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Deliverable D 7.3

Proceedings of the secondCO3AUVS Summer School

Contract number: FP7- 231378 Co3 AUVs

Cooperative Cognitive Control for Autonomous Underwater Vehicles

The research leading to these results has received funding from the European Communitys SeventhFramework Programme (FP7/2007-2013) under grant agreement number 231378.


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Identification sheet

Project ref. no. FP7-231378

Project acronym Co3AUVs

Status & version [Final] "1.0"Contractual date of delivery month 26

Actual date of deliver y December 2011

Deliverable number D 7.3

Deliverable title Proceedings of the second CO3AUVS Summer School

Nature Report

Dissemination level PU

WP contributing to the deliverable WP7

WP / Task responsible Jacobs

Editor Andreas Birk

Editor address Jacobs UniversityRobotics, School of Engineering and Science

Campus Ring 1, 28759 Bremen, Germany

EC Project Officer Franco Mastroddi

Keywords summer school

Abstract (for dissemination): The second summer school on Cooperative Cognitive Control forAutonomous Underwater Vehicles (Co3-AUVs) was held from 11.07.2011 until 17.07.2011 at Ja-cobs University in Bremen, Germany. Just like the first Co3-AUVs summer school, it had a stronghands-on character in combination with lectures where the underlying foundations were taught.The lectures were given by different representatives from all project partners. The scope of thesummer school concentrated in its sessions on multi-robot aspects of higher level control includingcooperative 2D and 3D SLAM, navigation, and control.Participants received in addition the opportunity to work with state of the art marine robotics equip-

ment and to engage in practical experiments. In addition, they were introduced to an Open Sourcemarine robotics simulator featuring among others 3D visualization and physics, which allowed themto continue work started in the summer school. The practical equipment used in the summer schoolincluded AUVs from the two partners Jacobs and GT in form of a Muddy Waters II and a Folagavehicle. The summer school also included a small workshop where the participants got the oppor-tunity to present their own research. This participant session was based on poster presentationsincluding discussions.

List of annexes (if any)

schedule and list of participants

photo impressions of the event

handout materials (slides and references)

1


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Schedule

and

Participant List


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Participants

NAME Affiliation1 Renato Caldas Porto University, Portugal

2

Enrique FernndezPerdomo

University of Las Palmas de Gran Canaria,Spain

3 Arun Ramakrishnan Katholieke Universiteit Leuven, Belgium

4 Syed Riaz un Nabi Jafri Italian Institute of Technology , Genoa-Italy

5 Gregory Skaltsas University of Plymouth

6 Jorge Santos LAAS-CNRS

7 Max Pfingsthorn Jacobs Univ.

8 Heiko Blow Jacobs Univ.

9 Ravi Rathnam Jacobs Univ.10 Sren Schwertfeger Jacobs Univ.

11 Kaustubh Pathak Jacobs Univ.

12 Narunas Vaskevicius Jacobs Univ.

13 Andreas Birk Jacobs Univ.

14 Giovanni Indiveri Uni Salento

15 Pedro Aguiar IST / ISR

16 Alessio Turetta Graaltech

17 Enrico Clerici Graaltech


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Photo Impressions


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Slides

from the Lecture Sessions


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Lecture layout

1) Geometricaldescription ofelementary robot tasks

2) Rotations and basicKinematics

3) A Glimpse onUnderactuation andNonholonomy (kinematicscase)

4) Dynamic Modeling ofMarine Vehicles

5) Specific modelexamples for ControlDesign (both dynamicand kinematics)

6) Discussion

Giovanni Indiveri,

Introduction to Marine Robot Modeling and Control

marted 12 luglio 2011

Main bibliography (books)[1] Richard M. Murray, Zexiang Liand S. Shankar Sastry, AMathematical Introduction toRobotic Manipulation, CRC Press,1994.

[2] Jorge Angeles, Fundamentalsof Robotic Mechanical Systems:Theory, Methods and Algorithms,Third Edition, Springer, 2007.

[4] Thor I. Fossen, Handbook of

Marine Craft Hydrodynamics andMotion Control, ISBN9781119991496, John Wiley &Sons Ltd, UK, 2011

[5] L. Sciavicco and B. Siciliano,Modelling and Control of RobotManipulators, Second Edition,Springer Verlag, 2000.

[6] Gianluca Antonelli,"Underwater Robots - Motionand Force Control of Vehicle-Manipulator Systems", ISBN-13978-3-540-31752-4 SpringerBerlin Heidelberg New York,2006.

Giovanni Indiveri,Introduction to Marine Robot Modeling and Control



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Main bibliography (conferences)

IEEE Oceans

IEEE ICRA

IEEE IROS

IFAC MCMC

IFAC CAMS

IFAC NGCUV

IFAC IAV

... many others

Giovanni Indiveri,


TC 7.2 Marine Systems

http://www.wikicfp.com/cfp/

TC 7.5 IntelligentAutonomous Vehicles


Main bibliography (journals)

IEEE Oceanic

Engineering

IEEE TAC

IEEE TRO

Elsevier (IFAC) CEP

Elsevier (IFAC)Automatica

Elsevier Robotics and

Autonomous Systems

Sage IJRR

Springer AutonomousRobots

... many (MANY) others




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Main motion tasks

TrajectoryTracking

Path Following

Pose Regulation(parking)

Giovanni Indiveri,



Basic Concepts

x= f(x,u,t) State equation

Output equationy=g(x,u,t)

y=g(x, u)x

=f

(x, u

) Time invariant

Dynamic System




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Basic Concepts

f(x,u,t)

+

x0

+u x g(x,u,t)y

deterministic, nonlinear, time varying,non autonomous dynamical system

Giovanni Indiveri,



Examples

x=u

y=x

x=u

y=x

x1 = u cos

x2 = u sin

=

y= (x1, x2)T

x1

x2




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Motion control problem

x= f(x,u,t)

e =xd x e= (x, xd,u ,t)

xd =

r(t) Trajectoryr0 Poser(s) Path

Problem: find u such that e=0 is a stableequilibrium. The problem is different accordingto the knowledge of the state or output only.

Giovanni Indiveri,



Modeling issues

Motion control synthesis and system analysisrequires to have a proper mathematical model.

f(x,u,t)

+

x0

+u x g(x,u,t)y




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Fundamental notation

Giovanni Indiveri,


x

z

y

x

z

y

inertial

body fixed

r

v = ddtr

Free vectors are independent fromany reference frame, but...

... one has to be very careful aboutwhich frame is being used forcomputations.


A first sight at rotations


x

z

y

x

z

y

inertial

body fixed

r

1

R0 =1i0

1

j01

k0IR

33

1r = 1R00r

0r = 0R11r

0r= 1r0 1



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Fundamental notation

Giovanni Indiveri,


x

z

y

x

z

y

inertial

body fixed

r v = ddtr

0v = ddt 0r

1v = 1R0d

dt

0r

absolute velocity in theinertial frame

absolute velocity in thebody frame


Rotations!




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A Glimpse on Underactuation andNonholonomy (kinematics case)

Giovanni Indiveri,


x= f(x,u,t) State equation

Output equationy=g(x,u,t)

dim x = dim u

dim x > dim u

dim x < dim u over actuated

fully actuated

under actuated


Holonomic == Integrable


Holonomic versus Nonholonomic constraints

Holonomic constraint h(q) = 0 geometric constraint, i.e. reduc-tion of the degrees of freedom.

Holonomic constraint h(q) = 0 hq (q)q = 0 a n integrable kine-matic constraint.

x= vx

y =vy

q (x, y)

T

q2 =x2 + y2 =r2 hol. cons.

2xx + 2yy = 0

qTq = 0 integrable kin. cons.

r

x

y

v



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Nonholonomic vs holonomic

Giovanni Indiveri,


Example of integrability of a holonomic constraints

2xx + 2yy = 0 = 2

tt0

xxd+ t

t0yyd

= 0 =

xtx0

xdx + yty0

ydy =1

2x2

t + y2

t (x2

0+ y2

0)= 0 =

x(t)2 + y(t)2 = const. xx + yy = 0




Holonomic versus Nonholonomic constraints

Nonholonomic constraint h(q, q) = 0 is a kinematic constraint thatcannot be integrated to yield a geometric constraint.

