Nonlinear PDE control of two-link flexible arm with nonuniform cross section

Nonlinear PDE Control of Two-Link Flexible Arm with

Nonuniform Cross Section

Mustafa Do˜gan

, ¨

Omer Morg¨ul

Department of Electrical and Electronics Engineering

1

_{Bas.kent University,}

_{Bo˜gazic¸i University,}

_{Bilkent University, Turkey}

Abstract

A two-link flexible arm with nonuniform or variable cross-section by design will be considered based on an exact PDE model with boundary conditions. In this re-search, the nonlinear controller is used to achieve set-point regulation of the rigid modes as well as suppression of elastic vibrations. The control laws are obtained by energy based Lyapunov approach.

I. INTRODUCTION

Two main advantages of flexible robot arms are less weight and low energy consumption. However, the struc-tural modelling and the control design of the flexible arms are much more complicated due to nonlinear coupling between elastic and rigid modes during the complex maneuvers especially with high angular velocities. Various methods have been proposed for control of flexible-link manipulators in the literature. Hybrid control of a single flexible-link manipulator using feedback linearization and singular perturbation approach has been used in [1]. Adaptive feedback linearization has been applied success-fully for a nonlinear discrete-time model of a single-link flexible manipulator [2]. Singular perturbation theory has also been used for position and force control in [3]. Strain feedback and active vibration control [4], [5] are other approaches different from the integrated structure-control for nonuniform flexible links in [6]. In order to improve the important features of flexible links such that low mass and moments of inertia and high natural frequencies [7], optimal shape design can be investigated. Furthermore, a high fundamental frequency is desired since it implies a large bandwidth that will allow for fast motion without causing serious vibration problems and stable endpoint control [8]. In this research, inspired by the last three approaches in [6], [7], [8], the control of a two-link flexi-ble arm with variaflexi-ble cross-section by design is improved by employing the Lyapunov method. Different from the energy based multi-link flexible robot control proposed in [9, ch.2], LaSalle’s invariance principle extended for infinite dimension [10] will be used in order to prove the asymptotic stability of the closed loop system without any modal truncation such that the higher order modes will not

Y0 Y1 Y2 X0 Oh1 X2 X1 Oh2 Ot1 _w 1 w2 T1 T2

Fig. 1. Arm configuration [4]

be ignored.

Referring to Figure 1 the various symbols represent the following;XoYo: global inertial system of coordinates;

X1Y1: body-fixed system of coordinates attached to unde-formed link 1, X2Y2: body-fixed system of coordinates attached to undeformed link 2, θ1, θ2: angular displace-ments of links 1 and 2,w1, w2: flexural displacements of links 1 and 2, Ohi: offset of the beam root to the center

of theith_{input torque motor,O}

ti: offset of the beam end

to the center of tip mass [4].

II. ANALYTICALMODEL

In this section, an exact model using partial differential equations with boundary conditions is derived using the Hamilton’s Principle based on Zhang’s work for uniform links [13], [4]. Since the ratio between the length of the beam and its thickness is sufficiently large as proposed in [14], [15], links can be modelled as Euler-Bernoulli beams, which can only be deformed in the flexural direc-tion. The links are modelled in clamped-free configuration, since natural modes of the separated clamped-free links agree very well with actual ones compared to pinned-free configuration [16]. Assuming the manipulator moves in the horizontal plane, in the absence of gravity the potential energy depends only on the flexural deflections.

In order to derive the PDE model, kinetic and potential energy expressions for the links and hubs are obtained.

Proceedings of the 2006 American Control Conference

Parameter Description

E Young’s Modulus

Ii(xi) Variable beam cross section moment to z axis at the locationxi

Ihi Inertia ofithhub

Iti Tip inertia ofithbeam

li Length ofithlink

mhi Mass ofithhub

mti Tip mass ofithbeam

xi Coordinate along the axial center of theithbeam

wi(xi, t) Transverse movement of pointi at the locationx_iofithbeam ˙wi(xi, t) Time rate of transverse movement

of pointi at the location xiofithbeam

wix(xi, t) Axial rate of transverse movement of pointi at the location x_iofithbeam

ρi(xi) Variable density of theithlink depends on the cross-sectional area atxi

τi Input torque atithmotor

θ1 Angular position of the first motor ˆθ2 Angular position of the second motor

θ2 θ2= ˆθ2+ w1x(l1, t) TABLE I

PARAMETERS FORPDEMODEL[4].

