stabilization of manually controlled sight-systems for ...1453591/fulltext01.pdf · stabilization...

IN DEGREE PROJECT ELECTRICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2019

Stabilization of Manually Controlled Sight-Systems for Tracking Tasks

STEFFANY REYNA MARQUEZ

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

Stabilization of Manually Controlled Sight-Systemsfor Tracking Tasks

Ste↵any Reyna Marquez

May 15, 2019

Master’s Thesis: Degree Project in Automatic Control

Supervisor: Daniel Hellberg

Examiner: Elling W. Jacobsen

EL205X - Degree Project in Automatic Control, Second Cycle

Abstract

In this thesis, we analyze the influence of an operator during manual control tasks in order toimprove her tracking ability. The operator sends control commands to a sight system so that shecan, among other things, track a target. This system is driven by a static non-linear function,also called joystick’s control characteristic which shapes the operator’s input.

To analyze the sight-system’s dynamics, we create a model of the operator based on theTustin-McRuer pilot model through measurements taken from the system itself. We use closed-loop identification to estimate the parameters of the operator model. During the measurements,it was seen that the operator has an adaptive behaviour and that her behaviour depends on thejoystick’s control characteristic. Therefore the model is updated so that her gain can be modeledusing a Gain-Scheduling controller.

Results show that the operator is able to adapt to the system fast and this fast adaptionmakes getting an accurate model di�cult. Since the target signal used during simulations wasderived from measurements of the error and output signal, we were not be able to simulate anon-linear function that deviated too far from the one that was built-in on the sight-system. Bydesigning a new joystick’s control characteristic we can improve the tracking error by 8% duringsimulations but to obtain a better improvement it will be needed to add a lead-filter to increasethe phase-margin of the open-loop system.

Sammanfattning

I detta arbete analyserar vi operatorens paverkan vid manual tracking tasks for att forbattrahans formaga att folja ett mal. Operatoren skickar kommandon till ett siktessystem sa att hon,bland annat, kan folja ett mal. Systemet drivs av en statisk olinjaritet, aven kallad joystikenskontroll karaktaristik, vilken modifierar operatorens input.

For att analysera siktessystemets dynamik sa skapar vi en modell av opeatoren baserat paTustin-McRuer pilotmodell genom matningar tagna fran systemet. Vi anvander closed-loopidentifiering for att estimera parametrarna for operatorsmodellen. Under matningarna sag viatt operatoren har ett adaptivt beteende och att hennes beteende beror pa joystikens kontrollkaraktaristik. Darfor updaterar vi modellen sa att hennes forstarkning kan styras genom Gain-Scheduling regulatorn.

Resultaten visar att operatoren snabbt anpassar sig till systemet och att denna adaptiongor det komplicerat att fa en exakt modell. Eftersom mal-signalen som vi anvande under simu-leringarna beraknades fran mattningar av fel- och utsignalen sa kunde vi inte simmulera en olinjarfunktion som divergerade mycket fran den som anvandes i siktessystemet. Genom att designa enny kontroll karaktaristik for joysticken sa kan vi minska foljefelet med 8% i simulationer. For attytterligare forbattra resultaten kravs dock att man adderar ett lead-filter for att oka det oppnasystemets fasmarginal.

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Acknowledgements

First, I will begin by thanking my supervisor Daniel Hellberg, for his truthful interest in mywork and continuous support throughout the execution of this thesis. I also want to thank NinaSoderstrom for giving me the opportunity of working at Saab and make this research possible.

Secondly, I want to thank Prof. Cristian Rojas for his interest, support and guidance in themodelling part of this thesis. His feedback was very appreciated and our discussions enriched thecontent of this thesis. I would also want to express my gratitude to my examiner Prof. Elling W.Jacobssen for guiding me in the writing of the report, his advice and for trusting in the makingof this thesis.

Last but not least, I am most grateful to my family. Starting with my sisters Roxana andChristine for their kind words of support. To my fiancee Oskar for encouraging me to never giveup, for his unwavering support during my research and the writing of this thesis. His supporthas sustained me in days of frustration and sadness. Finally, I want to dedicate this thesis to mygrandmother Petronila Marquez who wanted to see this thesis finished but is now watching overme from heaven.

This research could not have been done without you all, THANK YOU!

Ste↵any Reyna MarquezStockholm, May 2019

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Contents

1 Introduction 11.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Research Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background 52.1 Manual Tracking Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Fire Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 SAAB’s Fire Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Operator Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Theory and Methodology 83.1 Control Theory and Manual Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 The Tustin-McRuer Operator Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Modelling of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3.1 Closed-loop Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Gain Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 Software and Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.5.1 Simulink and Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5.2 System Identification Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5.3 CoCo80 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Modelling the Human Operator 124.1 Identifying Operator’s Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.1.1 Experimental Tracking Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.1.2 Operator Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.1.3 Operator’s Gain Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Mathematical Model of the MM-System . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 The MM-system’s Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3.1 Non-Minimum-Phase System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3.2 Nyquist Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Control Law Design 295.1 Joystick’s Control Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1.1 Circle Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.2 Stability Margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.1.3 Evaluation of Tracking Performance . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Filtering of Command Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6 Simulations and Analysis 476.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.1.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.1.2 Tracking Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.3 Input-output Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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7 Conclusions and Future Work 547.1 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Bibliography 56

ATracking Experiments 58

BGain-Scheduling Validation Tests 60

CStability Criteria Plots 65C.1 Nyquist Criterion for Operator Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 65C.2 Circle Criterion for Control Law Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 67C.3 Stability Margin for Designed Control Law . . . . . . . . . . . . . . . . . . . . . . . . 71

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1 IntroductionIn this section we will give a description of the problem at hand and explain why command shapingis of high importance for sight-systems. Later on, we will give a brief overview of the related workthat has been done in the field of operator-in-the-loop and tracking tasks. Finally, we will present thelayout of the remainder of this report.

In early man-machine systems (MM-systems), the human controlled the system directly throughimposing his strength on the system. This was the case for early cars and air-crafts, among others. Toreduce the amount of force needed to control the system, servo-controls were added so that the systemwas controlled indirectly through signals. These signals could then be filtered, modulated or regulatedbefore being sent to the servo-motor responsible for the movement of the system. Through analyzingthe signals sent by the operator and their output we gain insight in both the system dynamics andthe behaviour of the operator.

Fire-control systems are military systems operated by human operators in order to aim and fireat hostile targets. Due to their heavy armament they are usually mounted on vehicles on land orsea and due to the need for compensating the movement of the vehicles they have gyro-stabilization.A main component of a fire-control system is the sight system which is responsible for tracking thehostile target. When aiming at a target we usually use the term line-of-sight which is the invisibleline that connects the system to the point in the environment that the sight system is pointing at.The operator acts as a controller of the sight system, she receives input in the form of visual stimulifrom a periscope or a screen and attempts to adapt her control behaviour to make the system stableand also move the line of sight to minimize the tracking error. In this way, her manual operation canbe seen as a closed-loop in which the controlled element is the sight system.

The work of a sight system can be split into three tasks. The first task is target-tracking, wherethe goal is to follow a target as close as possible. The second task is target-capturing in which the goalis to point the system to the target as fast as possible. The third task is designation which is whenthe sight system is used to investigate the surrounding area either without focusing on any specifictarget or looking for a target.

1.1 Problem Formulation

In this thesis, the interaction between the operator and a sight-system is studied in the context oftarget-tracking and target-capturing. The main assumption is that the operator acts as a controllergiving inputs to the sight-system such that its output (line of sight) follows a particular reference path(the target) as accurately as possible.

Of the three tasks of the sight system, tracking and capturing both involve small movements whiledesignation involves larger movements thus the system has to be stable under both small and largemovements. To achieve these tasks, the sight-system is connected to a non-linear function which iscalled a control characteristic that links the joystick’s deflection to angular velocity.

Since the operator is continuously interacting with the sight-system, she has extensive knowledgeabout its dynamics and the specific maneuvers required to make the line of sight point to the target.On the other hand, during a high demanding tracking task the operator is not able to adapt completely.We can represent this interaction as a closed-loop system in which we need to guarantee stability atdi↵erent states (tasks).

From control theory, we know that systems including non-linearities can be very complex andstability can be di�cult to achieve. The loop’s gain has to be adjusted so that stability criteriaare achieved at all times. The operator and joystick’s control characteristic are tightly linked, bothinfluence the loop gain but only the joystick’s control characteristic can be modified and tuned. This

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is why the design of this control characteristic is of high importance. Not only for stability issues butalso so that we can make it easier for the operator to control the sight-system.

Another aspect of the sight-system is the time it takes to aim, fire and eliminate a hostile target.This time Ttot is the sum of the time it takes to aim Taim and the projectile’s flight time Tflight (seeeq. (1)).

Ttot = Taim + Tflight (1)

In combat between two equally designed systems (same Taim and Tflight for both systems), it isdesirable for one’s system to have a Ttot that is shorter than the opposing system’s Taim since otherwisethe opposing system will have time to fire it’s projectile. According to measurements performed bySAAB:

Tflight

Ttot⇡ 0.2 (2)

Therefore if we can reduce Taim by 20% through tuning the joystick’s control characteristic our systemwould be able to eliminate the hostile target before it has time to return fire.

1.1.1 Research Question

The research questions for this thesis are the following:

• Is it possible to build a control characteristic so that the operator-sight-system is stable underall relevant tracking and capturing conditions?

• Can this control characteristic be optimized so as to minimize the tracking error in tracking-tasksby 20% compared to the current control law?

• Can we make the settling-time for capture-tasks decrease by 20% compared to the current controllaw?

1.2 Limitations

Due to availability issues, the experiments in this thesis will be performed using the mirror-head ofthe sight-system instead of the whole system. Fortunately, the mirror head is highly representativeof the system since when the mirror moves, the gyro-stabilization immediately moves the rest of thesystem. In addition to this, we do not have access to experienced military operators and thereforewe use engineers who had experience working with the system as operators. This might have animpact on the results and models since the operator learns how to better control the system throughexperience, much like an aircraft pilot. For the modelling part of this thesis, the operator is seen aspart of the system to be modelled and any issues stemming from inexperienced operators will impactthe model and the parameters will then require re-tuning for more experienced operators.

Even though the goal of this thesis is to study the relation between the operator and the sight-system, all the experiments are performed on the same system with one type of joystick. It is believedthat di↵erent joysticks have di↵erent dynamics and it is unclear how the findings in this thesis wouldtranslate to other joysticks or other systems.

Due to time limitations, we only analyze the system for single-axis tracking tasks with the focuson azimuth tracking though data was recorded for both axes. It is known that there is a correlationbetween elevation and azimuth (dual-axis tracking) and the model might need to be updated whenintegrating the second axis.

1.3 Related Work

The human operator constitutes a part of most control systems such as airplanes, turrets, aimingguns and sight-systems. From the perspective of human factors and system dynamics applications,

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research has been made for the dynamical operator behaviour to produce a dynamic model that canbe as accurate as possible. These models were developed to investigate the capabilities and limitationsof pilot-in-the-loop and to evaluate how he interacts with the system so as to predict possible errors[17].

Most of the operator models derived over the years are focused on describing the operator’s be-haviour during flight operations. This is due to how big the resulting disaster may become if even theslightest control action fails. The first pilot model was proposed by professor A. Tustin in 1944, withthe assumption that the operator could be described by a linear mechanism with time-delays [14]. Heused a servo mechanism theory to analyze the operator’s behaviour in manual control. He proposeda model in which the operator acted as a pure integrator with a time-delay.

Another model was based on the assumption that the pilot’s behaviour could be described bycontrol theory with the Crossover law which stated that ”a human being (a pilot) adjusts his/hercontrol actions to comply with the controlled element” [11]. It was proposed by Robert McRuer in the1960s and this model became the basis of further model variants such as the Tustin-McRuer Modelor Gross Model were the latter is the most widely used at present [11].

The purpose of these models are to summarize behavioural data and provide a basis for under-standing the operator’s control actions [12].

