Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA...

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Scuola di Dottorato CIRA Controllo di Sistemi Robotici per la Manipolazione e la Cooperazione Bertinoro (FC), 14–16 Luglio 2003 Robots with Flexible Links: Modeling and Control Alessandro De Luca Dipartimento di Informatica e Sistemistica (DIS) Universit` a di Roma “La Sapienza”

Transcript of Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA...

Page 1: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Scuola di Dottorato CIRA

Controllo di Sistemi Robotici per la Manipolazione e la Cooperazione

Bertinoro (FC), 14–16 Luglio 2003

Robots with Flexible Links:

Modeling and Control

Alessandro De Luca

Dipartimento di Informatica e Sistemistica (DIS)

Universita di Roma “La Sapienza”

Page 2: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Outline

• Motivation for considering distributed link flexibility

• Dynamic modeling of FL robots: single link and multiple link cases

• Formulation of control problems

• Controllers for regulation tasks

• Controllers for joint and end-effector trajectory tracking tasks

• Controllers for rest-to-rest motion tasks

• Conclusions

• References

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Page 3: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Motivation

• distributed link deformation in robot manipulators arises when

– the design of very long and slender arms is needed for the application– lightweight materials are used (without special care)

• ‘link rigidity’ is always an ideal assumption and may fail when increasing

– payload-to-weight ratio– motion speed– control bandwidth

• flexible structures in motion are present in different domains: space manipulators,robots for underwater and underground waste sites, automated cranes, . . .

• as for joint elasticity, neglected link flexibility limits static (steady-state error) ordynamic (vibrations, poor tracking) task performance

• from the control point of view, there is an additional problem of non-colocationbetween input commands and typical outputs to be controlled

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Robots with link flexibility — SSRMS

• Space Shuttle Remote Manipulation System (Canadarm)→ telemanipulated byastronauts• link bending due to fast motion (not gravity!)

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Robots with link flexibility — Sam II

• developed at Georgia Tech (W. Book)

• macro-micro concept for remote exploration

and manipulation of nuclear waste sites

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Robots with link flexibility — prototypes in Roma

• DMA harmonic steel beam (0.5 kg): DD-DC motor, encoder, 7 strain gauges

• DIS/DIA FLEXARM: planar two-link with flexible forearm (1.8 kg), DD-DC

motors, encoders, on-board optical sensor measuring deformation at three points

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Robots with link flexibility — prototype in Waterloo

• WATFLEX planar 2R with both link flexible (each with 2 strain gauges), movingwith air bearings on a glass table to support the weight of the second motor;encoders, tachometers, overviewing CCD camera

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Page 8: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Robots with link flexibility — prototypes in Japan

• spatial 3R flexible arm at Kyoto • cooperating 6R flexible arms at Tohoku

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Frequency identification of a single flexible link

0 1 2 3 4 5 6 7 8 9 10 11−6000

−4000

−2000

0

2000

4000

6000

s

deg/

s2

10 20 30 40 50 60 70 80

30

40

50

60

70

80

90

100

110

[ left ] frequency sweep joint acceleration signal: plant vs. model

[ right ] joint acceleration frequency response: plant vs. model (matching within 1% of

resonances at f1 = 14.4, f2 = 34.2, and f3 = 69.3 Hz)

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Page 10: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Dynamic modeling of a single flexible link

• one-link flexible arm modeled as a Euler-Bernoulli beam in rotation

θJ

J

X

Y

x

y CoM

θ θct

pmp

• length , uniform density ρ, Young modulus · cross-section inertia EI

• actuator inertia J0, payload mass mp and inertia Jp

• reference frames: (X,Y ) absolute; (x, y) moving with instantaneous CoM

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Assumptions and definitions

• Euler-Bernoulli theory applies to slender arm design (length vs. section)

• beam undergoes small deformations of pure bending type in the plane of motion

(no torsion or compression)

• bending deformation w(x, t), with x ∈ [0, ], is directed along the y direction

(no shear)

• rotational inertia of beam sections is neglected (→ Timoshenko theory) as well

as the isoperimetric constraint (‘extension’ of beam neutral axis is negligible)

• definition of other relevant angular variables:

– position θ(t) of the CoM (not measurable, but convenient)

– position θc(t) of the tangent to the link base (measured by motor encoder)

– position θt(t) of a line pointing to the beam tip (measurable in several ways)

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• build the Lagrangian from kinetic and elastic potential energy of the beam

• using Hamilton principle and calculus of variations, the bending deformation

w(x, t) and angle θ(t) satisfy the linear differential equations

EIw′′′′(x, t) + ρ(w(x, t) + xθ(t)) = 0 τ(t)− Jθ(t) = 0

i.e., a PDE and an ODE (rigid motion), where J = J0+(ρ3)/3+Jp+mp2

and τ = torque input

• geometric/dynamic boundary conditions associated to the PDE

w(0, t) = 0

EIw′′(0, t) = J0(θ(t) + w′(0, t))− τ(t) (balance of moments at base)

EIw′′(, t) = −Jp(θ(t) + w′(, t)) (balance of moments at tip)

EIw′′′(, t) = mp(θ(t) + w(, t)) (balance of shear forces at tip)

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Page 13: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

• in free evolution (τ(t) ≡ 0⇒ θ(t) = 0), PDE solved by separation of variables

w(x, t) = φ(x)δ(t) ⇒ EI

ρ

φ′′′′(x)φ(x)

= − δ(t)

δ(t)

∆= ω2

for a positive constant ω2 (self-adjoint problem) to be determined

• time solution

δ(t) = −ω2δ(t) ⇒ δ(t) = c1 sinωt + c2 cosωt

with c1, c2 depending on the initial conditions δ(0) and δ(0)

• space solution (try it!)

