Nonlinear and linearized dynami models

The rst degree of freedom (# 1) of the arm is used for the vertial motion. The

seond and third ones (# 2 and # 3) are linked to the relative position of the

target enter of mass (blue ross in Figure 7.2b) and the laser spot (red ross in

Figure 7.2b). Forthis validationstep, the studied motionis restrited to these two

lastdegrees of freedom (# 2 and #3).

Asexplainedpreviously,the spotpositionmeasurementsaremadewiththe help

of a CCD amera. In pratie, this amera must be loated so that it does not

disturb the surgeon and his sta. Furthermore, it is diult to move during the

operation mainly for safety reasons. These pratial onditions highly redue the

workingeldoftherobot. Beauseoftheseonstraints,theuseofaglobaltehnique

[135℄requiringapersistentexitationoftheinputsaswellastheshedulingvariables

is not oneivable. On the ontrary, a loal approah [135℄ seems to be well-suited

for the LPV model identiationof suh asystem.

The

x

^and

y

^{positions of the end-eetoran bewrittenfrom the geometrimodel,}

resultingin the nonlinear measurement equation

z = g(q)

^,i.e.,

z 1 =(ℓ 1 − 2

3 ℓ ³ ₁ v ² ₁ ) cos(φ 1 ) − ℓ ² ₁ v 1 sin(φ 1 ) + (ℓ 2 − 2

3 ℓ ³ ₂ v ² ₂ ) cos(φ 12 ) − ℓ ² ₂ v 2 sin(φ 12 )

(7.6a)

z ₂ =(ℓ ₁ − 2

3 ℓ ³ ₁ v ² ₁ ) sin(φ ₁ ) + ℓ ² ₁ v ₁ cos(φ ₁ ) + ℓ ² ₂ v ₂ cos(φ ₁₂ ) + (ℓ ₂ − 2

3 ℓ ³ ₂ v ₂ ² ) sin(φ ₁₂ )

(7.6b)

where

φ 12 = φ 1 + φ 2 + 2ℓ 1 v 1

^.

In order tox the struture of the globalLPV modelrequired by the tehnique

introdued in Setion 5.5, a standard Jaobian linearization an be applied to the

generalized seond order model given in Eq. (7.4). More speially, for a set of

working points

(q 0 , q ˙ ₀ )

^{, weget3}

M(p)¨ q = D(p) ˙ q + K(p)v + Hu,

^(7.7)

where

p = p ( q ₀ , q ˙ ₀ )

^{isthe sheduling variable vetor and where}

M(p(q ₀ , q ˙ ₀ )) = M (q ₀ ),

^(7.8a)

D(p(q ₀ , q ˙ ₀ )) = ∂ F (q ₀ , q ˙ ₀ )

∂ q ˙ ,

^(7.8b)

K(p(q 0 , q ˙ ₀ )) = ∂ F (q 0 , q ˙ ₀ )

∂q − M ⁻¹ (q 0 ) M (q 0 )

∂q M ⁻¹ (q 0 ) F (q 0 , q ˙ ₀ ),

^(7.8)

are the inertia, the damping and the stiness of the linearized modelrespetively.

Notie that

H

^{does not depend on the sheduling variable}

p

^{. In this work, we}

fous onthe identiationofa modelwhihinludesthe variability of the behavior

with respet to the positions

φ k

^,

k = 1, 2

^{. Then, the other phenomena an be}

negleted and, onsequently 4

,

q ₀ =

φ ^⊤ ₀ 0 _{1×n v} ⊤

and

q ˙ ₀ = 0 _{n q ×1}

^{. Like for most of}

the methods onsidered inthe literature, the inertia matrix

M(p)

^{is inverted. This}

property is generally used in the literature (see, e.g., [54℄) leading to the following

loalequation

¨

q = M ⁻¹ ( p ) D ( p ) ˙ q + M ⁻¹ ( p ) K ( p ) q + M ⁻¹ ( p ) Hu .

