Experimental Investigation and Verification of Single-Motor Padlock Unlocking Control System for the...

8/9/2019 Experimental Investigation and Verification of Single-Motor Padlock Unlocking Control System for the EX-7 Bomb Di…

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Integrated Securities, Inc.1015 North Quincy Street

Arlington, VA 22201

November 5, 2013

To: David Levinson, Director, Research and Development

From: Ali Alhamaly

Subject: Experimental Investigation and Verification of Single-Motor Padlock

Unlocking control system for the EX-7 Bomb Disposal Robot

Summary and Introduction

As requested by the memo sent on September 2, 2013, several tests were performed toinvestigate the effectiveness of using control system with single-motor to unlock schoollooker padlocks. Experimental investigations suggest that using feedback control system witha DC motor is both quick and reliable in opening small padlocks.

The motivation behind the current study is to develop and verify a controller design

for a single motor to open a combination padlock. The single motor will be used as

attachment to the EX-7 Bomb Disposal Robot. The Market Research Department of ourcompany believes that if the motor attachment to the robot can open combination padlocksreliably and rapidly then the company has a big chance to expand its sales by capturing the

state-police market. The state-police forces will be interested in buying the EX-7 robot because it can operate in school scenarios where there are bomb threats.

The experimental setup that was used consists of a single DC motor that is connectedto a combination padlock. The motor is connected to a controller which controls the motion

of the motor. The response of the motor which is the position of the padlock dial is recordedas a function of time using integrated LabVIEW data collection software. The experimentalsetup was used to test and validate the response of the motor for different controllerconfigurations.

Dynamic model for the motor-lock system was derived to be used in analytical

controller design. The model was simulated in a negative feedback closed loop that replicatesthat actual physical test setup. The derived dynamic model was verified using experimentaldata. The comparison between model response and the experimental response suggests that

the model represents the actual physical system with high accuracy. The validated model was

used to design analytical PID controller that improves the dynamic response of the system.The PID gains for the final analytical PID controller were found to be: Kp= 1.5, Ki=1.4, andKd= .007. This particular PID configuration achieves large improvement in the transient

response compared to the system initial controller.The analytically determined PID gains were tested in the physical setup to validate the

performance of the system. Discrepancies were found between the analytical performanceand the actual performance. Due to these discrepancies, the PID gains were tuned

experimentally until the desired dynamic performance of the system achieved. The final PID

experimental gains were: Kp= 1.5, Ki=1.3, and Kd= .009. This PID configuration whentested on the system with 100 step input gave rise time of .04 seconds, settling time of .08seconds, and overshoot of 4%. The system was tested with these gains to see if it can open a

padlock with combination of 12-18-00. The system was successful in opening the lock each


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time it was tested. Reliability of the system was further investigated by measuring the truemean of the steady state error of the system response at a 360 degrees step input. The system

was tested for ten different trails and the results show that the true mean of the steady state

error is .02 ±.04 degrees. The error bound was found using 95% confidence interval.This report presents the detailed experimental and analytical approach that was used

to develop and verify controller design for the EX-7 robot attachment. The report starts bydescribing the basic control theory. The different control system terminology and components

are explained. In addition, step response performance parameters are explained and their physical interpretation is discussed as well. The control theory section concludes by

explaining the PID controllers and how they work. Next, the experimental setup is describedincluding the apparatus and data acquisition system. After that, the report discusses thederivation of the dynamic model of the system with experimental data verification. Next, theanalytical controller design procedure is explained and the results of the system response

using the designed controller are presented. The next section presents the experimental tuning

of the PID controller and the final PID gains that was used to open the padlock. Finally, thereport concludes by general comments about the performance of the system as a whole.

Introduction and Background of Control System Theory

This section presents brief overview of control system components and the feedback

control principles that will be used throughout this report. The section also discusses the basic

PID controller and how it is used to control a physical system.

Contr ol Systems Over view . Physical systems require some sort of process controlin order to achieve what the user wants in terms of operation performance. It is known that

the performance of physical dynamic systems without external control can be undesirable and

for this reason control systems are important to achieve desirable performance operation ofthe physical system. The physical system can be any device or process that performs certaintask when given a user input. The control system connected to the physical system insures

that the operation characteristics of the physical system are what the user wants. For instance,if the physical system is the car and the control system is the cruise control, then the cruisecontrol insures that the car follows the speed that the driver set as input.

Control systems can be divided into closed loop and open loop systems. The main

physical difference between the two divisions is that the closed loop systems have detectiondevice that tells the controller what the physical system response is whereas in the open loopsystems there is no detection device and hence the controller is clueless of what the physicalsystem response is. There are other differences between the two divisions based on the

response of the physical system the being controlled. In general sense, open loop systemsusually have slow response, very sensitive to external disturbances, might be unstable, andlimited in the ability of tweaking the transient response of the controlled systems. On theother hand, closed loop systems can be designed to achieve fast response and have fast

response. In addition, closed loop systems are very flexible in terms of tuning the transientresponse of the physical systems and can be designed to achieve high stability margins.