1.4 Scope

In this thesis, the control of a sight-system is evaluated with focus on analyzing the relation betweenan operator and the sight-system during manual control tasks. The sight-system contains a staticnon-linear function which is in charge of converting the input signals from the operator’s joystickdeflection to velocity.

We formulate the dynamics of the system as a continuous time-invariant system and considerscenarios where the operator performs tracking tasks. We base the operator model on the Tustin-McRuer behavioural model developed for aircraft control. Through linearization of the open-loopsystem we show that the operator model needs to be enhanced by using Gain-Scheduling on theoperator’s gain to take into account her adaptive behaviour. Due to the mirror-head of the sight-systemhaving a higher bandwidth than the joystick, we model it as a perfect integrator (servo-system).

The open-loop model is used for optimizing the static non-linear function with respect to trackingerror in position and in velocity. We also investigate how the design of the non-linear function a↵ectsthe stability of the system by using the Circle-Criterion which is a generalization of the Nyquistcriterion and assures asymptotic stability of the closed-loop if the Nyquist curve does not encircle agiven disk [6]. Based on analysis, we obtain a number of criteria which the non-linear function hasto fulfill in order for the system to be stable. For the optimization of the non-linear function, we usea Monte-Carlo algorithm running 10000 simulations taking into account the stability criteria. Earlyresults show that for increasing stability margins, a lead-filter needs to be added to the open-loop sothat it is easier for the operator to control the sight-system but due to time limitations, this filter cannot be implemented.

To acquire the data for building the mathematical model of the system, we performed trackingexperiments with human operators. Due to not having access to military people, we took help ofpeople from SAAB that had high knowledge about the sight-system and its dynamics.

1.5 Outline

The remaining of this thesis is organized in the following manner. Section 2, introduces the concept ofmanual tracking tasks and fire-control systems. We then describe the fire-control system investigatedin this thesis and the operator’s interaction with the system.

In section 3, we explain the basics of manual control and how the operator can be seen as acontroller. Secondly, we describe the Tustin-McRuer operator model which we use for modelling the

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operator. We also explain closed-loop identification and the approach used to identify the parametersof the operator’s dynamical model. Finally, we introduce the non-linear method gain-scheduling whichwe use for the operator’s gain.

In section 4, the operator and mirror-head models are derived. First, we explain the trackingexperiments used for obtaining the input and output signals and how we use them to estimate theoperator model parameters. Secondly, we investigate the relation between the obtained parametersand conclude that the operator’s gain is non-linear. We then explain how we use gain-scheduling tomodel this non-linearity. Finally we design the mirror-head model as a simple integrator and presentthe Nyquist stability criteria for the obtained model.

In section 5, we define the design requirements for the joystick’s control characteristic and explainthe three task areas of the fire-control system and how they relate to the joystick’s deflection signal.We use the circle criterion to obtain stability areas for the joystick’s control characteristic and finallyobtain numerical stability margins for the amplitude and phase-margin at di↵erent intervals. Finally,we evaluate the tracking performance of the designed joystick’s control characteristic and how we canfilter the operator’s command signal to improve the performance.

In section 6, numerical results of the models’ parameters and root-mean-square (RMS ) method arepresented. We also present simulations performed with the designed joystick’s control characteristicand finally we analyze the stability of the whole system. The thesis ends with a summarizing discussionand directions for future work in section 7.

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2 BackgroundIn this section, relevant background and definitions will be introduced to understand the relationbetween the operator and sight-system during tracking tasks. In the first section, we will covermanual-tracking tasks and what they involve. Later on, we will explain the main system of thisthesis: Fire-Control systems to have a better understanding about its functionality. Finally, we willcover the influence of the operator during feedback on manually-operated systems and extend it tofire-control systems.

2.1 Manual Tracking Tasks

In manual tracking tasks, the operator obtains information through her eyes: either by seeing a displayor his environment. When observing a display, two objects are visible: the target and a cursor. Theoperator uses the information displayed to make the cursor follow the target as close as possibletherefore this task is seen as tracking an object. Depending on the input device (also called actuator),the cursor can be moved using an input device in a single-axis or a double-axes. There is a large rangeof input devices: joysticks, a mouse, keyboard, hand gear, steering wheel, etc.

Figure 1: An example of pursuit and compensatory display [15] where y(t) is the output’s signal andr(t) the target’s signal

The two common tracking-displays studied are: compensatory-display and pursuit-display, whichcan be seen in fig. 1. Both displays only show the current values of the signals y(t), e(t), r(t) andno post or preview information is presented. In compensatory display, the operator concentrateson minimizing the error distance between the cursor and the target while in pursuit display sheconcentrates on following the target [15].

The most important class of situations for which operator-system models are useful are closed-loop compensatory tracking tasks. In this case, the operator acts on the displayed error between adesired command input and the comparable system output motion to produce a control action.[12]This control action can reduce or eliminate the error.

2.2 Fire Control Systems

Military fire-control (FC) systems such as those on combat vehicles or tanks have sight systems thatare normally stabilized through feedback with the help of a rate gyro or similar sensors. Since speed

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and accuracy are the main interests, often the vehicles’ line of sight is controlled by an automaticmanagement system which commands the sight system to point to a specific direction. This allowsthe operator to perform other actions simultaneously such as tracking a target. Even if the system isnot able to aim at a target itself, it can give a cue to the operator on how to aim.

On the other hand, many other FC systems are controlled manually which means that the operatorcontrols the line of sight to make the sight system point to an specific point through an actuator. Theoperator sends a reference signal to the sight system through moving the actuator side-wise (alsocalled Azimuth) or up-and-down (also called Elevation) so that the line of sight is directed to a pointof her choice.

Figure 2: An example of how the inside of a combat vehicle may look like (inspired by figures from[13])

Depending on the actuator, the reference signal sent by the operator, through the joystick, isdefined in angular velocity (rad/s), angle (rad) or angular acceleration (rad/s2).

2.3 SAAB’s Fire Control Systems

SAAB is a Swedish aerospace and defence company which mainly focuses on aircraft production, suchas the world-wide known JAS39 Gripen. But they also develop short range weapons, naval radarsand fire-control systems.

As explained in section 2.2, fire-control systems can be mounted on combat vehicles but also onships. This is the case for SAAB’s Trackfire which can be mounted in most types of military platforms.For this system, it has been implemented an stabilized independent line of sight: the line of sight isdecoupled from the weapons axes which gives the operator the possibility of maintaining the line ofsight on the target during the entire aiming. This system also has the help of a video tracker, which

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detects and presents moving objects (possible targets) from a video camera. Often, the video trackermakes use of an infra-red camera which is able to mark all possible targets and then lets the operatorchoose one target to, for example, start his tracking task.

Another FC-system that SAAB has developed is called UTAAS (Universal Tank and Anti-AircraftSystem), which can be mounted on tanks and combat vehicles. It works in the same way as theTrackfire: it has an independent line of sight so that the operator can retain the target in the centreof the reticle during the entire aiming and the gun-laying is automatically controlled by a computer.

2.4 Operator Feedback

The operator’s characteristic as a controller depends on four kinds of variables: task variables, en-vironment, operator-centered and procedural variables. During feedback through the operator, taskvariables such as: forcing function, displays, manipulator and the controlled element are the ones thata↵ect the operator dynamics most [18].

The existing feedback through the operator degrades the performance of the controlled system,specially in joystick-controlled systems. It was studied that this existing feedback can cause instabilityand possible oscillations [16]. Therefore, most of the work on operator feedback has been done in thisarea.

A Fire Control system is an integral part of a weapon system of any military vehicle and hasan integrated technology to stabilize the line-of-sight. The line-of-sight (LOS ) is the vector drawnbetween an electro-optical imaging sensor and the target. The sensors require a form of control tostabilize its pointing vector along the target on systems with movable carriers, this control is called:LOS Stabilization. It works on the measurement of the moving object orientation which is measuredby a gyroscope (the gyroscope indicates the rate of change of angle with time [18]). This stabilizationsystem limits the amount of image motion in the field of view during a frame. LOS stabilizationtherefore guarantees accurate aiming and tracking of the target in both elevation (along vertical-axis)and azimuth (along horizontal-axis).

For manual tracking systems, the operator controls the position of the LOS through a joystickbased on the image she observes via a video monitor. In this case, the control stabilization’s aim isto stabilize the system in the presence of disturbances and also follow the command signal given bythe operator.

In order for the operator to successfully track a target she can apply the following control tech-niques: closed-loop feedback, open-loop feedforward or a combination of both [17]. In closed-loopfeedback, the operator relies on her continuous fast sensing of the current system’s output, comparesit with the desired output (the target’s position) and then she acts on the di↵erence between the two(the error e(t)). When the operator is acting in open-loop feedforward, her commands are based onthe target only and the actual system’s output is not observed. This requires the operator to haveprevious knowledge of the system under her control.

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3 Theory and MethodologyThis section provides the theoretical background of manual control, closed-loop identification andgain scheduling. We also present the structure for the operator model and explain the meaning of itsparameters. At the end of this section, a brief description of the software and instruments used formodelling and identification of the system are also presented.

3.1 Control Theory and Manual Control

Automatic control has much similarity to Manual Control: in both we want the system to achieve acertain behaviour. We can thus see the human operator as a controller (trying to change the system’sbehaviour) and use control theory methods to model the operator’s controlling behaviour.

Modelling human action in dynamical environments has been studied for many years. And ascontrol theory advanced, it opened opportunities to better understand the human’s behaviour. Forexample, the researcher Tustin investigated the behaviour of an operator controlling the rational speedof an aircraft via a hand gear [2]. In this case, the operator had to compensate for the deviation ofthe speed which is seen as a typical tracking task (the operator follows a fly-path to reach her finaldestination). Apart from piloting an aircraft, there are many other situations where an operator wantsto make the output of a system follow a trajectory of her choice such as: recording an object whilekeeping it in the center of a camera, driving a car, aiming a tank turret, track a target, etc.

3.2 The Tustin-McRuer Operator Model

As explained in [9], operator models are typically represented in transfer function form. This transferfunction relates the operator’s control output in response to perceived error in the system’s responsecompared to the desired command.

The mathematical operator model most widely used is the Tustin-McRuer model which has theform presented in eq. (3). This model is based on Tustin’s precision model and provides less oscil-lating responses [11]. This model results from the hypothesis that the operator behaves like a linearcomponent.

FH(s) = KH .TLs+ 1

(TIs+ 1)(TNs+ 1).e

�⌧s (3)

KH denotes the operator’s gain. This variable determines the amount of control that the oper-ator commands, proportional to the perceived error. The term TLs + 1 denotes the lead dynamiccompensation produced by the operator and TIs+ 1 denotes the lag dynamic compensation which isassociated with her learned routine process. ⌧ denotes the transport (time) delay which representsthe delay from visual observation and information processing. Finally, the term TNs+ 1 denotes theoperator’s neuromuscular response. Her muscles do not respond instantaneously to a command moveand therefore she produces a lag-response, which is of (at most) approximately 0.1 seconds [9].

3.3 Modelling of Dynamical Systems

Mathematical models allow us to make predictions about how a system will behave and most of thework that will be done for this thesis implies having a good model of the operator-mirror system sincewe will be carrying out simulations. As explained in [8], we are mostly interested in modeling theinput-output behaviour of a system, specially of dynamical systems. Often in industry, the dynamics

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of the process can be too complex or we might only have partial knowledge of the process thus makingit di�cult to produce an exact dynamical model.

We use System Identification [8] to build a mathematical dynamical model of a system usingmeasurements of the system’s input and output. In System Identification we focus on determininga good model for a given system working in open-loop. All traditional identification methods used,such as: black-boxes, correlation and spectral analysis, work well and give good results for open-loopsystems. But when the measured data is collected from a feedback, the input and noise will becorrelated and thus the traditional methods fail when applied to closed-loop data. Therefore we usedi↵erent methods for closed-loop identification.

