Control problems for floating-base manipulators
Gianluca Antonelli
Universit`a di Cassino e del Lazio Meridionale
antonelli...
Outline
Introduction & variable definition
Inverse Kinematics
A possible kinematic solution: NSB behavioral control
A possi...
Space, aerial and underwater vehicle-manipulators
DLR
Canadian Space Agency
ALIVE
normal robots but
floating base
kinematic...
Cooperation
RedSea project
RedSea project
AMADEUS
ARCAS
Gianluca Antonelli Trondheim, 23 april 2014
Aerial Robotics Cooperative Assembly
Gianluca Antonelli Trondheim, 23 april 2014
Aerial Robotics Cooperative Assembly
ΣoΣe,1 Σe,2
Σv,1 Σv,2
Our task: cooperative control of the bar
Gianluca Antonelli Tro...
Marine Autonomous Robotics for InterventionS
Gianluca Antonelli Trondheim, 23 april 2014
Marine Autonomous Robotics for InterventionS
Our task: cooperative control of the bar (and logo design. . . )
Gianluca Ant...
Floating robots kinematics
Oi
η1
ηee
❅❅❘
end-effector velocities
❍❍
❍❍
❍❍
❍❍
❍❍
❍❍❥
Jacobian
system velocities˙ηee =
˙ηee1...
UVMS dynamics in matrix form
M(q)˙ζ + C(q, ζ)ζ + D(q, ζ)ζ + g(q, RI
B) = τ
formally equal to a ground-fixed industrial man...
Dynamics
Movement of vehicle and manipulator coupled
movement of the vehicle carrying the manipulator
law of conservation ...
Outline
Introduction & variable definition
Inverse Kinematics
A possible kinematic solution: NSB behavioral control
A possi...
Kinematic control in pills -1-
✛
✚
✘
✙
ζ
❘
✛
✚
✘
✙
˙σ
Starting from a generic m-dimensional task (e.g., the e.e. position)...
Kinematic control in pills -1-
✛
✚
✘
✙
ζ
❘
✛
✚
✘
✙
˙σ
✖✕
✗✔
■
Starting from a generic m-dimensional task (e.g., the e.e. p...
Kinematic control in pills -2-
˙σ = Jζ inverted by solving proper optimization problems
Pseudoinverse
ζ = J†
˙σ = JT
JJT −...
Kinematic control in pills -3-
A robotic system is kinematically redundant when it possesses more
degrees of freedom than ...
Kinematic control in pills -3-
A robotic system is kinematically redundant when it possesses more
degrees of freedom than ...
Kinematic control in pills -3-
A robotic system is kinematically redundant when it possesses more
degrees of freedom than ...
Kinematic control scheme
second. tasks
ηd, qd τ η, q
IK
main task
Control
Output of kinematic control is the variable to b...
A first kinematic solution
Assuming the vehicle in hovering is not the best strategy to e.e. fine
positioning3, better to ki...
Handling several tasks
Extended Jacobian4
Add additional (6 + n) − m constraints
h(η, q) = 0 with associated Jh
such that ...
Handling several tasks -2-
Augmented Jacobian
An additional task is given
σh = h(η, q) with associated Jh
such that the pr...
Handling several tasks -3-
Task priority redundancy resolution
σh = h(η, q) with associated Jh
further projected on the th...
Handling several tasks -4-
Singularity robust task priority redundancy resolution 5
σh = h(η, q) with associated Jh
furthe...
Handling several tasks -5-
AMADEUS
Agility task priority6
Task priority framework to handle both precision and set tasks
E...
Handling several tasks -6-
Behavioral algorithms (behavior=task), bioinspired, artificial
potentials, neuro-fuzzy, cognitiv...
But. . .
What are these tasks we are talking about ?
Gianluca Antonelli Trondheim, 23 april 2014
Tasks to be controlled
Given 6 + n DOFs and m-dimensional tasks: End-effector
position, m = 3
pos./orientation, m = 6
dista...
Tasks to be controlled
Manipulator joint-limits
several approaches proposed, m = 1 to n, e.g.
h(q) =
n
i=1
1
ci
qi,max − q...
Tasks to be controlled
Drag minimization, m = 1 7
h(q) = DT
(q, ζ)W D(q, ζ)
within a second order solution
˙ζ = J†
¨σ − ˙J...
Tasks to be controlled
Manipulability/singularity, m = 1
h(q) = det JJT
(In 8 priorities dynamically swapped between singu...
Tasks to be controlled
Restoring moments:
m = 3 keep close gravity-buoyancy of the overall system 9
m = 2 align gravity an...
Tasks to be controlled
Obstacle avoidance m = 1
Gianluca Antonelli Trondheim, 23 april 2014
Tasks to be controlled
Workspace-related variables
Vehicle distance from the bottom, m = 1
Vehicle distance from the targe...
Tasks to be controlled
Sensors configuration variables
Vehicle roll and pitch, m = 2
Misalignment between the camera optica...
Tasks to be controlled
Visual servoing variables
Features in the image plane 11
11
[Mebarki et al.(2013)Mebarki, Lippiello...
Outline
Introduction & variable definition
Inverse Kinematics
A possible kinematic solution: NSB behavioral control
A possi...
Behavioral control in pills
Inspired from animal behavior
sensors
behavior a
actuators
behavior b
actuators
behavior c
act...
Behavioral control in pills
Inspired from animal behavior
sensors
behavior a
actuators
behavior b
actuators
behavior c
act...
Competitive behavioral control
Behaviors are in competitions and the higher priority can subsume the
lower ones12
sensors
...
Cooperative behavioral control
Behaviors cooperate and the priority is embedded in the gains13
sensors
behavior b
ζ2
α2
be...
NSB
Null Space-based Behavioral control
Each action is decomposed in elementary behaviors/tasks
motion reference command f...
NSB: Merging different tasks
NSB inherits the approach of the singularity-robust task priority
inverse kinematics technique...
