Gearshift Mechanism Nonlinear Torque Control

8/14/2019 Gearshift Mechanism Nonlinear Torque Control

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BACKSTEPPING TORQUE CONTROL FOR

CYLINDRICAL CAM IN SEQUENTIAL TYPE GEAR

SHIFT MECHANISM

By

PRIYANSHU AGARWAL

DECEMBER 2009

Department of Mechanical and Aerospace Engineering

State University of New York at Buffalo

Buffalo, New York 14260


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i

Abstract

Faster and efficient gear shifts have always been the demand of automobile industry. Be

it an automatic transmission having distinctly low efficiency or an automated manual

transmission with combined benefits of automatic and manual gearboxes, the

phenomenon of gear shift has always been the area of interest. Sequential type gear shift

mechanisms have been increasingly employed in automated manual transmissions due to

their suitability to readily available (rotary) actuators. However, there is a dearth of

published literature addressing the issue of modeling the dynamics of the system and

developing a torque controller for such a gear shift mechanisms. The objective of this

work is to first model the dynamics of the mechanism and then to design a controller for

governing the torque applied on the cylindrical cam for the given mechanism parameters

and time varying reference input. Both the issues i.e. reducing shifting time and

improving efficiency of shifting can be addressed by the successful implementation of

such a torque controller.


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Acknowledgement

I am highly thankful to Professor Tarunraj Singh for his support during the execution of

this work. Not only did he equip me with the technical skills required to deal with the

challenges involved in the problem, but his motivating examples of the work done by

some of the brilliant students from previous batches proved to be a constant source of

motivation for accomplishing this work.

Finally, I am also thankful to my colleagues in the course of Applied Nonlinear Control

who had technical discussions at times on the subject and always motivated me to explore

the unexplored.


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Contents

Abstract

Acknowledgement

Contents

List of Symbols

1 Introduction1.1 Motivation

1.2 Necessity of Torque Control1.3 Problem Formulation

1.4 Challenges Involved

1.5 Report Organization

2 Background2.1 Mechanism description

2.1.1 Purpose

2.1.2 Construction

2.1.3 Working

2.2 Existing Controllers

3 Mathematical methods3.1 Mathematical Modeling

3.1.1 Fork Cantilever Beam Analogy

3.1.2 Drum Dynamics Equations

3.1.3 Fork Free Body Diagram

3.2 Governing Differential Equation and State-Space Representation

3.2.1 Governing Equation

3.2.2 System Analogy3.2.3 Shift Force Behavior

4 Implementation4.1 Basic Controller

4.1.1 Backstepping Control Law Derivation

4.1.2 Standard Least Squares Estimator

4.2 Tracking Controller



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4.3 Actual Controller

4.3.1 Backstepping Control Law

4.3.2Standard Least Squares Estimator4.3.3 State Estimator (Integrator)

4.4 Model Linearization for Parameter Estimation

4.5 Standard Least Squares Estimator with Linearized Model

5 Results5.1 Basic Controller

5.1.1 Simulation with known Coefficient of Friction

5.1.2 Simulation with unknown Coefficient of Friction

5.2 Tracking Controller

5.2.1 Simulation with known Coefficient of Friction5.2.2 Simulation with unknown Coefficient of Friction


5.4 Virtual Reality Simulation

6 DiscussionBibliography

Appendix

A.1 MATLAB code for solving equations involved in the Mathematical Model of

the Dynamics of the Gearshift Mechanism

A.2 MATLAB code for Basic Controller with parameter estimation and

asymptotically stabilizing origin (Known Coefficient of Friction)

A.3 MATLAB code for Tracking Controller with parameter estimation and

reference output tracking (Known Coefficient of Friction)

A.4 MATLAB code for Final Controller with parameter and state estimation,

noise, reference output tracking and Virtual Reality Simulation. (Known

Coefficient of Friction)A.5 MATLAB code for Basic Controller with parameter estimation using

linearized model for asymptotically stabilizing origin (Unknown Coefficient of

Friction)


reference output tracking (Unknown Coefficient of Friction)


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Chapter 1

Introduction

The problem of control of a gearshift mechanism in an automated manual transmission is a challenging

problem. Apart from developing a mathematically accurate model of the mechanism, the biggestchallenge towards developing a perfect controller is the lack of repeatability in the process of gearshift

itself. Every time a gearshift is effected there are numerous parameters involved in deciding the exact

effort required for the gearshift mechanism to bring this change. There are parameters that are constantand are inherently locked in the geometry of the components of the mechanism. And there are the ones

that vary with vehicle speed, gearshift mode, road load etc. And this becomes increasingly true for a

sequential type gearshift mechanism in an automated manual transmission where the lack of appropriate

control can lead to either quick wear of drive-train components such as clutch or synchronizers in caseof a synchromesh transmission.

1.1 MotivationThe direction, position and velocity of the fork during the shift can be governed by appropriate design and

control of the mechanism. What makes a mechanism proficient in gearshift is the ease with which it canachieve the required dynamically varying force on the shift fork during gearshift. However, what makes

designing a controller for the mechanism difficult is the fact that the phenomenon of gearshift lasts for afraction of a second. This demands that the response time of the controller be extremely low. Thus, there

exist apparently irreconcilable requirements for the controller to achieve all these objectives. This calls for adetailed study of the dynamics of the mechanism to make the system more amenable to control by

developing better controllers. The development of a controller in the context of sequential type mechanism becomes more important due to the inherent suitability of the mechanism for automation due to readily

available rotary actuators.

Figure 1-1 Gearshift mechanism as a component of the drive-train


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1.2 Necessity of Torque ControlThe following are the reasons why torque control of a gearshift mechanism becomes an absolute

necessity:

1. It will help in faster, more efficient and smoother gear shifts.2. There exists a dynamically varying force demand on Gearshift Forks.3. The extremely low gear shift time demands very-high response.4. There is also a variation of torque requirement between different gears.5. The mechanism has variation of torque requirement within same gear with variation in driving

conditions (vehicle speed, gear shift mode etc.).

1.3 Problem FormulationThe objective is to design a controller for governing the torque applied on the cylindrical cam of a

sequential type gear shift mechanism in presence of parametric uncertainties for the following given

inputs:Mechanism Parameters (dimensions and friction coefficients)Reference signal (in the form of output or states of the system) as a function of timeThe problem of designing a controller can be divided into two categories:

1. Controller to asymptotically stabilize the origin of the system states.2. Controller for tracking a given feasible trajectory. This requires zero tracking error to be an

equilibrium.

The report deals with both the type of controllers along with means to handle parametric uncertainty inthe model dynamics. However, the report does not deals with the optimization of the controller gains

which is necessary to ensure performance.

Both the issues i.e. reducing shifting time and improving efficiency of shifting can be addressed by the

successful implementation of such a torque controller.

Figure 1-2 Basic control block diagram

The system represented in Figure 1-2 illustrates the basic control block diagram of the system.

1.4 Challenges InvolvedThe following are the challenges involved in designing a controller for the gearshift mechanism:

1. Mathematical Modeling of the Dynamics of the Mechanism

Gearshift Mechanism

DynamicsController

Estimator

Reference

Signal


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A mathematical model need to be developed that can accurately represent the dynamics of the system.

The model will incorporate the effects of friction between various components. The presence of friction

makes the system highly nonlinear.

2. Model Parameter UncertaintiesThe model involves parameters such as sleeve force spring and damping coefficients that are not knownbeforehand and has to be estimated online. The presence of such uncertainties makes the controller and

estimator to run in tandem and still perform satisfactorily.

3. Controller DesignA controller that can handle the highly nonlinear system yet ensure successful tracking of reference

signal need to be developed. Also, in absence of a reference model some kind of a self-tuning techniqueneed to be employed for controlling the torque.

1.5 Report Organization

Chapter 1 deals with an introduction of the problem targeted in this work.

Chapter 2 deals with a basic overview of the mechanism.

Chapter 3 discusses the mathematical development involved in modeling the dynamics of the system.

Chapter 4 presents the various controllers developed for torque control.

Chapter 5 documents the results obtained by simulating the controllers developed.

Chapter 6 discusses the various results and concludes with the possible future work.

Appendix A contains a MATLAB code for the various controllers developed for the torque control ofthe sequential type gearshift mechanism.


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Shift Lever-1 (Guitar shaped with a welded splined shaft)

Shifter Lever-2 (Cross Lever)

Torsion Spring

Tension Spring

The drum, fork rail and splined shaft with shift lever-1 are supported in the casing. The shift lever-1 and shiftlever-2 are assembled along with the springs as shown in Figure 1 a). The star wheel is attached to the drum

with the help of a bolt and the drum is free to rotate about its axis. The drum is provided with grooves on itscircumference. Furthermore, the fork is provided with a lug that protrudes from its head. The lug on the fork

engages in the groove on the drum. The Fork legs rests on the gear collar (not shown in figure) and the forkis constrained to move axially on the fork rail. In addition to this, the star wheel is provided with projected

lugs on its front face on which the shift lever-2 rests. Shift lever-2 is connected to shift lever-1 through arevolute joint and tension spring between the two ensures that the shift lever-2 maintains contact with the star

wheel lugs. A torsion spring is provided on shift lever-1 to reposition the lever after shifting the gear to itsoriginal position. The entire assembly is inside a transmission casing. Figure 1 b) shows the automated

version of the same mechanism. In this case the drum is rotated using a motor through gearing. The detentlever is provided to keep the drum stationary on each engaged gear position.

2.1.3 WorkingThe shift lever-1 is rotated with the help of a push-pull type cable. As shift lever-1 rotates, it rotates shift

lever-2 which in turn rotates the star wheel. Since, the star wheel is rigidly connected with the drum, thedrum is also indexed by the same angle. The configuration of the two levers and star wheel is such that the

system is always indexed by a fixed angle for one stroke of the shift levers. Due to this fixed rotation of thedrum the forks engaged in the grooves on the drum surface move axially on the fork rail by a fixed distance.