φ′′′′(x) = β4φ(x) β4 =ρω2

EI

⇒ φ(x) = A sinβx + B cosβx + C sinhβx + D coshβx

with A,B,C,D obtained from the geometric/dynamic b.c.’s on w(x, t)

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• using w(x, t) = φ(x)δ(t) and δ = −ω2δ, and holding the b.c.’s for any δ(t),these are rewritten in terms of φ(x) only

φ(0) = 0

EIφ′′(0) + J0 ω2φ′(0) = 0

EIφ′′()− Jp ω2φ′() = 0

EIφ′′′() + mpω2φ() = 0

• using the general solution φ(x), a system of linear homogeneus equations follows

A(EI, ρ, , J0,mp, Jp, β)

ABCD

= 0 ()

to exclude the trivial solution, the determinant of matrix A should be zero

(eigenvalue problem)

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• det A(β) = 0 at infinite (but countable!) positive increasing roots βi of the

trascendental characteristic equation

(c sh− s ch)− 2mpρ βi s sh− mp

ρ2β4i (J0 + Jp)(c sh− s ch)− 2Jp

ρ β3i c ch

−J0ρ β3

i (1 + c ch) +J0Jpρ2

β6i (c sh + s ch)− J0Jpmp

ρ3β7i (1− c ch) = 0

where s = sinβi, c = cosβi, sh = sinhβi, ch = coshβi

• this is an exact result, that includes common physical approximations

– clamped-free model: mp = Jp = 0, J0 →∞ ⇒ 1 + c ch = 0

– pinned-free model: mp = Jp = J0 = 0 ⇒ c sh− s ch = 0

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Page 16: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

• associated to each root βi we have:

– an eigenfrequency ωi =√EIβ4

i /ρ, characterizing a system vibration

– an eigenvector φi(x) —a spatial deformation mode, defined up to a constant

– a deformation variable δi(t), oscillatory in time

• a finite-dimensional approximation of the distributed bending deformation is

obtained by truncation

w(x, t) =∞∑i=1

φi(x)δi(t) ≈ne∑i=1

φi(x)δi(t)

where ne is the (arbitrary) number of orthogonal modes that we wish to include

• normalization of the modes can be chosen in different ways (some integral of

the φi(x) equal to 1, to m, . . . )

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Page 17: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

• resulting dynamic model is particularly simple (after fairly complex analysis . . . )

Jθ = τ

δi + ω2i δi = φ′i(0)τ i = 1, . . . , ne

• remarkable properties:

– rigid motion θ(t) and each vibratory motion δi(t) are decoupled in free

evolution (τ(t) ≡ 0)

– all modes are excited by an input command τ(t) = 0, with a weighting that

depends on φ′i(0) —the tangent at the link base of each deformation mode

– arm ‘stiffness’ is summarized by the (squared) eigenfrequencies ωi

– each vibratory motion is persistent during free evolution, if initially excited

by δi(0) = 0 (absence of damping)

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Page 18: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

• modal damping can be easily included in the dynamic model

Jθ = τ

δi+2ζiωiδi + ω2i δi = φ′i(0)τ i = 1, . . . , ne

with coefficients ζi ∈ [0,1)

• its matrix version, with generalized coordinates q = ( θ δ1 . . . δne )T ∈ IRne+1,

shows the classical mass-spring-damper form

Mq + Dq + Kq = B τ

with

M =

[J 0

0 I

]D =

[0 0

0 2ZΩ

]K =

[0 0

0 Ω2

]B =

[1

Φ′

]

Ω = diag ω1, . . . , ωne, Z = diag ζ1, . . . , ζne, Φ′ = (φ′1(0) . . . φ′ne(0) )T

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• a different (but equivalent) choice of generalized coordinates may let the input

τ appear in just one equation

(θ, δ) = (θ, δ1, . . . , δne)⇓

(θc, δ) = (θ + δTΦ′, δ) = (θ +∑

φ′i(0)δi , δ1, . . . , δne)

leads to[J −JΦ′T

−JΦ′ I + J2Φ′Φ′T

] [θc

δ

]+

[0 0

0 2ZΩ

] [θc

δ

]+

[0 0

0 Ω2

] [θc

δ

]=

[1

0

with same (diagonal) damping D and stiffness K matrices, but full inertia matrix

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Choice of the controlled output

• joint level angular output (clamped angle)

