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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 549838, 21 pages
http://dx.doi.org/10.1155/2013/549838
Research Article

Design of an Automatically Tuned Fuzzy Controller for a Truck and Multitrailer System

School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney 2052, Australia

Received 9 April 2013; Revised 19 June 2013; Accepted 20 June 2013

Academic Editor: Shengyong Chen

Copyright © 2013 T. R. Ren. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The use of a truck and multitrailer system is advantageous because of its ability to transport heavy and large parts with a single powered vehicle. On the other hand, when the system is deployed in an autonomous and unmanned scenario, it remains a challenging task to design a drive controller. Since the drive is only applied to the truck and motivated by successful cases of human expert drivers, a fuzzy controller is developed to generate speed and turn rate commands in order to steer the multi-trailer system to reach the target position with minimum position error. Furthermore, in order to make the controller design efficient and effective, the parameters in the fuzzy controller including the membership functions and rules are automatically tuned using the implementation of efficient particle swarm optimization algorithm instead of relying solely on human expert knowledge. Near-optimal parameters are then derived and adopted in the controller, and drive commands are then generated. The performance of the truck-and-multi-trailer system under fuzzy control is verified through simulation studies, and satisfactory results are obtained.

1. Introduction

There has been a huge demand for the transportation of manufactured products from the plant to destinations for storage or dispatch. When the item to be carried is extremely large, for example, the components of an airplane [1], the use of a truck and multitrailer (TMT) would be advantageous against using a single vehicle. On the other hand, it is very difficult in driving such system due to the fact that only the truck can be manipulated while the trailers are passive. Skillful drivers are therefore always needed [2], but it is also desirable that automatic driving can be deployed while manipulation difficulties can be mitigated as much as possible [3].

The difficulties that hinder the operation of the truck system may be attributed to the complicated configuration of the trailer chain. In general, the trailer structure can be grouped according to the location of joints into the off-hooked, direct-hooked, and three-point structures. Trade-offs are required for mechanical simplicity and control limitations. It has been shown that the off-hooked configuration is able to provide a smaller steady state tracking error [4]. Moreover, based on the properties of the trailer system assembly [5], the steering of the truck has to be limited [6] in order to avoid collisions among the truck and the trailers. While attempting to achieve satisfactory control performance, it has also been shown that the problematic jackknife phenomenon can be avoided with dynamic output feedback with a reduction in the use of joint sensors [7]. Among the available control methods, nonlinear control approaches based on Lyapunov and back-stepping designs had been employed [8]. These methods are advantageous in tackling problems arising from the control of articulated and underactuated systems. In a similar context, the feedback stabilization scheme was adopted in controlling and stabilizing a truck-trailer system [9].

One of the common scenarios that a truck-trailer system employs for transporting parts is to follow a desired trajectory from the start to the final position [10]. As it can be seen, the control of the truck-trailer system to follow a desired path is nontrivial [11] due to nonholonomic constraints and limits in steering angles. This problem may become more severe if there are uncertainties in measuring the joint angles between trailers [12]. Furthermore, the effect of trailer configurations had been investigated for on-axle and off-axle hitching modes [13]. The feasibility of employing Lyapunov and back-stepping design was demonstrated. In the work reported in [14], a virtual articulated vehicle was inserted into the trailer chain to facilitate the control for reverse trajectory tracking. It has also been highlighted [15] that the trajectory curvature also plays an important role in the tracking performance of a truck-trailer system. Moreover, an alternative approach based on geometry features of the truck-trailer kinematics was proposed [16]. The method considered the cascaded kinematics and the propagation of the drive command through the system.

While model-based control theoretic approaches are able to derive controllers for the truck-trailer system with satisfactory performance, their design could be severely affected by the uncertainty in the model and unpredictable disturbances while driving the truck-trailer system. For these cases, soft computing techniques such as fuzzy theories may be useful in the design of mobile robots and truck-trailer system controllers [17, 18]. In the fuzzy controller, a rule base is formulated, and with the use of membership functions describing the input and output variables, an inference engine is then invoked to generate the desired output. In [19], the design of a fuzzy controller was accompanied with a fuzzy observer for better performance. Further examples of employing fuzzy control for truck-trailer systems had also been presented in [2022] which had incorporated the use of Lyapunov design, line-of-sight method, and constraints imposed from obstacle avoidance.