This axial movement of the fork leads to the axial movement of the shifting sleeve which leads to shifting ofthe corresponding gear. The fact to be noted here is that a single fork is manipulated by the mechanism

during a gearshift under load. The magnitude and pattern of required drum torque governs the gearshift feelin the manual version. However, the main concern in the automated version is minimization of the peaktorque due to actuator size limitations.

2.2 Existing Controllers

Figure 2-2 Gearshift Mechanism used in Mercedes Smart Fortwo.Though there are few products in the market that implement automated gearshift in the context of

synchromesh transmissions using sequential type gear shift mechanism. However, the reluctance of theirdevelopers to disclose the details of the mechanism and its control has led to unavailability of published

literature that can be referred. Also, whether such controllers handle parametric uncertainty is not known.


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Chapter 3

Mathematical Methods

This chapter deals with the development of the mathematical model for the dynamics of the gearshiftmechanism. Free body diagrams are developed for the various components in the mechanism and finally

a differential equation governing the dynamics of the system is obtained.

3.1 Mathematical ModelingThe objective of the mathematical model and the assumptions involved are as follows:

Objective: To develop a differential equation that can represent the dynamics of the mechanism. More precisely, to develop a relationship that can relate the torque applied on the cylindrical cam with its

angular acceleration and the axial force resisting this motion applied on the fork.

Assumptions:

1. While gearshift both the fork legs make continuous contact with the gearshift sleeve. This ensures thatwhen the fork is loaded both the legs experience equal deformation in axial direction. This is indeed

critical for the gearshift sleeve, as differential loading might tilt the sleeve and can even increase the fork

required to shift it during a gearshift. This assumption neglects the influence of variation of tolerancesinvolved during manufacturing of components.

2. The fork lug is always in contact with the drum groove wall. This assumption signifies that the model

does not take into account any nonlinearity in terms of contact between the fork and the drum. So, anyunique position of the drum signifies unique position of the fork provided that the drum groove

geometry is known.

3. Drum bearings have no frictional losses. This ensures no frictional moment is acting to diminish theapplied torque on the drum.

4.All bodies are rigid. No deformation of the bodies involved in the system is considered.5.Both the fork legs have same area moment of inertia. Though the moment of inertia of the fork legswill differ and will also vary with the distance from the fork center line. In order to avoid complexities,

the influence of these parameters on force distribution among the two fork legs has been neglected.

6.The coefficient of friction is same for all contacts. Usually the coefficient of friction between thecomponents is that of standard lubricated steel to steel contact. However, at times the fork is fixed with

the rail in which case the fork rail slides in the transmission casing which is usually a pressure die-castedaluminum component.

Figure 3-1 represents a simplified model of the mechanism to be used for developing the mathematical

model of the system. The torque TD is acting on the drum due to which drum experiences an angular

acceleration of which in turn accelerates the fork by . The gearshift sleeve also offers a force of Ffa1and Ffa2on the two fork legs which amounts to the total force acting on the fork axially.


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Figure 3-1 Simplified representation of the mechanism with external forces & moments acting on it

3.1.1 Fork Cantilever Beam Analogy

According to this analogy the fork leg is considered to be a simple cantilever beam which bends in the

presence of an external load. Figure 3-2(a) represents the formula for the maximum deflection of acantilever beam. It can be easily seen that for a given deformation the load acting on the beam isinversely proportional to the cube of the length of the beam.

As per the above mentioned analogy the two fork legs can be represented as two cantilever beams whichundergo same deflection under load. The condition of equal deflection comes from the assumption that

the two legs make continuous contact with the gearshift sleeve. So, the distribution of axial force on fork

legs is considered to be inversely proportional to the cube of their arm lengths from rail center line.

Hence, the fork acting on the fork can be distributed on the two legs accordingly.

(a)


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Figure 3-2

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Figure 3-3 represents a developed view of the drum obtained by unwrapping the drum groove profile.

Also shown is a zoomed view of the fork lug and the forces acting on the drum due it.

Now, from Newtons second law of motion

So, the net torque acting on the drum is given by

where,flrepresents the friction force developed between the fork lug and the drum and is given by

Further in Figure 3-3 it can be seen in the zoomed view of the fork lug traversing the drum groove thatthe drum groove ramp angle is related to the angular rotation of the drum and the axial movement of the

fork as given below

(A)

Also, the instantaneous axial velocity of the fork is related to the angular velocity of the drum by the

following equation

(B)

Differentiating the above equation we can obtain the following expression

(C)

Also, by integrating equation (A) we can get the following expression forx.

(D)

3.1.3 Fork Free Body Diagram

Figure 3-4 represents the free body diagram of the fork. The reaction of friction force fl and force R1aswas discussed in the previous section also acts on the fork lug. View U represents the top view of the

fork lug with these reactions acting on it. Now, in order to develop a better understanding of the system

in term so axial force and tangential force, these forces are distributed among the axial force F A,tangential force FT and a moment Mf. The axial force FA tries to tilt the fork due to which normal

reactions Ny1 and Ny2 from the rail are developed in order to avoid tilting. The tangential force tries to

move the component in its direction. However, the almost symmetric location of the fork lug avoidsdevelopment of substantial moment in one direction due to which the two reactions Nx1 and Nx2 from the

rail are acting in the same direction. The reactions Nx1, Nx2, Ny1 and Ny2 thus lead to the friction forces

fx1,fx2,fy1 andfy2 respectively. The gearshift sleeve also applies force Ffton the fork leg as the force FT


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Figure 3-4 Free-Body Diagram of Fork

also tries to rotate the fork about the rail center. Due to the force Fft a corresponding friction force alsoacts on the fork leg which is developed between the fork and gearshift sleeve.

The tangential force, axial force and the moment as discussed above are given by the following

equations

The friction forcesfx1,fx2,fy1 andfy2 are related to the corresponding normal reactions between the fork

and the rail by the following expressions

A coordinate system XFYFZF as represented in Figure 3-4 is also attached with the fork for convenientlyhandling forces and moments acting on the fork. It is also to be noted that it is only in the axial direction

that the fork experiences acceleration, in all other directions the forces acting on the fork are in

equilibrium.


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3.1.3.1 Equilibrium in XFYF Plane

Figure 3-5 Free-Body Diagram of Fork representing forces in XFYF plane.

Considering the forces acting on the fork in XF, YF direction and moment acting along ZF, we have

Since, the fork does not experience any acceleration (linear or angular) in these directions the net forcesand moments are equated to zero in the above equations.


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3.1.3.2 Equilibrium in XFZF Plane

Figure 3-6 Free-Body Diagram of Fork representing forces in XFZF plane.

Considering the fork in XFZF plane, the fork experiences an acceleration in the ZF direction. So, the

equations are:

3.1.3.3 Equilibrium in YFZF Plane

As forces along XF,YF, ZF have already been taken care of, the only quantity left to be considered is

moment about XF axis.

Thus, the moment equation is given by


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Figure 3-7 Free-Body Diagram of Fork representing forces in YFZF plane

3.2 Governing Differential Equation and State-Space RepresentationNow, we have all the equations that capture the dynamics of the system. All we need is to solve them in

order to obtain the final expression for the drum angular acceleration.

3.2.1 Governing Equation

All the above mentioned equations i.e. Equation (1) (13) & (A) (F) are solved using SymbolicToolbox in MATLAB to obtain the following expression:

(14)

where,f1(),f2() andf3() are given by the following expressions


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values for the unknown parameters Ks and Bs. The estimated parameters and the states are provided to the

controller for evaluating the input for the next instant.


It can be seen from the state-space representation developed in the previous section that system is instrict feedback form which is as follows:

Strict Feedback Form

So, the control law can be derived as follows:

where, u = TD is the torque input for the drum.

Since, the Lyapunov function is negative definite this ensures asymptotic convergence of the states to

zero.


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4.1.2 Standard Least Squares Estimator

A standard least squares estimator is developed for parameter estimation with the following cost

function.

Cost Function

The following are various update laws used for parameter updation and gain matrix updation:

Parameter Update Law

So, for the present system it becomes

Gain Matrix Update Law

where,

4.2 Tracking ControllerThe tracking controller has the capability of tracking a reference signal. It has the following control

objective and assumptions:

Control Objective: To design a control law to track the reference signal asymptotically.

Assumptions:

1. Full state feedback is available.

2. Parameters Ks and Bs are unknown and constant.

3. The reference signal is in the form of states of the system.


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Figure 4-2 represents the control block diagram for the tracking controller. Here the controller works on

the new states z1 and z2 which are the difference of the actual states and the reference signal. The

estimation process is the same as defined for the previous controller.

Figure 4-2 Control block diagram of Tracking Controller


The control law is now derived for the states z1 and z2 is as follows:


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where, u = TD

The negative definiteness of the Lyapunov function guarantees asymptotic convergence of the states to

the reference signal.


Figure 4-3 Actual mechanism with plan for state, output measurement and state estimation


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The actual controller is based on the fact that there will always be some noise in measurement.

Furthermore, in actual system a full state feedback can be avoided by measuring one of the states and

estimating the other one using the first one. Since, differentiating the signal to estimate a state leads tonoisy estimation, the technique of integration is used to estimate the angular position of the drum using

its angular velocity.

The control objective and assumptions of the controller are as follows:

Control Objective: To design a control law to track the reference signal asymptotically.

Assumptions:

1. Outputy and Statex2 is available as feedback.2. Parameters Ks and Bs are unknown and constant.

3. There is noise present in measurement.

Figure 4-4 Control block diagram of Actual Controller

Figure 4-4 illustrates the control block diagram of the actual controller. The noise is introduced in the

measured signals. The integrator is used to estimate the state x1 using state x2. Also, the estimated signal is

now used in parameter estimation.