The main components of a controlled system are: plant, actuator, controller, and

sensor. The plant is the physical system that needs to be controlled. The actuator is what

changes the state of the plant given controller input. The controller is the part where the userinput is converted into a proper signal to the actuator to achieve certain performance criteriaof the plant response. Finally, the sensor is the device that senses the response of the plant

and feed the information back to the controller for active control of the plant. Figure 1 showsa block diagram of both open and closed loop control system with the major components


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indicated. The block diagram also shows the flow of signals between the different parts of thesystem.

In terms of mathematical modeling of the control system, each component of the

system can be considered as a process that relates the input signal to the output signal. Eachcomponent can be described in terms of equations that relate the dynamics of that component

to the input signal. By coupling the entire system components with each other, thecharacteristics of the whole system can be described in terms of governing deferential

equations. Upon solving these equations, the response of the plant can be found and suitablecontroller can be designed to change the behavior of the plant response as desired.

As mentioned earlier, the main purpose of a controller is to control the response of the

plant to the desired characteristic that user wants. The performance of the controller is usuallymeasured against some design specifications for the plant response due to a sudden change inthe operating condition i.e. a step response. The response specifications due to a step

response contains: rise time, settling time, percent overshot, and the steady state error. Therise time is the time it takes the plant response to go from 10% to 90% of final desired stepvalue. The settling time is the time that the pant response remains between ± 2% of the finaldesired value. The percent overshoot is the maximum deviation in percent of the plant

response with respect to the desired final value. Finally, the steady state error is the actualdifference between the desired output and the plant output at long time after the initial

response has begun.Once the desired response specifications are obtained, then the controller need to be

designed to achieve these specifications. Depending on the plant that being controlled,

achieving all the design specifications might not be feasible and hence the design process of a

controller usually contains tradeoff between robustness, speed, and steady state errors. Figure2 shows a typical second order plant response with the several response specificationsindicated. From figure 2 it can be seen that the rise time and settling time are measure of thespeed of the response. The steady state error is a measure of the accuracy of the controller to

achieve the desired final output.

There are varieties of the mathematical forms that the controller can be represented by. Different forms of the controller depend on the type of the application and the designrequirements for a given application. One of the popular controller designs is the

Proportional Integral Derivative (PID). PID controller takes the error between the desiredoutput response and the actual plant output response as the input signal and performs some

Figure 1. Block diagram of open and closed loop control systems.


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mathematical operation on the error to obtain the output controlling signal to the actuator.The PID controller acts as a superposition of three main outputs. The proportional output

depends on the value of the current error. The Integral output depends on the sum of the

cumulative errors from the beginning of the input command to the current time. Thederivative output depends on the rate of change of the current error. The three outputs of the

PID controller establish the controlling signal at any given instant of time. The PID controllercan be tuned to achieve specific design characteristics of the plant response by changing the

different gains that are associated with each output. The gains are referred to as: Kp, Ki, andKd. These gains correspond to the proportional, integral, and derivative terms in the PIDcontroller respectively. Each gain is responsible of altering some of the dynamiccharacteristic of the plant response. By choosing the correct values of the gains, the desired

performance response can be obtained. Figure 3 shows the block diagram of a typical PIDcontroller with the mathematical operation that each part is performing on the error signal.

Figure 2. Step response of a typical second order plant showing the different performancemetrics

Figure 3. Block Diagram of a typical PID controller.


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Experimental Setup

This section presents the experimental apparatus and the test procedures that wereused to investigate the motor lock system for the purpose of designing a controller . The

section discusses the physical components in the system and the different experiments anddata collection methods.

Exper iment Apparatus . The main objective of this experimental investigation is todesign a controller that moves a padlock to a desired location in timely and consistent

manners. In order to test the controller performance, a test setup that contains a DC motorconnected to the padlock is used. The control system is connected in a closed loop setup asindicated in figure1. The system contains: the padlock which is the plant, DC motor withreduction gear box set which corresponds to the actuator, encoder (as a feedback sensor), NI

cRIO-9024 real time fast controller, NI 9505 H bridge DC motor drive and a 24V DC powersupply. Figure 4 shows a picture of the test setup with the different components labeled.

The encoder that was used is an optical incremental encoder in which it measures therelative position of the padlock for initial reference position. The reference position was set

always to be zero. LabVIEW software was used for data acquisition and for sendingcommands to the controller. The LabVIEW software can set the gain values of the PID

controller based on the user input. In addition, the software allows the test of any step inputvalues to the system. The LabVIEW was used as well to collect the data that corresponds tothe response of the padlock as function of time for different step inputs. All the data weresampled using a .01 seconds sampling time.

Test Procedu res . The tests that were conducted using the system mentioned earlier

serve two main purposes. The first is to collect data and use it to assess the derivation of the

system dynamic model and to verify quality of the model. The second is to test the system toevaluate its performance and its ability to satisfy the design requirements. For the first

purpose, the system was tested for five different step inputs correspond to: 90,180,360,720,and 1080 degrees. The response of the padlock as a function of time is collected and saved

Figure 4. Padlock test setup.


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for later processing. The values of the controller gains for this test were: Kp=1.2, Ki=2, andKd=.02. For the second purpose, the testing was about evaluating the final controller design

performance in opening the padlock successfully. Three trials were conducted to test the

reliability of the system to open the lock successfully. Finally, the system was tested for theevaluation of its repeatability. The test consists of measuring the steady state error for a step

input of 360 degrees ten times. The results of the different tests will be shown in latersections of this report.

