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H-infinity control of a SCARA robot using polytopic

LPV approach

Harouna Souley Ali, Latifa Boutat-Baddas, Yasmina Becis-Aubry, Mohamed

Darouach

To cite this version:

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H

control of a SCARA robot using

polytopic LPV approach

H. Souley Ali, L. Boutat-Baddas, Y. Becis-Aubry and M. Darouach

Abstract— This paper investigates the H∞ control of robotic

system presenting a Linear Parameter Varying (LPV) repre-sentation. From a usual Lagragian equation of the system, its LPV representation is given and is reduced to a poly-topic one. Then recent developments on polypoly-topic LPV H∞

approach are used to design a state feedback controller for the robotic system. The approach is extended to take into account pole placement requirements.

Keywords— SCARA robot manipulator, LPV system, Poly-topic representation, Robust design, Controller design, H-infinity optimization, pole placement, LMI.

I. Introduction

Many researches during the last decades concerns the attempts to apply linear techniques to control nonlinear systems. Thus, the gain scheduling and the polytopic H∞

approaches are used in many references [1], [2], [3], [4] to build controllers for nonlinear models. As it is well known, robotic manipulators constitute an important class of nonlinear systems; then these methods have been ap-plied specifically to them as shown, for example, by [5], [6], [7], [3], [8].

Here, we consider a robotic manipulator which is given with a LPV representation. Then, using a decomposition introduced in [8], the system is finally modelled with a polytopic structure. The main drawback of this approach should be the necessity to introduce a filter in order to have a constant control matrix for the LPV representation of the system [9], [8]. The order of the system is then increased and some implementation problems can also appear after the design of the control law.

The approach which is proposed in this paper does not need the introduction of the filter as all the system matrices may depend on the varying parameter. We extend some recent results concerning polytopic LPV H∞ control (see

[4]) to search for a control law that ensures pole placement requirements in addition to the H∞ performance.

II. Position of the problem

In this paper, the purpose is to put a robotic manip-ulator system in the following polytopic LPV form with respect tp the varying parameter ρ(t) [1], [8], [4]

˙x(t) = A(ρ(t))x(t)+B1(ρ(t))w(t)+B2(ρ(t))u(t) (1a) z(t) = C1(ρ(t))x(t)+D1(ρ(t))w(t)+D2(ρ(t))u(t) (1b)

where x(t) ∈ IR2n is the state vector (n is the number

of joints), z(t) ∈ IRq

is the controlled output, y(t) ∈ IRp

is the output vector, u(t) ∈ IRm is the control input and

w(t)∈ IRr is the disturbance vector. The disturbance w ∈ L2(0, ∞).

The authors are with Centre de Recherche en Automatique de Nancy (CRAN - UMR 7039), Nancy - Université − CNRS, IUT de Longwy, 186 rue de Lorraine 54400 Cosnes et Romain, FRANCE souley,baddas@iut-longwy.uhp-nancy.fr

As (1) is in polytopic form, it verifies (see [8], [4])

P := !" A(ρ(t)) B1(ρ(t)) B2(ρ(t)) C1(ρ(t)) D1(ρ(t)) D2(ρ(t)) # such that " A(ρ(t)) B1(ρ(t)) B2(ρ(t)) C1(ρ(t)) D1(ρ(t)) D2(ρ(t)) # = ν $ i=1 ρi(t) " Ai B1i B2i C1i D1i D2i # ; ρi(t)! 0; ν $ i=1 ρi(t) = 1 % (2)

P is a convex polytope with ν vertex, the ith vertex is

defined by (Ai, B1i, B2i, C1i, D1i, D2i) where each of these

matrices is constant.

Problem 1. The objective is to design the following

poly-topic LPV state feedback controller

u(t) = K(ρ(t))x(t) (3) where K(ρ(t)) = ν $ i=1 ρi(t)Ki, ρi(t)! 0, ν $ i=1 ρi(t) = 1. (4)

such that the closed loop (1)-(3) is quadratically stable, sat-isfies the H∞ performance

#z(t)#2" γ#w(t)#2 where γ > 0 (5) and the closed-loop poles are placed in the region S(α, r, θ). Note that having the poles of a system in S(α, r, θ) ensures a minimum decay rate α, a minimum damping ratio ξ =

cos(θ) and a maximum undamped natural frequency ωd =

r sin(θ) (see [8]). ❒

III. Model of the SCARA robotic manipulator under consideration

A robot manipulator is defined as an open kinematic chain of rigid links. Using the Euler-Lagrangian method, the equation of motion of a n degrees of freedom robot manipulator is given by

