Gearshift Mechanism Controllability and Observability Study

8/14/2019 Gearshift Mechanism Controllability and Observability Study

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Acknowledgement

I am highly thankful to Professor John L Crassidis for his support during the execution of

this work. The fruitful discussions I had with him often after the class session provided a

lot of useful information for the completion of this work.

Finally, I am also thankful to my colleagues in the course of Systems Analysis 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

1. Introduction

1.1 Mechanism description

1.2 Construction1.3 Working

1.4 Motivation

1.5 Expected Benefits of Study

2. Technical Plan

2.1 Mathematical Modeling

2.1.1 Fork Cantilever Beam Analogy

2.1.2 Drum Dynamics Equations

2.1.3 Fork Free Body Diagram

2.2 Governing Differential Equation and State-Space Representation

2.2.1 Governing Equation

2.2.2 System Analogy

2.2.3 Shift Force Behavior

2.3 Elimination of Dependent Design Variables

2.4 Measure of Controllability and Observability

2.5 Output Controllability

3. Results

3.1 Simulation Inputs3.2 System Response

3.3 Degree of Controllability and Observability Contour Plots


3.5 Condition Number improvement

4. Conclusion

5. Future Work


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Bibliography

Appendix

A.1 MATLAB code for condition number calculation of controllability and

observability graminas for drum ramp angle () and drum radius (RD)


observability graminas for angle () and angle ()


observability graminas for mean sleeve diameter (Ds) and distance Ac


observability graminas for length (O1) and length (O2)




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1. Introduction

Faster and efficient gear shifts have always been the demand of automobile industry [1]. Be it anautomatic transmission having distinctly low efficiency or an automated manual transmission withcombined benefits of automatic and manual gearboxes, the phenomenon of gear shift has always been

fascinating researchers [2]. 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 controllability of such gear shiftmechanisms. Both the issues i.e. reducing shifting time and improving efficiency of shifting can be

addressed by such a study. The study will also provide a rationale as to how parameters can be

calculated for designing more controllable mechanisms.

1.1 Mechanism descriptionThe mechanism is a cylindrical cam follower system. The basic purpose of the mechanism is to convertgiven torque input provided to it at the cylindrical cam into axial force on the fork for gearshift. The

mechanism is used universally in two-wheelers and occasionally in four-wheelers for gearshift.

(a) (b)Figure 1-1 An illustration of the Sequential type Gear Shift Mechanism (a) the manual version of the

mechanism, (b) the automated version of the mechanism

1.2 ConstructionThe mechanism shown in Figure 1 (a) consists of the following major components:

Drum (Cylindrical Cam) Forks Fork Rail


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Star Wheel 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 drumwith the help of a bolt and the drum is free to rotate about its axis. The drum is provided with grooves on its

circumference. Furthermore, the fork is provided with a lug that protrudes from its head. The lug on the forkengages in the groove on the drum. The Fork legs rests on the gear collar (not shown in figure) and the fork

is constrained to move axially on the fork rail. In addition to this, the star wheel is provided with projectedlugs on its front face on which the shift lever-2 rests. Shift lever-2 is connected to shift lever-1 through a

revolute 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 automatedversion 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.

1.3 WorkingThe shift lever-1 is rotated with the help of a push-pull type cable. As shift lever-1 rotates, it rotates shiftlever-2 which in turn rotates the star wheel. Since, the star wheel is rigidly connected with the drum, the

drum is also indexed by the same angle. The configuration of the two levers and star wheel is such that thesystem is always indexed by a fixed angle for one stroke of the shift levers. Due to this fixed rotation of the

drum 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 mechanismduring a gearshift under load. The magnitude and pattern of required drum torque governs the gearshift feel

in the manual version. However, the main concern in the automated version is minimization of the peaktorque due to actuator size limitations.

1.4 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 astudy of the controllability of the mechanism to make the system more amenable to control from design point

of view.

1.5 Expected Benefits of StudyThe benefits of the study are that once a complete picture of the controllability of the mechanism is

understood, the influence of design parameters on controllability can be evaluated. This will help indeveloping more controllable mechanism thus, realizing the objective of developing controllers with

faster response.


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2. Technical Plan

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 finallya differential equation governing the dynamics of the system is obtained. A discussion on the measure of

controllability of the system is then carried out.

2.1 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 itsangular 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 ensuresthat when 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 evenincrease the fork required to shift it during a gearshift. This assumption neglects the influence of

variation of tolerances involved during manufacturing of components.

2.The fork lug is always in contact with the drum groove wall. This assumption signifies that the modeldoes not take into account any nonlinearity in terms of contact between the fork and the drum. So,any unique 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 legs

will 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-casted aluminum component.

Figure 2-1 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 angularacceleration of which in turn accelerates the fork by . The gearshift sleeve also offers a force of Ffa1

and Ffa2on the two fork legs which amounts to the total force acting on the fork axially.


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Figure 2-1 Simplified representation of the mechanism with external forces & moments acting on it

2.1.1 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 2-2(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 is

inversely 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 which

undergo 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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(b) (c)

Figure 2-2 Gearshift fork legs represented as two cantilever beams. a) Formula for maximumdeflection of a cantilever beam, b) Forces acting on the two legs of gearshift fork, and c) Simplified

representation of fork legs as two cantilever beams

Considering the aforementioned analogy the force Ffa1can be deduced to be as follows:

( )

( ) ( )

32 2 2

2 2

1 3 32 2 2 22 2

1 1 2 2

fa

fa

F A EF

A E A E

+=

+ + +

The dimensions A1,A2, E1 and E2 can be referred in the Figures 2-5 and 2-6.