Derivation of the Lock-Motor Dynamic Model

This section presents the derivation of the transfer function for the padlock-DC motorsystem. The section discusses the governing equations of the system that describe thedynamics response. Comparison between the derived model response with experimental data

is shown to assess the accuracy of the overall model.

Open Loop Tr ansfer Fun ction . In order to quantify how the padlock-DC motorsystem (will be referred as the system from now and on) responds to a given controller signal,

a model describing the dynamic behavior and the interactions between the variouscomponents of the system is needed. The system we are dealing with has two main

interactions. The first is the electrical interaction between the controlling and energizingsource with the various electrical components in the DC motor such as the motor windings,

resistance, and inductance. The second is the mechanical interaction between the motor rotorand the connecting load which is in this case the gearbox and the attached padlock. Bothelectrical and mechanical characteristics of the system are coupled and their interactionsdetermine the overall performance of the system.

The system interaction between its components can be described using a set of twocoupled ordinary differential equations. The two equations describe the coupling between theelectrical and mechanical characteristic of the system. Figure 5 shows a schematic of thesystem that relates the electrical and mechanical parts together. As can be seen in the figure,

the applied voltage (V) causes a current (i) to flow throughout the various system componentssuch as the motor terminal resistance (R) and the motor terminal inductance (L). Due to the

current flowing in the motor internal windings, the padlock rotates to an angular position (θ)and a torque (τ) is developed to move the external load represented by the total mass moment

of inertia (J) and to overcome the internal fiction and viscous damping in the system (b).Because the rotor windings of the DC motor rotate through the magnetic field of the stator, a

back EMF (Vemf ) is developed across the motor terminal to resist the change in the magnetic

flux through the rotor windings.The model in figure 5 was chosen to represent the system for couple of reasons. First,the sets of equations that relate the voltage to the angular position in this particular model arelinear which simplifies the method of solutions. Second, the motor that was used in theexperiment has both terminal resistance and inductance and that is why the model in figure 5

includes resistance and inductance as system components. In addition, the resistance andinductance affect the transient response of the system and hence including them in the overall

model enhances the accuracy of the model. Third, any DC motor that is driven by a voltagesource has a back EMF resisting the applied voltage. This is due to the construction of themotor itself and how the magnetic flux is changed through the rotor windings. Having back

EMF in the model is important because it exists in the real motor under study and hence it

needs to be accounted for and because it represents a coupling between the rotation speed ofthe motor and the electrical load of the motor on the applied voltage. Finally, the mechanical


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load on the motor is related to the physical components that the motor needs to move androtate. The resistance of these components to the rotation can be represented by the mass

moment of inertia and the damping that results from friction between the different parts. So to

account for the mechanical load in the real system, the model in figure 5 includes the totalmass moment of inertia and the total damping of the system. Quantitative justification of the

model will be shown in the next subsection in which experimental response is compared withthe model response.

As mentioned earlier, there are two equations that relate the controlling signal which

is the applied voltage to the output angular position of the padlock. The two equations based

on the schematic of figure 5 are given by

Equations 1 and 2 represent the dynamic model of the whole system. These two equationscan relate the input of the system which is the applied to the output angular position of the

padlock directly if the equations are transformed to the Laplace domain. The relation betweenthe output to the input of the system in the Laplace domain is referred to as the open loop

transfer function. Full derivation of the open loop transfer function for the system can befound in appendix A. Table 1 shows a summary of the physical parameters of the systemalong with their symbols. The nominal numeric values of the parameters in table 1 can befound in appendix B.

Table1. Summary of the physical parameters of the system with their corresponding symbols

Paramete r name Symbol Paramete r name Symbol

Applied motor voltage V Gear box mass moment of inertia JgMotor terminal Resistance R Gear box ratio K gMotor terminal inductance L Motor constant K

Applied torque to the padlock τ Back EMF voltage VemfSystem total mass moment of inertia J Current iMotor mass moment of inertia Jm Angular position of the padlock θ

Viscous damping coefficient b Motor efficiency ηmGear box efficiency ηg

Figure 5. Schematic showing the system components and the interaction between the

mechanical part and the electrical part

(1)

2


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From the derivation in appendix A, the open loop transfer function for the system isgiven by

++++

The system open loop transfer function as shown in equation 3 relates the output to the input

through the internal properties of the physical system. Since the transfer function describesthe dynamics of the actual physical system, this means that the actual response of the physicalsystem to a controller signal can be simulated and hence a suitable controller can be designedto meet specific performance characteristics of the overall system. But before using the

transfer function to design the controller, validation of the accuracy of the transfer function

and how representative it is to the performance of the actual physical system is necessarily.For this reason, the simulated response using equation 3 is compared with experimentalsystem response. Discussion about the verification of the transfer function is included in the

next subsection. It is worth to notice that equation 3 describes the open loop transfer functionfor the actuator and the plant combined, the actuator being the motor with the gear box andthe plant being the padlock.

