Observer-Based Robot Arm Control System Nick Vogel, Ron Gayles, Alex Certa Advised by: Dr. Gary Dempsey

Outline Project Overview Project Overview Project Goals Project Goals Functional Description Functional Description Technical Background Information Technical Background Information Functional Requirements Functional Requirements Work Completed Work Completed Conclusions Conclusions 2

3 Project Overview Control of robot arms Control of robot arms Pendulum & 2 DOF arms Pendulum & 2 DOF arms Load Changes Load Changes Observer-based Observer-based Ellis's method Ellis's method

4 Pendulum Arm Configuration Pendulum Arm Configuration

5 2-DOF Arm Configuration

6 Project Goals Learn the Quanser software package Learn the Quanser software package Model the pendulum and horizontal arm Model the pendulum and horizontal arm Design controllers using classical control Design controllers using classical control Design controllers using observer-based control Design controllers using observer-based control Evaluate the relative performance of observers to classical controllers Evaluate the relative performance of observers to classical controllers

7 Equipment Used PC with Matlab, Simulink, and Real Time Workshop PC with Matlab, Simulink, and Real Time Workshop Motor with Quanser Control System Motor with Quanser Control System Linear Power Amplifier Linear Power Amplifier Robot arm with Gripper Robot arm with Gripper SRV-02 Rotary Servo Plant SRV-02 Rotary Servo Plant

8 Overall Block Diagram

9 Ellis's Observer-Based Controller

10 Situational Description Command of +-90 degrees Command of +-90 degrees Meet specifications for a load of up to 75 grams Meet specifications for a load of up to 75 grams Be able to pass a load back and forth between two systems Be able to pass a load back and forth between two systems Work with existing arm, sensor, and converters Work with existing arm, sensor, and converters

Technical Background Information % Overshoot – Amount the system advances past the target position % Overshoot – Amount the system advances past the target position Settling Time – Time it takes for the system to complete its response Settling Time – Time it takes for the system to complete its response Steady-State Error – Error of system after completely settling Steady-State Error – Error of system after completely settling 11

Technical Background Information Gain Margin – How much gain can be added without instability Gain Margin – How much gain can be added without instability Phase Margin – how much phase lag can be added to the system without instability Phase Margin – how much phase lag can be added to the system without instability PM=180-|system phase lag| PM=180-|system phase lag| 12

13 Product Specifications for 2-DOF Arm The overshoot of the arm shall be less than or equal to 15% The overshoot of the arm shall be less than or equal to 15% The settling time of the arm shall be less than or equal to 2s The settling time of the arm shall be less than or equal to 2s The phase margin shall be at least 50 deg The phase margin shall be at least 50 deg The gain margin shall be at least 3.5 dB The gain margin shall be at least 3.5 dB The steady state error of the system shall be at most 5 degrees The steady state error of the system shall be at most 5 degrees

14 Product Specifications For Pendulum Arm The overshoot of the arm shall be less than or equal to 15% The overshoot of the arm shall be less than or equal to 15% The settling time of the arm shall be less than or equal to 2s The settling time of the arm shall be less than or equal to 2s The phase margin shall be at least 50 deg The phase margin shall be at least 50 deg The gain margin shall be at least 3.5 dB The gain margin shall be at least 3.5 dB The steady state error of the system shall be at most 1 degree The steady state error of the system shall be at most 1 degree

15 Work Completed: Pendulum Arm Arm Modeling Arm Modeling Traditional Arm Control Traditional Arm Control Non-Linear Arm Modeling Non-Linear Arm Modeling Load Testing Load Testing Observer Design Observer Design

16 Modified Estimated DC gain vs Voltage

17 2 nd Order Pole Locations and Model System assumed to System assumed to be as shown to right Poles at -11, -2.6 Poles at -11, -2.6 Model resultsSystem results Model resultsSystem results

18 Frequency Response

19 Proportional Control Used control toolbox to find initial gain value Used control toolbox to find initial gain value Tuned gain: 0.14 Tuned gain: 0.14 For 20 degree input For 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 s ts= 0.41 s

20 PID controller Form: kp(0.09s+1)(0.4s+1)/[s(s/p1+1)] Form: kp(0.09s+1)(0.4s+1)/[s(s/p1+1)] Exact 2 nd order Exact 2 nd order Higher pole is faster Higher pole is faster D/A Converter saturates D/A Converter saturates Rate limitation needed Rate limitation needed

