Control problems for floating-base manipulators

Gianluca Antonelli

Universita 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

Outline

Introduction & variable definition

Inverse Kinematics

A possible kinematic solution: NSB behavioral control

A possible dynamic solution: Virtual decomposition

Simulation/experiments

Perspectives

Space, aerial and underwater vehicle-manipulators

DLRCanadian Space Agency

normal robots but

floating base

kinematic coupling

dynamic coupling

unstructured environment

Cooperation

RedSea project

AMADEUS

Aerial Robotics Cooperative Assembly

ΣoΣe,1 Σe,2

Σv,1 Σv,2

Our task: cooperative control of the bar

Marine Autonomous Robotics for InterventionS

Our task: cooperative control of the bar (and logo design. . . )

Floating robots kinematics

❅❅❘endeffector velocities

❍❍❍❍❍❍❍❍❍❍❍❍❥Jacobian

system velocitiesηee =

[ηee1

]= J(RI

B, q)ζ ζ =

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)]

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

Inverse Kinematics

Perspectives

Kinematic control in pills -1-

✛✚

✘✙

✛✚

✘✙

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

✛✚

✘✙

✛✚

✘✙

✖✕✗✔

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

σ = Jζ inverted by solving proper optimization problems

Pseudoinverseζ = J †σ = JT

)−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. . .

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

✛✚

✘✙

✛✚

✘✙

✖✕✗✔

Redundancy may be used to add additional tasks

✛✚

✘✙

✛✚

✘✙

✖✕✗✔

■ σa

✚✙✛✘

✖✕✗✔

Kinematic control scheme

second. tasks

ηd, qd τ η, q

IKmain task

Control

Output of kinematic control is the variable to be controlled by theactuators (vehicle thrusters and joints’ torques)

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 Trondheim, 23 april 2014

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

4[Chiaverini et al.(2008)Chiaverini, Oriolo, and Walker]Gianluca Antonelli Trondheim, 23 april 2014

Handling several tasks -2-

Augmented JacobianAn additional task is given

σh = h(η, q) with associated Jh

such that the problem is squared with

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

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 Trondheim, 23 april 2014

Behavioral algorithms (behavior=task), bioinspired, artificialpotentials, neuro-fuzzy, cognitive approaches, etc.

btw. . . mood ?

But. . .

What are these tasks we are talking about ?

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∑

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

7[Sarkar and Podder(2001)]Gianluca Antonelli Trondheim, 23 april 2014

Manipulability/singularity, m = 1

h(q) =∣∣det

)∣∣(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

Restoring moments:

m = 3 keep close gravity-buoyancy of the overall system 9

m = 2 align gravity and buoyancy (SAUVIM is 4 tons) 10

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

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 ones12

sensorsbehavior b

behavior a

behavior c

ζ3 ζd

12[Brooks(1986)]Gianluca Antonelli Trondheim, 23 april 2014

Cooperative behavioral control

Behaviors cooperate and the priority is embedded in the gains13

sensorsbehavior b

ζ2 ⊗

behavior a

supervisor

behavior c

ζ3 ⊗

∑ ζd

13[Arkin(1989)]Gianluca Antonelli Trondheim, 23 april 2014

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

NSB: Merging different tasks

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

ζd = J †a

(σa,d +Λaσa

︸︷︷︸+ J

(σb,d +Λbσb

︸︷︷︸primary secondary

Thus, defining:

ζa = J †a

(σa,d +Λaσa

(I − J†

ζb = J†b

(σb,d +Λbσb

ζd = J †a

(σa,d +Λaσa

︸︷︷︸+(I − J†

︸︷︷︸J†b

(σb,d +Λbσb

︸︷︷︸primary null space secondary

Thus, defining:

ζa = J †a

(σa,d +Λaσa

(I − J†

ζb = J†b

(σb,d +Λbσb

ζd = J †a

(σa,d +Λaσa

︸︷︷︸+(I − J†

︸︷︷︸J†b

(σb,d +Λbσb

Thus, defining:

ζa = J †a

(σa,d +Λaσa

(I − J†

ζb = J†b

(σb,d +Λbσb

ζd = J †a

(σa,d +Λaσa

︸︷︷︸+(I − J†

︸︷︷︸J†b

(σb,d +Λbσb

Thus, defining:

ζa = J †a

(σa,d +Λaσa

(I − J†

ζb = J†b

(σb,d +Λbσb

ζd = ζa +Naζb

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 †

N b =(I − J

†bJ b

ζd = ζa +Naζb +NaN bζc

Augmented projection approach

Nab =(In − J

†abJab

ζd = ζa +Naζb+Nabζc

ζa = J †a

(σa,d +Λaσ1

ζb = J†b

(σb,d +Λbσ2

ζc = J †c

(σc,d +Λcσ3

Na =(I − J †

N b =(I − J

†bJ b

Nab =(In − J

†abJab

ζa = J †a

(σa,d +Λaσ1

ζb = J†b

(σb,d +Λbσ2

ζc = J †c

(σc,d +Λcσ3

Na =(I − J †

N b =(I − J

†bJ b

Nab =(In − J

†abJab

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‖

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

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

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

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

Stability analysis

Lyapunov function14

V (σ) = 1

2σTσ > 0 where σ =

V = −σT

v = −σTMσ = −σ

Λa Oma,mbOma,mc

JbJ†aΛa JbNaJ

†bΛb JbNJ

†cΛc

JcJ†aΛa JcNaJ

†bΛb JcNJ

†cΛc

14[Antonelli(2009)]Gianluca Antonelli Trondheim, 23 april 2014

Stability results

V < 0 depending on the mutual relationships among the Jacobians:

dependent

ρ(J †x) + ρ(J †

y) > ρ([J†x J†

independent

ρ(J †x) + ρ(J †

y) = ρ([J†x J†

orthogonal

JxJ†y = Omx×my

(Dis)advantages

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

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 movement15

Within a weighted pseudoinverse framework

= 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

(σa +Λaσa) +(I − J

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

15[Antonelli and Chiaverini(2003)]Gianluca Antonelli Trondheim, 23 april 2014

= W−1JT(JW−1JT

)−1W−1(β) =

[(1− β)I6 O6×n

On×6 βIn

ζ = J†W

αiJ†s,iws,i

= W−1JT(JW−1JT

)−1W−1(β) =

[(1− β)I6 O6×n

On×6 βIn

ζ = J†W

αiJ†s,iws,i

Dynamic programming to balance the movement16

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

Dynamic programming to balance the movement16

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

Outline

Inverse Kinematics

Perspectives

Dynamic control

Classical vs modular approaches (both compatible with NSB)

second. tasks

ηd, qd τ η, q

IKmain task

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

Dynamic control

second. tasks

ηd, qd τ η, q

IKmain task

Control

Dynamic control

second. tasks

ηd, qd τ η, q

IKmain task

Control

Virtual Decomposition in a nutshell

based on Newton-Euler formulation

forward propagation of kinematicerrors assuming 6 DOFs

backward computation and projectionof control forces/torques

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

6-DOF kinematicerrors νi ∈ R

each link consideredas a 6 DOFs body

propagation

forward

νi+1i+1

= U iT

i+1νii + qi+1z

propagation

forward

νi+1i+1

= U iT

i+1νii︸︷︷︸+ qi+1z

velocity propagation

propagation

forward

νi+1i+1

= U iT

i+1νii + qi+1z

i+1i︸︷︷︸

joint contribution

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

6-DOF generalizedcontrol forceshc ∈ R

hi+1τi

propagation

backward

hic,i = Y

(Ri,ν

ir,i, ν

)θi +Kv,is

ii +U i

i+1hi+1c,i+1

hi+1τi

propagation

backward

hic,i = Y

(Ri,ν

ir,i, ν

)θi︸︷︷︸

+Kv,isii +U i

i+1hi+1c,i+1

model-based compensation

hi+1τi

propagation

backward

hic,i = Y

(Ri,ν

ir,i, ν

)θi +Kv,is

ii︸︷︷︸+U i

i+1hi+1c,i+1

feedback term

hi+1τi

propagation

backward

hic,i = Y

(Ri,ν

ir,i, ν

)θi +Kv,is

ii +U i

i+1hi+1c,i+1︸︷︷︸

force propagation

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,n+1 hc,n+1

Outline

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: underwater6-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 jointlimits

animation ⇒

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

Inverse Kinematics

Perspectives

Perspectives: the underactuated case

Motivated by quadrotors control

Roll and pitch functional to the horizontalmovement ⇒ kinematic disturbances!

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

[f c,v

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

Y (1) · θ1 + U12h

+. . .=[

Y (0) U01Y (1) U

12Y (2) . . . U

12 . . .Un−1

nY (n)

︸︷︷︸

θ0θ1θ2

Perspectives: 2-arm-equipped vehicle

Proposal to be submitted in few hours to H2020-ICT23

FadaCatec

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

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