4.3.1 Backstepping Control Law

The control law for this controller remains same as derived for the previous controller with the following

modifications:

1. The measured value of x1 is now replaced with the estimated value obtained using an integrator.2. High frequency noise is introduced in the measured states.


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where, z1 and z2 includes the high frequency noise signal.

The negative definiteness of the Lyapunov function as derived in the control law derivation of thetracking controller still holds good and so the states will converge to the reference signal asymptotically.

4.3.3 Standard Least Squares EstimatorThe estimator for the actual system is the same with a modification that the estimated value of x1 is now

used in the estimator to estimate the parameters.

4.3.3 State Estimator (Integrator)

The state x1 is estimated by integrating the state x2 which also contains the noise signal. So, the

differential equation for estimator becomes

where, n2 represents the high frequency noise signal introduced in the state x2 during measurement.

4.4 Model Linearization for Parameter Estimation

The above developed parameter estimators are based on the assumption that the coefficient of frictionbetween the various components is precisely known. However, the coefficient of friction is not known

always. Since, the system is highly nonlinear the influence of coefficient of friction plays a vital role in

determining the behavior of the system. In actual vehicle there is always an uncertainty involved indetermining the value of coefficient of friction due to variation in lubrication. Even the viscosity of the

lubricant varies with temperature which in turn varies the coefficient between the interacting surfaces.

Thus, there is always a variation in the coefficient of friction which cannot be predicted, though the

range of variation might be narrow. Thus, this demands an estimation of the coefficient of friction byusing an estimator. However, considering the highly nonlinear dynamics of the system the estimation of

the coefficient of friction would be difficult.

The functions f1, f2 and f3 are functions of state and the coefficient of friction. Now, the Taylor seriesexpansion for a function about a point 0 is given by


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Now, neglecting higher order terms, we can obtain

where, f11(0) represents the value of the function at 0 andf12represents the value of the derivate of the

function with respect to at 0.

Thus, each of the nonlinear function can be replaced by its linear approximation to give rise to the

following state-space representation of the system:

The unknown parameters in the model are then combined in such a manner that the reduced systems will

have unknowns that appear linearly. The value of functionsf11

,f

12, f

21, f

22,f

31andf

32is mentioned in

Appendix A.1

4.5 Standard Least Squares Estimator with Linearized ModelA standard least squares estimator is then developed using this linearized model.

The parameter update law is given by

where,

The gain matrix update law is given by

where, P is a 5x5 gain matrix. Also, the input will now have the values of functionf1, f2 and f3 evaluated

for the estimated value of the coefficient of friction.


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Table 4-4 Mechanism Parameters used in Simulation

5.1.1.2 Simulation Results

0 1 2 3 4 5 6 7 8 9 100

5

10

15

time (s)

Y

(deg)

Y vs t

0 1 2 3 4 5 6 7 8 9 10-14

-12

-10

-8

-6

-4

-2

0

time (s)

dY/dt(deg/s)

dY/dt vs t

0 1 2 3 4 5 6 7 8 9 10-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

time (s)

Ffa(kg)

Ffa vs t


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Figure 5-1 Simulation results for basic controller with known coefficient of friction

The simulation results showed the following:1. Both the states i.e. drum angular position and velocity, converge to zero asymptotically as is

expected.

2. Output i.e. the axial force developed on the fork also converges asymptotically to zero.3. However, due to decay of states to zero the signal matrix of the parameter estimator is notpersistently exciting and it converges to zero in the course of time.4. As a result of signal matrix converging to zero, the gain matrix for the estimator doesnt converge to

zero, which is obvious from the following equation for gain updation.

5. The unknown parameters do not converge to their true value. However, they become constant aftercertain time, which can be explained by the following equation for parameter updation.

0 1 2 3 4 5 6 7 8 9 10-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

time (s)

W

W (Signal Matrix) vs t

W1

W2

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

time (s)

P

P (Gain Matrix) vs t

P11

P12

P21

P22

0 1 2 3 4 5 6 7 8 9 102

4

6

8

10

12

14

16

18

20

time (s)

Ks(kg/mm)

Ks vs t

Ksreal

Kshat

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

time (s)

Bs(kgs/m)

Bs vs t

Bsreal

Bshat


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27

As the signal matrix converged to zero the updation of parameters stop due to which they remain

constant.

5.1.2 Simulation with unknown Coefficient of Friction (Linearized model for Parameter

Estimation)

This section presents the result for the unknown coefficient of friction case.

5.1.2.1 Simulation Inputs

The following additional inputs (different values) are used for the simulation of this controller.

real = 0.14

(0) = 0.50 = 0.16

K (backstepping) = 1000


0 1 2 3 4 5 6 7 8 9 100

5

10

15

time (s)

Y(deg)

Y vs t

0 1 2 3 4 5 6 7 8 9 10-14

-12

-10

-8

-6

-4

-2

0

time (s)

dY/dt(deg/s)

dY/dt vs t

0 1 2 3 4 5 6 7 8 9 10-15

-10

-5

0

5

10

time (s)

Ffa(kg)

Ffa vs t


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28

Figure 5-2 Simulation results for basic controller with unknown coefficient of friction

The simulation results showed the following:1. Both the states i.e. drum angular position and velocity; converge to zero asymptotically in almost the

same time as was for the known coefficient of friction case.

2. Output i.e. the axial force developed on the fork also converges asymptotically to zero in almost thesame time.

3. However, there is appreciable difference in the signal matrix of the estimator between the two cases.The signal matrix converged to zero in the unknown coefficient of friction case much rapidly.

4. In this case the estimation of the coefficient of friction also takes place. There is considerable errorin the estimated value of the coefficient. However, the error became constant in the course of time.

This error in estimation can be attributed to the fact that the signal matrix decayed to zero rapidly inthis case.

5. The unknown parameters also become constant after certain time, which can be explained by thefollowing equation for parameter updation.

0 1 2 3 4 5 6 7 8 9 10-5

0

5

10

15

20

25

30

35

40

time (s)

W


W1

W2

W3

W4

W5

0 1 2 3 4 5 6 7 8 9 10-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

time (s)

vs t

ureal

uhat

0 1 2 3 4 5 6 7 8 9 10-5

0

5

10

15

20

25

time (s)

Ks(kg/mm)

Ks vs t

Ksreal

Kshat

0 1 2 3 4 5 6 7 8 9 10-10

-5

0

5

10

15

20

25

30

35

time (s)

Bs(kgs/m)

Bs vs t

Bsreal

Bshat


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However, the steady-state error in estimation is more in this case then the known coefficient case.

5.2Tracking ControllerThe tracking controller as discussed in the previous chapter is also simulated for the two cases discussedin the previous section.

5.2.1 Simulation with known Coefficient of Friction

This section presents the results of the tracking controller for the known coefficient of friction case.


xr = sin(t)


0 1 2 3 4 5 6 7 8 9 10-60

-40

-20

0

20

40

60

80

time (s)

Y

(deg)

Y vs t

Yr

Y

0 1 2 3 4 5 6 7 8 9 10-60

-40

-20

0

20

40

60

time (s)

dY/dt(deg/s)

dY/dt vs t

Yrdot

Ydot


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30

0 1 2 3 4 5 6 7 8 9 10-5

0

5

10

15

20

25

30

35

40

45

time (s)

Ffa(kg)

Ffa vs t

Ffar

Ffa

0 10 20 30 40 50 60 70 80 90 100-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

time (s)

P

P(Gain Matrix) vs t

P11

P12

P21

P22

0 5 10 15 20 25-8

-6

-4

-2

0

2

4

6

8

10

time (s)

W


W1

W2

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time (s)

Bs(kgs/m)

Bs vs t

Bsreal

Bshat

0 100 200 300 400 500 600 700 800 900 10004

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

time (s)

Ks(kg/mm)

Ks vs t

Ksreal

Kshat


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31

Figure 5-3 Simulation results for tracking controller with known coefficient of friction

The simulation results showed the following:1. Both the states i.e. drum angular position and velocity, tracks the reference signal asymptotically.2. Output i.e. the axial force developed on the fork which is combination of the two states and the

unknown parameters also tracks the reference signal asymptotically.3. The signal matrix of the estimator is persistently exciting in this case.4. Due to persistently exciting signal matrix, the gain matrix for the estimator converges to zero.5. The unknown parameters also converge to their true value asymptotically.5.2.2 Simulation with unknown Coefficient of Friction (Linearized model for Parameter

Estimation)

The simulation results for the unknown coefficient of friction are presented in this section.


The following additional inputs (different values) are used for the simulation of this controller.real = 0.14

(0) = 0.40 = 0.16

K(backstepping) = 10;

P0 = [0.001:0.001:0.025];

0 1 2 3 4 5 6 7 8 9 10-60

-40

-20

0

20

40

60

time (s)

Y

(deg)

Y vs t

Yr

Y

0 1 2 3 4 5 6 7 8 9 10-60

-40

-20

0

20

40

60

time (s)

dY/dt(deg/s)

dY/dt vs t

Yrdot

Ydot


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32

Figure 5-4 Simulation results for tracking controller with unknown coefficient of friction

0 1 2 3 4 5 6 7 8 9 10-5

0

5

10

15

20

25

30

35

time (s)

Ffa(kg)

Ffa vs t

Ffar

Ffa

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2

3

4

5

6

time (s)

W


W1

W2

W3

W4

W5

0 1 2 3 4 5 6 7 8 9 100.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

time (s)

vs t

ureal

uhat

0 1 2 3 4 5 6 7 8 9 103.8

4

4.2

4.4

4.6

4.8

5

time (s)

Ks(kg/mm

)

Ks vs t

Ksreal

Kshat

0 1 2 3 4 5 6 7 8 9 10-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

time (s)

Bs(kgs/m)

Bs vs t

Bsreal

Bshat


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33

The simulation results showed the following:

1. The controller had hard time tracking both the states i.e. drum angular position and velocity.2. The output i.e. the axial force developed on the fork also shows large variations between the actual

and the reference signal.

