Geometric Methods for Modelling and Control of Shape-Actuated - PDF document

Geometric Methods for Modelling and Control of Shape-Actuated Underwater Vehicles Kristi A. Morgansen Department of Aeronautics and Astronautics University of Washington Nonlinear Dynamics and Control Lab Modeling and control of shape-actuated immersed mechanical systems Coordinated control with limited communication Bioinspired system modeling for coordinated control Integrated communication and control 2 1

Modeling and control of fin-actuated underwater vehicles Free-swimming agile maneuverability Tail locomotion and pectoral fin maneuverability • Near shore (littoral) data collection • Operation in caves/canyons • Agile maneuverability • Prototype platform for autonomous system development • Platform for nonlinear control system design • Control with limited communication/bandwidth • Coordinated control and sensing Why fins? •Low-speed maneuverability •Low-noise (bubbles in water) •Low-drag 3 Objectives Nonlinear dynamics and control of underwater bioinspired systems •Shape actuated mechanical system •Coupled mechanical-fluid dynamics •Inertial effects not negligible •Controllable system model but uncontrollable linearization •Model predictive but not accurate •Higher order effects necessary for full range of motion 4 2

Outline • UW fin-actuated UAV • Vehicle control – Modeling – Optimality – Motion generation – Feedback • Actuation surface modeling 5 Outline • UW fin-actuated UAV • Vehicle control – Modeling – Optimality – Motion generation – Feedback • Actuation surface modeling 6 3

UW Fin-Actuated UAV Tail locomotion and pectoral fin maneuvering servo motors microcontroller 3D digital compass acoustic modem batteries pressure sensor and buoyancy control 7 Outline • UW fin-actuated UAV • Vehicle control – Modeling – Optimality – Motion generation – Feedback • Actuation surface modeling 8 4

Modeling Geometric techniques •Control theoretic model of shape-actuated mechanical system immersed in fluid •Momentum equations of body and shape variables •Added mass •Simplified lift and drag •Geometric nonlinear control methods •Controllability •Motion generation •Feedback 9 Modeling Momentum and shape equations •Unforced equations (rigid body) in body-fixed coordinates 10 5

Modeling Momentum and shape equations • Added mass •Potential forces (gravity, buoyancy) Motion of foil results in acceleration of fluid •Lift on airfoils (quasisteady) •Drag (quasisteady) 11 Modeling Momentum and shape equations •Potential forces •Lift on airfoils •Drag 12 6

Modeling Momentum and shape equations Body-fixed coordinates of the main body Kirchoff’s equations Shape equations 13 Modeling Simplified model for pectoral fin actuation 14 7

Modeling Control affine equations Control affine model Constructive controllability: How to determine if arbitrary motion can be achieved, and if so, how to construct the control functions u I 15 Outline • UW fin-actuated UAV • Vehicle control – Modeling – Optimality – Motion generation – Feedback • Actuation surface modeling 16 8

Optimality Biological and optimal results indicate nonlinear frequencies • Wing beat trajectories of dragonflies, and tail sweep patterns in fish follow a trajectory that is nonlinear in frequency rather than linear [Wang 2000] • Numerical optimization results for robot fish tail propulsion demonstrate nonlinear frequencies [Saimek/Li 2003] • These fluid dynamic effects can be modeled by a class of second order nonlinear systems with uncontrollable linearization [Morgansen et al 2001] 17 Optimality Geometric techniques •Analyze optimal control of class of dynamic systems •Construct amplitude-modulated nonlinear frequency oscillations The values of the states are respectively determined by the integral of u 1 , the integral of u 2 and the area swept out by the integrals of u 1 and u 2 . 18 9

Optimality Point to point optimal control Simplify using an integration by parts 19 Optimality Point to point optimal control Proposition: Given the dynamic system above, the controls which minimize the cost function and meet the endpoint constraints satisfy where M(t) = -M T (t), N = -N T and 20 10

Optimality Point to point optimal control •Full solution does not exist in closed form for general problem, sample case here is hypergeometric •Solution was derived for t ∈ [0,1] •Frequency increases in time nonlinearly for all time 21 Optimality Motion generation - Requirements Goal: Construct control input functions that are cyclic and that can be incorporated into averaging methods Fresnel Sine Requirements: •Integral of the inputs must have closed form solution •Inputs should start and end at the same value •Over a cycle, the inputs should integrate to zero Fresnel Cosine 22 11