A (first order) nonholonomic constraint does not reduce the geo-

metrical degrees of freedom.

x= u cos q(x , y , )T

y =u sin v (u, )T

=

dydx

= tan y cos x sin = 0

h(q, q) =A(q)q =

= (sin , cos , 0)(x, y, )T = 0

x

y u



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Giovanni Indiveri,


Pfaffian Kinematic Constraints

A(q)q = 0

integrable holonomic constraint

nonintegrable nonholonomic constraint

If the constraint is holonomic, q must satisfy some equation h(q) = 0

and thus the degrees of freedom are reduced.

If the constraint is nonholonomic, q is free and q must belong to

Ker(A(q)), i.e. it is constrained.




Integrability of Kinematic Constraints

Given the scalar constraint a(q)T q = 0 is it holonomic or nonholo-nomic? The answer is by no means trivial ...

Holonomic means that there h(q) : Rn R : h(q) = 0 h(q) =ni=1

hqi

qi= 0 such that

ni=1

ai(q)qi = 0 n

i=1

h

qiqi = 0

In turn this requires that (q) : Rn

R (integrating factor) suchthat

h

qj=(q)aj(q) j



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Giovanni Indiveri,


Integrability of Kinematic Constraints

h

qj=(q)aj(q) j

Remembering that for aC2 function 2f

xy=

2fyx

(Schwartz Theorem)

2h

qiqj

= 2h

qjqi

= (q) ai(q)

qj

=(q) aj(q)

qi

i, j = 1, 2, . . . , n

The test of integrability can be made checking the existence of the

solution of a PDE (differential geometry tools).




Kinematic Models of Wheeled Mobile Robots

v =J(q)q : qR n1

A(q)q = 0 : A(q) R mn, m < n Rank(A) =m

S(q) = [s1(q) s2(q) . . . snm(q)] Rn(nm) :

{si(q) Rn1 i= 1, 2, . . . , n m} is a basis of ker(A)

q =S(q)u is by construction a feasible velocity as

A(q)S(q)u= 0 u R (nm)1 pseudovelocity



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Giovanni Indiveri,


Kinematic Models of Wheeled Mobile Robots

As wheeled vehicles are affected by nonholonomic constraints, their

kinematics can be always written as

q =S(q)u

with

q R n1 : n# of degrees of freedom

u Rp1 : p dimension of input vector

p n : underactuated system




Ideal Unicycle

q = (x , y , )T

u= (v, )T

x= v cos y =v sin =

S(q) =

cos 0sin 00 1

q =S(q)u

vt= 0 = y cos x sin = 0

A(q) = ( sin , cos , 0) : A(q)S(q) = 0

x

yv

vt



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Giovanni Indiveri,


Pioneer - Like Kinematics

q = (x,y,)T

u= (vR, vL)T

q =

cos /2 cos /2sin /2 sin /21

2r 1

2r

u

q =S(q)u

vR =r R

vL =r L

L

R

v

r


Dynamic Modeling of Marine Vehicles


http://www.itk.ntnu.no/ansatte/Fossen_Thor/

Thor I. Fossen, Handbook of Marine

Craft Hydrodynamics and MotionControl, ISBN 9781119991496, JohnWiley & Sons Ltd, UK, 2011

Lecture Notes: TTK4190 Guidance and Controlhttp://www.itk.ntnu.no/fag/gnc/Wiley/slides.html



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Specific model examples for ControlDesign

Giovanni Indiveri,







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Giovanni Indiveri,







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Giovanni Indiveri,






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Introduction to SLAM:Particle Filters and Pose Graph

Max Pfingsthorn

Jacobs University Bremenhttp://robotics.jacobs-university.de

Tuesday, July 12, 2011

Jacobs University 2

SLAM problem definition

Given an initially unknown environment, generate amapof the robots surroundings and localizethe robotin it at the same time.

Simultaneous Localization and Mapping

Probabilistically:Given sensor observations and control inputs, find themost likely pose and map at the same time.

(xt ,m) = argmax

xt,m

p(xt,m|z1:t, u1:t)



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Jacobs University

Potential SLAM Solutions

(Extended/Unscented) Kalman Filter- Great if easily identifiable features are used in maps

- Computationally infeasible with very large maps- Linearizes usually non-linear motion and measurement models

Particle Filter- Generally applicable, even with occupancy grid maps

- Suffers from particle starvation after long trajectories withoutloops, only remedy to use more memory

Pose Graph

- Conceptually simple, easy to implement- Locally linear, globally non-linear

- Low memory requirement

3


Jacobs University



- Computationally infeasible with very large maps

- Linearizes usually non-linear motion and measurement models Particle Filter

- Generally applicable, even with occupancy grid maps

- Suffers from particle starvation after long trajectorieswithout loops, only remedy to use more memory

Pose Graph- Conceptually simple, easy to implement- Locally linear, globally non-linear

- Low memory requirement

4



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Jacobs University

SLAM problem definition

PDF can be factored into- Trajectory probability- Map Probability

Map (and its parts, such as landmarks) conditionallyindependent given the robot path!

Therefore, focus on estimating the robot path

One particle m: (weight, pose, map)

5

p(x1:t,m|z1:t, u1:t) = p(x1:t|z1:t, u1:t)p(m|x1:t, z1:t)

= p(x1:t|z1:t, u1:t)n

p(mn|x1:t, z1:t)

w

[m]t , x

[m]t , m

[m]t


Jacobs University

Particle Filter Algorithm

6

From: Probabilistic Robotics. S. Thrun, W. Burgard, and D. Fox. MIT Press 2005

Input: Xt1, ut, ztXt= Xt= for m=1 to M do

sample x

[m]

t p(xt|ut, x

[m]

t

1)w

[m]t

=p(zt|x[m]t

)

Xt = Xt+x

[m]t , w

[m]t

end

for m=1 to M do

draw index i with a probability w[i]t

from Xt

add x[i]t

to Xtend

return Xt

Here, , , and are thecurrent pose, control input,

and observation, respectively.

utxt zt



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Jacobs University

Particle Filter

basic steps:- for each particle i

sample normal distribution around registration result add sample to map in particle i

- choose N particles from old particle set according to itsprobability

Samples from posterior probability distribution withoutexplicitly computing it

Many tradeoffs and improvements:- when to resample (second step) and when not to- particle representation (exploit common ancestry)- how many particles to use

7


Jacobs University

Map Representation

Usually Occupancy Grid

Each cell containsprobability of beingoccupied by something

Corridors, doors vs walls,other objects, etc.

Memory scales linearlywith covered area

8



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Jacobs University

Examples of 2D Occupancy Grids

9

http://radish.sourceforge.net


Jacobs University

Examples of 3D Occupancy Grids

10

http://octomap.sourceforge.net



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Jacobs University

Particle Filter

Maintains estimateof probability

density bycollection ofhypotheses

Very easy to applyto occupancy grid

One possible gridper particle(hypothesis)

11

An Efficient FastSLAM Algorithm for Generating Maps of Large-scale Cyclic Environments From Raw Laser RangeMeasurements. D. Haehnel, W. Burgard, D. Fox, and S. Thrun. IROS 2003


Jacobs University



- Computationally infeasible with very large maps

Particle Filter- Generally applicable, even with occupancy grid maps- Suffers from particle starvation after long trajectories without

loops, only remedy to use more memory

Pose Graph- Conceptually simple, easy to implement

- Locally linear, globally non-linear- Low memory requirement

12



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Jacobs University

Pose Graph Intuition

Robot moves through environment

... makes sensor observation and stores it

... compares it with previous observations

... relates new observation to previous ones

... stores graph of relations

... finds best fit of all observations given the relations

Gives rise to a graph of pose relations

13


Jacobs University

Example

14

Example pose graph,local covariance matrices

registrationresult = edge



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Jacobs University 15

Pose Graph Data Structure

Graph G = (E,V) where- Vertices: Places where sensor observations were recorded

e.g. sonar scans, laser scans, camera frames, acoustic range readings 2D or 3D data leads to 2D or 3D maps

- Edges: Relative pose estimates between vertices usually from registration, but also from odometry, DVL, IMU, GPS/Compass

Can be 2D or 3D, does not change underlying data structure

Primarily an alternative map representation

More flexible than occupancy grid Occupancy grid / image mosaic easily obtainable by

rendering sensor observations


Jacobs University

Pose Graph Pros vs Cons

No information loss

Optimization can bedeferred

Can be pruned easily

Can contain diverse sensorobservations (evenincompatible ones)

16

May contain globallyinconsistent information

Needs to be optimized

May grow with time, notarea

Occupancy grid or similar

not readily available



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Jacobs University

Pose Graph Optimization

Full SLAM problem estimates trajectory and map.Map (and its parts, s.a. landmarks) conditionallyindependent given trajectory!