Then, Extended Hamilton’s Principle [14] is applied such that

 t2

t1

(δT − δVs + δWnc) dt = 0 (1)

where δT and δVs are the variation of total kinetic and

potential energy respectively,δWncis the variation of

non-conservative work. Thus, the governing equations for a two-link flexible arm with nonuniform cross-section are derived by the variational method and integration by parts and listed below with the notation given in TABLE I:

¨ w1+(EI1(x1)w1xx)xx ρ1(x1) = −x1¨θ1 (2) ¨ w2+(EI2(x2)w2xx)xx ρ2(x2) + Cosθ2  ¨ w1(l1, t) + l1¨θ1  − ˙θ2Sinθ2  ˙w1(l1, t) + l1˙θ1  = −x2¨θ2 (3) Ih1¨θ1− EI1(0)w1xx(0, t) = τ1 (4) It1Ih2¨θ2− Ih2EI1(l1)w1xx(l1, t) −It1EI2(l2)w2xx(l2, t) = (It1+ Ih2) τ2 (5) −  l2 0 ρ2(x2) x2dx2 ˙θ2Sinθ2  ˙w1(l1, t) + l1˙θ1  +  l2 0 x2(EI2(x2)w2xx)xxdx2 −Ih2[¨θ1+ ¨θ2] − It2[¨θ1+ ¨θ2+ ¨w2x(l2, t)] − Sinθ2  ˙w1(l1, t) + l1˙θ1   l2 0 ρ2(x2) ˙w2dx2 −mt2l2  l2¨θ2+ ¨w2(l2, t) + Cosθ2  ¨ w1(l1, t) + l1¨θ1  −mt2Sinθ2  ˙w1(l1, t) + l1˙θ1  ˙w2(l2, t) −It1[¨θ1+ ¨w1x(l1, t)] − EI1(l1)w1xx(l1, t) = 0 (6) −  l2 0 ρ2(x2)  ¨ w1(l1, t) + l1¨θ1  +Cosθ2  ¨ w2+ x2¨θ2  − ˙θ2Sinθ2  ˙w2+ x2˙θ2  dx2 −(mt1+ mh2+ mt2)  ¨ w1(l1, t) + l1¨θ1  + [(EI1(x1)w1xx)x]x1=l1 −mt2  Cosθ2  ¨ w2(l2, t) + l2¨θ2  − ˙θ2Sinθ2  ˙w2(l2, t) + l2˙θ2  = 0 (7) [(EI2(x2)w2xx)x]x2=l2 −mt2  ¨ w2(l2, t) + l2¨θ2  + Cosθ2  ¨ w1(l1, t) + l1¨θ1  − ˙θ2Sinθ2  ˙w1(l1, t) + l1˙θ1  = 0 (8) It2  ¨θ1+ ¨θ2+ ¨w2x(l2, t)  + EI2(l2)w2xx(l2, t) = 0 (9) w1(0, t) = w1x(0, t) = w2(0, t) = w2x(0, t) = 0 (10)

where all offset values (Ohi, Oti) can be omitted by design

and it is assumed thatθ2≈ ˆθ2since the slope at the end of the first link is relatively small. The equations ( 2) and ( 3) are the main PDEs(balance of forces) and the equations ( 4) and ( 5) are ODEs (conservation of momentum) for rigid coordinates. Equations ( 6 - 10) are the boundary conditions where equation ( 10) includes the four boundary conditions at the clamped end of the links [17].