θc = θ +ne∑i=1

φ′i(0) δi

!! is always minimum phase (zeros in left-hand side of complex plane)

• tip level angular output

θt = θ +ne∑i=1

φi()

δi

!! is typically non-minimum phase (at least for no tip payload)

• output at a point x ∈ [0, ] along the link

θx = θ +ne∑i=1

φi(x)

xδi

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Page 21: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Transfer functions

• torque τ → clamped joint angle θc

Pc(s) =1

Js2+

ne∑i=1

φ′i(0)2

s2 + 2ζiωis + ω2i

=1J

∏nei=1(s

2 + 2ζiωis + ω2i ) + s2

∑nei=1 φ′i(0)

2∏nej =i(s

2 + 2ζjωjs + ω2j )

s2∏ne

i=1(s2 + 2ζiωis + ω2

i )

• torque τ → tip angle θt

Pt(s) =1

Js2+

ne∑i=1

φ′i(0)φi()

s2 + 2ζiωis + ω2i

=1J

∏nei=1(s

2 + 2ζiωis + ω2i ) + s2

∑nei=1 φ′i(0)

φi()

∏nej =i(s

2 + 2ζjωjs + ω2j )

s2∏ne

i=1(s2 + 2ζiωis + ω2

i )

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Page 22: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

A numerical example

• physical data

J0 = 0.002, = 1, ρ = 0.5, EI = 1 (mp = Jp = 0)

• by considering up to ne = 5 modes (and no damping), we obtain

Ω2 = diag 421.585, 3122.603, 10273.194, 31562.286, 82049.350

Φ′T =[7.8259 14.6803 12.1284 6.4761 3.7648

]

ΦT =

[−2.6954 2.3268 −2.4970 2.7380 −2.7982

]

!! note the alternating signs of φi()’s . . .

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First four mode shapes

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x [m]

phi_

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

x [m]

phi_

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x [m]

phi_

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

2

3

x [m]

phi_

4

at f1 = 3.2678, f2 = 8.8936, f3 = 16.1314, and f4 = 28.2751 [Hz]

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Page 24: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Pole-zero patterns

-1 -0.5 0 0.5 1-60

-40

-20

0

20

40

60

Real Axis

Imag

Axi

s

poli-zeri FdT di giunto (2 modi)

-40 -20 0 20 40-60

-40

-20

0

20

40

60

Real Axis

Imag

Axi

s

poli-zeri FdT di tip (2 modi)

-1 -0.5 0 0.5 1-150

-100

-50

0

50

100

150

Real Axis

Imag

Axi

s

poli-zeri FdT di giunto (3 modi)

-100 -50 0 50 100-150

-100

-50

0

50

100

150

Real Axis

Imag

Axi

s

poli-zeri FdT di tip (3 modi)

two modes clamped joint and tip output three modes

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Frequency responses

10-1

100

101

102

103

-100

-50

0

50

100

Frequency (rad/sec)

Mag

nitu

de (

dB)

10-1

100

101

102

103

-90

-180

0

Frequency (rad/sec)

Pha

se (

deg)

10-1

100

101

102

103

-100

-50

0

50

100

Frequency (rad/sec)

Mag

nitu

de (

dB)

10-1

100

101

102

103

-360

-720

0

Frequency (rad/sec)

Pha

se (

deg)

clamped joint three modes tip

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Page 26: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Useful control-oriented remarks

• in pole-zero patterns of Pc(s), zeros precede and alternate with poles on the

imaginary axis ⇒ passivity property

• zero patterns of Pt(s) are symmetric w.r.t. the imaginary axis⇒ non-minimum

phase property ⇒ no direct system inversion is feasible

• while moving the output along the link (Px(s)), zeros migrate along imaginary

axis and several phenomema occurr:

– total pole-zero cancellation when pointing at CoM (vibrations unobservable

from rigid motion variable θ)

– for a particular x∗ ∈ (0, ), all zeros vanishes together at infinity (Px∗(s)

has maximum relative degree equal to 2(ne + 1))

– beyond x∗ (e.g., for x = ), all pairs of zeros reappear in Re+/Re−

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• modal damping destroys perfect symmetry in zeros and poles (system is anyway

asymptotically stable), but not the non-minimum phase property of Pt(s)

• from the Bode plots, it follows that classical controller synthesis in the frequency

domain is harder for the tip output

– multiple crossing of 0dB axis of |Pt(jω)| —especially for high control gain

– increasing phase lag when adding modes

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Page 28: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Dynamic modeling of robots with multiple flexible links

• a convenient kinematic description should be used, both for rigid body motion

and flexible deformation

• recursive procedures can be set up for open chains with flexible links, similarly

to the rigid case

• differential kinematic relationships are needed for computing kinetic and poten-

tial energy, within a Lagrangian approach

• modeling results from the one-link case can be embedded (with caution) in the

description of each flexible link of the robot

• to limit complexity, we sketch here only the planar case (with gravity) of robots

with N flexible links undergoing small bending deformations

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Page 29: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Kinematics

θ1

θ2

X0

X1

Y1

Y0w1(x1)

