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

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Geometric Methods for Modelling and Control of Shape-Actuated Underwater Vehicles

Kristi A. Morgansen

Department of Aeronautics and Astronautics University of Washington

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Nonlinear Dynamics and Control Lab

Coordinated control with limited communication Bioinspired system modeling for coordinated control Integrated communication and control Modeling and control of shape-actuated immersed mechanical systems

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Modeling and control of fin-actuated underwater vehicles

Free-swimming agile 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

Tail locomotion and pectoral fin maneuverability

Why fins?

• Low-speed maneuverability
• Low-noise (bubbles in water)
• Low-drag

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Objectives

• 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

Nonlinear dynamics and control of underwater bioinspired systems

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Outline

• UW fin-actuated UAV
• Vehicle control

– Modeling – Optimality – Motion generation – Feedback

• Actuation surface modeling

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Outline

• UW fin-actuated UAV
• Vehicle control

– Modeling – Optimality – Motion generation – Feedback

• Actuation surface modeling

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UW Fin-Actuated UAV

Tail locomotion and pectoral fin maneuvering

servo motors microcontroller pressure sensor and buoyancy control batteries 3D digital compass acoustic modem

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Outline

• UW fin-actuated UAV
• Vehicle control

– Modeling – Optimality – Motion generation – Feedback

• Actuation surface modeling

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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

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Modeling

Momentum and shape equations

• Unforced equations (rigid body) in body-fixed coordinates

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Modeling

Momentum and shape equations

• Potential forces (gravity, buoyancy)
• Lift on airfoils (quasisteady)
• Drag (quasisteady)
• Added mass

Motion of foil results in acceleration of fluid

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Modeling

Momentum and shape equations

• Potential forces
• Lift on airfoils
• Drag

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Modeling

Momentum and shape equations

Body-fixed coordinates

• f the main body

Kirchoff’s equations Shape equations

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Modeling

Simplified model for pectoral fin actuation

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Modeling

Control affine model

Control affine equations Constructive controllability: How to determine if arbitrary motion can be achieved, and if so, how to construct the control functions uI

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Outline

• UW fin-actuated UAV
• Vehicle control

– Modeling – Optimality – Motion generation – Feedback

• Actuation surface modeling

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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]

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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 u1, the integral of u2 and the area swept out by the integrals of u1 and u2.

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Optimality

Point to point optimal control

Simplify using an integration by parts

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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) = -MT(t), N = -NT and

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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

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Optimality

Goal: Construct control input functions that are cyclic and that can be incorporated into averaging methods Motion generation - Requirements 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 Sine Fresnel Cosine

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Optimality

Motion generation - Linear frequencies Proposition Given three times differentiable functions xd(t) and a controllable dynamic system of the form discussed here, there exist ω-parameterized controls of the form such that

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Proposition: Given three times differentiable functions xd(t) and a controllable dynamic system of the form discussed here, there exist ω- parameterized controls of the form with such that

Optimality

Motion generation - Symmetric nonlinear frequencies

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Optimality

Motion generation - Symmetric nonlinear frequencies

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Optimality

The values of ν and µ determine the relative importance

• f x3 and its derivative in the cost function

Linear frequency Nonlinear frequency Finite cost comparison

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Outline

• UW fin-actuated UAV
• Vehicle control

– Modeling – Optimality – Motion generation – Feedback

• Actuation surface modeling

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Motion Generation

Constructive controllability - Nonlinear controllability Controllability For a given x0, xf , there exists a time T and a control law u(t) such that x(0)=x0 and x(T)=xf. How to determine if the system can be moved in arbitrary directions The Lie bracket

−ε ε ε −ε f 1 f 2 f 1 f 2 f 2 f 1

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Motion Generation

Constructive controllability - Nonlinear controllability

Small time local configuration controllability

span forward and turning directions at all points x ∈ M, and all brackets with an odd occurrence of f0 and even

• ccurrence of each fi must be linear

combinations of lower order brackets. The Lie brackets

−ε ε ε −ε f 1 f 2 f 1 f 2 f 2 f 1

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Motion Generation

Constructive controllability - Nonholonomic systems

Differential drive mobile robot

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Motion Generation

Constructive controllability - Nonholonomic systems

The nonholonomic integrator

The values of the states are respectively determined by the integral of u1, the integral of u2 and the area swept out by the integrals of u1 and u2.

