Adaptive control of track tension estimation using radial basis function neural network

Ping-xin Wng ,Xio-ting Rui ,Hi-long Yu ,,Guo-ping Wng ,Dong-yng Chen

a Institute of Launch Dynamics,Nanjing University of Science and Technology,Nanjing,210094,China

b College of Electrical,Energy and Power Engineering,Yangzhou University,Yangzhou,225000,China

Keywords: Track tension Monitor Multibody dynamics Neural network Anti-disturbance ability

ABSTRACT Track tension is a major factor in fluencing the reliability of a track.In order to reduce the risk of track peel-off,it is necessary to keep track tension constant.However,it is dif ficult to measure the dynamic tension during off-road operation.Based on the analysis of the relation and external forces depending on free body diagrams of the idler,idler arm,road wheel and road arm,a theoretical estimation model of track tension is built.Comparing estimation results with multibody dynamics simulation results,the rationality of track tension monitor is validated.By the aid of this monitor,a track tension control system is designed,which includes a self-tuning proportional-integral-derivative(PID)controller based on radial basis function neural network,an electro-hydraulic servo system and an idler arm.The tightness of track can be adjusted by turning the idler arm.Simulation results of the vehicle starting process indicate that the controller can reach different expected tensions quickly and accurately.Compared with a traditional PID controller,the proposed controller has a stronger anti-disturbance ability by amending control parameters online.

1.Introduction

Tracked vehicles,especially armored military tanks,work in extremely harsh conditions[1].When a vehicle is running,its track tension varies dramatically on account of the contact and collision among tracks,wheels and the ground[2].Track pre-tension is usually adjusted by moving the position of idler before the vehicle starts.Then tensioners are locked and idlers are fixed relative to the chassis.This limits the ability of the tensioner to control the tension.An ideal track dynamic tensioning control system should be able to maintain the track with relatively stable dynamic tension under all driving conditions,and maintain a relatively small tension as far as possible.This system is conducive to enhance the combat readiness,reliability and maintainability of the track over its life cycle.

Track tension may change violently with the change of driving condition[3].There are two main movement states of a track:driving mode and braking mode[4].For a tracked vehicle with sprockets rear,along the direction of forward motion,the track segment between the sprocket and the idler is the slack side under driving mode[5].However,this segment will be on tension side in braking mode.Steering operation will also convert the tension distribution.In addition,changes in tension may be exacerbated when vehicles pass through obstacles at high speed.Therefore,if not under control,changes in track tension may provoke a range of malfunctions.For example,when a vehicle moves in a straight line,if the driving torque of the sprocket is too large,the road wheel closest to this sprocket will lift upward,which reduces the wheel envelope perimeter.It makes the track loose and partially detached from the idler.If the vehicle suddenly encounters an obstacle or attempts to swerve,this situation may result in track peel-off[6].

Many scholars have studied on the adjustment strategy of a tensioner.Ketting[7]designed an elastic connection device between the idler and chassis.It can stabilize track tension by setting a reasonable spring stiffness.If the track around the idler becomes loose,this device will drive the idler forward.Conversely,if the track becomes excessively tight,the idler will be driven backward.But this device can not realize the real-time control of tension.Matej[8]proposed that track tension was a function of terrain.The motion of a tensioner is adjusted based on displacements of all road wheels.Taking the variance of tension as an optimization objective,he obtained the relationship between the displacement of hydraulic piston rod and that of road wheels.However,he did not give a speci fic implementation solution.By establishing a mechanicalelectrical-hydraulic integrated model of a tracked vehicle,Myk[9]simulated obstacle surmounting processes with different piston rod elongations.The motion of an idler is adjusted with a proportional-integral-derivative(PID)hydraulic controller.Simulation results indicate that the extension of piston rod will increase the suspension stiffness of a tracked vehicle,and thus aggravate the vertical and pitching vibration of the chassis.Nevertheless,he has not probed track tension control in much detail.

