Robust Non-Linear Observer for a Non-collocated Flexible System

Robust Non-Linear Observer for a Non-collocated Flexible System
Mohsin Waqar Intelligent Machine Dynamics Lab Georgia Institute of Technology December 12, 2007

Agenda Project Motivation and Goals
Non-collocated Flexible System and Non-minimum Phase Behavior Control Overview Test-bed Overview Plant Model Optimal Observer – The Kalman Filter Robust Observer – Sliding Mode Project Roadmap REFER BACK TO THIS SLIDE DURING PRESENTATION!

Motivation for Research – Flexible Robotic Arms
1) Manipulators with very large workspaces (long reach): Example - handling of nuclear waste. 2) Manipulators with constraint on mass: Example – space manipulators. 3) Manipulators with constraint on cost: Example – Camotion Depalletizer 4) Manipulators with Actuator/Sensor Non-collocation: Collocation can be impossible. Source: NASA.gov 2) Space Station carbon fiber 7d0f robotic manipulator, 60ft long extended, 4000lbm, moves 260,000lbm (60+ times its own weight), loaded EEF moves 0.8” per sec 3) A lightweight depalletizer allows mounting rails to be much less bulky as well. Whole system has reduced cost. Collocating actuators and sensors can be very costly. 4) These are manipulators with truly high performance: Truly high precision results from minimizing error at end point of tip, not at the joint. We can do better than 20 micron repeatability of modern robots. Its sometimes impossible to try and collocate sensors and actuators. No choice but to apply a control torque far away from end-point of flexible arm. Source: camotion.com

Problem Statement Contribute to the field of active vibration suppression in motion systems Examine the usefulness of the Sliding Mode Observer as part of a closed-loop system in the presence of non-collocation and model uncertainty. Point out that the SMO is to be compared to the KF. That’s what determines usefulness.

What is a Flexible Robotic Arm?
Source: Shabana, A. A. Vibration of Discrete and Continuous Systems Robotic arm is subject to torsion, axial compression, bending. Structural stiffness, natural damping, natural frequencies and boundary conditions are important to consider. A flexible arm is one in which deflections are large enough or persist long enough to disrupt some task. Energy is stored in the form of kinetic and potential energy when a robotic arm is twisted, compressed or bent. A good model will consider energy storage from all three of these deformations. There are an infinite number of natural frequencies and corresponding mode shapes. Any free vibration consists of superposition of individual mode shapes. Increasing damping makes control easier. Boundary conditions basically tell you the state of displacements, forces, moments, and stress at the ends of a beam element for example. This is done by identifying the ends as fixed or free. Note: pole = eigenvalue = mode = natural frequency

What is a Flexible Robotic Arm? - References
W.J. Book, “Modeling, Design, and Control of Flexible Manipulator Arms: A Tutorial Review,” Proceedings of the 29th Conference on Decision and Control, Dec W.J. Book, “Structural Flexibility of Motion Systems in Space Environment,” IEEE Transactions on Robotics and Automation, Vol. 9, No. 5, pp , Oct W.J. Book, “Flexible Robot Arms,” Robotics and Automation Handbook., pp , CRC Press, Boca Raton, FL, 2005. A flexible arm is one in which deflections are large enough or persist long enough to disrupt some task. Energy is stored in the form of kinetic and potential energy when a robotic arm is twisted, compressed or bent. A good model will consider energy storage from all three of these deformations. There are an infinite number of natural frequencies and corresponding mode shapes. Any free vibration consists of superposition of individual mode shapes. Increasing damping makes control easier. Boundary conditions basically tell you the state of displacements, forces, moments, and stress at the ends of a beam element for example. This is done by identifying the ends as fixed or free.

Non-Minimum Phase Behavior (in continuous time system)
Causes: Combination of non-collocation of actuators and sensors and the flexible nature of robot links Detection: System transfer function has positive zeros. Effects: Limited speed of response. Initial undershoot (only if odd number of pos. zeros). Multiple pos. zeros means multiple direction reversal in step response. PID control based on tip position fails. Source: Cannon, R.H. and Schmitz, E. “Initial Experiments on the End-Point Control of a Flexible One-Link Robot.” 1984. Zeros of a system depend on A, B and C matrix. Hence the zeros depend on placement of actuators and sensors. Noncollocated sensors used in past: accelerometers, machine vision, strain gauges, photodiodes (vibration sensor), laser measurement devices (for tip), Cause for limited speed of response is in our case: time delay between torque input and link tip displacement from finite wave propagation speed. Zero flipping occurs as the transfer function changes when a parameter is changed. Once this occurs, system will likely become unstable. Other examples of non-min system: inverted pendulum, steering a car while driving in reverse.