^(7.9)

By looking loser at the equations available in [68℄, it is lear that the matrix

M

is an ane funtion of

cos(φ 2 )

^and

sin(φ 2 )

^{. Then, the sheduling variable an be}

hosen as

cos(φ 2 )

^{, i.e.,}

p = cos(φ 2 )

^{. Now, the inner matries an be modeled [68℄}

asfollows,

M(p) = M ₀ + M ₁ p,

^(7.10)

while the other matries

D

^and

K

^{are onstant. Notie that this angular position}

is easy to measure on a exible robot beause an enoder is generally loated at

the motor side of the joints. This availabilityis paramount when the experimental

3. Inthefollowing,theequilibriumvaluesareomittedin ordertoshortenthenotations.

4. This hoie means that the Coriolis eets are negleted. The validity of this assumption

modeling of the LPV model is onsidered. So, the matrix inversion involved in

Eq. (7.9)leadstomatries

M ⁻¹ (p)D(p)

^,

M ⁻¹ (p)K(p)

^and

M ⁻¹ (p)H

^whihsatisfy

a frational dependeny on the shedulingvariable

p = cos(φ 2 )

^{. This hint justies}

the use of alinear frationalLPV desription(see Eq. (2.15))of thesystem instead

of a more standard ane LPV model(see Eq. (2.21)). Now, by onsidering

χ =

q ^⊤ q ˙ ^⊤ ⊤

∈ R ⁸

(7.11)

as state vetor, the following loal linearized dark-gray-box state-spae model an

bededued 5

˙

χ = A ₀ (p)χ + B ₀ (p)u

^(7.12)

with

A ₀ ( p ) =

0 _{n q ×n q} I _{n q} M ⁻¹ (p)K(p) M ⁻¹ (p)D(p)

(7.13a)

B ₀ (p) =

0 _{n q ×n u} M ⁻¹ (p)H

.

^(7.13b)

Forthisstate-spaeform,thestateequationonlydependson

φ 2

6

whereastheoutput

equation depends onboth

φ 1

^and

φ 2

^{. As shown in Eq. (7.6), the output equations}

Imageplane

rigidarms

exiblearms

Figure 7.3: Flexible manipulator(gray) and virtual rigid manipulator (blak) with

the same imagepositionof the end-eetor.

are highly nonlinear with respet to

φ ₁

^and

φ ₂

^{. This omplexity an be redued}

by using eiently the Jaobian of the rigid geometri model (see Eq. (7.6) for a

denition of

g

^{), i.e.,}

J(φ 0 ) = ∂ g

φ ^⊤ ₀ 0 _{1×n v} ⊤

∂φ .

^(7.14)

Exept for the singular positions, this Jaobian is invertible and it is possible to

dene a new measurement vetor

α = α ₀ + J ⁻¹ (φ 0 )(z − z ₀ )

^(7.15)

5. Inthesequel,

z •

^,

φ •

^,

v •

^,

q •

^{,...,areonsideredasvariationsaroundanominalvalue.}

6. Thesystem behavioris invariantby arotationof angle

φ 1

^{. Therefore, thematries}

A

^and

B

^{donotdependon}

φ 1

^.

where, in the following,

α ₀

^and

z ₀

^{are assumed to be equal to 0 without any}

lim-itation for the appliability of the approah. The entries of this new measurement

vetor

α

^{are the angular positions of a titious rigid arm whih would have the}

same geometryand the samemeasurement

z

^{than the exible one (see Figure7.3).}

The use of

α

^{instead of}

z

^{allows the} simpliation of the measurement equation.

Letus nowonsider

y = φ ˙ α ˙ ⊤

,

^(7.16)

assystem outputs, i.e., the jointveloities (

φ ˙

^{)measured by enoders and the}

ti-tious joint veloities (

α ˙

^{). On top of that, the industrial}manipulatorsare equipped with low-level joint-veloity ontrol loops in order to redue the eets of the

fri-tions whih our in the gear-boxes and therefore obtain a simpler behavior (see

Controller in Figure7.2a). In this partiular ase, the inner loop is assumed to be

asa standard stati output feedbak. More preisely,

u = Λ

φ ˙ ^∗ − φ ˙

,

^(7.17)

where

φ ˙ ^∗ = φ ˙ ^∗ ₁ φ ˙ ^∗ ₂ ⊤

is the vetor of the speed referenes. The state-spae

repre-sentationgiven in Eq. (7.12)beomes

˙ χ =

0 _{n q ×n φ} M ⁻¹ (q)HΛ

φ ˙ ^∗ +

0 _{n q ×n q} I _{n q} M ⁻¹ (q)K(p) M ⁻¹ (p) D(p) − HΛ

I _{n φ} 0 _{n φ ×n v}

χ,

^(7.18)

where

Λ = diag(λ 1 , λ 2 )

^{. By analyzing the relations given in Eq. (7.18), it appears}

thatthisstate-spae formisnotminimalbeausethetworst states

φ 1

^and

φ 2

^have

no eets on the output. The hange of output signal allows the redution of the

modelorder. Thus, aminimalrealizationoforder6anbeextrated by onsidering

¯ χ =

v ^⊤ φ ˙ ^⊤ v ˙ ^⊤ ⊤

.