To a large extent, the design of fuzzy controllers originated and relied on the availability of expert knowledge of the behavior of the system under consideration. Hence, the design of controllers may be time consuming and tedious. Therefore, techniques were developed for setting fuzzy rules by learning from test examples [23]. This approach had been well received, and the incorporation of neural network based methods was reported in [24] with the aim to facilitate training the fuzzy controller. Further to effects laid in releasing the burdens in building the fuzzy system, the use of intelligent [25] and evolutionary computing methods is also a very attractive approach [26]. These attempts include the genetic algorithm [27] which was inspired by the process in which living species evolve according to the principle of the survival of the fittest. Due to its generality, the algorithm is able to tune the fuzzy controller parameters to near-optimal values. Recently, the quantum-inspired evolutionary algorithm has been used to tune the parameters of a fuzzy controller [28]. This algorithm possesses a wider diversity of potential solutions which may be advantageous in searching the optima.

Though these evolutionary and intelligent computing based methods produce fuzzy controllers for truck-trailer systems with satisfactory performances, their implementation may not be efficient. It is mainly because the algorithmic complexity and overheads are imposed in addition to problem dependent evaluation of objective functions. To this end, the particle swarm optimization algorithm (PSO) [29, 30] could be employed in tuning fuzzy controllers for its implementation simplicity. The use of the PSO algorithm, inspired from mimicking swarms of agents moving through the solution space, has been very attractive because of its generic applicability to a wide domain of problems. For example, it was used in the coordination of a swarm of mobile robots [31], robot motion planning [32], and in the design of fuzzy controllers [33]. Additional examples on the use of PSO in fuzzy controller designs can be found in [34, 35]. The fuzzy membership functions and fuzzy rules were tuned using PSO, and satisfactory results had been obtained in the control of industrial processes.

In this work, a fuzzy controller is designed for a truck-trailer system to reach a desired position. In order to be released from depending solely on human expert knowledge in the controller design an intelligent computation algorithm, the particle swarm optimization, is adopted for its implementation efficiency. In particular, fuzzy membership function coefficients and fuzzy rules are tuned simultaneously using the PSO algorithm. The generality of the proposed approach is demonstrated when the method is used to tune fuzzy controllers for truck-trailer systems with single and multiple trailers. The effectiveness of the developed controller is illustrated using simulation studies.

The rest of this paper is organized as follows. In Section 2, the kinematic model of the truck-trailer system is given. The fuzzy controller design is detailed in Section 3. Simulations are described in Section 4 with results presented and discussed. Finally, a conclusion is drawn in Section 5.

2. System Model

A system model is developed here describing the kinematic relationships between the driving truck and the passive trailers. The positions of the centers of the trailers are derived from the position of the truck. The translation and angular velocities of the trailers are then formulated with respect to the drive commands applied at the truck. Finally, the construction of the overall truck-and-multi-trailer system is presented.

2.1. Positional Model

Let the truck-and-multi-trailer system consist of an actively driven truck and a number of passible trailers. The truck and trailers, collectively named as vehicles, are connected at joints along the front-to-end axis of each vehicle [4, 5]. Figure 1 shows a diagram of the typical connection between two vehicles and .

549838.fig.001
Figure 1: Truck-and-multi-trailer system consisting of connected vehicles.

The truck-and-multi-trailer system is deployed in a flat terrain described by a world coordinate . The center of the wheels is taken as the vehicle position labeled as , and the vehicle is inclined to an angle with respect to the world coordinate -axis. The size of the vehicle is denoted as the length from the wheel center to the front and to the rear. The angular difference between two consecutive vehicles is given by . The indices increase towards the last trailer.

By making use of the definitions of the parameters of the TMT system, the general positions of the vehicles can be determined from

2.2. Kinematic Model

For the vehicles inclination angles , they are determined by considering the effect of the driving efforts applied to the leading vehicle. Let the th vehicle be driven to a translational velocity and an angular velocity ; see Figure 2.