Non-Minimum Phase Behavior (in continuous time system)
Effects: Limited gain margin (limited robustness of closed-loop system) Model inaccuracy (parameter variation) becomes more troubling (Zero- flipping). Im X X X Re Limited gain margin is due to the fact that closed-loop poles made up of open loop poles and zeros. This limits use of high gain feedback. As loop gain increases, poles move towards zeros. Min phase systems can have infinite gain margin. In certain non-collocated systems, poles/zeros are alternate as one moves up the imaginary axis in s-plane. For these transfer functions, Zero flipping occurs as parameters vary. Zero moves in between plant vibration poles. Zeros are actually more sensitive than poles to param. Variation. Zero flipping leads to instability.

Non-Minimum Phase Behavior - References
R.H. Cannon and D. E. Rosenthal, “Experiments in Control of Flexible Structures with Noncolocated Sensors and Actuators,” J. Guidance, Vol. 7, No. 5, Sept.-Oct R.H. Cannon and E. Schmitz, “Initial Experiments on the End-Point Control of a Flexible One-Link Robot,” International Journal of Robotics Research, 1984. D.L. Girvin, “Numerical Analysis of Right-Half Plane Zeros for a Single-Link Manipulator,” M.S. Thesis, Georgia Institute of Technology, Mar J.B. Hoagg and D.S. Bernstein, “Nonminimum-Phase Zeros,” IEEE Control Systems Magazine, June 2007.

Control Overview + u F δ y -
Noise V + Commanded Tip Position u F δ y Feedforward Gain F Linear Motor Flexible Arm Sensors - Feedback Gain K Observer Emphasize that state feedback control requires knowledge of all states and that’s where an observer comes in. State feedback has been shown to be effective but it is sensitive to parameter variation. I am separating estimation and control. I am using a simpler two step approach which works for linear systems. Design objective: Accuracy, repeatability and steadiness of the beam end point.

Test-Bed Overview - + + - R PCB 352a Accelerometer PCB Power Supply C
LS7084 Quadrature Clock Converter Anorad Encoder Readhead Anorad Interface Module - + LV Real Time 8.2 Target PC w/ NI-6052E DAQ Board + NI SCB-68 Terminal Board Anorad DC Servo Amplifier 160VDC Linear Motor - PWM -10 to +10VDC Accelerometer bw=5-8000hz After RC filtering: signal noise ~6mV Phase delay Linear motor range workspace = 5 ft DCmotor.

Test-Bed Rigid Sub-system ID
Starting Parameters: Km=8.17; % overall motor gain [N/V] M=9.6; % base mass [kg] b=50; % track-base viscous damping [N*s/m] Using fmincon in Matlab with bounds: Km: +/- 25% M: +/- 10% b: (0,inf) Final Parameters with Step Input: [Km,M,b] = [6.9, 10.52, 38.97] With ramp input (0-5V over 2 sec): [Km,M,b] = [6.13, 10.56, 35.16] Assume type-1 motion system model Initial Km found using spring scale an applying voltage from 3-6V. Accelerometer bw=5-8000hz Linear motor range workspace = 5 ft 3phase motor.

Flexible Arm Modeling Lumped Parameter System (or Discrete System)
Distributed Parameter System (or Continuous System) Finite degrees of freedom. Described by one second-order ODE per degree/order of the system. Mashner (2002) Beargie (2002) 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. Lumped mass systems composed of lumped masses, springs and dampers. Finite degree of freedom means a finite number of modes. In a distributed parameter system, location of a point on the link is described by rigid body motion of link and elastic deflection of link. Lagrangian and newton euler are both used to model rigid body motion of links as well as vibration of the link. Approximate methods neglect high frequency modes and represent solution as in terms of finite number of modes. TMM is similar to FEA because the links are discretized and represented by idealized elements. For AMM, you predict the mode shape beforehand using a polynomial function. The catch with using a model with finite number of modes is that we are now prone to control spillover (exciting unmodeled higher modes) and measurement spillover (measurements corrupted by unmodeled higher modes). Added obstacles to control. Approaches: 1) Lagrangian: Obergfell (1999) 2) Newton Euler: Girvin (1992) Approximate methods: 3) Transfer Matrix Method: Krauss (2006), Girvin (1992) 4) Assumed Modes Method: Sangveraphunsiri (1984), Huggins (1988), Lane (1996)