Optimality Motion generation - Linear frequencies Proposition Given three times differentiable functions x d (t) and a controllable dynamic system of the form discussed here, there exist ω -parameterized controls of the form such that 23 Optimality Motion generation - Symmetric nonlinear frequencies Proposition: Given three times differentiable functions x d (t) and a controllable dynamic system of the form discussed here, there exist ω - parameterized controls of the form with such that 24 12

Optimality Motion generation - Symmetric nonlinear frequencies 25 Optimality Finite cost comparison The values of ν and µ determine the relative importance of x 3 and its derivative in the cost function Linear frequency Nonlinear frequency 26 13

Outline • UW fin-actuated UAV • Vehicle control – Modeling – Optimality – Motion generation – Feedback • Actuation surface modeling 27 Motion Generation Constructive controllability - Nonlinear controllability Controllability For a given x 0 , x f , there exists a time T and a control law u(t) such that x(0)=x 0 and x(T)=x f . How to determine if the system can be moved in arbitrary directions The Lie bracket f 2 −ε f 1 f 1 −ε f 1 ε f 2 f 2 ε 28 14

Motion Generation Constructive controllability - Nonlinear controllability Small time local configuration controllability The Lie brackets f 2 −ε f 1 f 1 −ε f 1 ε f 2 span forward and turning directions at all points x ∈ M, and all brackets with f 2 ε an odd occurrence of f 0 and even occurrence of each f i must be linear combinations of lower order brackets. 29 Motion Generation Constructive controllability - Nonholonomic systems Differential drive mobile robot 30 15

Motion Generation Constructive controllability - Nonholonomic systems The nonholonomic integrator The values of the states are respectively determined by the integral of u 1 , the integral of u 2 and the area swept out by the integrals of u 1 and u 2 . 31 Motion Generation Constructive controllability - Motion generation 32 16

Motion Generation Planar locomotion from tail actuation 33 Motion Generation Free locomotion from tail actuation and pectoral maneuvering 34 17

Motion Generation UW fin-actuated UAV - Tail locomotion and pectoral fin maneuvering servo motors microcontroller 3D digital compass acoustic modem batteries pressure sensor and buoyancy control 35 Motion Generation Forward locomotion simulation 36 18

Motion Generation Forward locomotion Experiment Simulation 37 Motion Generation Tail yaw actuation simulation 38 19

Motion Generation Turning locomotion Experiment Simulation 39 Motion Generation Pectoral kicking and “dolphin” locomotion 40 20

Motion Generation Oscillatory pectoral yaw actuation 41 Motion Generation Motion from nature - Turning locomotion To map curves to joint angles, each curve was digitized and reoriented with the head directed along the horizontal axis and scaled to have the same length as the robot. A least squares fit was then used to determine joint angles matching the curve. Backbone curve of a goldfish during a snap turn (images at 0.4s). [D. Weihs, Proc. R. Soc. London B, 1972] 42 21

Motion Generation Motion from nature - Experimental results 43 Motion Generation Comparison of experiment and simulation Experiment Simulation While the simulated response does demonstrate turning, it does not agree well with experiment either numerically or qualitatively. Most likely need more exact calculation of added mass and possibly unsteady lift and drag calculations. 44 22

Outline • UW fin-actuated UAV • Vehicle control – Modeling – Optimality – Motion generation – Feedback • Actuation surface modeling 45 Feedback Modify oscillation amplitudes over whole periods to stabilize Theorem Given a mechanical system controllable with first and second order symmetric products, there exist feedback gains such that the averaged system is stable. Evaluate system error at end of each input cycle and update input amplitudes Possible problem : may require large actuator bandwidth 46 23

Feedback Underactuated mechanical systems - Oscillatory control Equations of motion for a mechanical system High frequency, high amplitude averaging Correspondence 47 Feedback System motion can be approximated over input oscillation cycles Given system shape inputs The system response is approximately Problem : fish robot requires high order averaging 48 24