NSB: Merging different tasks
NSB inherits the approach of the singularity-robust task priority
inverse kinematics technique...
NSB: Merging different tasks
NSB inherits the approach of the singularity-robust task priority
inverse kinematics technique...
NSB: Merging different tasks
NSB inherits the approach of the singularity-robust task priority
inverse kinematics technique...
NSB: Three-task example
ζa = J†
a ˙σa,d + Λaσ1
ζb = J†
b ˙σb,d + Λbσ2
ζc = J†
c ˙σc,d + Λcσ3
Successive projection approac...
NSB: Three-task example
ζa = J†
a ˙σa,d + Λaσ1
ζb = J†
b ˙σb,d + Λbσ2
ζc = J†
c ˙σc,d + Λcσ3
Successive projection approac...
NSB: Three-task example
ζa = J†
a ˙σa,d + Λaσ1
ζb = J†
b ˙σb,d + Λbσ2
ζc = J†
c ˙σc,d + Λcσ3
Successive projection approac...
From behaviors to actions
sensing/perception
elementary behaviors actions
commands
supervisor
Gianluca Antonelli Trondheim...
Simple comparison: move to goal with obstacle
avoidance
obstacle avoidance
σ1 = p − po ∈ R
σ1,d = d
J1 = ˆrT
∈ R1×2
ˆr =
p...
Simple comparison: competitive approach
Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: competitive approach
❆
❆
❆
❆
❆❯
only move-to-goal
Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: competitive approach
❄
only obstacle avoidance
Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: competitive approach
❄
only move-to-goal
Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: cooperative approach
Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: cooperative approach
❆
❆
❆
❆❯
only move-to-goal
Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: cooperative approach
❇
❇
❇❇◆
linear combination: higher task is corrupted
Gianluca Antonelli Trondheim,...
Simple comparison: cooperative approach
❄
only move-to-goal
Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: NSB
Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: NSB
❆
❆
❆
❆❯
only move-to-goal
Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: NSB
❇
❇
❇◆
null-space-projection: higher task is fulfilled
Gianluca Antonelli Trondheim, 23 april 2014
Simple comparison: NSB
❄
only move-to-goal
Gianluca Antonelli Trondheim, 23 april 2014
Gain tuning
Cooperative
task a b c
situation 1 α1,1 α1,2 α1,3
sit. 2 α2,1 α2,2 α2,3
sit. 3 α3,1 α3,2 α3,3
sit. 4 α4,1 α4,2...
Gain tuning
Cooperative
task a b c d
situation 1 α1,1 α1,2 α1,3 α1,4
sit. 2 α2,1 α2,2 α2,3 α2,4
sit. 3 α3,1 α3,2 α3,3 α3,4...
Gain tuning
Cooperative
task a b c
situation 1 α1,1 α1,2 α1,3
sit. 2 α2,1 α2,2 α2,3
sit. 3 α3,1 α3,2 α3,3
sit. 4 α4,1 α4,2...
Stability analysis
Lyapunov function14
V (˜σ) = 1
2 ˜σT
˜σ > 0 where ˜σ = ˜σT
a ˜σT
b ˜σT
c
T
˙V = −˜σT


Ja
Jb
Jc

 v...
Stability results
˙V < 0 depending on the mutual relationships among the Jacobians:
dependent
ρ(J†
x) + ρ(J†
y) > ρ J†
x J...
(Dis)advantages
PROS
Priorities strictly satisfied
Analitical convergence results
Easiest gain tuning
Clear handling of the...
However. . .
End effector going out of the workspace and one (eventually weighted)
task always leads to singularity
❅
❅
❅❘
...
Balance movement between vehicle and manipulator
Need to distribute the motion e.g.:
move mainly the manipulator when targ...
Fuzzy logic to balance the movement15
Within a weighted pseudoinverse framework
J†
W = W −1
JT
JW −1
JT −1
W −1
(β) =
(1 −...
Fuzzy logic to balance the movement15
Within a weighted pseudoinverse framework
J†
W = W −1
JT
JW −1
JT −1
W −1
(β) =
(1 −...
Fuzzy logic to balance the movement15
Within a weighted pseudoinverse framework
J†
W = W −1
JT
JW −1
JT −1
W −1
(β) =
(1 −...
Dynamic programming to balance the movement16
Freeze, as a free parameter, the vehicle velocity ν and implement
the agilit...
Dynamic programming to balance the movement16
Freeze, as a free parameter, the vehicle velocity ν and implement
the agilit...
Outline
Introduction & variable definition
Inverse Kinematics
A possible kinematic solution: NSB behavioral control
A possi...
Dynamic control
Classical vs modular approaches (both compatible with NSB)
second. tasks
ηd, qd τ η, q
IK
main task
Contro...
Dynamic control
Classical vs modular approaches (both compatible with NSB)
second. tasks
ηd, qd τ η, q
IK
main task
Contro...
Dynamic control
Classical vs modular approaches (both compatible with NSB)
second. tasks
ηd, qd τ η, q
IK
main task
Contro...
Virtual Decomposition in a nutshell
based on Newton-Euler formulation
forward propagation of kinematic
errors assuming 6 D...
VD: forward propagation, from base to e.e.
6-DOF kinematic
errors ˜νi ∈ R6
each link considered
as a 6 DOFs body
νi
νi+1
˙...
VD: forward propagation, from base to e.e.
6-DOF kinematic
errors ˜νi ∈ R6
each link considered
as a 6 DOFs body
νi
νi+1
˙...
VD: forward propagation, from base to e.e.
6-DOF kinematic
errors ˜νi ∈ R6
each link considered
as a 6 DOFs body
νi
νi+1
˙...
VD: backward propagation, from e.e. to base
6-DOF generalized
control forces
hc ∈ R6 hi
hi+1
τi
propagation
backward
hi
c,...