549838.fig.002
Figure 2: Propagation of translational velocity.
2.2.1. Translational Velocity

The resolved translational velocity along the front-to-end axis of vehicle due to the translational velocity of the leading vehicle is

Moreover, the translational velocity component; arising from the angular velocity applied to vehicle and effected on th vehicle, is first obtained from the rotation of the link, giving , It is then resolved along the direction of the front-to-end axis of the th vehicle as Hence, the total translational velocity of the th vehicle becomes

2.2.2. Angular Velocity

For the angular velocity of the th vehicle, due to the drive from the th vehicle, is obtained by balancing the rotations (Figure 3).

549838.fig.003
Figure 3: Propagation of angular velocity.

The component from the rotation of th vehicle is The element arising from the translational velocity of the th vehicle is Furthermore, the portion of the angular velocity due to the link is obtained from the effect caused by the rotation of the th vehicle and then resolved along the perpendicular to the front-to-end axis of the th vehicle; that is Hence, the resultant angular velocity of the th vehicle is

2.3. Motion of the Truck and Multitrailer System

Now (4) and (8) have provided the translational and angular velocities of a generic vehicle with regard to the velocities of a leading vehicle in the TMT system. Also note that the truck is the head of the truck-trailer chain, and its motion is driven by given commands as its translational and angular velocities, namely, and (Figure 4). Then by iterating through the TMT system chain, the velocities can be found. Based on the translational and angular velocities applied to the truck which is equipped with a steering wheel, the angle sustained with respect to the truck front-to-end axis is

549838.fig.004
Figure 4: Truck-and-multi-trailer system with trailers.

Furthermore, by invoking (1), the positions of the centers of the vehicles can be found while the dimensions of the links and had been given as system parameters, and the initial intervehicle angles are also known. We have In addition, The velocities of other trailers in the system can then be obtained iteratively by referencing to an ordered indexing convention. Together with the rate of change of vehicle orientation, the positions and orientations of the vehicles can be determined.

3. Controller Design

In order to steer the truck-and-multi-trailer system to a desired target position, a controller is designed to generate the drive commands. In particular, to simplify the design, a direct-hooked model is used. Further, to cater for the nonlinearity and complexity, a fuzzy logic controller is also proposed.

3.1. Control Strategy

The desired motion is obtained by driving the pair of truck wheels with a commanded linear velocity and turn rate . Let the system state be parameterized by where , , and are the position and orientation of the truck with respect to a world coordinate, is the angle of the vehicle with a vehicle in front, and is the number of trailers. It is also assumed that there is no slip on any of the wheels.

There are three possible configurations of truck-trailer systems, which are off-hooked, direct-hooked, and three-point trailer systems. From the previous work [4], the mechanical structure of three-point trailer system is very complicated. Direct-hooked trailer system is a special condition of off-hooked trailer system. In such scenario, the length of the rear link becomes diminished; that is, .

In order to avoid the jackknife problem, the angle is limited to within where . The range of turning rate in rad/s is

Furthermore, the positional errors are given by where , , and define the desired ending system position and , , and represent the distance error in the direction parallel to the - and -axes and the desired truck orientation. The tracking errors are calculated by

A fuzzy logic controller is able to transfer uncertain experience and knowledge about the behaviour of observed objects into the process of decision making based on fuzzy rules and membership functions. Therefore, how to generate both rules and membership functions plays a very important role in the fuzzy controller design. If the exact knowledge about the dynamics or behaviour of the observed system is available, the rule and membership function generations are straightforward. However, if knowledge and experience are limited, the controller design task will be very difficult. In this section, the generation of fuzzy logic controller rules and membership functions is formulated as an optimization problem, whereby the objective is to specify the controller parameters to the satisfaction of all constraints. The particle swarm optimization algorithm is employed to solve this optimization problem.

3.2. Particle Swarm Optimization

The particle swarm optimization algorithm belongs to an agent-based heuristic search method used to find the maxima/minima of an objective function in a search space, whose potential solutions are coded as particles. The algorithm contains a recursive loop of iterations and could be described by Algorithm 1. In addition, the PSO algorithm can be described by the following expressions: where is a weighting factor, is the velocity of the practical in the solution space. and are random numbers in , ; here, presents the uniform distribution and is the maximum value for and for . Position represents the global best in the group of particles, and denotes a record of the individual particle's best solution over the past iterations, in which is the index of the particle, and is the iteration count.

alg1
Algorithm 1: The PSO algorithm.

