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### Submitted on 18 Dec 2006

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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:**

*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)+B*1*(ρ(t))w(t)+B*2*(ρ(t))u(t) (1a)*
*z(t) = C*1*(ρ(t))x(t)+D*1*(ρ(t))w(t)+D*2*(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) _{∈ IR}r*

*is the disturbance vector. The disturbance w ∈*

*L*2

*(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))* *B*1*(ρ(t)) B*2*(ρ(t))*
*C*1*(ρ(t)) D*1*(ρ(t)) D*2*(ρ(t))*
#
such that
"
*A(ρ(t))* *B*1*(ρ(t)) B*2*(ρ(t))*
*C*1*(ρ(t)) D*1*(ρ(t)) D*2*(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) =* &*q*1 *... qk* *... qn*

'*T*

*, ˙q(t), ¨q(t) ∈ IRn* _{are}

joints position, velocity and acceleration vectors,
*respec-tively. M(q) ∈ IRn×n*_{is 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.}

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

rewritten in the following form [10], [11]
"
*M*11*(q*2*) M*12*(q*2)
*M*21*(q*2*) M*22*(q*2)
# "
*¨q*1
*¨q*2
#
+
"
*−C*12*(q*2*) ˙q*2 *−C*12*(q*2*)( ˙q*1*+ ˙q*2)
*C*12*(q*2*) ˙q*1 0
# "
*˙q*1
*˙q*2
#
=
"
*τ*1
*τ*2
#
(7)
with
*M*11*(q*2*) = (m*1*+m*2*)r*12*+m*2*r*22*+2m*2*r*1*r*2*cos(q*2) (8a)
*M*12*(q*2*) = M*21*(q*2*) = m*2*r*22*+ m*2*r*1*r*2*cos(q*2) (8b)
*M*22*(q*2*) = m*2*r*22 (8c)
*C*12*(q*2*) = m*2*r*1*r*2*sin(q*2) (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* *∈ IR*2*×2* *and bi* *∈ IR*2*×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 ˙q*1*(t) and ˙q*2*(t) are bounded as *

fol-lows

*˙q*1*(t)" ν*1*,* *˙q*2*(t)" ν*2 (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 A*33 *A*34
*0 0 A*43 *A*44
*, B(q) =*
0 0
0 0
*m*11 *m*12
*m*21 *m*22
(15)
where
*A*33*= m*11*C*12*(q*2*)ν*2*− ν*1*m*12*C*12*(q*2) (16)
*A*34*= m*11*C*12*(q*2*)(ν*1*+ ν*2) (17)
*A*43*= m*21*C*12*(q*2*)ν*2*− ν*1*m*22*C*12*(q*2) (18)
*A*44*= m*21*C*12*(ν*1*+ ν*2) (19)
and
*m*11= *M*22
*(q*2)
*D*1*(q*2) *, m*
12*= m*21*= −M*12
*(q*2)
*D*1*(q*2)
*m*22=
*M*11*(q*2)
*D*1*(q*2) (20)
with
*D*1*(q*2*) = [M*11*(q*2*)M*22*(q*2*) − M*12*(q*2*)M*21*(q*2)] (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}

*∈*

IR*2n×2n _{, Z}*

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

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

*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}= −2γ*2

*I,*

*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 θ*1

*AiW + W AiT+ B2iZi+ ZiTB2iT*2 (31)

*Nθ12= cos θ*1

*AiW + B2iZi− W AiT*

*− ZiTB2iT*2 (32) "

*Pθ11*

*Pθ12*

*PT*

*θ12*

*Pθ11*#

*< 0*(33)

*with (for i = 1, ..., N − 1, j = i + 1, ..., N)*

*Pθ11= sin θ*1

*(Ai+ Aj)W + W (Ai+ Aj*)

*T*

*+ B2iZj*

*+B2jZi+ ZjTB2iT*

*+ ZiTB2jT*2 (34)

*Pθ12= cos θ ((Ai+ Aj)W + B2iZj+ B2jZi*

*−W (Ai+ Aj*)

*T*

*− ZjTB2iT*

*− ZiTB2jT*2 (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 θ*1

*Acl(ρ)W + W AclT(ρ)*2 (40)

*Mcl12= cos θ*1

*Acl(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*
*ρi*2*(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

*| ˙q*2*(t)| " ˙qmax,* *| ˙q*2*(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

*r*1*= 0.50m, r*2*= 0.35m,*

*m*1*= 54.21kg, m*2*= 17.99kg (45)*

*The system (1)-(2) matrices are given by (15) for Ai*and

*B2i*. The other matrices are taken as follow

*Taking α = 0.5, r = 3 and θ =* *π*_{4} for the pole placement
*requirements, the different gains Kiare given by (with γ =*

*2.71)*
*K*1=
"
*−37.654 −7.590 −71.987 −14.511*
*−7.590* *−3.125 −14.511* *−5.975*
#
*K*2=
"
*−17.003 −3.428 −34.103 −6.874*
*−3.427* *−1.411* *−6.874* *−2.830*
#
*K*3=
"
*−8.938* *0.605* *−17.928* *1.212*
*0.604* _{−1.411}*1.213* * _{−2.830}*
#

*K*4= "

*−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 q*1.

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 q*2.

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 ˙q*1.
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.*