M (q)¨q + C(q, ˙q) ˙q + G(q) = τ (6)

where q(t) = &q1 ... qk ... qn

'T

, ˙q(t), ¨q(t) ∈ IRn are

joints position, velocity and acceleration vectors, respec-tively. M(q) ∈ IRn×nis the symmetric and positive definite

inertia matrix, C(q, ˙q) ∈ IRn×n represents the centrifugal

and Coriolis forces, G(q) is the vector of gravitational forces and τ ∈ IRn is the torque input vector.

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assumption that the centers of mass of links 1 and 2 are at the distant ends (r1 = )1, r2 = )2) , the model (6) is

rewritten in the following form [10], [11] " M11(q2) M12(q2) M21(q2) M22(q2) # " ¨q1 ¨q2 # + " −C12(q2) ˙q2 −C12(q2)( ˙q1+ ˙q2) C12(q2) ˙q1 0 # " ˙q1 ˙q2 # = " τ1 τ2 # (7) with M11(q2) = (m1+m2)r12+m2r22+2m2r1r2cos(q2) (8a) M12(q2) = M21(q2) = m2r22+ m2r1r2cos(q2) (8b) M22(q2) = m2r22 (8c) C12(q2) = m2r1r2sin(q2) (8d)

Note that no gravitational effects are considered in this model. Gravity is acting along the vertical direction, so it does not work in the directions where the manipulator can move. Then, the gravitational forces vector G(q) is identically zero for all configurations of the manipulator.

The model (7) does not take into account actuators dy-namics (motors and gearboxes), as well as friction forces and external disturbances which will be added in the sim-ulation model. Then (7) can be written as

d d t " q ˙q # = " ˙q −M−1(q)C(q, ˙q) ˙q # + " 0 M−1(q) # τ (9)

We assume that the nonlinear system (9) can be repre-sented by a polytopic form as in [3], [8], [12]; that is

d d t " q ˙q # = ν $ i=1 ρi(t) (" 0 I 0 ai # " q ˙q # + " 0 bi # τ ) (10) which can be rewritten in the compact form

˙x = ν $ i=1 ρi(t) (Aix + Biu) (11) where *ν

i=1ρi(t) = 1, ρi(t) > 0 for i = 1, ..., ν, and with

Ai= " 0 I 0 ai # , Bi= " 0 bi # , x = " q ˙q # and u = τ. (12) where ai ∈ IR2×2 and bi ∈ IR2×2 are obtained from

−M−1(q)C(q, ˙q) and M−1(q) respectively. A

i and Bi are

the vertexes of the polytopic system so defined. Then sys-tem (1) is finally put in the polytopic LPV form of (7).

Based on [8], we then suppose that the joints positions

qk are such that qmin" qk" qmax, for k = 1, ..., n where n

is the number of joints of the manipulator. So, the motion range has N = 2n vertices given by the different

combina-tions Si for i = 1, ..., N between

((q1min)OR(q1max), (q2min)OR(q2max),

..., (qnmin)OR(qnmax)) (13)

Applying this to (11), the polytopic system has ν =

N = 2n vertexes obtained from each of the combinations

of (13).

By considering that ˙q1(t) and ˙q2(t) are bounded as

fol-lows

˙q1(t)" ν1, ˙q2(t)" ν2 (14)

each vertex i is obtained by replacing a combination of (13) in the following matrices (see [8] for details)

A(q) =       0 0 1 0 0 0 0 1 0 0 A33 A34 0 0 A43 A44      , B(q) =       0 0 0 0 m11 m12 m21 m22       (15) where A33= m11C12(q22− ν1m12C12(q2) (16) A34= m11C12(q2)(ν1+ ν2) (17) A43= m21C12(q22− ν1m22C12(q2) (18) A44= m21C121+ ν2) (19) and m11= M22 (q2) D1(q2) , m 12= m21= −M12 (q2) D1(q2) m22= M11(q2) D1(q2) (20) with D1(q2) = [M11(q2)M22(q2) − M12(q2)M21(q2)] (21)