X2 = X3

Y2 = Y3

Y1

X1

X2

Y2

w2(x2 )

• link i: rigid motion by clamped angle θi(t); lateral bending wi(xi, t), xi ∈[0, i]• position vectors and (rigid/flexible) rotation matrices (w′ie = ∂wi

∂xi

∣∣∣xi=i

)

ipi(xi) =

[xi

wi(xi)

]iri+1 = ipi(i) Ai =

[cos θi − sin θisin θi cos θi

]Ei =

[1 −w′iew′ie 1

]

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Page 30: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

• recursive equations for absolute quantities in (X0, Y0)

pi = ri + Wiipi ri+1 = ri + Wi

iri+1 Wi = Wi−1Ei−1Ai

• differential kinematics

– absolute angular velocity of frame (Xi, Yi)

αi =i∑

j=1

θj +i−1∑k=1

w′ke

– absolute linear velocity of a point on link i

pi = ri + Wiipi + Wi

ipi

where ipi = [0 wi(xi) ]T (link extension is neglected)

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Page 31: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Kinetic energy

T =N∑

i=1

Thi +N∑

i=1

Ti + Tp

• hub i

Thi =1

2mhir

Ti ri +

1

2Jhiα

2i

• link i

Ti =1

2

∫ i

0ρi(xi)pi(xi)

T pi(xi)dxi

• payload

Tp =1

2mpr

TN+1rN+1 +

1

2Jp(αN + w′Ne)

2

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Potential energy

U =N∑

i=1

Uei +N∑

i=1

Ughi +N∑

i=1

Ugi + Ugp

• elastic energy of link i

Uei =1

2

∫ i

0(EI)i(xi)

(d2wi(xi)

dx2i

)2

dxi

• gravitational energy of hub i and of link i

Ughi = −mhigT0 ri Ugi = −gT0

∫ i

0ρi(xi)pi(xi)dxi

• gravitational energy of payload

Ugp = −mpgT0 rN+1

where g0 is the gravity acceleration vector

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Page 33: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Euler-Lagrange equations

• introduce any finite-dimensional discretization for wi(xi, t)

wi(xi, t) =nei∑j=1

ϕij(xi)δij(t) i = 1, . . . , N

• Lagrangian is in terms of N + M generalized coordinates, M =∑N

i=1nei,

L = T − U = L(θi(t), δij(t), θi(t), δij(t)

)and satifies to

d

dt

(∂L

∂θi

)− ∂L

∂θi= τi i = 1, . . . , N

d

dt

(∂L

∂δij

)− ∂L

∂δij= 0 j = 1, . . . , nei i = 1, . . . , N

being τi the torque delivered by the actuator at joint i

33

Page 34: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

• the general dynamic model of flexible robots (including modal damping) is then

[Mθθ(θ, δ) Mθδ(θ, δ)

MTθδ(θ, δ) Mδδ(θ, δ)

] [θ

δ

]+

[cθ(θ, δ, θ, δ)

cδ(θ, δ, θ, δ)

]+

[gθ(θ, δ)

gδ(θ, δ)

]+

[0

Dδ + Kδ

]=

0

]

with blocks of suitable dimensions (e.g., Mθδ in the inertia matrix is (N×M)),or in the more compact form

M(q)q + c(q, q) + g(q) +

[0

Dδ + Kδ

]=

0

]

with q := (θ, δ) ∈ IRN+M

• vector of centrifugal/Coriolis terms can be factorized using Christoffel symbols

c(q, q) = C(q, q)q =

[Cθθ(q, q) Cθδ(q, q)

Cδθ(q, q) Cδδ(q, q)

] [θ

δ

]

34

Page 35: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Model properties

• as usual, matrix M−2C is skew-symmetric — also blockwise, e.g. Mδδ−2Cδδ

• spatial dependence in the kinetic and potential energy of the links can be resolved

introducing a set of dynamic coefficients so that (De Luca, Siciliano, 1991)

Y (θ, δ, θ, δ, θ, δ)a =

0

]

where constant vector a summarizes the mechanical (rigid + flexible) properties

of the links and can be computed off-line (or identified experimentally)

• choice of assumed modes —basis functions ϕij(xi) for bending deformation

− admissible functions satisfy only geometric b.c.’s

− comparison functions (FE method, Ritz-Kantorovich) satisfy also natural b.c.’s

− orthonormal eigenfunctions (links modeled as Euler-Bernoulli beams) leads to

simplifications in inertia submatrix Mδδ (block diagonal, constant)

35

Page 36: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

• a common approximation evaluates total kinetic energy in the undeformed arm

configuration δ = 0

⇒ M = M(θ) and thus c = c(θ, θ, δ)

⇒ cδ loses quadratic dependence on δ

• moreover, if Mδδ is constant

⇒ cθ loses quadratic dependence on δ

⇒ cδ is a quadratic function of θ only

• if also Mθδ is constant

⇒ cδ ≡ 0

⇒ cθ is a quadratic function of θ only

• finally, small deformation of each link implies gδ = gδ(θ)

