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Observer-Based Robot Arm Control System
Nick Vogel, Ron Gayles, Alex CertaAdvised by: Dr. Gary Dempsey
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OutlineOutline
Project OverviewProject Overview Project GoalsProject Goals Functional DescriptionFunctional Description Technical Background InformationTechnical Background Information Functional RequirementsFunctional Requirements Work CompletedWork Completed ConclusionsConclusions
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Project OverviewProject Overview Control of robot armsControl of robot arms Pendulum & 2 DOF armsPendulum & 2 DOF arms Load ChangesLoad Changes Observer-based Observer-based Ellis's methodEllis's method
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Pendulum Arm Pendulum Arm ConfigurationConfiguration
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2-DOF Arm Configuration2-DOF Arm Configuration
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Project GoalsProject Goals
Learn the Quanser software package Learn the Quanser software package Model the pendulum and horizontal Model the pendulum and horizontal
armarm Design controllers using classical Design controllers using classical
control control Design controllers using observer-Design controllers using observer-
based control based control Evaluate the relative performance of Evaluate the relative performance of
observers to classical controllersobservers to classical controllers
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Equipment UsedEquipment Used
PC with Matlab, Simulink, and PC with Matlab, Simulink, and Real Time WorkshopReal Time Workshop
Motor with Quanser Control Motor with Quanser Control SystemSystem
Linear Power AmplifierLinear Power Amplifier Robot arm with GripperRobot arm with Gripper SRV-02 Rotary Servo PlantSRV-02 Rotary Servo Plant
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Robot Arm Controller
Robot ArmCommand
Signal
Sensors
PositionPosition
Command
Motor Command
Overall Block Diagram
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GC(s) GPC(s) GP(s) GS(s)
GCO(s)
GPEst(s) GSEst(s)
+
+
+
+
-
-
R(s) PR(s)E(s) PC(s) C(s) Y(s)
EO(s)
CO(s)
YO(s)
Ellis's Observer-Based Controller
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Situational DescriptionSituational Description Command of +-90 degreesCommand of +-90 degrees
Meet specifications for a load of up to 75 Meet specifications for a load of up to 75 gramsgrams
Be able to pass a load back and forth Be able to pass a load back and forth between two systemsbetween two systems
Work with existing arm, sensor, and Work with existing arm, sensor, and convertersconverters
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Technical Background Technical Background InformationInformation
% Overshoot – Amount the system % Overshoot – Amount the system advances past the target positionadvances past the target position
Settling Time – Time it takes for the Settling Time – Time it takes for the system to complete its responsesystem to complete its response
Steady-State Error – Error of system Steady-State Error – Error of system after completely settlingafter completely settling
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Technical Background Technical Background InformationInformation
Gain Margin – How much gain can Gain Margin – How much gain can be added without instabilitybe added without instability
Phase Margin – how much phase lag Phase Margin – how much phase lag can be added to the system without can be added to the system without instabilityinstability
PM=180-|system phase lag|PM=180-|system phase lag|
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Product Specifications for Product Specifications for 2-DOF Arm2-DOF Arm
The overshoot of the arm shall be less The overshoot of the arm shall be less than or equal to than or equal to 15%15%
The settling time of the arm shall be less The settling time of the arm shall be less than or equal to than or equal to 2s2s
The phase margin shall be at least The phase margin shall be at least 50 50 degdeg
The gain margin shall be at least The gain margin shall be at least 3.5 dB3.5 dB The steady state error of the system The steady state error of the system
shall be at most shall be at most 5 degrees5 degrees
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Product Specifications Product Specifications For Pendulum ArmFor Pendulum Arm
The overshoot of the arm shall be less The overshoot of the arm shall be less than or equal to than or equal to 15%15%
The settling time of the arm shall be The settling time of the arm shall be less than or equal to less than or equal to 2s2s
The phase margin shall be at least The phase margin shall be at least 50 50 degdeg
The gain margin shall be at least The gain margin shall be at least 3.5 3.5 dBdB
The steady state error of the system The steady state error of the system shall be at most shall be at most 1 degree1 degree
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Work Completed: Work Completed: Pendulum Arm Pendulum Arm
Arm ModelingArm Modeling
Traditional Arm ControlTraditional Arm Control
Non-Linear Arm ModelingNon-Linear Arm Modeling
Load TestingLoad Testing
Observer DesignObserver Design
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Modified Estimated DC Modified Estimated DC gain vs Voltagegain vs Voltage
Modified Estimated DC gain vs Voltage
0
10
20
30
40
50
60
70
0 0.5 1 1.5 2Voltage in(V)
Esti
mate
d D
C g
ain
(d
eg
rees/v
olt
)
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22ndnd Order Pole Locations Order Pole Locations and Modeland Model