3.3.1 Closed-loop Identification

According to Lennart Ljung, ”the fundamental problem with closed-loop data is the correlation betweenthe unmeasurable noise and the input”. Due to this correlation, the obtained estimates will be biased[7]. To overcome this issue, there are three known methods:

1. Direct Approach: We use a prediction error method using measurements of the input andoutput signals to identify the open-loop.

2. Indirect Approach: We use measurements of the reference and output signals to identify theclosed-loop system and then solve for the open-loop system using the knowledge that we haveof the controller.

3. Joint Input-Output Approach: We regard the input and output of the system as the outputof a system driven by an external input or reference signal. Then we determine the open-loopparameters using the measurements from this new system.

The problem in industrial practice is that few controllers have a simple linear form and that non-linearities in the open-loop will make the input deviate from the reference. As explained in section 1,in our system the operator is working in closed-loop and the mirror-head is a non-linear system (dueto the joystick’s control characteristic), additionally, we cannot measure the signals produced by theoperator alone. Since we can manipulate the target and measure the visual error (operator’s input)through the mirror-head we can use the direct approach for identifying the parameters of the operatormodel.

3.4 Gain Scheduling

Gain Scheduling is a controller design approach for non-linear systems and has a wide range of usein industrial applications. It consists of building linear controllers depending on time-varying systemvariables (scheduling variables). This requires knowledge of how the dynamics of the system changewith their operating point. The resulting linear controllers are designed to achieve a desired stabilityaround the selected operating point. The linearization of the closed-loop system under one gain-scheduling controller is equivalent to linearization under a fixed-gain controller [6].

One advantage of gain scheduling is that it enables the controller to respond very fast to changingthe operating point’s condition. It is therefore important that the selected scheduling variables reflectthe changes in the system’s dynamics as the operating points change.

In section 1.1 we explained that the operator is continuously interacting with the sight-system andshe can control it through the joystick’s control characteristic. Therefore we will attempt to producea linear operator-model but if her dynamics change according to the joystick’s dynamics then we willuse gain scheduling to enhance the model.

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3.5 Software and Instruments

Since the foundation of this thesis is modelling and control theory, we need a software in which wecan use transfer functions, build models, perform simulations, work with vectors and analyze dataand non-linear systems. Therefore this thesis will be developed in MATLAB.

3.5.1 Simulink and Matlab

MATLAB is a computing environment that allows, among other things, plotting of functions anddata, implementation of algorithms and signal processing [21]. It also has a built-in environmentcalled Simulink which is a graphical environment for model-based design, simulation and analyze ofdynamical systems.

Simulink has many interfaces such as: graphical block diagrams and a set of block libraries thereforeit is widely used in Automatic Control. It also allows simulations of both linear and non-linear systems,as well as, connections between systems and sub-systems.

For this thesis we will use MATLAB and Simulink version R2016b with their included Toolboxes.

3.5.2 System Identification Toolbox

One of the toolboxes included in MATLAB is the System Identification toolbox in which we can esti-mate linear and non-linear dynamic models from measured input-output data. The data is importedand preprocessed to choose an estimating model and validate the results. This toolbox performsblack-box system identification to estimate the parameters of a given model [20].

In this thesis, we will use time-domain input-output data to estimate the parameters of the oper-ator’s continuous-time model. We will also use the Linear Analysis tool to plot bode diagrams of theopen-loop system.

3.5.3 CoCo80

CoCo80 is a signal analysis instrument widely used in industry, especially in aerospace and military,that is specially designed to record and present dynamical data. The data is recorded in real-time andthe analyzer is equipped with an LCD screen and a physical keypad. The recorded data is presentedin the LCD screen as a bode-diagram and can be saved to an SD-card for later use [22].

For this thesis, the analyzer is used to estimate the open-loop system Go (for a given control lawH) with data recorded in closed-loop. This is used to allow us to validate our operator model and itsparameters for di↵erent working points. The exciting signal is injected to the system between signalsu1 and u2 to the mirror-head as seen in fig. 3, the analyzer then presents the open-loop system Go(s)which is given by dividing these two signals and analyzing them at di↵erent frequency.

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Figure 3: Block-diagram of the sight-system. The operator moves the joystick, this signal is thenmodulated by the control law, H(s), and then sent to the sight-system, G(s). The output, y(t), is theline-of-sight which is observed by the operator. The exciting signal is connected and adjusted throughport 1. The signals u1 and u2 are measured and plotted as a Bode-diagram for further analysis.

From the block-diagram in fig. 3, we can obtain the relations in eq. (4) to eq. (6). The operator’shand and joystick can be put together as one transfer function denoted as Gop(s). The exciting signalis denoted as d. By using eq. (4) and eq. (5) one can thus derive mathematically the open-loop systemGo(s) = H(s)Gop(s)G(s), that the analyzer CoCo80 estimates.

y = G(s)u2 (4)

u1 = H(s)Gop(s)y (5)

u2 = u1 + d (6)

Inserting eq. (4) into eq. (5) we obtain:

u1 = H(s)Gop(s)hG(s)u2

i(7)

we then multiply the terms, which yields:

u1 = H(s)Gop(s)G(s)u2 (8)

The transfer function from u2 to u1 in eq. (8) denotes the open-loop system Go which means that byplotting these signals in terms of phase-margin and amplitude-margin the analyzer can estimate andpresent the bode-diagram for Go while measuring the signals in closed-loop.

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4 Modelling the Human OperatorIn this section, we will start by identifying the operator model from experimental operator-in-the-loop data directly. Then we will adjust relevant parameters to capture the operator’s behaviour asaccurately as possible. Finally, we will prove that the model obtained is stable according to Nyquistcriterion and present its Bode diagram.

4.1 Identifying Operator’s Behaviour

From previous research, it has been shown that the operator’s control behaviour becomes more erraticand non-linear when the dynamics of the controlled system are di�cult to predict [9]. This is why, toacquire relevant measurement data, we build experimental tracking tasks based on non-constant targetpaths. After that, we analyze all the data and estimate the parameters of the operator’s mathematicalmodel.

4.1.1 Experimental Tracking Set-Up

Measurements of the operator’s control action as a response to a visual stimulus, the target, areacquired from human operators. As explained in section 3.3, the data measured is acquired in closed-loop. Therefore we have to make sure that the chosen measurement signals reflect the true behaviourof the operator. This is achieved by making the target’s path unpredictable to the operator thusmaking sure that the measurements capture her dynamics. The target will be moved so as to mimica parabola-segment signal and we will also change its velocity, at random, throughout the trackingexperiment. This gives us the dynamics of the operator since he will not be able to easily predict themovement or velocity of the target.

Two quasi-random variations in the target are done: short and fast versus long and slow parabola-like. An example of these recorded variations is shown in fig. 4.

Figure 4: An example of the target’s position signal (measured by a video tracker) for a given tracking-trial test.

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Since the operator obtains an input (visual stimuli) and produces an output (her command throughthe joystick) the Direct Approach presented in section 3.3.1 is chosen for the identification of her modelparameters. Two signals are recorded and used: the joystick’s movements (operator’s output) andthe visual error (operator’s input). The joystick movements, aka deflections, are measured at 32Hzvia an analog-digital converter which is connected to a PC. The visual error is also measured at 32Hzwith the help of a video-tracker.

The tracking experiments lasts about 3 minutes and human subjects are told to wait for the targetto move before they can make any movement on the joystick.

As mentioned in previous sections, the operator is tightly linked to the joystick’s control character-istic. Therefore, we use a linear function, see eq. (9), as control characteristic for these experiments.In this way, we can isolate the joystick’s deflections and how they are a↵ected by the joystick’s gainKjoy. Measurements are done for di↵erent Kjoy-values.

f(u) = Kjoy · u (9)

The variable u denotes the joystick’s deflection signal, f(u) denotes the velocity (in rad/s) sent tothe mirror-head and Kjoy is a constant value in the scale [0, 1].

A more in depth description of the experiments done, data recorded and tables can be found inappendix A.

4.1.2 Operator Mathematical Model

Since the operator only responds to the visual error (has a compensatory control behaviour) she canbe modeled as pure feedback control. This model is a continuous-time model whose structure is basedon the Tustin-McRuer operator model introduced in section 3.2. With the Tustin-McRuer model, theoperator’s control behaviour can be entirely described by a linear, time-invariant model.

We estimate the operator parameters KH , TL, TI , TN and ⌧ for the di↵erent Kjoy-values recordedduring the tracking experiments by using black-box identification. This method consists on choosingthe model structure, determine the size of the model and use observed data to identify the modelparameters [8]. Generally the model structure for observation data can be denoted by eq. (10) inwhich y denotes the prediction value of observation data y described by vector ✓ and g denotes theunknown function.

y(k|✓) = g(xk, ✓) (10)

The measured data is generated according to eq. (11) where q denotes the shift operator.

y(t) = G(q, ✓)u(t) +H(q, ✓)e(t) (11)

When the model structure is found, we use a Maximum Likelihood estimator (MLE) to find theparameters of the model. The MLE finds the values of the model parameters that maximizes thelikelihood that the data was generated by the model. It also gives the ”fit” of the prediction in formof a probability (percentage value) [8].

The data obtained from the tracking experiments is first normalized through subtracting the mean,then we divide the resulting data into a training set and a validation set. With the training set we fitthe model parameters to the data and to avoid overfitting we use the validation set. Each set consistsof 50% of the data. Since we know the transfer function structure to use for our operator model,the training will consist of identifying the parameters of the process model. This training consists ofiterative minimization of the error between the estimated output and the data’s output. The wholeprocedure is performed using MATLAB’s System Identification Toolbox.

From the estimations of the operator model parameters, we can seen that TL, TI , TN and ⌧ do notvary based on the joystick’s gain Kjoy, however the operator’s gain KH does vary. This reflects ourprevious assumption in section 1.1 about the operator-joystick link: the operator senses the joystick’scontrol characteristic’s gradient and applies a gain accordingly to make the loop gain stable. Theoperator’s gain (KH) is plotted against the joystick’s gain (Kjoy) to analyze their relationship.

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Figure 5: Plot of the operator’s gain for two human operators

Figure 6: Plot of the open-loop gain for two human operators

From fig. 5 it can be seen that for Kjoy-values bigger than 0.04, KH decreases as Kjoy increasesfor both operators. For Kjoy-values below 0.04, the estimation for the operator’s gain KH gives low

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values. This is because the target’s velocity is higher than what the operator can send to the mirror-head and therefore he does not try to follow the target completely and his gain is lower. He adjustshis gain so as to keep up with the target’s velocity at di↵erent intervals but not during the wholetracking task (for more details, see appendix A). On the other hand, from fig. 6, it can be seen thatthe loop-gain Ktot is kept the same for Kjoy between 0.06 and 0.10. This means that the operatordoes not have to apply much gain to the loop for it to be stable.

From these two figures, we conclude that the operator’s gain varies as a function of the joystick’sgain. To reflect this operator behaviour in our model, we will use the non-linear method: GainScheduling.

The operator parameters are related to each other, which means that one parameter will a↵ect theothers. The value of his gain KH will compensate to variations in TL, if TI is small then the time-delay⌧ and KH will be adjusted to compensate and adjust the closed-loop stability of the system. Sincewe want to build one operator model then we analyze changes on each of its remainder parameters todecide on their values.

Figure 7: Plot of the neuromuscular constant TN for two di↵erent operators. Points with TN -valueequals to zero, are measurements not performed for the respective operator.

The TN -values for both operators have the same tendency as can be seen in fig. 7. There is oneoutlier from the measurements in operator 2 which has a value over 0.04, this might be due to himnot being accustomed to the linear joystick’s control characteristic at that time. The TN points atzero are those measurements which are not performed for the respective operator.

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Figure 8: Plot of the lead time constant TL for two di↵erent operators. Points with TL-value equalsto zero, are measurements not performed for the respective operator.

As for the TL-values seen in fig. 8, they are slightly di↵erent for both operators. This is becausethe lead constant might be di↵erent for individual operators [10]. Excluding three measurements ofoperator 2, the TN -values for both operators are kept between 0.25 and 0.5.

