TRIDENT school 2012

Introduction to modeling and control
of underwater vehicle-manipulator systems

Gianluca Antonelli

Universit` di Cassino e del Lazio Meridionale
a
antonelli@unicas.it
http://webuser.unicas.it/lai/robotica
http://www.eng.docente.unicas.it/gianluca antonelli

TRIDENT school

Gianluca Antonelli TRIDENT school, Mallorca, 1 october 2012

Targeted audience and talk’s shape

50 minutes talk about the mathematical foundations of
Underwater Vehicle Manipulator Systems (UVMS)
Educational shape (entry level)
knowledge of
mathematics, physics
control
basic robotics

equations, equations still equations. . .

SAUVIM

Outline

UVMSs
Introduction
Mathematical modeling
Two words about dynamic control
Kinematic control

ALIVE

(semi)autonomus UVMSs

Use of a manipulator is common for ROV, mainly in remotely
controlled 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

PETASUS

Notation1

φ (roll)
υ (surge)
θ (pitch) xb
υ (sway) ψ (yaw) η1
yb

ω (heave) x
zb y
Forces and ν 1, ν 2 η1, η2
z moments
Motion 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)]

Rigid body attitude

yaw
roll

Euler angles commonly used

pitch

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

η1 ν1
η= ∈ R6 ν= ∈ R6
η2 ν2
I
and by deﬁning the matrix J e (RB ) ∈ R6×6
B
RI O 3×3
J e (RI ) =
B I
O 3×3 J k,o (RB )

it is
I
˙
ν = J e (RB )η

✠

body-fixed velocities


Rigid body dynamics

moving in the free space

body-fixed acceleration
✒

˙
M RB ν + C RB (ν)ν = τ v
❅
❅
❅
❘
6-dof force/moment at the body

✠

mI 3 −mS(r b )
M RB = b )
C ∈ R6×6
mS(r C I Ob


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:

M A = − diag {Xu , Yv , Zw , Kp , Mq , Nr }
˙ ˙ ˙ ˙ ˙ ˙
0 0 0 0 −Zw w
 
˙ Yv v
˙
 0 0 0 Zw w
˙ 0 −Xu u
˙ 

 0 0 0 −Yv v ˙ Xu u
˙ 0 
CA =  0

 −Zw w ˙ Yv v
˙ 0 −Nr r ˙ Mq q 
˙ 
 Zw w
˙ 0 −Xu u Nr r
˙ ˙ 0 −Kp p 
˙
−Yv v Xu u
˙ ˙ 0 −Mq q Kp p˙ ˙ 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
D RB (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
 
νc,x
νc,y 
 
I
νc,z 
νc = 
  νI = 0
˙c
 0 

 0 
0

eﬀects added considering the relative velocity in body-ﬁxed frame

ν r = ν − RB ν I
I c

in the Coriolis/centripetal and damping


Current

νI
c

o x
y

ob xb
yb

ψ

xb
ob yb

intuitively, the current is pushing the vehicle


Gravity and buoiancy


0
gI f G (RB ) = RB  0 
I I
W
ox  
z 0
ob fb fb Mr B B 
xb rb rb f B (RI ) = −RI 0
zb r B
rg θ fg
fg g
obxb
zb
MR = r G × f G (RB ) + r B × f G (RB )
B
I
B
I

linear in the 3 parameters: W r B − Br B constant in body-fixed
G B


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 fine
positioning to avoid disturbance caused by the controller 2

2
[Antonelli(2007)]

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 fine
positioning to avoid disturbance caused by the controller 2

νI νI current
c c
compensation
during a 90◦
rotation

inertial body-fixed
2
[Antonelli(2007)]

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
 
ν1
˙
η ee1 I
˙
η ee = = J w (RB , q)ζ ζ= ν 2  system velocities
˙
η ee2 ❍
❅ ❍ q˙
❘
❅ ❍❍
end-effector velocities ❍❍
❍❍
η1 ❍❍
❍❍
❥
Jacobian

η ee
Oi


UVMS dynamics

Dynamics via classical Newton-Euler equations by propagating the
velocities and forces

−ρ∇i g
f i+1 , µi+1

r i−1,B Bi
Oi−1
r i−1,i Oi
r i,C
r i−1,C
Ci
f i , µi
di mi g


UVMS dynamics in matrix form

˙
M (q)ζ + C(q, ζ)ζ + D(q, ζ)ζ + g(q, RI ) = τ
B

formally equal to a ground-fixed 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]

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? Space manipulator literature?
Need to exactly compensate for The assumption of the
the dynamics, not appropriate momentum conservation is not
for the underwater environment valid

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. ﬁne
positioning6 , better to kinematically compensate with the manipulator

6
[Hildebrandt et al.(2009)Hildebrandt, Christensen, Kerdels, Albiez, and Kirchner]

Kinematic control in pills

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

Redundancy may be used to add additional tasks and to handle
singularities

Example for the sole end-eﬀector trajectory
η ee,d ηd , qd τ η, q
IK control

off-line 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 invert
˙
σ = 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 can
correspond large joint velocities



˙
σ = Jζ inverted by solving proper optimization problems
Pseudoinverse
−1
ζ = J †σ = J T J J T
˙ ˙
σ
Transpose-based
ζ = J Tσ
˙
Weighted pseudoinverse
−1
ζ = J † σ = W −1 J T J W −1 J T
W ˙ ˙
σ

Damped Least-Squares
−1
ζ = J T J J T + λ2 I m ˙
σ

need for closed-loop also. . .