2.1.2 Drum Dynamics Equations

Figure 2-3 Developed View of Drum Profile & Force Components acting on it


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Figure 2-3 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,fl represents the friction force developed between the fork lug and the drum and is given by

Further in Figure 2-3 it can be seen in the zoomed view of the fork lug traversing the drum groove that

the drum groove ramp angle is related to the angular rotation of the drum and the axial movement of thefork as given below

(A)

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

2.1.3 Fork Free Body Diagram

Figure 2-4 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 normalreactions 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 avoids

development of substantial moment in one direction due to which the two reactions Nx1 and Nx2 from the


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rail are acting in the same direction. The reactions Nx1, Nx2, Ny1 and Ny2 thus lead to the friction forcesfx1,fx2,fy1 andfy2 respectively. The gearshift sleeve also applies force Ffton the fork leg as the force FT

Figure 2-4 Free-Body Diagram of Fork

also tries to rotate the fork about the rail center. Due to the force Fft a corresponding friction force also

acts 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


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A coordinate system XFYFZF as represented in Figure 2-4 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.

2.1.3.1 Equilibrium in XFYF Plane

Figure 2-5 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


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Since, the fork does not experience any acceleration (linear or angular) in these directions the net forces

and moments are equated to zero in the above equations.

2.1.3.2 Equilibrium in XFZF Plane

Figure 2-6 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:

2.1.3.3 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.


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Thus, the moment equation is given by

Figure 2-7 Free-Body Diagram of Fork representing forces in YFZF plane

2.2 Governing Differential Equation and State-Space Representation

Now, 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.

2.2.1 Governing Equation

All the above mentioned equations i.e. Equation (1) (13) & (A) (F) are solved using Symbolic

Toolbox in MATLAB to obtain the following expression:


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(14)

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

where,

ki sare constants which depend on various mechanism parameters and which depends on

the drum groove profile.

2.2.2 System Analogy

Figure 2-8 Gearshift Mechanism analogy with a mass spring-damper system

It can be easily seen from the system equations that the gearshift mechanism dynamics is analogous to a

mass spring-damper system having time varying forces acting on the mass in opposite directions and adamping coefficient that varies as a function of the displacement of the mass.


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2.2.3 Shift Force Behavior

A precise model for the shift force behavior will involve mathematical modeling of the entire drive-train

along with characteristics of the clutch. This in itself is another area of research. So, in order to avoidcomplexities the shift force is assumed to behave as a combination of a spring with Ks as the spring

constant and a damper system with Bs as the damping coefficient as shown in Figure 2-9. However, the

value of Ks and Bs is assumed to be unknown. The axial force from the fork Ffa acts on the sleeve acomponent of which compresses the spring and another component is dissipated due to the damping

behavior of the shift force.

Figure 2-9 Shift force behavior representation as a spring-damper system

So, mathematically the axial force can be represented as follows:

Substituting the expression for the axial force in the final equation (14), we get the following equation

(15)

Now, the state-space representation of the system is as follows:

Now, in order simplify the system to convert it into a LTI system the drum ramp-angle is assumed to be

constant for the system. The above system can then be reduced to the following state-spacerepresentation:

( )2 1 2 1 2 3 2tan tan.

s D s Dx f u f K R x f B R f x = + + +

11

2 2 3 2 12

0 1 0

tan tan

.

s D s D

xx uf K R f B R f x f x

= + +


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[ ] [ ]1

2

tan tan 0s D s Dx

y K R B R ux

= +

Please note the value of the functionsf1.f2 andf3 is not mentioned here explicitly in terms of the designparameter due the sheer size of the results. All the computation that has been performed is done after

solving the equations symbolically in MATLAB and then substituting the desired values for the

parameters.

2.3 Elimination of Dependent Design Variables

Considering the number of design variables involved it is advisable to eliminate the dependent variables soas to effectively deal with the problem. However, the above mathematical model is developed without anysuch optimization so as to establish a more generic approach for solving the problem in case any of the

following mentioned relationships ceases to exist.

It was observed that the design variables A1, A2, E1 & E2 are related and that by introducing two morevariables the total number of design variables can be optimized.

Figure 2-10 Design variables defining the fork geometry

S NO. DESIGN VARIABLES DESIGNATION

1 Sleeve Mean Diameter (excluding chamfer & fillet) Ds

2 Distance of Fork Leg Contact Point Circle Center from Rail Center Ac

Table 2-1 Geometric design variables added in the mathematical model


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The value of, Ac and Ds can be substituted in the following equations to obtain the values of A1, A2, E1 andE2.

The following design variables are then eliminated:


1 Arm Length of Fork Leg Force Opposite to Lug Side A1

2 Offset Distance of Fork Leg opposite to Lug Side from Shifting Sleeve Center E1

3 Offset Distance of Fork Leg on Lug Side from Shifting Sleeve Center E2

4 Arm Length of Fork Leg Force on Lug Side A2

Table 2-2 Geometric design variables eliminated from the mathematical model

The following are the constraints for the design variables:

The fork leg offset from fork lug center should always lie betweenB1 andB2.

The fork leg length A2 should satisfy the following constraint:

Table 2-3 Final geometric design variables in the mathematical model

This study focuses on ten variables out of the variables mentioned in Table 2-3. The fork lug diameter


1

Angle between 'Perpendicular to Radial Reaction from Shifting Sleeve' and

Line joining Fork Leg Center to Rail Center'

2 Fork Support Base Length from Fork Leg to opposite to Lug Side end B1

3 Fork Support Base Length from Fork Leg to Lug Side end B2

4 Distance of Lug from Rail Center Line C

5 Fork Lug Diameter dl

6 Fork Leg offset from Lug Center O3

7 Fork Lug Angle from the YF-axis as viewed in XY Plane.

8 Fork Rail Diameter d

9 Groove Ramp Angle

10 Drum Radius RD

11 Sleeve Mean Diameter (excluding chamfer & fillet) Ds

12 Distance of Sleeve Center from Rail Center Ac


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and fork rail diameter are not considered since these parameters have little scope for variation.