This paper proposes a constant track tension estimation control strategy for tensioner.Because the track motion is complicated,it is dif ficult to monitor track tension in real time using force sensors.According to the structural characteristics of tracked vehicles,the forces on the idler,idler arm,road wheel and road arm are analyzed.Then a track tension monitor(TTM)around the idler is designed.Considering the time-varying and nonlinear properties of the vehicle system,a radial basis function neural network PID(RBFNNPID)controller is adopted.RBFNN is a local approximation neural network,which has fast convergence speed and can effectively avoid the problem of local minimum[10].Combining it with PID control can achieve online parameter tuning.

The novelties of the proposed control strategy are:

(1)The RBFNNPID controller has good robustness and strong anti-jamming capability.

(2)The control system can be applied in practice by the aid of TTM.

(3)Compared with a traditional PID controller,the adaptive control system can availably decrease the maximum between the tension estimation and expected value.

The main contributions of this paper are summarized as follows.

(1)A mathematical model of a tracked vehicle system with electro-hydraulic tensioners is established.

(2)The track tension estimation obtained by TTM can well re flect the variation of actual tension around the idler.

(3)The proposed control approach can effectively maintain the dynamic track tension constant,which enhances the reliability of tracks.

2.Numerical model of tracked vehicle

A tracked vehicle can be considered as a complex multibody system.In this paper,Transfer Matrix Method for Multibody Systems(MSTMM)[11,12]is employed for dynamical modeling.It has virtues of low system matrix order,high programming,and high computational ef ficiency.

2.1.Dynamical model

Fig.1.Tracked vehicle model(a)and corresponding topology figure(b).

The dynamical model of a tracked vehicle is exhibited in Fig.1a,which consists of a chassis subsystem and two track subsystems[13,14].According to MSTMM,its topology is illustrated in Fig.1b.The chassis subsystem is comprised of a hull(7),a revolved body(5),a pitch body(3),a barrel(1),two sprockets(60,72),twelve road arms(32-37,39-44),twelve road wheels(61-66,73-78),eight support rollers(67-70,79-82)and two idlers(71,83)with tensioners(38,45).On each side,support rollers,road arms and wheels are numbered in order in the running direction,with the number closest to the idler being 1.

The hull is modeled as a spatial rigid body with multiple input and single output,and other body components are rigid bodies with single input and single output.Their connections are shown in Fig.2.The hull and road wheels are connected by road arms as the guide mechanism.The torsion bar and damper are simpli fied as a spring damping unit.A rubber bushing on a track pin is simpli fied as a spring damping unit,and two adjacent track links are connected with a single pin[15].The interactions between wheels,the ground and tracks are modeled as contact force units.

2.2.Dynamics equations

In MSTMM,the state vector of a connection point between two elements is defined as

Based on the kinematics information of each element,their transfer matrices are deduced.Then according to the topological structure of the chassis subsystem,its transfer equation is

where,zallis the overall state vector which includes state vectors of all inputs and outputs of this system

Its overall transfer matrix isUall,which represents the relationship between each state vector

Fig.2.Connections of each body elements.

The generalized coordinates of this system include position coordinates and Euler parameters of element 1 output in the global coordinate system.Also including relative rotation angles of all revolute joints,i.e.,

where,jis the number of a revolute joint.At timeti,the generalized position and velocity coordinatesare known quantities.By solving Eq.(2),we can obtain accelerations of all connection points.Then generalized accelerations coordinatescan be acquired.Combined with a numerical integration method,at next momentti+1can be fi gured out.

3.Track tension monitor around the idler

The driving condition of a tracked vehicle is complicated.There are many factors affecting its track tension,such as the engagement between sprocket and track[16,17],and the collision between track and ground.Therefore,it is dif ficult to measure the dynamic tension[18].In order to evaluate the dynamic performance of track,a theoretical estimation formula for track tension around the idler is deduced,combining the geometric parameters of the tension device.