We first define the following variables fori = 1, 2 τbi = −EIi(0)wixx(0, t) (11)

τei = −EIi(li)wixx(li, t) (12)

whereτb1, τb2 represent the base strain measured torques

and τe1, τe2 represent the end strain measured torques

for the first and second beams respectively. Boundary conditions ( 6), ( 8) and ( 9) can be simplified further

to obtain the following one Sinθ2  ˙w1(l1, t) + l1˙θ1   − ˙θ2  l2 0 ρ2(x2) x2dx2 −l2 ˙θ2mt2 −  l2 0 ρ2(x2) ˙w2dx2 − mt2 ˙w2(l2, t)  −τb2+ τe1− Ih2[¨θ1+ ¨θ2] − It1[¨θ1+ ¨w1x(l1, t)] = 0 (13)

where the new set of boundary conditions will be ( 13), ( 7), ( 8) and ( 10).

III. CONTROLLERDESIGN

The total energy of the system ( 2 - 10) is calculated to get the total energy rate of the system. Then with this insight, the control laws are obtained by Lyapunov approach as given below

τ11= τb1+ Ih1(− ¨w1(l1, t)

l1 − K1(α ˙θ1+ β(θ1− θ1d))) (14)

τ21= (Ih2τe1+ It1τe2+ It1Ih2(− ¨w2(l2, t)

l2

−K2(α ˙θ2+ β(θ2− θ2d))))/(It1+ Ih2) (15)

where K1, K2, α, β are positive constants and θ1d, θ2d

are the desired positions. Besides,w¨i(li, t) signals in the

control laws can be obtained after filtering the output of the wireless accelerometers which are known to be vibration measurement tools. On the other hand, the control laws ( 14 - 15) should be augmented with a parallel controller to ensure asymptotic stability of the closed loop system. In addition to ( 14 - 15), we also propose the following control laws τ12 = τb2− τe2− K ˙θ1 (16) τ22 = It1Ih2(¨θ1+ ¨θ2) + It1τe2 /(It1+ Ih2)(17) τ1 = τ11+ γ1τ12 (18) τ2 = τ21+ γ2τ22 (19)

where K, γ1, γ2 > 0 are proportional gains that are

adjusted to make the control lawsτ11, τ21 dominant. This

is achieved by choosing small γ1 and γ2. The main controller ( 18 - 19) consists of the parallel connection of two separate controllers. Since the system ( 2 - 10) became decoupled and linear by the dominant part of the controller, then additivity and homogeneity properties of linear systems can be applied for the closed-loop system. Thus, the effects of the control laws τ12, τ22 should be considered independently from the others. Consequently, the main theorem can be introduced as follows

Theorem 3.1: The control laws ( 18 - 19) achieve

set-point regulation of the rigid modes and asymptotic stability of the two-link flexible arm with variable cross-section.

Proof:

The total energy V of the system is a good candidate

for Lyapunov function and can be expressed in terms

of kinetic and potential energy terms with additional correction term such that

V = T1+ T2+ T3+ T4+ T5+ T6+ Vs+ ¨θ21 ≥ 0 (20)

whereTis are the kinetic energy terms of beams, tips and

hubs for link 1 and 2 [13]. The total strain potential energy for both links is represented as Vs. After applying the

control laws ( 18 - 19) to the system ( 2 - 10) and after some lengthy but straightforward calculations, the total energy rate of the system is obtained forM > 0 as

follows ˙V < − 2K1β  K1(α + β M) + K γ1 Ih1  ˙θ2 1 − 2  K1α +Kγ1 Ih1  ¨θ2 1< 0 . (21)

Note that the equation ( 21) only assures the stability of the closed loop system but does not prove the asymptotic stability. The latter can be shown by using LaSalle’s in-variance principle extended to infinite dimensional spaces [10]. In order to initiate this part of the proof, equations ( 2)- ( 5) should be rewritten in the new coordinates such that

z = [z1 z2 z3 z4 z5 z6 z7 z8]T

= [w1 ˙w1 w2 ˙w2 θ1− θ1d ˙θ1 θ2− θ2d ˙θ2]T

The system ( 2)- ( 5) can be decomposed to linear and nonlinear parts separately, see [11], such as