2.4 Measure ofControllability and Observability

The rank test of controllability matrix only provides binary (yes/no) information and does not provide a

graduated measure of how controllable a system is. A quantification of controllability can provide

inputs to design components so as to make the mechanism more controllable.

The condition number can be defined, using the L2 norm, as the ratio of the maximum singular value of

a matrix to the minimum singular value. The application of this to the controllability matrix relates to the

standard definition of on-off controllability, namely that if the rank of C is equal to the order of thesystem, then the system is controllable. Consider a diagonalized 3x3 controllability matrix. If the

diagonal entries are all equal to one, we can see that this system passes the controllability test. This

matrix would have a condition number equal to one, as the maximum and minimum singular values are

both one. If the first entry is increased while decreasing the last entry, it can be seen that controllabilityis being lost until a certain threshold is reached when the maximum entry is much larger than the

smallest entry. At this point, control over the state corresponding to that smallest entry is lost. A matrix

with this property would have a very large condition number, because the largest singular value is muchgreater than the smallest singular value. Using the condition number on the controllability matrix is

therefore a measure of controllability that gives information as to partial, or relative, control over a

system. This is advantageous compared to the on-off measure given by the rank test of thecontrollability matrix.[3]

Although in theory the rank conditions of the controllability and observability matrices are easy tocheck, these can be poorly conditioned computational operations to perform. A better way to determine

these properties is via the controllability and observability gramians. [4] However, the system should bestable for calculating gramians.

Another important consideration is that if reciprocal of the condition number of the gramian is

considered it would convert the measure to a value that lies between 0 and 1, with 0 being the poorly

controllable and 1 being highly controllable. This would help in directly accessing the improvementachieved by varying a certain parameter.

The system developed in the previous section will be studied for the observability and controllability.The controllability for the system under consideration can be defined as the ability to achieve the desired

drum angular velocity and position by providing a torque input to the drum within a finite time. The

observability can be defined as the ability to determine uniquely the drum angular velocity and positionfor a given axial force on the fork. The observability and controllability gramians are calculated. The

reciprocal of condition number of the gramians is then evaluated and plotted as a function of the

parameters.


Output controllability is the related notion for the output of the system; the output controllability

describes the ability of an external input to move the output from any initial condition to any final

condition in a finite time interval. A controllable system is not necessarily output controllable, and anoutput controllable system is not necessarily controllable.


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For a linear continuous-time system, like the example above, described by matrices A, B, C, and D,

the output controllability matrix

must have full row rank (i.e. rank m) if and only if the system is output controllable. [5]

The output controllability in this context is defined as the ability of the torque applied at the drum to

achieve the desired axial force.


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3. Results

This chapter presents the results for the variation of condition number of the controllability andobservability gramians with the variation in mechanism parameters.

3.1 Simulation InputsThe following inputs are used to calculate the controllability and observability gramians. The following

inputs are for an existing mechanism.

Constants

= 0.16

Independent Design Variables

= 42*pi/180

RD= 24.5;

= 31.61*pi/180;= 62.08*pi/180;

Ds = 79.63;Ac = 66.22;

O1 = 85.44;

O2 = 16.55;O3 = 67.8;

C = 14.95;

Ks = 5;Bs = 1;

dl = 8.3;

d = 13;ID = 200;

mf= 0.2;

3.2 System Response

0 1 2 3 4 5 6 7 8 9 10-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06Impulse Response

Time (sec)

Amplitude

0 1 2 3 4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

0.025

0.03Step Response

Time (sec)

Amplitude

Figure 3-1 System response for step and impulse input


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The system response is plotted for the step and impulse input which shows that the system is stable forthe above mentioned value of variables. Also the same can be proved by calculating the eigen values of

the A matrix which lies in the Left Half Plane. Hence, the gramians for the system can be calculated.

3.3 Degree of Controllability and Observability Contour PlotsThis section deals with the analysis of the effect of parametric variation on degree of controllability and

observability of the system.

Drum Ramp Angle () Analysis

The contour plot in Figure 3-1 depicts that the reciprocal of condition number of the controllabilitygramian increases with increase in drum ramp angle (). This implies that the degree of controllability of

the system reduces with increasing drum ramp angle. Also, the scale in the figure shows that the number

varies considerably with the variation of the parameter. The decrease in controllability of the states with

increase in drum ramp angle can be attributed to the fact that increase in ramp angle leads to an increasein the reaction from the fork lug on the drum, for the same axial force developed on the fork, due to

which the drum angular acceleration decreases. This reduction in acceleration also leads to a decrease in

angular velocity and hence, angular displacement than was possible with lower ramp angle. Thus, theease with which any given angular velocity and displacement of the drum can be achieved decreases

with increase and ramp angle. Hence, the controllability of the system reduces with increase in drum

ramp angle.

The second plot in Figure 3-1 shows that the degree of observability decreases with increase in drum

radius (RD). The decrease in observability of the states with increase in drum ramp angle can be

Drum Ramp Angle (deg)

DrumR

adius(mm)

Degree of Controllability

25 30 35 40 45 50 55 6020

25

30

35

40

45

50

0.05

0.1

0.15

0.2

0.25

0.3

Drum Ramp Angle (deg)

DrumR

adius(mm)

Degree of Observability

25 30 35 40 45 50 55 6020

25

30

35

40

45

50

0.05

0.1

0.15

0.2

0.25

Figure 3-2 Contour plots of the reciprocal of condition number of controllability and observability

gramians as a function of drum radius and drum ramp angle.

interpreted mathematically. The increase in the drum ramp angle leads to the increase in the value of C

matrix, which can be seen from the following equation.