IV. Theoretical results

In this section, we give sufficient conditions to ensure quadratic stability, H∞ performance and pole placement

requirement for system (1)-(2) by using the state feedback (3). These conditions are given as follows

Theorem 1. Problem 1 is resolved if, for a γ > 0 and

a region S(α, r, θ), there exist matrices W = WT > 0

IR2n×2n, Z

i ∈ IRm×2n, Zj ∈ IRm×2n such that the LMI

(22),(23), (25), (26), (27), (28), (30) and (33) are verified

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M∞112= B2iZj+ B2jZi+ ZjTB2iT + ZiTB2jT (24c) M∞12= B1i+ B1j (24d) M∞13= W (Ci+ Cj)T + ZjTD2iT + ZiTD2jT (24e) M∞21= M∞12T (24f) M∞22= −2I (24g) M∞23= D1iT + D1jT (24h) M∞31= M∞13T (24i) M∞32= M∞23T (24j) M∞33= −2γ2I, i = 1, ..., N− 1, j = i + 1, ..., N (24k) AiW + W AiT + B2iZi+ ZiTB2iT+ 2αW < 0, i = 1, ..., N; (25) (Ai+ Aj)W + W (Ai+ Aj)T+ B2iZj+ ZjTB2iT+ B2jZi+ ZiTB2jT + 2αW < 0, i = 1, ..., N− 1, j = i + 1, ..., N (26) " −rW AiW + B2iZi W AiT + ZiTB2iT −rW # < 0 i = 1, ..., N ; (27) " −rW Mr12 MT r12 −rW # < 0 (28) with (for i = 1, ..., N − 1, j = i + 1, ..., N) Mr12= ((Ai+ Aj)W + B2iZj+ B2jZi) (29) " Nθ11 Nθ12 Nθ12T Nθ11 # < 0 (30) with (for i = 1, ..., N) Nθ11= sin θ1AiW + W AiT+ B2iZi+ ZiTB2iT2 (31) Nθ12= cos θ1AiW + B2iZi− W AiT − ZiTB2iT2 (32) " Pθ11 Pθ12 PT θ12 Pθ11 # < 0 (33) with (for i = 1, ..., N − 1, j = i + 1, ..., N) Pθ11= sin θ1(Ai+ Aj)W + W (Ai+ Aj)T + B2iZj +B2jZi+ ZjTB2iT + ZiTB2jT2 (34) Pθ12= cos θ ((Ai+ Aj)W + B2iZj+ B2jZi −W (Ai+ Aj)T − ZjTB2iT − ZiTB2jT2 (35) ■

Proof. The pole placement requirements (for a fixed ρ) can

be expressed through the following inequalities [13], [14], [8]

Acl(ρ)W + W AclT(ρ) + 2αW < 0 (36) " −rW Acl(ρ)W W AclT(ρ) −rW # < 0 (37) " Mcl11 Mcl12 MT cl12 Mcl11 # < 0 (38) where Acl(ρ) = ((N $ i=1 ρi(t)Ai ) + (N $ i=1 ρi(t)B2i ) ( N $ i=1 ρi(t)ZiW−1 )) (39) Mcl11= sin θ1Acl(ρ)W + W AclT(ρ)2 (40) Mcl12= cos θ1Acl(ρ)W − W AclT(ρ)2 (41)

Using the fact that*N

i=1ρi(t) = 1, inequality (36) can

be rewritten as Acl(ρ)W + W AclT(ρ) + ($N i=1 ρi(t) ) ($N i=1 ρi(t) ) 2αW < 0 (42) which is equivalent to N $ i=1 ρi2(t)Li+ N$−1 i=1 N $ j=i+1 ρi(t)ρj(t)Lij < 0 (43)

where Li and Lij are respectively the left part of

inequali-ties (25) and (26). Then it is obsvious if (25) and (26) are verified, the closed loop has its eigenvalues verifying the α stability. The other LMI are proven in a similar way (see [13], [14]).

Note that the LMI (22) and (23) expressing the quadratic stability and the H∞ performance are obtained

in [8] by rewritting the well-known bounded real lemma for the closed loop system (1)-(3).

V. Numerical example

We consider the following bounds on the velocities

| ˙q2(t)| " ˙qmax, | ˙q2(t)| " ˙qmax with ˙qmax= 10[rad/s]

(44) and the positions are such that q1min = 0, q1max = π, q2min= 0 and q2max= π.

