Control problems for floating-base manipulators

Control problems for floating-base manipulators

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

Universita di Palermo, 27 February 2014

Gianluca Antonelli Palermo, 27 february 2014

mailto:[email protected]

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

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

Outline

Introduction

Modeling

Inverse Kinematics

A possible kinematic/behavioral solution: NSB

A possible dynamic solution: Virtual decomposition

Simulation/experiments

Perspectives


Space, aerial and underwater vehicle-manipulators

DLRCanadian Space Agency

ALIVE

normal robots but

floating base

kinematic coupling

dynamic coupling

unstructured environment


Cooperation

RedSea project

RedSea project

AMADEUS

ARCAS


Aerial Robotics Cooperative Assembly


Aerial Robotics Cooperative Assembly

ΣoΣe,1 Σe,2

Σv,1 Σv,2

Our task: cooperative control of the bar


Marine Autonomous Robotics for InterventionS


Marine Autonomous Robotics for InterventionS

Our task: cooperative control of the bar


Outline

Introduction

Modeling

Inverse Kinematics




Perspectives


Floating robots kinematics

Oi

η1

ηee

❅❅❘endeffector velocities

❍❍❍❍❍❍❍❍❍❍❍❍❥Jacobian

system velocitiesηee =

[ηee1

ηee2

]= J(RI

B, q)ζ ζ =

ν1

ν2

q


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

Kinematic redundancy of the system

1[Siciliano et al.(2009)Siciliano, Sciavicco, Villani, and Oriolo][Fossen(2002)] [Schjølberg and Fossen(1994)]


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]


Outline

Introduction

Modeling

Inverse Kinematics




Perspectives


A first kinematic 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 Palermo, 27 february 2014

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


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

main task

ηd, qd τ η, q

IKsecond. tasks

Control

offline planning not appropriate for unstructured environments


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

)−1σ

Transpose-basedζ = JTσ

Weighted pseudoinverse

ζ = J†W

σ = W−1JT(JW−1JT

)−1σ

Damped Least-Squares

ζ = JT(JJT + λ2Im

)−1σ

need for closedloop also. . .



Handling several tasks4

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

4[Chiaverini et al.(2008)Chiaverini, Oriolo, and Walker]Gianluca Antonelli Palermo, 27 february 2014


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 5


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

ζ = J †σ +(I − J†J

)J†

hσh

5we are talking about algorithmic singularitieshere. . . [Chiaverini(1997)]



AMADEUS

Agility task priority6

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

6[Casalino et al.(2012)Casalino, Zereik, Simetti, Sperinde, and Turetta]Gianluca Antonelli Palermo, 27 february 2014


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 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 Palermo, 27 february 2014


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)]



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]



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



Visual servoing variables

Features in the image plane 11

11[Mebarki et al.(2013)Mebarki, Lippiello, and Siciliano,Mebarki and Lippiello(in press, 2014)]


Outline

Introduction

Modeling

Inverse Kinematics




Perspectives


Behavioral control in pills

Inspired from animal behavior

sensorsbehavior a

actuators

behavior bactuators

behavior cactuators

How to combine them in one single behavior?


Behavioral control in pills

Inspired from animal behavior

sensorsbehavior a

actuators

behavior bactuators

behavior cactuators

How to combine them in one single behavior?


Competitive behavioral control

Behaviors are in competitions and the higher priority can subsume thelower ones

sensorsbehavior b

v2

behavior av1

behavior cv3 vd


Cooperative behavioral control

Behaviors cooperate and the priority is embedded in the gains

sensorsbehavior b

v2 ⊗

α2

behavior av1

supervisor

⊗

α1

behavior cv3 ⊗

α3

∑ vd


NSB for robotic systems

A geometric-based approach: based on the null-space projection

Idea inherited from task priority inverse kinematics


NSB control

Each action is decomposed in elementary behaviors/tasks

σ = f(p1, . . . ,pn)

σ =n∑

i=1

∂f(p)