36

Page 37: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Control problems

• regulation to a constant equilibrium configuration (θ, δ, θ, δ) = (θd, δd,0,0)

– only the desired joint position θd is assigned, while δd has to be determined

– θd may come from the kineto-static inversion of a desired cartesian pose xd,

but no closed-form solution exists (see De Luca, Panzieri, 1994)

– direct kinematics of FL robots is in fact a complete function of rigid and

flexible variables: x = kin(θ, δ)

• tracking of a joint trajectory θd(t) —the easy case

• tracking of a end-effector trajectory xd(t) —the difficult case

• rest-to-rest motion in given time T (a trajectory planning problem in first place)

∗ in tracking problems, controllers try to stiffen the flexible arm at a point in a

way or the other

37

Page 38: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Sensing requirements

• full state feedback requires sensing of joint/motor variables (θ, θ), deflections

δ, and deflection rates δ (no direct sensor available)

• at least encoder + tachometer on the motor axis (sometimes is enough . . . )

• a range of sensors for measuring δ (or deformation related quantities), each

with pros and cons: strain gauges, accelerometers, optical sensors, video camera

(on-board or fixed in workspace), piezoelectric actuating/sensing devices, . . .

38

Page 39: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

• problems with camera: frame rate, field of view/accuracy

39

Page 40: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Regulation with joint PD + feedforward

• for regulation tasks, consider the control law

τ = KP (θd − θ)−KDθ + gθ(θd, δd)

with symmetric (diagonal) KP > 0, KD > 0, and the associated link deflection

δd := −K−1gδ(θd)

Theorem (De Luca, Siciliano, 1993) If

λmin

([KP 0

0 K

])> α

then the closed-loop equilibrium state (θd, δd,0,0) is asymptotically stable

40

Page 41: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Remarks

• Lyapunov-based proof similar to the joint elastic case (not repeated here)

• determination of α

– in view of small deformation

Ue =1

2δTKδ ≤ Ue,max ⇒ ‖δ‖ ≤

√2Ue,max

λmax(K)

– bound on the gradient of the gravitational term∥∥∥∥∥∂g∂q∥∥∥∥∥ ≤ α0 + α1‖δ‖ ≤ α0 + α1

√2Ue,max

λmax(K)=: α

• in absence of modal damping D = 0, special care in LaSalle analysis

• for tip regulation, compute θd by solving via iterative techniques

kin(θ,−K−1gδ(θ)

)= xd

41

Page 42: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Numerical results

• a two-link flexible arm with two bending modes for each link with f11 = 1.4,

f12 = 5.1, f21 = 5.2, f22 = 32.4 [Hz]

• point-to-point motion: θ(0) = (−90,0)→ θd = (−45,0)

-100

-50

0

50

0 1 2 3 4

joint angles

sec

deg

-10

0

10

20

0 1 2 3 4

joint torques

sec

Nm

-0.2

-0.1

0

0.1

0 1 2 3 4

1st link deflections

sec

m

-0.01

-0.005

0

0.005

0 1 2 3 4

2nd link deflections

sec

m

42

Page 43: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Joint trajectory tracking

• given a desired θd(t) ∈ C2, assuming that the state is measurable and the

dynamic model of FL robot is available, we proceed by system inversion from

the joint position output θ

• a nonlinear static state feedback is obtained that decouples and linearizes the

input-output behavior, leaving an unobservable internal (nonlinear) dynamics

• exponential stabilization of the output tracking error is performed on the linear

side of the problem

• some stability/boundedness of the internal system dynamics should be enforced

• original results in (De Luca, Siciliano, 1993b)

43

Page 44: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

System inversion

• from second set of M equations in the dynamic model, solve (globally) for δ

δ = −M−1δδ (cδ + gδ + Kδ + Dδ + MT

θδθ)

and plug in first set of N equations ⇒ effects of flexible dynamics onto rigid

dynamics(Mθθ −MθδM

−1δδ MT

θδ

)θ+cθ+gθ−MθδM

−1δδ

(cδ + gδ + Kδ + Dδ

)= τ

• Matrix Mθθ −MθδM−1δδ MT

θδ has always full rank, since[Mθθ Mθδ

MTθδ Mδδ

] [I 0

−M−1δδ MT

θδ I

]=

[Mθθ −MθδM

−1δδ MT

θδ Mθδ

0 Mδδ

]

44

Page 45: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

• system output θ has uniform vector relative degree 2,2, . . . ,2 (θ depends onτ in a nonsingular way)

• define the nonlinear control law

τ =(Mθθ −MθδM

−1δδ MT

θδ

)a+ cθ + gθ−MθδM

−1δδ (cδ + gδ +Kδ +Dδ)

in which only the inversion inertia block M−1δδ is required

• the closed-loop system is

θ = a

δ = −M−1δδ

(MT

θδa + cδ + gδ + Dδ + Kδ)

• for stabilizing the output tracking error e = θd − θ, choose

a = θd + KD(θd − θ) + KP (θd − θ)

with (diagonal) KP > 0, KD > 0

45

Page 46: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Analysis of internal dynamics