Model Ver if ic ation . The approach to verify the derived transfer function is tocompare the system closed loop experimental response to the modeled closed loop response.The experimental setup shown in figure 4 is used in closed loop with a PID controller to testthe system response for multiple of step inputs. The inputs were 90, 180, 360, 720, and 1080

degrees. The PID gains for the controller were: Kp= 1.2, Ki= 2, Kd=.02. The experimental

step response for each input was recorded using LabVIEW.The closed loop analytical response is obtained by using the analytical transfer

function in a closed loop model like the one shown in figure 1. Simulink was used toimplement the transfer function in a closed loop model that has a PID controller and a unitygain negative feedback loop. Simulink is able to obtain the step response of the overallsystem as a function of time which simulates the actual physical system. The closed loop

block diagram Simulink model is shown in figure 6.

Numeric values of the parameters in the transfer function can be found in AppendixB. using these values equation 3 becomes

∗2∗.24.∗.85∗.755.72∗− ∗2.8∗−+(2.8∗−∗2.7∗−..+5.72∗− ∗2.2Ω)+.85∗.75∗2∗.24.

+2.2Ω∗2.7∗−..

(3)

Figure 6. Block diagram of the Simulink model that used to simulate the system response


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The damping coefficient (b) was used as the inverse of the speed/torque constant in appendixB. The mass moment of inertia for the padlock was not given to us and hence it is not

included in the total moment of inertia for the system. The fact that the inertia of the padlock

is negligible compared to the motor and gear box inertias makes the effects of neglecting itsinertia on the system response very minimal. Figure 7 shows a plot of the experimental

response and the modeled response for a step input of 1080 degrees. The figure shows thatthe modeled response is in great agreement with the experimental response. It can be seen

also that the modeled response has a lower percent overshoot than the experimental data andalso has some oscillations near the peak of the response that are not present in theexperimental data. The good match between the experimental data and the model suggeststhat the overall model captures the dominant dynamics of the system and hence the derived

transfer function represents the system very well.

The differences in the response between the model and the actual system can be

reduced by changing the values of some parameters in the transfer function. The parametersthat can be changed to increase the quality of the model are: the damping coefficient (b), themotor efficiency ηm, and the gear box efficiency ηg. The reason for choosing these parametersin particular is because damping coefficient is given only for the motor and not for the whole

system and the efficiencies are given in terms of their maximum values and not the value

representing the operating condition that we are dealing with. The three parameters werechanged to investigate the effects on the analytical response and its relation with theexperimental data. It was found after analyzing the analytical response that changing the

damping coefficient has very negligible effect on the overall response while changing thevalues of the efficiencies have a relatively larger effects on the response. After several trialsof changing the parameters and analyzing the response in comparison with experimental data ,

it was found that setting the motor efficiency to .45 and gear box efficiency to .15 increased

the accuracy of the model response and reduced the differences between the model andexperimental data. Figure 8 shows a plot similar to figure 7 but with the new values of

efficiencies. It can be seen that the model response match the experimental data better thanthe response in figure 8. In addition, the oscillations near the peak response that exists in

Figure 7. Experimental and analytical step response of the system


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figure 7 is now suppressed which means that the new model with the tuned parameters is better in estimating the performance of the system from the model in figure 7.

In order to quantitatively assess the quality of the model response with respect to theexperimental data, performance parameters like the rise time and percent overshoot need to

be compared. Since the performance parameters characterize the response of the system ingeneral sense, they are good indicators of how the model represent the general characteristics

of the actual system being modeled. Built in function in Matlab was used to obtain the performance parameters of the experimental and modeled response. Table 1 shows a

comparison between the experimental performance parameters and the analytical performance parameters. The table shows parameter for model of figures 7 and 8 to show the

advantages of changing efficiencies.

Table1. Performance parameters comparison for 1080 degrees step input

Performance parameter Experimental data Model (figure 7) Model (figure 8)

Rise Time (s) .346 .339 .346

Settling Time (s) 1.67 1.52 1.60Percent Overshoot (%) 12.9 11.2 12.8

Steady state error (Degrees) .31 .008 .01

Results of table 1 show that the modified model (figure 8) performance parametersare closer to the experimental data from model (figure 7). Hence, the model in figure 8 will

be used as the model that represents the actual physical system and will be referred to as themodel in the rest of this report. The response of the system to the other step inputs (90, 180,

360, and 720) were analyzed. The analytical response for all these step inputs shows a good

match to the experimental data. Table 2 shows a comparison between the model andexperimental performance parameters for step inputs of 360 and 720. Tables 1 and 2 showthat the model is a good representative of the actual physical system which means that the

Figure 8. Experimental and analytical step response of the system after changingefficiency values


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model can be used to investigate the effects of changing the controller gains on the performance of the actual physical system.

Table2. Performance parameters comparison for 720 and 360 degrees step inputs

720 Step inputPerformance paramete r Experimental data Model

Rise Time (s) .231 .235

Settling Time (s) 1.43 1.32

Percent Overshoot (%) 10.6 10.4

Steady state error (Degrees) .31 .004

360 Step input

Performance paramete r Experimental data Model

Rise Time (s) .117 .125

Settling Time (s) 1.18 .956Percent Overshoot (%) 7.27 6.85


Very important point to notice about the performance of the system to different stepinput values is that the performance parameter changes between the different inputs. This isin contrast with linear control theory in which the performance of the system doesn’t change

with the value of the input. The reason for this variation is that the motor has limits on its

maximum torque and speed and hence higher step inputs will have longer rise and settlingtimes. In the Simulink model, the motor limitation is taken into account by limiting the

voltage output from the PID controller to be ±24V which is the actual motor voltage rating.Analytical Controller Design

This section presents the approach to design a PID controller for the system to meetcertain design specifications about the system step response. The section starts by defining

the design goals that need to be met. Then, controller design procedure is explained and thefinal design results are presented. Lastly, the section concludes by commenting on thedifferences between the analytical performance and the performance of the actual systemusing the designed controller.

