.17 | 0.288 | 0.87 |

Damping ratio is evaluated for the dry asphalt condition.

Compared to the initial design, design *O* presented significantly improved error tracking performance and reduced control effort, as demonstrated by the corresponding criteria of $\epsilon \delta $ and $Ee$, which were decreased by 48% and 21%, respectively. In contrast to design *I*, which failed to satisfy the friction utilization constraint, with $\mu u$ > 0.2, the optimized design was able to successfully generate the necessary braking forces to steer without exceeding the limiting friction circle.

To aid in understanding the performance tradeoffs between the two designs, time histories of the trailer axle steer angles during the chosen cornering maneuver and the corresponding brake force hysteresis loops are plotted in Figs. 11 and 12, respectively. As shown in Fig. 11, both axle designs exhibit accurate tracking of the steer angle demands for most of the maneuver across all three axles, owing to the selection of optimal controller gains based on Step 1 of the optimization process. This is shown by the values of $\zeta $, which in the case of an ideal high friction road is close to 0.9 for both designs. However, during the exit of the roundabout, the tracking accuracy of the rear axle suddenly degrades for design *I* due to a brake-lock up experienced at the right wheel between 40 and 45 s (see inset plot of Fig. 11). The brake lock-up causes the wheel to fully utilize the available friction at the road interface, which diminishes its ability to generate lateral tire forces, resulting in trailer swinging out and deviating from the reference steer angle. When the reference steer angle returns to zero, the controller can demand a control input that is below the maximum admissible brake force, allowing the vehicle to regain directional control.

The behavior described above is reflected in the brake force hysteresis loops shown in Fig. 12. Note that due to the brake distribution logic adopted, positive values of $\Delta Fx$ correspond to the left brake being active, whereas negative values of $\Delta Fx$ are associated with braking of the right wheel. The hatched areas show the limiting friction forces. Figure 12(a) shows that during the last part of the maneuver (point A), the right rear wheel requires a brake force that is higher than the available friction to steer the vehicle and meet the reference steer angle. Consequently, the brake force saturates up to the maximum admissible force, $Fmax,r$. The hysteresis loops also reveal differences in the maximum brake forces between the left and right sides of the vehicle ($Fmax,l$ and $Fmax,r)$. These arise from the lateral load transfer during cornering. Since the load on each wheel changes throughout the maneuver, the admissible brake forces (hatched boundaries in Fig. 12) are calculated based on the minimum values of normal load experienced on each side of the vehicle. Additionally, the area of the hysteresis loop can be used to infer information about the required control work for each axle design, similar to the metric $Ee$.

From Fig. 12(b), it is clear that the design *0* shows a smoother and more consistent brake force distribution at all wheels throughout the entire maneuver, and that the generated brake forces for all wheels are within the limits imposed by the friction force envelope, i.e., the criterion in Eq. (11) is met. This leads to a reduced area of the hysteresis plot and, as a result, reduced control effort, consistent with the lower value of $Ee$ for design 0 compared to that achieved by design *I.*

#### 6.1.2 Parameter Selection.

The set of optimized parameters shown in Tables 4 and 5 can be further investigated to explain the superior performance of design 0. The increased value of scrub radius $d$ enables lower forces to be used to steer, resulting in reduced energy consumption and friction utilization to generate similar brake torques as design *I*. This is achieved for design 0 by selecting a negative value of kingpin inclination $\varphi $, along with a reduced value of kingpin offset at wheel center $rw$. Note that the optimization ensures that the constraint on that maximum scrub radius, which is set to $dmax$ = 0.29, is still satisfied.

Figure 13 shows the steering torque resulting from the jacking forces due to kingpin inclination during a steady-state cornering maneuver, when the vehicle is moving forward at 1.5 m/s, with a constant 12 deg steer angle at the rear-steered axle, and a normal axle load of 3*t*. Note that high values of positive jacking torque result in a tendency of the steering system to return-to-center, which is undesirable for the BAS application. For simplicity, the torque due to the differential braking forces is assumed to be equal for both designs. It can be seen that by selecting the optimized values of kinematic parameters for design 0, the steering effort at the rear-steered axle of the vehicle is significantly reduced compared to the case with the initial design *I*.