21 PID Controller Continued Pole Location Gain Value Overshoot % Settling Time Rate Limitation Rate Limited Settling Time Rad/s Rad/ssdeg/s 1 deg input 180 deg input

22 PID Results 45 deg input 45 deg input % OS=3.3% % OS=3.3% Ts=0.4 s Ts=0.4 s

23 Non-Linear Modeling

24 Loaded Testing Tested Loaded DC gain: approximately 27 degrees/volt (compared to 50 for unloaded model) Tested Loaded DC gain: approximately 27 degrees/volt (compared to 50 for unloaded model) Performed Frequency Response and compared to original model with adjusted DC gain Performed Frequency Response and compared to original model with adjusted DC gain

25 Observer Controller Design

26 Observer Feedback Controller used: Parallel PI controller Feedback Controller used: Parallel PI controller Linear System Model Used Linear System Model Used

Controller Used PID Controller with disturbance rejection

Unloaded Results

Loaded Results

Disturbance Rejection Observer Specifications Phase Margin = 50 degrees Gain Margin = 3.5 Steady state error < 1 degree Rise Time = 1.17 s % Overshoot = 3%

How the Others Fail All: good rise time and overshoot Proportional controller: bad steady state error Observer and PID: insufficient phase margin

32 Work Completed: 2-DOF Arm Base Modeling Spring Modeling Sample Rate Controller Design

33 Base Modeling Model of arm without effect of springs Model of arm without effect of springs T s =4/( ζ ω n ) T s =4/( ζ ω n ) ζ ω n is the real part of poles ζ ω n is the real part of poles Gp=1500/(s 2 +10s) Gp=1500/(s 2 +10s)

34 Spring Modeling Reran test and plotted arm displacement Reran test and plotted arm displacement Frequency of oscillation is imaginary part Frequency of oscillation is imaginary part Settling time is real part Settling time is real part G D =G Ddc s/(s 2 +8s+289) G D =G Ddc s/(s 2 +8s+289)

35 Spring Modeling Spring effect is instantaneous Spring effect is instantaneous Springs have no steady state effect Springs have no steady state effect Behaves like differentiator Behaves like differentiator G D =0.42s/(s 2 +8s+289) G D =0.42s/(s 2 +8s+289)

36 Spring and Arm Together Modeled as a minor loop disturbance Modeled as a minor loop disturbance Positive feedback because of increasing overshoot and settling time Positive feedback because of increasing overshoot and settling time Actual Arm Position Base transfer function remains unchanged Spring Displacement depends on base movement

37 Model and Plant Comparison ArmModel

38 Model and Plant Comparison PlantModel PlantModel %os=41.7%os=37.4% %os=41.7%os=37.4% T s =1.12sT s =1.21s T s =1.12sT s =1.21s

39 System Root Locus

40 New Sample Rate For smooth operation of motor, ω s ≥ 6ω c For smooth operation of motor, ω s ≥ 6ω c ω c =10.7rad/s : T c = 0.587s ω c =10.7rad/s : T c = 0.587s T sam max ≈0.0978s T sam max ≈0.0978s T sam chosen to be 0.1s T sam chosen to be 0.1s Largest sample time spreads out root locus Largest sample time spreads out root locus Complex poles and zeros don’t affect response Complex poles and zeros don’t affect response

41 New Plant Root Locus

42 Proportional Control KP = KP = 0.024Unloaded 0.27% OS 0.27% OS ζ= 0.88 ζ= 0.88 T s = 1.1s T s = 1.1s KG = KG = PM = 70.5 deg PM = 70.5 deg GM = 20.5dB GM = 20.5dBLoaded 3.91% OS 3.91% OS ζ= 0.72 ζ= 0.72 T s = 1.9s T s = 1.9s KG = KG = PM= 72 deg PM= 72 deg GM= 21dB GM= 21dB

43 PID Control

44 PID Control KP = KP = KI = 0.01 KI = 0.01 KD = 0.01 KD = 0.01Unloaded 0% OS 0% OS ζ= 1 ζ= 1 T s = 1.1s T s = 1.1s PM≈ 75 deg PM≈ 75 degLoaded 3.3% OS 3.3% OS ζ= 0.74 ζ= 0.74 T s = 1.8s T s = 1.8s PM≈ 75 deg PM≈ 75 deg

