Small-Scale Robotic Arm

Senior Capstone Project

Ben Boyle and Kitera Hayes

Project Advisor: Dr. Gary Dempsey

April 29, 2004

Outline Objectives Equipment List System Specifications Functional Description Block Diagram System Parameters System Identification Implementation of Controllers Flexible Rotary Joint System Limitations Conclusion Completed Tasks Questions

Objectives

Determination of Plant ModelFast System ResponseWide Command Range (± 90 degrees)High Stability Margin (GM, PM)User-friendly Software InterfaceLow Resonant Frequency Mode with New Arm

Equipment List

200 MHz Pentium-based computerQuanser System

Robotic Arm with Flexible Rotary Joint Power Amplifier

Software MATLAB (SIMULINK) Borland C

Lab Workstation

Robotic Arm

System Specifications

Command: ± 90 set points, ± 40 deg/sec velocityPercent Overshoot = 0 %Steady-State Error = ± 2 degrees Phase Margin 70 degrees

Functional Description

Positioning

Figure 1 - Input/Output Description

Command Input

Small Scale Robotic Arm

Control

Functional Description

Software InterfacePositioning

Modes of Operation

Block Diagram

System (Plant)Software

Figure 2 - Block Diagram of Robotic Arm

System Parameters

System (Plant) Amplifier 5 [V] @ 3 [A] Position Sensor 180 of travel DC motor 5 [V] External Gears 5:1 velocity reduction Internal Gears 14.1:1 velocity reduction Antialiasing Filter first-order low-pass with pole @ 163 [rad/sec]

Software 200 [MHz] PC A/D converter 12 bit plus sign, 5 [V] D/A converter 12 bit, 5 [V]

System Identification

Closed-loop ResultsOpen-loop ResultsPlant Model Equation Plant Model Verification

System Identification

Closed-loop Results Gain k = 0.025 Best Fit

Close to 0% overshoot

Step input of ±20° DC Gain

Gp(0) = 27°/[V]

System Identification

k=0.025 D/A GpR=20

E=12

Controller voltage=0.2954

C=8

Controller Voltage = (12°)(.025) = 0.295 [V]

DC Gain [Gp(0)] = 8°/0.295 [V] = 27°/[V]

Figure 3 – DC Gain Calculation of System

System Identification

Figure 4 - Gain k = 0.025, Step input of ±20°, Closed-loop

(Experimental Results)

System Identification

Open-loop Results Verify DC gain of plant Calculate accurate time delay Help to determine plant model

System Identification

Figure 5 - k = 1.0, Step input voltage of 0.74 [V], Open-loop

(Experimental Results)

System Identification

Input Voltage = 20°/(27°/[V])

= 0.74 [V] (Open-loop)

Command Degree Calculation:

(K)(Command Voltage)(DC Gain) = Command Degrees

Theoretical Command Degrees 20°

Experimental Command Degrees 17°

Percent Error = 17.6%

System Identification

Plant Gp = k[a/(s+a)2]c(t) = k[1-e-at - at(e-at)]

@ k = 1.0 and t = 2.86 seconds, c = 11.352° Double Pole @ a = -0.76

Pole Identification using Laplace Transform

System Identification

TypicalOpen-loop

Poles

Figure 6 – Second Order System (Poles = -0.76)

ActualOpen-loop

Double Pole

-0.76

System Identification

Plant Model Equation:

27e-0.0562s

(s/0.76 + 1) 2

(OPEN-LOOP)

System Identification

20.48º

Figure 7 - SIMULINK Scope Output for Open-loop System = 20.48º

Plant Model Verification

System Identification

8.38º

Figure 8 - SIMULINK Scope Output for Closed-loop System = 8.38º

Plant Model Verification

P Controller

Figure 9 - Theoretical P Controller Output Figure 10 - P Controller System Output

PI Controller

Figure 12 - PI Controller System OutputFigure 11 - Theoretical PI Controller Output

PID Controller

Figure 13 - Theoretical PID Controller Output Figure 14 - PID Controller System Output

Feed-Forward/PI Controller

Figure 15 - Feed-Forward/PI Controller Block Diagram

Feed-Forward/PI Controller

Figure 16 - Theoretical FF/PI Controller Output Figure 17 - FF/PI Controller System Output

Controller Comparison

P Controller FF/PI Controller

Figure 19 - FF/PI Controller System OutputFigure 18 - P Controller System Output

Flexible Rotary Joint

Flexible Rotary Joint

Figure 20 - P Controller System Output Figure 21 - P Controller Flex Joint System Output

System Limitations

D/A Converter ± 5 [V] Static Friction

Just matches the applied force to try and prevent motion

Modeling Time delay e-std (linear) Kinetic Friction

Moving friction with respect to speeds Inertia

J = (mass)(radius2) Gravity

System Limitations

(a) With Friction (b) Without Friction

Figure 22(a-b) – Friction Characteristics for Pendulum System

-B/2J

PENDULUM

System Limitations

Figure 23 - Closed-loop Time Delay and % Overshoot Calculations for Varying Gain k

Tdavg = 56.2 [ms]

Time Delay Gain k Percent Overshoot Time Delay (Td) 0.01 0.64 % 80 ms 0.015 0.76 % 66 ms 0.02 0.91 % 80 ms 0.025 2.29 % 57 ms 0.03 10.45 % 58 ms 0.035 8.07 % 19 ms 0.04 28.00 % 50 ms 0.045 26.70 % 47 ms 0.05 33.48 % 48 ms

Conclusion

PI Controller is slowPID Controller does not workSolution is FF/PI Controller

Completed Tasks

Plant Model and Validation Proportional, PI, and PID Controllers FF Controller with PI

User-friendly Software Interface

Future Work Plant Model for Flexible Rotary Joint Gripper Motor with Varying Loads Notch Filter Incorporation

Questions?