underwater manipulation
TRANSCRIPT
Underwater manipulation
Gianluca Antonelli
Universita di Cassino & ISME
http://webuser.unicas.it/lai/robotica
http://www.isme.unige.it
http://www.eng.docente.unicas.it/gianluca antonelli
Gianluca Antonelli BiogradNaMoru, 8 October 2015
ISME in brief
Italian Joint Research Unit established in 1999
Sites:
AnconaCassinoFirenzeGenovaLeccePisa
Gianluca Antonelli BiogradNaMoru, 8 October 2015
ISME in brief
SEA Lab
Joint Italian Navy/ISME located in La Spezia
No need of advance area clearance
Availability of Navy support personnel
Some restrictions (activities/personnel to be listed in advance, no working at nights. . . )
Gianluca Antonelli BiogradNaMoru, 8 October 2015
ISME in brief
A selected map of projects logos. . .
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Marine Autonomous Robotics for InterventionS
PRIN20102011
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Marine Autonomous Robotics for InterventionS
PRIN20102011
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Effective Dexterous ROV Operations in Presence of
Communications Latencies
H2020BG2014
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Effective Dexterous ROV Operations in Presence of
Communications Latencies
H2020BG2014
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Robotic subsea exploration technologies
H2020SC52014
mineral and raw material exploration and recovery (in negotiation)
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Outline
Motivation
Inverse Kinematics
A possible kinematic solution: NSB behavioral control
Simulation/experiments
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Applications
Where uw manipulation is used/needed:
Oil & gas industry
Renewable energy
Power/communication cables
Fisheries & aquaculture
Archaeology
Security
Natural science/biology
Decommissioning
Diver assistance
Gianluca Antonelli BiogradNaMoru, 8 October 2015
State of the art
aged approach
off-shore operator acts on the vehicleoff-shore operator acts on the arm motors (!)voice coordination between the twomanned visual feedback
Gianluca Antonelli BiogradNaMoru, 8 October 2015
State of the art
Recent approach
vehicle in automatic station keeping or dockedoff-shore operator with a master/slave architecture
Gianluca Antonelli BiogradNaMoru, 8 October 2015
State of the art
Effort for working class vehicles
13 peoples on 24 hours4 to 6 weeksROV crew work 12 hours a day - 7/71 day of operation costs 100÷ 300 ke
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Objective
Autonomously (as much ass possible. . . ) achieve complex operations
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Space, aerial and underwater vehicle-manipulators
DLRCanadian Space Agency
ALIVE
normal robots but
floating base
kinematic coupling
dynamic coupling
unstructured environment
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Floating robots kinematics
Oi
η1
ηee
❅❅❘endeffector velocities
❍❍❍❍❍❍❍❍❍❍❍❍❥Jacobian
system velocitiesηee =
[ηee1
ηee2
]= J(RI
B, q)ζ ζ =
ν1
ν2
q
Gianluca Antonelli BiogradNaMoru, 8 October 2015
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 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 BiogradNaMoru, 8 October 2015
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 BiogradNaMoru, 8 October 2015
A first solution
Assuming the vehicle in hovering is not the best strategy to e.e. finepositioning3, better to kinematically compensate with the manipulator
3[Hildebrandt et al.(2009)Hildebrandt, Christensen, Kerdels, Albiez, and Kirchner]Gianluca Antonelli BiogradNaMoru, 8 October 2015
Outline
Motivation
Inverse Kinematics
A possible kinematic solution: NSB behavioral control
Simulation/experiments
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Kinematic control scheme
second. tasks
ηd, qd τ η, q
IKmain task
Control
Output of IK (Inverse Kinematics) is the position/velocity to becontrolled by the actuators (vehicle thrusters and joints’ torques)
Btw, torque level usually not available ⇒ kinematic controller
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Kinematic control in pills
✛✚
✘✙
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 BiogradNaMoru, 8 October 2015
Kinematic control in pills
✛✚
✘✙
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 BiogradNaMoru, 8 October 2015
Kinematic control in pills
A robotic system is kinematically redundant when it possesses moredegrees of freedom than those required to execute a given task
Gianluca Antonelli BiogradNaMoru, 8 October 2015
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
✛✚
✘✙
ζ
❘
✛✚
✘✙
σ
✖✕✗✔
■
Gianluca Antonelli BiogradNaMoru, 8 October 2015
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
✛✚
✘✙
ζ
❘
✛✚
✘✙
σ
✖✕✗✔
■ σa
✚✙✛✘
σb
✙
✖✕✗✔
✶
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Kinematic control in pills
Classical example: control e.e. position while reconfiguring thestructure with internal motion
Kuka Iiwa
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Kinematic control in pills
In the redundant case, the equation
σ = Jζ
is solved by
ζ = JT(JJT
)−1
︸ ︷︷ ︸J
†
σ +(I − J †J
)
︸ ︷︷ ︸N
ζo
i.e., by a pseudoinverse and an arbitrary vector projected onto thenull-space
need for closedloop also. . .