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[ ] [ ]1

2

tan tan 0s D s Dx

y K R B R ux

= +

The problem of observability is similar to determining the value ofx1 andx2 for a given value ofy. Thisincrease in value of C matrix makes the controllability gramian more biased in terms of singular values

i.e. the maximum singular value increases and the minimum singular value decreases thus, leading to

increase in condition number or decrease in degree of observability and making system less observable.

Drum Radius (RD)AnalysisFigure 3-2 also shows the variation of degree of controllability as a function of drum radius. The

increase in drum radius leads to a decrease in the degree of controllability of the system. The scaleshows that the effect of drum radius on controllability is also considerable though less than that of the

drum ramp angle. Hence, drum radius also becomes a critical parameter in deciding controllability of

the system.

The observability of the system also shows a similar trend though the improvement in the degree is a

little lesser in this case. However, the improvement on a practical system in controllability andobservability cannot be compared as these two represent totally different quantities.

Angle (deg)

Angle(deg

)


15 20 25 30 35 40 45 50

45

50

55

60

65

70

75

80

0.0765

0.077

0.0775

0.078

0.0785

0.079

0.0795

0.08

Angle (deg)

Angle(deg

)


15 20 25 30 35 40 45 50

45

50

55

60

65

70

75

80

0.0705

0.071

0.0715

0.072

0.0725

0.073

0.0735

Figure 3-3Contour plots of the reciprocal of condition number of controllability and observabilitygramians as a function of fork lug angle and fork leg angle .

Fork Lug Angle ()AnalysisThe variation of degree of controllability with the fork lug angle is shown in Figure 3-3. The plot showsan interesting trend this time. It seems that the controllability is poor around a specific value of the angle

i.e. around 65 to 70 degrees. As one moves away from this angle the controllability improves. However,

it would be difficult to provide a practical explanation as to why this happens.

The degree of observability also varies in a similar fashion with variation in fork lug angle. However,

the rate at which the degree improves as one moves away from the poor observability region is slower as


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compared to the rate of improvement shown by controllability. This again cannot be compared inabsolute terms.

An improvement fact to be noted here is that the improvement achievable both in controllability andobservability of the system is not much by varying this parameter.

Fork Leg Angle ()AnalysisHowever, the plot in Figure 3-3 shows that the controllability improves with the increase in fork leg

angle. This means that an asymmetric fork will lead to a better controllability. This however, cannot be

questioned as it is the complex interaction of forces and reactions induced in the system that is leading to

determination of controllability and so an asymmetric configuration might improve controllability.

The observability shows a very similar trend with improving degree against increase in angle.

Mean Sleeve Diameter (mm)

DistanceAc(mm)


60 65 70 75 80 85 90 95

50

55

60

65

70

75

80

85

0.066

0.068

0.07

0.072

0.074

0.076

0.078

Mean Sleeve Diameter (mm)

DistanceAc(mm)


60 65 70 75 80 85 90 95

50

55

60

65

70

75

80

85

0.06

0.062

0.064

0.066

0.068

0.07

0.072

Figure 3-4Contour plots of the reciprocal of condition number of controllability and observabilitygramians as a function of distance Ac and mean sleeve diameter.

Distance AcAnalysisThe plot in Figure 3-3 shows that the controllability increases with increase in value of Ac. This means

the farther the shifting sleeve is from the fork rail the more controllable the system will be. The plot for

observability also shows a similar trend. However, the improvement is not significant.

Mean Sleeve Diameter (Ds) Analysis

However, the mean sleeve diameter shows opposite trend. The controllability decreases with increase in

the diameter. Also, the effect is much more prominent as compared to the one with decrease in distanceAc. The observability follows a similar trend.

Length O2 Analysis

The controllability of the system increases with increase in length O2 as can be seen in Figure 3-5. For

the considered mechanism the fork lug is highly biased on one side due to which the increase in O2 lead

to a symmetrically placed fork lug. This implies that controllability increases with symmetrically located


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Length O1 (mm)

LengthO2(mm)


70 75 80 85 90 95 100 105

5

10

15

20

25

30

0.075

0.0755

0.076

0.0765

0.077

0.0775

0.078

0.0785

0.079

0.0795

0.08

Length O1 (mm)

LengthO2(mm)


70 75 80 85 90 95 100 105

5

10

15

20

25

30

0.069

0.0695

0.07

0.0705

0.071

0.0715

0.072

0.0725

0.073

0.0735

Figure 3-5Contour plots of the reciprocal of condition number of controllability and observabilitygramians as a function of length O1 and O2.

fork lug. The same it true for the observability of the system.

Length O1 Analysis

The increase in length O1 leads to an asymmetric fork lug due to which the controllability decreases with

increase in length O1. The same holds good for the observability.

Length O3 (mm)

DistanceC

(mm)


50 55 60 65 70 75 80 8510

12

14

16

18

20

22

24

0.07

0.071

0.072

0.073

0.074

0.075

0.076

0.077

0.078

0.079

0.08

Length O3 (mm)

DistanceC

(mm)


50 55 60 65 70 75 80 8510

12

14

16

18

20

22

24

0.065

0.066

0.067

0.068

0.069

0.07

0.071

0.072

0.073

Figure 3-6Contour plots of the reciprocal of condition number of controllability and observability

gramians as a function of distance C and length O3.

Length O3 Analysis

The fork lug offset from the fork leg i.e. O3 does not affect the controllability of the system much as is

obvious from Figure 3-6. Still, the controllability reduces with increase in its value. The same is valid

for its influence on observability.