˙z = A z + f(z) (22)

whereA represents the infinite dimensional linear operator

andf represents the nonlinear operator. For the linear part

of the system ( 2)- ( 3) which is dissipative by feedback controls ( 18 - 19), will be considered in the energy Hilbert space Hi = HE2(0, li) × L2(0, li), HE2(0, li) =

{wi ∈ H2(0, li) | wi(0) = wix(0) = 0}, in which the

inner product induced norm is defined by [12]

(wi, gi)2Hi =  li

0 [ρi(x)|gi(x)|

2_{+ EI}

i(x)|wixx(x)|2] dx

for i = 1, 2 and ∀ (wi, gi) ∈ Hi, 0 < x < li. Define

operatorAi: D(Ai)(⊂ Hi) → Hi as

Ai(wi, gi) = ( gi, −1

ρi(x)(EIi(x)wixx(x))xx)

(23)

D(Ai) = {(wi, gi) ∈ (HE2 ∩ H4) × HE2}

whereA−1_i is compact on Hi [12, Lemma 2.1]. We also

define the function space ˆ

Using the set of equations ( 2 - 10) and integration by parts, we have the linear operator A : ˆH → ˆH and the

nonlinear operatorf : ˆH → ˆH are given in equation ( 22)

such as: A = ⎡ ⎣ ₀A1_2×2 0_A22×2 0_B11×4 04×2 04×2 B2 ⎤ ⎦ (24) f(z) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 −x1W1−Ixt11 (τe1− τb2+ τe2) 0 −x2W2+Ixt12 (τe1− τb2) + x2 Ih2+It1 Ih2It1 τe2+ f0 0 W1+_I1_t1(τe1− τb2+ τe2) 0 W2−_I1_t1(τe1− τb2) −IIh2h2+IIt1t1τe2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ where A1, A2 are given in equation ( 23); τei, τbi are

defined by equations ( 11)-( 12), and B1, B2, W1, W2, f0

are given below

B1 = ⎡ ⎢ ⎣ 0 0 x1Ih2β˜ Ih2+It1 x1Ih2α˜ Ih2+It1 0 0 0 0 0 0 −x2˜β −x2˜α ⎤ ⎥ ⎦ B2 = ⎡ ⎢ ⎢ ⎣ 0 1 0 0 0 0 −Ih2β˜ Ih2+It1 −Ih2α˜ Ih2+It1 0 0 0 1 0 0 ˜β ˜α ⎤ ⎥ ⎥ ⎦ W1 = Ih2 γ2It1 (EI2(x2)w2xx)xx l2ρ2(x2) |x2=l2 W2 = I_−γh2+ It1 2It1 (EI2(x2)w2xx)xx l2ρ2(x2) |x2=l2 f0 = Cosθ2  K1l1(α ˙θ1+ β(θ1− θ1d)) − γ1l1 Ih1 (τb2− τe2− K ˙θ1)  + ˙θ2Sinθ2  ˙w1(l1, t) + l1˙θ1  ˜β = Ih2+ It1 γ2It1 K2β ˜α = Ih2+ It1 γ2It1 K2α

where B1, B2 are finite dimensional linear bounded op-erators, f(0) = 0 and f(z) is differentiable. Since

(λ I − A)−1 is compact for λ > ˜α, (see e.g. [12]), then

it follows that the solutions of ( 22) locally exists in; moreover if z(0) ∈ D(A), then z(t) ∈ D(A) as well. Since the solutions are bounded, see ( 21), it can easily be shown that local solutions can be extended globally as well. In order to complete the proof, we should show that ˙V = 0 implies z = 0. If ˙V = 0 then δ Wnc= 0 since the

power associated with nonconservative forces is equal to the time rate of change of the total energy [14]. For the system ( 2 - 10),

δ Wnc= τ1δθ1+ τ2δ(θ2− w1x(l1, t))

where δθ1 and δ(θ2− w1x(l1, t)) are arbitrary nonzero

variations by definition thenτ1= 0 and τ2= 0. Besides,

˙V = 0 implies ˙θ1 = ¨θ1 = 0 directly and since τ1 = 0

by ( 4) and ( 11) we obtain τb1 = 0 as well. Using the

boundary condition ( 8) we getW1= W2= 0 for f(z).