Figure 9: Plot of the lag time constant TI for two di↵erent operators. Points with TI -value equals tozero, are measurements not performed for the respective operator.

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As we can see in fig. 9, the TI -values for both operators are kept between 1 and 2.5. Even thoughthere is one outlier from the measurements in operator 2 which has a value of 0.044. In this case, theoperator experiences that she can not compensate enough for such a low Kjoy-gain.

Figure 10: Plot of the time-delay ⌧ for two di↵erent operators. Points with ⌧ -value equals to zero, aremeasurements not performed for the respective operator.

The ⌧ -values shown in fig. 10 are around 0.10 for operator 1 while for operator 2 they vary. In thecase of operator 1: he’s reaction time is less for low Kjoy-values since he has had a lot of experiencewith the system before and is able to adjust faster. For operator 2: his reaction time is almost 0.18 fora very lowKjoy-value, this is due to him experiencing compensating issues and therefore his time-delaywas estimated to 0.18.

As explained and seen from fig. 7, fig. 8, fig. 9 and fig. 10, the remaining parameters do not varymuch for the di↵erent operators. Therefore, for each parameter, we will remove the outliers andthen set the parameter to the mean of its remaining values weighted towards operator 1 since hermeasurements are more similar to military operators. This yields the results presented in table 1.

TI TN [sec] TL ⌧ [sec]

1.85 0.003 0.38 0.10

Table 1: The remaining operator parameters and their respective final value

4.1.3 Operator’s Gain Scheduling

As discussed before, the operator’s gain KH has to be adjusted so that it can vary according tothe derivative of the joystick’s function f(u) in eq. (9). Therefore we use Kjoy as the schedulingvariable. We select the majority of operating points in the tracking area because this area is of thehighest interest. No points will be selected in the designation area since this is not the focus of thesimulations. Then the gain of the system is adjusted at the selected operating points and we obtain

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a series of operator gains Kgs which are gathered in table 2. To smooth the gain transitions betweenoperating points we use the numerical method cubic spline [19] as seen in fig. 11.

Kjoy 0.05 0.065 0.075 0.085 0.10 0.15

Kgs 188 118 104 93 76 43

Table 2: The operating points Kjoy and their respective operator gain Kgs

Figure 11: Gain Scheduling curve for the values in table 2. We see that for Kjoy-values larger than0.05, the operator’s gain decreases with the Kjoy value.

In fig. 11 we can see that the estimation for the operator gain at low joystick’s gain is 188 andthen it decreases as Kjoy increases. The lowest operator gain obtained by the estimation is 43. Track-ing experiments tend to fail when using Kjoy values higher than 0.15 because the operators cannotcompensate for such a high joystick’s velocity. Therefore we use the analyzer CoCo80, introduced insection 3.5, to also measure the loop’s gain at high Kjoy-values.

Finally, we simulate the closed-loop system and analyze the bode-diagram of the open-loop Go(s) =GH(s)f(u)Gmirror(s) to see that the interpolation process between one operating point to anotherworks as intended and also that the behavioural pattern is still the same.

The bode-diagram of two operators for di↵erent scheduled-variables can be seen in fig. 12 andfig. 13.

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Figure 12: Bode diagram of the open-loop system Go for operator 1 in the frequency interval up to10Hz

Figure 13: Bode diagram of the open-loop system Go for operator 2 in the frequency interval up to10Hz

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From fig. 12 and fig. 13, it can be seen that we have achieved an amplitude slope of approximately�20dB per decade which according to [12] is an expected characteristic of the rule Equalization Selec-tion in which ”seek or create (by equalization) a fair stretch of -20 dB/decade slope for the amplituderatio and adjust the loop gain so as to put the unity-amplitude crossover frequency near the higheredge of this region”. To provide a good low-frequency response in the closed-loop system Gc we haveto achieve |Go| >> 1 at those frequencies, which is in shown in both figures.

To prove that the gains for the operator’s gain-scheduling are accurate and to obtain measurementsat very low/high joystick gain values, we use the dynamic signal analyzer CoCo80 (which was intro-duced in section 3.5). This analyzer induces a disturbance signal (sinus signal) to the sight-systemwith a gain of Ksinus = 0.05 making the line-of-sight move to the right and to the left. The operatoris told to compensate for the sinus deviation by moving the line-of-sight back to the centre of thescreen. A more detailed description of this test can be found in appendix B along with data obtainedand plots.

We analyze the bode-diagram for the di↵erent scheduled-variables recorded by CoCo80 and fromthem we extract the amplitude Am at frequency w = 0i. Finally we use eq. (12), to calculate theoperator’s gain through solving for KH (see appendix B for proof of the calculation).

|G(0i)| = 20log10(Kjoy ·KH) = Am (12)

The KH -values are gathered and yield the gain-plots presented in fig. 14 and fig. 15.

Figure 14: Plot of the Gain-Scheduling obtained by CoCo80 (blue dots) in comparison to the oneobtained by Linear Analysis (red crosses).

In fig. 14 we can see that the operator gain obtained by CoCo80 has the same pattern and similarvalues as the gain-scheduling derived analytically. Both start above 180 for low Kjoy-values and end

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at around 15, which means that the operator will continue having a gain of at least 15 for Kjoy-valuesup to 1 (the highest joystick’s gain) and it will not go down to zero since some loop gain is needed forthe line-of-sight to move (see appendix B for more details).

Figure 15: Plot of the open-loop gain obtained by CoCo80 (blue dots) in comparison to the oneobtained by Linear Analysis (red crosses).

As for the open-loop gain, we can see in fig. 15 that CoCo80 gave resembling values as the open-loop gain derived analytically. The final values of the gain-scheduling (KH = Kgs) for the operatormodel can be seen in eq. (13).

KH = Kgs = [180, 176, 155, 142, 126, 110, 80, 61, 40, 25, 25, 25] (13)

4.2 Mathematical Model of the MM-System

As explained in section 1, the closed-loop system is composed of a controller and a controlled system.The controller is, in this case, the operator himself while the system to be controlled is the mirror-headof the sight-system. This closed-loop system is known as man-machine system or MM-system. Theblock diagram for the MM-system can be seen in fig. 16.

Since the mathematical model of the operator was successfully derived in the previous section, wewill now proceed to derive a mathematical model for the mirror-head.

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Figure 16: Block Diagram of the manual-control system with compensatory display where GH(s) isthe operator, f(u) the joystick’s control characteristic and G(s) the mirror-head

The mirror-head of the sight-system is equipped with line-of-sight stabilization (as explained insection 2). This means that it is able to take an input signal, u(t), and send the exact input as outputsignal, y(s), without delays. Also the mirror-head has a higher bandwidth than the rest of the system,this is why its mathematical model is chosen as a perfect-integrator. In this way, the mirror-headbecomes a Servo-system. The transfer function of the mirror-head is presented in eq. (14) below.

G(s) =1

s· P (s) =

1

s· 1 (14)

To have a more accurate model, there are two things left to add to the block-diagram in fig. 16:

1. Saturation at operator’s output: The saturation can be defined as an undesired limitationwhich restricts the operator’s intervention in the system. This is to reflect the behaviour oftrying to press the joystick to the right/left bottom. The joystick’s deflection signal can only beup to ± 1 but the operator can be allowed to press the joystick to the bottom.

2. Video Chain as time-delay: A time-delay is added before the error signal e(t) since the signalreceived by the operator does not correspond to the contemporary process information. Thisis to reflect the time it takes for the video chain to present an image. There is a video chainconnected to the sight-system and computer which allows the operator to observe the target andthe track through a screen. Since the video chain has to process and present the image thenthere’s a delay in the signals presented to the operator (the target’s path and line-of-sight).

The saturation is set up to saturate the operator’s signal at �1 and 1 like shown in fig. 17.

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Figure 17: Saturation function plot from [5]

The video chain has a time-delay of at most 2 images per frame. With the help of a high-velocitycamera, the time-delay for the video chain (⌧2) is estimated to be in the interval [0.05, 0.10]. Themathematical equation for the video chain, Gvc(s), then became like in eq. (15) below with ⌧2 = 0.07.

Gvc(s) = e�⌧2s (15)

The block-diagram in fig. 16 is updated with the saturation and time-delay which yields the finalblock-diagram model for the operator-mirror system presented in fig. 18.

Figure 18: Final block-diagram for the closed-loop system of the MM-system. The variable ed(t)denotes the delayed error signal sent to the operator, ujoy(t) denotes the deflection signal sent from theoperator/joystick and u(t) denotes the converted velocity sent from the joystick’s control characteristicto the mirror-head.

4.3 The MM-system’s Stability Criteria

To analyze the stability of the operator-mirror system or MM-system, we use the Nyquist Criterion.According to this criterion, the closed-loop system Gc(s) is asymptotically stable if the open-loopsystem Go(s) does not have poles in the right-half plane and the point �1 is not encircled by theNyquist curve and lies to the left of the curve [4].

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4.3.1 Non-Minimum-Phase System

We start by analyzing the poles of the open-loop Go (from e(t) to u(t), shown in fig. 19) with f(u) = 1and without the saturation, since the system will only saturate at 100% joystick’s deflection (whenuop(t) = ±1):

Go(s) = Gvc(s)GH(s)f(u) = e�⌧2s ·KH

TLs+ 1

(TIs+ 1)(TNs+ 1)e�⌧s · 1 (16)

Figure 19: Block-diagram of the operator-mirror system derived in section 4.2. The marked blocksare the open-loop system Go analyzed in this section

From eq. (16), it can be seen that the poles of Go(s) are given by the pole-polynomial

p(s) = (TIs+ 1)(TNs+ 1)

solving for p(s) = 0 yields the following poles:

s1 = �1/1.85

s2 = �1/0.003(17)

To analyze the zeros of Go, first we approximate the total time-delay, e�Ts = e�(⌧2+⌧)s, by a rational

function as shown below:

e�Ts ⇡

1� sT2

1 + sT2

(18)

substituting this into eq. (18) leads to the following equation:

Go(s) = Gvc(s)GH(s)f(u) =1� sT

2

1 + sT2

·KHTLs+ 1

(TIs+ 1)(TNs+ 1)· 1 (19)

From eq. (19) it can be seen that the zero-polynomial is given by:

z(s) =⇣1� sT

2

⌘(TLs+ 1) (20)

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solving for z(s) = 0 yields the following zeros:

z1 = 0.17/2

z2 = �1/0.38(21)

Since all the poles are real and lie in the left-half plane, then Go(s) is stable. Because of the zero(which comes from the time-delay) in the right-half plane then G(s) is also non-minimum-phase.The poles s1 and s2 will influence the speed of the step response. Plots of the step response for theopen-loop system Go(s) and the closed-loop system Gc =

Go1+Go

can be seen in fig. 20 and fig. 21.

Figure 20: Step response of the open-loop in fig. 19 with f(u) = 1, from error signal e(t) to controlsignal u(t). The operator’s gain for this case is KH = 176

As we can see from fig. 20, there is a time-delay of 0.07 seconds before our model produces a signalbut then it reaches a steady-state with no overshoots or undershoots. This means that our model iswell-damped. It can also be seen that our model reaches a steady-state value of 176. It is known thatfor PID-controllers (such as our operator model), the steady-state value is the gain of the controller.[4]

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Figure 21: Step response of the closed-loop operator model with KH = 25 and f(u) = 1, from errorsignal e(t) to target’s position r(t)

From the step response plot in fig. 21, it can be seen that the time-delay is present. The closed-loop is not well-damped and the settling time (Ts) is around 5 seconds. For feedback systems withtime-delays, we expect a decrease in the phase-margin [4] but if the time-delay is too big comparedto the desired cross-over frequency wc then this decrease will impose a challenge on the stability ofthe whole operator-mirror system. This problem will need to be taken into account when designingthe joystick’s control characteristic.

