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Introduction to modeling and controlof underwater vehicle-manipulator systems

Gianluca Antonelli

Universita di Cassino e del Lazio Meridionale

[email protected]

http://webuser.unicas.it/lai/robotica

http://www.eng.docente.unicas.it/gianluca antonelli

TRIDENT school

Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012

mailto:[email protected]

http://webuser.unicas.it/lai/robotica

http://www.eng.docente.unicas.it/gianluca_antonelli

Targeted audience and talk’s shape

SAUVIM

50 minutes talk about the mathematical foundations ofUnderwater Vehicle Manipulator Systems (UVMS)

Educational shape (entry level)

knowledge of

mathematics, physicscontrolbasic robotics

equations, equations still equations. . .


Outline

ALIVE

UVMSs

Introduction

Mathematical modeling

Two words about dynamic control

Kinematic control


(semi)autonomus UVMSs

PETASUS

Use of a manipulator is common for ROV, mainly in remotelycontrolled or in a master-slave configuration

Among the first autonomus modes:

AMADEUS I & II before 2000, EU

SAUVIM 1997–, USA

PETASUS, Korea

ALIVE 2000-2003, EU

Twin Burger + manipulator, Japan

TRIDENT 2010-2012, EU


Notation1

yzb

η1

x

z

θ (pitch)

φ (roll)

ψ (yaw)

ω (heave)

υ (sway)yb

υ (surge)

xb

Forces and ν1,ν2 η1,η

2

momentsMotion along x Surge X u x

Motion along y Sway Y v y

Motion along z Heave Z w z

Rotation about x Roll K p φ

Rotation about y Pitch M q θ

Rotation about z Yaw N r ψ

1[Fossen(1994)]Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012

Rigid body attitude

Euler angles commonly used

roll

pitch

yaw

ok for the vehicle, designed stable in roll and pitch

For the end-effector possible issues of representation singularities→ non-minimal representations (quaternions)


Rigid body kinematics

η =

[

η1

η2

]

∈ R6 ν =

[

ν1

ν2

]

∈ R6

and by defining the matrix Je(RIB) ∈ R

6×6

Je(RIB) =

[

RBI O3×3

O3×3 Jk,o(RIB)

]

it isν = Je(R

IB)η

��

��

��✠

bodyfixed velocities


Rigid body dynamics

moving in the free space

MRBν +CRB(ν)ν = τ v

��

��

��✠

MRB =

[

mI3 −mS(rbC)mS(rbC) IOb

]

∈ R6×6

��✒

bodyfixed acceleration

❅❅❅❘

6dof force/moment at the body


Added mass and inertia

A body moving in a fluid accelerates it (ρ ≈ 1000 kg/m3)Need to account for an additional inertia(the added mass is not a quantity to be added to the body such that

it has an increased mass)

For submerged bodies, with common AUV shape at low velocities:

MA = − diag {Xu, Yv, Zw,Kp,Mq, Nr}

CA =

0 0 0 0 −Zww Yvv

0 0 0 Zww 0 −Xuu

0 0 0 −Yvv Xuu 00 −Zww Yvv 0 −Nrr Mqq

Zww 0 −Xuu Nrr 0 −Kpp

−Yvv Xuu 0 −Mqq Kpp 0


Damping

Viscosity of the fluid causes dissipative drag and lift forces to the body

lift

drag

relative flow

The simplest model is drag-only, diagonal, linear/quadratic in velocity

DRB(ν)ν

DRB(ν) = − diag {Xu, Yv, Zw,Kp,Mq, Nr}+

− diag{

Xu|u| |u| , Yv|v| |v| , Zw|w| |w| ,Kp|p| |p| ,Mq|q| |q| , Nr|r| |r|}


Current

Assume a current constant and irrotational in the inertial frame

νIc =

νc,xνc,yνc,z000

νIc = 0

effects added considering the relative velocity in body-fixed frame

νr = ν −RBI ν

Ic

in the Coriolis/centripetal and damping


Current

ob

ob

xb

xb

yb

yb

o x

y

νIc

ψ

intuitively, the current is pushing the vehicle


Gravity and buoiancy

ob

ob

xb

xb

zb

zb

oxz

rgrg fgfg

rbrbf bf b Mr

gI

θ

fG(RBI ) = RB

I

00W

fB(RBI ) = −RB

I

00B

MR = rBG × fG(RBI ) + rBB × fG(R

BI )

linear in the 3 parameters: WrB

G−BrB

Bconstant in bodyfixed


Some dynamic considerations

Considering the sole vehicle two effects affects steady state

current effect, constant in the inertial frame

restoring forces, (depends on) constant in the body-fixed frame

Proper integral/adaptive actions need to be designed for finepositioning to avoid disturbance caused by the controller 2