Design Specif ic ation . Opening the padlock using the system we have requires somespecification on the step response. As can be seen in figure 8, the system desired position was1080 degrees, however, the system overshot 1080 degrees by large amount (about 13%)

which indicates that the current system controller is unable to control the transient behaviorof the system as necessarily. In addition, the current system response suffers from relatively

large settling time which causes long time to open the padlock. In order to use the currentsystem to open a padlock reliably and in a timely manner, the system step response need to be

changed. For this reason, we were given set of design goals that the system should achieve toopen the lock accurately. The design goals were based on the response of the system to a stepinput. The design goals are: rise time of about .05 seconds, settling time of .2 seconds and

percent overshoot of less than 10%. In addition, the steady state error of the system should beas small as possible. The Simulink model shown in figure 6 was used to design a controller


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that meets the mentioned design goals. The Simulink model allows for changing the PIDgains in the controller block to change the dynamic response of the system.

Contr ol ler Design . The approach to change the dynamic response of the system is based on changing the PID gains in the PID controller. Changing the PID gains can make

wide changes to the response of the system and depending on the choice of several gainvalues, the desired dynamic response of the system can be obtained.

The Simulink model in figure 6 was used to investigate the behavior of the systemwhile changing the PID gains. Having the design goals in mind, built-in Matlab function wasused to tune the PID controller to achieve the design goals. The Matlab function “pidtune”

can change the response of the system to the desired characteristic by changing the multiplegains in the PID block in the Simulink model. pidtune was used initially to obtain the PIDgain values that satisfy the design goals. pidtune analyzes the unit step response of the systemand allows the user to change performance parameters like the rise time and the overshoot

until the desired performance is achieved, from there pidtune outputs the values of thedifferent PID gains that cause the system response to behave as desired. Pidtune is very

efficient and fast tool to design a controller because it takes into account the tradeoffs between changing different gains and hence the function changes all different gains

simultaneously to achieve the desired performance. The only disadvantage of pidtune is thatthe PID controller it produces is based only on the unit step response of the system and the

step input value cannot be changed from unity. It was shown earlier in the previous sectionthat the response of the system is slightly dependent on the value of the step input which

means that designing a controller based on the unit step input might not be adequate forhigher value of the step input and in the application we are interested in the values of thecommand step input is always high compare to unity. For this reason, the results of the

pidtune produce unwanted dynamic behavior for the higher value of the step response. This

problem was tackled through using the PID gain values obtained from pidtune as only aninitial estimate of the final PID gains. This means that pidtune was used initially to improvethe response of the system by making it faster and having low percent overshoot, and then amanual tuning of the PID gains based on the initial estimate of the results from pidtune was

used to obtain the desired performance for the higher value of the step input. This approach is better and more efficient than tuning the PID gains manually from the beginning because

pidtune has optimization algorithms that produce an excellent estimate of the PID gains thatrequire only few modifications to be applicable for high values of the step input.

The manual tuning of the PID gains from the initial results of the pidtune functionwas based on some rules of thumb on how each PID gain alter the response of the system.Table 3 shows the effect of increasing each gain on the response of the system. These general

rules were used to tune the PID gains manually to achieve the desired performance.

Table3. PID gains effect on the system response parameters

Gain Rise Time Overshoot Settling Time Steady State Error

↑Kp Decrease Increase Small Change Decrease↑Ki Decrease Increase Increase Eliminate↑Kd Small Change Decrease Decrease Small Change

After several trial and error and tradeoff decisions between the different design goals, a finalset of analytical gains was chosen. The gains are: Kp = 1.5, Ki= 1.4, and Kd = .007. The resultingresponse of the system suing the final PID gains for a step input of 1000 degrees is shown in figure 9.

It can be seen in figure 9 that the response time is much faster than the initial response shown infigure 8. The percent overshoot also decreased a lot in comparison with figure 8.


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As discussed earlier, the system response is slightly sensitive to the value of the stepinput, for this reason the performance of the system with the new controller need to beinvestigated across multiple values of the step input. Table 4 shows a comparison between

the system response parameters for different values of the step input.

