A H-Infinity Control for Path Tracking with Fuzzy Hyperbolic Tangent Model

To achieve the goal of driver-less underground mining truck, a fuzzy hyperbolic tangent model is established for path tracking on an underground articulated mining truck. Firstly, the sample data of parameters are collected by the driver controlling articulated vehicle at a speed of 3m/s, including both the lateral position deviation and the variation of heading angle deviation.Then, according to the improved adaptive BP neural network model and deriving formula of mediation rate of error estimator by the method of Cauchy robust, the weights are identified. Finally,H-infinity control controller is designed to control steering angle. The results of hardware-in-the-loop simulation show that lateral position deviation, heading angle deviation, and steering angle of the vehicle can be controlled, respectively, at 0.024m, 0.08 rad, and 0.21 rad. All the deviations are asymptotically stable, and error control is in less than 2%. The method is demonstrated to be effective and reliable in path tracking for the underground vehicles.


Introduction
Articulated vehicle is widely used in underground mining.Researches on driver-less articulated vehicle have been carried out for many years to prompt efficiency and safety in underground mine.A key principle of autonomous driving in underground tunnel is to find a path for articulated vehicle to track at a reasonable high speed and avoiding crash into sidewall.Many automatic control algorithms can be applied in this filed.
In the field of fuzzy model, Zeng and Singh [1] and Wang [2] have theorized a fuzzy relational model, and its purpose is to build a fuzzy model to approximate the ideal control behavior.This fuzzy relationship can be seen as a fuzzy mapping from input space to the output space.Its main disadvantage is that many important dynamic information systems are ignored during the modeling process, and it is difficult to obtain good performances of controller.Takagi and Sugeno [3,4] have proposed a fuzzy model named T-S, aiming to construct a series of linear equations to express each subsystem, then to make a global model through the membership functions of these subsystems.Its main drawback is that the complexity of the structure itself to establish a model requires lots of work offline.Fuzzy hyperbolic tangent model is better than the above models [5,6].
Path tracking problem is a very important issue in the course of unmanned field.The former studies are based on articulated vehicle kinematics, and many scholars have done a lot of researches.Nayl et al. [7], Lee and Yoo [8], Ridley and Corke [9], and Xuan et al. [10] derived kinematic models of articulated vehicle and the reference model errors of path, and they used method of model predictive control for tracking control with simulation.The work efficiency and safety are decided by the accuracy of path tracking.Path tracking control design based on articulated vehicle kinematic characteristics has no good adaptability in underground mining environment.Dynamics characteristics are significant for path tracking control design [11].Literature [12] considers the influence of articulated vehicle dynamics on sideslip, but there would be a big deviation as they only consider one aspect and the lack of system modeling.
To solve these problems above and to deal with this multivariable, strong coupling, highly complex nonlinear dynamical systems of underground mining vehicles, we use fuzzy hyperbolic model and design a nonlinear quadratic controller through testing field experiments by hardware-in-the-loop (HIL) simulation to ensure the quality of control, aiming at achieving the goal of unmanned underground mining articulated vehicles.This method is to obtain kinematic relations of vehicle via the driver information, and the fact that driver repeatedly driving the process can make the relationship dynamics of the vehicle which are included in those data becomes increasingly evident.This method can also be considered to be intrinsically linked with a data mining to reduce the complexity of the modeling process, which greatly reduces the system error caused by the dynamics modeling.The coefficient matrix of fuzzy hyperbolic model nonlinear systems becomes constant matrix.The parameters of the identification via neural network, supervised learning methods, make the model more close to the real model.Also it is convenient to the control algorithm design.Young [13] has proposed variable structure controller for the first time, which was applied to robot control.Jafarov et al. [14] have proposed a new type of sliding PID controller.Although these methods can solve the control problem, they could not consider the robustness of the system.-infinity control is a good robustness design method, with clear design ideas, good control effect, and advantages, especially on model perturbation of multiple input multiple output (MIMO) systems.Finally, hardware verification control which results in the final loop (HIL) simulation is done in order to ensure quality control.