VD: backward propagation, from e.e. to base
6-DOF generalized
control forces
hc ∈ R6 hi
hi+1
τi
propagation
backward
hi
c,...
VD: backward propagation, from e.e. to base
6-DOF generalized
control forces
hc ∈ R6 hi
hi+1
τi
propagation
backward
hi
c,...
VD: backward propagation, from e.e. to base
6-DOF generalized
control forces
hc ∈ R6 hi
hi+1
τi
propagation
backward
hi
c,...
VD: cooperative control
1 Forward propagation until the object
2 Object control forces for the movement+internal
3 Forces ...
Outline
Introduction & variable definition
Inverse Kinematics
A possible kinematic solution: NSB behavioral control
A possi...
Numerical simulation: underwater
6-DOF vehicle + 6-DOF manipulator
Reach a pre-grasp configuration in terms of end-effector ...
Numerical simulation: underwater
6-DOF vehicle + 7-DOF manipulator
Cameraman action: keep the object in the field of view
i...
Numerical simulation:
cooperative transportation of a bar
The final objective for ARCAS
kinematics-only
no perception probl...
First HW tests
Tests made in Seville, November 2013
vehicle controller not implemented
task: e.e. position+orientation
sim...
Additional HW tests
Tests made in Seville, February 2014
vehicle controller to be tuned/improved
vehicle obstacle avoidanc...
Outline
Introduction & variable definition
Inverse Kinematics
A possible kinematic solution: NSB behavioral control
A possi...
Perspectives: the underactuated case
Motivated by quadrotors control
Roll and pitch functional to the horizontal
movement ...
Perspectives: Null base reaction forces
Dynamic model
Mv M12
MT
12 Mq
˙ν
¨q
+
fc,v
fc,q
+
fg,v
fg,q
=
τv
τq
for the sole v...
Perspectives: arm’s motors not controllable in torque
Several manipulators on the market equipped with position/velocity
f...
Perspectives: 2-arm-equipped vehicle
Proposal to be submitted in few hours to H2020-ICT23
Fada-Catec
Gianluca Antonelli Tr...
Thanks for the attention
The presented results are the outcome of the work of several
colleagues from the University of Ca...
Bibliography I
G. Antonelli.
Stability analysis for prioritized closed-loop inverse kinematic algorithms for
redundant rob...
Bibliography II
R.A. Brooks.
A robust layered control system for a mobile robot.
IEEE Journal of Robotics and Automation, ...
Bibliography III
P. Chiacchio, S. Chiaverini, L. Sciavicco, and B. Siciliano.
Closed-loop inverse kinematics schemes for c...
Bibliography IV
T.I. Fossen.
Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and
Underwater Vehicl...
Bibliography V
J. Kim, G. Marani, WK Chung, and J. Yuh.
Kinematic singularity avoidance for autonomous manipulation in und...
Bibliography VI
R. Mebarki and V. Lippiello.
Image-based control for aerial manipulation.
Asian Journal of Control, in pre...
Bibliography VII
B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo.
Robotics: modelling, planning and control.
Springe...
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Control problems for floating-base manipulators

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Control problems for floating-base manipulators

  1. 1. Control problems for floating-base manipulators Gianluca Antonelli Universit`a di Cassino e del Lazio Meridionale antonelli@unicas.it http://webuser.unicas.it/lai/robotica http://www.eng.docente.unicas.it/gianluca antonelli Trondheim, 23 April 2014 Gianluca Antonelli Trondheim, 23 april 2014
  2. 2. Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
  3. 3. Space, aerial and underwater vehicle-manipulators DLR Canadian Space Agency ALIVE normal robots but floating base kinematic coupling dynamic coupling unstructured environment Gianluca Antonelli Trondheim, 23 april 2014
  4. 4. Cooperation RedSea project RedSea project AMADEUS ARCAS Gianluca Antonelli Trondheim, 23 april 2014
  5. 5. Aerial Robotics Cooperative Assembly Gianluca Antonelli Trondheim, 23 april 2014
  6. 6. Aerial Robotics Cooperative Assembly ΣoΣe,1 Σe,2 Σv,1 Σv,2 Our task: cooperative control of the bar Gianluca Antonelli Trondheim, 23 april 2014
  7. 7. Marine Autonomous Robotics for InterventionS Gianluca Antonelli Trondheim, 23 april 2014
  8. 8. Marine Autonomous Robotics for InterventionS Our task: cooperative control of the bar (and logo design. . . ) Gianluca Antonelli Trondheim, 23 april 2014
  9. 9. Floating robots kinematics Oi η1 ηee ❅❅❘ end-effector velocities ❍❍ ❍❍ ❍❍ ❍❍ ❍❍ ❍❍❥ Jacobian system velocities˙ηee = ˙ηee1 ˙ηee2 = J(RI B, q)ζ ζ =   ν1 ν2 ˙q   Gianluca Antonelli Trondheim, 23 april 2014
  10. 10. UVMS dynamics in matrix form M(q)˙ζ + C(q, ζ)ζ + D(q, ζ)ζ + g(q, RI B) = τ formally equal to a ground-fixed industrial manipulator 1 however. . . Model knowledge Bandwidth of the sensor’s readings Vehicle hovering control Dynamic coupling between vehicle and manipulator External disturbances (current) Kinematic redundancy of the system 1 [Siciliano et al.(2009)Siciliano, Sciavicco, Villani, and Oriolo] [Fossen(2002)] [Schjølberg and Fossen(1994)] Gianluca Antonelli Trondheim, 23 april 2014