Table4. Comparison between the system performance for different step input values

StepInput

(degrees)

RiseTime(s)

SettlingTime(s)

Overshoot(%)

SteadyState Error

(degrees)

100 .047 .28 22.1 Zero

180 .071 .27 14.1 Zero

360 .125 .26 6.99 Zero

720 .235 .39 3.22 Zero

1080 .347 .45 1.8 Zero

Table 4 shows that for low step input the value of the overshoot is relatively highwhich means that the designed controller has better performance at higher values of the step

input. This result is expected because it is hard to have a fast rise response and still achievelow overshoot for small inputs, because the system needs to accelerate very sharply and thenin very short time needs to decelerate to prevent the overshoot. It is possible to design a

controller to achieve fast response and low overshoot for the small input values, but this

controller will have undesired dynamic response for any relatively large input.The motivation behind choosing this particular configuration of the controller has to

do with the low value of the percent overshoot among wide range of the step inputs. Inaddition, the resulting rise and settling times of the system are very short. Most importantly,

the chosen controller always yields a zero steady state error which is important characteristicin order to be able to open the lock successfully. The major disadvantage of the chosen

Figure 9. Analytical system response for a step input of 1000 degrees using the tuned PIDgains.


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controller is the large overshoot for the low step input values which might cause a problemwhen attempting to open the lock.

Experiment al Result s Using th e Designed Cont rol ler . The designed PIDcontroller was tested using the actual physical system to assess the performance of the system

and to investigate how well the controller achieves the desired dynamic response. Thedesigned controller was tested using a step input of 100 and 1000 degrees. Figure 10 shows a

plot for the experimental and analytical system response to a 1000 degrees step input usingthe same PID gain values. Table 5 presents a comparison summary between the modelresponse and the experimental response for the 100 and 1000 degrees step input.

Table5. Experimental and model performance parameters comparison for 1000 and 100 degrees stepinputs using same PID gain values

1000 Step input


Rise Time (s) .319 .322

Settling Time (s) .402 .423


Steady state error (Degrees) .4 zero

100 Step input


Rise Time (s) .038 .047

Settling Time (s) 2.1 .28


Steady state error (Degrees) .03 zero

Figure 10 and table 5 show that there are differences between the responses of theanalytical and the actual system. Discrepancies between the model and the actual system are

Figure 10. Experimental and analytical system response for a step input of 1000 degreesusing the same PID gains.


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clearly apparent from table 5 for the step 100 case. The 1000 step analytical and experimentalresponses agree with each other in terms of the performance parameters but figure 10 shows

that the actual transient dynamic behavior of the two responses are different after the peak

time. These differences are caused by several factors.First, the model that was chosen to represents the system (which is figure 5 and

equations 1 and 2) doesn’t take into account all the interactions and the physical processesthat happen between the motor and the energizing circuit. In addition, the relations that were

used to relate the mechanical behavior of the motor to the electrical voltage and current areall simple linear relations, where in fact it is expected that nonlinear terms be present in theserelations. Not accounting for nonlinear system interactions cause the analytical response tohave some discrepancies in compared with the actual system response. Example for nonlinear

process in the system is the friction between several system components. The friction ismodeled as linear viscous damping and the value of the damping coefficient was given onlyfor the motor. In reality, the friction exists between all the system components that rotate and

it is nonlinear and cannot be described simply by a viscous damping term.

Second, in modeling the closed loop system it was assumed that the output angular

position when sent back through the feedback loop is known precisely and instantly.However, in the actual system the output is measured by a sensor so depending on theaccuracy of the sensor and the speed in which it processes the measuring signals, the output

might not be known instantly or precisely. Inaccuracies in the measured output will cause the

error signal going to the controller to be different from the modeled error signal, and hencethe PID controller output signal for the case of actual system will be different from the modelsystem and this might cause some of the discrepancies between the two responses.

Third, the system internal parameters such as the values of the inertia and theelectrical components that was used in the transfer function is based on the manufacturer datasheet for the motor. These values for the system that is being tested might not be exactly the

same as the data sheet suggests because the specifications given on data sheet are not 100%

accurate. Any discrepancy between the value of the parameters between what is used in thetransfer function and what is actually presents in the real system will cause differences

between the model and experimental responses.Lastly, noise generated by the power supply and the DC motor H bridge drive can

affect the controller signal in the actual system. The controller signal in the model is assumed

to be a pure controlling signal with zero noise which is not the situation in the real system.Any of the above mentioned factors can be a cause to discrepancies between the model andexperimental responses. The list of possible factors that might cause differences between the

two responses is not limited to what is mentioned here, but these reasons are believed to bethe most dominant among all factors.

Results of Implementing the System in Opening the Padlock

This section presents the results of using the system with appropriate PID controller toopen the padlock. The section starts by describing the experimental tuning of the PID gains to

improve the response of the system compared with the analytically found PID gains. The

section concludes by presenting the results of using the final gains to open the padlock giventhe lock three combinations.

Exper imenta l Tu ning of t he Contr o l ler . It was shown in the last section that the

analytically determined PID gains for the system controller produce slightly differentresponse in reality that what is expected. The differences in the expected response and the


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real response get larger for low input values as explained earlier. In addition, the analytic PIDgains led the response of low input values to have large overshoot which is undesirable

characteristic for the purpose of opening the padlock. For these reasons, the analytically

determined PID gains when used in the physical system to control the DC motor led toresponse characteristics that were not successful in opening the padlock.

The padlock that was used in the testing system is shown in figure 11. The padlockhas 40 digit combinations which means that each digit is spaced by 9 degrees from each

other. The lock combinations for the padlock that was used were set to 12-18-00. In order toopen the padlock, the motor needs to turn the padlock clockwise for two full revolutions andthen stop at the digit 12. Then, the motor needs to turn the padlock counter -clockwise for onefull revolution while passing the digit 12 and then stop at the digit 18. Finally, the motor

needs to turn the padlock clockwise from the digit 18 until it reaches the digit 00. This process can be interpreted as a set position sequence in degrees of: (-972) – (-558) – (-720).This set position sequence requires that the padlock starts at zero.