Estimate trajectory only, map easily computed given

17

p(x1:t,m|z1:t, u1:t) = p(x1:t|z1:t, u1:t)p(m|x1:t, z1:t)

= p(x1:t|z1:t, u1:t)n

p(mn|x1:t, z1:t)

x1:t

x1:t = argmaxx1:t

[p(x1:t|G)]

p(x1:t|G) =

(vi,vj ,ck)E

p(xj xi|ck)

x1:t


Jacobs University


Constraint is a multivariate normal distribution:

For numerical reasons, minimize negative log-probability

18

p(tj

i

|ck) = |2k|1/2 exp1

2(tj

i

k)T1

k

(tj

i

k)ck

lnp(x1:t|G) =

(vi,vj ,ck)E

ln|2k|

1/2

+

1

2

(vi,vj ,ck)E(tji k)T1k (tji k)



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Jacobs University


Dropping the normalizing constant, we get the sum ofsquared Mahalanobis distances

All Pose Graph optimization methods use this costfunction

Only difference is the specific non-linear optimizationmethod used, generally least-squares methods- e.g. Gauss-Seidel, Gauss-Newton,LevenbergMarquardt, etc.

- also, more problem-specific Gradient Descent, or others

19

cost(x1:t|G) =

(vi,vj ,ck)E

(tji k)T1k (t

ji k)


Jacobs University 20

Lu & Milios Relaxation

Minimizes the above squared Mahalanobis distanceincrementally, based on Newtons method

where is the adjacency matrix of G, is theblock-diagonal matrix of all constraint covariances,and is the vector of residuals

Big problem: Nonlinearity because of rotationalcomponents

Therefore needs to be solved repeatedly Solves for global poses simultaneously

x= (HTC1H)1HTC1rH C

r

Globally Consistent Range Scan Alignment for Environment Mapping. F. Lu and E. Milios. Autonomous Robots 4, 4.Springer, 1997



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Jacobs University

Freses Multilevel Relaxation

Very similar to Lu&Milios

Solves it one vertex at a time, at different scales- Based on Gauss-Seidel relaxation

21

A multilevel relaxation algorithm for simultaneous localization and mapping. U. Frese, P. Larsson, and T. Duckett. IEEETransactions on Robotics 21, 2. IEEE Press, 2005


Jacobs University

Olsons Relaxation with Grisettis Extensions

Also computed one constraint at a time

Parameterizes graph as a tree

Does not involve inverting a matrix

Modified Stochastic Gradient Descent- Stochastic: Randomly choose a constraint- Modified: All constraints will be chosen, in random order

22

A Tree Parameterization for Efficiently Computing Maximum Likelihood Maps using Gradient Descent. G. Grisetti, C.Stachniss, S. Grzonka, and W. Burgard. Proceedings of Robotics: Science and Systems 2007



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Jacobs University

iSAM - Incremental Smoothing and Mapping

Bayesian networkinterpretation of trajectory

dependence on controlsand observations

Variable reordering tomake global covariancematrix sparser

Uses COLAMD/CSparselibrary as backend

23

iSAM: Incremental Smoothing and Mapping. M. Kaess, A. Ranganathan, F. Dellaert. IEEE Transactions on Robotics 24, 6.IEEE Press 2008


Jacobs University

HogMan

Uses a hierarchicalsubdivision of the graph,like Multilevel Relaxation

before

Errors and corrections areimplicitly linearized at theconstraint level

Allows for better global

convergence

24

Hierarchical Optimization on Manifolds for Online 2D and 3D Mapping. G. Grisetti, R. Kmmerle, C. Stachniss, U. Frese,and C. Hertzberg. ICRA 2010

x[k1]i x

[k1]j

x[k]i x

[k]je

[k]ij

G[k1]i G

[k1]j



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Cooperative Cognitive Control for Autonomous Underwater Vehicles (Co3AUVs)

2nd Summer School, Bremen, Jul. 11-16, 2011

Part I: Nonlinear Motion Controlof Marine Robotic Vehicles

Antnio Pedro Aguiar

http://users.isr.ist.utl.pt/[email protected]

In collaboration with:

Antnio M. Pascoal

Institute for Systems and Robotics (ISR)

Instituto Superior Tecnico (IST) Tech. Univ. of Lisbon, Portugal

1. Practical motivation and mission scenarios

Ocean Exploration: Scientific andTechnological Challenges.

Brief Overview of Robotic Marine Vehicles

GNC Architecture: basic building blocks

Outline


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Outline

2. Control: Linear versus Nonlinear

(a brief review)

When should we go nonlinear?

Lyapunov theory of stability

Input to State Stability (ISS)

Lyapunov based analysis and design tools

Outline

3. Dynamic Positioning of Autonomous Vehicles

Pose stabilization (position and attitude)

4. Trajectory Tracking (TT) and Path Following (PF)


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Motivation:

Ocean exploration: scientific challenges

The need for marine robots: technicalchallenges

Theory and Practice: single and multiplerobotic vehicle navigation and control (anoverview)

Robots for Ocean Exploration

Key Objectives:

To better know planet Earth

To promote the rationalexploitation of marine

resources

To ensure sustainable development

Ocean Exploration and Exploitation


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A small surface terrestrial area: 2,400 km2A wide EEZ: 1,000,000 km2

The Azores

A chain of mountains at the bottom of the Atlantic ocean

Azores

The Middle Atlantic Ridge and the Azores


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The Azores Triple Junction (ATJ)

Rainbow 2300 m

Lucky Strike 1700 m

Menez Gwen 850m

ATJ: where three tectonic plates meet

Hydrothermal fields

The Azores Triple Junction (ATJ)

The region harbours a great variety ofseamounts, active underwater volcanoes,chemosynthetic ecosystems, and extreme life forms (extremophyles)

Black smoker


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Deep water hydrothermal vents

Challenges:

Accurate 3D-mapping,geological surveying, andmonitoring of biological andphysical-chemical activityat the vents andsurrounding areas

High-resolution 3D mapping of ahydrothermal vent (Fuca Ridge[Yoerger et al., Woods Hole, USA. ] )using a Mesotech horizontal scanner.

Perform data acquisition at

the appropriate spatial

and temporal scales!

Acoustic image of hydrothermal vent

Lucky Strike, August 2006

ISR-IST / VICTOR ROV (IFREMER)

Shallow-Water Hydrothermalism

Hydrothermal activity at the D. Joo de Castro seamount, Azores


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Hydrothermal Activity

Gas collection Detail of a gas bubble

Collecting bacteria

Depth: 35m

Study the spatial extent of shallow water hydrothermalism.

Determine the patterns of community diversity at the vents.

Challenges


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Deep water hydrothermal vents

In situ observations

The need for technology

Vents are very hard to study:

Large depth (pressure is high)

Highly corrosive environment

Lack of optical visibility

Navigation is a challenge (lack ofa GPS-like system)

Submersibles: place human lifesat risk


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Monte da Guia,Faial Island, Azores

Marine Reserves

Algae Codium elisabethae. Algae Sargassum sp.

Codium elisabethae(video image)

Marine Reserves


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Inspection site(inside the crater)

Mosaic

Marine Habitat Mapping

Challenges

Afford scientistsadvanced instruments for automatic marine data

acquisition over wide areas

GIS (Geographical Information System) based habitat maps

Science Technology

Educated marine management


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Technological Challenges

Afford marine scientistsINSTRUMENTS and METHODOLOGIES for

Marine data acquisition(tri-dimensional world)

Signal processing and datafusion

Management andDissemination of Information

Scientific and CommercialCommunities; Civilian

Authorities

Marine Board,European ScienceFoundation, Dec. 2000http://www.esf.org/marineboard

Marine data acquisition

Adequate 3-Dtemporal and spatial sampling

Coastal areas

Deep ocean

Open sea


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Classical Methods

Divers

Divers - restricted coverage; dangerous.Hard to georeference data.

Classical Methods

Vessels (tool par excellence) -Poor maneuverability; poor 3-D + time coverage.

High operation costs.