45 Lead Network Pole-zero cancellation Pole-zero cancellation Lead pole chosen to be at zero for fastest settling time Lead pole chosen to be at zero for fastest settling time

46 Lead Network Gain of 0.06 should give T s of 0.72s with 15%OS Gain of 0.06 should give T s of 0.72s with 15%OS

47 Lead Network KP = 0.09 KP = 0.09 G c =z-0.458/z G c =z-0.458/zUnloaded %OS = 0% %OS = 0% ζ=1.0 ζ=1.0 T s = 0.9s T s = 0.9s PM = 75 deg PM = 75 deg GM =21.3 dB GM =21.3 dBLoaded %OS = 0% %OS = 0% ζ=1.0 ζ=1.0 T s = 1.1s T s = 1.1s PM = 76 deg PM = 76 deg GM = 22.2 dB GM = 22.2 dB

48 Minor Loop With PI Control Diagram Position Velocity PI Control

49 Minor Loop With PI Control KP = 6.0 KP = 6.0 KI = 0.05 KI = 0.05Unloaded %OS = 7.0% %OS = 7.0% T s = 1.0s T s = 1.0s PM = 50 deg PM = 50 degLoaded %OS = 10% %OS = 10% T s = 1.0s T s = 1.0s PM ≈ 61 deg PM ≈ 61 deg

Classical Control Conclusions Proportional and PID control did not handle loads very well Proportional and PID control did not handle loads very well Minor Loop Performed well but is close to instability Minor Loop Performed well but is close to instability Lead Network was the best choice by far Lead Network was the best choice by far 50

51 Observer Controller G C (s) = Our Lead Network = (0.2)(z )/z G PEst (s) = Plant Estimator = (3.127z )/(z z ) G CO (s) = Observer Compensator (Lead-Network Controller) = (0.06)(z – 0.4)/z

52 Estimator Output

53 The Observer gave us no overshoot and a settling time of 0.9 seconds.

Observer Controller With Disturbance Rejection 54

Observer With Disturbance Rejection KDD = 2 KDD = 2 G CO = (z-0.4)/z G CO = (z-0.4)/z No Load No Load %OS = 9% %OS = 9% T s = 1.2s T s = 1.2s PM ≈ 63 deg PM ≈ 63 deg Loaded Loaded %OS = 12% %OS = 12% T s = 1.4s T s = 1.4s PM ≈ 55 deg PM ≈ 55 deg 55

Comparison Of No Load Results 56

Comparison Of Loaded Results 57

Spring Inaccuracy 58

Results

2-DOF Arm Conclusions Observer works best if there is no need for disturbance rejection Observer works best if there is no need for disturbance rejection With disturbance rejection, observer was not better than classical controller methods With disturbance rejection, observer was not better than classical controller methods Lead Network Controller proved to be the most effective overall for both loaded and unloaded conditions Lead Network Controller proved to be the most effective overall for both loaded and unloaded conditions 60

Inverted Arm Conclusions Encoder used was very accurate Results mildly are improved Useful if computational complexity is cheap

62 Questions

63 Root Locus with Graphical K Proportional control

64 Lead Network Root Locus

65 Minor Loop Graphical Gain

66 Minor Loop Bode Plot

67 2-DOF Arm Configuration

68 Inverted Arm Configuration

69 2 nd Order Step Response Proportional gain of 0.45 Proportional gain of 0.45 %O.S.=46% %O.S.=46% Ts=0.58 s Ts=0.58 s Tr=0.06 s Tr=0.06 s Tp=0.14 s Tp=0.14 s

70 Q8 Acquisition Board Specs 14 bit A/D converter +-10V 14 bit A/D converter +-10V mV resolution - Maximum conversion time = 5.2μs - Maximum Sample Frequency = 192kHz

71 Q8 Acquisition Board Specs 12 bit D/A converter +- 5V 12 bit D/A converter +- 5V mV resolution - Slew rate = 2.5V/μs - Max voltage change is from -5 to 5, or 10V - Max conversion time = 4μs - Max sample frequency = 250kHz