Kinematic Model of Underground Articulated Vehicle
2.1.Steering Deviation Model. Figure 1 shows the errors between the real and reference path.The circle centered by  is the reference path.Ideally, the vehicle should pass  1 ,  2 , and  3 .The variables are defined as follows [5,15]: (1) lateral displacement error: the lateral displacement error between the vehicle reference point and the corresponding point  (the nearest point on the reference path); (2) orientation error: the orientation error between the velocity orientation of  and the tangential orientation of ; (3) steering angle: relative rotation angle of underground mining articulated vehicle body before and after the horizontal plane.
These two variables can basically reflect the position and posture of articulated underground mining vehicles in the tunnel, and building the trajectory of articulated vehicles model can be achieved by the fuzzy hyperbolic method.
Figure 2(a) is the sketch of an artificial tunnel for testing of articulated vehicle driver-less control system.The truck used in this test is shown in Figure 2(b).
As can be seen in Figures 3-5, at speed of 3 m per second, by driver controlling the truck in the process of turning, the lateral deviation is stable at 1.5 m.Heading angle deviation remains stable at around 0 rad, while body roll is obvious.The steering angle remains stable at 0.2 rad with obvious adjustments.As shown in Figures 6 and 7, at speed of 3 m per second, rate of change of lateral deviation and the heading angle also is 0, but there are more noise and severe jitter.

Kinematic Models Based on Fuzzy Hyperbolic Model.
Near the origin, fuzzy hyperbolic model is a global model [6] with a global approximation performance.And it is not only an essentially nonlinear model, but also easy to express the dynamic characteristics of nonlinear systems.Therefore, according to this model, the controller can be designed so that the whole system can achieve the optimal performances.Compared with other fuzzy models, fuzzy hyperbolic model is more suitable for the control object known as multivariable nonlinear finite object.
Given a plant  state variables number:  = ( 1 (), . . .,   ()) T , and  input variables numbers: u = (u 1 (), . . ., u  ()) T , and the fuzzy rules are used to describe this system by the following conditions, so this group of fuzzy rules is termed as the hyperbolic tangent type of fuzzy rules.
There are certain mathematical relationships between manned lateral deviation, heading angle deviation, the steering angle, and rate of change of the lateral deviation and heading angle deviation from the data observed from relevant papers [10].And they can be roughly described as the following information through the professional experiences so as to  construct the corresponding fuzzy hyperbolic model of fuzzy rules: ( Then where Bku are linearized to Bu: When the absolute value of  is smaller, tanh() ≈ , fuzzy hyperbolic model can be written as ẋ = A tanh(k  ) + Bu that the system is a linear model when it is close to the equilibrium point.

Parameter Identification Based on Improved Adaptive BP Neural Network
Recognition technology evaluation includes two indicators.One is the identification accuracy, and the other is speed identification.Compared to other models in terms of fuzzy, topology of the neural network can be used to optimize the parameters for FHM [16].The  weights of mediation rate of adaptive BP neural network are deduced by Cauchy robust error, to eliminate the effect of outliers in the data and to fit the original data better.Neural network topology is shown in Figure 8 of FHM.Controlled object state variables are  = ( 1 (), . . .,   ()) T and input variables u = (u 1 (), . . ., u  ()) T as the input of the neural network topology, and model output is the rate of change of the state variables k  ( = 1, . . ., ),   ( = 1, . . ., ),   (,  = 1, . . ., ), and   ( = 1, . . ., ,  = 1, . . ., ) as the connection weights which need to be trained.
A and B are constant matrix, whose elements are the weights comprised of   and   .If the hidden layer function  1 () = tanh() and output layer activation function  2 () = , then the following models can be obtained: Obviously, when making u linearize, variables can be obtained Thus, FHM may be built by the neural network model, because it can train FHM model parameters [6].Since there are many jitters in the data of rate of change of horizontal and heading angle, the samples show a large number of outliers.So from the statistical view of robust, the traditional MSE aggravate the "outliers" of samples.Therefore, Cauchy robust error estimator can be used.
Let error of outputs be where   k is ideal output for the network and  k is actual output.
Cauchy error estimator obviously does not depend on the initial weights and thresholds transition, and it can effectively eliminate the negative impact of "outliers," while retaining the main characteristics of the value of the output error and speeding up the convergence rate.
By the steepest descent method, it can obtain weights iterative equations of each layer of neurons: where Δ = /;  = {  }.So get a rate adjustment based on the following model parameters above BP algorithm: In this paper, learning rate  is considered as the adaptive scheme, in which its main idea is that the initial value of  being set higher, generally about 0.7-0.9, with the increase or decrement of the number of learning, may change in a law.When  decreases to a certain extent, if  has not still become convergence or error has still no improvement,  is set again at about 0.5-0.7. enters the learning process again, until the end of the run.The learning rate is associated with the error function: when the error is reduced, increase the learning rate; when the error increases, reduce the learning rate: Then The fitting results of Figures 9 and 10 substantially eliminate the effect of outliers with relative error of control in less than 1% and less than 10%.The trend is essentially coincident with the sample data.
The existing experimental data can be applied to the above-described method for fuzzy hyperbolic model parameter optimization in order to be close to the actual model.It provides the basis to the next controller design.