Thus,W1= W2= 0, ¨θ1= 0 and equations ( 13), ( 22)

implies that τbi = τei = 0. Using these results and the

control laws ( 18 - 19) for the boundary condition ( 6) with equation ( 13), we have τb2 = −Ih2¨θ2. Equations

( 5) and ( 9) give respectively

τe2= −Ih2¨θ2= It2¨θ2 (25)

that verifies τe1 = τb2 = τe2 = ¨θ2 = 0. Thus, the main

PDEs ( 2 - 3) become homogeneous such as

ρ1(x1) ¨w1+ (EI1(x1)w1xx)xx = 0 (26)

ρ2(x2) ¨w2+ (EI2(x2)w2xx)xx = 0 (27)

where flexural deflection wi(x, t) can be expressed by

separation of variables such that [18]

wi(x, t) = φi(x)eλit (28)

where i = 1, 2 ; λi’s are nonzero complex eigenvalues

andφi’s are eigenfunctions for equations ( 26 - 27) with

the above boundary conditions that are derived for ˙V = 0.

Thus, equations ( 26 - 27) become variable coefficient, ordinary differential equations of order four, such as [18]

λ2₁ρ1(x1) φ1(x1) + d dx2₁(EI1(x1) d dx2₁φ1(x1)) = 0 (29) λ2₂ρ2(x2) φ2(x2) + d dx22(EI2(x2) d dx22φ2(x2)) = 0 (30)

For nonuniformith _link,φ

i(x) can be defined as

φi(x) = pi(x)ri(x)

φix(x) = pix(x)ri(x) + pi(x)rix(x)

φixx(x) = pixx(x)ri(x) + 2pix(x)rix(x)

+ pi(x)rixx(x)

φixxx(x) = pixxx(x)ri(x) + 3pixx(x)rix(x)

+ 3pix(x)rixx(x) + pi(x)rixxx(x)

φixxxx(x) = pixxxx(x)ri(x) + 4pixxx(x)rix(x)

+ 6pixx(x)rixx(x) + 4pix(x)rixxx(x)

+ pi(x)rixxxx(x)

wherepi(x) = aix4+ bix3+ cix2+ dix + ei andri(x)

is possibly nonlinear function which is the fourth order differentiable at least and satisfies the conditions such that ri(0) = 0, ri(li) = 0. Then the coefficients of

the polynomialpi(x) are obtained by using the boundary

conditions and the dominant control laws such as

φi(0) = 0 → ei= 0 from ( 10)

φix(0) = 0 → di = 0 from ( 10)

φi(li) = 0

φix(li) = 0 see Remark 3.2

φixx(0) = 0 → ci= 0 from τbi = 0

φixx(li) = 0 → 12ail2i + 6bili= 0 from τei= 0

φixxx(li) = 0 → 24aili+ 6bi= 0

φixxxx(li) = 0 → 24ai= 0

thus ai = bi = 0 while ρi(xi), EIi(xi), EIix(xi) and

EIixx(xi) are assumed to be nonzero at xi= 0, and xi=

li. Consequently,pi(x) = 0, φi(x) = 0, then wi= ˙wi= 0

for ˙V = 0. On the other hand, using the dominant control

laws ( 14 - 15) and the above results that include(τbi =

0, τei= 0), and equations ( 4), ( 5) yields ¨wi(li, t) = 0,

we also have ˙θ1 = ˙θ2 = 0 and θ1 = θ1d, θ2 = θ2d.

Consequently, ˙V = 0 really implies z = 0.

In the light of [10, Theorem 3.64 and 3.65], the closed-loop system ( 2 - 19) is asymptotically stable. 

Remark 3.2: Since we have already neglectedw1x, we

have also omitted w¨1x in the final calculations. Note that

such terms are also omitted in [13]. Note that although in [13] two damping terms are introduced in the simulations, we do not use any damping term in our case.