System assumed to System assumed to
be as shown to rightbe as shown to right Poles at -11, -2.6Poles at -11, -2.6 Model resultsModel results System System
resultsresults
k/[(s/(p1+1)* (s/(p2+1)]
KP+-
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Frequency ResponseFrequency Response
0.01
0.1
1
10
0.01 0.10 1.00 10.00SystemResponse
CorrectedmodelresponseModelresponse
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Proportional ControlProportional Control Used control toolbox to find initial Used control toolbox to find initial
gain valuegain value Tuned gain: 0.14Tuned gain: 0.14 For 20 degree inputFor 20 degree input % O.S.=15% % O.S.=15% ess= 2.5 degrees ess= 2.5 degrees tr=0.12 s tr=0.12 s ts= 0.41 sts= 0.41 s
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PID controllerPID controller Form: Form:
kp(0.09s+1)(0.4s+1)/[s(s/p1+1)]kp(0.09s+1)(0.4s+1)/[s(s/p1+1)] Exact 2Exact 2ndnd order order Higher pole is fasterHigher pole is faster D/A Converter saturatesD/A Converter saturates Rate limitation neededRate limitation needed
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PID Controller ContinuedPID Controller ContinuedPole Pole LocatioLocationn
Gain Gain ValuValuee
OvershoOvershoot %ot %
SettliSettling ng TimeTime
Rate Rate LimitatiLimitationon
Rate Rate Limited Limited Settling Settling TimeTime
-40-40 0.750.75 14.914.9 0.200.20 155155 1.161.16
-80-80 1.51.5 1515 0.100.10 148148 1.201.20
-60-60 1.11.1 14.914.9 0.140.14 151151 1.181.18
Rad/sRad/s ss deg/sdeg/s
1 deg 1 deg inputinput
180 deg input180 deg input
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PID ResultsPID Results 45 deg input45 deg input
% OS=3.3%% OS=3.3%
Ts=0.4 sTs=0.4 s
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Non-Linear ModelingNon-Linear Modeling
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Loaded TestingLoaded Testing Tested Loaded DC gain: Tested Loaded DC gain:
approximately 27 degrees/volt approximately 27 degrees/volt (compared to 50 for unloaded model)(compared to 50 for unloaded model)
Performed Frequency Response and Performed Frequency Response and compared to original model with compared to original model with adjusted DC gainadjusted DC gain
0.001
0.01
0.1
1
10
0.01 0.10 1.00 10.00
Freqency(Hz)
Att
enu
atio
n
SystemResponse
Modelresponse
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Observer Controller Observer Controller DesignDesign
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ObserverObserver
Feedback Controller used: Parallel PI Feedback Controller used: Parallel PI controllercontroller
Linear System Model UsedLinear System Model Used
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Controller
Used PID Controller with disturbance rejection
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Unloaded Results
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Loaded Results
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Disturbance Rejection Observer Specifications
Phase Margin = 50 degrees Gain Margin = 3.5 Steady state error < 1 degree Rise Time = 1.17 s % Overshoot = 3%
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How the Others Fail
All: good rise time and overshoot
Proportional controller: bad steady state error
Observer and PID: insufficient phase margin
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Work Completed: 2-DOF Work Completed: 2-DOF ArmArm
•Base Modeling
•Spring Modeling
•Sample Rate
•Controller Design
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Base ModelingBase Modeling
• Model of arm without effect of springsModel of arm without effect of springs
• TTss=4/(=4/(ζζωωnn))
• ζζωωn n is the real part of polesis the real part of poles
• Gp=1500/(sGp=1500/(s22+10s)+10s)
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Spring ModelingSpring Modeling
• Reran test and plotted arm displacementReran test and plotted arm displacement
• Frequency of oscillation is imaginary Frequency of oscillation is imaginary partpart
• Settling time is real partSettling time is real part
• GGDD=G=GDdcDdcs/(ss/(s22+8s+289)+8s+289)
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Spring ModelingSpring Modeling
• Spring effect is instantaneousSpring effect is instantaneous
• Springs have no steady state effectSprings have no steady state effect
• Behaves like differentiatorBehaves like differentiator
• GGDD=0.42s/(s=0.42s/(s22+8s+289)+8s+289)
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Spring and Arm Spring and Arm TogetherTogether
• Modeled as a minor loop disturbanceModeled as a minor loop disturbance• Positive feedback because of Positive feedback because of
increasing overshoot and settling increasing overshoot and settling timetime
Actual Arm Position
Base transfer function remains unchanged
Spring Displacement depends on base movement
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Model and Plant Model and Plant ComparisonComparison
Arm Model
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Model and Plant Model and Plant ComparisonComparison
PlantPlant ModelModel %os=41.7%%os=41.7% %os=37.4%%os=37.4% TTss=1.12s=1.12s TTss=1.21s=1.21s
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System Root LocusSystem Root Locus
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New Sample RateNew Sample Rate
• For smooth operation of motor, For smooth operation of motor, ωωs s ≥≥ 6 6ωωcc
• ωωc c =10.7rad/s : =10.7rad/s : TTcc= 0.587s= 0.587s
• TTsam maxsam max≈0.0978s≈0.0978s
• TTsamsam chosen to be 0.1s chosen to be 0.1s• Largest sample time spreads out root Largest sample time spreads out root