4.3.2 Nyquist Criterion

As shown in section 4.3.1, the open-loop Go(s) = Gvc(s)GH(s)f(u) does not have unstable poles.Therefore, we plot the Nyquist curve of Go(s) to see if the criterion is fulfilled for the operator’sgain-scheduling with a linear f(u) (which means f(u) = Kjoyu). The value of Kjoy is chosen as theworst-case slope-value that it can have for the given operator gain.

Nyquist plots for three di↵erent operating points (velocities) can be seen in fig. 22 to fig. 24. Thesefigures show that the Nyquist curve does not encircle -1 and that this point lies to the left of the curvefor the whole gain-scheduling interval. These plots are kept the same at di↵erent operating points(Nyquist plots for other fixed operating points can be found in appendix C).

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Figure 22: Nyquist curve (blue line) of the open-loop with KH = 176 and Kjoy = 0.023. The redcross shows the point -1.

Figure 23: Nyquist curve (blue line) with KH = 155 and Kjoy = 0.038. The red cross shows the point-1.

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Figure 24: Nyquist curve (blue line) with KH = 25 and Kjoy = 0.48. The red cross shows the point-1.

Since the Nyquist criterion is fulfilled (the curve does not encircle -1), we can guarantee stabilityfor a linear joystick’s control characteristic (f(u) = Kjoyu) and thus the operator model derived inthis section is asymptotically stable. According to a theorem of Stability of Linear Systems: ”Alinear, time invariant system is input-output stable if and only if its poles are inside the stabilityregion” Theorem 3.9 from [5]. Because there is no pole/zero cancellation in the open-loop and due toasymptotic stability then, by using this theorem, the open-loop system Go(s) is also input-outputstable.

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5 Control Law DesignIn this section, we will proceed to the design of the joystick’s control characteristic and go throughrequirements for implementing a shaping filter. We will first design the joystick’s control characteristicto achieve some stability requirements, then use non-linear methods to prove that the system is stableand evaluate its tracking performance. Finally, we will give directions on how to design the shapingfilter to increase stability margins for the capture/designation area and increase the phase-margin ofthe operator-mirror system.

5.1 Joystick’s Control Characteristic

The joystick is linked to the mirror-head system through a control characteristic. This control char-acteristic takes the joystick’s deflection as input and converts it to rad/s as output. As mentioned insection 1.1, the joystick’s deflection signal is converted to velocity by using a static non-linear functionf(u) in the form presented in eq. (22) below.

f(u) = C1u+ C2u2 + C3u

3 + ...+ Cnun (22)

The u-value represents the joystick’s deflection (signal sent by the operator) scaled on the interval[0, 1]. The degree of the polynomial, n, is chosen so that the tracking, capture and designation tasksare achieved.

The static non-linearity has to satisfy the following physical requirements:

• The output has to be zero at joystick’s deflection equals to zero: If the operator doesnot move the joystick then the joystick’s control characteristic has to send a velocity signal ofvalue zero.

• At 100% deflection (scale = 1), the joystick’s control characteristic has to output a velocity ofat most 1 rad/s which is the maximum velocity of the sight-system.

• The joystick’s control characteristic should be an increasing continuous function:This is important since the operator has an expectation that the higher the angle of the joystick,the faster the movement of the system. Therefore the first derivative of f(u) should be positivefor all points.

These requirements can be translated into mathematical form as:

f(u) = 0, when u = 0

f(u) 1, when u = 1

f(u) < 1, for |u| < 1

f0(u) > 0, f

0(u) f0(u+ "), for 8|u| > 0 and 9" > 0

(23)

From the tracking experiments done for estimating the parameters of the operator-model (seesection 4) and previous knowledge of sight-systems (see section 2), the deflection span for the threetasks can be determined. The tracking area is set from 0% up to around 30% deflection, the capturearea from around 30% up to around 80% deflection and the designation area from around 80% up to100% deflection. These areas are presented in fig. 25.

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Figure 25: Plot of regions for the three tasks and their mapping to an arbitrary f(u)

The operator tracks a target of her choice in the tracking area and when she wants to changetarget she will increase the joystick’s deflection so that her signal will be in the capture area. We needto guarantee stability of the whole system in these two areas since they are the most important tasksof the mirror-head and sight-system.

During designation tasks, the operator is not as interested in stability but rather in moving themirror-head as fast as possible to start tracking another target or point to another direction. Thereforewe need to guarantee stability up to when she is right above the capture/designation area (yellow linein fig. 25).

In order to analyze the operator-mirror system with a static non-linearity, we re-form the blockdiagram in fig. 18 so that we can isolate the non-linear function and study the influence of the signals.Following the steps in [5] Chapter 11, first the summation point of the reference signal r(t) is movedafter the blocks Gvc(s) and GH(s). Then the linear blocks are merged obtaining Gvc(s)GH(s)G(s)which allows us to finally merge the non-linear blocks into one block: sat(u)f(u). This re-shapingyields the block-diagram in fig. 26.

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Figure 26: The non-linear block-diagram for the servo operator-mirror system where r(t) = 0 ande1(t) is the signal u(t).

As explained in section 4 the saturation sat(u) will only saturate the operator/joystick’s signalwhen she is in the designation area’s upper limit. The system will therefore be unstable in the endof the designation area but this will not impact the stability during tracking or capture tasks sincesat(u) is inactive in these two areas. Therefore it is decided to focus on finding the stability regionsfor f(u).

To analyze the stability of the system for a given f(u), we use the Circle Criterion. Accordingto this criterion, the servo system in fig. 26 is input-output stable if the operator-mirror systemGom = Gvc(s)GH(s)G(s) does not have any poles in the right-half plane and its Nyquist curve doesnot enclose a circle that goes through the points [�1

k1,�1k2

].[5]Furthermore the non-linearity f(u) has to fulfill the following requirement:

f(0) = 0, k1 f(u)

u k2 for u 6= 0 and k1, k2 > 0 (24)

which means that the graph of f(u) will be enclosed in a cone-shaped sector with lower-bound valuek1 and upper-bound value k2.

5.1.1 Circle Criterion

Consider the feedback connection of a system G(s) with a static and unique non-linearity describedby the function '(u) as seen in fig. 27. The graph of this function satisfies a sector condition if '(u)fulfills the requirement in eq. (24) and we say that '(u) belongs to the sector [k1, k2] [6]. Mappingthis sector to the imaginary axis we obtain a circle with diameter

h�1

k1,�1

k2

i

if G(s) is stable then its Nyquist curve does not penetrate this circle. This gives the Circle Criteriondescribed in theorem 1.

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Figure 27: Linear system G(s) with non-linear feedback given by '(u)

Theorem 1 Assume that G has no poles in the right-half plane and that the non-linearity ' satisfiesthe sector condition in eq. (24). The system in fig. 27 is global asymptotically stable if the Nyquist

curve of G does not encircle or enter the circle D

⇣�1k1

,�1k2

⌘[6].

In this thesis, we aim to design a static non-linear function f(u) that ensures stability of the mirror-head during the three tasks described in section 5.1: tracking, capture and designation. Therefore, weuse the Circle Criterion to find the sector bounds of f(u) for the tracking and capture regions. For thedesignation region, we know that the system will be unstable for very high velocities and therefore weuse the Circle Criterion to find the maximum upper-bound that f(u) can have for this region. Thesebounds will be found using fixed operating points of the operator’s gain-scheduling.

To apply the Circle Criterion to our system in fig. 26, we need to ensure that the open-loop systemGom = Gvc(s)GH(s)G(s) is stable. It was concluded in section 4.3 that the system Go = Gvc(s)GH(s)has stable poles so before applying the criterion we need to ascertain the following:

1. The open-loop system Gom has also stable poles

2. The Nyquist curve of Gom does not enclose the circle D(k1, k2).

The open-loop system Gom is given by:

Gvc(s)H(s)G(s) = e�⌧2s ·KH

TLs+ 1

(TIs+ 1)(TNs+ 1)e�⌧s · 1

s(25)

From eq. (25) it can be seen that the poles of Gom(s) are given by the pole-polynomial:

p(s) = (TIs+ 1)(TNs+ 1)s (26)

solving for p(s) = 0 yields the following poles:

s1 = �1/1.85

s2 = �1/0.003

s3 = 0

(27)

Since there are no poles in the right-half plane then the open-loop system Gom has stable poles.Having a pole at origin means that the system’s output does not change faster than its input [4] andthe system is still stable. The dominating poles s1 and s2 will influence the speed of the system’s stepresponse.

There are two versions of the Circle Criterion: the first version requires the system to be intransfer function form while the second version requires the system to be in state-space form [5]. Dueto continuous systems with time-delays not having a common state-space form, the first version ofthe Circle Criterion (given in theorem 1) is applied for this thesis.

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A circle and the Nyquist curve of Gom for each of the operator gains derived in section 4.1.3 wereplotted. The diameter of the circle denotes the requirements on the joystick’s control characteristicf(u) specified in eq. (24). The values k1 and k2 are chosen as working points.

Representative plots of the Circle Criterion for the three di↵erent regions of f(u) can be seen infig. 28, fig. 29 and fig. 31. On each plot, we can see the circle (on red) along with the values for k1 (tothe left of the circle) and k2 (to the right of the circle). Circle Criterion plots for other fixed operatorgains can be found in appendix C.

Figure 28: Circle Criterion showing the least and maximum k-value that the system could have atoperator gain KH = 176. This represents the beginning of the tracking area. The blue line is theNyquist curve of the system Gom.

In fig. 28, we can see that there is a very small margin between the circle and the Nyquist curve(the curve almost touches the right side of the circle). The maximum value of k2 at this operator gainis 0.035 which represents a 13% deflection angle of the joystick.

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Figure 29: Circle Criterion showing the least and maximum k-value that the system can have atoperator gain KH = 126. This represent the transition between the tracking and capture area. Theblue line is the Nyquist curve of Gom

Figure 30: Circle Criterion showing the least and maximum k-value that the system can have atoperator gain KH = 80. This represents the transition between the capture and designation area.The blue line is the Nyquist curve of Gom

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In fig. 29, we can see an increase in the margin between the circle and the Nyquist curve comparedto fig. 28. The maximum value of k2 at this operator gain is 0.13 which represents a 34% deflectionangle of the joystick. In fig. 30, the margin between the circle and the Nyquist curve has also increased.The maximum value of k2 at this operator gain is 0.24 which represents a 46% deflection angle of thejoystick.

Figure 31: Circle Criterion showing the least and maximum k-value that the system can have atKH = 25 This represents remaining designation area. The blue line is the Nyquist curve of Gom

In fig. 31, we can see that the maximum value of k2 at this operator gain is 0.70 which representsa 81% deflection angle of the joystick. This is the maximum upper bound that f(u) can have sinceincreasing k2’s value will make the Nyquist curve of Gom enter the circle and we will not be able toguarantee the stability of the system. In our operator model designed in section 4, the operator gainfor the whole designation area is 25 and with k2 = 0.70 we can guarantee stability of the system forup to 0.52 rad/s.

As we can see from the plots above, the Circle Criterion is fulfilled for the whole gain-schedulingat each region. We can guarantee stability of the system in the whole tracking and capture area. Atvery high operator gain we have small margins which is expected since the joystick’s deflection is lessthan 5% and the loop gain will be dominated by the operator’s gain. When increasing the joystick’sdeflection, the operator’s gain will decrease since the operator does not need to add much gain to theloop and therefore we will have slightly larger marginals in the remaining tracking area and capturearea.

To quantify how much margin we have at the di↵erent operator gains, we plot the bode diagram ofthe open-loop Gom and read out both the amplitude margin Am and phase-margin 'm of the system.Bode diagram plots for the respective sector bounds shown in the previous figures are presented infig. 32, fig. 33 and fig. 35 (plots for other fixed operator gains can be found in appendix C).