Controller Design
Based on this model, we can design conventional linear controllers or other nonlinear controllers.In this paper the controller is designed with hyperbolic tangent function of the state variables, so we can use language to describe the information of the controller.Thus this given controller is a fuzzy one [16].The system can be stabilized by nonquadratic performance index function [16].System performance can get to a minimum with u through the given Q and R ẋ = A () + Bu. ( Definition 1 (see [17]).Let   satisfy the following conditions: all sets of (⋅) include R → R: (1)  is continuous; (2) (0) = 0, for other  ∈ R, () > 0; (3) when || → ∞, ∫  0 () → ∞.
), and there is L( 0 ) with boundary.Then, If there is a diagonal positive definite matrix P min satisfying Riccati equation then [, , Q, R] is optimized diagonal matrix.
Given nonlinear system and nonlinear quadratic performance index function, [, , Q, R] is an optimized diagonal matrix.The optimal control vector is * u where ().Thus, tanh(k) ∈   , and for fuzzy hyperbolic tangent, the quadratic performance index function is

HIL Simulation
The difference between simulation and hardware-in-the-loop simulation is that there is a real-time simulation of the circuit's physical hardware.Hardware-in-the-loop simulation is intended to provide real-time incentives as the true signal to the controller so that the controller is connected to its real accused equipment and test its performance.V = 3 m/s.So as to test this robust of control, there is one pit per 20 m on the road.The reference trajectory is set as a round whose center is (0, 0), and R is 25 m and an initial parameter setting is simulation time 200 s, and the starting point coordinates (−3, −25).
Derived from the simulation results, Figure 12 is the comparison chart of the actual trajectory and the reference trajectory.As can be seen from the simulation curve, the articulated vehicle traveling trace and the reference circular path are consistent, and the trajectory is relatively smooth.The speed which is controlled by PID from Figure 13 becomes finally steady at 30 seconds.The lateral displacement error is gradually flat from the beginning of violent shaking in Figure 14.The overshoot amount is 4.28%, and it is at most about 0.024 m at 18 seconds, with respect to the tread based on 2.280 m, and the error is 1.7% and then close to the origin step by step.The orientation error is gradually flat from the beginning of violent shaking in Figure 15.The overshoot amount is 4.9%, and it is at the top at about 0.08 rad at 18 seconds and then close to the origin step by step.The articulated angle is gradually flat from the beginning of violent shaking in Figure 16.The overshoot amount is 3.5%, and it is eventually stabilized at 0.21 rad at 22 seconds.Compared with the articulated vehicle steering based on angle 45 ∘ , the error is 1.2%.And the peaks are caused by pits, but then the system immediately recovers to the right path.It has better robustness than literature [12], in which there are a lot of jitters to adapt to the path as a similar situation.The results of literature [10] are as follows: lateral deviation was 1m and heading angle deviation is 0.1 rad.Literature [19] also analyzed the models of articulated vehicle to control it with synovial control method, whose results of the simulation are as follows: the lateral deviation is 0.1 m and heading angle deviation was 0.17 rad.So there exists a large gap when compared with the results of this paper.

Conclusions
This article provides a method used in unmanned systems based on the way of fuzzy hyperbolic pole control act for articulated vehicle trajectory tracking accurately.Conclusions are as follows.be reduced by the influence of singular error of neural network learning and fitting error, and relative error below expectations can be decreased so that it can achieve the system of identification.
(3) By the -infinity controller design, controller has a good control performance, and it makes the system keep a better performance about robustness in order to achieve the purpose of the comprehensively optimal control error.
(4) This method can be used in articulated vehicle path tracking effectively based on the -infinity controller.Simulation in hardware-in-the-loop shows that overshoot and the response time are less than expectations and are eventually to stabilize.The controller has met the requirement of the real-time control performance.

Figure 1 :
Figure 1: Trajectory curve and parametric model image of underground mining articulated vehicle.

Figure 2 :Figure 3 :
Figure 2: (a) The plan view of the tunnel.(b) The real truck.

FittingFigure 9 :
Figure 9: Fitting of change rate of the lateral deviation.

Fitting 1 )Figure 10 :
Figure 10: Fitting of change rate of the heading deviation.