2[Antonelli(2007)]Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012

Some dynamic considerations

Considering the sole vehicle two effects affects steady state

current effect, constant in the inertial frame

restoring forces, (depends on) constant in the body-fixed frame

Proper integral/adaptive actions need to be designed for finepositioning to avoid disturbance caused by the controller 2

νIcνI

c

inertial body-fixed

current

compensation

during a 90◦

rotation

2[Antonelli(2007)]Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012

Thrusters

6 or more for full vehicle control (thrust required also in hovering)force/moment (nonlinear) function of

propeller revolution

fluid speed

input torque

affected by several parameters

fluid density

tunnel cross-sectional area

tunnel length

propeller diameter and input-output volumetric flowrate

main cause of bandwidth constraints and limit cycles


Some references

For modeling and control of marine vehicles in a control perspective:

[Fossen(1994)]

[Fossen(2002)]

[Antonelli et al.(2008)Antonelli, Fossen, and Yoerger]

[Fossen(2011)]


UVMS kinematics

Oi

η1

ηee

❅❅❘endeffector velocities

❍❍❍❍❍❍❍❍❍❍❍❍❥Jacobian

system velocitiesηee =

[

ηee1

ηee2

]

= Jw(RIB , q)ζ ζ =

ν1

ν2

q


UVMS dynamics

Dynamics via classical Newton-Euler equations by propagating thevelocities and forces

Bi

Ci

Oi−1

Oiri−1,i

ri−1,C

ri−1,B

ri,C

f i,µi

f i+1,µi+1

−ρ∇ig

migdi


UVMS dynamics in matrix form

M(q)ζ +C(q, ζ)ζ +D(q, ζ)ζ + g(q,RIB) = τ

formally equal to a groundfixed industrial manipulator 3

however. . .

Uncertainty in the model knowledge

Low bandwidth of the sensor’s readings

Difficulty to control the vehicle in hovering

Dynamic coupling between vehicle and manipulator

Kinematic redundancy of the system

3[Siciliano et al.(2008)Siciliano, Sciavicco, Villani, and Oriolo]Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012

UVMS dynamics

Movement of vehicle and manipulator coupled

movement of the vehicle carrying the manipulator

law of conservation of momentum

Need to coordinate

at velocity level ⇒ kinematic control

at torque level ⇒ dynamic control 4

4[McLain et al.(1996b)McLain, Rock, and Lee][McLain et al.(1996a)McLain, Rock, and Lee]


Need for coordination

Coordination and redundancy exploitation is required5:

Redundancy at torque level?

Need to exactly compensate forthe dynamics, not appropriatefor the underwater environment

Space manipulator literature?

The assumption of themomentum conservation is notvalid

5[Khatib(1987), Sentis(2007),Nenchev et al.(1992)Nenchev, Umetani, and Yoshida]


Needs for coordination

let us move to the kinematical level

What is coming next

an example

a short review

algorithms & tasks for UVMSs

balance movement between vehicle/manipulator


A first kinematic solution

Hoping the vehicle in hovering is not the best strategy to e.e. finepositioning6, better to kinematically compensate with the manipulator

6[Hildebrandt et al.(2009)Hildebrandt, Christensen, Kerdels, Albiez, and Kirchner]Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012

Kinematic control in pills

A robotic system is kinematically redundant when it possesses moredegrees of freedom than those required to execute a given task

Redundancy may be used to add additional tasks and to handlesingularities

Example for the sole end-effector trajectory

ηee,d ηd, qd τ η, q

IK control

offline trajectory planning not appropriate underwater


Kinematic control in pills -2-

Starting from a generic m-dimensional task

σ = f(η, q) ∈ Rm

it is required to invertσ = J(η, q)ζ

The configurations at which J ∈ Rm×6+n is rank deficient are

kinematic singularities

The mobility of the structure is reduced

Infinite solutions to the inverse kinematics problem might exist

Close to a kinematic singularity at small task velocities cancorrespond large joint velocities



σ = Jζ inverted by solving proper optimization problems

Pseudoinverseζ = J †σ = JT

(

JJT)−1

σ

Transpose-basedζ = JTσ

Weighted pseudoinverse

ζ = J†W

σ = W−1JT(

JW−1JT)−1

σ

Damped Least-Squares

ζ = JT(

JJT + λ2Im

)−1σ

need for closedloop also. . .