3.3. PSO for Fuzzy Logic Controller Tuning

In the Mamdani type of controller, the conventional fuzzy rules can be represented as where terms    represent the input variables and    represent the output variables. The numbers and together represent the number of input and output variables. , denote the corresponding linguistic values (fuzzy sets), respectively. Furthermore, and . , , is the th rule; is the number of rules.

3.3.1. Fuzzy Rules Tuning

The relationship of input and output can be expressed using numerical values. Therefore, fuzzy rules could be tuned automatically by employing the PSO algorithm. Let the fuzzy rule be expressed by numerical values as follows: In (20), the first three digits represent the membership function associated with three inputs (1 denotes large, 2 denotes middle, and 3 denotes small). The two digits after the comma indicate two outputs (1 denotes large; 2 denotes middle). The value in the bracket is the weighting factor of this rule while the last digit after the colon represents the relationship between the three inputs (1 denotes AND; 2 denotes OR). Therefore, it is possible to find the appropriate relationship between the rules and to determine if the rules are adequate.

3.3.2. Fuzzy Membership Functions Tuning

The membership functions employed in this fuzzy logic controller are all of Gaussian forms. The Gaussian (the normal distribution) is a continuous probability distribution that has a bell-shaped appearance. It is generally considered as the most prominent probability distribution in statistics. The parameters that define the membership functions are the mean and deviation of each membership function, which is defined as: Figure 5 shows the strategy of using a PSO for rule tuning in fuzzy logic controller. In the proposed PSO process, each particle is formed to represent the rule of the fuzzy logical controller. The purpose of the PSO is to find the set of rules which minimize the response error of the truck and trailer system. The objective function is defined as where is the total running time of the fuzzy logic controller, is the error between the reference inputs and actual system outputs.

549838.fig.005
Figure 5: The proposed fuzzy logical controller based on PSO.

4. Simulation

In this section, a PSO tuned fuzzy logic controller is designed for steering the truck-and-multi-trailer system to its desired position. The objective function that evaluates the fitness of each practice was given in (22). After tuning membership functions and fuzzy rules, the fuzzy logic controller is able to provide a minimized response error. In order to verify the effectiveness of the proposed method, computer simulation results of the TMT system control are shown in this section. Simulations are conducted with both one-trailer system and multiple trailer system.

In practical applications the truck could start from arbitrary locations and orientations and is steered by the drive commands to the desired position. In the simulation studies, it is assumed that the truck starts from the origin of the world coordinate and parallel with -axis; that is, . The truck and trailer are initialized with an orientation of to the world coordinate. The desired position is selected randomly, and the desired orientations for both the truck and the trailer is . Furthermore, the maximum linear velocity of the truck is  m/s. The lengths of the vehicles are  m.

4.1. Case Study: One-Trailer System

For the one-trailer system, there are three inputs to the controller. The first input, labeled Distance, is the distance between the reference input and the current truck position. The second input, labeled TruckO (truck orientation) is the orientation of the truck with respect to -axis. The third input represents the orientation of trailer (TruckO1) with respect to -axis. The output of the fuzzy logical controller is the linear velocity and turn rate while moving the system.

The initial controller is set to have 10 rules: : IF (Distance is Small) AND (TruckO is Small) AND (TrailerO1 is Large) THEN (Velocity is Small) AND (Turning rate is Mid).: IF (Distance is Mid) AND (TruckO is Small) AND (TrailerO1 is Large) THEN (Velocity is Small) AND (Turning rate is Mid).: IF (Distance is Small) AND (TruckO is Large) AND (TrailerO1 is Mid) THEN (Velocity is Mid) AND (Turning rate is Large).: IF (Distance is Large) and (TruckO is Mid) AND (TrailerO1 is Mid) THEN (Velocity is Large) AND (Turning rate is Small) : IF (Distance is Large) AND (TruckO is small) AND (TrailerO1 is Large) THEN (Velocity is Large) AND (Turning rate is Small).: IF (Distance is Large) AND (TruckO is Mid) AND (TrailerO1 is Large) THEN (Velocity is Large) AND (Turning rate is Mid). : IF (Distance is Large) AND (TruckO is Large) AND (TrailerO1 is Mid) THEN (Velocity is Large) AND (Turning rate is Large) : IF (Distance is Large) AND (TruckO is Small) AND (TrailerO1 is Small) THEN (Velocity is Large) AND (Turning rate is Small). : IF (Distance is Mid) AND (TruckO is Mid) AND (TrailerO1 is Mid) THEN (Velocity is Mid) AND (Turning rate is Mid). : IF (Distance is Mid) AND (TruckO is Small) AND (TrailerO1 is Small) THEN (Velocity is Mid) AND (Turning rate is Small).