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Motion Generation

Constructive controllability - Motion generation

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Motion Generation

Planar locomotion from tail actuation

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Motion Generation

Free locomotion from tail actuation and pectoral maneuvering

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Motion Generation

UW fin-actuated UAV - Tail locomotion and pectoral fin maneuvering

servo motors microcontroller pressure sensor and buoyancy control batteries 3D digital compass acoustic modem

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Motion Generation

Forward locomotion simulation

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Motion Generation

Forward locomotion Experiment Simulation

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Motion Generation

Tail yaw actuation simulation

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Motion Generation

Turning locomotion Experiment Simulation

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Motion Generation

Pectoral kicking and “dolphin” locomotion

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Motion Generation

Oscillatory pectoral yaw actuation

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Motion Generation

Motion from nature - Turning locomotion

Backbone curve of a goldfish during a snap turn (images at 0.4s).

[D. Weihs, Proc. R. Soc. London B, 1972]

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.

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Motion Generation

Motion from nature - Experimental results

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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.

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Outline

• UW fin-actuated UAV
• Vehicle control

– Modeling – Optimality – Motion generation – Feedback

• Actuation surface modeling

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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

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Feedback

Equations of motion for a mechanical system High frequency, high amplitude averaging Correspondence Underactuated mechanical systems - Oscillatory control

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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

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Feedback

Averaged approximation can be extended for higher order effects

Theorem [Vela, Morgansen, Burdick ACC 2002]

Given admissible oscillatory controls and the averaged system the error between the original and averaged systems is

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Feedback

For bandwidth limitations can use low-order effects Turning in the fish can be accomplished by biasing the tail oscillations.

Restriction of stabilization to first order effects

Goal: stabilization to forward motion down center of tank Solution: superimpose turn over forward motion to correct lateral motion

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Feedback

System model captures qualitative effects

simulation experiment

Open loop trajectory tracking requires precise knowledge of system model and does not correct disturbances.

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Feedback

Implementation of feedback stabilization allows tracking

simulation experiment

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Feedback

Heading control

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Feedback

Depth control

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Feedback

Pectoral fin depth control

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Outline

• UW fin-actuated UAV
• Vehicle control

– Modeling – Optimality – Motion generation – Feedback

• Actuation surface modeling

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Actuation Surface Modeling

Motion control in unsteady flow conditions

• Active flow control
• Flexible control surface actuators
• Energy efficient actuation
• Minimize cavitation

Control theoretic modeling of fluid-actuator dynamics

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Actuation Surface Modeling

Strain gauges lateral motor (y) tail motor (θ) encoders reflection shield

x y θ

Flow visualization experiment

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Actuation Surface Modeling

Motion

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Actuation Surface Modeling

Flow visualization experiment

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Actuation Surface Modeling

Digital Particle Image Velocimetry (DPIV)

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Actuation Surface Modeling

Model extensions - Vortex effects on fluid dynamics Experimental thrust measurement

• Inclusion of vortices in model gives more accurate results

Add vortex dynamics to model based on actuation frequency and amplitude

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Conclusions and Ongoing Work

• Results extendable to many fluid-body models
• Rigorous mathematics with simple implementation
• Experimental stabilization robust

ºIncorporate vortex dynamics and unsteady effects

into model

ºOptimal motion generation ºExtension to flexible actuators

• Custom pool: 8ft x

8ft x 20ft

• Acoustic modem
• Underwater

cameras

• Transducer

localization system

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Acknowledgements

In collaboration with

• Paul Belitz
• Zac Berkowitz
• Shane Cantrell
• Ben Deagan
• Stephen Frister
• Carlos Gonzalez
• Laura Grupp
• Dan Klein
• Alice Kunkel

This work supported in part by University of Washington Royalty Research Fund Grant number 3029, by NSF grant CMS-028461, by NSF grant SBE-0123552 and by AFOSR.

• Timothy La Fond
• Kudah Mushambi
• Jamie Petranek
• Kalun Schmidt
• Ben Triplett
• Dana Wen
• Jennifer Zhang
• Andy Crick
• Dana Dabiri