As mentioned earlier, the analytic PID controller when used in the system to control

the motor for the purpose of opening the lock results in failure to open the lock. The reason behind the failure is that the motor overshot the last digit (which is zero) by almost a twodigit. This is actually expected because the motor in the last set position needs to move from

-558 to -720 degrees which corresponds to a step input of 162 degrees. Table 5 shows that forlow step inputs the overshoot is about 10% which leads for a step of 162 degrees to overshootof about two digits.

The PID gains need to be tuned slightly to compensate for the large overshoot that

causes system failure. Information in table 3 was used as a guideline to make the necessarilychanges on the analytical gains. Since the desired change in the performance is to decreasethe overshoot, the derivative gain (Kd) was increased. It was found also that the settling timefor the system need to be decreased and hence the integral gain (Ki) was decreas ed. After

some trial and error with changing Kd and Ki, a final set of PID gains was chosen. The gainswere: Kp = 1.5, Ki = 1.3, and Kd = .009. These final gains when used in the system controllerallow the motor to open the lock successfully and repeatedly in each trial. In addition, thenew gains improve the transient response characteristics of the system for both the high and

low inputs. It is worth to notice that differences between the analytical and final experimentalcontroller gains are in Ki and Kd only. The changes made to these gains were for the purpose

of decreasing the overshoot and settling time. Table 6 shows a comparison between theanalytical and final experimental controller gains.

Figure 11. Picture of the padlock used in testing the system.


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Table6. Comparison between the analytical and final experimental controller gains

Gain Analytical Experimental

Kp 1.5 1.5

Ki 1.4 1.3

Kd .007 .009

Figure 12 shows a plot comparing the experimental responses for a step input of 100

degrees before and after the experimental tuning of the PID gains. The figure shows clearlyhow the transient response of the system improved in terms of both the overshoot and thesettling time. Table 7 shows the performance parameters comparison between theexperimental responses for a step input of 100 and 1000 degrees before and after the

experimental tuning of the PID gains. Table 7 shows quantitatively how the response of thesystem improved in terms of its performance parameters especially for the case of 100 stepinput.

Figure 12. Comparison between experimental responses before and after experimentaltuning of controller gains for100 degrees step input.


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Table7. Experimental performance parameters comparison for 1000 and 100 degrees step inputs before and after experimental tuning of the PID gains.

1000 Step input

Performance parameter Before Tuning After Tuning

Rise Time (s) .319 .321

Settling Time (s) .402 .405

Percent Overshoot (%) 1.81 .407


100 Step input

Performance parameter Before Tuning After Tuning

Rise Time (s) .038 .039

Settling Time (s) 2.1 .08



The system with the final experimental controller gains were tested for several trialsto insure repeatability in opening the padlock. The padlock was successfully opened for all

the trials that were tested. The average time for opening the padlock using the system is 2.81seconds. This opening time was compared with manually opening the padlock using hand itwas found that the system is faster in opening the lock than hand by at least 2.5 seconds.Figure 13 shows the experimental response of the system while opening the lock

successfully.

Evaluation of Using the System to Open a Padlock for Bomb Disposal Robot

This section aims to discuss the feasibility of using the system as attachment to bomb

disposal robot. The section starts with evaluation of the repeatability and reliability of the

system by considering the random error in the response. The section concludes with brief

Figure 13. Response of the system while opening the lock.


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discussion about the system advantages and its performance when used with different lockcombinations.

Repeatabi l i t y of th e System . The use of the current system in bomb disposal robotfor purpose of opening school looker padlocks requires high reliability in the system.

Reliability means that the system works as designed for every time it is used. In other words,the system operation needs to be repeatable within certain allowable limits.

The repeatability of the system can be determined by measuring the change in certain parameter during a repeated process of the system. The system parameter that was chosen tomeasure the repeatability is the steady state error of the angular position of the lock. The

steady state error was measured ten times for a step input of 360 degrees. From the tenmeasurements that were taken, we are interested in finding the true mean of the steady stateerror. The true mean will be given in terms of upper and lower limits that are based on thestandard error. The width of the limits of the true mean can be used to estimate how

repeatable the system is.The steady state error was calculated as the difference between the input set point and

the value of the response after 8 seconds from the start of the input step. In terms of equation,the steady state error is given by

360 is the steady state error in degrees. Equation 4 was applied to the entire ten measuredtrials t0 find the individual steady state error for each measurement. The mean value of thesteady state error from the ten trials is given by

̅ ∑ == Where ̅ is the mean value of the steady state error from the ten trials and is the steadystate error for each individual trial. The standard error in the ten trials is given by

̅ ̅ √ ∗9 ∗ ∑ ( ̅ )2==

Finally the true mean of the steady state error is given by

′ ̅ ± ̅ ̅ Where ′ is the true mean of the steady state error and t is value of the student t distributionthat is based on the number of measurements and the confidence interval that the true mean is

wanted to be estimated in. the value of t in our case is dependent of the confidence intervalonly because the number of the measurements is known which is ten. The confidence intervalwas chosen to be 95%. The choice of the confidence interval was based on the idea that withten measurements it is possible to estimate true mean with high confidence without causing

the upper and lower limits of the true mean to be large. The value of t in equation 7 with 95%confidence interval is 2.262. Table 8 shows a summary of the statistical quantities ofinterests. Figure 14 presents the information in table 8 pictorially with the measured raw data.