The fuzzy rule surfaces are showed in Figure 6. Initial fuzzy membership functions are showed in Figure 7.

fig6
Figure 6: Fuzzy rule surfaces based on human expert knowledge.
fig7
Figure 7: Fuzzy membership functions.

In the particle swarm optimization algorithm, the total searching iterations are set to be 200. It is also defined in the searching process that if the error is not reduced for 10 iterations then the search should be terminated. The inertia factor was set to be 0.5, and weighting factors and were set to be 1 and 0.2, respectively. At the completion of the PSO tuning, the rules of the fuzzy controller have attended a reduced number. The final rules are given by the following: : IF (Distance is Large) OR (TruckO is Large) OR (TrailerO1 is Large) THEN (Velocity is Small) AND (Turning rate is Small). : IF (Distance is Large) OR (TruckO is Mid) OR (TrailerO1 is Mid) THEN (Velocity is Large) AND (Turning rate is Small). : IF (Distance is Small) OR (TruckO is Large) OR (TrailerO1 is Mid) THEN (Velocity is Large) AND (Turning rate is large). : IF (Distance is Large) AND (TruckO is Small) AND (TrailerO1 is Small) THEN (Velocity is Large) AND (Angle Turn is Large). : IF (Distance is Small) AND (TruckO is Small) AND (TrailerO1 is Large) THEN (Velocity is Large) AND (Angle Turn is Small).: IF (Distance is Small) OR (TruckO is Mid) OR (TrailerO1 is Large) THEN (Velocity is Large) AND (Angle Turn is Large).

Generally, the number of fuzzy rules that have to be processed for fuzzy controller to make decision mainly depends on the number of input variables, the number of output variables, and the number of predefined linguistic values for an individual input variable. Therefore, the total number of the fuzzy rules can be calculated by the following equation: where and denote the number of inputs and outputs respectively, and and are the linguistic values of the input and output. Based on (23), the total rule number of the fuzzy controller is 243. However, after PSO tuning, only 6 rules are employed in this fuzzy controller if weighting factor approaches zero; this rule is deleted. Figure 8 shows the rule surfaces which are generated by the PSO rule tuned fuzzy logic controller.

fig8
Figure 8: Fuzzy rule surfaces tuned by PSO algorithm.

At the completion of the PSO membership functions tuning, the membership functions for the inputs and outputs of the fuzzy logic controller have been modified. They are shown in Figure 9.

fig9
Figure 9: PSO tuned fuzzy membership functions.

Based on the rules which are generated by the PSO algorithm, the control error is shown in Figure 10. The maximum error has been reduced from 229.45 to be around 96.02 after 200 iterations.

549838.fig.0010
Figure 10: Control error of the truck and trailer system.

Figure 11 is the simulation result of the truck and multiple trailer system. The truck is initially located at the origin of the coordinate system as shown in Figure 11(a) and then moves to the desired position which is shown in Figures 11(b) and 11(c). In Figure 11(d), the system arrived at the end point with the minimum control error.

fig11
Figure 11: Motion of truck and multiple trailer system.
4.2. Case Study: Multiple Trailer System

For a system with three trailers, there are five inputs to the controller. The first input, labeled Distance, is the distance between the reference input and the current truck position. The second input, labeled TruckO (truck orientation) is the orientation of the truck with respect to -axis. The rest of inputs present the orientation of three trailers (TrailerO1, TrailerO2, and TrailerO3) with respect to -axis. The output of the fuzzy logical controller is the linear velocity and turn rate to drive the system.