3. The signal matrix of the estimator remains persistently exciting in this case.4. Even when the signal matrix is persistently exciting the coefficient of friction do not converge to its

true value.

5. The unknown parameters also do not converge to their respective true values.5.3 Actual ControllerThe actual controller discussed in the previous chapter is simulated in the following section. Due to theinability of the tracking controller to track the reference signal for the unknown coefficient of friction

case, it becomes pointless to simulate the same for the actual controller which includes noise and state

estimation. The following section presents the result with only the known coefficient of friction.

5.3.1 Simulation Inputs

Apart from the simulation inputs of the tracking controller with known coefficient of friction, thefollowing inputs are used for the simulation.

Noise (x2)n2 = 0.2sin(20*t)

Noise (y)

ny = sin(20*t)

The noise signal is modeled as a high frequency periodic disturbance from a rotating source which might

be the engine or the rotating driveline with huge inertia.

5.3.2 Simulation Resutls

The simulation results showed the following:

0 1 2 3 4 5 6 7 8 9 10-60

-40

-20

0

20

40

60

80

time (s)

Y

(deg

)

Y vs t

Yr

Y

Yhat

0 1 2 3 4 5 6 7 8 9 10-80

-60

-40

-20

0

20

40

60

80

time (s)

dY/dt(deg

/s)

dY/dt vs t

Yrdot

Ydot

Ydot measured


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34

Figure 5-5 Simulation results for actual controller with known coefficient of friction

0 1 2 3 4 5 6 7 8 9 10-5

0

5

10

15

20

25

30

35

40

45

time (s)

Ffa(kg)

Ffa vs t

Ffar

Ffa

Ffa measured

0 5 10 15 20 25-10

-8

-6

-4

-2

0

2

4

6

8

10

time (s)

W


W1

W2

0 10 20 30 40 50 60 70 80 90 100-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

time (s)

P

P(Gain Matrix) vs t

P11

P12

P21

P22

0 100 200 300 400 500 600 700 800 900 10004

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

time (s)

Ks(kg/mm)

Ks vs t

Ksreal

Kshat

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time (s)

Bs(kgs/m)

Bs vs t

Bsreal

Bshat


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35

1. The estimated drum angular position tracks the actual position and in due course of time both ofthem track the reference signal.

2. The measured state follows the actual state though with lesser deviations, however there are slightdeviations of the two from the reference signal due to high frequency noise present in the

measurement. Such deviations can be dealt with by appropriate filtering of the measured state.3. Output i.e. the axial force developed on the fork also tracks the reference signal asymptotically with

small deviations present due to noise in the measured signal.

4. The signal matrix of the parameter estimator is persistently exciting in this case.5. Due to persistently exciting signal matrix, the gain matrix for the estimator converges to zero

asymptotically.

6. The unknown parameters also converge to their true value asymptotically.5.4 Virtual Reality Simulation

In order to understand the physical behavior of the system a simulation is developed in MATLAB using

Virtual Reality Toolbox for the results obtained for the actual controller. The variation in angular

velocity was appreciable in the simulation of the 3d-model.


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36

Chapter 6

Discussion

The basic controller with known coefficient of friction worked satisfactorily for asymptoticallyconverging states to zero. The basic controller with unknown coefficient of friction worked as good as

the one with known value. The tracking controller worked satisfactorily for the known coefficient of

friction case. However, the tracking controller with unknown coefficient of friction was unable to trackthe reference signal. The actual controller worked well for the known coefficient for friction case.

The results showed that the unknown coefficient of friction case is difficult to handle with, in real world.

When it comes to determining the asymptotic stability using a controller the parameter estimator with alinearized model worked as good as a known parameter case, with states decaying to zero in almost the

same time for the two cases. However, the parameter estimation failed to track the reference signal withthe linearized model against the perfect tracking achieved by the known parameter case controller. The

results also showed that the standard least squares estimator is immune to noise in the signal matrix.

Thus, a first order linearization of the system for parameter estimation is not sufficient for a gearshift

mechanism which is a highly nonlinear system to not only estimate the unknown parameters accurately

but also for reference signal tracking.

The future work for dealing with the problem of control of a gearshift mechanism might involve

improving the model that has been developed and to improve the controller. The model can be modifiedto take into account the nonlinearity in the model due to deformation of the fork. Also, the variation inarea moment of inertia of the two legs of the fork can be taken into account for more realistic

distribution of forces. The controller for the unknown coefficient of friction case can be developed

considering the second order approximation of the nonlinear functions involved. However, it will lead tomore number of unknown parameters in the model to be estimated which need to be dealt with

accordingly.


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37

Bibliography

[1] Chaouch S., Nait-Sait M (2006), Backstepping control design for position and speed trackingof DC motors.

[2] Aneke N. P. I., Nijmeijerz H. and Jager A. G. (2003), Tracking control of second-orderchained form systems by cascaded backstepping, International journal of robust and nonlinear

control.

[3] Xie Chengkang, (2004), Nonlinear Output Feedback Control: An Analysis of Performance and

Robustness, Phd thesis, UNIVERSITY OF SOUTHAMPTON.

[4] Faa J L, Rong J W(2001), A Hybrid Computed Torque Controller Using Fuzzy Neural Network for

Motor-Quick-Return Servo Mechanism, IEEE/ASME TRANSACTIONS ON MECHATRONICS,

VOL. 6, NO. 1, MARCH 2001

[5] Yang Z; Wu J; Mei J; Gao J and Huang T (2008), Mechatronic Model Based Computed Torque

Control of a Parallel Manipulator, International Journal of Advanced Robotic Systems, Vol. 5, No. 1ISSN 1729-8806, pp.123-128.

[6] Yang Z; Wu J; Mei J (2007), Motor-mechanism dynamic model based neural network optimizedcomputed torque control of a high speed parallel manipulator, Mechatronics 17 (2007) 381390.

[7] Weiwei S., Shuang C (2009), Nonlinear computed torque control for a high-speed planar parallelmanipulator, Mechatronics 19 (2009) 987992.

[8] Rothbart H. A., Cam design handbook, McGraw-Hill Handbooks, 2004, pp. 27-55.

[9] Norton R. L., Cam design and manufacturing handbook, Industrial Press, Inc., pp. 177-312.


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Appendix A

A.1 MATLAB code for solving equations involved in the Mathematical Model of the

Dynamics of the Gearshift Mechanismclc;clear all;close all;

syms R1FAFTFft_fNx1Nx2Ny1Ny2FfaTDthetaRDuIDmfzeta_angCA2A1B1B2

E1E2O1O2O3ddlphigammadotgammadot2K1K2dthetadgammaDsi;

Eqn = 'FA -R1*(cos(theta)-u*sin(theta)) ,

FT -R1*(sin(theta)+u*cos(theta)),

FT+Fft_f*cos(zeta_ang)-Nx1-Nx2,FT*sin(phi)+Ny1-Ny2-Fft_f*sin(zeta_ang),

-FT*C+Fft_f*A2/cos(zeta_ang),

FA*C*sin(phi)+Ffa*((A2^2+E2^2)^(3/2)/((A1^2+E1^2)^(3/2) +(A2^2+E2^2)^(3/2))*E1-(1-(A2^2+E2^2)^(3/2)/((A1^2+E1^2)^(3/2)+ (A2^2+E2^2)^(3/2)))*E2)+

Fft_f*cos(zeta_ang)*O3+Nx2*O2-Nx1*O1+u*(Nx1+Nx2)*d/2+u*R1*dl/2*cos(phi)-

u*Fft_f*Dsi*cos(zeta_ang),

-FA*C*cos(phi)-FT*sin(phi)*O3-Ffa*((A2^2+E2^2)^(3/2)/((A1^2+E1^2)^(3/2)+

(A2^2+E2^2)^(3/2))*A1-(1-(A2^2+E2^2)^(3/2)/((A1^2+E1^2)^(3/2)+

(A2^2+E2^2)^(3/2)))*A2)+Ny1*B1+Ny2*B2-u*Ny1*d/2+u*Ny2*d/2+u*R1*dl*sin(phi)/2-

u*Fft_f*((A1+A2)/2-Dsi*sin(zeta_ang)),

FA-u*(Ny1+Ny2+Nx1+Nx2)-Ffa-u*Fft_f-mf*RD*(tan(theta)*gammadot2+

sec(theta)^2*dthetadgamma*gammadot^2),

TD -R1*(sin(theta)+u*cos(theta))*RD-ID*gammadot2';

S = solve(Eqn,FA,FT,Fft_f,Nx1,Nx2,Ny1,Ny2,R1,gammadot2)

gammadot2_val = S.gammadot2fun_sim = vpa(simplify(subs(simplify(gammadot2_val),{dl,RD, zeta_ang, C,A2, A1, B1,

B2, E1, E2, O1, O2, O3, d, phi, ID, mf, Dsi} ,{8.3, 24.5, 31.61*pi/180, 14.95,

45.35, 87.09, 17.64, 84.35, 33.91,33.91, 85.44, 16.55, 67.8, 13, 62.08*pi/180, 200,

0.2, 74.63})))f1 = vpa(simplify(subs(fun_sim,{TD, Ffa, gammadot,u} ,{1,0,0,0.16})),4)f2 = vpa(simplify(subs(fun_sim,{TD, Ffa, gammadot,u} ,{0,1,0,0.16})),4)f3 = vpa(simplify(subs(fun_sim,{TD, Ffa, gammadot,u} ,{0,0,1,0.16})),4)

-----------------------------------------------------------------------------------

The expressions obtained for f1, f2 and f3f1 = .3906e-2*cos(theta)*(-.2294e21*cos(theta)-.4584e18+.1275e21*sin(theta))/