ResearchVessels


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Semi-Classical Methods

Manned Submersibles

(direct observation ofthe deep sea)

Nautile, IFREMER, FR

LULA, Rebikoff Foundation,

Azores, PT

Limited ocean coverageJeopardize human lives

High operation costs

Glimpses of amazingundersea

adventures

Semi-Classical Methods


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Modern Methods

ROVs Remotely Operated Vehicles

TITANIC

The small companion ROV

(carrying an umbilical)

Joint operation of ALVINand ROBIN

Modern Methods


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ROVs Remotely Operated Vehicles

Present trend:

To free the end-user from thetedious task of direct vehicleoperation.

Modern Methods

Romeo (IT)

Modern Methods

AUVs - Autonomous UnderwaterVehicles (cut the umbilical!)

High maneuverability

Autonomy

Automatic execution of tedious tasks


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GLIDER TECHNOLOGY

Example of high resolution results obtainedFaial, Azores August 2008

Seeing with acoustic eyes

Multibeam sonar, Azores, PT


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Sampling networks fixed and movingunits (Divers, Floating devices, Moored

equipment, Inhabited submersibles,Ocean vessels, ROVs, AUVs, ASCs,Benthic stations).

Future Trends

Sampling Networks

Coriolis Initiative, FR

Marthas Vineyard coastalobservatory, USA

AUVs deployed to monitorepisodic events.

Marine Board, ESC

Marine Technoloy Frontiers forEurope, Brest, April 2001

Key Issues: Communications, Information, Decision, Control.


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Lucky Strike1700 m

Explore the Ocean

Advanced technology is mandatory

Future: Networked Mobile/Fixed

Sensors (Moored systems, AUVs, ASCs,

Gliders, Submersibles, Vessels, etc)

Sea: the Ultimate Frontier

Sampling Networks

Key Issues: Communications, Information,

Decision, Control.Stepping stones:

Single and coordinated

vehicle control

The ASIMOV concept

MAST-III, EC

MONAZ, PT-USA

Office of Naval Research


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RepresentativeAutonomous Underwater Vehicles (AUVs)

AUTOSUB (SOC, UK)

ARIES (NPS, USA)ODYSSEY, MIT

Representative AUVs

HUGIN (Norway)

Ocean Explorer (FAU, USA)

PTEROA (Japan)

REMUS (WHole, USA)


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Tests with theMARIUS AUV

SINES, Portugal

The MARIUS AUV

The INFANTE AUV

2 back thrusters

2 bow planes

2 stern planes

2 stern rudders

Streamlined

Designed to fly above the seabed

No hovering capabilitiesTrajectory Tracking and Path

Following required.


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2 back thrusters1 vertical thruster

NO side thruster

Bluff body Depth Control

Point stabilization required

The SIRENE ROV-LIKE AUV

Tests with theSIRENEshuttle

TOULON, France

The SIRENE ROV-LIKE AUV


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Autonomous Surface Craft (ASC)

2 back thrusters

2 bow planes

2 stern planes

2 stern rudders

AutoCat (AUV Lab, MIT)Protector (Israel)

Atlantis

(Stanford Univ. USA)

ThrustersTypical configuration:

2 Back Thrusters

(underactuated)

Streamlined

. Course control in the presence of waves,

currents, and wind

. Trajectory Tracking and Path

Following required.

The DELFIM ASC

Trajectory Tracking and Path

Following required.

2 back thrusters (common anddifferential modes)


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The DELFIMx ASC

2 back thrusters (common and

differential modes)

Trajectory Tracking and Path

Following required.

The CARAVELA 2000 ocean vessel

Designed to acquire ocean data autonomously.Large autonomy; operation over VERY large areas.


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CARAVELA 2000

FCT / ADI Caravela Project (PT)

Display model (scale 1/10)

CARAVELA 2000

The CARAVELA ASC

Construction phase


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Technology Matured

over the years

New Challenges

(Marine Science, Industry)

The quest for mid-water columnhydrothermal vents, Azores, PT

AUV Fleet Methane gradient descent

Deep water hydrothermal vent

Methane plume


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Very high-resolution / accurate bathymetric and visual maps

are required

ArchaeologicalData Fusion

Multiple platforms (ASC, ROVs,ASVS)equipped with complementary sensors

Preservation of cultural heritage /Virtual underwater trips (museums)

ROMAN SHIPWRECK200 AD

AMPHORAS

Make data availableto

a wide audience!


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Acoustic Positioning

Underwater vehicle equipped with PINGER

GIB System ACSA/ORCA(GPS Intelligent Buoys)

re-engineered by IST

surface buoys withAcoustic receivers(hydrophones)

Multibeam Images

High resolution15m

Low resolution60 m


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Building blocks for a final demo

Scouts

DIVER

Diver position estimation

Optimal vehicleconfiguration

Motion towardsdesired configuration

Acoustics play a very important role


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The MEDUSA vehicles

Robot companions


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Vehicle systems: basic building blocks

Hull andEnergy

Hull and Energy

The INFANTEAUV

(streamlined)

The SIRENE underwater shuttle

The CARAVELA

and DELFIM ASCs

(streamlined)


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Vehicle systems: a brief review

Hull andEnergy

Propulsion andActuators

Propulsion and actuators

Propellers(The MARIUS AUV)

Speed ofRotation

of the

Propellers

ThrustForce


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Propellers(The SIRENEShuttle)


AUV Motion in the Vertical Plane

LIFT (L1)

Torque

LIFT (L2)

Speed - V


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Navigation

Hull andEnergy

Propulsion andActuators

Navigation

Where am I?

Where am I going?


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{U}

{B}

Navigation

Basic variables

Position

Orientation

Linear Velocity

Rotational Velocity

SENSORS

COMPUTER

+NAVIGATION

ALGORITHMS

Linear Position, Linear Velocity

Orientation, Rotational Velocity

Navigation System


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THENAUTILE

SUBMARINE :

GUIDANCE and

CONTROL

AH! NOW IM ON THE

RIGHT PATH!

Guidance and Control


Navigation

Guidance andControl

Hull andEnergy Propulsion andActuators


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Guidance andControl

Simple controlobjective: drive the orientationerror (desired actual) to zero.

desired orientation

actual orientation

Guidance andControl

Computer +Control

Algorithm

Vehicle +HeadingSensor

Desiredheading

Measured heading(from navigation system)

Heading error

Stern

rudder


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OTHER CONTROL SYSTEMS

SpeedOrientation

Depth

Altitude above the seabed

Key step in the design process:QUANTIFY THE VEHICLE BEHAVIOUR

IN THE WATER (MODELING)

HYDRODYNAMIC TESTS

Guidance and Control

SIRENE

(scaled vehicle)

CIRCULATING WATERTUNNEL

VWS, BERLIM

HYDRODYNAMIC TANK TESTS


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HYDRODYNAMIC TANK TESTS

SIRENE

coupled to aBENTHIC

LABORATORY

Present trend: Carry out tests with the REAL vehicle

at sea; run identification experiments

Guidanceand Control

desired heading

actual heading

visibility distance

GUIDANCE FOR PATH FOLLOWING

Closest point

on the Path

Path


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Integrated Navigation, Guidanceand Control

GUIDANCE

NAVIGATION VEHICLE

CONTROL

Path to

be followed

Commanded

heading

Actual path

Vehicle motion data

Auxiliary data


Navigation

Guidance andControl

Communications



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Communications

Underwater Communications very hard!

Communications

(multipath, failures, latency, asynchronous comms, reduced bandwidth...)


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Mission Sensors

Sidescanunit

Acoustic

Profiler

Major challenge: autonomous sensoroperation

Echosounder data over the seamount


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Sidescan data / acoustic map building

Mission Control

Navigation

MISSIONCONTROL

Guidance andControl

Communications


MissionSensors


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Mission Control

Mission specifications

Automatic CodeGeneration

Real-Time VehicleSystem Control

MissionPerformanceAssessment

Mission Control

Navigation

MISSIONCONTROL

Guidance andControl

Communications


MissionSensors

CONCERTED SYSTEM OPERATION


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Control Objectives

Controller K

Sensor

Design a controller K that stabilizes the plantand such that r- is small in spite of external disturbances

d, noise n,and plant parameter uncertainty (robuststability and performance)

Linear versus Nonlinear control

Controller K

Sensor

Modeling: Interactions with Control

Linear or Nonlinear?