For multiple trailer system, it is extremely difficult to build fuzzy rules based on human expert knowledge. Therefore, the initial 10 fuzzy rules are randomly set as follows:: IF (Distance is Large) AND (TruckO is Large) AND (TrailerO1 is Large) AND (TrailerO2 is Small) AND (TrailerO3 is Small) THEN (Velocity is Small) AND (Turning rate is Small) : IF (Distance is Large) AND (TruckO is Large) AND (TrailerO1 is Large) AND (TrailerO2 is Small) AND (TrailerO3 is Large) THEN (Velocity is Large) AND (Turning rate is Large) : IF (Distance is Small) AND (TruckO is Mid) AND (TrailerO1 is Mid) AND (TrailerO2 is Small) AND (TrailerO3 is Small) THEN (Velocity is Small) AND (Turning rate is Small) : IF (Distance is Small) AND (TruckO is Mid) AND (TrailerO1 is Mid) AND (TrailerO2 is Large) AND (TrailerO3 is Small) THEN (Velocity is Large) AND (Turning rate is Small) : IF (Distance is Small) AND (TruckO is Small) AND (TrailerO1 is Small) AND (TrailerO2 is Mid) AND (TrailerO3 is Large) THEN (Velocity is Small) AND (Turning rate is Large) : IF (Distance is Small) AND (TruckO is Large) AND (TrailerO1 is Large) AND (TrailerO2 is Small) AND (TrailerO3 is Large) THEN (Velocity is Small) AND (Turning rate is Large) : IF (Distance is Small) AND (TruckO is Small) AND (TrailerO1 is Small) AND (TrailerO2 is Small) AND (TrailerO3 is Small) THEN (Velocity is Mid) AND (Turning rate is Mid) : IF (Distance is Mid) AND (TruckO is Large) AND (TrailerO1 is Large) AND (TrailerO2 is Large) AND (TrailerO3 is Mid) then (Velocity is Small) AND(Turning rate is Large) : IF (Distance is Small) AND (TruckO is Small) AND (TrailerO1 is Small) AND (TrailerO2 is Small) AND (TrailerO3 is Mid) then (Velocity is Small) AND (Turning rate is Mid) : IF (Distance is Mid) AND (TruckO is Small) AND (TrailerO1 is Small) AND (TrailerO2 is Large) AND (TrailerO3 is Small) THEN (Velocity is Mid) AND (Turning rate is Small).

Rule surfaces are shown in Figure 12 and Initial fuzzy membership functions are shown in Figure 13.

fig12
Figure 12: Randomly chosen fuzzy rule surfaces of truck and multiple trailer system.
fig13
Figure 13: Fuzzy membership functions of truck and multiple trailer system.

At the completion of the PSO membership functions tuning, the fuzzy rule surface are shown in Figure 14. The membership functions for the inputs and outputs of the fuzzy logic controller have been modified. They are shown in Figure 15.

fig14
Figure 14: PSO tuned fuzzy membership functions.
fig15
Figure 15: PSO tuned fuzzy membership functions of TMT system.

Based on the rules which are generated by the PSO algorithm, the control error is shown in Figure 16. The maximum error has been reduced from 729.18 to be around 330.14 after 15 iterations.

549838.fig.0016
Figure 16: Control error of the truck and trailer system.

Simulation results of the truck-and-multi-trailer system are shown below. Figure 17(a) indicates the TMT system which is initially located at the origin of the coordinate system. In Figures 17(b) and 17(c), the system is driven to the desired position. Figure 17(d) shows the system arrived to the end point with the minimum control error.

fig17
Figure 17: Motion of truck and multiple trailer system.

5. Conclusion

In fuzzy logic controller design, there are large numbers of fuzzy rules that have to be processed in order to produce decisions. The number of rules in a fuzzy controller primarily originates from the number of input variables that are entering the decision process, and one possible solution for decreasing it is to use the method of decomposition. In addition, the process of tuning fuzzy membership functions is also time consumed and often frustrated. In this paper a particle swarm optimization based automatic fuzzy rules and fuzzy membership function tuning method in designing fuzzy logic controller is introduced to control a truck-and-multi-trailer system. Simulation results for both one-trailer system and multiple trailer system have shown that the fuzzy rules have been decreased, and tuned membership function is performing satisfactorily result.

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