  11. 11. 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 2 2 [McLain et al.(1996b)McLain, Rock, and Lee] [McLain et al.(1996a)McLain, Rock, and Lee] Gianluca Antonelli Trondheim, 23 april 2014
  12. 12. Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
  13. 13. Kinematic control in pills -1- ✛ ✚ ✘ ✙ ζ ❘ ✛ ✚ ✘ ✙ ˙σ Starting from a generic m-dimensional task (e.g., the e.e. position) σ = f(η, q) ∈ Rm ˙σ = J(η, q)ζ An inverse mapping is required Gianluca Antonelli Trondheim, 23 april 2014
  14. 14. Kinematic control in pills -1- ✛ ✚ ✘ ✙ ζ ❘ ✛ ✚ ✘ ✙ ˙σ ✖✕ ✗✔ ■ Starting from a generic m-dimensional task (e.g., the e.e. position) σ = f(η, q) ∈ Rm ˙σ = J(η, q)ζ An inverse mapping is required Gianluca Antonelli Trondheim, 23 april 2014
  15. 15. Kinematic control in pills -2- ˙σ = Jζ inverted by solving proper optimization problems Pseudoinverse ζ = J† ˙σ = JT JJT −1 ˙σ Transpose-based ζ = JT ˙σ Weighted pseudoinverse ζ = J† W ˙σ = W −1 JT JW −1 JT −1 ˙σ Damped Least-Squares ζ = JT JJT + λ2 Im −1 ˙σ need for closed-loop also. . . Gianluca Antonelli Trondheim, 23 april 2014
  16. 16. Kinematic control in pills -3- A robotic system is kinematically redundant when it possesses more degrees of freedom than those required to execute a given task Gianluca Antonelli Trondheim, 23 april 2014
  17. 17. Kinematic control in pills -3- 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 ✛ ✚ ✘ ✙ ζ ❘ ✛ ✚ ✘ ✙ ˙σ ✖✕ ✗✔ ■ Gianluca Antonelli Trondheim, 23 april 2014
  18. 18. Kinematic control in pills -3- 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 ✛ ✚ ✘ ✙ ζ ❘ ✛ ✚ ✘ ✙ ˙σ ✖✕ ✗✔ ■ ˙σa ✚✙ ✛✘ ˙σb ✙ ✖✕ ✗✔ ✶ Gianluca Antonelli Trondheim, 23 april 2014
  19. 19. Kinematic control scheme second. tasks ηd, qd τ η, q IK main task Control Output of kinematic control is the variable to be controlled by the actuators (vehicle thrusters and joints’ torques) Gianluca Antonelli Trondheim, 23 april 2014
  20. 20. A first kinematic solution Assuming the vehicle in hovering is not the best strategy to e.e. fine positioning3, better to kinematically compensate with the manipulator 3 [Hildebrandt et al.(2009)Hildebrandt, Christensen, Kerdels, Albiez, and Kirchner] Gianluca Antonelli Trondheim, 23 april 2014
  21. 21. Handling several tasks Extended Jacobian4 Add additional (6 + n) − m constraints h(η, q) = 0 with associated Jh such that the problem is squared with ˙σ 0 = J Jh ζ 4 [Chiaverini et al.(2008)Chiaverini, Oriolo, and Walker] Gianluca Antonelli Trondheim, 23 april 2014
  22. 22. Handling several tasks -2- Augmented Jacobian An additional task is given σh = h(η, q) with associated Jh such that the problem is squared with ˙σ ˙σh = J Jh ζ Gianluca Antonelli Trondheim, 23 april 2014
  23. 23. Handling several tasks -3- Task priority redundancy resolution σh = h(η, q) with associated Jh further projected on the the null space of the higher priority one ζ = J† ˙σ + Jh I − J† J † ˙σh − JhJ† ˙σ Gianluca Antonelli Trondheim, 23 april 2014
  24. 24. Handling several tasks -4- Singularity robust task priority redundancy resolution 5 σh = h(η, q) with associated Jh further projected on the the null space of the higher priority one ζ = J† ˙σ + I − J† J J† h ˙σh 5 we are talking about algorithmic singularities here. . . [Chiaverini(1997)] Gianluca Antonelli Trondheim, 23 april 2014
  25. 25. Handling several tasks -5- AMADEUS Agility task priority6 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 6 [Casalino et al.(2012)Casalino, Zereik, Simetti, Sperind`e, and Turetta] Gianluca Antonelli Trondheim, 23 april 2014
  26. 26. Handling several tasks -6- Behavioral algorithms (behavior=task), bioinspired, artificial potentials, neuro-fuzzy, cognitive approaches, etc. btw. . . mood ? Gianluca Antonelli Trondheim, 23 april 2014
  27. 27. But. . . What are these tasks we are talking about ? Gianluca Antonelli Trondheim, 23 april 2014
  28. 28. 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 Gianluca Antonelli Trondheim, 23 april 2014
  29. 29. Tasks to be controlled 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) Gianluca Antonelli Trondheim, 23 april 2014
  30. 30. Tasks to be controlled Drag minimization, m = 1 7 h(q) = DT (q, ζ)W D(q, ζ) within a second order solution ˙ζ = J† ¨σ − ˙Jζ − k I − J† J ∂h ∂η ∂h ∂q + ∂h ∂ζ 7 [Sarkar and Podder(2001)] Gianluca Antonelli Trondheim, 23 april 2014
  31. 31. Tasks to be controlled Manipulability/singularity, m = 1 h(q) = det JJT (In 8 priorities dynamically swapped between singularity and e.e.) joints inhibited direction singularity singularity setclose to 8 [Kim et al.(2002)Kim, Marani, Chung, and Yuh, Casalino and Turetta(2003)] [Chiacchio et al.(1991)Chiacchio, Chiaverini, Sciavicco, and Gianluca Antonelli Trondheim, 23 april 2014
  32. 32. Tasks to be controlled Restoring moments: m = 3 keep close gravity-buoyancy of the overall system 9 m = 2 align gravity and buoyancy (SAUVIM is 4 tons) 10 fb fg τ 2 9 [Han and Chung(2008)] 10 [Marani et al.(2010)Marani, Choi, and Yuh] Gianluca Antonelli Trondheim, 23 april 2014