4

(5)

(6)

(7)


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Table8. Summary of the statistical parameters for determining the true mean of the steady state error.

Table 8 and figure 14 show that the true mean of the steady state error with 95%

confidence interval is .02 ± .04 degrees. This means that the true steady state error mean forthe system is always known to be within .02 ± .04 with a 95% confidence interval. Note thatthis interval represents only the range in which the mean of the steady state error lays in but

not the individual measurement of the steady state error.The low value of the true mean range which is much less than a degree tells that the

system is reliable and accurate when it comes to moving the lock to a certain position. It isimportant for the system to be highly accurate and repeatable within tight limits to open

padlocks for bomb disposal robot application. The system given the information figure 14 is believed to be both reliable and repeatable which is important advantage in the current

system.

Overal l System Performan ce Evaluation . The performance of the system wastested with different step input values ranging from low inputs to high inputs. The overall

performance of the system among the range of input values that was tested seems to be veryconsistent and provide desirable results. That being said, the system response overshoot forlow step inputs seems to get higher as the input value gets smaller. It was explained in earliersection that large overshoot leads to error in opening the lock because the motor miss the

desired digit by more than the allowable limit. For this reason, the current system might be

unable to open padlocks in which their combinations require low value of the command stepinput. The current system has an overshoot of about 4% for a 100 degrees step input whichmeans a 4 degrees overshoot in total. 4 degrees overshoot is acceptable because the digits in

the tested padlock are spaced with 9 degrees so as long as the lock position is within 4.5

Mean (degrees) .0195

Standard Error (degrees) .0192

True Mean Upper Limit (degrees) .0629

True Mean Lower Limit (degrees) -.0239

Figure 14. Steady state error for ten trials for a 360 degrees step input.


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degrees from the desired final value the lock will still open, but if a padlock has combinationsthat require a step input of less than 100 degrees, problems in opening the lock might be

expected. The current system controller needs some modifications that aim to decrease the

overshoot for low inputs. These modifications are believed to be possible and achievable withcarful investigation of the response of the system to low input values.

There are couples of advantages of using the current system as attachment to bombdisposal robots. The main advantage is that the system is very reliable and accurate in doing

what is being asked. In addition, the system transient response is very short for the entire stepinputs of interest in real application which insures fast opening time of the lock. In terms ofease of modifications to the system controller, the system gain parameters can be changedand tested easily using the test setup presented in the report. So any modifications in the

response of the system can be made easily and can be tested to grantee the performance andreliability as shown in the report.

Conclusions

The main focus of this experimental work was to investigate the effectiveness ofusing control system with single-motor to unlock school looker padlocks. Experimental and

analytical results prove that the current system with the designed PID controller is both quick

and reliable in opening small padlocks.Analytical modeling of the system was found to be useful in obtaining the final PID

controller configuration. The analytically determined PID gains produce response

characteristics that are closely related to what the physical response characteristics are. Thismeans that the analytical model can be used as a design tool to get a good estimate of the PIDconfiguration. From the PID configuration determined analytically, it was shown that only

small changes need to be made on the controller configuration in order to get the desired

response characteristics in actual system.The final PID gains that were determined experimentally are: Kp= 1.5, Ki=1.3, and

Kd= .009. This PID configuration when tested on the system with 100 step input gave risetime of .04 seconds, settling time of .08 seconds, and overshoot of 4%. These performance

characteristics are considered to be rapid due to the low rise and settling times and robust due

to the low overshoot. In addition, the system with this PID configuration was able to open the padlock with combinations 12-18-00 in each time it was tested which proves the reliability ofthe system. It was found also that true mean of the steady state error with 95% confidence

interval is .02 ±.04 degrees. This low value of the true mean insures the reliability of thesystem as well.

Attachments:

Appendix A: Open loop transfer function derivationAppendix B: Data sheet of system components


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Appendix A: Open loop transfer function derivation

This appendix presents the detailed derivation of the open lop transfer function for the

motor-lock system.

The derivation starts with equations 1 and 2 which are repeated here for convenience

Three extra equations are needed to relate different parameters with each other

(1 2)

Substituting equation A-4 into A-1 yields


Taking Laplace transform of equation A-6 yields

Taking Laplace transform of equation A-7 yields

2

Solving for in equation A-9 yields

(+)

(A-1)

(A-2)

(A-3)

A-4

(A-5)

(A-6)

(A-7)

(A-8)

A-9

(A-10)


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+(+)

Rearranging equation A-11 yields

+ +(+)

Expanding the numerator in equation A-12 yields

++++

Flipping equation A-13 and multiplying by

8

to convert from radians to degrees yields

the final results of the open loop transfer function

++++

(A-11)

(A-12)

(A-13)

(A-14)


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Appendix B: Data Sheet of System Components

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