∂pi

vi = J(p)v

motion reference command for each task

vd = J†(σd +Λσ

)σ = σd−σ


NSB: Merging different tasks

NSB inherits the approach of the singularity-robust task priorityinverse kinematics technique

vd = J †a

(σa,d +Λaσa

)

︸︷︷︸+ J

†b

(σb,d +Λbσb

)

︸︷︷︸primary secondary

Thus, defining:

va = J†a

(σa,d +Λaσa

)Na =

(I − J †

aJa

)

vb = J†b

(σb,d +Λbσb

)




vd = J †a

(σa,d +Λaσa

)

︸︷︷︸+(I − J †

aJa

)

︸︷︷︸J

†b

(σb,d +Λbσb

)

︸︷︷︸primary null space secondary

Thus, defining:

va = J†a

(σa,d +Λaσa

)Na =

(I − J †

aJa

)

vb = J†b

(σb,d +Λbσb

)




vd = J †a

(σa,d +Λaσa

)

︸︷︷︸+(I − J †

aJa

)

︸︷︷︸J

†b

(σb,d +Λbσb

)


Thus, defining:

va = J†a

(σa,d +Λaσa

)Na =

(I − J †

aJa

)

vb = J†b

(σb,d +Λbσb

)




vd = J †a

(σa,d +Λaσa

)

︸︷︷︸+(I − J †

aJa

)

︸︷︷︸J

†b

(σb,d +Λbσb

)


Thus, defining:

va = J†a

(σa,d +Λaσa

)Na =

(I − J †

aJa

)

vb = J†b

(σb,d +Λbσb

)

vd = va +Navb


NSB: Three-task example

va = J†a

(σa,d +Λaσ1

)

vb = J†b

(σb,d +Λbσ2

)

vc = J†c

(σc,d +Λcσ3

)

Successive projection approach

Na =(I − J †

aJa

)

N b =(I − J

†bJ b

)

vd = va +Navb +NaN bvc

Augmented projection approach

Jab =

[Ja

J b

]

Nab =(In − J

†abJab

)

vd = va +Navb+Nabvc



va = J†a

(σa,d +Λaσ1

)

vb = J†b

(σb,d +Λbσ2

)

vc = J†c

(σc,d +Λcσ3

)


Na =(I − J †

aJa

)

N b =(I − J

†bJ b

)



Jab =

[Ja

J b

]

Nab =(In − J

†abJab

)

vd = va +Navb+Nabvc



va = J†a

(σa,d +Λaσ1

)

vb = J†b

(σb,d +Λbσ2

)

vc = J†c

(σc,d +Λcσ3

)


Na =(I − J †

aJa

)

N b =(I − J

†bJ b

)



Jab =

[Ja

J b

]

Nab =(In − J

†abJab

)

vd = va +Navb+Nabvc


From behaviors to actions

sensing/perception

elementary behaviors actions

commands

supervisor


Simple comparison: move to goal with obstacleavoidance

obstacle avoidance

σ1 = ‖p− po‖ ∈ R

σ1,d = d

J1 = rT ∈ R1×2

r =p− po

‖p− po‖

v1 = 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

v2 = Λ2

(pg − p

)


Simple comparison: competitive approach



❆❆❆❆❆❯

only movetogoal



❄

only obstacle avoidance



❄

only movetogoal


Simple comparison: cooperative approach



❆❆❆❆❯

only movetogoal



❇❇❇❇◆

linear combination: higher task is corrupted



❄

only movetogoal


Simple comparison: NSB



❆❆❆❆❯

only movetogoal



❇❇❇◆

nullspaceprojection: higher task is fulfilled



❄

only movetogoal


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


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



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



Stability analysis

Lyapunov function12

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

12[Antonelli(2009)]Gianluca Antonelli Palermo, 27 february 2014

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


(Dis)advantages

PROS

Priorities strictly satisfiedAnalitical convergence resultsEasiest gain tuningClear handling of the DOFsCompetitive and cooperative as particular cases

CONS

Couples all the behaviorsImpedance control not possible


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 movement13

Within a weighted pseudoinverse framework

J†W

= W−1JT(JW−1JT

)−1W−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 dichotomy fuzzy/probability theory very effective intransferring ideas