While the caster angle $\nu $ and trail $nc$ have similar values for both designs, the improved axle performance of design 0 can be further explained by the selected optimized values of $c\delta \u2009$ and $k\delta \u2009.$ By considering the steady-state and transient parts of the steering maneuver separately, it can be assumed that the energy is consumed mainly during the constant steer angle phase, which occupies most of the maneuver.

where $Mat$ is the aligning torque.

Note that $\delta \u02d92,max$ is negative at the exit of the cornering maneuver and, therefore, larger values of $k\delta $ are needed for increasing values of $c\delta \u2009$ to minimize $\Delta Fx$. This analysis is consistent with the selection of $k\delta $ and $c\delta \u2009$ for the optimized design 0. Additionally, the decreased value of $c\delta $ compared with design *I* allows for lower controller gains in design 0, resulting in reduced control effort and energy consumption.

### 6.2 Effect of Vehicle Speed.

This section investigates the effect of varying vehicle speed on the optimization results of the MOO problem formulated in Eq. (12). Figure 14 shows the resulting Pareto-optimal fronts for vehicle speeds $u2$ = 10 km/h and 15 km/h for the dry asphalt road (*μ* = 0.8). When simulating vehicle cornering on a slippery road at speeds above $u2$ = 10 km/h, the friction utilization constraint was often exceeded, resulting in the optimization routine failing to generate a feasible set of candidate axle designs.

From Fig. 14, it can be seen that both objectives $(\epsilon \delta $ and $Ee$) increase with the speed of the vehicle. This causes the tradeoff between the performance metrics to become more evident at higher vehicle speeds, indicating that a significant increase in control effort is required. Note that the BAS was designed for low-speed maneuvering and, as shown in Fig. 9, the system becomes less stable at higher speeds. As a result, higher controller gains are required at these speeds, which, together with the noise introduced due to the higher values of steer angle rates, contribute to larger controller outputs. Consequently, this leads to higher braking torques and forces needed to steer the trailer. There is scope for improving the controller to handle this tradeoff.

Figure 15 shows the values of design parameters selected from different parts of the pareto front at different vehicle speeds. The points labeled 1-4 are also shown in Fig. 14. It can be seen that similar ranges of axle mechanical parameters are effective in optimizing the axle performance, irrespective of speed.

Some of the Pareto-optimal solutions are highlighted here for comparison. Points 1 and 2 (at 10 km/h) and points 3 and 4 (at 15 km/h) primarily focus on optimizing one of the objectives. By contrast, points depicted by *O* demonstrate a better balance between the two objectives and are computed at both vehicle speeds.

From Figs. 15(b), 15(c), 15(e), and 15(f), a general trend regarding $\varphi $ and $rw$ can be inferred, where lower values of $rw$ are paired with negative values of $\varphi $ (see points 2 and 4), while larger values of $rw$ are paired with positive values of $\varphi $ (see points 1 and 3). This means that the designs in the pareto front have all a similar value of scrab radius *d*.

### 6.3 Global Sensitivity Analysis.

To determine the influence of each design parameter on the design objectives $(\epsilon \delta $, $Ee$), a global sensitivity analysis was conducted. In contrast to a local approach, which studies the variations of the objectives based on small perturbations of the model inputs about their nominal values, a global approach was chosen to test wide ranges of parameters and their simultaneous effects on variations in the design objectives [33,34]. This is expected to result in a more reliable and robust design.