^(7.19)

More preisely,

˙¯

χ = A(p, θ ^g ) ¯ χ + B(p, θ ^g ) ˙ φ ^∗ ,

^(7.20)

y =

˙ α φ ˙ ⊤

= C χ, ¯

^(7.21)

where 7

A(p, θ ^g ) =

0 _{n v ×n φ} 0 _{n v ×n v} I _{n v} (M ⁻¹ (p, θ ^g )K(p, θ ^g )) (:, n φ + 1 : end) − M ⁻¹ (p, θ ^g )HΛ 0 n q ×n v

,

^(7.22a)

B(p, θ ^g ) =

0 _{n v ×n φ} M ⁻¹ (p, θ ^g )HΛ

, C =

0 _{2∗n φ ×n v} C ₁

,

^(7.22b)

7. For oursystem,

D = 0 _n

q ×n _q

^.

with

C ₁ =



 



1 0 0 0 0 1 0 0 1 0 ℓ 1 0 0 1 ℓ ₁ ℓ ₂



 

 .

^(7.22)

with

θ ^g ∈ R ¹⁶

^{ontaining the unknown gray-box} parameters. This linearized state-spae representation an now be relatedtothe LPV given inEq. (2.13).

Indeed, with straightforward alulations, it an be shown that suh a

dark-gray-box LPVdesriptionan be transformedinto anLFR suhasthe one given in

Eq.(2.15)byapplyingthestruturedmatriesofEq.(7.22)andtheinnerstrutures

dened by Eq. (7.10) asfollows,

M(θ ^g ) =





D _zw (θ ^g ) C _z (θ ^g ) D _zu (θ ^g ) B _w (θ ^g ) A(θ ^g ) B _u (θ ^g ) D _yw (θ ^g ) C _y (θ ^g ) D _yu (θ ^g )



 =



 



− M ⁻¹ ₀ M ₁ M ⁻¹ ₀ M ₁ − M ⁻¹ ₀ M ₁ 0 _2×2 M ⁻¹ ₀ M ₁ 0 _2×2 0 _2×2 0 _2×2 I ₂ 0 _2×2

− M ⁻¹ ₀ K M ⁻¹ ₀ K − M ⁻¹ ₀ HΛ 0 _4×2 M ⁻¹ ₀ HΛ 0 _2×2 0 _2×2 I ₂ 0 _2×2 0 _2×2



 

 ,

^(7.23)

where

θ ^g ∈ R ²⁸

. More preisely,

M ⁻¹ ₀ K ∈ R ^4×2 , M ⁻¹ ₀ M ₁ ∈ R ^2×2 , M ⁻¹ ₀ HΛ ∈ R ^4×2 .

Notie that the matrix produts dened above ontains the unknown parameters

to estimate. In this ase, the

∆ _p

blok-diagonal matrix ontaining the sheduling variable(

p = cos(φ 2 )

^{) has the following form}

∆ _p = I ₂ cos(φ 2 ).

^(7.24)

Moreover, in the blak-box ase, one an onsider

M _{LP V} (θ ^b ) =





D _zw (θ ^b ) C _z (θ ^b ) D _zu (θ ^b ) B _w (θ ^b ) A (θ ^b ) B _u (θ ^b ) D _yw (θ ^b ) C _y (θ ^b ) D _yu (θ ^b )



 ∈ R ^{(n x +n w +n u )×(n x +n z +n y )} ,

^(7.25)

where

θ ^b ∈ R ^{(n x +n w +n u )×(n x +n z +n y )}

with

n u = n y = 2

^{. As a onsequene, the}

gray-box LPV model an be alulated by the following upperLFT, i.e.,

F ^u ( M _{LP V} (θ ^b ), ∆ _p ),

^(7.26)

while the blak-box one by using,

F ^u (M(θ ^g ), ∆ _p ).

^(7.27)

Notie that, beause of the spei test-bed set-up presented in Setion 7.1, the

amera measurements are available only loally. So, in the LTI framework, the

system understudy has fouroutputs

y _{LT I} = φ ˙ α ˙ ⊤

.

^(7.28)

Namely,thejointveloitiesofthetitiousrobotiarmand themeasured realones.

However, whentherobothastomoveinalargerworkspae,likeintheLPVase,the

outputsofthe systemare restritedtothe jointveloitiesmeasuredby themounted

enoders and

y _{LP V} = ˙ φ.

^(7.29)

In document H ∞-norm for the (Pldal 112-117)