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22

Distance C Analysis

The distance C cannot be varied much as is limited by the length of the fork lug which cannot be

increased beyond a certain value due to its small cross-section and the load being transferred through it.

For the given variation the controllability decreases with increase in its value. Observability follows thesame trend.


For the input values mentioned previous in the chapter the following A, B, C and D matrices are

obtained.

[ ]

0 1.0000=

-12.9094 -2.5819

0

0.0022

110.2995 22.0599

[0]

A

B

C

D

=

=

=

The output controllability matrix is then given by

[ ]CB CAB D [0.0493 0.1192 0]=

From the above equation it follows that the above matrix always has row rank as 1 until the first to termsreduce to zero which for the given parametric variation is never true. Also, the condition number for the

matrix is k = 1. Hence, the system is always output controllable. This seems very plausible practically asthe torque applied at the drum can always be used to control the force developed at the fork in the axial

direction.

3.5 Condition Number improvement

The value of the drum ramp angle and drum radius is reduced to the following: = 22*pi/180

RD= 20.5;

The degree of controllability improves from 0.0775 to a value of 0.3634. Also the degree of

observability of the system improves from 0.0712 to 0.3229. Thus, the study has a good potential to

improve the controllability and observability of the system.


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23

4. Conclusion

In general, controllability and observability show similar trends with variation in parameters. This

means design for improved controllability will also lead to design for improved observability. This in

fact is good for designing a controller for the mechanism since a controller almost always work intandem with an observer.

The following conclusions can be made about choosing the design variables for sequential type gearshiftmechanism for improved controllability and observability:

1. Drum ramp angle and Drum radius should be kept as low as possible2. Fork lug angle should be kept as low as possible.3. Fork leg angle should be kept as high as possible.4. Distance Ac should be as large as possible.5. The mean sleeve diameter should be as small as possible.6. The length O1 should be kept as small as possible.7. The length O2 should be kept as large as possible.8. The length O3 and distance C should be kept as small as possible.Also, it can be concluded that the following parameters have a major role in determining the

controllability and observability of the system and should be given attention during the design stage.1. Drum ramp angle2. Drum radiusPlease note that the parameters are listed in the order of their decreasing influence on controllability andobservability for the given range of parameter variation.

Furthermore, the system is found to be always output controllable.

The above conclusions can give rise to an altogether new area of design i.e. design for controllability

and design for observability. Such an awareness of the controllability and observability of the systemduring the design stage itself can lead to much improved controller performance in later stages.


29/40


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31/40

- 1 -

Appendix

Please note that dynamics equations are solved for each parametric variation due the inability ofMATLAB (unable to reshape) to handle bulky symbolic calculations iteratively.



clc;clear all;close all;

syms R1FAFTFft_fNx1Nx2Ny1Ny2FfaTDthetaRDuIDmfzeta_angCA2A1B1B2

E1E2O1O2O3ddlphigammadotgammadot2K1K2dthetadgammaDsi;

%Constantsu_val = 0.16;

% Independent Design Variablestheta_val = 42*pi/180;RD_val = 24.5;zeta_ang_val = 31.61*pi/180;phi_val = 62.08*pi/180;Ds_val = 79.63;Ac_val = 66.22;O1_val = 85.44;O2_val = 16.55;O3_val = 67.8;C_val = 14.95;Ks_val = 5;Bs_val = 1;dl_val = 8.3;d_val = 13;ID_val = 200;mf_val = 0.2;

%Dependent Design VariablesA1_val = Ac_val+Ds_val*sin(zeta_ang_val)/2; % A1_val = 87.09;A2_val = Ac_val-Ds_val*sin(zeta_ang_val)/2; % A2_val = 45.35;E1_val = Ds_val*cos(zeta_ang_val)/2; % E1_val = 33.91;

E2_val = E1_val; % E2_val = 33.91;B1_val = O1_val - O3_val; %B1_val = 17.64;B2_val = O2_val + O3_val; %B2_val = 84.35;Dsi_val = Ds_val-5;

%System Statesgammadot_val = 0;

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)


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- 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.gammadot2;gammadot2_val = vpa(simplify(subs(simplify(gammadot2_val),{gammadot},{0})));

%Degree of Controllability & Observability variation with Drum Radius and%Drum Ramp Anglefun_sim = vpa(simplify(subs(simplify(gammadot2_val),{dl,u, zeta_ang, C,A2, A1, B1,

B2, E1, E2, O1, O2, O3, d, phi, ID, mf, Dsi} ,{dl_val, u_val, zeta_ang_val, C_val,

A2_val, A1_val, B1_val, B2_val, E1_val, E2_val, O1_val, O2_val, O3_val, d_val,

phi_val, ID_val, mf_val, Dsi_val})));f1 = vpa(simplify(subs(fun_sim,{TD, Ffa} ,{1,0})),4);f2 = vpa(simplify(subs(fun_sim,{TD, Ffa} ,{0,1})),4);i=0;X = (42-20)*pi/180:pi/180:(42+20)*pi/180;Y = 20:1:50;for RD_val = Y

i= i+1;j=0;for theta_val = X

j = j+1;f1_val = double(subs(f1,{RD,theta},{RD_val,theta_val}));f2_val = double(subs(f2,{RD,theta},{RD_val,theta_val}));A = [0 1; f2_val*Ks_val*RD_val*tan(theta_val)

f2_val*Bs_val*RD_val*tan(theta_val)];B = [0; f1_val];C = [Ks_val*RD_val*tan(theta_val) Bs_val*RD_val*tan(theta_val)];D = [0];sys = ss(A,B,C,D);CG = gram(sys,'c'); % Computing Controllability GramianOG = gram(sys,'o'); % Computing Observability GramianCC(i,j) = 1/cond(CG); %Condition Number Controllability GramianCO(i,j) = 1/cond(OG); %Condition Number Observability Gramian

endendfigurecontourf(X*180/pi,Y,CC);colormap(gray);xlabel('Drum Ramp Angle (deg) \rightarrow','FontSize',12);ylabel('Drum Radius (mm) \rightarrow','FontSize',12);title('Degree of Controllability','FontSize',15);figure;contourf(X*180/pi,Y,CO);colormap(gray);xlabel('Drum Ramp Angle (deg) \rightarrow','FontSize',12);ylabel('Drum Radius (mm) \rightarrow','FontSize',12);