Control Objectives


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The MARIUS AUV

Vehicle is streamlined. Small angles of attack and

sideslip: linearizations are justified

Separate control systems (assuming lightinteraction between the horizontal and verticalplanes)

Speed Control

Horizontal Plane control (roll is left uncontrolled)

Vertical Plane Control

Motivating example

AUV speed control

Practical problem: design a control law so that the torpedo speed ufollows a desired speed envelope.

First step: modeling

External Thrust (generated by propeller)

- Speed

- Mass

Hydrodynamic Damping:

Linear + quadratic terms

- Added mass


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Motivating example

Newton Law

- Sum of all external forces;

External

thrust force

HydrodinamicDamping

Added mass effectInertial term

Nonlinear Control

Control Objectives

Controller K

Sensor

Design a controller K that stabilizes the plantand such that r-u is small in spite of external disturbancesd, noise n,and plant parameter uncertainty (robuststabilty and performance)


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A Linear Approximation

Dependence on trimming condition

LINEAR

SYSTEM

taking Laplace transforms (zero initial conditions)

Linear Controller Design

LINEAR

SYSTEM

Controller K

Sensor

Linear ControlLaw


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Loop Shaping golden rules for controlsystem design

_

Controller Plant

r reference signal ( to be tracked by y)

d external disturbance (referred to the output)

n sensor noise

e tracking error

y output (controlled variable)

u actuation signal

_

Controller Plant

nominal model

Bounded

perturbations

Goal: achieve robust stability and performance(by proper choice of K(s) a hard GAME)

Loop Shaping golden rules for controlsystem design


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The SIRENE Shuttle

Vehicle is a bluff body. May exhibit large angles ofattack and sideslip as well as considerable parameteruncertainy.

No preferred motions for linearizations to be applied

Nonlinear Design

Non-preferred directions of motion(highly nonlinear models);

underactuated vehicles

Dealing with large hydrodynamicparameter uncertainty

Globally attracting path followingcontrol laws; coordinated vehicle

control


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Nonlinear Control: (a brief review)

Lyapunov theory

Input-to-State Stability (ISS)

Lyapunov based design

AUV speed control

Dynamics

NonlinearPlant

Objective: generate T(t) so that track the reference speed

Tracking error

Error Dynamics


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Error Dynamics

Nonlinear Control Law

Tracking error tends to

zero exponentially fast.

Simple and elegant!

Catch: the nonlinear dynamics are known EXACTLY.

Key idea: i) use simple concepts, ii) deal withrobustness against parameter uncertainty.

New tools are needed: LYAPUNOV theory


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(free mass, subjected to a simple motion resisting force)

vfv

v

m/f

0 v

t

v=0 is an equilibrium point; dv/dt=0 when v=0!

v=0 is attractive

(trajectoriesconverge to 0)

0v

How can one prove that the trajectories go to the equilibrium point

WITHOUT SOLVINGthe differential equation?

(energy function)

V positive and bounded below by zero;

dV/dt negative implies convergence

of V to 0!

dV((t))

dt =

V

(t)

.d(t)

dt= m=f 2


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What are the BENEFITS of this seemingly strange approach to investigate

convergence of the trajectories to an equilibrium point?


dV/dt negative implies convergence of V to 0!

vf(v)

f a general dissipative force

v0

Q-I

Q-III

e.g. v|v|

Very general form of nonlinear equation!

State vector

Q-positive definite

2-D case


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1x2 2x11x1


dV/dt negative implies convergence of V to 0!x tends do 0!

2-D case

ShiftingIs the origin always the TRUE origin?

mg

y

y-measured from spring at rest

Examine if yeq is attractive!

Equilibrium point xeq: dx/dt=0

Examine the

ZERO eq. Point!


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Is the origin always the TRUE origin?

Examine if xref(t) is attractive!

xref(t) is a solution

Examine the

ZERO eq. Point!

Shifting

Control Action

Nonlinear

plant

yu

Static

control law

Investigate if 0is attractive!


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(the two conditions are required for

Asymptotic Stability!)

There are at least three ways of gaging the stability (of

an equilibrium point of a system:

Solve the differential equation (brute-force)

Linearize the dynamics and examine the behaviour

of the resulting linear system (local results for hyperbolic

eq. points only)

Use Lypaunovs direct method (elegant and powerful,

may yield global results)


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then the origin is globallyasymptotically stable (GAS)

dV(x)

dt


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What happens when

Is the situation hopeless? No!

If the only trajectory of the system entirely

contained in is the null trajectory. Then,the origin is asymptotically stable

(Let M be the largest invariant setcontained in . Then all solutionsconverge to M. If M is the origin, theresults follows)

Krazovskii-LaSalle

y


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y

f(.), k(.) 1st and 3rd quadrants

f(0)=k(0)=0

V(x)>0!

y

Examine dynamics here!

Trajectory

leaves

unless x1=0!

M is the origin.

The origin is

asymptotically

stable!


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Similar results can be derived for the time-varying case

However, K-LaSalle DOES NOT APPLY!

and the equilibrium point

Saved by Barbalats Lemma:

Let: R+

R be a uniformly continuous function and

suppose that limtt0() dexists and is finite.

Then, limt (t) = 0

How can Barbalats Lemma be used?

Uniform continuity of the derivate of V follows

from boundedness of its second derivative.

Explore in an intelligent manner to show that thetrajectories converge to the equilibrium point!

Suppose there exists a uniformly continuous functionV

:R

R

R

satisfyingi) V(x, t) 0

ii) dVdt

0

iii) dVdt

is uniformly continuous

Then, limtdV(x)dt

= 0.


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(r) = tan1(r) (r) =r2

Definition: A continuous function : [0, a) [0, ) is said to belong to class

K if it is strictly increasing and (0) = 0. It is said to belong to class K ifa= and (r) as r .

Comparison Functions

(r, s) = r

sr+ 1

Definition: A continuous function : [0, a) [0, ) [0, ) is said to belongto class KL if, for each fixed s, the mapping (r, s) belongs to class K withrespect tor and, for each fixedr, the mapping (r, s) is decreasing with respect

to s and (r, s) 0 as s .

(r, s) =r2es

Comparison Functions


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x= Ax+Bu, x(t0) =x0

x(t) =eA(tt0)x(t0) +

tt0

eA(t)Bu() d

x(t) ke(tt0)x(t0) +kB

supt0t

u()

Linear time-invariant system

Solution

eA(tt0) ke(tt0)

If A is Hurwitz

Thus, it follows that

x= Ax+Bu, x(t0) =x0


tt0

eA(t)Bu() d


supt0t

u()


Solution

eA(tt0) ke(tt0)

If A is Hurwitz


Converges to zero


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x= Ax+Bu, x(t0) =x0


tt0

eA(t)Bu() d


supt0t

u()


Solution

eA(tt0) ke(tt0)

If A is Hurwitz


Bounded andproportional to thebound of the input

x

x= 3x+ (1 + 2x2)u

However, ifx(0) = 2 and u(t) = 1, the solution is

x(t) =3 et

3 2et

whenu= 0, x= 0 is GES

Does this hold for a general nonlinear system ?

ux= f(t,x,u)

In what conditions?

Is it sufficient to have GUAS of the unforced system?

NO!

Unbouded (finite escape time)!


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x(t) (x(t0), t t0) +

supt0t u()

Definition [ISS] The system x = f(t,x,u) is said to be input-to-state stableif there exist functions KL and Ksuch that for any initial state x(t0)and any bounded input u(t), the solution x(t) exists for all t t0 and satisfies

xu

x= f(t,x,u)

ISS

x(t) (x(t0), t t0) +

supt0t

u()

- For any bounded input u(t), the state x(t) is bounded

- Ifu(t) 0 as t =x(t) 0 as t

- As t increases, the state will be ultimately bounded by a class K functionof suptt0 u(t), that is, lim supt x(t) (suptt0 u(t))

xux= f(t,x,u)

Remarks:

- For u(t) = 0 = x(t) (x(t0), t t0)

The unforcedsystem is GUAS


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TheoremLet V : [0, ) Rn R be a continuously differentiable functionsuch that

1(x) V(t, x) 2(x)

V

t +

V

xf(t,x,u) W3(x), x (u)> 0

(t,x,u) [0, ) Rn Rm, where 1, 2 K, K, and W3(x) > 0.Then, the system is ISS with =1 2 .