Flexible Arm Modeling – Assumed Modes Method
E, I, ρ, A, L m F w(x,t) x Assumptions: Uniform cross-section 3 flexible modes + 1 rigid-body mode Undergoes flexure only (no axial or torsional displacement) Linear elastic material behavior Horizontal Plane (zero g) No static/dynamic friction at slider Light damping (ζ << 1) A rigid body mode is one in which the elastic element undergoes no deformation. Use n shape functions or basis functions to approximate how a continuous system deforms. Basis functions must be independent and continuous. They must satisfy boundary conditions (vanish where beam cannot deflect). If rigid body motion is possible, a basis function for free-free boundary conditions of beam must be present. Kinetic energy and strain energy written with finite number of terms. Then Lagrange’s equation is written which leads to n ODEs. The equation of motion is written where M is nxn mass/inertia matrix, K is nxn stiffness matrix, q are generalized coordinates along the length of beam and Q are generalized forces. This result is solved like a lumped parameter multi-DOF system. The general eigenvalue problem is solved using Matlab and eigenvalues (nat. freqs) and eigenvectors (mode shapes) are found. The eigenvectors are normalized. The normalized eigenvectors are used to transform from generalized coordinates to mode coordinates. This allows us to decouple the equation of motion. The N dof system is not N single dof systems so we can now write it easily in state space. The mode coordinates are not the quanities we need. To get displacements at points of interest we use the mode coordinates and the basis functions. Note that the damping terms in the decoupled eom are only valid under assumption of light damping (damping ratios <<1).

E, I, ρ, A, L F w(x,t) Geometric Boundary Conditions: For i = 1 to 4 Ritz Basis Functions: Source: J.H. Ginsberg, “Mechanical and Structural Vibrations,” 2001 A rigid body mode is one in which the elastic element undergoes no deformation. Use n shape functions or basis functions to approximate how a continuous system deforms. Basis functions must be independent and continuous. They must satisfy boundary conditions (vanish where beam cannot deflect). If rigid body motion is possible, a basis function for free-free boundary conditions of beam must be present. Kinetic energy and strain energy written with finite number of terms. Then Lagrange’s equation is written which leads to n ODEs. The equation of motion is written where M is nxn mass/inertia matrix, K is nxn stiffness matrix, q are generalized coordinates along the length of beam and Q are generalized forces. This result is solved like a lumped parameter multi-DOF system. The general eigenvalue problem is solved using Matlab and eigenvalues (nat. freqs) and eigenvectors (mode shapes) are found. The eigenvectors are normalized. The normalized eigenvectors are used to transform from generalized coordinates to mode coordinates. This allows us to decouple the equation of motion. The N dof system is not N single dof systems so we can now write it easily in state space. The mode coordinates are not the quanities we need. To get displacements at points of interest we use the mode coordinates and the basis functions. Note that the damping terms in the decoupled eom are only valid under assumption of light damping (damping ratios <<1).

E, I, ρ, A, L F w(x,t) A rigid body mode is one in which the elastic element undergoes no deformation. Use n shape functions or basis functions to approximate how a continuous system deforms. Basis functions must be independent and continuous. They must satisfy boundary conditions (vanish where beam cannot deflect). If rigid body motion is possible, a basis function for free-free boundary conditions of beam must be present. Kinetic energy and strain energy written with finite number of terms. Then Lagrange’s equation is written which leads to n ODEs. The equation of motion is written where M is nxn mass/inertia matrix, K is nxn stiffness matrix, q are generalized coordinates along the length of beam and Q are generalized forces. This result is solved like a lumped parameter multi-DOF system. The general eigenvalue problem is solved using Matlab and eigenvalues (nat. freqs) and eigenvectors (mode shapes) are found. The eigenvectors are normalized. The normalized eigenvectors are used to transform from generalized coordinates to mode coordinates. This allows us to decouple the equation of motion. The N dof system is not N single dof systems so we can now write it easily in state space. The mode coordinates are not the quanities we need. To get displacements at points of interest we use the mode coordinates and the basis functions. Note that the damping terms in the decoupled eom are only valid under assumption of light damping (damping ratios <<1).