Gianluca Antonelli BiogradNaMoru, 8 October 2015
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 BiogradNaMoru, 8 October 2015
Handling several tasks
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
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Handling several tasks
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)
Also known as the exact solution with close similarities to the
convexoptimizationbased methods
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Handling several tasks
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†
hσh
5algorithmic singularities here. . . [Chiaverini(1997)]Gianluca Antonelli BiogradNaMoru, 8 October 2015
Handling several tasks
Behavioral algorithms (behavior=task), bioinspired, artificialpotentials, neuro-fuzzy, cognitive approaches, etc.
btw. . . mood ?
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Geometrical meaning of the null-space
σ = Jζ with m = 1 and n = 2 is a line! (left)
Range of the pseudoinverse and the null spaces are orthogonal (right)
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Comparison between exact and robust solutions
ζ = J†σ +
[
Jh
(
I − J†J)]† (
σh − JhJ†σ
)
ζ = J†σ +
(
I − J†J)
J†
hσh
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Comparison between exact and robust solutions
ζ = J†σ +
[
Jh
(
I − J†J)]† (
σh − JhJ†σ
)
ζ = J†σ +
(
I − J†J)
J†
hσh
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Comparison between exact and robust solutions
ζ = J†σ +
[
Jh
(
I − J†J)]† (
σh − JhJ†σ
)
ζ = J†σ +
(
I − J†J)
J†
hσh
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Some issues
Kinematic singularities
Damped least squareSingular-value-decomposition-based filteringOther kind of filtering
Algorithmic singularities
Two different-priority tasks are achievable alone but not together:
ranks of both J and Jh is full but not of
[J
Jh
](still the
inversion of a singular matrix)
Set-based/inequality control 6
Task transition vs continuity/priority
6[Escande et al.(2013)Escande, Mansard, and Wieber,Simetti et al.(2013)Simetti, Casalino, Torelli, Sperinde, and Turetta,Antonelli et al.(2015)Antonelli, Moe, and Pettersen]
Gianluca Antonelli BiogradNaMoru, 8 October 2015
But. . .
What are these tasks we are talking about ?
Gianluca Antonelli BiogradNaMoru, 8 October 2015
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 BiogradNaMoru, 8 October 2015
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 BiogradNaMoru, 8 October 2015
Tasks to be controlled
Drag minimization, m = 1 7
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
)
7[Sarkar and Podder(2001)]Gianluca Antonelli BiogradNaMoru, 8 October 2015
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
singularitysingularity
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 BiogradNaMoru, 8 October 2015
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
f b
f g
τ 2
9[Han and Chung(2008)]10[Marani et al.(2010)Marani, Choi, and Yuh]
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Tasks to be controlled
Workspace-related variablesVehicle distance from the bottom, m = 1Vehicle distance from the target, m = 1
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Tasks to be controlled
Sensors configuration variables
Vehicle roll and pitch, m = 2Misalignment between the camera optical axis and the target lineof sight, m = 2
Gianluca Antonelli BiogradNaMoru, 8 October 2015
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 BiogradNaMoru, 8 October 2015
Outline
Motivation
Inverse Kinematics
A possible kinematic solution: NSB behavioral control
Simulation/experiments
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Behavioral control in pills
Inspired from animal behavior
sensorsbehavior a
actuators
behavior bactuators
behavior cactuators
How to combine them in one single behavior?