Handling several tasks7
Extended Jacobian
Add additional (6 + n) − m constraints

h(η, q) = 0 with associated J h

such that the problem is squared with

˙
σ J
= ζ
0 Jh

7
[Chiaverini et al.(2008)Chiaverini, Oriolo, and Walker]


Augmented Jacobian
An additional task is given

σh = h(η, q) with associated J h

such that the problem is squared with

σ˙ J
= ζ
˙
σh Jh


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
✗✔✘
✛ ✗✔ ✛✘
❘
✚ ✖✕
✖✕✙ ✚✙
✶ ■ ˙
σa
✛✘
✙
˙
σb
✚✙




Task priority redundancy resolution

σh = h(η, q) with associated J h

further projected on the the null space of the higher priority one
†
ζ = J †σ + J h I − J †J
˙ σh − J hJ †σ
˙ ˙



Singularity robust task priority redundancy resolution 8

σ h = h(η, q) with associated J h

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

ζ = J † σ + I − J † J J † σh
˙ h
˙

8
we are talking about algorithmic singularities here. . .


Agility task priority9
Task priority framework to handle both precision and set tasks
Each task is the norm of the corresponding error (i.e., mi = 1)
Recursive constrained least-squares within the set satisfying
higher-priority tasks

AMADEUS
9
[Casalino et al.(2012)Casalino, Zereik, Simetti, Sperind`, and Turetta]
e


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

supervisor
α1 α2 α3

ζ1
behavior a

sensors ζ2 ζ
behavior b

ζ3
behavior c


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.
n
1 qi,max − qi,min
h(q) =
ci (qi,max − qi )(qi − qi,min )
i=1



Drag minimization, m = 1 10

h(q) = D T (q, ζ)W D(q, ζ)

within a second order solution
∂h
˙ ˙ ∂η ∂h
ζ = J † σ − Jζ − k I − J † J
¨ ∂h +
∂q ∂ζ

10
[Sarkar and Podder(2001)]


Manipulability/singularity, m = 1
h(q) = det J J T
(In 11 priorities dynamically swapped between singularity and e.e.)

close to singularity
singularity set

inhibited direction

joints

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

fb
τ2

fg

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


Obstacle avoidance m = 1



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



Sensors conﬁguration variables
Vehicle roll and pitch, m = 2
Misalignment between the camera optical axis and the target line
of 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

−1 (1 − β)I 6 O 6×n
J † = W −1 J T JW −1 J T
W W −1 (β) =
O n×6 βI n

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

ζ = J † (xE,d + K E eE ) + I − J † J W
W ˙ W αi J † ws,i
s,i
i

Only one αi active at once
Need to be complete, distinguishable, consistent and compact
Beyond the dicotomy fuzzy/probability theory very eﬀective in
transferring ideas

14
[Antonelli and Chiaverini(2003)]

Dynamic programming to balance the movement15

Freeze, as a free parameter, the vehicle velocity ν and implement
˙
the agility task priority to the sole manipulator ⇒ q d
˙
Freeze the manipulator velocity q d and then ﬁnd the vehicle
velocity ν d needed for the remaining tasks components not
˙
satisﬁed ⇒ ζ d

ν

νe

15
[Casalino et al.(2012)Casalino, Zereik, Simetti, Sperind`, and Turetta]
e

Acknowledge

Several researchers kindly provided the materials/video (or the
explications...) for this talk
In 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, 2nd
edition, June 2006.

G. Antonelli.
On the use of adaptive/integral actions for 6-degrees-of-freedom control of
autonomous 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 underwater
vehicle-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 and
Systems, pages 2203–2210, Las Vegas, NE, Oct. 2003.

G. Casalino, E. Zereik, E. Simetti, S. Torelli A. Sperind`, and A. Turetta.
e
Agility for underwater ﬂoating manipulation: Task & subsystem priority based
control strategy.
In 2012 IEEE/RSJ International Conference on Intelligent Robots and
Systems, Vilamoura, PT, october 2012.

S. Chiaverini, G. Oriolo, and I. D. Walker.
Springer Handbook of Robotics, chapter Kinematically Redundant
Manipulators, 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 and
Underwater 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 with
minimizing restoring moments.
In Intelligent Robots and Systems, 2008. IROS 2008. IEEE/RSJ International
Conference 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 underwater
manipulators.
In IEEE OCEANS 2009-Europe, pages 1–6, 2009.

O. Khatib.
A uniﬁed approach for motion and force control of robot manipulators: The
operational 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 autonomous
underwater 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 Robotics
and 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 underwater
vehicle-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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