13[Antonelli and Chiaverini(2003)]Gianluca Antonelli Palermo, 27 february 2014

Dynamic programming to balance the movement14

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


Dynamic programming to balance the movement14

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


Outline

Introduction

Modeling

Inverse Kinematics




Perspectives


Dynamic control

Classical vs modular approaches (both compatible with NSB)

main task

ηd, qd τ η, q

IKsecond. tasks

Control

Classical model-based

natural extension ofwell-known control laws

adaptive and robustversions

Virtual decomposition

modular ⇒ same controllaw for all rigid bodies

adaptive ⇒ integral-likeaction


Virtual Decomposition in a nutshell

based on Newton-Euler formulation

forward propagation of kinematicerrors assuming 6 DOFs

ν1

ν2

backward computation and projectionof control forces/torques


VD: forward propagation, from base to e.e.

6-DOF kinematicerrors νi ∈ R

6

each link consideredas a 6 DOFs body

νi

νi+1

qi+1

propagation

forward

νi+1i+1

= U iT

i+1νii + qi+1z

i+1i


VD: backward propagation, from e.e. to base

6-DOF generalizedcontrol forceshc ∈ R

6 hi

hi+1τi

propagation

backward

hic,i = Y

(Ri,ν

ii,ν

ir,i, ν

ir,i

)θi +Kv,is

ii +U i

i+1hi+1c,i+1


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


Outline

Introduction

Modeling

Inverse Kinematics




Perspectives


Numerical simulation: underwater6-DOF vehicle + 6-DOF manipulator

Reach a pre-grasp configuration in terms of end-effector position andorientation

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 ⇒


Numerical simulation:cooperative transportation of a bar

The final objective for ARCAS

kinematics-only

no perception problems

switch among actions


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


Additional HW tests

Tests made in Seville, February 2014

vehicle controller to be tuned/improved

vehicle obstacle avoidance

e.e. + arm reconfiguration


Outline

Introduction

Modeling

Inverse Kinematics




Perspectives


Perspectives: the underactuated case

Motivated by quadrotors control

Roll and pitch functional to the horizontalmovement ⇒ kinematic disturbances!

pitch

end effector

Uncontrolled DOF affect the tasks

Possible to handle in a task-priority approach?


Perspectives: Null base reaction forces

Dynamic model

[Mv M12

MT12 M q

] [ν

q

]+

[f c,v

f c,q

]+

[fg,v

f g,q

]=

[τ v

τ q

]

for the sole vehicle:

Mvν +M12q + f c,v︸︷︷︸try to zeroing

+fg,v = τ v,

achievable with:q = −M

†12f c,v + Nqo︸︷︷︸

yet another null space. . .


Perspectives: arm’s motors not controllable in torque

Several manipulators on the market equipped with position/velocityfeedback control loopsNeed to design a proper controller for the sole vehicle

Partially coupled model

Iterative formulation

Adaptive

Simplest version: only gravity

Mvν +M 12q + f c,v + f g,v = τ v,

compensate only for f g,v

h00=Y (0)·θ0+U

01

(

Y (1) · θ1 + U12h

22

)

+. . .=[

Y (0) U01Y (1) U

01U

12Y (2) . . . U

01U

12 . . .Un−1

nY (n)

]

︸︷︷︸

Y

θ0θ1θ2

.

.

.

θn


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


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 and S. Chiaverini.

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

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


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.

Singularity-robust task-priority redundancy resolution for real-time kinematiccontrol of robot manipulators.

IEEE Transactions on Robotics and Automation, 13(3):398–410, 1997.


Bibliography III

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.

T.I. Fossen.

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

Marine Cybernetics, Trondheim, Norway, 2002.

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.

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.

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.


Bibliography V

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.

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 underwatervehicle-manipulator systems subject to drag optimization.

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


Bibliography VI

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.

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

Robotics: modelling, planning and control.

Springer Verlag, 2009.


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Control problems for floating-base manipulators

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