As we consider the simulation example of the SCARA robot manipulator in [11], the parameter values are

r1= 0.50m, r2= 0.35m,

m1= 54.21kg, m2= 17.99kg (45)

The system (1)-(2) matrices are given by (15) for Aiand

B2i. The other matrices are taken as follow

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Taking α = 0.5, r = 3 and θ = π4 for the pole placement requirements, the different gains Kiare given by (with γ =

2.71) K1= " −37.654 −7.590 −71.987 −14.511 −7.590 −3.125 −14.511 −5.975 # K2= " −17.003 −3.428 −34.103 −6.874 −3.427 −1.411 −6.874 −2.830 # K3= " −8.938 0.605 −17.928 1.212 0.604 −1.411 1.213 −2.830 # K4= " −19.794 1.339 −37.842 2.561 1.339 −3.125 2.560 −5.975 #

Notice that the simulations are performed using Matlab LMI Control Toolbox.

Positions and velocities are then given in the figures be-low. 0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 1. The first joint position q1.

0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 2. The second joint position q2.

0 2 4 6 8 10 12 14 16 18 20 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Fig. 3. The velocity ˙q1. VI. Conclusion

Although the calculations can be cumbersome due to the huge number of LMI (especially when there are many vertexes), the method proposed in this paper seems to be more convenient than that in [8] where the initial system must be transformed. In fact with the recent development of microprocessor, the difficulty of our method can be avoid whereas for the method of [8], the difficulty is in the im-plementation.

References

[1] P. Apkarian, P. Gahinet, and G. Becker, “Self-scheduled H∞

con-trol of linear parameter-varying systems,” in Proc. IEEE

Amer-ican Contr. Conf., (Baltimore, USA), pp. 856–860, 1994.

[2] P. Apkarian and R. Adams, “Advanced gain-scheduling tech-niques for uncertain systems,” IEEE Trans. Contr. Syst. Techn., vol. 6, pp. 21–32, 1997.

[3] G. Angelis, System Analysis, modelling and control with

poly-topic linear models. PhD thesis, Technische Universiteit

Eind-hoven, The Netherlands, 2001.

[4] V. Montagner, R. Oliveira, V. Leite, and P. Peres, “LMI ap-proach for H∞linear parameter-varying state feedback control,”

IEE Proc. - Part D, Contr. Theory & Applications, vol. 152,

2005.

[5] T. Namerikawa, M. Fujita, and F. Matsumura, “H∞control of

robot manipulator using a linear parameter varying represen-tation,” in Proc. IEEE American Contr. Conf., (Albuquerque, USA), 1997.

[6] Z. Kang, Y. Yin, S. Fujii, and T. Chai, “Gain-scheduling H∞

vibration control for scara type robot manipulators via lmis,” in

Proc. IEEE American Contr. Conf., (Albuquerque, USA), 1997.

[7] Z. Kang, T. Chai, K. Oshima, J. Yang, and S. Fujii, “Robust vi-bration control for scara-type robot manipulators,” Control

En-gineering Practice, vol. 5, pp. 907–917, 1997.

[8] Z. Yu, H. Chen, and P. Woo, “Gain scheduled LPV H∞control

based on lmi approach for a robotic manipulator,” Journal of

Robotic Systems, vol. 19, pp. 585–593, 2002.

[9] P. Apkarian, P. Gahinet, and G. Becker, “Self-scheduled H∞

control of linear parameter-varying systems: a design example,”

Automatica, vol. 31, pp. 1251–1261, 1995.

[10] F. Lewis, C. Abdallah, and D. Dawson, Control of Robot

Ma-nipulators. New York: Macmillan, 1993.

[11] S. Arisariyawong and S. Charoenseang, “Reducing steady-state errors of a direct drive robot using neurofuzzy control,” in

Proc. Asian Symposium on Industrial Automation and Robotics,

(Bangkok, Thailand), 2001.

[12] N. Arrifano and V. Oliveira, “Guaranteed cost fuzzy controllers for a class of uncertain nonlinear dynamic systems,”

Computa-tional & Applied Mathematics, vol. 24, 2005.

[13] M. Chilali and P. Gahinet, “H∞design with pole placement

con-straints: an LMI approach,” IEEE Trans. Aut. Contr., vol. 41, pp. 358–367, 1996.

[14] C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective output-feedback control via LMI optimization,” IEEE Trans. Aut.

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