3.1.Estimation formula

The track dynamic tensioning device is installed between the idler and hull,and is assembled by articulating an idler arm and a hydraulic actuator.Fig.3 represents the free body diagram of the idler.The hull is connected with idler arm atP0,and hydraulic actuator atP3.These two are connected at pointP2.In addition,the idler is connected with idler arm atP1.All connections are revolute joints.The in fluence of track gravity is ignored due to its small magnitude.The tension on the upper track isTi1,and its horizontal de flection angle isθ1.The tension on the nether track isTi2,and its vertical de flection angle isθ2.The dynamic equations of the idler are follows:

Fig.3.Free body diagram of the idler.

where,Fceis a centrifugal force generated by the track rotation around the idler,and its horizontal de flection angle isθce.The components of the force exerted by the idler arm on the idler inxandy-directions areRixandRiy,respectively.Giis the gravity of the idler,Jithe moment of inertia,rithe radius andωithe angular velocity.The linear density of the track isρ.The formulas forFceand θceare follows:

Anglesθ1andθ2can be determined from the geometric relation between the idler,support roller and road wheel,as shown in Fig.4.

Fig.4.Geometric relation between the idler,support roller and road wheel.

Fig.5.Free body diagram of the idler arm.

where,rris the radius of support roller.In addition,lΔx3andlΔy3can expressed by the position of road arm

where,rwis the radius of road wheel.

The free body diagram of the idler arm is illustrated in Fig.5.It is mainly subject to reaction forcesRixandRiyby the idler through a revolute joint and a driving forceFpexerted by the hydraulic actuator.The rotational equation of the idler arm aroundP0is

Available from Sine Law

where,Jiais the moment of inertia of idler arm,Giathe gravity,l0the distance between the center of gravity of idler arm toP0.Combining Eqs.(8)-(10)and(17)and(18)results in calculation formulas ofTi1andTi2:

It can be seen fromthe above formulas that rack tensions around the idler are only related to the idler’s angular velocity,angular acceleration,rotation angle of idler arm,driving force and length of hydraulic actuator.Considering the frequency response of the hydraulic system,the acceleration of the idler arm is generally small.In addition,during the driving operation,inertia terms are negligible compared to other terms,so the track tension estimation around the idler can be simpli fied into:

3.2.Multibody dynamics simulation verification

The track tension estimation around the idler is obtained by assuming that the track is a flexible belt.Compared with this theoretical model,a multibody dynamics(MBD)model of a tracked vehicle takes into account such factors as the plate structure of track,the engagement between track and sprocket,and the contact between track and rollers,which is closer to the actual situation[19].Therefore,the effectiveness of the estimation formula is veri fied with the results acquired from MBD simulations under different operating conditions.

3.2.1.Static condition

A track pre-tension should be set before the vehicle starts.Different hydraulic driving forces result in different pre-tension.Simulation results from two methods are shown in Table 1.

According to Table 1,it can be seen that the estimations are basically consistent with MBD simulation results under the static state,and all deviations are within 3%.The reason why the former is greater than the latter is that the track gravity is neglected.

Table 1 Track pre-tension with different driving forces.

3.2.2.Acceleration operation

The vehicle is accelerated from 0 m/s to 10 m/s within 10 s on a flat road.Track tension around the idler is shown in Fig.6.It suggests that the variation patterns of the estimation and MBD simulation are approximately the same.The maximum deviation is 12.6%and the average is 5.4%.

3.2.3.Off-road operation

Fig.6.Track tension under acceleration condition.