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- 3 -

title('Degree of Observability','FontSize',15);


observability graminas for angle () and angle ()clc;clear all;close all;syms R1FAFTFft_fNx1Nx2Ny1Ny2FfaTDthetaRDuIDmfzeta_angCA2A1B1B2

E1E2O1O2O3ddlphigammadotgammadot2K1K2dthetadgammaDsi;%Constantsu_val = 0.16;% Independent Design Variablestheta_val = 42*pi/180;RD_val = 24.5;zeta_ang_val = 31.61*pi/180;phi_val = 62.08*pi/180;

Ds_val = 79.63;Ac_val = 66.22;O1_val = 85.44;O2_val = 16.55;O3_val = 67.8;C_val = 14.95;Ks_val = 5;Bs_val = 1;dl_val = 8.3;d_val = 13;ID_val = 200;mf_val = 0.2;%Dependent Design Variables

A1_val = Ac_val+Ds_val*sin(zeta_ang_val)/2; % A1_val = 87.09;A2_val = Ac_val-Ds_val*sin(zeta_ang_val)/2; % A2_val = 45.35;E1_val = Ds_val*cos(zeta_ang_val)/2; % E1_val = 33.91;E2_val = E1_val; % E2_val = 33.91;B1_val = O1_val - O3_val; %B1_val = 17.64;B2_val = O2_val + O3_val; %B2_val = 84.35;Dsi_val = Ds_val-5;%System Statesgammadot_val = 0;Eqn = 'FA -R1*(cos(theta)-u*sin(theta)) , FT -




+(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-




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.gammadot2;gammadot2_val = vpa(simplify(subs(simplify(gammadot2_val),{gammadot},{0})));%Degree of Controllability & Observability variation with Angle Zeta and


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- 4 -

%Phiclear fun_simf1f2f1_valf2_valCGOGCCCOfun_sim = vpa(simplify(subs(simplify(gammadot2_val),{dl,u, RD, C,A2, A1, B1, B2,

E1, E2, O1, O2, O3, d, theta, ID, mf, Dsi} ,{dl_val, u_val, RD_val, C_val, A2_val,A1_val, B1_val, B2_val, E1_val, E2_val, O1_val, O2_val, O3_val, d_val, theta_val,

ID_val, mf_val, Dsi_val})));f1 = vpa(simplify(subs(fun_sim,{TD, Ffa} ,{1,0})),4);f2 = vpa(simplify(subs(fun_sim,{TD, Ffa} ,{0,1})),4);i=0;X = zeta_ang_val-20*pi/180:pi/180:zeta_ang_val+20*pi/180;Y = phi_val-20*pi/180:pi/180:phi_val+20*pi/180;for Y_val = Y

i= i+1;j=0;for X_val = X

j = j+1;f1_val = double(subs(f1,{zeta_ang,phi},{X_val,Y_val}));

f2_val = double(subs(f2,{zeta_ang,phi},{X_val,Y_val}));A = [0 1; f2_val*Ks_val*RD_val*tan(theta_val)

f2_val*Bs_val*RD_val*tan(theta_val)];B = [0; f1_val];C = [Ks_val*RD_val*tan(theta_val) Bs_val*RD_val*tan(theta_val)];D = [0];sys = ss(A,B,C,D);CG = gram(sys,'c'); % Computing Controllability GramianOG = gram(sys,'o'); % Computing Observability GramianCC(i,j) = 1/cond(CG); %Condition Number Controllability GramianCO(i,j) = 1/cond(OG); %Condition Number Observability Gramian

endend

figure;contourf(X*180/pi,Y*180/pi,CC);colormap(gray);xlabel('\zeta Angle (deg) \rightarrow','FontSize',12);ylabel('\phi Angle (deg) \rightarrow','FontSize',12);title('Degree of Controllability','FontSize',15);figure;contourf(X*180/pi,Y*180/pi,CO);colormap(gray);xlabel('\zeta Angle (deg) \rightarrow','FontSize',12);ylabel('\phi Angle (deg) \rightarrow','FontSize',12);title('Degree of Observability','FontSize',15);

A.3 MATLAB code for condition number calculation of controllability andobservability graminas for mean sleeve diameter (Ds) and distance Acclc;clear all;close all;syms R1FAFTFft_fNx1Nx2Ny1Ny2FfaTDthetaRDuIDmfzeta_angCA2A1B1B2

E1E2O1O2O3ddlphigammadotgammadot2K1K2dthetadgammaDsi;%Constantsu_val = 0.16;% Independent Design Variablestheta_val = 42*pi/180;RD_val = 24.5;


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- 5 -

zeta_ang_val = 31.61*pi/180;phi_val = 62.08*pi/180;Ds_val = 79.63;

Ac_val = 66.22;O1_val = 85.44;O2_val = 16.55;O3_val = 67.8;C_val = 14.95;Ks_val = 5;Bs_val = 1;dl_val = 8.3;d_val = 13;ID_val = 200;mf_val = 0.2;%Dependent Design VariablesA1_val = Ac_val+Ds_val*sin(zeta_ang_val)/2; % A1_val = 87.09;A2_val = Ac_val-Ds_val*sin(zeta_ang_val)/2; % A2_val = 45.35;