IV. SIMULATIONRESULTS

Parameter Value

Length of links, l1= 0.5 m, l2= 0.6 m Time step Δt = 3e − 5 sec Spatial steps Δx1= l1/20, Δx2= l2/20 Young’s Modulus,E 70 GP a Density,ρ 2742 kgm−3 Thickness of links (m) c1= 0.003175, c2= 0.00238 Maximum height for tapering bo= 0.0654 m at the root of the link

Linear slope for tapering a₁= 0.04 Hub inertias (kgm2) I_h1= 0.0055, Ih2= 0.0068 Tip inertias, (kgm2) I_t1= 0.0139, It2= 0.00024 Hub mass (kg) mh2= 0.678 Tip mass, (kg) mt1= 0.981, mt2= 0.204 θ1d(desired) π/2 rad θ2d(desired) -π/2 rad K1= .007, K2= 0.01 α = 600, β = 800 γ1= 1e − 3, γ2= 1e − 4 K = 9 TABLE II

PARAMETERS OF THEFLEXIBLEARM.

The proposed control scheme is tested with the simu-lation program implemented in MATLAB. The PDEs are discretized in the space domain by the finite difference

0 2 4 6 8 10 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Bending Strain of Link 1 [1/m]

time [sec]

Fig. 2. Bending strain at the end of the link 1

0 2 4 6 8 10 −2 0 2 4 6 8x 10 −4

Bending Strain of Link 2 [1/m]

time [sec]

Fig. 3. Bending strain at the end of the link 2

method, to obtain ODEs at each of the nodes. Then, ODEs are solved numerically. Instead of dealing with complexity of the fourth order derivative approximation, the second order derivative approximation has been used with the help of auxiliary states. Those states are more meaningful in a real problem as well since they corresponds to physical variables such as deflections, velocity and bending mo-ments [20]. However, the number of ODEs to solve and the computation time are increased in return of the robust stability of the numeric scheme. The parameters used in the model for system ( 2 - 10), partially given in [13] are listed in TABLE II.

Although the control setup in the previous section is given for links that can have any kind of variable cross-section; in this particular simulation, rectangular cross-sections of given uniform thickness are used. For small values of tip mass and tip inertia moment relative to the ones for beam, the optimum shape is approximately a linearly tapered beam [8]. Therefore, the height bi(x), density ρi(x)

and cross-section area momentIi(x) at any point can be

calculated with the parameters given in TABLE II such as 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 θ1 [radian] time [sec]

0 2 4 6 8 10 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 θ2 [radian] time [sec]

Fig. 5. Joint Angle of the link 2

0 2 4 6 8 10 −0.01 0 0.01 0.02 0.03 0.04 0.05 τ1 [Nm] time [sec]

Fig. 6. Control Torque of Joint 1

bi(x) = bo− 2lia1+ 2(li− x)a1 ρi(x) = ρ bi(x) ci

Ii(x) = bi(x) c3i/12.

The simulation results are presented in Figures 2 - 7. Smooth time histories of all variables of interest without overshoot show the effectiveness of the controller perfor-mance especially with such demanding desired positions (θ1d = π/2 and θ2d = −π/2). Comparing the time

responses for FEM (Finite Element Method) case in [21] with the simulation results for PDE, it is observed that the required control energy in PDE cases is much less than the one in FEM due to the exactness of PDE. Also, PDE responses are smoother than the ones in FEM approach, and have no overshoot or no chattering for all states.

V. CONCLUSIONS

It has been shown by the simulation results that the new controller design method can provide asymptotic stability of flexural modes and set-point regulation of rigid

0 2 4 6 8 10 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 τ2 [Nm] time [sec]

Fig. 7. Control Torque of Joint 2

modes simultaneously. In the proof of the main theorem, infinite dimensionality of the problem has been retained as opposed to other energy-based approaches for multi-link robot arms in the literature. In future, compensation of variable tip mass would be an extension to the proposed controller that already manages the nonuniform flexible robot arms successfully.

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