In order to verify the general applicability of TTM,a random offroad running is tested.In order to establish the change of road surfaceh(x)with running distancex,the harmonic superposition method is employed[20,21].It assumes that the road surface is a stationary Gaussian process with zero mean.By means of trigonometric series superposition of phase sinusoidal waves,a set of random displacementsh(xi)is obtained.From the generated displacement matrix,a deterministic arrayh(xi)is singled out as the law of road elevation changes with horizontal displacement.According to B-to G-grade of road roughness,six roads with 300 m are simulated.The C-and E-grade road pro files are shown in Fig.7.

A tracked vehicle is accelerated from 0 km/h to 30 km/h,and then driven at this constant speed.In the process of uniform driving,track tension is shown in Fig.8.Although there are deviations in both,especially the estimated tension is generally a little larger,their variation patterns are the same.Table 2 shows the standard deviation(SD)of track tension with the two methods on different road grades.

Table 2 SD of track tension on different roads unit:kN.

According to simulation comparisons of the above three operating conditions,it indicates that the proposed estimation can re flect the variation of track tension around the idler.This veri fies the validity of the theoretical estimation formula.

4.Track tension control system based on RBFNNPID control

The track tension control system is shown in Fig.9,which includes a RBFNNPIDcontroller,an electro-hydraulic servo system,an idler arm and a track tension estimator.The angle and force sensor monitor the rotation angleθiof idler arm and the hydraulic driving forceFp,respectively.Then the track tension estimationTiis acquired.By calculating the error between it and the expected tensionTr,the flow of the hydraulic cylinder is adjusted to keepTi=Tr.

4.1.Mathematical model of a hydraulic system

An electro-hydraulic servo system consists of a hydraulic cylinder,a servo valve,a servo ampli fier and sensors.When the natural frequency of servo valve is far greater than that of the cylinder,a first-order inertia element[22]can be used to describe the relationship between spool displacementxvand input voltageu:

where,Kais the servo ampli fier gain,Ksvthe servo valve gain and ωsvthe natural frequency of servo valve.

The flow equation and continuity equation of the valve are as follows[23]:

Fig.7.C-(a)and E-grade(b)road pro files.

Fig.8.Track tension:(a)C-grade,(b)E-grade.

Fig.9.Track tension control system.

where,Kqis the flow-gain coef ficient,Kcthe flow-pressure coef ficient,Athe surface area of piston,βethe bulk modulus,Vtthe total oil volume andCtcthe leakage coef ficient.The driving forceFpis

Available from Cosine Law

By differentiation w.r.t.time,

Combining Eqs.(23)-(25)and(27)results in

4.2.RBFNNPID controller design

RBFNN is a kind of neural network structure that simulates the local adjustment and mutual coverage of the receptive field in human brain[24].It is a three-layer feedforward network with a single hidden layer,as shown in Fig.10.

4.2.1.Identification algorithm of controlled object Jacobian information

In a RBFNN,X=［x1，x2，…，xn］Tis its input vector andH=［h1，…，hj，…，hm］Tis the radial basis vector,wherehjis Gaussian basis function:

where,Cjis the central vector of thejth node and can be described asCj=［cj1，cj2，…，cjn］T[25,26].The baseband parameter of thejth node isbjand the baseband vector isB=［b1，b2，…，bm］T,wheremis the number of hidden layer nodes.In our case,we use empirical formulas below to determine the number ofm.

Fig.10.RBFNN structure diagram.

where,nis the number of neurons in the input layer,λis a constant between 1 and 10.

For this network,its weight vector isW=［w1，…，wj，…，wm］T,so its output is

4.2.2.Self-tuning PID control based on RBFNN

A self-tuning PID control structure model based on RBFNN is shown in Fig.11,whereZ-1denotes a delay link and RBFNNI is the RBFNN identi fication.RBFNNI is used to obtain Jacobian information and its input vector is

where,u（k）andTi（k）are the control voltage and track tension estimation at thekth step.The approximation error of RBFNN isem（k）=Ti（k）-Tm（k）,and the performance indicator of RBFNNI is selected as

The parameters ofCj,BandW.are trained by supervised learning.A gradient descent method is adopted in the parameter learning.

where,ηis the learning rate(η>0),andαis the momentum factor(α∈［0，1）)[25].