*y*=$\u2009\phi (x)$, where $x=[x1,\u2009x2,\u2009\u2026,\u2009xk]T$ represents a vector composed of k design parameters, and y is the vector of design objectives. It is assumed that the model inputs are uniformly distributed within the unit hypercube. Consequently, the function $\phi (x)$ can be decomposed into a sum of $2k$ terms with increasing dimensionality, as follows:

where $Vi$ measures the variation of the output due to a single parameter $xi$, $Vi,j$ refers to the contribution of two parameters ($xi$,$\u2009xj$) to the total model output variance, and so on, to include higher-order effects.

where the total sensitivity index $STi$ considers all the effects of each parameter $xi$, including its higher-order interaction with other parameters.

Figure 16 shows the results of the sensitivity analysis carried out in this study. The first-order and total sensitivity indices were evaluated for the mechanical design parameters $x=[nc,\u2009\nu ,\u2009\varphi ,\u2009rw,c\delta \u2009,k\delta ]T$ and the two optimization objectives *y* = $[\epsilon \delta ,\u2009Ee]T$. It can be seen that both the energy use and the RMS error of the steering angle are more sensitive to $nc,rw,$ and $\varphi $ than the other design parameters. Variation in these three parameters from their expected values all have the highest impact on fluctuations in the objective values. By contrast, variations in $c\delta $ and $\nu $ have less influence on the optimization objectives. Additionally, for both $\varphi $ and $rw$, large differences exist between the total and first-order indices, indicating higher-order interactions of these parameters. This is consistent with the fact that $\varphi $ and $rw$, are dependent design variables and both contribute to determining the scrub radius *d*, which has a significant impact on the BAS axle performance. The results of the sensitivity analysis can be utilized to simplify the axle design problem by fixing the less influential design parameters to their nominal values, and prioritizing design efforts for the BAS system on the parameters that have greater impact on the optimization objectives.

### 6.4 Experiments.

The model of Sec. 2 was experimentally validated in full-scale vehicle tests on an experimental tractor semitrailer combination, during low-speed cornering in low tire-road friction conditions (i.e., wet basalt tile road surface) on the Horiba-MIRA test track in Nuneaton, UK.

Figure 17(a) shows the experimental tractor-semitrailer, which was equipped with the BAS system. The hardware was configured to design *I* (Sec. 4). The axle centering and locking mechanism discussed in detail in Sec. 4 is shown fitted to the test vehicle in Fig. 17(b). Six prototype fast-acting brake valve actuators [38] (one on each trailer axle wheel) were used for steering actuation.

An RT3022 inertial and GPS Navigation System from OxTS [39] was installed above the trailer's middle axle and used to measure the vehicle's motion. The prototype braking system provided information about the wheel speeds, which were combined with the vehicle speed from the RT3022 to calculate the wheel slips on each wheel. The tractor steer angle was measured using a string potentiometer, which was fitted to one of the axle-steering arms. Rotary sensors were fitted to the trailer axle kingpins and used to measure the wheel steering angles. A calibrated rotary potentiometer [40] was fitted to the 5th wheel kingpin and measured the articulation angle between the tractor and the trailer. A real-time program was run on a dSpace Micro-autobox located inside the trailer, which was used to individually control the brake actuators and log data at 100 Hz from various sensors via CANbus.

#### 6.4.1 Tire Model Fitting.

Estimation of the tire adhesion-slip curve, to fit the Fancher tire model [41], was conducted using the approach by Henderson [42]. Coast-down tests from 40 km/h were performed in a straight line on a wet-basalt tile road surface, with the trailer's axles locked in the straight-ahead direction. Once the vehicle reached a predefined steady-state speed $vi\u2009$ of approximately 35 km/h, the controller demanded a constant value of brake pressure on one of the trailer wheels. Constant-pressure braking tests were repeated with gradual increments in brake pressure until complete wheel-lock up was detected. Note that the anti-lock braking system was not active, allowing wheels to lock during the tests. Tests were repeated for two brake actuators, with one acting on the left (L) and the other on the right (R) sides of the vehicle.