  33. 33. Tasks to be controlled Obstacle avoidance m = 1 Gianluca Antonelli Trondheim, 23 april 2014
  34. 34. Tasks to be controlled Workspace-related variables Vehicle distance from the bottom, m = 1 Vehicle distance from the target, m = 1 Gianluca Antonelli Trondheim, 23 april 2014
  35. 35. Tasks to be controlled Sensors configuration variables Vehicle roll and pitch, m = 2 Misalignment between the camera optical axis and the target line of sight, m = 2 Gianluca Antonelli Trondheim, 23 april 2014
  36. 36. Tasks to be controlled Visual servoing variables Features in the image plane 11 11 [Mebarki et al.(2013)Mebarki, Lippiello, and Siciliano, Mebarki and Lippiello(in press, 2014)] Gianluca Antonelli Trondheim, 23 april 2014
  37. 37. Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
  38. 38. Behavioral control in pills Inspired from animal behavior sensors behavior a actuators behavior b actuators behavior c actuators How to combine them in one single behavior? Gianluca Antonelli Trondheim, 23 april 2014
  39. 39. Behavioral control in pills Inspired from animal behavior sensors behavior a actuators behavior b actuators behavior c actuators How to combine them in one single behavior? Gianluca Antonelli Trondheim, 23 april 2014
  40. 40. Competitive behavioral control Behaviors are in competitions and the higher priority can subsume the lower ones12 sensors behavior b ζ2 behavior a ζ1 behavior c ζ3 ζd 12 [Brooks(1986)] Gianluca Antonelli Trondheim, 23 april 2014
  41. 41. Cooperative behavioral control Behaviors cooperate and the priority is embedded in the gains13 sensors behavior b ζ2 α2 behavior a ζ1 supervisor α1 behavior c ζ3 α3 ζd 13 [Arkin(1989)] Gianluca Antonelli Trondheim, 23 april 2014
  42. 42. NSB Null Space-based Behavioral control Each action is decomposed in elementary behaviors/tasks motion reference command for each task ζd = J† ˙σd + Λσ σ = σd−σ Gianluca Antonelli Trondheim, 23 april 2014
  43. 43. NSB: Merging different tasks NSB inherits the approach of the singularity-robust task priority inverse kinematics technique ζd = J† a ˙σa,d + Λaσa + J† b ˙σb,d + Λbσb primary secondary Thus, defining: ζa = J† a ˙σa,d + Λaσa Na = I − J† aJa ζb = J† b ˙σb,d + Λbσb Gianluca Antonelli Trondheim, 23 april 2014
  44. 44. NSB: Merging different tasks NSB inherits the approach of the singularity-robust task priority inverse kinematics technique ζd = J† a ˙σa,d + Λaσa + I − J† aJa J† b ˙σb,d + Λbσb primary null space secondary Thus, defining: ζa = J† a ˙σa,d + Λaσa Na = I − J† aJa ζb = J† b ˙σb,d + Λbσb Gianluca Antonelli Trondheim, 23 april 2014
  45. 45. NSB: Merging different tasks NSB inherits the approach of the singularity-robust task priority inverse kinematics technique ζd = J† a ˙σa,d + Λaσa + I − J† aJa J† b ˙σb,d + Λbσb primary null space secondary Thus, defining: ζa = J† a ˙σa,d + Λaσa Na = I − J† aJa ζb = J† b ˙σb,d + Λbσb Gianluca Antonelli Trondheim, 23 april 2014
  46. 46. NSB: Merging different tasks NSB inherits the approach of the singularity-robust task priority inverse kinematics technique ζd = J† a ˙σa,d + Λaσa + I − J† aJa J† b ˙σb,d + Λbσb primary null space secondary Thus, defining: ζa = J† a ˙σa,d + Λaσa Na = I − J† aJa ζb = J† b ˙σb,d + Λbσb ζd = ζa + Naζb Gianluca Antonelli Trondheim, 23 april 2014
  47. 47. NSB: Three-task example ζa = J† a ˙σa,d + Λaσ1 ζb = J† b ˙σb,d + Λbσ2 ζc = J† c ˙σc,d + Λcσ3 Successive projection approach Na = I − J† aJa Nb = I − J† bJb ζd = ζa + Naζb + NaNbζc Augmented projection approach Jab = Ja Jb Nab = In − J† abJab ζd = ζa + Naζb+Nabζc Gianluca Antonelli Trondheim, 23 april 2014
  48. 48. NSB: Three-task example ζa = J† a ˙σa,d + Λaσ1 ζb = J† b ˙σb,d + Λbσ2 ζc = J† c ˙σc,d + Λcσ3 Successive projection approach Na = I − J† aJa Nb = I − J† bJb ζd = ζa + Naζb + NaNbζc Augmented projection approach Jab = Ja Jb Nab = In − J† abJab ζd = ζa + Naζb+Nabζc Gianluca Antonelli Trondheim, 23 april 2014
  49. 49. NSB: Three-task example ζa = J† a ˙σa,d + Λaσ1 ζb = J† b ˙σb,d + Λbσ2 ζc = J† c ˙σc,d + Λcσ3 Successive projection approach Na = I − J† aJa Nb = I − J† bJb ζd = ζa + Naζb + NaNbζc Augmented projection approach Jab = Ja Jb Nab = In − J† abJab ζd = ζa + Naζb+Nabζc Gianluca Antonelli Trondheim, 23 april 2014
  50. 50. From behaviors to actions sensing/perception elementary behaviors actions commands supervisor Gianluca Antonelli Trondheim, 23 april 2014
  51. 51. Simple comparison: move to goal with obstacle avoidance obstacle avoidance σ1 = p − po ∈ R σ1,d = d J1 = ˆrT ∈ R1×2 ˆr = p − po p − po ζ1 = J† 1λ1 (d − p−po ) N(J1) = I − J† 1J1 = I − ˆrˆrT move to goal σ2 = p ∈ R2 σ2,d = pg J2 = I ∈ R2×2 ζ2 = Λ2 pg − p Gianluca Antonelli Trondheim, 23 april 2014
  52. 52. Simple comparison: competitive approach Gianluca Antonelli Trondheim, 23 april 2014
  53. 53. Simple comparison: competitive approach ❆ ❆ ❆ ❆ ❆❯ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
  54. 54. Simple comparison: competitive approach ❄ only obstacle avoidance Gianluca Antonelli Trondheim, 23 april 2014