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Figure 32: Bode diagram for Circle Criterion in fig. 28 at KH = 176. This shows the beginningof the tracking area in which we have an amplitude margin of Am = 18.7 dB and phase-margin of'm = 34.4�

Figure 33: Bode diagram for Circle Criterion in fig. 29 at KH = 126. This shows the transitionbetween the tracking and the capture area in which we have an amplitude margin of Am = 9.5 dBand phase-margin of 'm = 28.8�

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Figure 34: Bode diagram for Circle Criterion at KH = 40. This shows the last part of the capturearea in which we have an amplitude margin of Am = 9.9 dB and phase-margin of 'm = 29.2�

Figure 35: Bode diagram for Circle Criterion in fig. 30 at KH = 25. This shows the designation areain which we have an amplitude margin of Am = 9.3 dB and phase-margin of 'm = 28.6�

From fig. 32 to fig. 35, the open-loop’s phase-margin 'm is kept between 28� and 34� for thethree di↵erent areas. By designing the joystick’s control characteristic f(u) we want to improve the

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phase-margin of the system in the tracking and capture area (which, as explained in section 1.1, arethe most important tasks of the fire-control system). This means that we need to ensure that thelowest point in this interval is not decreased. Therefore we have to ensure that the designed f(u)has at least a phase-margin of 30� for the tracking, capture and designation areas. As explained insection 4.3, we have a restriction in how much the system’s phase-margin can be increased due to thetime-delays (from the operator and the video-chain model) over the whole system.

As for the amplitude margin Am, it can be seen that it decreases as KH decreases. This is alsoshown in the Circle Criterion plots and means that if f(u) is not kept within the sector boundswith a maximum k2-value of 0.70 the system will become unstable (the system will have reached anamplitude margin of Am = 0). So we need to ensure that the designed f(u) has an amplitude of atleast 9dB for high joystick deflection values, which also applies to the transition between capture anddesignation area.

5.1.2 Stability Margins

A Monte-Carlo based optimization algorithm is produced to find the values for the C-constants C1,C2, ..., Cn in eq. (22) that meet the stability and margin requirements obtained in section 5.1.1 whichare stated below:

• Keep f(u) inside the sector bounds for the three areas.

• Have an amplitude margin Am of at least 9dB at high joystick’s deflection values.

• Have a phase-margin 'm of at least 30� at low joystick’s deflection values (tracking area).

The optimized joystick’s control characteristic f(u) has to also fulfill the physical requirements spec-ified in eq. (23).

After 10000 iterations, our algorithm managed to find a joystick’s control characteristic of the formpresented in eq. (28) which is a third degree non-linear function.

f(u) = C1u+ C2u2 + C3u

3 (28)

Bode-diagrams of the open-loop system with the designed f(u) for interesting working points(linearization at interesting velocities) can be seen in the figures below (Bode-diagrams for otherworking points can be found in appendix C).

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Figure 36: Bode Diagram for the obtained joystick’s control characteristic in eq. (28) with Kop = 176.This represents the beginning of the tracking area in which we have achieved an Am = 14.4dB and a'm = 32.2�.

Figure 37: Bode Diagram for the obtained joystick’s control characteristic in eq. (28) with Kop = 126.This represents the transition between the tracking and capture area in which we have kept the sameamplitude and phase-margin as in fig. 33

In fig. 36, we can see that for the tracking area we have achieved an amplitude margin ofAm > 10dBand phase-margin of 'm > 30�. In the transition area between tracking and capture we have achievedan amplitude margin of Am = 9.1dB and a phase-margin of 'm = 28.4� as seen in fig. 37. As explainedin section 5.1, in this area the joystick’s deflection d is kept in the interval 25% < d < 35%. Sincethe operator oscillates between these two areas then the algorithm cannot find a non-linear functionwith a higher increase in the tracking and capture areas but with this f(u), we keep the amplitude

2019 39


and phase-margin that we had before (see fig. 33). From the two figures, it can also be seen that thecross-over frequency wc for the whole tracking area is kept between 2 and 3 rad/s.

Figure 38: Bode Diagram for the obtained joystick’s control characteristic in eq. (28) with Kop = 61.This represents most part of the capture area in which we have achieved an Am = 9.1dB and a'm = 28.4�

For the capture area, we can see in fig. 38 that we have a phase-margin of 'm = 28.4� and anamplitude margin of Am = 9.1dB. In this area, the operator is likely to make large movements givingfaster inputs to the mirror-head system. The input velocity in this area is more than 0.05 rad/s

which is regarded as fast. From this plot (and similar plots found in appendix C), we can see that thecross-over frequency in the capture area is kept between 3 and 4 rad/s.

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Figure 39: Bode Diagram for the obtained joystick’s control characteristic in eq. (28) with Kop = 25.This represents the designation area in which have achieved an Am = 10.2dB and a 'm = 29.5�

As for the transition area between capture and designation, we can see in fig. 39 that we haveachieved an amplitude margin of Am > 9dB and a phase-margin of 'm > 29�. We have thus increasedthe amplitude and phase-margin compared to fig. 35. With these margins, we can guarantee stabilityof the system for up to 83% deflection which allows a maximum joystick’s velocity of 0.53 rad/s.

We can conclude that the designed joystick’s control characteristic f(u) in eq. (28) fulfills thestability and margin requirements. As for the physical requirements stated in eq. (23), the designedjoystick’s control characteristic yields the following:

At u = 0:f(0) = C1 · 0 + C2 · (0)2 + C3 · (0)3 = 0 (29)

At u = 1:f(1) = C1 · 1 + C2 · (1)2 + C3 · (1)3 = 0.923 (30)

For 8|u| > 0:f0(u) = C1 + 2C2 · u+ 3C3 · (u)2 > 0 (31)

which means that the designed f(u) also fulfills the physical requirements.From the bode-diagrams of f(u), it can be seen that there is a trade-o↵ between trying to design

a joystick’s control characteristic with more amplitude in the tracking area and losing amplitude inthe capture area. It can also be seen that the phase-margin is kept at around 30�. It is thereforenecessary to build a lead-filter to increase the phase-margin of the system at these two areas.

5.1.3 Evaluation of Tracking Performance

To quantify the tracking performance of the closed-loop system, we calculate the root mean square(RMS ) of the errors. This is used to evaluate the tracking error of the system in both position andvelocity (rmspos, rmsvel respectively).

The tracking and position errors are calculated according to eq. (32). The variable T denotesthe trial duration, ep(t) is the instantaneous tracking error in position at time t and ev(t) is the

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instantaneous tracking error in velocity at time t.

rmspos =

s1

T

Z 0

T[ep(t)]2 =

s1

T

Z 0

T[r(t)� y(t)]2

rmsvel =

s1

T

Z 0

T[ev(t)]2 =

s1

T

Z 0

T[r0(t)� u(t)]2

(32)

According to [10], the tracking error in man-machine (MM) systems can be due to three aspects:

• Operator delay-time.

• Deficiencies of the operator in the control process.

• Operator ignoring part of the signal that she considers is not feasible to track.

To have a realistic simulation, the target’s path is reconstructed from measured signals obtained fromthe mirror-head system. The reference signal of any system is given by r(s) = e(t)�y(s) [4], thereforewe derive the target’s signal by subtracting the line-of-sight ylos(t) (output of the mirror-head) fromthe error signal evt(t) (measured by the video tracker) as seen in eq. (33).

rtarget(t) = evt(t)� ylos(t) (33)

For obtaining the measured signals we use the joystick’s control characteristic forig(u) which thesystem originally has. This function is given by

forig(u) = Au(t) +Bu(t)3 (34)

We simulate the closed-loop system with forig(u) in eq. (34) and calculate the RMS according toeq. (32) which yields the values in table 3 and the output’s simulation plot in fig. 40 (see next page).

T [sec] rmspos rmsvel

290 0.0050 0.0856

Table 3: The trial duration T and RMS error in position and velocity for the original joystick’s controlcharacteristic forig(u) in eq. (34) during simulation.

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Figure

40:Sim

ulation

obtained

fortheoriginal

joystick’s

controlcharacteristic

forig(u).

Thebluesign

alis

themeasuredline-of-sight

from

themirror-head,theredsign

alis

theline-of-sight

producedby

ourmod

el(derived

insection4.1.2)

andtheyellow

sign

alis

thereconstructed

target’s

path.

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Figure 41: The designed joystick’s control characteristic f(u) found by the algorithm which fulfillsthe stability requirements for tracking tasks.

The Monte-Carlo based optimization algorithm is also run with the aim at finding the combinationsof C-values (see eq. (22) in section 5.1) and stability requirements which, compared to the originaljoystick’s control characteristic forig(u), gives less RMS error in position and velocity.

For each iteration, the algorithm tries to find a function f(u) with a set of C-values that meets thestability requirements (explained in section 5.1.2), simulates the closed-loop system with the foundjoystick’s control characteristic f(u) and then calculates its RMS error in position and velocity. TheRMS obtained is compared to the values in table 3.

T [sec] rmspos rmsvel

290 0.0046 0.0871

Table 4: The trial duration T and RMS error in position and velocity for the designed controlcharacteristic f(u) in fig. 41 during simulation.

The simulation of the designed joystick’s control characteristic f(u) yields the RMS error in positionand velocity presented in table 4. Through comparing this table with table 3, we can see that there isa trade-o↵ between error in position and velocity (decreasing the error in position leads to an increasein the error in velocity). By having less error in position, we know that the line-of-sight will be closerto the target’s position and that the operator does not need to adjust her command as much, thereforeit is more important to have less error in position than in velocity.

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Figure

42:Sim

ulation

obtained

forthedesigned

joystick’scontrolcharacteristic

ineq.(28).Theyellow

sign

alisthereconstructed

target’spath

andtheredsign

alis

theLOSproducedby

ourmod

el

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5.2 Filtering of Command Signal

As seen in section 5.1.2, we cannot increase the phase-margin of the system over the whole capturearea by simply designing a joystick’s control characteristic f(u). The designed f(u) increases theamplitude of the system but does not increase the phase-margin at the capture and designation area(in which instability can occur since the velocities are high). This is why we need a method to increasethe phase-margin of the system in these areas.

It was discussed in section 4 that a limitation to the operator-mirror system is the decrease inphase-margin due to the time-delays over the whole chain (video tracker and operator models). Thislimitation imposes a constraint on how much the cross-over frequency wc for the open-loop can beincreased. Having a system with time delay Td means that a control action at time t won’t take e↵ectuntil time t+ Td therefore it is not possible to counter act reference signal variations in less than Td

[5]. This corresponds to angular frequencies higher than 1Td

. So wc can be increased according toeq. (35).

wc <1

Td(35)

Our operator-mirror system (with the designed joystick’s control characteristic in section 5.1) has atotal time-delay of Td = 0.17 which yields:

wc <1

0.17wc < 5.88

(36)

The cross-over frequency of the open-loop system should therefore be less than 5.88 rad/s. To increasethe phase-margin of the system at a certain frequency range, we need a lead controller of the formpresented in eq. (37). A lead controller increases the phase-margin 'm of the system through thechoice of � value. Having a small � value yields a higher increase to 'm but also increases theloop gain at high frequencies which amplifies high-frequency noise [4]. This controller also improvesthe control performance when the error signal e(t) changes fast since it helps decrease the system’srise-time Tr and settling-time Ts.

Flead(s) = Klead⌧Ds+ 1

�⌧Ds+ 1(37)

From fig. 38 and fig. 39 we see that the cross-over frequency for the open-loop system is wc ⇡ 4 rad/s inthe capture area and wc ⇡ 3 rad/s in the designation area. Compared to the measurements performedwith CoCo80 during the tracking experiements, we see that our model has a lower phase-margin thanthe actual system. Therefore we would like to increase the 'm in the capture and designation areasto at least 30� while keeping the wc below 5.88 which can be accomplished by using Flead in eq. (37).

In order to track the target exactly we want the transfer function between the desired and actualline-of-sight to be 1. Therefore Flead(s) needs to be a feedforward controller positioned between f(u)and the mirror-head system G(s).