• zero dynamics, when output θ(t) ≡ 0 (or constant):

δ = −M−1δδ

(cδ + gδ + Dδ + Kδ

)asymptotically stable (via Lyapunov argument)⇒ whole closed-loop system too

• clamped dynamics, when output θ(t) ≡ θd(t):

δ = −A2(t)δ −A1(t)δ + fδ(t)

where (in the case M independent of δ)

fδ(t) = −M−1δδ (θd)(M

Tθδ(θd)θd + cδ(θd, θd) + gδ(θd))

A1(t) = M−1δδ (θd)K

A2(t) = M−1δδ (θd)D

all time-varying functions are bounded ⇒ closed-loop stability is ensured

46

Page 47: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Remarks on joint trajectory tracking

• the input-output linearization result is the nonlinear/MIMO counterpart of the

transfer function τ → θc with minimum phase zeros (stable zero dynamics)

• the more ‘rigid’ is the tracking of a desired joint trajectory, the less vibration

energy is taken out from (or the more is injected into) the rest of flexible arm!!

• a nominal feedforward is computed by forward integration of flexible dynamics

δ = −M−1δδ (θd, δ)(cδ(θd, δ, θd, δ) + gδ(θd) + Dδ + Kδ + MT

θδ(θd, δ)θd)

from δ(0) = δ0, δ(0) = δ0 ⇒ nominal (and bounded) evolution δd(t), δd(t)

• substitution of (θd(t), δd(t), θd(t), δd(t)) in the expression of the control law

(w/out feedback) yields τd(t) and the simple local tracking controller

τ = τd(t) + KP (θd − θ) + KD(θd − θ)

47

Page 48: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

End-effector trajectory tracking

• accurate end-effector trajectory tracking is the toughest control problem for

flexible robots

• direct extension of inversion strategies to end-effector output ⇒ closed-loop

instabilities

– linear (single-link) case: non-minimum phase tip transfer function

– nonlinear (multilink) case: unstable zero dynamics for end-effector motion

• main ideas proposed in the literature:

– resort to suitable feedforward strategy (non-causal solutions)

– use feedback, but avoid cancellation (causal solutions)

∗ selection of suitable end-effector trajectories that induce smaller arm deflections

is of interest in any case

48

Page 49: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Inversion in frequency domain

• idea: desired motion trajectory as being part of a periodic profile⇒ use Fourier

transforms (Bayo, 1987)

• single-link flexible arm (with generic variables)[mθθ mT

δθ

mδθ Mδδ

] [θ

δ

]+

[0 0

0 D

] [θ

δ

]+

[0 0

0 K

] [θ

δ

]=

0

]

• tip position output

y(t) = [ 1 cTe ]

δ

]

• rewrite in terms of (y, δ)[mθθ mT

δθ −mθθcTe

mδθ Mδδ −mδθcTe

] [y

δ

]+

[0 0

0 D

] [y

δ

]+

[0 0

0 K

] [y

δ

]=

0

]

49

Page 50: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

• take bilateral Fourier transforms

Y (ω) =∫ ∞−∞

exp(jωt)y(t)dt ∆(ω) =∫ ∞−∞

exp(jωt)δ(t)dt

T (ω) =∫ ∞−∞

exp(jωt)τ(t)dt

and obtainmθθ mT

δθ −mθθcTe

mδθ Mδδ −mδθcTe +

1

jωD − 1

ω2K

[Y (ω)

∆(ω)

]=

[T (ω)

0

]

• solve for accelerations[Y (ω)

∆(ω)

]=

[g11(ω) gT12(ω)

g21(ω) G22(ω)

] [T (ω)

0

]

50

Page 51: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

• torque is obtained through inversion (in the frequency domain)

T (ω) =1

g11(ω)Y (ω) = r(ω)Y (ω)

• for a given zero-mean yd(t), with yd(t) = 0 for t ≤ −T/2 and t ≥ T/2, this

can be embedded into a periodic signal from (−∞,+∞)

• yd(t)→ Yd(ω)→ Td(ω)→ τd(t) from finite inverse Fourier transform

τd(t) =∫ ∞−∞

r(t− τ)yd(τ)dτ =∫ T/2

−T/2r(t− τ)yd(τ)dτ

expanding beyond [−T/2, T/2] (non-causal inverse system)

51

Page 52: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Remarks

• outside the given interval T of output motion, the computed input torque has a

– precharging action, bringing internal flexible state from rest to a suitable

initial state at t = −T/2

– discharging action, bringing internal flexible state from the final state at

t = T/2 to rest

• obtained initial condition is the unique state from which inversion control does

lead to bounded internal evolution for the desired end-effector output trajectory!

