modeling and control robots with flexible links · 2004. 10. 26. · scuoladidottoratocira...
TRANSCRIPT
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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)
Universita di Roma “La Sapienza”
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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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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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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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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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• 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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• 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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• 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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• 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
]τ
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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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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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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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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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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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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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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• 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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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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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
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• 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
]=
[τ
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
]=
[τ
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
]
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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 =
[τ
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)
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• 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)
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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
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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, . . .
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• problems with camera: frame rate, field of view/accuracy
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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
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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
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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
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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)
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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
]
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• 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
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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
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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)
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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
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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
]=
[τ
0
]
• tip position output
y(t) = [ 1 cTe ]
[θ
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
]=
[τ
0
]
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• 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
]
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• 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)
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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)
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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
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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
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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
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• 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
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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
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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
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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)
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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)
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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)
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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
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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)
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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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