Handling several tasks7

Extended JacobianAdd additional (6 + n)−m constraints

h(η, q) = 0 with associated Jh

such that the problem is squared with

[

σ

0

]

=

[

J

Jh

]

ζ

7[Chiaverini et al.(2008)Chiaverini, Oriolo, and Walker]Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012


Augmented JacobianAn additional task is given

σh = h(η, q) with associated Jh

such that the problem is squared with

[

σ

σh

]

=

[

J

Jh

]

ζ



✛✚

✘✙

ζ

❘

✛✚

✘✙

σ

A mapping from the controlled variable to the task space

An inverse mapping is required

Additional tasks may be considered (e.g. task priority)



✛✚

✘✙

ζ

❘

✛✚

✘✙

σ

✖✕✗✔

■






✛✚

✘✙

ζ

❘

✛✚

✘✙

σ

✖✕✗✔

■ σa

✚✙✛✘

σb

✙

✖✕✗✔

✶






Task priority redundancy resolution


further projected on the the null space of the higher priority one

ζ = J†σ +[

Jh

(

I − J †J)]† (

σh − JhJ†σ)



Singularity robust task priority redundancy resolution 8


further projected on the the null space of the higher priority one

ζ = J †σ +(

I − J†J)

J†

hσh

8we are talking about algorithmic singularities here. . .Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012


AMADEUS

Agility task priority9

Task priority framework to handle both precision and set tasksEach task is the norm of the corresponding error (i.e., mi = 1)Recursive constrained least-squares within the set satisfyinghigher-priority tasks

9[Casalino et al.(2012)Casalino, Zereik, Simetti, Sperinde, and Turetta]Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012


Behavioral algorithms (behavior=task), bioinspired, artifical potentials

sensorsbehavior b

ζ2⊗

α2

behavior a

ζ1

supervisor

⊗

α1

behavior c

ζ3⊗

α3

∑

ζ


Tasks to be controlled

Given 6 + n DOFs and m-dimensional tasks: End-effector

position, m = 3

pos./orientation, m = 6

distance from a target, m = 1

alignment with the line of sight, m = 2



Manipulator joint-limits

several approaches proposed, m = 1 to n, e.g.

h(q) =

n∑

i=1

1

ci

qi,max − qi,min

(qi,max − qi)(qi − qi,min)



Drag minimization, m = 1 10

h(q) = DT(q, ζ)WD(q, ζ)

within a second order solution

ζ = J †(

σ − Jζ)

− k(

I − J†J)

([

∂h∂η∂h∂q

]

+∂h

∂ζ

)

10[Sarkar and Podder(2001)]Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012


Manipulability/singularity, m = 1

h(q) =∣

∣det(

JJT)∣

∣

(In 11 priorities dynamically swapped between singularity and e.e.)

joints

inhibited direction

singularitysingularity

setclose to

11[Kim et al.(2002)Kim, Marani, Chung, and Yuh,Casalino and Turetta(2003)]



Restoring moments:

m = 3 keep close gravity-buoyancy of the overall system 12

m = 2 align gravity and buoyancy (SAUVIM is 4 tons) 13

f b

f g

τ 2

12[Han and Chung(2008)]13[Marani et al.(2010)Marani, Choi, and Yuh]



Obstacle avoidance m = 1



Workspace-related variablesVehicle distance from the bottom, m = 1Vehicle distance from the target, m = 1



Sensors configuration variables

Vehicle roll and pitch, m = 2Misalignment between the camera optical axis and the target lineof sight, m = 2


However. . .

End effector going out of the workspace and one (eventually weighted)task always leads to singularity

❅❅❅❘

manipulator stretched


Balance movement between vehicle and manipulator

Need to distribute the motion e.g.:

move mainly the manipulator when target in workspace

move the vehicle when approaching the workspace boundaries

move the vehicle for large displacement

Some solutions, among them dynamic programming or fuzzy logic


Fuzzy logic to balance the movement14

Within a weighted pseudoinverse framework

J†W

= W−1JT(

JW−1JT)−1

W−1(β) =

[

(1− β)I6 O6×n

On×6 βIn

]

with β ∈ [0, 1] output of a fuzzy inference engineSecondary tasks activated by additional fuzzy variables αi ∈ [0, 1]

ζ = J†W

(xE,d +KEeE) +(

I − J†W

JW

)

(

∑

i

αiJ†s,iws,i

)

Only one αi active at onceNeed to be complete, distinguishable, consistent and compactBeyond the dicotomy fuzzy/probability theory very effective intransferring ideas

14[Antonelli and Chiaverini(2003)]Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012

Dynamic programming to balance the movement15

Freeze, as a free parameter, the vehicle velocity ν and implementthe agility task priority to the sole manipulator ⇒ qd

Freeze the manipulator velocity qd and then find the vehiclevelocity νd needed for the remaining tasks components notsatisfied ⇒ ζd

ν

νe

15[Casalino et al.(2012)Casalino, Zereik, Simetti, Sperinde, and Turetta]Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012

Acknowledge

Several researchers kindly provided the materials/video (or theexplications...) for this talkIn casual order:

ISME (Pino Casalino, . . . )

TRIDENT partners (Pedro Sanz, Pere Ridao, . . . )

SAUVIM partners (Junku Yuh, Giacomo Marani, . . . )

DFKI (Frank Kirchner)

OTTER (Tim McLain)


Bibliography I

G. Antonelli.

Underwater robots. Motion and force control of vehicle-manipulator systems.