• truncations (in time or frequency domain) inherent to actual computations (FFT)

• can be extended to the nonlinear (multilink flexible) setting, by repeated linear

approximations along nominal trajectory (starting from rigid body motion)

52

Page 53: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Extension to nonlinear case: End-effector bang-bang trajectory

0 0.5 1 1.5 2 2.5 3 3.5 4-100

-80

-60

-40

-20

0

20

40

60

80

100

sec

grad

i / s

ec ^

2

0 0.5 1 1.5 2 2.5 3 3.5 4-4

-3

-2

-1

0

1

2

3

sec

Nm

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

x (m)

y (m

)

acceleration profiles command torques FLEXARM motion

53

Page 54: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Extension to nonlinear case: End-effector sinusoidal trajectory

0 0.5 1 1.5 2 2.5 3 3.5 4-150

-100

-50

0

50

100

150

sec

grad

i / s

ec ^

2

0 0.5 1 1.5 2 2.5 3 3.5 4-1.5

-1

-0.5

0

0.5

1

1.5

2

sec

Nm

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x (m)

y (m

)

acceleration profiles command torques FLEXARM motion

54

Page 55: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Regulation theory

• end-effector trajectory tracking in robots with flexible links is an instance of

asymptotic output tracking with internal state stability (regulation problem)

• well-established technique in linear case and, by now, also in nonlinear case

• idea: compute the (bounded!) state trajectory associated to the desired output

trajectory (generated by an autonomous antistable system, the exosystem)

• in linear case, write state-space equations (with x = (q, q)) for the flexible arm

x = Ax + Bτ e = y − yd

and for the generator of desired output

w = Sw yd = −Qw

55

Page 56: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

• when (A,B) is stabilizable, the problem has a solution if and only if the following

regulator equations are solvable in Π and Γ

ΠS = AΠ + BΓ CΠ + Q = 0

• a state feedback + feedforward controller is then

τ = F (x−Πw) + Γw

with feedback matrix F such that (A + BF ) is Hurwitz

• the computed Πw is the desired state trajectory; xd(0) = Πw(0) is the unique

initial state from which inversion control does lead to bounded internal evolution!

• control solutions with dynamic output feedback are also available

56

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Numerical results for sinusoidal end-effector trajectory

0 1 2 3 4 5 6 7 8 9 10-100

-80

-60

-40

-20

0

20

40

60

80

100autovalori in -10 (quattro) e -10 (quattro)

sec

usci

ta ti

p (d

eg)

0 1 2 3 4 5 6 7 8 9 10-10

0

10

20

30

40

50

60

70autovalori in -10 (quattro) e -10 (quattro)

sec

erro

re a

l tip

(de

g)tip output tip error

57

Page 58: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

0 1 2 3 4 5 6 7 8 9 10-0.03

-0.02

-0.01

0

0.01

0.02

0.03autovalori in -10 (quattro) e -10 (quattro)

sec

varia

bili

defo

rmaz

ione

del

ta (

m)

0 1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5autovalori in -10 (quattro) e -10 (quattro)

sec

cont

rollo

(N

m)

deformation variables torque input

58

Page 59: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

0 1 2 3 4 5 6 7 8 9 10-100

-80

-60

-40

-20

0

20

40

60

80

100autovalori in -10 (quattro) e -10 (quattro)

sec

usci

ta a

l giu

nto

(deg

)

clamped joint angle (and desired tip output)

59

Page 60: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Rest-to-rest motion

• task: execute a slew motion with a FL robot arm between two undeformed

configurations in given time

• problem: fast transfers induce residual oscillations extending the actual task

completion time

• strategy: design a suitable output and plan output trajectories (and associated

torque profiles) inducing complete absence of vibrations at the desired final time

• solution: output with maximum relative degree (no zeros); closed-form algorithm

in the linear case; direct extension to MIMO nonlinear case (DFL or flat output,

no zero dynamics)

60

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Time-based algorithm for a single flexible link

• choose a parametric output y (unknown ci’s) (De Luca, Di Giovanni, 2001)

y = θ +ne∑i=1

ciδi = θ + cT δ

• impose τ -independence of (even) output derivatives

y = (1

J+

ne∑i=1

ciφ′i(0))τ −

ne∑i=1

ciω2i δi ⇒ ∑

ciφ′i(0) = −1/J

y[4] =:d4y

dt4= −

ne∑i=1

ciω2i φ′i(0) τ +

ne∑i=1

ciω4i δi ⇒ ∑

ciω2i φ′i(0) = 0

and so on, until a set of ne equations are generated (torque τ appears in the

2(ne + 1)-th output derivative)

61

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• solve for the coefficients c = (c1, . . . , cne)

V · diagφ′1(0), . . . , φ′ne(0) c = [−1/J 0 . . . 0 ]T

with Vandermonde matrix V generated by (ω21, . . . , ω

2ne

)

• nominal torque τd(t) computed by inversion on highest derivative imposing

y[2(ne+1)] = y[2(ne+1)]d

for a suitably planned output trajectory yd(t), t ∈ [0, T ] (given trasfer time)

• for the output trajectory yd(t), solve a simple interpolation problem

yd(0) = θi yd(T ) = θfdiyddti

(0) =diyddti

(T ) = 0 i = 1, . . . ,2ne + 1

e.g., a polynomial of degree 4ne + 3 will be sufficient

62

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Laplace-based algorithm (only in the linear case)