  55. 55. Simple comparison: competitive approach ❄ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
  56. 56. Simple comparison: cooperative approach Gianluca Antonelli Trondheim, 23 april 2014
  57. 57. Simple comparison: cooperative approach ❆ ❆ ❆ ❆❯ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
  58. 58. Simple comparison: cooperative approach ❇ ❇ ❇❇◆ linear combination: higher task is corrupted Gianluca Antonelli Trondheim, 23 april 2014
  59. 59. Simple comparison: cooperative approach ❄ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
  60. 60. Simple comparison: NSB Gianluca Antonelli Trondheim, 23 april 2014
  61. 61. Simple comparison: NSB ❆ ❆ ❆ ❆❯ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
  62. 62. Simple comparison: NSB ❇ ❇ ❇◆ null-space-projection: higher task is fulfilled Gianluca Antonelli Trondheim, 23 april 2014
  63. 63. Simple comparison: NSB ❄ only move-to-goal Gianluca Antonelli Trondheim, 23 april 2014
  64. 64. Gain tuning Cooperative task a b c situation 1 α1,1 α1,2 α1,3 sit. 2 α2,1 α2,2 α2,3 sit. 3 α3,1 α3,2 α3,3 sit. 4 α4,1 α4,2 α4,3 NSB Each behavior tuned as if it was alone but in each situation the priority needs to be designed Gianluca Antonelli Trondheim, 23 april 2014
  65. 65. Gain tuning Cooperative task a b c d situation 1 α1,1 α1,2 α1,3 α1,4 sit. 2 α2,1 α2,2 α2,3 α2,4 sit. 3 α3,1 α3,2 α3,3 α3,4 sit. 4 α4,1 α4,2 α4,3 α4,4 NSB Each behavior tuned as if it was alone but in each situation the priority needs to be designed Gianluca Antonelli Trondheim, 23 april 2014
  66. 66. Gain tuning Cooperative task a b c situation 1 α1,1 α1,2 α1,3 sit. 2 α2,1 α2,2 α2,3 sit. 3 α3,1 α3,2 α3,3 sit. 4 α4,1 α4,2 α4,3 NSB Each behavior tuned as if it was alone but in each situation the priority needs to be designed Gianluca Antonelli Trondheim, 23 april 2014
  67. 67. Stability analysis Lyapunov function14 V (˜σ) = 1 2 ˜σT ˜σ > 0 where ˜σ = ˜σT a ˜σT b ˜σT c T ˙V = −˜σT   Ja Jb Jc   v = −˜σT M ˜σ = −˜σT       Λa Oma,mb Oma,mc JbJ† aΛa JbNaJ† bΛb JbNJ† cΛc JcJ† aΛa JcNaJ† bΛb JcNJ† cΛc       ˜σ 14 [Antonelli(2009)] Gianluca Antonelli Trondheim, 23 april 2014
  68. 68. Stability results ˙V < 0 depending on the mutual relationships among the Jacobians: dependent ρ(J† x) + ρ(J† y) > ρ J† x J† y independent ρ(J† x) + ρ(J† y) = ρ J† x J† y orthogonal JxJ† y = Omx×my Gianluca Antonelli Trondheim, 23 april 2014
  69. 69. (Dis)advantages PROS Priorities strictly satisfied Analitical convergence results Easiest gain tuning Clear handling of the DOFs Competitive and cooperative as particular cases CONS Couples all the behaviors Impedance control not possible Gianluca Antonelli Trondheim, 23 april 2014
  70. 70. However. . . End effector going out of the workspace and one (eventually weighted) task always leads to singularity ❅ ❅ ❅❘ manipulator stretched Gianluca Antonelli Trondheim, 23 april 2014
  71. 71. 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 Gianluca Antonelli Trondheim, 23 april 2014
  72. 72. Fuzzy logic to balance the movement15 Within a weighted pseudoinverse framework J† W = W −1 JT JW −1 JT −1 W −1 (β) = (1 − β)I6 O6×n On×6 βIn with β ∈ [0, 1] output of a fuzzy inference engine Secondary tasks activated by additional fuzzy variables αi ∈ [0, 1] ζ = J† W ( ˙σa + Λa ˜σa) + I − J† W JW i αiJ† s,iws,i Only one αi active at once Need to be complete, distinguishable, consistent and compact Beyond the dichotomy fuzzy/probability theory very effective in transferring ideas 15 [Antonelli and Chiaverini(2003)] Gianluca Antonelli Trondheim, 23 april 2014
  73. 73. Fuzzy logic to balance the movement15 Within a weighted pseudoinverse framework J† W = W −1 JT JW −1 JT −1 W −1 (β) = (1 − β)I6 O6×n On×6 βIn with β ∈ [0, 1] output of a fuzzy inference engine Secondary tasks activated by additional fuzzy variables αi ∈ [0, 1] ζ = J† W ( ˙σa + Λa ˜σa) + I − J† W JW i αiJ† s,iws,i Only one αi active at once Need to be complete, distinguishable, consistent and compact Beyond the dichotomy fuzzy/probability theory very effective in transferring ideas 15 [Antonelli and Chiaverini(2003)] Gianluca Antonelli Trondheim, 23 april 2014
  74. 74. Fuzzy logic to balance the movement15 Within a weighted pseudoinverse framework J† W = W −1 JT JW −1 JT −1 W −1 (β) = (1 − β)I6 O6×n On×6 βIn with β ∈ [0, 1] output of a fuzzy inference engine Secondary tasks activated by additional fuzzy variables αi ∈ [0, 1] ζ = J† W ( ˙σa + Λa ˜σa) + I − J† W JW i αiJ† s,iws,i Only one αi active at once Need to be complete, distinguishable, consistent and compact Beyond the dichotomy fuzzy/probability theory very effective in transferring ideas 15 [Antonelli and Chiaverini(2003)] Gianluca Antonelli Trondheim, 23 april 2014
  75. 75. Dynamic programming to balance the movement16 Freeze, as a free parameter, the vehicle velocity ν and implement the agility task priority to the sole manipulator ⇒ ˙qd Freeze the manipulator velocity ˙qd and then find the vehicle velocity νd needed for the remaining tasks components not satisfied ⇒ ˙ζd ν νe 16 [Casalino et al.(2012)Casalino, Zereik, Simetti, Sperind`e, and Turetta] Gianluca Antonelli Trondheim, 23 april 2014