E1_val = Ds_val*cos(zeta_ang_val)/2; % E1_val = 33.91;E2_val = E1_val; % E2_val = 33.91;B1_val = O1_val - O3_val; %B1_val = 17.64;B2_val = O2_val + O3_val; %B2_val = 84.35;Dsi_val = Ds_val-5;% System Statesgammadot_val = 0;Eqn = 'FA -R1*(cos(theta)-u*sin(theta)) , FT -




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


+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-




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.gammadot2;gammadot2_val = vpa(simplify(subs(simplify(gammadot2_val),{gammadot},{0})));%Degree of Controllability & Observability variation with Ds and Ac%Calculating Condition Numberfun_sim = vpa(simplify(subs(simplify(gammadot2_val),{zeta_ang,phi,dl,u, RD, C,B1,

B2, O1, O2, O3, d, theta, ID, mf} ,{zeta_ang_val,phi_val,dl_val, u_val, RD_val,C_val, B1_val, B2_val, O1_val, O2_val, O3_val, d_val, theta_val, ID_val,

mf_val})));f1 = vpa(simplify(subs(fun_sim,{TD, Ffa} ,{1,0})),4);f2 = vpa(simplify(subs(fun_sim,{TD, Ffa} ,{0,1})),4);i=0;X = Ds_val-20:1:Ds_val+20;Y = Ac_val-20:1:Ac_val+20;for Ac_val = Y

i= i+1;j=0;for Ds_val = X

j = j+1;


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- 6 -

%Dependent Design VariablesA1_val = Ac_val+Ds_val*sin(zeta_ang_val)/2;

A2_val = Ac_val-Ds_val*sin(zeta_ang_val)/2;

E1_val = Ds_val*cos(zeta_ang_val)/2;E2_val = E1_val;

Dsi_val = Ds_val-5;f1_val = double(subs(f1,{A2, A1, E1, E2, Dsi},{A2_val, A1_val, E1_val,

E2_val, Dsi_val}));f2_val = double(subs(f2,{A2, A1, E1, E2, Dsi},{A2_val, A1_val, E1_val,

E2_val, Dsi_val}));A = [0 1; f2_val*Ks_val*RD_val*tan(theta_val)

f2_val*Bs_val*RD_val*tan(theta_val)];B = [0; f1_val];C = [Ks_val*RD_val*tan(theta_val) Bs_val*RD_val*tan(theta_val)];D = [0];sys = ss(A,B,C,D);CG = gram(sys,'c'); % Computing Controllability Gramian

OG = gram(sys,'o'); % Computing Observability GramianCC(i,j) = 1/cond(CG); %Condition Number Controllability GramianCO(i,j) = 1/cond(OG); %Condition Number Observability Gramian

endendfigure;contourf(X,Y,CC);colormap(gray);xlabel('Mean Sleeve Diameter (mm) \rightarrow','FontSize',12);ylabel('Distance Ac (mm) \rightarrow','FontSize',12);title('Degree of Controllability','FontSize',15);figure;contourf(X,Y,CO);

colormap(gray);xlabel('Mean Sleeve Diameter (mm) \rightarrow','FontSize',12);ylabel('Distance Ac (mm) \rightarrow','FontSize',12);title('Degree of Observability','FontSize',15);


observability graminas for length (O1) and length (O2)

clc;clear all;close all;syms R1FAFTFft_fNx1Nx2Ny1Ny2FfaTDthetaRDuIDmfzeta_angCA2A1B1B2

E1E2O1O2O3ddlphigammadotgammadot2K1K2dthetadgammaDsi;%Constantsu_val = 0.16;% Independent Design Variablestheta_val = 42*pi/180;RD_val = 24.5;zeta_ang_val = 31.61*pi/180;phi_val = 62.08*pi/180;Ds_val = 79.63;Ac_val = 66.22;O1_val = 85.44;O2_val = 16.55;O3_val = 67.8;


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- 7 -

C_val = 14.95;Ks_val = 5;Bs_val = 1;

dl_val = 8.3;d_val = 13;ID_val = 200;mf_val = 0.2;%Dependent Design VariablesA1_val = Ac_val+Ds_val*sin(zeta_ang_val)/2; % A1_val = 87.09;A2_val = Ac_val-Ds_val*sin(zeta_ang_val)/2; % A2_val = 45.35;E1_val = Ds_val*cos(zeta_ang_val)/2; % E1_val = 33.91;E2_val = E1_val; % E2_val = 33.91;B1_val = O1_val - O3_val; %B1_val = 17.64;B2_val = O2_val + O3_val; %B2_val = 84.35;Dsi_val = Ds_val-5;%System Statesgammadot_val = 0;

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-



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




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-




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.gammadot2;gammadot2_val = vpa(simplify(subs(simplify(gammadot2_val),{gammadot},{0})));% %Degree of Controllability & Observability variation with O1 and O2% %Calculating Condition Numberfun_sim = vpa(simplify(subs(simplify(gammadot2_val),...