(.8091e20*sin(theta)*cos(theta)-.6243e20*cos(theta)^2-.3581e18*cos(theta)-.1168e21)

f2 =.8896e18*cos(theta)*(4.*cos(theta)+25.*sin(theta))/(.8091e20*sin(theta)*cos(theta)-

.6243e20*cos(theta)^2-.3581e18*cos(theta)-.1168e21)

f3 = .4671e19*dthetadgamma*(4.*cos(theta)+25.*sin(theta))/cos(theta)/

(.8091e20*sin(theta)*cos(theta)-.6243e20*cos(theta)^2-.3581e18*cos(theta)-.1168e21)

-----------------------------------------------------------------------------------------------------------The expressions obtained for f11, f12, f21, f22, f31 and f32.

f11 =.1562e-1*cos(theta).*(.2293e9*cos(theta)+.4584e6-

.1275e9*sin(theta))./(.2497e9*cos(theta).^2+.4670e9+.1432e7*cos(theta)-

.3236e9*sin(theta).*cos(theta))


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- 2 -

f12 =-4.882*cos(theta).*(.3153e15*cos(theta).^3-.3155e15*cos(theta)-

.8563e13*cos(theta).^2+.7925e15*sin(theta).*cos(theta).^2+.8563e13+.6851e12*sin(the

ta).*cos(theta)-.1186e16*sin(theta))./(.4239e17*cos(theta).^4-

.3379e18*cos(theta).^2-.7153e15*cos(theta).^3+.1616e18*cos(theta).^3.*sin(theta)-

.2181e18-

.1338e16*cos(theta)+.3023e18*sin(theta).*cos(theta)+.9271e15*sin(theta).*cos(theta)

.^2)

f21 =-

.3558e7*cos(theta).*(4.*cos(theta)+25.*sin(theta))./(.2497e9*cos(theta).^2+.4670e9+

.1432e7*cos(theta)-.3236e9*sin(theta).*cos(theta))

f22 =-500.*cos(theta).*(.3237e15*cos(theta).^3-

.4609e15*cos(theta)+.2731e12*cos(theta).^2-

.1016e15*sin(theta).*cos(theta).^2+.3711e14*sin(theta)+.3299e13*sin(theta).*cos(the

ta))./(.4239e17*cos(theta).^4-.3379e18*cos(theta).^2-

.7153e15*cos(theta).^3+.1616e18*cos(theta).^3.*sin(theta)-.2181e18-


.^2)

f31 =-

.1868e8*dtdg*(4.*cos(theta)+25.*sin(theta))./cos(theta)./(.2497e9*cos(theta).^2+.46

70e9+.1432e7*cos(theta)-.3236e9*sin(theta).*cos(theta));

f32 =-.4670e9*dtdg*(.1657e10*cos(theta).^2-.2479e10+.1432e7*cos(theta)-

.6595e9*sin(theta).*cos(theta)+.1790e8*sin(theta))./(.4239e17*cos(theta).^4-

.3379e18*cos(theta).^2-.7153e15*cos(theta).^3+.1616e18*cos(theta).^3.*sin(theta)-

.2181e18-


.^2)

A.2 MATLAB code for Basic Controller with parameter estimation andasymptotically stabilizing origin (Known Coefficient of Friction)function Basic_Controllerclc;clear all;global Ksreal Bsreal ktheta RDktheta = 0.5;RD = 24.5;sgn =1;Ksreal = 20;Bsreal = 30;Kshat0 = 3;Bshat0 = 4;

P0 = [40 50;60 70];

y0 = [0.25 0 Kshat0 Bshat0 P0(1,1) P0(1,2) P0(2,1) P0(2,2)]; %gamma gammadot Kshat

Bshat P(1,1) P(1,2) P(2,1) P(2,2)

[T Y] = ode45(@mysystem,linspace(0,5),y0); %gamma gammadot Kshat Bshat P(1,1)

P(1,2) P(2,1) P(2,2)figure;plot(T,Y(:,1)*180/pi,'-');grid onxlabel('time (s) \rightarrow','FontSize',12);ylabel('Y (deg)\rightarrow','FontSize',12);title('Y vs t','FontSize',15);


45/59


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- 4 -

P = [y(5) y(6);y(7) y(8)];

u = (-y(2)-y(1)-K_bs*(y(1)+y(2))-f2_val*Kshat*RD*log(sec(theta_val))/(ktheta)-

(f2_val*Bshat*RD*tan(theta_val)+f3_val)*y(2))/f1_val;dy(1) = y(2); %gamma gammadot Kshat Bshat P(1,1) P(1,2) P(2,1) P(2,2)dy(2) =

f1_val*u+f2_val*(Ksreal*RD*log(sec(theta_val))/(ktheta)+Bsreal*RD*tan(theta_val)*y(

2))+f3_val*y(2);dy(3) = -

(P(1,1)*RD*log(sec(theta_val))/(ktheta)+P(1,2)*RD*tan(theta_val)*y(2))*(W*[Kshat

Bshat]'-(Ksreal*W(1)+Bsreal*W(2)));dy(4) = -

(P(2,1)*RD*log(sec(theta_val))/(ktheta)+P(2,2)*RD*tan(theta_val)*y(2))*(W*[Kshat

Bshat]'-(Ksreal*W(1)+Bsreal*W(2)));dy(5) = -(P(1,1)*W(1)+P(1,2)*W(2))*(W(1)*P(1,1)+W(2)*P(2,1));dy(6) = -(P(1,1)*W(1)+P(1,2)*W(2))*(W(1)*P(1,2)+W(2)*P(2,2));dy(7) = -(P(2,1)*W(1)+P(2,2)*W(2))*(W(1)*P(1,1)+W(2)*P(2,1));dy(8) = -(P(2,1)*W(1)+P(2,2)*W(2))*(W(1)*P(1,2)+W(2)*P(2,2));


reference output tracking (Known Coefficient of Friction)function Tracking_Controllerclc;clear all;close all;global Ksreal Bsreal ktheta RDktheta = 0.5;RD = 24.5;sgn =1;Ksreal = 5;Bsreal = 0.1;Kshat0 = 4;Bshat0 = 0.01;P0 = [0.02 0.01;

0.03 0.04];ref = @(T) sin(T);refdot = @(T) cos(T);y0 = [0.5 -0.5 Kshat0 Bshat0 P0(1,1) P0(1,2) P0(2,1) P0(2,2)]; %z(gamma-gammaref)

zdot(gammadot-gammadotref) Kshat Bshat P(1,1) P(1,2) P(2,1) P(2,2)[T Y] = ode45(@mysystem,[0,10],y0); %z(gamma-gammaref) zdot(gammadot-gammadotref)

Kshat Bshat P(1,1) P(1,2) P(2,1) P(2,2)figure;p = plot(T,ref(T)*180/pi,'-b',T,(Y(:,1)+ref(T))*180/pi,'-r');grid on

legend(p,'Yr','Y',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('Y (deg)\rightarrow','FontSize',12);title('Y vs t','FontSize',15);

figurep = plot(T,refdot(T)*180/pi,'-b',T,(Y(:,2)+refdot(T))*180/pi,'-r');grid onlegend(p,'Yrdot','Ydot',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('dY/dt (deg/s) \rightarrow','FontSize',12);title('dY/dt vs t','FontSize',15);figure


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- 5 -

p =

plot(T,Ksreal*RD*log(sec(ktheta*(ref(T))))/ktheta+Bsreal*RD*tan(ktheta*(ref(T))).*r

efdot(T),...'-

b',T,Ksreal*RD*log(sec(ktheta*(Y(:,1)+ref(T))))/ktheta+Bsreal*RD*tan(ktheta*(ref(T)

)).*Y(:,2),'-r');grid onlegend(p,'Ffar','Ffa',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('Ffa (kg) \rightarrow','FontSize',12);title('Ffa vs t','FontSize',15);figurep = plot(T,Ksreal*ones(size(T)),'-b',T,Y(:,3),'-r');grid onlegend(p,'Ksreal','Kshat',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('Ks (kg/mm) \rightarrow','FontSize',12);title('Ks vs t','FontSize',15);

figurep = plot(T,Bsreal*ones(size(T)),'-b',T,Y(:,4),'-r');grid onlegend(p,'Bsreal','Bshat',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('Bs (kgs/m) \rightarrow','FontSize',12);title('Bs vs t','FontSize',15);figurep = plot(T,Y(:,5),'-b',T,Y(:,6),'-r',T,Y(:,7),'-y',T,Y(:,8),'-k');grid onlegend(p,'P11','P12','P21','P22',1);xlabel('time (s) \rightarrow','FontSize',12);ylabel('P \rightarrow','FontSize',12);title('P(Gain Matrix) vs t','FontSize',15);figurep = plot(T,RD*(log(sec(ktheta*(Y(:,1)+ref(T)))))/(ktheta) ,'-

b',T,RD*tan(ktheta*(Y(:,1)+ref(T))).*(Y(:,2)+refdot(T)),'-r');grid onlegend(p,'W1','W2',1);xlabel('time (s) \rightarrow','FontSize',12);ylabel('W \rightarrow','FontSize',12);title('W (Signal Matrix) vs t','FontSize',15);clear all;

function dy = mysystem(t,y)global Ksreal Bsreal ktheta RDK_bs = 100;

dy = zeros(8,1);xr = sint(t);xrdot = cos(t);xrdot2 = -sin(t);dtdg = ktheta;theta_val = ktheta*(y(1)+sin(t));f1_val = .3906e-2*cos(theta_val)*(-

.1275e21*sin(theta_val)+.2294e21*cos(theta_val)+.4584e18)/(.6243e20*cos(theta_val)^

2+.3581e18*cos(theta_val)-.8091e20*sin(theta_val)*cos(theta_val)+.1168e21);f2_val =-