How can we prove ISS ?

x(t) (x(t0), t t0) +

supt0t

u()

xu

x= f(t,x,u)

x= d|x|x+u

V =1

2x2

V =d|x|x2 +xu

=(1 )d|x|x2 d|x|x2 +xu, 0<


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x1 =f1(x1, x2)

x2 =f2(x2, u)

x1(t) 1(x1(t0), t t0) +1

supt0t x2()

x1 is ISS with respect to input x2

x2 is ISS with respect to input u

x2(t) 2(x2(t0), t t0) +2

supt0t

u()

Cascade system

x1x2u

x1 =f1(x1, x2)

x2 =f2(x2, u)

Cascade system

x1x2u

x=

x1x2

x1 ISS + x2 ISS = coupled system is ISS

x(t) (x(t0), t t0) +

supt0t

u()


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w1

w2

x2 x1

if1 2(r)< r, r >0, then the interconnected system is ISSwith respect to state x= (x1, x2) and input w= (w1, w2)

x1(t) 1(x1(t0), t t0) +1

supt0t

x2()

+w1

supt0t

w1()

x2(t) 2(x2(t0), t t0) +2

supt0t

x1()

+w2

supt0t

w2()

ISS Small-Gain Theorem

x2 x1

x1 = f1(t,x1,x2,w1)

x2 = f2(t,x1,x2,w2)

w1

w2

x2 x1

x1(t) 1(x1(t0), t t0) +1

supt0t

x2()

x2(t) 2(x2(t0), t t0) +2

supt0t

u()

if1 2(r)< r, r >0, then the interconnected system is ISSwith respect to state x= (x1, x2) and input w= (w1, w2)

ISS Small-Gain Theorem

x2 x1

x1 = f1(t,x1,x2,w1)

x2 = f2(t,x1,x2,w2)

if1(r) =1r, 2(r) =2r

Then 12


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x(t) max

(x(t0), t t0),

supt0t

u()

Restrict ISS to a neighborhood x(t) B(0) and to a restrictedset of perturbations supt0t u() R

Variants of ISS

Local ISS

ISS can be viewed as an L L stability property

How about a (weaker) L2 L stability property?

x(t) (x(t0), t t0) +1 t

t0 2(u())d

K L, 1, 2K

Variants of ISS:

x(t) (x(t0), t t0) +

supt0t

u()

integral ISS (iISS)


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y

Definition [IOS]The system is said to be input-to-output-stable (IOS) if thereexist functions KL and K such that for any initial state x(t0) andany bounded input u(t), the solution y(t) exists for all t t0 and satisfies

y(t) (x(t0), t t0) +

supt0t u()

u

IOS

x = f(x, u)

y = h(x)

The small-gain theorem can be easily generalized do IOS systems

y

Definition [IOSS] The system is said to be input-output-to-state-stable (IOSS)if there exist K L and 1, 2K such that for any initial state x(t0) andany bounded input u(t), the following inequality holds for all t t0

x(t) (x(t0), t t0) +1

supt0t

u()

+2

supt0t

y()

u

IOSS

x = f(x, u)

y = h(x)

This is a nonlinear version of zero detectability


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How can we explore the Lyapunov tools forcontrol design?

M+ C() + D() + g() =

=J()

= (u, v, w, p, q, r)

= (u,v,w,p,q,r)

= (x , y, z, , , )

Goal: Design a state feedback control so that (t) converges to adesired position and attitude d (Pose stabilization)

Fully actuated AUV/ROV

Model:


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M+ C() + D() + g() =

=J()

e(t) = (t) d e = = J()

V(, e) = 12

T M + eT KPe

Model:

Error dynamics:

Control Lyapunov function:

V(, e) = 1

2

T M +eT KPe

V = T M+ eT KPe

V = T

M+ JT() KPe

V = T

D() g() +JT() KPe

T C()

= JTKPe(t) KD+ g()

Computing the time derivative with respect to the trajectory of the system...

Assign a feedback law...

V = T [D() +KD] 0

We havestability


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E = {(, e) R12 := 0}

Can we prove Asymptotic Stability ?

Use LaSalles invariance principle...

limt

(t) =d

The largest invariant set M in E is the origin, thus we have asymptotic stability!

Therefore....

0 = JT() KPe

x1 = x21 x

31+x2

x2 = u

V1 =1

2x21

V1 =x1(x21 x

31+x2) =x

41+x1(x

21+ x2)

x2 =(x1) =x21 x1

V1 =x41 x

21

Backstepping

Example

Consider

Choose


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z2 =x2 (x1) =x2+x21+x1

V2 =V1+1

2z22

V2 = x1(x31 x1+z2) +z2(u+ (2x1+ 1)(x

31 x1+z2))

= x41 x21+z2((u+ (2x1+ 1)(x

31 x1+z2) +x1)

u= (2x1+ 1)(x31 x1+ z2) x1 z2

V1 =x41 x

21 z

22

x= 0 is GAS

Backstepping

Consider the augmented Lyapunov

Choose

x1 = x2+ f1(x1)

x2 = x3+ f2(x1, x2)...

xi = xi+1+fi(x1, x2, . . . , xi)

xn = fn(x1, x2, . . . , xn) +u

z1 = x1

z2 = x2 (x1)

V1 = 1

2z21

(x1) =x1 f1(x1)

Recursive application of Backstepping

Strict-feedback form

Define


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zi+1 = xi+1 i(z1, , zi)

Vi = 1

2(z21 +z

22+ +z

2i)

zi = zi+1+i(z1, , zi) + fi(z1, , zi)

Vi = z21 z

2i1+zi1zi+zi(zi+1+i

z1, , zi) + fi(z1, , zi)

i

z1, , zi) =zi1 zi fi(z1, , zi)

zi = zi1 zi+zi+1

Vi = z21 z

2i +zizi+1


i-th step:

Set

zn= fn(z1, , zn) +u

u= n(z1, , zn) =zn1 zn fn(z1, , zn)

Vn=1

2(z21+ +z

2n)

zn = zn1 zn

Vn = z21 z

22 z

2n


last step:

Choose


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Consider the following system

x1 =x2x2 =h(x) +g(x)u

whereh, g are unknownnonlinear functions and g(x)g0 >0,x.Goal: Design a state-feedback control law to stabilize the origin.

Idea: Design a control law that restrict the motion of the system to the

manifold or surfaces= a1x1+x2 = 0, a1 >0

Note that the motion on the manifold s= 0 satisfies

x2 =a1x1 x1 =ax1 x= (x1, x2)0

and furthermore the motion is independent ofh andg!

Now the question is how can we bring the trajectory to the manifold?

How can we deal with parametric uncertainty?

Let

V = 1

2s2

and therefore

V =ss

=s(a1x1+ x2)

=s(a1x2+h(x)) +sg(x)u

Suppose thata1x2+h(x)g(x)

(x), x 2 and assume that (x) is known.Then,

V |s|(x)g(x) +su

=g(x)|s|[(x) + sgn(s)u]

Letu= (x)sgn(s) with (x) (x) +0, 0 >0.

V g(x)|s|0 g00|s|


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LetW =

2V =|s| (Note thatu = u2

u)

The upper right-hand derivative is given by

D+W = 2V

2

2V=

V

W g00 W

W

By the comparison lemma

W(s(t))W(s(0)) g00t

Thus, the trajectory reaches the manifold s= 0 in finite time.Moreover, once it reaches the manifold we have V

g00

|s

|= 0, which means

that it cannot leave from it.

In summary, for the example above, the sliding mode control strategy is com-posed by two phases:

1. reaching phase: the trajectory starting off the manifolds = 0 move toward

it and reach it in finite time.2. sliding phase: the motion is confined to the manifold s = 0 and the

dynamics of the system are represented by the reduced-order model x1=a1x1.

Remark: The control lawu= (x)sgn(s) is called a sliding mode control law.Note that it is robust with respect to uncertainty on h and g. We only need toknow the upper form (x).

Furthermore, ifa1x2+h(x)

g(x)

k1, x D then u = ksgn(s), k > k1 and if kcan be chosen arbitrarily large, it can achieve semi-global stability.


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x= f(x, ) +u, x R

V =1

2x2

u= f(x, ) x V =x2

Rp unknown parameter vector

u= f(x,) x

V =x

f(x, ) +u

Adaptive nonlinear control (a simple approach)

Example:

Consider

If the parameter vector is known...