{zeta_ang,phi,dl,u, RD, C,A2, A1, E1, E2, Dsi, O3, d, theta, ID, mf} ,...{zeta_ang_val,phi_val,dl_val, u_val, RD_val, C_val, A2_val, A1_val,...E1_val, E2_val, Dsi_val, O3_val, d_val, theta_val, ID_val, mf_val})));

f1 = vpa(simplify(subs(fun_sim,{TD, Ffa} ,{1,0})),4);f2 = vpa(simplify(subs(fun_sim,{TD, Ffa} ,{0,1})),4);i=0;X = O1_val-20:1:O1_val+20;

Y = O2_val-15:1:O2_val+15;for O2_val = Y

i= i+1;j=0;for O1_val = X

j = j+1;%Dependent Design VariablesB1_val = O1_val - O3_val;

B2_val = O2_val + O3_val;

f1_val = double(subs(f1,{B1, B2, O1, O2},{B1_val, B2_val, O1_val,

O2_val}));f2_val = double(subs(f2,{B1, B2, O1, O2},{B1_val, B2_val, O1_val,

O2_val}));


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A = [0 1; f2_val*Ks_val*RD_val*tan(theta_val)

f2_val*Bs_val*RD_val*tan(theta_val)];B = [0; f1_val];

C = [Ks_val*RD_val*tan(theta_val) Bs_val*RD_val*tan(theta_val)];D = [0];sys = ss(A,B,C,D);CG = gram(sys,'c'); % Computing Controllability GramianOG = gram(sys,'o'); % Computing Observability GramianCC(i,j) = 1/cond(CG); %Condition Number Controllability GramianCO(i,j) = 1/cond(OG); %Condition Number Observability Gramian

endendfigure;contourf(X,Y,CC);colormap(gray);xlabel('Length O1 (mm) \rightarrow','FontSize',12);ylabel('Length O2 (mm) \rightarrow','FontSize',12);

title('Degree of Controllability','FontSize',15);figure;contourf(X,Y,CO);colormap(gray);xlabel('Length O1 (mm) \rightarrow','FontSize',12);ylabel('Length O2 (mm) \rightarrow','FontSize',12);title('Degree of Observability','FontSize',15);


observability graminas for drum ramp angle () and drum radius (RD)clc;clear all;

close all;syms R1FAFTFft_fNx1Nx2Ny1Ny2FfaTDthetaRDuIDmfzeta_angCA2A1B1B2E1E2O1O2O3ddlphigammadotgammadot2K1K2dthetadgammaDsi;%Constantsu_val = 0.16;% Independent Design Variablestheta_val = 42*pi/180;RD_val = 24.5;zeta_ang_val = 31.61*pi/180;phi_val = 62.08*pi/180;Ds_val = 79.63;Ac_val = 66.22;O1_val = 85.44;O2_val = 16.55;

O3_val = 67.8;C_val = 14.95;Ks_val = 5;Bs_val = 1;dl_val = 8.3;d_val = 13;ID_val = 200;mf_val = 0.2;%Dependent Design VariablesA1_val = Ac_val+Ds_val*sin(zeta_ang_val)/2; % A1_val = 87.09;A2_val = Ac_val-Ds_val*sin(zeta_ang_val)/2; % A2_val = 45.35;E1_val = Ds_val*cos(zeta_ang_val)/2; % E1_val = 33.91;E2_val = E1_val; % E2_val = 33.91;


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B1_val = O1_val - O3_val; %B1_val = 17.64;B2_val = O2_val + O3_val; %B2_val = 84.35;Dsi_val = Ds_val-5;

%System Statesgammadot_val = 0;Eqn = 'FA -R1*(cos(theta)-u*sin(theta)) , FT -




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




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-



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.gammadot2;gammadot2_val = vpa(simplify(subs(simplify(gammadot2_val),{gammadot},{0})));%Degree of Controllability & Observability variation with O3 and C%Calculating Condition Numberfun_sim = vpa(simplify(subs(simplify(gammadot2_val),...

{zeta_ang,phi,dl,u, RD, A2, A1, E1, E2, Dsi, O1, O2, d, theta, ID, mf} ,...{zeta_ang_val,phi_val,dl_val, u_val, RD_val, A2_val, A1_val,...E1_val, E2_val, Dsi_val, O1_val, O2_val, d_val, theta_val, ID_val, mf_val})));

f1 = vpa(simplify(subs(fun_sim,{TD, Ffa} ,{1,0})),4);f2 = vpa(simplify(subs(fun_sim,{TD, Ffa} ,{0,1})),4);

i=0;X = O3_val-20:1:O3_val+20;Y = C_val-5:1:C_val+10;CC = zeros(length(Y),length(X));CO = zeros(length(Y),length(X));for C_val = Y

i= i+1;j=0;for O3_val = X

j = j+1;%Dependent Design VariablesB1_val = O1_val - O3_val;

B2_val = O2_val + O3_val;

f1_val = double(subs(f1,{B1, B2, O3, C},{B1_val, B2_val, O3_val, C_val}));

f2_val = double(subs(f2,{B1, B2, O3, C},{B1_val, B2_val, O3_val, C_val}));A = [0 1; f2_val*Ks_val*RD_val*tan(theta_val)

f2_val*Bs_val*RD_val*tan(theta_val)];B = [0; f1_val];C_M = [Ks_val*RD_val*tan(theta_val) Bs_val*RD_val*tan(theta_val)];D = [0];sys = ss(A,B,C_M,D);CG = gram(sys,'c'); % Computing Controllability GramianOG = gram(sys,'o'); % Computing Observability GramianCC(i,j) = 1/cond(CG); %Condition Number Controllability GramianCO(i,j) = 1/cond(OG); %Condition Number Observability Gramian

endend


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figure;contourf(X,Y,CC);colormap(gray);

xlabel('Length O3 (mm) \rightarrow','FontSize',12);ylabel('Distance C (mm) \rightarrow','FontSize',12);title('Degree of Controllability','FontSize',15);figure;contourf(X,Y,CO);colormap(gray);xlabel('Length O3 (mm) \rightarrow','FontSize',12);ylabel('Distance C (mm) \rightarrow','FontSize',12);title('Degree of Observability','FontSize',15);O3_val = 67.8;C_val = 14.95;

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