.8896e18*cos(theta_val)*(25.*sin(theta_val)+4.*cos(theta_val))/(.6243e20*cos(theta_

val)^2+.3581e18*cos(theta_val)-.8091e20*sin(theta_val)*cos(theta_val)+.1168e21);


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f3_val =.4671e19*dtdg*(25.*sin(theta_val)+4.*cos(theta_val))/cos(theta_val)/(-

.6243e20*cos(theta_val)^2-

.3581e18*cos(theta_val)+.8091e20*sin(theta_val)*cos(theta_val)-.1168e21);Kshat = y(3);Bshat = y(4);W = [RD*log(sec(theta_val))/(ktheta) RD*tan(theta_val)*(y(2)+xrdot)];P = [y(5) y(6);

y(7) y(8)];u = (-y(2)-y(1)-K_bs*(y(1)+y(2))-f2_val*Kshat*RD*log(sec(theta_val))/ktheta -

(f2_val*Bshat*RD*tan(theta_val)+f3_val)*(y(2)+xrdot)-xrdot2)/f1_val;dy(1) = y(2); %z(gamma-gammaref) zdot(gammadot-gammadotref) Kshat Bshat P(1,1)

P(1,2) P(2,1) P(2,2)dy(2) = f1_val*u+f2_val*Ksreal*RD*log(sec(theta_val))/ktheta +

(f2_val*Bsreal*RD*tan(theta_val)+f3_val)*(y(2)+xrdot)-xrdot2;dy(3) = -

(P(1,1)*RD*log(sec(theta_val))/(ktheta)+P(1,2)*RD*tan(theta_val)*(y(2)+xrdot))*(W*[

Kshat Bshat]'-(Ksreal*W(1)+Bsreal*W(2)));dy(4) = -

(P(2,1)*RD*log(sec(theta_val))/(ktheta)+P(2,2)*RD*tan(theta_val)*(y(2)+xrdot))*(W*[Kshat Bshat]'-(Ksreal*W(1)+Bsreal*W(2)));dy(5) = -(P(1,1)*W(1)+P(1,2)*W(2))*(W(1)*P(1,1)+W(2)*P(2,1));dy(6) = -(P(1,1)*W(1)+P(1,2)*W(2))*(W(1)*P(1,2)+W(2)*P(2,2));dy(7) = -(P(2,1)*W(1)+P(2,2)*W(2))*(W(1)*P(1,1)+W(2)*P(2,1));dy(8) = -(P(2,1)*W(1)+P(2,2)*W(2))*(W(1)*P(1,2)+W(2)*P(2,2));

A.4 MATLAB code for Final Controller with parameter and state estimation, noise,

reference output tracking and Virtual Reality Simulation. (Known Coefficient of

Friction)function Actual_Backstepping_Controllerclc;

clear all;close all;global Ksreal Bsreal ktheta RDktheta = 0.5;RD = 24.5;sgn =1;Ksreal = 5;Bsreal = 0.1;Kshat0 = 4;Bshat0 = 0.01;P0 = [0.02 0.01;

0.03 0.04];ref = @(T) sin(T);refdot = @(T) cos(T);n2 = @(T) 0.2*sin(20*T);ny = @(T) sin(20*T);y0 = [0.5 -0.5 Kshat0 Bshat0 P0(1,1) P0(1,2) P0(2,1) P0(2,2) 0.5]; %z(gamma-

gammaref) zdot(gammadot-gammadotref) Kshat Bshat P(1,1) P(1,2) P(2,1) P(2,2)[T Y] = ode45(@mysystem,[0,10],y0); %z(gamma-gammaref) zdot(gammadot-gammadotref)

Kshat Bshat P(1,1) P(1,2) P(2,1) P(2,2)

figure;p = plot(T,ref(T)*180/pi,'-b',T,(Y(:,1)+ref(T))*180/pi,'-

.r',T,(Y(:,9)+ref(T))*180/pi,'--k');grid onlegend(p,'Yr','Y','Yhat',4);xlabel('time (s) \rightarrow','FontSize',12);


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dy(5) = -(P(1,1)*W_e(1)+P(1,2)*W_e(2))*(W_e(1)*P(1,1)+W_e(2)*P(2,1));dy(6) = -(P(1,1)*W_e(1)+P(1,2)*W_e(2))*(W_e(1)*P(1,2)+W_e(2)*P(2,2));dy(7) = -(P(2,1)*W_e(1)+P(2,2)*W_e(2))*(W_e(1)*P(1,1)+W_e(2)*P(2,1));dy(8) = -(P(2,1)*W_e(1)+P(2,2)*W_e(2))*(W_e(1)*P(1,2)+W_e(2)*P(2,2));dy(9) = y(2)+n2;

A.5 MATLAB code for Basic Controller with parameter estimation using linearized

model for asymptotically stabilizing origin (Unknown Coefficient of Friction)

function Basic_Controller_u_unknownclc;clear all;close all;global Ksreal Bsreal ktheta RD ureal K_bsK_bs = 1000;ktheta = 0.5;RD = 24.5;

ureal = 0.14;Ksreal = 20;Bsreal = 30;u0 = 0.5;ubar = 0.16;Kshat0 = 3;Bshat0 = 4;a1hat0 = u0-ubar;a2hat0 = a1hat0*Kshat0;a3hat0 = a1hat0*Bshat0;P0 = [0.1:0.1:2.5];y0 = [0.25 0 a1hat0 Kshat0 a2hat0 Bshat0 a3hat0 P0]; %gamma gammadot a1hat Kshat

a2hat Bshat a3hat P11->P15, P21->P25, P31->P35, P41->P45, P51->P55[T Y] = ode45(@mysystem,linspace(0,10),y0); %gamma gammadot a1hat Kshat a2hat Bshat

a3hat P11->P15, P21->P25, P31->P35, P41->P45, P51->P55figure;plot(T,Y(:,1)*180/pi,'-');grid onxlabel('time (s) \rightarrow','FontSize',12);ylabel('Y (deg)\rightarrow','FontSize',12);title('Y vs t','FontSize',15);figureplot(T,Y(:,2)*180/pi,'-r');grid onxlabel('time (s) \rightarrow','FontSize',12);ylabel('dY/dt (deg/s) \rightarrow','FontSize',12);title('dY/dt vs t','FontSize',15);

figureplot(T,Ksreal*RD*log(sec(ktheta*(Y(:,1))))/ktheta+Bsreal*RD*tan(ktheta*(Y(:,1))).*Y(:,2),'-r');grid onxlabel('time (s) \rightarrow','FontSize',12);ylabel('Ffa (kg) \rightarrow','FontSize',12);title('Ffa vs t','FontSize',15);figurep = plot(T,Ksreal*ones(size(T)),'-b',T,Y(:,4),'--r');grid onlegend(p,'Ksreal','Kshat',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('Ks (kg/mm) \rightarrow','FontSize',12);title('Ks vs t','FontSize',15);


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figurep = plot(T,Bsreal*ones(size(T)),'-b',T,Y(:,6),'--r');grid onlegend(p,'Bsreal','Bshat',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('Bs (kgs/m) \rightarrow','FontSize',12);title('Bs vs t','FontSize',15);u = Y(:,3)+ubar;figurep = plot(T,ureal*ones(size(T)),'-b',T,u,'--r');grid onlegend(p,'ureal','uhat',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('\mu \rightarrow','FontSize',12);title('\mu vs t','FontSize',15);dtdg = ktheta;Kshat = Y(:,4);Bshat = Y(:,6);

theta_val = ktheta*Y(:,1);f1_val =.7812e-

2*cos(theta_val).*(.3528e16*u.^3.*cos(theta_val)+.1411e18*u.^2.*cos(theta_val)-

.4584e16*u.^2+.8750e16*u.*cos(theta_val)-.6374e17*cos(theta_val)-

.5222e16*u.^2.*sin(theta_val)+.2048e18*u.*sin(theta_val))./(.5513e16*u.^3.*cos(thet

a_val).^2+.2204e18*u.^2.*cos(theta_val).^2-.7162e16*u.^2.*cos(theta_val)-

.5978e17+.1367e17*cos(theta_val).^2.*u-

.3981e17*cos(theta_val).^2+.2602e18*u.*sin(theta_val).*cos(theta_val)-

.8160e16*u.^2.*sin(theta_val).*cos(theta_val));f2_val =-8.*cos(theta_val).*(.6356e15*u.^2.*cos(theta_val)-.1525e16*sin(theta_val)-

.1525e16*u.*cos(theta_val)+.6356e15*u.*sin(theta_val))./(.5513e16*u.^3.*cos(theta_v

al).^2+.2204e18*u.^2.*cos(theta_val).^2-.7162e16*u.^2.*cos(theta_val)-

.5978e17+.1367e17*cos(theta_val).^2.*u-


.8160e16*u.^2.*sin(theta_val).*cos(theta_val));f3_val

=.5978e17*dtdg*(sin(theta_val)+u.*cos(theta_val))./cos(theta_val)./(.5513e16*u.^3.*

cos(theta_val).^2+.2204e18*u.^2.*cos(theta_val).^2-.7162e16*u.^2.*cos(theta_val)-

.5978e17+.1367e17*cos(theta_val).^2.*u-


.8160e16*u.^2.*sin(theta_val).*cos(theta_val));f11 =.1562e-1*cos(theta_val).*(.2293e9*cos(theta_val)+.4584e6-

.1275e9*sin(theta_val))./(.2497e9*cos(theta_val).^2+.4670e9+.1432e7*cos(theta_val)-

.3236e9*sin(theta_val).*cos(theta_val));f12 =-4.882*cos(theta_val).*(.3153e15*cos(theta_val).^3-.3155e15*cos(theta_val)-

.8563e13*cos(theta_val).^2+.7925e15*sin(theta_val).*cos(theta_val).^2+.8563e13+.685