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Behavioral control in pills
Inspired from animal behavior
sensorsbehavior a
actuators
behavior bactuators
behavior cactuators
How to combine them in one single behavior?
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Competitive behavioral control
Behaviors are in competitions and the higher priority can subsume thelower ones12
sensorsbehavior b
ζ2
behavior a
ζ1
behavior c
ζ3 ζd
12[Brooks(1986)]Gianluca Antonelli BiogradNaMoru, 8 October 2015
Cooperative behavioral control
Behaviors cooperate and the priority is embedded in the gains13
sensorsbehavior b
ζ2 ⊗
α2
behavior a
ζ1
supervisor
⊗
α1
behavior c
ζ3 ⊗
α3
∑ ζd
13[Arkin(1989)]Gianluca Antonelli BiogradNaMoru, 8 October 2015
Competitive-cooperative and tasks conflicting
Cooperative always owns an error, can we inherit the benefit ofGianluca Antonelli BiogradNaMoru, 8 October 2015
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 BiogradNaMoru, 8 October 2015
NSB: Merging different tasks
NSB inherits the approach of the singularity-robust task priorityinverse 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 BiogradNaMoru, 8 October 2015
NSB: Merging different tasks
NSB inherits the approach of the singularity-robust task priorityinverse 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 BiogradNaMoru, 8 October 2015
NSB: Merging different tasks
NSB inherits the approach of the singularity-robust task priorityinverse 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 BiogradNaMoru, 8 October 2015
NSB: Merging different tasks
NSB inherits the approach of the singularity-robust task priorityinverse 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 BiogradNaMoru, 8 October 2015
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
)
N b =(I − J
†bJ b
)
ζd = ζa +Naζb +NaN bζc
Augmented projection approach
Jab =
[Ja
J b
]
Nab =(In − J
†abJab
)
ζd = ζa +Naζb+Nabζc
Gianluca Antonelli BiogradNaMoru, 8 October 2015
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
)
N b =(I − J
†bJ b
)
ζd = ζa +Naζb +NaN bζc
Augmented projection approach
Jab =
[Ja
J b
]
Nab =(In − J
†abJab
)
ζd = ζa +Naζb+Nabζc
Gianluca Antonelli BiogradNaMoru, 8 October 2015
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
)
N b =(I − J
†bJ b
)
ζd = ζa +Naζb +NaN bζc
Augmented projection approach
Jab =
[Ja
J b
]
Nab =(In − J
†abJab
)
ζd = ζa +Naζb+Nabζc
Gianluca Antonelli BiogradNaMoru, 8 October 2015
From behaviors to actions
sensing/perception
elementary behaviors actions
commands
supervisor
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Simple comparison: move to goal with obstacle
avoidance
obstacle avoidance
σ1 = ‖p− po‖ ∈ R
σ1,d = d
J1 = rT ∈ R1×2
r =p− po
‖p− po‖
ζ1 = J†1λ1 (d− ‖p−po‖)
N (J1) = I − J†1J1 = I − rrT
move to goal
σ2 = p ∈ R2
σ2,d = pg
J2 = I ∈ R2×2
ζ2 = Λ2
(pg − p
)
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Simple comparison: competitive approach
❆❆❆❆❆❯
only movetogoal
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Simple comparison: competitive approach
❄
only obstacle avoidance
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Simple comparison: competitive approach
❄
only movetogoal
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Simple comparison: cooperative approach
❆❆❆❆❯
only movetogoal
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Simple comparison: cooperative approach
❇❇❇❇◆
linear combination: higher task is corrupted
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Simple comparison: cooperative approach
❄
only movetogoal
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Simple comparison: NSB