As a summary, we suggest updating the block-model in fig. 18 with a lead filter situated betweenthe joystick’s control characteristic and the mirror-head system in order to increase the phase-marginof the system. The lead filter parameters can be calculated according to the equations in [4] Chapter5.4 using wc ⇡ 4 rad/s and $ ⇡ 30� though additional simulations and measurements will be neededto choose the exact value of wc. Unfortunately due to time constraints this will not be implementedin the thesis.

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6 Simulations and AnalysisIn this section, we will start by presenting the numerical results obtained for the operator-mirror modeland the joystick’s control characteristic. We will also present stability plots according to the CircleCriterion using the designed joystick’s control characteristic. Next, we will analyze the simulationsfrom the joystick’s control characteristic that the sight-system had originally and compare it with theone obtained in this research. Finally, the root mean square (RMS ) value in position and velocity ofboth joystick’s control characteristics will be analyzed.

6.1 Numerical Results

The numerical results derived and discussed in previous sections are summarized in this section. Intotal, we produced three transfer functions: operator, video-chain, mirror-head and one static non-linear function: joystick’s control characteristic. The operator and video-chain models as well as thestatic non-linear function have numerical values.

6.1.1 Models

The transfer function of the operator model designed in section 4 with numerical values is presentedbelow:

FH(s) = Kgs.TLs+ 1

(TIs+ 1)(TNs+ 1).e

�⌧s = Kgs.0.38s+ 1

(1.85s+ 1)(0.003s+ 1).e

�0.10s (38)

where the operator gain is given by the gain-scheduling:

Kgs =⇥180, 176, 155, 142, 126, 110, 80, 61, 40, 25, 25, 25

⇤

The transfer function for the video chain obtained in section 4.2 has the following time-delay:

Gvc(s) = e�0.07s (39)

The joystick’s control characteristic designed in section 5.1 is a static non-linear function of thirddegree with the following values:

f(u) = au+ bu2 + cu

3 (40)

where a = 0.0345, b = �0.0925 and c = 0.9806

6.1.2 Tracking Error

The RMS (root mean square) error in position and velocity for the designed joystick’s control char-acteristic f(u) in eq. (40) during simulations are found in table 5 below:

Signal Type rmspos rmsvel T [sec]

Simulation 0.0046 0.0871 290

Table 5: The trial duration T and RMS error in position rmspos and velocity rmsvel for the designedjoystick’s control characteristic f(u)

For the original joystick’s control characteristic forig(u), the RMS error values for the measuredsignals and simulation yield the results in table 6.

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Signal Type rmspos rmsvel T [sec]

Measurement 0.0053 0.0164 290Simulation 0.0050 0.0856 290

Table 6: The trial duration T and rms error in position rmspos and velocity rmsvel for the originaljoystick’s control characteristic forig(u)

From table 5 we can see a decrease of 8% on rmspos during simulations compared to the originaljoystick’s control characteristic forig(u).

6.2 Simulations

Simulations of the operator-mirror system with the designed joystick’s control characteristic obtainedin section 5 are performed using Simulink. The input given to the system is the target’s path recordedduring a regular tracking-task experiment: constant target’s velocity and path. The simulation lasts290 seconds and we plot the bode-diagram of the open-loop system Go = Gvc(s)GH(s)sat(u)f(u)G(s)at the end of the simulation which can be seen in fig. 43.

Figure 43: Bode diagram of the open-loop system Go with the designed joystick’s control characteristicf(u)

In fig. 43, we see that the system Go has an amplitude margin of Am = 28.9 dB and a phase-margin of 'm = 47.1�. The cross-over frequency is of wc = 1.83 rad/s. Since this is a regular trackingtask the operator’s signal is kept in the middle of the tracking area yielding a higher increase of thesystem’s amplitude and phase-margin than when analyzing the system at a working point (velocity).

Simulations of the operator-mirror model with the original joystick’s control characteristic forig(u)are also performed to compare its results with fu designed in this thesis. The bode-diagram of thisopen-loop Gorig = Gvc(s)GH(s)sat(u)forig(u)G(s) can be seen in fig. 44.

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Figure 44: Bode diagram from simulations with the original joystick’s control characteristic forig(u)

In fig. 44 we see that the system has an amplitude margin of Am = 35.1 dB and a phase-marginof 'm = 46.2�. It can be seen that even though the designed joystick’s control characteristic has areduced rms in position it has a lower amplitude than the original control characteristic.

6.3 Input-output Analysis

We will analyze the step response of the operator-mirror model simulated with the original and thedesigned joystick’s control characteristic, f(u) and forig respectively, to see how fast the system reactsto input signals. The two step-response plots can be found in fig. 45 and in fig. 46. To analyse theinput-output stability in the capture area for both control characteristics we use the Circle Criterionwhich can be found in fig. 50 and fig. 49.

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Figure 45: Impulse response of the system using the original joystick’s control characteristic forig(u)

The step response of the system simulated with the original joystick’s control characteristic forig(u)can be seen in fig. 45. The system produces an output after 0.17s which is the model’s time-delay andis well-damped since there is no overshoots. We can also see that the operator-mirror system has arise-time of Tr = 5s and a settling-time of Ts = 7s while reaching an amplitude margin of Am = 2.6.

Figure 46: Impulse response of the system using the designed joystick’s control characteristic f(u)

As for the designed joystick’s control characteristic f(u), we can see in fig. 46 that the system isstill well-damped and that the model’s time-delay is present. We can also see that we have achieveda rise-time of Tr = 3.25s and a settling-time of Tr = 6.84s while reaching an amplitude margin of

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Am = 5.14. This means a reduction of 2,2% in settling-time for tracking tasks.To see how disturbances are attenuated we plot the sensitivity function S(s) of the linearized

operator-mirror system for f(u) and forig(u).

Figure 47: Sensitivity function of the system using the designed joystick’s control characteristic f(u)

Figure 48: Sensitivity function of the system using the original joystick’s control characteristic forig(u)

From the S(s) plot with the designed joystick’s control characteristic in fig. 47, we can see thatdisturbances are attenuated for low-frequencies. We can also see an overshoot from around 1.41 rad/s

which gives the largest amplification of the disturbances. The same can be seen from the S(s) plot

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with the original joystick’s control characteristic in fig. 48 but there is an overshoot from around0.96 rad/s. We have achieved a larger frequency range of disturbance attenuation with f(u).

To analyze the stability of the system in the capture area and to know how much velocity thesight-system is allowed to produce we use the Circle Criterion since it ensures input-output stability.

Figure 49: Circle Criterion of the system using the original joystick’s control characteristic forig(u)

Figure 50: Circle Criterion of the system using the designed joystick’s control characteristic f(u)

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For the designation area of the original joystick’s control characteristic forig(u), it can be seenin fig. 49 that it allows a maximum velocity of 0.49 rad/s. While for the designed joystick’s controlcharacteristic in fig. 50, it can be seen that it allows a maximum velocity of 0.51 rad/s. This meansthat we have provided a higher velocity range to the system with f(u).

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7 Conclusions and Future WorkIn this section, we will discuss the results obtained in previous sections by drawing parallels to theliterature-research that has been previously done. We will also answer the research questions andpresent our conclusions. Finally, we will give directions for what can be done from here on-wards.

7.1 Discussion and Conclusion

In this thesis, by optimizing the joystick’s control characteristic we have obtained a decrease of RMSerror in position by 8% (during tracking tasks) compared to the original joystick’s control charac-teristic. Through this optimization, the settling-time for tracking tasks has been reduced by 2.2%.For capture tasks it has not been reduced but it is kept the same as the original joystick’s controlcharacteristic.

Due to time-constraints, we could not implement the designed joystick’s control characteristicin the sight-system but based from the inputs obtained during the tracking experiments we havereason to believe that this designed joystick’s control characteristic will be preferred. This is dueto the operators experiencing a better tracking performance while having a linear joystick’s controlcharacteristic response in the tracking area and a slow increase in velocity in the transition areabetween tracking and capture which could be achieved by joystick’s control characteristic obtained inthis thesis.

The major focus of this research has been on building an operator model to evaluate the sight-system’s dynamics and its relation to her. During the modelling of the operator-mirror system wehave discovered several issues. Early time-delay experiments show that the system’s time-delay ismuch higher than what our model-estimations yield (around 0.24 seconds). The reason for the lowvalue in our model-estimations might have been due to the target signal being identical throughoutthe entire experiment which might have enabled the operator to learn and memorize the target’srelevant features and use these for a more ”e↵ective” control. That might be the reason why theoperator’s time-delay is estimated as 0.10 seconds, we might have a larger time-delay if the target’spath is totally di↵erent for the whole tracking experiment.

From the bode-diagrams in section 5, it can be concluded that a system with time-delay su↵ersfrom a decrease in phase-margin and therefore there is a limit as to how big the time-delay can beincreased before falling into instability. Since as the phase-margin decreases, the system’s responsebecomes more oscillatory.

In section 4 we have seen that the human operator is an adaptive and learning controller capableof exhibiting a variety of behaviour but when a task is regarded as too di�cult then the operatorstarts accepting larger error and does not represent a tracking-behaviour. This needs to be taken intoregard when building a model for the operator because the measurements might not be fully reliable.

The aim of this research has been to design a joystick’s control characteristic since it is throughit that the operator can send inputs to the sight-system. Our results indicate that we need a phase-increasing filter to increase the phase-margin of the system and improve the step-response of thesystem during capture tasks. Even so, a well tuned joystick’s control characteristic signal-responseon the tracking error allows the operator to achieve basic stability of the system and prevents theline-of-sight signal from moving away from the reference path in an uncontrolled manner. To improvethe stability margin of the system, it would be needed to construct a phase-increasing filter which willcompensate for the time-delays in the operator-mirror system.

Initially it was believed that the joystick’s control characteristic measurements could be totallyseparated from the operator in order to manipulate the joystick’s control characteristic alone. How-ever simulation’s results contradict this hypothesis. With the operator-mirror model designed in this

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research, simulations failed whenever we diverged too far from the original joystick’s control charac-teristic. This is probably due to the model being derived for a joystick’s control characteristic of orderthree and made it correlated to how this static non-linearity looked like when the measurements wereobtained. Through the tracking-experiments performed, it has been seen that the operator prefers tohave a joystick’s control characteristic of the form f(u) = Au + Bu

n where n = [0, 5] since we wantto have a linear increase in velocity for tracking and capture tasks while having a faster increase fordesignation tasks. A fourth-degree joystick’s control characteristic was not achievable since we couldnot simulate it with the measured signals from the original joystick’s control characteristic.

7.2 Future Work

From Gain-Scheduling analysis performed for the operator model, it can be seen that the operatortries to adapt to the joystick’s control characteristic therefore she might be modelled as an AdaptiveController to reflect his adaptive capacity.

The operator-mirror model produced in this research is categorized as a single-axis pursuit model inwhich the focus is on the target’s motion relative to the horizon. It is known that for certain tracking-tasks there will be a correlation between azimuth and elevation. Therefore, to further improve thismodel, we will need to provide a dual-axis pursuit model to evaluate motions in diagonal. In thisthesis, only the velocity of the target was varied but we did not independently vary the amplitudeor time duration of these changes which might have yielded other results for the operator modelvariables. It might also be needed to take measurements from real tracking-capture experiments tosee the dynamics of the operator-mirror system during higher demanding tasks.

There are limitations in the operator model derived in this thesis. While the linearization of thesystem yielded a good insight of the dynamics of the system it is only valid in the presence of small-signal disturbances. Having a large disturbance might lead to PIOs (pilot induced oscillations) in thesystem and yield constant amplitude oscillations. It has also been studied that the pilot model isregarded as the weakest point in manual-control systems due to the pilot’s behaviour complexity andthat it can also introduce PIOs.

Implementation of the designed joystick’s control characteristic in the sight-system should beperformed to see the real interaction of the operator with the improved mirror-head system. Also,the lead-filter should be implemented to compensate for the time-delays from the operator and thevideo-chain which might yield a higher decrease in root mean square (RMS ) error in position and inthe settling-time.