1e12*sin(theta_val).*cos(theta_val)-

.1186e16*sin(theta_val))./(.4239e17*cos(theta_val).^4-.3379e18*cos(theta_val).^2-

.7153e15*cos(theta_val).^3+.1616e18*cos(theta_val).^3.*sin(theta_val)-.2181e18-

.1338e16*cos(theta_val)+.3023e18*sin(theta_val).*cos(theta_val)+.9271e15*sin(theta_

val).*cos(theta_val).^2);f21 =-

.3558e7*cos(theta_val).*(4.*cos(theta_val)+25.*sin(theta_val))./(.2497e9*cos(theta_

val).^2+.4670e9+.1432e7*cos(theta_val)-.3236e9*sin(theta_val).*cos(theta_val)); f22 =-500.*cos(theta_val).*(.3237e15*cos(theta_val).^3-

.4609e15*cos(theta_val)+.2731e12*cos(theta_val).^2-

.1016e15*sin(theta_val).*cos(theta_val).^2+.3711e14*sin(theta_val)+.3299e13*sin(the

ta_val).*cos(theta_val))./(.4239e17*cos(theta_val).^4-.3379e18*cos(theta_val).^2-



val).*cos(theta_val).^2);


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f31 =-

.1868e8*dtdg*(4.*cos(theta_val)+25.*sin(theta_val))./cos(theta_val)./(.2497e9*cos(t

heta_val).^2+.4670e9+.1432e7*cos(theta_val)-

.3236e9*sin(theta_val).*cos(theta_val));f32 =-.4670e9*dtdg*(.1657e10*cos(theta_val).^2-.2479e10+.1432e7*cos(theta_val)-

.6595e9*sin(theta_val).*cos(theta_val)+.1790e8*sin(theta_val))./(.4239e17*cos(theta

_val).^4-.3379e18*cos(theta_val).^2-



val).*cos(theta_val).^2);U = (-Y(:,2)-Y(:,1)-K_bs*(Y(:,1)+Y(:,2))-

f2_val.*Kshat*RD.*log(sec(theta_val))./(ktheta)-

(f2_val.*Bshat*RD.*tan(theta_val)+f3_val).*Y(:,2))./f1_val;W = [f12.*U+f32.*Y(:,2), f21*RD.*log(sec(theta_val))./(ktheta),

f22*RD.*log(sec(theta_val))./(ktheta), f21*RD.*tan(theta_val).*Y(:,2),

f22*RD.*tan(theta_val).*Y(:,2)];figurep = plot(T,W(:,1),'-b',T,W(:,2),'--r',T,W(:,3),'-.y',T,W(:,4),':g',T,W(:,5),'-

.ok');grid onlegend(p,'W1','W2','W3','W4','W5',1);xlabel('time (s) \rightarrow','FontSize',12);ylabel('W \rightarrow','FontSize',12);title('W (Signal Matrix) vs t','FontSize',15);clear all;

function dy = mysystem(t,y)global Ksreal Bsreal ktheta RD ureal K_bsdy = zeros(8,1);dtdg = ktheta;u = y(3)+0.16;

theta_val = ktheta*y(1);f1_val =.7812e-2*cos(theta_val)*(.3528e16*u^3*cos(theta_val)+.1411e18*u^2*cos(theta_val)-

.4584e16*u^2+.8750e16*u*cos(theta_val)-.6374e17*cos(theta_val)-

.5222e16*u^2*sin(theta_val)+.2048e18*u*sin(theta_val))/(.5513e16*u^3*cos(theta_val)

^2+.2204e18*u^2*cos(theta_val)^2-.7162e16*u^2*cos(theta_val)-

.5978e17+.1367e17*cos(theta_val)^2*u-

.3981e17*cos(theta_val)^2+.2602e18*u*sin(theta_val)*cos(theta_val)-

.8160e16*u^2*sin(theta_val)*cos(theta_val));f2_val =-8.*cos(theta_val)*(.6356e15*u^2*cos(theta_val)-.1525e16*sin(theta_val)-

.1525e16*u*cos(theta_val)+.6356e15*u*sin(theta_val))/(.5513e16*u^3*cos(theta_val)^2

+.2204e18*u^2*cos(theta_val)^2-.7162e16*u^2*cos(theta_val)-

.5978e17+.1367e17*cos(theta_val)^2*u-


.8160e16*u^2*sin(theta_val)*cos(theta_val));f3_val

=.5978e17*dtdg*(sin(theta_val)+u*cos(theta_val))/cos(theta_val)/(.5513e16*u^3*cos(t

heta_val)^2+.2204e18*u^2*cos(theta_val)^2-.7162e16*u^2*cos(theta_val)-

.5978e17+.1367e17*cos(theta_val)^2*u-


.8160e16*u^2*sin(theta_val)*cos(theta_val));f11 =.1562e-1*cos(theta_val)*(.2293e9*cos(theta_val)+.4584e6-

.1275e9*sin(theta_val))/(.2497e9*cos(theta_val)^2+.4670e9+.1432e7*cos(theta_val)-

.3236e9*sin(theta_val)*cos(theta_val));f12 =-4.882*cos(theta_val)*(.3153e15*cos(theta_val)^3-.3155e15*cos(theta_val)-

.8563e13*cos(theta_val)^2+.7925e15*sin(theta_val)*cos(theta_val)^2+.8563e13+.6851e1

2*sin(theta_val)*cos(theta_val)-


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- 13 -

dy(2) =

f1_val*U+f2_val*(Ksreal*RD*log(sec(theta_val))/(ktheta)+Bsreal*RD*tan(theta_val)*y(

2))+f3_val*y(2);ahat = [y(3:7)];ahatdot = -P*W'*(W*ahat-dy(2)-f31*y(2)-f11*U);for i=1:1:5

dy(i+2) = ahatdot(i);endPdot = P*W'*W*P;k = 8;for i=1:1:5

for j=1:1:5dy(k) = Pdot(i,j);k = k+1;

endend

A.6 MATLAB code for Tracking Controller with parameter estimation andreference output tracking (Unknown Coefficient of Friction)function Tracking_Controller_u_unknownclc;clear all;close all;global Ksreal Bsreal ktheta RD ureal K_bsK_bs = 10;ktheta = 0.5;RD = 24.5;ureal = 0.14;Ksreal = 5;Bsreal = 0.1;u0 = 0.4;ubar = 0.16;Kshat0 = 4;Bshat0 = 0.01;a1hat0 = u0-ubar;a2hat0 = a1hat0*Kshat0;a3hat0 = a1hat0*Bshat0;P0 = [0.001:0.001:0.025];ref = @(T) sin(T);refdot = @(T) cos(T);y0 = [0.5 -0.5 a1hat0 Kshat0 a2hat0 Bshat0 a3hat0 P0]; %gamma gammadot a1hat Kshat

a2hat Bshat a3hat P11->P15, P21->P25, P31->P35, P41->P45, P51->P55[T Y] = ode45(@mysystem,[0,10],y0);

figure;

p = plot(T,ref(T)*180/pi,'-b',T,(Y(:,1)+ref(T))*180/pi,'--r');grid onlegend(p,'Yr','Y',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('Y (deg)\rightarrow','FontSize',12);title('Y vs t','FontSize',15);figurep = plot(T,refdot(T)*180/pi,'-b',T,(Y(:,2)+refdot(T))*180/pi,'--r');grid onlegend(p,'Yrdot','Ydot',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('dY/dt (deg/s) \rightarrow','FontSize',12);title('dY/dt vs t','FontSize',15);figure


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- 14 -

p =

plot(T,Ksreal*RD*log(sec(ktheta*(ref(T))))/ktheta+Bsreal*RD*tan(ktheta*(ref(T))).*r

efdot(T),...'-

b',T,Ksreal*RD*log(sec(ktheta*(Y(:,1)+ref(T))))/ktheta+Bsreal*RD*tan(ktheta*(Y(:,1)

+ref(T))).*(Y(:,2)+refdot(T)),'--r');grid onlegend(p,'Ffar','Ffa',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('Ffa (kg) \rightarrow','FontSize',12);title('Ffa vs t','FontSize',15);figurep = plot(T,Ksreal*ones(size(T)),'-b',T,Y(:,4),'--r');grid onlegend(p,'Ksreal','Kshat',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('Ks (kg/mm) \rightarrow','FontSize',12);title('Ks vs t','FontSize',15);

figurep = plot(T,Bsreal*ones(size(T)),'-b',T,Y(:,6),'--r');grid onlegend(p,'Bsreal','Bshat',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('Bs (kgs/m) \rightarrow','FontSize',12);title('Bs vs t','FontSize',15);u = Y(:,3)+ubar;figurep = plot(T,ureal*ones(size(T)),'-b',T,u,'--r');grid onlegend(p,'ureal','uhat',4);xlabel('time (s) \rightarrow','FontSize',12);ylabel('\mu \rightarrow','FontSize',12);title('\mu vs t','FontSize',15);Y(:,1)=Y(:,1)+ref(T);Y(:,2)=Y(:,2)+refdot(T);dtdg = ktheta;Kshat = Y(:,4);Bshat = Y(:,6);theta_val = ktheta*Y(:,1);f1_val =.7812e-

2*cos(theta_val).*(.3528e16*u.^3.*cos(theta_val)+.1411e18*u.^2.*cos(theta_val)-

.4584e16*u.^2+.8750e16*u.*cos(theta_val)-.6374e17*cos(theta_val)-

.5222e16*u.^2.*sin(theta_val)+.2048e18*u.*sin(theta_val))./(.5513e16*u.^3.*cos(thet

a_val).^2+.2204e18*u.^2.*cos(theta_val).^2-.7162e16*u.^2.*cos(theta_val)-

.5978e17+.1367e17*cos(theta_val).^2.*u-


.8160e16*u.^2.*sin(theta_val).*cos(theta_val));f2_val =-8.*cos(theta_val).*(.6356e15*u.^2.*cos(theta_val)-.1525e16*sin(theta_val)-

