1. Feedback Control of Flexible Robotic Arms Mohsin Waqar Intelligent Machine Dynamics Lab Georgia Institute of Technology January 26, 2007

2. My Background in Brief Education: BS Degree in Mechanical Engineering, Cum Laude, May 2006 San Jose State University, San Jose, California Senior Design Project: Injection Molding Machine Tending Robot Work Experience: Automation Intern RAININ INSTRUMENT LLC, Oakland, CA, June 2005 – July 2006 Truck Assembly Maintenance Co-op NEW UNITED MOTOR MANUFACTURING INC, Fremont, CA, January – August 2004

3. Presentation Overview Project Goals and Motivation What is a Flexible Robotic Arm? Design Challenges Modeling of Flexible Robotic Arms State Observer Simulation in MATLAB Tentative Experimental Setup Project Roadmap

4. Problem Statement Expand field of active vibration suppression (Ryan Krauss, Spring 2006) Implement novel control approaches (Ryan Krauss, Spring 2006) What suitable sensors can be used to receive vibration feedback from a single link flexible robotic arm? What suitable observer can be used to estimate the position of the end point of a flexible robotic arm, based on corrupted measurements? What suitable control scheme can be used to make this closed-loop system robust to parameter uncertainty?

5. Motivation for Research 1) Manipulators with very large workspaces (long reach): Example - handling of nuclear waste. 2) Manipulators with constraints on mass: Example – space manipulators. 3) Manipulators with improved performance: Examples - “truly high precision,” quicker motion, less energy requirement, and lower cost. Source: http://archives.cnn.com/2000/TECH/space/08/21/canada.hand/index.html

6. What is a Flexible Robotic Arm? Robotic arm is subject to torsion, axial compression, bending. Structural stiffness, natural damping, natural frequencies and boundary conditions are important to understand. It’s NOT that we want to design flexible robotic arms, but we do have to deal with them. Design focus: Accuracy, repeatability and steadiness of the beam end point. Source: Shabana, A. A. Vibration of Discrete and Continuous Systems. 1997.

7. Design Challenges Accurate Modeling of Flexible Structure and Actuator System Nonlinearities Non-minimum Phase Behavior Parameter Uncertainty Corrupted Sensor Measurements

8. Non-Minimum Phase Behavior Causes: 1) positioning of sensors (non-collocation) and 2) flexible nature of robot links Detection: System transfer function has zeros in right half plane. Poles and zeros in S-plane are not interlaced. Effects: Limited speed of response. End point of flexible arm initially moves in wrong direction. Unstable in closed loop with increasing controller gain. Parameter variation becomes more troubling (Zero-flipping). X X X Re Im Source: Cannon, R.H. and Schmitz, E. “Initial Experiments on the End-Point Control of a Flexible One-Link Robot.” 1984. Accurate knowledge of natural frequencies and damping ratios becomes a requirement.

9. Flexible Arm Modeling Lumped Mass System (or Discrete System) For approximating a distributed parameter system. Results in multi-degree of freedom system. Finite degrees of freedom. Described by one second-order ODE per degree/order of the system. Distributed Parameter System (or Continuous System) Symbolic form retains infinite degrees of freedom and non-minimum phase characteristics. Describes rigid body motion of link and elastic deflection of link. Described by second order PDE. Several Approaches: Lagrangian: Obergfell (1999) Newton Euler: Girvin (1992) Approximate methods: Transfer Matrix Method: Krauss (2006), Girvin (1992) Assumed Modes Method: Sangveraphunsiri (1984), Huggins (1988), Lane (1996) Mashner (2002) Beargie (2002)

10. Simple Model of Single Link Recall Project Goal: What suitable observer can be used to estimate the position of the end point of a flexible robotic arm, based on corrupted measurements? F M1 M2 k

11. Overview of Kalman Filter Why Use? Needed when internal states are not measurable directly (or costly). Needed in presence of noise: process noise input noise Notable Aspects: Recursive Nature Optimal – chooses estimate which minimizes sum of squares of error (like least squares estimation). Predictor-Corrector Nature

12. Overview of Kalman Filter How it works: Step 1. Declare Initial Conditions: Error Covariance Matrix P, initial state guess x. Step 2. Declare Filter Parameters: Noise Covariance Matrices Q and R. Step 3. Predict States x based on Plant Dynamics Step 4. Update Error Covariance Matrix P (increase) Step 5. Update Kalman Matrix K Step 6. Correct State Estimate x based on measurement Step 7. Update Error Covariance Matrix P (decrease) Iterate through Steps 3 – 7 …. Noise Covariance Matrices Q and R – measure of uncertainty in plant and in measurements, respectively. Higher values for a matrix element means lots of uncertainty in a process state or in a measurement. Error Covariance Matrix P - measure of uncertainty in state estimates. Higher elements mean high uncertainty in pre- measurement estimate so weight measurements heavier. Depends on process noise. Kalman Matrix K - determines how much to weight fresh estimates based on a recent measurement (correcting estimate). Depends on Error Covariance Matrix P. At Steady State: P, K become constant. If Q, R and system matrices A, B, C already constant: can use steady state Kalman filter.

13. Steady State Kalman Filter Simulation

14. R = 1 R = 0.1 R = 0.01

15. Tentative Experimental Setup Controller Linear Servomotor Flexible Arm Sensors Estimator Commanded Tip Position Estimate for Tip Position + - Other State Estimates This has been done before by Beargie and Mashner in 2002!

16. Roadmap Phase I. Analysis and Simulation: System Modeling Simulate Noisy Conditions Observer Design Controller Design Phase II. Experimental: Familiarize with testbed + customize Sensor Experiments Observer Design Controller Design

17. Roadmap Phase I. Analysis and Simulation: System Modeling Simulate Noisy Conditions Observer Design Controller Design Phase II. Experimental: Familiarize with testbed + customize Sensor Experiments Observer Design Controller Design Questions?