  76. 76. Dynamic programming to balance the movement16 Freeze, as a free parameter, the vehicle velocity ν and implement the agility task priority to the sole manipulator ⇒ ˙qd Freeze the manipulator velocity ˙qd and then find the vehicle velocity νd needed for the remaining tasks components not satisfied ⇒ ˙ζd ν νe 16 [Casalino et al.(2012)Casalino, Zereik, Simetti, Sperind`e, and Turetta] Gianluca Antonelli Trondheim, 23 april 2014
  77. 77. Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
  78. 78. Dynamic control Classical vs modular approaches (both compatible with NSB) second. tasks ηd, qd τ η, q IK main task Control Classical model-based natural extension of well-known control laws adaptive and robust versions Virtual decomposition modular ⇒ same control law for all rigid bodies adaptive ⇒ integral-like action Gianluca Antonelli Trondheim, 23 april 2014
  79. 79. Dynamic control Classical vs modular approaches (both compatible with NSB) second. tasks ηd, qd τ η, q IK main task Control Classical model-based natural extension of well-known control laws adaptive and robust versions Virtual decomposition modular ⇒ same control law for all rigid bodies adaptive ⇒ integral-like action Gianluca Antonelli Trondheim, 23 april 2014
  80. 80. Dynamic control Classical vs modular approaches (both compatible with NSB) second. tasks ηd, qd τ η, q IK main task Control Classical model-based natural extension of well-known control laws adaptive and robust versions Virtual decomposition modular ⇒ same control law for all rigid bodies adaptive ⇒ integral-like action Gianluca Antonelli Trondheim, 23 april 2014
  81. 81. Virtual Decomposition in a nutshell based on Newton-Euler formulation forward propagation of kinematic errors assuming 6 DOFs ˜ν1 ˜ν2 backward computation and projection of control forces/torques Gianluca Antonelli Trondheim, 23 april 2014
  82. 82. VD: forward propagation, from base to e.e. 6-DOF kinematic errors ˜νi ∈ R6 each link considered as a 6 DOFs body νi νi+1 ˙qi+1 propagation forward νi+1 i+1 = UiT i+1νi i + ˙qi+1zi+1 i Gianluca Antonelli Trondheim, 23 april 2014
  83. 83. VD: forward propagation, from base to e.e. 6-DOF kinematic errors ˜νi ∈ R6 each link considered as a 6 DOFs body νi νi+1 ˙qi+1 propagation forward νi+1 i+1 = UiT i+1νi i + ˙qi+1zi+1 i velocity propagation Gianluca Antonelli Trondheim, 23 april 2014
  84. 84. VD: forward propagation, from base to e.e. 6-DOF kinematic errors ˜νi ∈ R6 each link considered as a 6 DOFs body νi νi+1 ˙qi+1 propagation forward νi+1 i+1 = UiT i+1νi i + ˙qi+1zi+1 i joint contribution Gianluca Antonelli Trondheim, 23 april 2014
  85. 85. VD: backward propagation, from e.e. to base 6-DOF generalized control forces hc ∈ R6 hi hi+1 τi propagation backward hi c,i = Y Ri, νi i, νi r,i, ˙νi r,i ˆθi + Kv,isi i + Ui i+1hi+1 c,i+1 Gianluca Antonelli Trondheim, 23 april 2014
  86. 86. VD: backward propagation, from e.e. to base 6-DOF generalized control forces hc ∈ R6 hi hi+1 τi propagation backward hi c,i = Y Ri, νi i, νi r,i, ˙νi r,i ˆθi + Kv,isi i + Ui i+1hi+1 c,i+1 model-based compensation Gianluca Antonelli Trondheim, 23 april 2014
  87. 87. VD: backward propagation, from e.e. to base 6-DOF generalized control forces hc ∈ R6 hi hi+1 τi propagation backward hi c,i = Y Ri, νi i, νi r,i, ˙νi r,i ˆθi + Kv,isi i + Ui i+1hi+1 c,i+1 feedback term Gianluca Antonelli Trondheim, 23 april 2014
  88. 88. VD: backward propagation, from e.e. to base 6-DOF generalized control forces hc ∈ R6 hi hi+1 τi propagation backward hi c,i = Y Ri, νi i, νi r,i, ˙νi r,i ˆθi + Kv,isi i + Ui i+1hi+1 c,i+1 force propagation Gianluca Antonelli Trondheim, 23 april 2014
  89. 89. VD: cooperative control 1 Forward propagation until the object 2 Object control forces for the movement+internal 3 Forces projected to the link n of the two robots 4 Backward propagation hc,o hc,n+1 hc,n+1 Gianluca Antonelli Trondheim, 23 april 2014
  90. 90. Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
  91. 91. Numerical simulation: underwater 6-DOF vehicle + 6-DOF manipulator Reach a pre-grasp configuration in terms of end-effector position and orientation initial configuration priority-1 task: e.e. configuration priority-2 task: vehicle roll+pitch priority-3 task: position of joint 2 only e.e. ⇒ complete solution ⇒ Gianluca Antonelli Trondheim, 23 april 2014
  92. 92. Numerical simulation: underwater 6-DOF vehicle + 7-DOF manipulator Cameraman action: keep the object in the field of view initial configuration priority-1 task: field of view priority-2 task: vehicle roll+pitch priority-3 task: mechanical joint limits animation ⇒ Gianluca Antonelli Trondheim, 23 april 2014
  93. 93. Numerical simulation: cooperative transportation of a bar The final objective for ARCAS kinematics-only no perception problems switch among actions Gianluca Antonelli Trondheim, 23 april 2014