.1525e16*u.*cos(theta_val)+.6356e15*u.*sin(theta_val))./(.5513e16*u.^3.*cos(theta_v

al).^2+.2204e18*u.^2.*cos(theta_val).^2-.7162e16*u.^2.*cos(theta_val)-

.5978e17+.1367e17*cos(theta_val).^2.*u-


.8160e16*u.^2.*sin(theta_val).*cos(theta_val));f3_val

=.5978e17*dtdg*(sin(theta_val)+u.*cos(theta_val))./cos(theta_val)./(.5513e16*u.^3.*

cos(theta_val).^2+.2204e18*u.^2.*cos(theta_val).^2-.7162e16*u.^2.*cos(theta_val)-

.5978e17+.1367e17*cos(theta_val).^2.*u-


.8160e16*u.^2.*sin(theta_val).*cos(theta_val));


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- 15 -

f11 =.1562e-1*cos(theta_val).*(.2293e9*cos(theta_val)+.4584e6-

.1275e9*sin(theta_val))./(.2497e9*cos(theta_val).^2+.4670e9+.1432e7*cos(theta_val)-

.3236e9*sin(theta_val).*cos(theta_val));f12 =-4.882*cos(theta_val).*(.3153e15*cos(theta_val).^3-.3155e15*cos(theta_val)-

.8563e13*cos(theta_val).^2+.7925e15*sin(theta_val).*cos(theta_val).^2+.8563e13+.685

1e12*sin(theta_val).*cos(theta_val)-

.1186e16*sin(theta_val))./(.4239e17*cos(theta_val).^4-.3379e18*cos(theta_val).^2-



val).*cos(theta_val).^2);f21 =-

.3558e7*cos(theta_val).*(4.*cos(theta_val)+25.*sin(theta_val))./(.2497e9*cos(theta_

val).^2+.4670e9+.1432e7*cos(theta_val)-.3236e9*sin(theta_val).*cos(theta_val)); f22 =-500.*cos(theta_val).*(.3237e15*cos(theta_val).^3-

.4609e15*cos(theta_val)+.2731e12*cos(theta_val).^2-

.1016e15*sin(theta_val).*cos(theta_val).^2+.3711e14*sin(theta_val)+.3299e13*sin(the

ta_val).*cos(theta_val))./(.4239e17*cos(theta_val).^4-.3379e18*cos(theta_val).^2-


.1338e16*cos(theta_val)+.3023e18*sin(theta_val).*cos(theta_val)+.9271e15*sin(theta_val).*cos(theta_val).^2);f31 =-

.1868e8*dtdg*(4.*cos(theta_val)+25.*sin(theta_val))./cos(theta_val)./(.2497e9*cos(t

heta_val).^2+.4670e9+.1432e7*cos(theta_val)-

.3236e9*sin(theta_val).*cos(theta_val));f32 =-.4670e9*dtdg*(.1657e10*cos(theta_val).^2-.2479e10+.1432e7*cos(theta_val)-

.6595e9*sin(theta_val).*cos(theta_val)+.1790e8*sin(theta_val))./(.4239e17*cos(theta

_val).^4-.3379e18*cos(theta_val).^2-



val).*cos(theta_val).^2);U = (-Y(:,2)-Y(:,1)-K_bs*(Y(:,1)+Y(:,2))-

f2_val.*Kshat*RD.*log(sec(theta_val))./(ktheta)-

(f2_val.*Bshat*RD.*tan(theta_val)+f3_val).*Y(:,2))./f1_val;W = [f12.*U+f32.*Y(:,2), f21*RD.*log(sec(theta_val))./(ktheta),

f22*RD.*log(sec(theta_val))./(ktheta), f21*RD.*tan(theta_val).*Y(:,2),

f22*RD.*tan(theta_val).*Y(:,2)];figurep = plot(T,W(:,1),'-b',T,W(:,2),'--r',T,W(:,3),'-.y',T,W(:,4),':g',T,W(:,5),'-

.ok');grid onlegend(p,'W1','W2','W3','W4','W5',1);xlabel('time (s) \rightarrow','FontSize',12);ylabel('W \rightarrow','FontSize',12);title('W (Signal Matrix) vs t','FontSize',15);clear all;

function dy = mysystem(t,y)global Ksreal Bsreal ktheta RD ureal K_bsdy = zeros(8,1);dtdg = ktheta;u = y(3)+0.16;xr = sin(t);xrdot = cos(t);xrdot2 = -sin(t);dtdg = ktheta;theta_val = ktheta*(y(1)+sin(t));f1_val =.7812e-

2*cos(theta_val)*(.3528e16*u^3*cos(theta_val)+.1411e18*u^2*cos(theta_val)-


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- 16 -




.5978e17+.1367e17*cos(theta_val)^2*u-





.5978e17+.1367e17*cos(theta_val)^2*u-





.5978e17+.1367e17*cos(theta_val)^2*u-


.8160e16*u^2*sin(theta_val)*cos(theta_val));

f11 =.1562e-1*cos(theta_val)*(.2293e9*cos(theta_val)+.4584e6-.1275e9*sin(theta_val))/(.2497e9*cos(theta_val)^2+.4670e9+.1432e7*cos(theta_val)-

.3236e9*sin(theta_val)*cos(theta_val));f12 =-4.882*cos(theta_val)*(.3153e15*cos(theta_val)^3-.3155e15*cos(theta_val)-

.8563e13*cos(theta_val)^2+.7925e15*sin(theta_val)*cos(theta_val)^2+.8563e13+.6851e1

2*sin(theta_val)*cos(theta_val)-

.1186e16*sin(theta_val))/(.4239e17*cos(theta_val)^4-.3379e18*cos(theta_val)^2-

.7153e15*cos(theta_val)^3+.1616e18*cos(theta_val)^3*sin(theta_val)-.2181e18-

.1338e16*cos(theta_val)+.3023e18*sin(theta_val)*cos(theta_val)+.9271e15*sin(theta_v

al)*cos(theta_val)^2);f21 =-

.3558e7*cos(theta_val)*(4.*cos(theta_val)+25.*sin(theta_val))/(.2497e9*cos(theta_va

l)^2+.4670e9+.1432e7*cos(theta_val)-.3236e9*sin(theta_val)*cos(theta_val));f22 =-500.*cos(theta_val)*(.3237e15*cos(theta_val)^3-

.4609e15*cos(theta_val)+.2731e12*cos(theta_val)^2-

.1016e15*sin(theta_val)*cos(theta_val)^2+.3711e14*sin(theta_val)+.3299e13*sin(theta

_val)*cos(theta_val))/(.4239e17*cos(theta_val)^4-.3379e18*cos(theta_val)^2-


.1338e16*cos(theta_val)+.3023e18*sin(theta_val)*cos(theta_val)+.9271e15*sin(theta_v

al)*cos(theta_val)^2);f31 =-

.1868e8*dtdg*(4.*cos(theta_val)+25.*sin(theta_val))/cos(theta_val)/(.2497e9*cos(the

ta_val)^2+.4670e9+.1432e7*cos(theta_val)-.3236e9*sin(theta_val)*cos(theta_val)); f32 =-.4670e9*dtdg*(.1657e10*cos(theta_val)^2-.2479e10+.1432e7*cos(theta_val)-

.6595e9*sin(theta_val)*cos(theta_val)+.1790e8*sin(theta_val))/(.4239e17*cos(theta_v

al)^4-.3379e18*cos(theta_val)^2-


.1338e16*cos(theta_val)+.3023e18*sin(theta_val)*cos(theta_val)+.9271e15*sin(theta_val)*cos(theta_val)^2);Kshat = y(4);Bshat = y(6);U = (-y(2)-y(1)-K_bs*(y(1)+y(2))-f2_val*Kshat*RD*log(sec(theta_val))/(ktheta)-

(f2_val*Bshat*RD*tan(theta_val)+f3_val)*(y(2)+xrdot)-xrdot2)/f1_val;W = [f12*U+f32*(y(2)+xrdot), f21*RD*log(sec(theta_val))/(ktheta),

f22*RD*log(sec(theta_val))/(ktheta), f21*RD*tan(theta_val)*(y(2)+xrdot),

f22*RD*tan(theta_val)*(y(2)+xrdot)];P = [y(8) y(9) y(10) y(11) y(12);

y(13) y(14) y(15) y(16) y(17);y(18) y(19) y(20) y(21) y(22);y(23) y(24) y(25) y(26) y(27);y(28) y(29) y(30) y(31) y(32);];


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u = ureal;f1_val =.7812e-

2*cos(theta_val)*(.3528e16*u^3*cos(theta_val)+.1411e18*u^2*cos(theta_val)-




.5978e17+.1367e17*cos(theta_val)^2*u-





.5978e17+.1367e17*cos(theta_val)^2*u-





.5978e17+.1367e17*cos(theta_val)^2*u-


.8160e16*u^2*sin(theta_val)*cos(theta_val));dy(1) = y(2); %z(gamma-gammaref) zdot(gammadot-gammadotref) a1hat Kshat a2hat Bshat

a3hat P11->P15, P21->P25, P31->P35, P41->P45, P51->P55dy(2) = f1_val*u+f2_val*Ksreal*RD*log(sec(theta_val))/ktheta +

(f2_val*Bsreal*RD*tan(theta_val)+f3_val)*(y(2)+xrdot)-xrdot2;ahat = [y(3:7)];ahatdot = -P*W'*(W*ahat-(dy(2)+xrdot2)-f11-f31*(y(2)+xrdot));for i=1:1:5

dy(i+2) = ahatdot(i);endPdot = P*W'*W*P;k = 8;for i=1:1:5

for j=1:1:5dy(k) = Pdot(i,j);k = k+1;

endend

Gearshift Mechanism Nonlinear Torque Control

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