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.

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

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

System Model

System Model Equations of Motion Small Angle Approximation

System Model

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

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

VSC: Sliding Surface Design Regular Form Dynamics of state feedback structure

VSC: Sliding Surface Design Transformation to Regular Form

VSC: Control Design Use Lyapunov stability theory Positive Definite Lyapunov Function Want Derivative to be Negative Definite for Stability

VSC: Control Design Control Structure Resulting Equation

VSC: Generalizing Gain Calculation

Control System Overview Desired Trajectory System Dynamics Control Algorithm RASID Motor & Amp Kalman Filter Encoder Meas. Accelerometer Meas. Vision Meas. Computer @ 1kHz RASID: internal PID control @ 10kHz

Outer Loop Simulation Used LQR for Sliding Surface Design Error used in Control Calculation

Outer Loop Simulation Max error: 0.015mm

Inner Loop Simulation Force converted into Position Signal PD Equations Discrete Position Calculation

Inner Loop Simulation Max error: 0.02mm

Simulation using Estimated States Developed by Mashner Vision Measurement Frequency of 30 Hz Delay of 5 ms Covariance Accelerometer: std. deviation squared Vision/Encoder:

Simulation using Estimated States Max error: 0.2mm

Simulation: Penalty on xtip and vbase Max error: 0.5mm

Robustness Simulation: 50% of mtip Max error: 0.3mm

Robustness Simulation: 110% of mtip Max error: 0.5mm

Experimental Set-up

Experimental Results: VSC w/ Kalman Filter MSE = 1.3620e-6 m2

Experimental Results: Robustness Mean Squared Error 0%: 1.4170e-6 m2 10%: 1.6309e-6 m2 16%: 1.8068e-6 m2

Experimental Results: Comparison of Control Methods Mean Squared Error PD: 9.7750e-7 m2 LQR: 1.8366e-5 m2 VSC: 1.3620e-6 m2

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

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

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