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References[1] F. Marguliest, H. Zemanek, Man’s Role in Man-Machine Systems, Automatica Vol.19 No.6, Great

Britain, 1983.

[2] Thomas Jurgensohn, Control Theory Models of the Driver, pp277-292, 2007.

[3] P. Carlo Cacciabue, Modelling Driver Behaviour in Automotive Environments, ISBN-13: 978-1-84628-617-9 , Springer Edition, London, 2007.

[4] Torkel Glad and Lennart Ljung, Reglerteknik: Grundlaggande tori, Studentliteratur, 4th Edition,Lund, 2006.

[5] Torkel Glad and Lennart Ljung, Reglerteori: Flervariabla och olinjara system, Studentliteratur,Lund, 2003.

[6] Hassan, K. Khalil, Nonlinear Systems, International Edition, Pearson, 2013.

[7] Urban Forssell, Lennart Ljung, Closed-loop Identification Revisited, Division of Automatic Con-trol, Linkoping University, Linkoping, 1999.

[8] Torkel Glad, Lennart Ljung, Modeling and Identification of Dynamical Systems, ISBN:9789144116884, Studentlitteratur, August 2016.

[9] Constantin Rotaru, Simona Roatesi, Raluca Ioana Edu, Ionica Cırciu, Human Pilot’s DynamicResponse Characteristics, International Conference on E-Health and Bioengineering, Romania,2015.

[10] Duanne T. McRuer, Human Dynamics in Man-Machine Systems, Automatica Vol.16, pp237-253,Pergamon Press Limited, England 1980.

[11] Miroslav Jirgl, Marie Havlikova, Zdenek Bradac, The Dynamic of Pilot Behavioral Models, De-partment of Control and Instrumentation, Brno University of Technology, Czech Republic, 2015.

[12] Duanne T. McRuer and Henry R. Jex, A Review of Quasi-Linear Pilot Models, Transactions onHuman Factors in electronics Vol HFE-8, pp231-249, 1967.

[13] Duanne McRuer and Dunstan Graham, Human Pilot Dynamics in Compensatory Systems, USA.

[14] A.Tustin, The Nature of the Operator’s Response in Manual Control and its implications ForController Design, Journal of the Institution of Electrical Engineers - Part I: General Vol.94,pp.532-533, 1947.

[15] James J. Potter, William E. Singhose, E↵ects of Input Shaping on Manual Control of Flexibleand Time-Delayed Systems, Human Factors Vol.56, pp 1284-1295, USA, 2014.

[16] Chuan Wang, Michael Santone and Chengyu Cao, Pilot-Induced Oscillation Suppression by UsingL1 Adaptive Control, Department of Mechanical Engineering, University of Connecticut, USA,2012.

[17] Frank M. Drop, Theoretic Models of Feedforward in Manual Control, ISBN 978-94-6186-728-5,2016

[18] Michael K. Masten, Henry R. Sebesta, Line-of-Sight Stabilization/Tracking Systems - AnOverview, Conference on American Control, pp 1477-1482, USA, 1987.

[19] Gerd Eriksson, Kompendium i Tillampade Numeriska Metoder, Kungliga Teknikska HogskolanCSC, ISBN: 91-7178-258-3, Stockholm Sweden, 2006.

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[20] Matworks Documentation, System Identification Toolbox, https://se.mathworks.com/help/ident

[21] Matworks Documentation, MATLAB, https://se.mathworks.com/help/matlab/index.html

[22] Crystal Instruments website, CoCo-80X Dynamic Signal Analyzer,https://www.crystalinstruments.com/coco80x-dynamic-signal-analyzer

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ATracking Experiments

To build the operator model we need to log how she reacts to a visual stimulus therefore the experi-mental data is going to be collected from a realistic control tracking task. For this task, we will choosepeople that have worked with the sight-system before and have good knowledge about the system.Each participant will use a one-hand joystick to give their control inputs u(t) to the mirror-head ofthe sight-system. The joystick has a maximum deflection of around ±60� and it is connected to acomputer so that we can monitor its signal.

The control tracking task consists on pointing the line-of-sight to the target and follow it as closeas possible through moving the joystick side-wise. As target, we choose a small can filled with warmwater since the video tracker which detects and present moving objects, has an infra-red camera. Thetarget followed a sinus-like path but changed velocity at di↵erent intervals so that the operator willnot predict the movements. Since the joystick’s deflection are converted to velocity by the joystick’scontrol characteristic we will use a linear joystick’s control characteristic of the form: f(u) = Kjoyu

for these experiments. An example of the target’s position signal rtarget(t) recorded by the videotracker is presented in fig. 51 below.

Figure 51: An example of the target signal obtained from measurements with Kjoy = 0.060

The individual tracking runs of the experiment last around 120 seconds, of which the first 10frames are removed and the rest is used as the measurement data. This is because the participantshave to become familiar with the joystick and cursor dynamics and these 10 frames will be regardedas ”training”. Tracking performance is monitored by the experimenter which also is in charge ofchanging the velocity of the target during the experiment. After each round, the participants givefeedback on the joystick’s sensitivity and on how well they could perceive the target’s path or velocityfor a given Kjoy.

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Figure 52: The joystick’s deflection (blue line) and error signal (red line) measured by the videotracker for one tracking-task.

During the experiment, the time traces of the error signal e(t), the joystick’s deflection signal sendby the operator uop(t), the target’s position rtarget(t) and the line-of-sight signal yLOS(t) are recordedfor three repetitions of each experimental condition. An example of the signals uop and e(t) recordedcan be seen in fig. 52. A total of 14 tracking experiments are done and the data will be processedusing System Identification Toolbox.

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BGain-Scheduling Validation Tests

We perform some tracking tests with two human operators to validate the operator’s gain-schedulingand also to estimate the open-loop at very high/low joystick’s gain. The signals are measured bythe analyzer CoCo80 which was introduced in section 3 and it is set to record signals from 0.1Hzto 1Hz. In these tests we use the same linear joystick’s control characteristic f(u) as in the trackingexperiments in appendix A (f(u) = Kjoyu) and also the same one-hand joystick to make these testscomparable.

We send an exciting signal with a gain corresponding to 0.5 Volts and a gain to the sight-systemcorresponding to 0.025 rad/s. The exciting signal induces a disturbance in the system making theline-of-sight move to the right and to the left in a sinus-like manner. The operator’s task is thereforeto compensate for the disturbance by keeping the line-of-sight in the middle of the screen at all times.At the end of each session the signals recorded by CoCo80 are saved in a mat file for later analyze.

The open-loop signal recorded by CoCo80 is plotted as a bode-diagram. Through inspecting thisdiagram, the amplitude margin Am of the open-loop at frequency w = 0i can be extracted and isgiven by eq. (41).

|G(0i)| = 20 log10

�Kjoy · x

�= Am (41)

By using eq. (41), the operator’s gain at a given Kjoy-value can be calculated through solving for x:

Am = 20 log10(Kjoy · x)10Am/20 = Kjoy · x

x =10Am/20

Kjoy

(42)

which is the operator’s gain at joystick’s gain Kjoy.

Figure 53: The open-loop system for operator 1 with Kjoy = 0.03. The operator gain for this case isestimated as 194.03

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Figure 55: The open-loop system for operator 1 with Kjoy = 0.075. The operator gain for this caseis estimated as 93.31

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Figure 56: The open-loop system for operator 1 with Kjoy = 0.10. The operator gain for this case isestimated as 100


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The bode-diagrams for relevant Kjoy-values can be seen in fig. 53 to fig. 59 which show the samepattern as the gain-scheduling obtained in section 4. In these figures we can also see the phase-margin

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'm of the open-loop system and the cross-over frequency wc at two di↵erent regions of the joystick(tracking and capture). It can also be seen that the operator’s gain will never decrease to zero sincethe operator-mirror system will need to have some gain value so that the line-of-sight keeps moving.

Kjoy Am [dB] wc [rad/s] 'm [degree]

0.03 15.3 2.4 91.30.05 14.7 2.76 780.075 16.9 3.19 75.90.10 20 2.88 85.80.20 18.7 2.94 78.50.30 22.1 2.95 90.20.40 14.8 2.34 76

Table 7: Amplitude margin Am, cross-over frequency wc and phase-margin 'm from measurementsdone with CoCo80 for operator 1

Kjoy Am [dB] wc [rad/s] 'm [degree]

0.05 15.4 3.26 75.90.10 21.7 2.99 94.70.40 18 4.19 54.8

Table 8: Amplitude margin Am, cross-over frequency wc and phase-margin 'm from measurementsdone with CoCo80 for operator 2

In table 7 and table 8 we summarize the results obtained for each operator. The tables showsimilar amplitude values for the two operators and relate to the same pattern as the gain-schedulingderived analytically in section 4.

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CStability Criteria Plots

C.1 Nyquist Criterion for Operator Model

As explained in section 4, we apply the Nyquist Criterion to show stability of the closed-loop systemat di↵erent velocities which we call operating points. We plot the Nyquist curve at the three regionsof the joystick: tracking, capture and designation. On each plot, the blue line denotes the Nyquistcurve and the red cross shows the point -1.

Figure 60: Nyquist curve of the open-loop with KH = 126 and Kjoy = 0.098

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The plots above show the remaining operating-point plots. In fig. 60, fig. 61 and fig. 62 we seethe tracking, capture and designation areas respectively. These plots show that the Nyquist criterionis fulfilled for all operating points since the Nyquist curve does not encircle -1 and therefore we canguarantee stability of the closed-loop system with the operator model derived in this thesis.

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C.2 Circle Criterion for Control Law Bounds

In section 5, we use the Circle Criterion to ensure stability of the man-machine (MM ) system Gom =Gvc(s)GH(s)G(s) by finding the sector bounds that the joystick’s control characteristic can have atfixed operator gains. For each plot, the circle (on red) along with the values for k1 (to the left of thecircle) and k2 (to the right of the circle) can be found. The Nyquist curve of Gom is represented bythe blue line.

In fig. 63 and fig. 64 we can see plots of the remaining tracking area while in fig. 65 and fig. 66 wesee plots of the remaining capture area.

Figure 63: Circle Criterion showing the least and maximum k-value that the system can have atoperator gain KH = 155

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The plots above show that the Circle Criterion is fulfilled for the whole gain-scheduling at eachregion and therefore we can guarantee stability of the system in the whole tracking and capture area.

Figure 67: Bode diagram for Circle Criterion in fig. 63 at operator gain KH = 155

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We also plot the bode-diagram of the linearized open-loop Gom to quantify how much margin wehave at the remaining operator gains. These plots can be found in fig. 67, fig. 69 and fig. 68 below.From these plots we can read out both the amplitude margin Am and phase-margin 'm of the system.We can see that the the open-loop’s phase-margin 'm is kept between 28� and 34�.

C.3 Stability Margin for Designed Control Law

In section 5.1.2, we plot the system’s Bode diagram with the designed joystick’s control characteristicto see if the stability and margin requirements are fulfilled. Here we present plots for the remainingworking points (velocities) in the tracking and capture area.

Figure 70: Bode Diagram for the obtained joystick’s control characteristic in eq. (28) with Kop = 155.This is part of the tracking area in which we have achieved an Am = 13.4dB and a 'm = 31.7�

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Figure 71: Bode Diagram for the obtained joystick’s control characteristic in eq. (28) with Kop = 142.This is part of the capture area in which we have achieved an Am = 11.1dB and a 'm = 30.3�.

Figure 72: Bode Diagram for the obtained joystick’s control characteristic in eq. (28) with Kop = 110.This is part of the capture area in which we have achieved Am = 6.5dB and a 'm = 24.6�

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Figure 73: Bode Diagram for obtained joystick’s control characteristic in eq. (28) with Kop = 80. Thisis part of the capture area in which we have achieved Am = 7.2.5dB and a 'm = 25.9�

From the figures above we can see that the stability and margin requirements are fulfilled by thedesigned joystick’s control characteristic. We can also see that we have achieved a higher increase inthe tracking area than in the capture area.

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