Sliding Mode Control of a Non-Collocated Flexible System

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Sliding Mode Control of a Non-Collocated Flexible
System

• Aimee Beargie
• November 13, 2002
• Committee
• Dr. Wayne Book, Advisor
• Dr. Nader Sadegh
• Dr. Stephen Dickerson
• Sponsor
• CAMotion, Inc.

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Problem Statement

• Develop an algorithm to control the tip position
  of a mechanism that is actuated at the base
  (non-collocated problem)
• Recently developed algorithms generally deal with
  collocated problems
• Sensors Encoder, Accelerometer, Machine Vision
• State Feedback Control
• Kalman Filter
• Robust to parameter variations

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Variable Structure Control Research

• Model using Assumed Mode Method
• Qian Ma Tracking Control
• Chang Chen Force Control
• Comparison to other Methods
• Hisseine Lohmann Singular Perturbation
• Chen Zhai Pole Placement
• Robustness
• Iordanou Surgenor using inverted pendulum
• Combined with Other methods
• Romano, Agrawal, Bernelli-Zazzera Input
  Shaping
• Li, Samali, Ha Fuzzy Logic

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System Model
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System Model

• Equations of Motion
• Small Angle Approximation

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System Model
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System Model

• System Parameters
• m1 8 kg
• m2 2.55 kg
• L 0.526 m
• r 0.377 m
• I 0.4367 kg-m2
• k 32,199 N-m
• b 9.8863 N-m-s

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Variable Structure Control (VSC)

• Also called Sliding Mode Control
• Switched feedback control method that drives a
  system trajectory to a specified sliding surface
  in the state space.
• Two Part Design Process
• Sliding Surface (s) desired dynamics
• Controller Lyapunov analysis

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VSC Sliding Surface Design

• Regular Form
• Dynamics of state feedback structure

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VSC Sliding Surface Design

• Transformation to Regular Form

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VSC Control Design

• Use Lyapunov stability theory
• Positive Definite Lyapunov Function
• Want Derivative to be Negative Definite for
  Stability

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VSC Control Design

• Control Structure
• Resulting Equation

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VSC Generalizing Gain Calculation
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Control System Overview
Desired Trajectory
System Dynamics
Control Algorithm
RASID
Motor Amp
Encoder Meas.
Kalman Filter
Accelerometer Meas.
Vision Meas.
Computer _at_ 1kHz

• RASID internal PID control _at_ 10kHz

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Outer Loop Simulation

• Used LQR for Sliding Surface Design
• Error used in Control Calculation

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Outer Loop Simulation
Max error 0.015mm
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Inner Loop Simulation

• Force converted into Position Signal
• PD Equations
• Discrete Position Calculation

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Inner Loop Simulation
Max error 0.02mm
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Simulation using Estimated States

• Developed by Mashner
• Vision
• Measurement Frequency of 30 Hz
• Delay of 5 ms
• Covariance
• Accelerometer std. deviation squared
• Vision/Encoder

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Simulation using Estimated States
Max error 0.2mm
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Simulation Penalty on xtip and vbase
Max error 0.5mm
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Robustness Simulation 50 of mtip
Max error 0.3mm
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Robustness Simulation 110 of mtip
Max error 0.5mm
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Experimental Set-up
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Experimental Results VSC w/ Kalman Filter
MSE 1.3620e-6 m2
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Experimental Results Robustness

• Mean Squared Error
• 0 1.4170e-6 m2
• 10 1.6309e-6 m2
• 16 1.8068e-6 m2

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Experimental ResultsComparison of Control
Methods

• Mean Squared Error
• PD 9.7750e-7 m2
• LQR 1.8366e-5 m2
• VSC 1.3620e-6 m2

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Conclusions

• Developed method results in acceptable tracking
  of tip position
• Verified through simulations and experiments
• Method generalized for LTI systems
• Better performance than other control methods
• Robust to parameter variations
• Choice of Cost function critical
• Verified experimentally for tip mass

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Further Work

• Desired Trajectory
• Currently designed for rigid system
• Possible use trajectory that is continuous in
  fourth derivative
• Adaptive Learning
• Input Shaping

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QUESTIONS
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