Springer Tracts in Advanced Robotics, Springer-Verlag, Heidelberg, D, 2ndedition, June 2006.

G. Antonelli.

On the use of adaptive/integral actions for 6-degrees-of-freedom control ofautonomous underwater vehicles.

IEEE Journal of Oceanic Engineering, 32(2):300–312, April 2007.

G. Antonelli and S. Chiaverini.

Fuzzy redundancy resolution and motion coordination for underwatervehicle-manipulator systems.

IEEE Transactions on Fuzzy Systems, 11(1):109–120, 2003.

G. Antonelli, T. Fossen, and D. Yoerger.

Springer Handbook of Robotics, chapter Underwater Robotics, pages 987–1008.

B. Siciliano, O. Khatib, (Eds.), Springer-Verlag, Heidelberg, D, 2008.


Bibliography II

G. Casalino and A. Turetta.

Coordination and control of multiarm, nonholonomic mobile manipulators.

In Proceedings IEEE/RSJ International Conference on Intelligent Robots andSystems, pages 2203–2210, Las Vegas, NE, Oct. 2003.

G. Casalino, E. Zereik, E. Simetti, S. Torelli A. Sperinde, and A. Turetta.

Agility for underwater floating manipulation: Task & subsystem priority basedcontrol strategy.

In 2012 IEEE/RSJ International Conference on Intelligent Robots andSystems, Vilamoura, PT, october 2012.

S. Chiaverini, G. Oriolo, and I. D. Walker.

Springer Handbook of Robotics, chapter Kinematically RedundantManipulators, pages 245–268.

B. Siciliano, O. Khatib, (Eds.), Springer-Verlag, Heidelberg, D, 2008.


Bibliography III

T.I. Fossen.

Guidance and Control of Ocean Vehicles.

Chichester New York, 1994.

T.I. Fossen.

Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs andUnderwater Vehicles.

Marine Cybernetics, Trondheim, Norway, 2002.

T.I. Fossen.

Handbook of marine craft hydrodynamics and motion control.

Wiley, 2011.

J. Han and W.K. Chung.

Coordinated motion control of underwater vehicle-manipulator system withminimizing restoring moments.

In Intelligent Robots and Systems, 2008. IROS 2008. IEEE/RSJ InternationalConference on, pages 3158–3163. IEEE, 2008.


Bibliography IV

M. Hildebrandt, L. Christensen, J. Kerdels, J. Albiez, and F. Kirchner.

Realtime motion compensation for ROV-based tele-operated underwatermanipulators.

In IEEE OCEANS 2009-Europe, pages 1–6, 2009.

O. Khatib.

A unified approach for motion and force control of robot manipulators: Theoperational space formulation.

IEEE Journal of Robotics and Automation, 3(1):43–53, 1987.

J. Kim, G. Marani, WK Chung, and J. Yuh.

Kinematic singularity avoidance for autonomous manipulation in underwater.

Proceedings of PACOMS, 2002.

G. Marani, S.K. Choi, and J. Yuh.

Real-time center of buoyancy identification for optimal hovering in autonomousunderwater intervention.

Intelligent Service Robotics, 3(3):175–182, 2010.


Bibliography V

T.W. McLain, S.M. Rock, and M.J. Lee.

Coordinated control of an underwater robotic system.

In Video Proceedings of the 1996 IEEE International Conference on Roboticsand Automation, pages 4606–4613, 1996a.

T.W. McLain, S.M. Rock, and M.J. Lee.

Experiments in the coordinated control of an underwater arm/vehicle system.

Autonomous robots, 3(2):213–232, 1996b.

D. Nenchev, Y. Umetani, and K. Yoshida.

Analysis of a redundant free-flying spacecraft/manipulator system.

Robotics and Automation, IEEE Transactions on, 8(1):1–6, 1992.

N. Sarkar and T.K. Podder.

Coordinated motion planning and control of autonomous underwatervehicle-manipulator systems subject to drag optimization.

Oceanic Engineering, IEEE Journal of, 26(2):228–239, 2001.


Bibliography VI

L. Sentis.

Synthesis and Control of Whole-Body Behaviors in Humanoid Systems.

PhD thesis, Stanford University, 2007.

B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo.

Robotics: modelling, planning and control.

Springer Verlag, 2008.


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