If not, choose

f(x, ) = (x)

f(x, ) =a1x+a2x2

f(x, ) =a1x+a2x2 = [a1a2]

xx2

V =x2 +xf(x, ) f(x,)

Suppose that f(x, ) can be written in the form

Adaptive nonlinear control (a simple approach)

e.g.


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Speed, Heading, and Depth Control

Bottom Following (Terrain Contouring)

Point Stabilization, Hovering, Manipulation

Trajectory tracking and Path Following

Motion Control Problems

Point Stabilization

Objective: steer an underwater vehicle to a target point,with a desired orientation


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Point StabilizationRegulation of a nonholonomic dynamic wheeled mobile robot

Step 5. Switching control law

When e= 0 do

Point StabilizationDynamic positioning of an underactuated AUV

Objective

Derive a feedback control law for u and r to regulate the AUV to a desired

target point in the presence of a constant, unknown ocean current disturbance

and parametric model uncertainty.

In the presence of ocean currents, the problem

of regulating an underactuated AUV to a

desired point with a an arbitrary desired

orientation does not have a solution.

Possible behaviors:

The vehicle will diverge from the desired target position.The controller will keep the vehicle moving around a neighborhood of the

desired position, trying insistently to steer it to the given point, and

consequently inducing an oscillatory behavior.


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e=

(x xd)2 + (y yd)2

x xd =e cos(+)

y yd =e sin(+)

+= tan1

(y yd)

(x xd)

Point StabilizationDynamic positioning of an underactuated AUV

A. Pedro Aguiar and Antnio M. Pascoal,Dynamic Positioning and Way-Point Tracking ofUnderactuated AUVs in the Presence of Ocean Currents.

International Journal of Control, Vol. 80, No 7, pp. 1092-1108, July 2007.

Point Stabilization




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Point Stabilization with currents

Vc



Way-Point Tracking with Currents

Vc




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Reference-tracking versus path-following

Additional design of freedom

The reference-tracking problem is subjected to the limitations

imposed by the unstable zero-dynamics.

The path-following problem is not subjected to these limitations

The freedom to design a timing law is a major advantage of path-

following over reference tracking.

A. Pedro Aguiar, Joo P. Hespanha, and Petar Kokotovi,Path-Following for Non-Minimum PhaseSystems Removes Performance Limitations.

IEEE Transactions on Automatic Control, Vol. 50, No. 2, pp. 234-239, Feb. 2005.

due to side-slip the velocity of the hovercraft is not tangent to the trajectory

Goal: force the hovercraft to track acircular trajectory (black)

Position tracking of an underactuated Hovercraft


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Problem statement

Consider an underactuated vehicle modeled as a rigid body subject to external

forces and torques

Kinematics

Dynamics

Trajectory-tracking problem

Given a trajectorypd:[0,)R3, we want the tracking errorp(t)-pd(t)

to converge to a neighborhood of the origin that can be madearbitrarily small

The solution should be robust with respect toparametric modeling uncertainty

later extended to path following

due to side-slip the velocity of the hovercraft is not tangent to the trajectory

Goal: force the hovercraft to track a

circular trajectory (black)

Position tracking of an

underactuated Hovercraft


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Step 1. Coordinate Transformation

Controller design

tracking error inbody frame

Step 2. Convergence of eerror only in position!


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Step 4.Backstepping for z2

Controller design

2nd control signalhas been assigned


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What happens if there is parametric modeling uncertainty?

Coefficient of viscous friction assumed used by the controller is 10% of the real value

Closed-loop system still stable but considerable performance degradation

Model parameter uncertainty

Supervisory control

supervisor

process

controller 1

controller n

yu

w

bank of candidatecontrollers

measuredoutput

controlsignal

exogenousdisturbance/

noiseswitching signal

Key ideas:1. Build a bank of alternative controllers

(one for each possible value/range of the unknown parameter)

2. Supervisor places in the feedback loop the controller that seems

more promising based on the available measurements


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Estimator-based supervisions setup

Processis assumed to be in a family

parametric uncertainty

For each admissible process model Mp,

there is one candidate controller Cpthat provides adequate performance.

processucontrolsignal measuredoutput

w exogenousdisturbance/noise

How to determine which admissible model matches the real process?

y

Estimator-based supervisor

epsmallprocess is

likely to be Mp

should

use Cp

Multi-estimator

ypestimate of the process output ythat would be correct if the process was Mpepoutput estimation error that would be small if the process was Mp

Processis assumed to be in family

process is

Mp,pPcontroller Cpprovides

adequate performance

Decision logic:

multi-

estimatoru

measuredoutput

control

signal

y

y

decision

logic

switchingsignal

+

+

Certainty equivalence inspired


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Multi-estimator for the vehicle model

Process model

Family of estimator equations (pP)

estimation errors

scalar positive functions

The correct estimation error ep*satisfies

uncertainty in the dynamic equations through: M, J, fv, f

convergence to zero

small integral norm

Multi-estimator

ypestimate of the process outputythat would be correct if the process was Mpepoutput estimation error that would be small if the process was Mp


Certainty equivalence inspired


process is



Decision logic:

multi-

estimator

measuredoutput

controlsignal

decision

logic

switchingsignal

+

+

epsmallprocess is

likely to be Mpuse Cp

u

y

y

A stability argument cannot be

based on this because typically

process is Mpepsmall

but not the converse


177/336


Multi-estimator

ypestimate of the process outputythat would be correct if the process was Mpepoutput estimation error that would be small if the process was Mp


process is



Decision logic:

multi-

estimator

measuredoutput

controlsignal

decision

logic

switchingsignal

+

+

overall state

is small

overall system

is detectable

through ep

Certainty equivalence inspired, but formally justified by detectability

detectable meanssmall epsmall state

epsmall use Cpprocess is

likely to be Mp

u

y

y

Detectability property

Using the original Lyapunov function and

in L1constantpositiveconstant

overall state

is small

overall system

is detectable

through ep

detectable meanssmall epsmall state

epsmall use Cp

in L1in L2

When epis small, all signals remain bounded

& e converges to ball of radius proportional to

process is

likely to be Mp


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Path Following

Inspired by the work of Claude Samson et al. for wheeled robots

A. Micaelli and C. Samson (1992). Path following and time-

varying feedback stabilization of a wheeled robot. In Proc.International Conference ICARCV92, Singapore.

. Useforward motionto make the robot track adesired speed profile.

. Compute the closest pointon the path.

. Compute theSerret-Frenet (SF)frame at that point.

. Use rotational motionto align the body-axis with the SFframe and reduce the distance to closest point to zero.

Path Following (control strategy)

Rabbit movingalong the path

path curvature at

control signals

exogenoussignal

- Lyap. func.

- approach angle

- do back stepping to deal with the dynamics


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guide (rabbit) movingalong the path a mind

of its own (control variable)

This will makethe vehicle follow the path

1. Drive the distance from Q to the rabbit to zero;2. Align the flow frame with the Serret-Frenet(align total velocity with the tangent to the path).

Path Following (AUV)

Path-following

Consider an underactuated vehicle modeled as a rigid body subject to

external forces and torques

Kinematics

Dynamics

Path-following problem

Given a geometric path {pd()R3 : [0,)} and speed assignment vr()

R, we want

the position to converge and remain inside a tube centered around thedesired path than can be made arbitrarily thin, and

satisfy (asymptotically) the desired speed assignment, i.e., vras

t


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The Sirene AUV developed

for Deep Sea Intervention on

Future Benthic Laboratories

Goal: force the underactuated

AUV to track a desired

helix trajectory

Tracking and path-following ofan underwater vehicle (3-D space)

Path-following

Goal Given a geometric path {pd()R

3 : [0,)} and speed assignment vr() R, we want

the position to converge and remain inside a tube centered around the desired path

than can be made arbitrarily thin, and

satisfy (asymptotically) the desired speed assignment, i.e., vras t

Define

speed error

Choosing

Same conclusions as before


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Simulation results

0 50 100 150 200 250 300-5

0

5

time [s]

r

oll[degree]

0 50 100 150 200 250 300-20

0

20

time [s]

pitch[degree]

0 50 100 150 200 250 300-200

0

200

time [s]

yaw[degree]

0 50 100 150 200 250 300-5

0

5

time [s]

r

oll[degree]

0 50 100 150 200 250 300-20

0

20

time [s]

pitch[degree]

0 50 100 150 200 250 300-200

0

200

time [s]

yaw[degree]

Trajectory tracking Path-following


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Observability Issues for

Cooperative Marine RobotsGiovanni Indiveri,

Dipartimento Ingegneria Innovazione,Universit del Salento, Lecce, Italy

[email protected]

2nd Co3-AUVs Summer School, Bremen, 11.-17.07.2011

mercoled 13 luglio 2011

Lecture layout1) Introduction to the

concept of Observability

2) The LTI (Linear TimeInvariant) case

3) Observability andLocalization forCooperative Marine

Robots

4) Some CO3AUVs

results

5) Discussion

Giovanni Indiveri,Observability Issues for Cooperative Marine Robots



191/336

Main bibliography

Wilson J. Rugh, "Linear System Theory", Prentice Hall,1993

CO3AUVs results

Giovanni Indiveri,

Observability Issues for Cooperative Marine Robots


Lets start!