*m*is the gross combination vehicle mass,

*u*is the vehicle speed,

*t*is the time, and the subscripts

*i*and

*f*refer to the initial and final data points, respectively. $Fd$ and $Frr$ are the aerodynamic drag and rolling resistance acting on the vehicle, whose estimated values were taken from [42]. The brake adhesion force was also estimated using the measured chamber brake pressure $Pc$ from the braked wheel during the same time interval, as follows:

where $Kb$ is the brake gain, $Pcr$ is the cracking pressure, and $Rr$ is the wheel rolling radius.

Figure 18 shows the estimated adhesion forces $Fx$ versus the measured wheel slips *ξ* resulting from the test results and compared to the fitted Fancher adhesion-slip curve model. Good agreement is seen between the test data and the fitted model, particularly for small values of slip. Negligible differences were observed in the calculated adhesion forces from the two sides of the vehicle. A list of the brake and tire model parameters fitted for the wet-basalt tile surface is reported in Table 7.

Symbol | Parameter | Value |
---|---|---|

$Vf$ | Shaping factor (Fancher combined-slip tire model) | 5 m/s |

$\mu 0$ | Coefficient of static friction (Fancher combined-slip tire model) | 0.148 |

$\mu f$ | Coefficient of dynamic friction (Fancher combined-slip tire model) | 0.135 |

$Pcr$ | The threshold chamber pressure at which the braking force is nonzero | 1 bar |

$Kb$ | Brake gain | 1.70 kNm//bar |

$Rr$ | Wheel rolling radius (unladen) | 0.528 m |

Symbol | Parameter | Value |
---|---|---|

$Vf$ | Shaping factor (Fancher combined-slip tire model) | 5 m/s |

$\mu 0$ | Coefficient of static friction (Fancher combined-slip tire model) | 0.148 |

$\mu f$ | Coefficient of dynamic friction (Fancher combined-slip tire model) | 0.135 |

$Pcr$ | The threshold chamber pressure at which the braking force is nonzero | 1 bar |

$Kb$ | Brake gain | 1.70 kNm//bar |

$Rr$ | Wheel rolling radius (unladen) | 0.528 m |

#### 6.4.2 Inner Loop Controller Validation.

Low-speed weave maneuvers with a trailer steering frequency of 0.25 Hz were performed on the 7 m wide wet-basalt test track. The tractor unit was driven forward with the front wheels straight at a constant speed of approximately 1.5 m/s. The BAS axles were activated to ensure tracking of the weaving steering demand once all trailer wheels were on the basalt tiles. Figure 19(d) illustrates the simulated trajectories of the fifth wheel and of the rear of the trailer, and the boundaries (hatched) delimiting the basalt tile surface.

The measured tractor steering angle was set as input to the vehicle simulation and fitted tire model parameters for the wet-basalt tile were applied. Figures 19(a)–19(c) shows a selection of test results from the validation of the BAS vehicle model. The trailer front, middle and rear axle steering angles are shown. Despite the tires operating near the limits of adhesion, reasonably good agreement can be seen between the measured data and the predicted values from the simulations. However, some discrepancies are noticeable during changes in steering directions, and at start of the steering maneuver, indicating additional unmodelled dynamics, which could be mainly attributed to friction and backlash in the prototype BAS axles.

#### 6.4.3 Overall Model Validation.

The overall BAS controller, including the global and local controllers described in Sec. 2.3, was validated for the test vehicle performing a standard UK roundabout maneuver on dry asphalt, and at a speed of approximately 1.5 m/s. The tractor front axle steering angle and vehicle speed were used as input to the simulation model. The tractor longitudinal speed was computed from the position measurements of the trailer center of gravity given by the RT3022, and from the sensed articulation angle.

Representative values of delays, sensor noise, steer, and angle rate limits were considered for the simulation. To simulate measurement noise in the trailer angle sensor signals, band-limited white noise was added to the modeled signals. The magnitude of the noise was determined based on the computed covariance of the sensor, which was averaged over multiple test runs.