locuslocus• Complex poles and zeros don’t affect Complex poles and zeros don’t affect
responseresponse
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New Plant Root LocusNew Plant Root Locus
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Proportional ControlProportional Control
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PID ControlPID Control
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PID ControlPID Control
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Lead NetworkLead Network
Pole-zero cancellationPole-zero cancellation Lead pole chosen to be at zero for Lead pole chosen to be at zero for
fastest settling timefastest settling time
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Lead NetworkLead Network
Gain of 0.06 Gain of 0.06 should give should give TTss of 0.72s of 0.72s with 15%OSwith 15%OS
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Lead NetworkLead Network
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Minor Loop With PI Minor Loop With PI Control DiagramControl Diagram
PositionVelocity
PI Control
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Minor Loop With PI Minor Loop With PI ControlControl
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Classical Control Classical Control ConclusionsConclusions
Proportional and PID control did not Proportional and PID control did not handle loads very wellhandle loads very well
Minor Loop Performed well but is Minor Loop Performed well but is close to instabilityclose to instability
Lead Network was the best choice by Lead Network was the best choice by farfar
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Observer ControllerObserver ControllerGC(s) GPC(s) GP(s) GS(s)
GCO(s)
GPEst(s) GSEst(s)
+
+
+
+
-
-
R(s) PR(s)E(s) PC(s) C(s) Y(s)
EO(s)
CO(s)
YO(s)
• GC(s) = Our Lead Network = (0.2)(z - 0.458)/z
•GPEst(s) = Plant Estimator = (3.127z + 2.246)/(z2 - 1.368z + 0.3679)
•GCO(s) = Observer Compensator (Lead-Network Controller) = (0.06)(z – 0.4)/z
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Estimator OutputEstimator Output
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The Observer gave us no The Observer gave us no overshoot and a settling overshoot and a settling time of 0.9 seconds.time of 0.9 seconds.
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Observer Controller With Observer Controller With Disturbance RejectionDisturbance Rejection
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Observer With Observer With Disturbance RejectionDisturbance Rejection
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Comparison Of No Load Comparison Of No Load ResultsResults
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Comparison Of Loaded Comparison Of Loaded ResultsResults
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Spring InaccuracySpring Inaccuracy
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ResultsController Ts (unloaded) Ts (loaded) % Overshoot (unloaded) % Overshoot (loaded)
Lead-Network 0.9 1.1 0.0 0.0Minor-Loop 1.0 1.0 7.0 10.0Observer 1.2 1.4 9.0 12.0PID Controller 1.1 1.8 0.0 3.3Proportional 1.1 1.9 0.3 3.9
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2-DOF Arm Conclusions2-DOF Arm Conclusions
Observer works best if there is no Observer works best if there is no need for disturbance rejectionneed for disturbance rejection
With disturbance rejection, observer With disturbance rejection, observer was not better than classical was not better than classical controller methodscontroller methods
Lead Network Controller proved to Lead Network Controller proved to be the most effective overall for both be the most effective overall for both loaded and unloaded conditionsloaded and unloaded conditions
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Inverted Arm Conclusions
Encoder used was very accurate
Results mildly are improved
Useful if computational complexity is cheap
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QuestionsQuestions
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Root Locus with Graphical Root Locus with Graphical KK
Proportional controlProportional control
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Lead Network Root Lead Network Root LocusLocus
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Minor Loop Graphical Minor Loop Graphical GainGain
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Minor Loop Bode PlotMinor Loop Bode Plot
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2-DOF Arm Configuration
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Inverted Arm Inverted Arm ConfigurationConfiguration
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22ndnd Order Step Response Order Step Response
Proportional gain Proportional gain of 0.45of 0.45
%O.S.=46%%O.S.=46% Ts=0.58 sTs=0.58 s Tr=0.06 sTr=0.06 s Tp=0.14 sTp=0.14 s
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Q8 Acquisition Board Q8 Acquisition Board SpecsSpecs
14 bit A/D converter +-10V14 bit A/D converter +-10V
- 1.22 mV resolution- 1.22 mV resolution
- Maximum conversion time = - Maximum conversion time = 5.25.2μμss
- Maximum Sample Frequency = - Maximum Sample Frequency = 192kHz192kHz
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Q8 Acquisition Board Q8 Acquisition Board SpecsSpecs
12 bit D/A converter +- 5V12 bit D/A converter +- 5V- 2.44 mV resolution- 2.44 mV resolution
- Slew rate = 2.5V/- Slew rate = 2.5V/μμss
- Max voltage change is from -5 to - Max voltage change is from -5 to 5, or 10V5, or 10V
- Max conversion time = 4- Max conversion time = 4μμss
- Max sample frequency = 250kHz- Max sample frequency = 250kHz