Fig.11.Self-tuning PID control structure model based on RBFNN.

When the identi fication is within the allowable error range,the sensitivity information (Jacobian information)of the object’s output to its input can be obtained as

4.2.3.PID tuning principle based on RBFNN

An incremental PID controller is implemented in the control system and the control error is

Three inputs of this PID controller are

Its output is

A single neural network controller(NNC)is adopted to updatekP,kIandkDonline through the identi fication model.The tuning index is defined as

Three parameterskP,kIandkDare amended with gradient descent method[26,27],which is defined as

Then the learning algorithm of PID parameters is[28].

where,ηcandαcare the learning rate and momentum factor of PID parameters[29].

Now that the control process steps based on RBFNNPID can be summarized as follows:

Step 1:Input the initial data of the system and set the initial values ofC,B,W,m,η,α,ηcandαc.

Step 2:The actual outputTi（k）and expected inputTr（k）of the system are obtained by sampling,and thenu（k）is calculated from Eqs.29-31.

Step 3:The current network outputTm（k）is calculated from Eqs.(23)and(24),and Jacobian information is calculated from Eq.(28).

Step 4:PID parameters are amended using Eqs.(40)and(41).

Step 5:Adjust network parameters with Eq.(34).

Step 6:Return to Step 2(k→k+1)and continue the loop until the end.

5.Simulation results

During the running process,the dynamic tension should be retained stable as far as possible.Two simulation scenarios are performed,including vehicle starting and passing obstacles procedures.By setting different control signals and adding disturbance,the control effect of RBFNNPID controller is observed.

5.1.Vehicle starting process

When the tracked vehicle starts,in order to adjust track tension to a required value quickly,it is necessary to open the hydraulic valve of the tensioner.The number of hidden layer nodes is 10 and parameters of RBFNN areη=0.2,α=0.02.All PID parameters are initialized to 0.Their learning rate and momentum factor are ηc=0.001 andαc=0.0005.Sampling period is 0.01 s.The expected input tensionTris 30 kN,40 kN and 50 kN,respectively.Their PID self-tuning control results based on RBFNN are shown in Figs.12-14.

Fig.12.Expected tension is 30 kN:(a)control effect and network fitting effect,(b)changes of PID parameters.

For different control signals,the controller can reach expected values quickly and accurately,indicating that it has good robustness.According to curves ofkP,kI,andkD,during the system operation,the controller has been learning online to realize the online adjustment of PID parameters,which is adaptive.Finally,the vehicle reaches a static equilibrium state and the system disturbance is 0,accordingly PID parameters tend to be stable.With different expected inputs,final values of PID parameters are shown in Table 3.

Table 3Final values of PID parameters.

Fig.13.Expected tension is 40 kN:(a)control effect and network fitting effect,(b)changes of PID parameters.

Fig.14.Expected tension is 50 kN:(a)control effect and network fitting effect,(b)changes of PID parameters.

Fig.15.Rotation angle of idler arm.

Fig.16.Locations and sizes of triangular obstacles.

Fig.17.Block diagram of the whole control system.

With different expected tensions,rotation angles of the idler arm are presented in Fig.15.The smaller the rotation angle,the longer the distance between the idler and sprocket.It follows that the wheel envelope perimeter becomes longer and the track elongation increases,resulting in increased track tension.

Fig.19.Changes of PID parameters.

5.2.Passing obstacles process

A customized road is adopted for testing the running conditions of a tracked vehicle.As exhibited in Fig.16,it consists of seven triangular obstacles,of which the third and fourth are continuous and the fifth to seventh are also.The vehicles pass through these obstacles at a constant speed of 2 m/s.