Flexible Arm Model vs Experimental
AMM Model with Optimization Bounds: (0,inf) Length and Tip Mass +/- 25% All Others AMM Model with Optimization Bounds: +/- 25% On All Parameters Experimental Data Length (m) .32 Width (m) 0.035 (1 3/8”) Thickness (m) (1/8”) Material AISI 1018 Steel Density (kg/m^3) 7870 Young’s Modulus (GPa) 205 Tip Mass (kg) .110 Length (m) .4 Width (m) 0.0412 Thickness (m) .0024 Material AISI 1018 Steel Density (kg/m^3) 9838 Young’s Modulus (GPa) 205 Tip Mass (kg) .1375 Length (m) .707 Width (m) 0.0262 Thickness (m) .0037 Material AISI 1018 Steel Density (kg/m^3) 5903 Young’s Modulus (GPa) 205 Tip Mass (kg) 1.76 For flexible system system ID: Ts was 1khz and Kp, Ki, Kd = 4, .1, 2 First Mode 5.5 hz Second Mode 49.5 hz Third Mode 130.5 hz First Mode 14.3 hz Second Mode 81.5 hz Third Mode 331.4 hz First Mode 5.83 hz Second Mode 45.2 hz Third Mode 186.5 hz

Flexible Arm Model vs Experimental
System ID setup: Loop rate 1khz Closed-loop PID 0-40hz Chirp Reference Signal with cm p-p amplitude 15 data sets averaged Kp, Ki, Kd = 4, .1, 2

Flexible Arm Model – Root Locus

Performance Criteria for Observer Study
What is a useful observer anyway? Robust (works most of the time) Accuracy not far off from optimal estimates Not computationally intensive Straightforward design Uses simple rather than a complex plant model Focus is on parameter uncertainty which means changes to mass, stiffness, damping. Not computationally intensive means we can use it for real-time control of fast motion systems.

A Hypothesis for Observer Study
WHOOPS…PARABOLAS SHOULD BE FACING UPPPP!!!!!!! SMO Estimate Mean Square Error (MSE) Kalman Filter (KF) Focus is on parameter uncertainty which means changes to mass, stiffness, damping. -50% % % % % Deviation in some Beam Parameter

Overview of Steady State Kalman Filter
Why Use? Needed when internal states are not measurable directly (or costly). Sensors do not provide perfect and complete data due to noise. No system model is perfect Notable Aspects: Designed off-line (constant gain matrix) and reduced computational burden Minimizes sum of squares of estimate error (optimal estimates) Predictor-Corrector Nature Shortcomings: Limited robustness to model parameter variation Steady State KF gives sub-optimal estimates at best Process noise due to any model uncertainty such as unmodeled behavior or A/D quantization. Input noise due to measurement uncertainty in sensor signals, for example by electrical interference and sensors own dynamics and distortions. Elements for noise cov. Matrices are found by collecting measurement data offline, doing statistical analysis to find mean, variance. The diagonal elements of the matrices are variance squared for the corresponding sensor. Recursive nature means past measurements are not stored. Only thing we need to make new estimate is past estimate and an index, not the actual measurements.

How it works - Kalman Filter
Plant Dynamics Kalman Filter State Estimates with minimum square of error Measurement & State Relationships Noise Statistics Initial Conditions Filter Parameters: Noise Covariance Matrix Q – measure of uncertainty in plant. Directly tunable. Noise Covariance Matrix R – measure of uncertainty in measurements. Fixed. Error Covariance Matrix P – measure of uncertainty in state estimates. Depends on Q. Kalman Gain Matrix K – determines how much to weight model prediction and fresh measurement. Depends on P. Process noise due to any model uncertainty such as unmodeled behavior or A/D quantization. Input noise due to measurement uncertainty in sensor signals, for example by electrical interference. Elements for noise cov. Matrices are found by collecting measurement data offline, doing statistical analysis to find mean, variance, etc. The diagonal elements of the matrices are variance squared for the corresponding sensor.

+ - v 1/s A B C ~A L K F u r x y Filter Design: Find R and Q 1a) For each measurement, find μ and σ2 to get R 1b) Set Q small, non-zero 2. Find P using Matlab CARE fcn Find L=P*C'*inv(R) Observer poles given by eig(~A-LC) 5. Tune Q as needed

+ - v 1/s A B C ~A L K F u r x y

Kalman Filter – LabVIEW Simulation
Observer model = Plant model KF, feedback controller (including SRL) and feedforward gain all based on false model.

Kalman Filter – LabVIEW Simulation
Plant model: A + Δ A This shows us that an optimal observer works very well for certain conditions that its designed for but poorly in other situations. Δ A may be from system wear and tear or change in tip mass

Kalman Filter – Testbed On-Line Estimation
Tip Acceleration (m/s^2) Base Position (m) Mean E-06 Variance E-10 Note: Accelerometer DC Bias of volts or m/s^2 Accel low pass filtered with wc=480hz Loop rate = 1khz PID control with low frequency chirp Filter based on AMM model: decent position estimates but poor acceleration. Q>5e-6 gave unstable estimates while Q<1e-8 gave poor estimates.