• impose that the transfer function has no zeros

y(s)

τ(s)=

1

Js2+

ne∑i=1

ciφ′i(0)

s2 + ω2i

∆=

K

s2∏ne

i=1(s2 + ω2

i )

• partial fractions expansion yields closed-form expressions

K =1

J

ne∏i=1

ω2i ci = −

1

Jφ′i(0)

ne∏j=1j =i

ω2j

ω2j − ω2

i

(i = 1, . . . , ne)

• set y = yd and invert in the transformed domain (then back to time → τd(t))

τd(s) =J∏ne

i=1 ω2i

s2 ne∏

i=1

(s2 + ω2i )

yd(s)

63

Page 64: Modeling and Control Robots with Flexible Links · 2004. 10. 26. · ScuoladiDottoratoCIRA ControllodiSistemiRoboticiperlaManipolazioneelaCooperazione Bertinoro(FC),14–16Luglio2003

Remarks

• method applies to any linear (controllable) model of a single-link flexible arm

• output structure for modal damping (De Luca, Caiano, Del Vescovo, 2003)

y = θ +ne∑i=1

ciδi + γθ +ne∑i=1

diδi

• design output is (in the limit) a specific point x∗ on the physical beam: for a

given ne, ci = φi(x∗ne

)/x∗newhile limne→∞ x∗ne

= x∗

• for improved torque/time performance, modified method generates smoothed

bang-bang/bang-coast-bang torque profiles, with polynomial interpolating phases

• trajectory planning (feedforward) combined with feedback control

τ = τd(t) + KP (θc,d(t)− θc) + KD(θc,d(t)− θc)

with clamped joint reference θc,d(t) computed from the algorithm

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Numerical results

• ne = 3 modes with f1 = 4.05, f2 = 12.34, f3 = 22.87 [Hz]

• θf − θi = 90 in T = 2 s

• 19th degree polynomial (continuous torque derivative)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

70

80

90

100

s

deg

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3

-2

-1

0

1

2

3

s

Nm

output trajectory rest-to-rest torque

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

70

80

90

100

s

deg

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

s-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

m

m

clamped (—), tip angle (- -) deformation variables arm motion

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Experiment on DMA single flexible link

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Extension to nonlinear case: Rest-to-rest motion

• time-based algorithm for two-link with flexible forearm (De Luca, Di Giovanni,

2001b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025delta

s0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-8

-6

-4

-2

0

2

4

6

8u1, u2

s

Nm

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

m

m

first deformation mode command torques FLEXARM motion

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Conclusions

• extra effort in dynamic modeling pays off

– model-based controllers for accurate trajectory tracking

– proof of stability for model-inpedendent regulation controllers

• conventional control strategies tend to suppress vibrations wherever they arise

– outcome of the analysis: controlled system should be brought to a vibratory

behavior compatible with the given output task

• EJ robots are similar to FL robots in mechanical modeling, but intrinsically

different from the control point of view

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What has been left out . . .

• singular perturbation modeling and control (for joint or link stiffness K →∞),

including corrective and invariant manifold controllers

• iterative learning control that yields same accuracy (for all types of tasks) without

using a dynamic model but assuming repetitive tasks

• model uncertainties, disturbances, . . .

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References

(Bayo, 1987) “A finite-element approach to control the end-point motion of a single-link flexible

robot,” JRS, 4, 63–75

(De Luca, Caiano, Del Vescovo, 2003) “Experiments on rest-to-rest motion of a flexible arm,” in

Experimental Robotics VIII (Siciliano, Dario Eds), STAR 5, Springer, 338–349

(De Luca, Di Giovanni, 2001) “Rest-to-rest motion of a one-link flexible forearm,” IEEE/ASME

AIM 01, 923–928

(De Luca, Di Giovanni, 2001b) “Rest-to-rest motion of a two-link robot with a flexible forearm,”

IEEE/ASME AIM 01, 929–935

(De Luca, Lanari, Ulivi, 1991) “End-effector trajectory tracking in flexible arms: Comparison of

approaches based on regulation theory,” in Advanced Robot Control (Canudas de Wit Ed), LNCIS

162, Springer, 190–206

(De Luca, Panzieri, 1994) “An iterative scheme for learning gravity compensation in flexible robot

arms,” Automatica, 30(6), 993–1002

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(De Luca, Panzieri, Ulivi, 1998) “Stable inversion control for flexible link manipulators,” IEEE

ICRA, 799–805

(De Luca, Siciliano, 1991) “Closed-form dynamic model of planar multi-link lightweight robots,”

IEEE SMC, 21(4), 826–839

(De Luca, Siciliano, 1993) “Regulation of flexible arms under gravity,” IEEE TRA, 9(4), 463–467

(De Luca, Siciliano, 1993b) “Inversion-based nonlinear control of robot arms with flexible links,”

AIAA JGCD, 16(6), 1169–1176

(De Luca, Siciliano, 1996) “Flexible links,” in Theory of Robot Control (Canudas de Wit, Siciliano,

Bastin Eds), Springer, 219–261

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