  94. 94. First HW tests Tests made in Seville, November 2013 vehicle controller not implemented task: e.e. position+orientation simple e.e. hold no priority yet Gianluca Antonelli Trondheim, 23 april 2014
  95. 95. Additional HW tests Tests made in Seville, February 2014 vehicle controller to be tuned/improved vehicle obstacle avoidance e.e. + arm reconfiguration Gianluca Antonelli Trondheim, 23 april 2014
  96. 96. Outline Introduction & variable definition Inverse Kinematics A possible kinematic solution: NSB behavioral control A possible dynamic solution: Virtual decomposition Simulation/experiments Perspectives Gianluca Antonelli Trondheim, 23 april 2014
  97. 97. Perspectives: the underactuated case Motivated by quadrotors control Roll and pitch functional to the horizontal movement ⇒ kinematic disturbances! pitch end effector Uncontrolled DOF affect the tasks Possible to handle in a task-priority approach? Gianluca Antonelli Trondheim, 23 april 2014
  98. 98. Perspectives: Null base reaction forces Dynamic model Mv M12 MT 12 Mq ˙ν ¨q + fc,v fc,q + fg,v fg,q = τv τq for the sole vehicle: Mv ˙ν + M12¨q + fc,v try to zeroing +fg,v = τv, achievable with: ¨q = −M† 12fc,v + N ¨qo yet another null space. . . Gianluca Antonelli Trondheim, 23 april 2014
  99. 99. Perspectives: arm’s motors not controllable in torque Several manipulators on the market equipped with position/velocity feedback control loops Need to design a proper controller for the sole vehicle Partially coupled model Iterative formulation Adaptive Simplest version: only gravity Mv ˙ν + M12¨q + fc,v + fg,v = τv, compensate only for fg,v h 0 0=Y (0)·θ0+U 0 1 Y (1) · θ1 + U 1 2h 2 2 +. . .= Y (0) U0 1Y (1) U0 1U1 2Y (2) . . . U0 1U1 2 . . . Un−1 n Y (n) Y         θ0 θ1 θ2 . . . θn         Gianluca Antonelli Trondheim, 23 april 2014
  100. 100. Perspectives: 2-arm-equipped vehicle Proposal to be submitted in few hours to H2020-ICT23 Fada-Catec Gianluca Antonelli Trondheim, 23 april 2014
  101. 101. Thanks for the attention The presented results are the outcome of the work of several colleagues from the University of Cassino, the Consortium ISME and PRISMA, the projects ARCAS and MARIS Filippo Arrichiello, Khelifa Baizid, Elisabetta Cataldi, Stefano Chiaverini, Amal Meddahi Gianluca Antonelli Trondheim, 23 april 2014
  102. 102. Bibliography I G. Antonelli. Stability analysis for prioritized closed-loop inverse kinematic algorithms for redundant robotic systems. IEEE Transactions on Robotics, 25(5):985–994, October 2009. G. Antonelli. Underwater robots. Springer Tracts in Advanced Robotics, Springer-Verlag, Heidelberg, D, 3rd edition, January 2014. 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. R.C. Arkin. Motor schema based mobile robot navigation. The International Journal of Robotics Research, 8(4):92–112, 1989. Gianluca Antonelli Trondheim, 23 april 2014
  103. 103. Bibliography II R.A. Brooks. A robust layered control system for a mobile robot. IEEE Journal of Robotics and Automation, 2(1):14–23, 1986. 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`e, and A. Turetta. 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. Gianluca Antonelli Trondheim, 23 april 2014
  104. 104. Bibliography III P. Chiacchio, S. Chiaverini, L. Sciavicco, and B. Siciliano. Closed-loop inverse kinematics schemes for constrained redundant manipulators with task space augmentation and task priority strategy. The International Journal Robotics Research, 10(4):410–425, 1991. S. Chiaverini. Singularity-robust task-priority redundancy resolution for real-time kinematic control of robot manipulators. IEEE Transactions on Robotics and Automation, 13(3):398–410, 1997. 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. Gianluca Antonelli Trondheim, 23 april 2014
  105. 105. Bibliography IV T.I. Fossen. Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics, Trondheim, Norway, 2002. 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. 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. Gianluca Antonelli Trondheim, 23 april 2014
  106. 106. Bibliography V 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. 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. Gianluca Antonelli Trondheim, 23 april 2014
  107. 107. Bibliography VI R. Mebarki and V. Lippiello. Image-based control for aerial manipulation. Asian Journal of Control, in press, 2014. R. Mebarki, V. Lippiello, and B. Siciliano. Exploiting image moments for aerial manipulation control. In ASME Dynamic Systems and Control Conference, Palo Alto, CA, USA, 2013. 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. I. Schjølberg and T. Fossen. Modelling and control of underwater vehicle-manipulator systems. In in Proc. 3rd Conf. on Marine Craft maneuvering and control, pages 45–57, Southampton, UK, 1994. Gianluca Antonelli Trondheim, 23 april 2014
  108. 108. Bibliography VII B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo. Robotics: modelling, planning and control. Springer Verlag, 2009. Gianluca Antonelli Trondheim, 23 april 2014
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