When the vehicle is driven on a rough road,its road arms will swing,causing the track tension to change.In this paper,it is assumed that road wheels are close to the ground,and the road arm swing caused by road undulations is regarded as system disturbance.At the same time,it can be seen from Sec.3.2 that there is a certain deviation between the estimationTiand actual tensionT.Moreover,this deviation is dif ficult to measure accurately.Therefore,a white noise is added on the estimation to simulate the actual tension,so as to calculate the rotation angle of idler arm.The block diagram of the whole control system is shown in Fig.17.

The track tension estimation around the idler in the process of passing obstacles is delineated in Fig.18.The expected tensionTr=40 kN.The control effect of the proposed controller is compared with that of a traditional PIDcontroller.When driving on the flat road,the two curves roughly coincide.From the second obstacle,the adaptive controller can effectively reduce the tension variation.This is due to the self-tuning of PID parameters,making the control effect more obvious.Over the second obstacle,the maximum difference betweenTiandTrdecreases by 19%,the third to fourth 31%and the fifth to seventh 44%.The changes of PID parameters are shown in Fig.19.The results show that the proposed control system can adaptively and quickly adjust PID parameters according to the system change.

6.Conclusions

In this paper,a track dynamic tensioning device is designed,which is composed of an idler arm and an electro-hydraulic servo system.Track tension can be tailored by changing the elongation of the piston rod.Based on MSTMM,a mathematical model of a mechanical-electrical-hydraulic coupling tracked vehicle system is established,which can be utilized for evaluating the control system.Based on the analysis of the relation and external forces depending on free body diagrams of the idler,idler arm,road and road arm,a theoretical estimation model of track tension is built.Comparing estimation results with MBD simulation results,the rationality of TMM is validated.

A track tension control system is adopted to maintain track tension estimation constant,which includes an RBFNNPID controller,a tensioner and a TTM.Robustness and anti-interference ability of the controller are enhanced by means of the on-line identi fication of RBFNN and self-tuning of PID parameters.Compared with a traditional PID controller,it can effectively keep track tension constant,which enhances the combat readiness,reliability and maintainability of the track over its life cycle.In future research,the neural network parameters can be initialized in combination with optimization algorithms to avoid blindness.In this way,the approximation degree and accuracy of neural network can be improved effectively.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to in fluence the work reported in this paper.

Acknowledgment

The authors gratefully acknowledge the Natural Science Foundation of Jiangsu Province(No.BK20190871)and Natural Science Foundation of Jiangsu Province(No.BK20190438)for the financial support of this research.

Appendix A

Acronym PID proportional-integral-derivative TTM track tension monitor RBFNN radial basis function neural network RBFNNPID radial basis function neural network PID MSTMM transfer matrix method for multibody systems MBD multibody dynamics SD standard deviation RBFNNI RBFNN identi fication

Appendix B

Nomenclature

x y,z translational accelerations Ωx Ωy,Ωz angular accelerations mx my,mz internal moments qx qy,qz internal forces Uall overall transfer matrix zall overall state vector I12 an identity matrix with the dimension of 12×12 O zero matrix U j transfer matrix of element j Hi,j geometric relationship matrix between input ends i and j y generalized position coordinates y generalized velocity coordinates images/BZ_255_2109_1805_2139_1843.png generalized acceleration coordinates ε1，ε2，ε3，ε4 Euler parameters θ α,βrotation angle F external force G gravity J the moment of inertia ω angular velocity ρ linear density r the radius of wheel Ti track tension estimation Tr expected tension u input voltage xv spool position of the servo valve Ka servo ampli fier gain Ksv servo valve gain

ωsv natural frequency of servo valve Kq flow-gain coef ficient Kc flow-pressure coef ficient A surface area of piston βe bulk modulus Vt total oil volume Ctc total leakage coef ficient kP kI,kD PID parameters λ a constant between 1 and 10 η learning rate of RBFNN α momentum factor of RBFNN ηc learning rate of PID parameters αc momentum factor of PID parameters