Figure 20 shows how accurately the BAS axle angles (in dark) track the reference steering angles (shown in gray), which are computed from the path following control. A good agreement is shown between the simulated (dashed lines) and measured results (solid lines), particularly regarding the steer angle demands. Discrepancies between the measured and simulated steer angles (dark) are consistent with those evaluated during the inner loop controller validation above. This indicates that the decreased accuracy in tracking performance at the start of the maneuver is not due to the reduced friction utilization when maneuvering on low-friction surfaces, but is likely caused by the unmodelled friction in the axle.

Figure 21 shows the brake torques developed at each wheel of the tri-axle semitrailer to actuate the BAS axles during the same roundabout maneuver. For each axle, the left brake is shown in dark, while the right brake is shown in gray.

Note that the brake torques during the steady-state phase of the maneuver are moderate, with a maximum value of 2 kNm at the rear axle, which corresponds to the largest steer angle of approximately 24 deg. Peak brake torques can be seen at 5 s and 75 s, as the vehicle enters and exits the roundabout, respectively. As shown in Figs. 21(a) and 21(b), the brake torque distribution logic alternates the wheel brake actuation between the left and right sides of the vehicle, following the change in direction of the steer angle requests (see Figs. 20(a) and 20(b)). At the rear axle (Fig. 20(c)), only one brake is actuated during the entry phase of the maneuver, as the steer angle demand remains steady or increases during this stage.

Comparison between the simulated (dashed) and measured (solid) results show a good agreement at the front axle (see Fig. 21(a)), where the magnitude of the brake torques, and the left/right actuation logic of the brakes is consistent. Conversely, larger deviations are shown in Figs. 21(b) and 21(c), when comparing the simulated and measured inner-loop data at the middle and rear axles. These can be attributed to the reduced accuracy in steer angle tracking for the outer loop, as shown in Figs. 20(b) and 20(c).

Overall, the experiments validated the ability of the BAS system to perform accurate path-following during low-speed cornering by individually controlling the brake actuators. While the validated vehicle model showed good agreement to the measured data, there is scope to model additional nonlinear axle dynamics, or to compensate their unwanted effects using the controller.

## 7 Conclusions

The design of BAS axles for a semi-trailer has been optimized to minimize energy consumption and the RMS error of steering angle. A tradeoff was found to exist between these two conflicting optimization objectives.

To fully utilize the potential of the BAS system, both mechanical and control parameters were chosen as design variables. A MOO approach was used with two steps to reduce computation costs.

Simulations were performed using a validated model of a tractor-semitrailer to compare the performance of the vehicle with optimized and nonoptimized BAS axles. Results showed that the optimized design allowed a reduction of 48% and 21% in RMS error of the steering angle and energy consumption, respectively, when cornering on low-friction surfaces.

Sensitivity analysis showed that the mechanical trail, kingpin inclination, and kingpin offset at the wheel center significantly impact the performance of the BAS system. These finding can aid in the design process.

A prototype triaxle BAS group was manufactured and tested for low-speed maneuvering on low-friction surfaces, verifying that the model can predict the behavior of the system under challenging operating conditions.

The overall vehicle model and the BAS controller were validated through full-scale testing of the vehicle performing low-speed cornering on dry asphalt.

BAS was shown to be an effective way to control trailer axle steering, offering maneuverability performance comparable to active steering systems with path-following control, while reducing additional mass and costs relative to standard fixed axles by approximately 50%.

## Funding Data

Engineering and Physical Sciences Research Council on the Centre for Sustainable Road Freight 2018–2023 (Grant No. EP/R035199/1; Funder ID: 10.13039/501100000266).

Cambridge Vehicle Dynamics Consortium (CVDC)

.

## Conflict of Interest

There is no conflict of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding authors upon reasonable request and available at https://doi.org/10.17863/CAM.109720.