Stop me any timeQuestions are welcomeAlways remember that the devil isin the details



192/336

Basic Concepts

f(x,u,t)

+

x0

+u x g(x,u,t)y

deterministic, nonlinear, time varying,non autonomous dynamical system

Giovanni Indiveri,



Basic Concepts

f(x,u,t)

+

x0

+u x g(x,u,t)y


x=f(x,u,t) State equation

Output equationy=g(x,u,t) What preventsfrom being known?x



193/336

Basic Concepts

f(x,u,t)

+

x0

+u x g(x,u,t)y

Giovanni Indiveri,


x(t) =x0+

t

0

f(x(), u(), )d


Basic Concepts

f(x,u,t)

+

x0

+u x g(x,u,t)y


Two distinct (initial) states are said to be

indistinguishable (i.e. non observable) if for thesame input they produce the same output.



194/336

The Linear Case

Giovanni Indiveri,


x=Ax+Bu

y=Cx+Du

A IRnn B IRnp

C IRmn D IRmp

A IRnn B IRn1

C IR1n D IR

Single Input SingleOutput SISO case

Usually m < n, i.e. the output is smaller than thestate.


The Linear Case


x=Ax+Bu

y=Cx+Du

x(t) =eA(tt0)x0+

tt0

eA(t)Bu()d

y(t) =CeA(tt0)x0+ tt0

CeA(t)B+D(t )

u()d

M IRnn

eMt :=+l=0

Ml tll!

nn

Mt

l=0

l



195/336

Giovanni Indiveri,


x=Ax+Bu

y=Cx+Du

x(t) =eA(tt0)x0+ tt0

eA(t)Bu()d

y(t) =CeA(tt0)x0+

tt0

CeA(t)B+D(t )

u()d

+

x0

+u xy

A

B +

+

C

D

+

+



Discrete time version

x(k+ 1) =Ax(k) +Bu(k)

y(k) =Cx(k) +Du(k)

x=Ax+Bu

y=Cx+Du

x x((k+ 1)T) x(kT)

T =Ax(kT) + Bu(kT)

x((k+ 1)T) = (Inn+ AT) x(kT) + BT u(kT)

(Inn+ AT) A

BT B

kT k



196/336

Giovanni Indiveri,


Discrete time versionx

(k

+ 1) =Ax

(k

) +Bu

(k

)y(k) =Cx(k) +Du(k)

+

x0

+u x

yA

B

+

+ C

D

+

+1



x(k+ 1) =Ax(k) +Bu(k)

y(k) =Cx(k) +Du(k)

x(1) =Ax(0) +Bu(0)

x(2) =Ax(1) +Bu(1) =A2x(0) +ABu(0) +Bu(1)

x(3) =Ax(2) +Bu(2) =A3x(0) +A2Bu(0) +ABu(1) +Bu(2)

...

x(n 1) =An1x(0) +n2

l=0

AlBu(n 2 l)

x(n) =Anx(0) +n1l=0

AlBu(n 1 l)



197/336

Giovanni Indiveri,


x(n) =Anx(0) +n1l=0

AlBu(n 1 l)

Cayley - Hamilton Theorem

A IRmm

p() = det ( Imm A)

then

p(A) = 0mm

p() =m +m1k=0

ckk = 0:

A IRnn = An = n1k=0

ckAknn n

1

k=0

kk



x(k+ 1) =Ax(k) +Bu(k)

y(k) =Cx(k) +Du(k)

x(n 1) =An1

x(0) +

n2l=0

Al

Bu(n 2 l)

y(0) =Cx(0) + Du(0)

...

y(n 1) =CAn1x(0) +n

2l=0

CAlBu(n 2 l) + Du(n 1)



198/336

Giovanni Indiveri,


x(k+ 1) =Ax(k) +Bu(k)

y(k) =Cx(k) +Du(k)

y(0)

Du(0) =Cx(0)...

y(n 1) n2l=0

CAlBu(n 2 l) Du(n 1) =CAn1x(0)

y(0) =Cx(0)

y(1) =CAx(0)

...

y(n 1) =CAn1x(0)



y(0) =Cx(0)

y(1) =CAx(0)

...

y(n 1) =CAn1

x(0)

Assume y to bea scalar (SI), i.e.

Y = (y(0),y(1), . . . ,y(n 1))T IRn1

O =

C

CA..

.CAn1

IRnn

Y = O x00

(Kalman) Observabilitymatrix

C IR1n



199/336

Giovanni Indiveri,


Y = O x00O =

C

CA...

CAn

1

IR

nn

If all pairs of (initial) states are distinguishablethe system is said to be completely observable.

The system is completely observable iff theobservability matrix has full rank.



Y = O x00O =

C

CA...

CAn1

IRnn


xo = O1 Y

xo = (OT

O)1 OTYFor multi-output systems:

O IRmnn



200/336

Giovanni Indiveri,


Y = O x00O =

C

CA...

CAn

1

IR

nn


Assumex1 =x2 such that

Y1= Ox1 and Y2= Ox2

with Y1= Y2

O (x1 x2) = 0

Absurdmercoled 13 luglio 2011


Y = O x00O =

C

CA...

CAn1

IR

nn

kerO= unobservable states

Unobservable states represent information thatcannot possibly be recovered based on the model,input and output only (structural obstruction).

Its not a matter of how good your filter is ... itssimply an ill posed problem that does not admitany solution.



201/336

Giovanni Indiveri,


Canonical observability form

xuo

xo

=

Auo A12

0 Ao

xuo

xo

+

B1B2

u

y =

0Co

xuo

xo

+ Du



xuo

xo

=

Auo A12

0 Ao

xuo

xo

+

B1B2

u y=

0Co

xuo

xo

+ Du

+

+u

y

+

+ +

+B1

Ao

Co

D

xo(0)

xo

+

++

+

xuo(0)

Auo

B2

12

+

xo

xuo



202/336

Giovanni Indiveri,


xuo

xo

=

Auo A12

0 Ao

xuo

xo

+

B1B2

u y=

0Co

xuo

xo

+ Du

+

+u

y

+

+ +

+

B1

Ao

Co

D

xo(0)

xo

+

++

+

xuo(0)

Auo

B2

12

+

xo

xuo



Example

J+=ki+l

L didt

+Ri+k=V

d

dt

i

=

0 1 00 /J k/J

0 k/L R/L

i

+

0 00 1

1/L 0

V

l

y=C xC= (1 0 0)

DC motor



203/336

Giovanni Indiveri,


J+=ki+l

Ldi

dt+Ri+k=V

d

dt

i

=

0 1 00 /J k/J

0 k/L R/L

i

+

0 00 1

1/L 0

V

l

y=C xC= (1 0 0) Absolute encoder ....

O =

C

CA...

CAn1

= 1 0 00 1 0

0 /J k/J

in state space form:



J+=ki+l

Ldi

dt+Ri+k=V

Tachometric dynamo or relative encoder

ddt

i

= 0 1 00 /J k/J 0 k/L R/L

i

+ 0 00 1

1/L 0

Vl

y=C xC= (0 1 0)

O =

C

CA

...CAn1

= 0 1 0

0 /J k/J 0 [(/J)2 k2/(JL)] [(k)/J2 kR/(JL)]

kerO= unobservable states unobservable



204/336

Giovanni Indiveri,


J+=ki+l

Ldi

dt+Ri+k=V

d73-summerschool2

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