❇❇❇◆
nullspaceprojection: higher task is fulfilled
Gianluca Antonelli BiogradNaMoru, 8 October 2015
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 butin each situation the priority needs to be designed
Gianluca Antonelli BiogradNaMoru, 8 October 2015
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 butin each situation the priority needs to be designed
Gianluca Antonelli BiogradNaMoru, 8 October 2015
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 butin each situation the priority needs to be designed
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Stability analysis
Lyapunov function14
V (σ) = 1
2σTσ > 0 where σ =
[σT
a σT
b σT
c
]T
V = −σT
Ja
Jb
Jc
v = −σTMσ = −σ
T
Λa Oma,mbOma,mc
JbJ†aΛa JbNaJ
†bΛb JbNJ
†cΛc
JcJ†aΛa JcNaJ
†bΛb JcNJ
†cΛc
σ
V < 0 depending on the mutual relationships among the Jacobians
14[Antonelli(2009)]Gianluca Antonelli BiogradNaMoru, 8 October 2015
Outline
Motivation
Inverse Kinematics
A possible kinematic solution: NSB behavioral control
Simulation/experiments
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Numerical simulation on MARIS model:
underwater 6-DOF vehicle + 7-DOF manipulator
Reach a pre-grasp configuration in terms of end-effector position andorientation
priority-1 task: e.e. configuration (m = 6)
priority-2 task: vehicle roll+pitch (m = 2)
priority-3 task: position of joint 2 (m = 1)
only e.e. ⇒
complete solution ⇒
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Numerical simulation on MARIS model:
underwater 6-DOF vehicle + 7-DOF manipulator
Cameraman action: keep the object in the field of view
priority-1 task: field of view (m = 2)
priority-2 task: vehicle roll+pitch (m = 2)
priority-3 task: arm manipulability (m = 1)
priority-4 task: mechanical joint limits (m = 7)
animation ⇒
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Simulations and experiments within TRIDENT
[Simetti et al.(2013)Simetti, Casalino, Torelli, Sperinde, and Turetta]
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Numerical simulation on MARIS model: interaction
within the task-priority approach
An impedance external loop is designed to push a button
Σ0
ΣI
Σee
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Numerical simulation on MARIS model: interaction
within the task-priority approach
An impedance external loop is designed to turn a valve
Σ0
ΣI
Σee
have a look at the experiments made by Pedro Sanz, Pere Ridao
and colleagues within TRIDENT
Gianluca Antonelli BiogradNaMoru, 8 October 2015
The presented results are the outcome of the work of several
colleagues from the University of Cassino, the Consortium ISME
and PRISMA, the projects DEXROV and MARIS
Filippo Arrichiello, Elisabetta Cataldi, Stefano Chiaverini, Paolo Di Lillo
ISME PRISMA
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Bibliography I
G. Antonelli.
Stability analysis for prioritized closed-loop inverse kinematic algorithms forredundant 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, 3rdedition, January 2014.
G. Antonelli, S. Moe, and K. Pettersen.
Incorporating set-based control within the singularity-robust multipletask-priority inverse kinematics.
In 23th Mediterranean Conference on Control and Automation, pages1132–1137, Torremolinos, S, June 2015.
Gianluca Antonelli BiogradNaMoru, 8 October 2015
Bibliography II
R.C. Arkin.
Motor schema based mobile robot navigation.
The International Journal of Robotics Research, 8(4):92–112, 1989.
R.A. Brooks.
A robust layered control system for a mobile robot.
IEEE Journal of Robotics and Automation, 2(1):14–23, 1986.
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