Sliding Mode Observer – Lit. Review
Slotine et al. (1987) – Suggests a general design procedure. Simulations shows superior robustness properties. Chalhoub and Kfoury (2004) – 4th order observer with single measurement. Adapts Slotine’s design approach with modifications to observer structure. Presents a unique method for selecting switching gains. Simulations of a single flexible link with observer in closed-loop. Shows KF unstable in presence of uncontrolled modes while SMO remains stable. Chalhoub and Kfoury (2006) – 6th order observer with 3 measurements. Same approach as earlier paper. Simulations of the rigid/flexible motion in IC engine. Depending on design, K may be constant or time-varying.

Sliding Mode Observer – Lit. Review
Kim and Inman (2001) – SMO design based on Lyapunov equation. Unstable estimates by KF in presence of uncontrolled modes while SMO remains stable. Simulations and experimental results of closed-loop active vibration suppression of cantilevered beam (not a motion system). Zaki et al. (2003) – 14th order observer with 3 measurements, with design based on Lyapunov equation. Experimental results (including parameter variation studies) from three flexible link testbed with PD control. Observer in open loop. Elbeheiry and Elmaraghy (2003) – 8th order observer with two measurements, with design based on Lyapunov equation. Simulations and experimental results from 2 link flexible joint testbed with PI control. Observer in open loop. Depending on design, K may be constant or time-varying.

Sliding Mode Observer – Definitions
Sliding Surface – A line or hyperplane in state-space which is designed to accommodate a sliding motion. Sliding Mode – The behavior of a dynamic system while confined to the sliding surface. Signum function (Sgn(s)) if Reaching phase – The initial phase of the closed loop behaviour of the state variables as they are being driven towards the surface.

Sliding Mode Observer – 3 Basic Design Steps
Design a sliding surface. One surface per measurement. Design a sliding condition to reach the sliding surface in finite time. Design sliding observer gains to satisfy the sliding condition.

Sliding Mode Observer – Overview
Example: Sliding Surface Single Sliding Surface Dynamics on Sliding Surface Sliding Condition Error Vector Trajectory (0,0)

Sliding Mode Observer Form
Example: Observer Error Dynamics: Luenberger Observer: Is due to model imperfection. Has potential to destabilize observer error dynamics. Sliding Mode Observer:

Sliding Mode Observer Form
General Form: Error Dynamics: + bounded nonlinear perturbations Slides modes arise in systems with discontinuous control action (the switching term). Sliding mode action occurs once switching term is zero which means estimate error has reached zero. Motion is pre-sliding mode and then sliding mode. Dimension of sliding mode is the number of sliding surfaces = number of measurements. All sliding surfaces pass through origin in phase plane (state plane). States slide along one surface than meets other surface and beings sliding along it towards origin. Manifold is intersection of sliding surfaces. Surfaces are planes and manifold is a line passing through origin. Order of new observer dynamics = n – m Order of sliding mode motion equations = order of orig. system – number of measurements …confirm this. SMO Design: first select discontinuity surface and then design discontinuous control to enforce sliding mode. L assures asymptotic stability of estimate error while G takes care of model errors. Adding sliding mode gains to Kf alone will not improve estimates under nominal operating conditions. Only under parameter variation will role of sliding mode gains show. What I have realized is that since the control action is u=-k(xhat)…all the jitter in xhat gets passed to u which is bad!! I will need to smooth it out using boundary layer. With proper selection of K which is based on some P, the Lyapunov function candidate can be used to show that is negative definite and so error dynamics are stable.

Sliding Mode Observer LabVIEW Simulation
Simplified Flexible Arm Model: n=4, y=x1 (tip position) Plant zeros: -0.7+i1.3e7, -0.7-i1.3e7, 2.75 Simulation Parameters: Parameter Mismatch: 100% (spring constant) deltaF for SMO Design: 20% Eta for SMO Design: 0.05 No K No L L + K MSE: Just L MSE: Loop rate 100hz Ke=2*K Eta=.05 deltaF=20%

Roadmap December 2007: Extend SMO to AMM Model and 2 measurements
January 2008: SMO with Closed Loop Control Simulation KF with Closed Loop Control on Testbed SMO with Closed Loop Control on Testbed February 2008: Conduct Parameter Studies

Roadmap December 2007: Extend SMO to AMM Model and measurements January 2008: SMO with Closed Loop Control Simulation KF with Closed Loop Control on Testbed SMO with Closed Loop Control on Testbed February 2008: Conduct Parameter Studies Questions?

Robust Non-Linear Observer for a Non-collocated Flexible System

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