Journal of the Franklin Institute 338 (2001) 353–370

Robust fuzzy logic tracking control of mechanical systems Sylvia Kohn-Rich, Henryk Flashner* Department of Aerospace and Mechanical Engineering, University of Southern California, University Park, 3650 McChintock, 430 Olin Hall, Los Angeles, CA 90089-1453, USA

Abstract A methodology for the design of fuzzy control laws for tracking control of mechanical systems is described. The approach uses Lyapunov’s stability theory to formulate a class of control laws that guarantee convergence of the tracking errors to within specification limits in presence of bounded parameter uncertainties and input disturbances. The proposed methodology results in control laws that possess a large number of parameters and functional relationships to be chosen by the designer. The flexibility of the approach makes it suitable for fuzzy logic implementation. Different fuzzy implementations of the proposed control methodology are described. All implementations guarantee tracking error convergence to within prespecified performance limits. Simulations using a model of a two-degree-of-freedom robot manipulator were performed to investigate fuzzy and non-fuzzy implementations of the proposed methodology. The study demonstrates better performance of the fuzzy control implementation compared to its non-fuzzy counterpart. # 2001 Published by Elsevier Science Ltd. on behalf of The Franklin Institute.

1. Introduction Since its inception by Zadeh [1], fuzzy logic theory has attracted diehard supporters and antagonists. In the context of control engineering, the former group advocates the universal aptness of fuzzy logic theory to all control problems based

*Corresponding author. Tel.: +1-213-740-0489; fax: +1-213-740-8071. E-mail address: hfl[email protected] (H. Flashner). 0016-0032/01/$20.00 # 2001 Published by Elsevier Science Ltd. on behalf of The Franklin Institute. PII: S 0 0 1 6 - 0 0 3 2 ( 0 0 ) 0 0 0 9 3 - 4

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on its successful application to consumer appliances and industrial processes [2]. The diehard antagonists of fuzzy logic control (FLC) claim that conventional control based on rigorous mathematics can outperform any FLC. However, in the last decade a third point of view has emerged which seeks consensus from a synthesis of both methodologies. The emerging hybrid control methodology combines the conventional control theory with soft-computing theories to create controllers for dynamical systems [3]. Mechanical systems with large uncertainties, nonlinearities and bounded external disturbances have been successfully controlled using Lyapunovbased robust controllers (see the survey in [4]). In the past few years, the major focus of research in control of conventional mechanical systems has been on robust and adaptive controllers [5–8] where a nominal model is assumed and uncertainty accounts for all aspects of the system not accounted for by this nominal model. Some properties of the nominal system and uncertainty are assumed, such as the boundedness of the inertia matrix, the structure of the bound of the Coriolis/centrifugal forces, and the boundedness of the gravitational forces. Considerable research effort has been directed toward laying the foundation of a systematic design methodology of FLC systems [9]. In the last decade, there have been several research directions in fuzzy logic control. One deals with modeling and stability of Takagi–Sugeno–Kang (TSK) linear-time-varying-type systems using Lyapunov, linear matrix inequalities and H1 methodologies [10–14]. The corresponding control strategy is closely related to traditional gain scheduling. Another research direction deals with stability of PID-like fuzzy logic controllers (PID-FLCs) [15–17] based on classical control theory techniques. A third research direction focuses on the application of Lyapunov, variable structure (sliding mode) and adaptive control techniques to nonlinear systems for which the nominal dynamics (analytic functions or TSK models) and the bounds on the model uncertainties are known [18–21]. This paper presents a new robust control design approach for tracking control of mechanical systems. The approach is based on Lyapunov stability theory and extends the work of Corless and Leitmann [22,23] on robust control of nonlinear systems. It also extends the work of Efrati and Flashner [24]. The proposed control law defines a large class of robust tracking controllers allowing a great deal of flexibility for the designer, thus making it appropriate for fuzzy logic implementation to enhance the overall system performance. The importance of the new methodology is that it provides a general procedure for the design of robust-fuzzy hybrid (analog+fuzzy) control laws for a large class of mechanical systems. The paper is organized as follows. In Section 2, the characteristics of mechanical systems and fuzzy logic controller (FLC) are presented, and the new robust tracking law is formulated. Three tracking FLCs are presented in Section 3. The performance of the proposed approach is demonstrated in a simulation example for a rotational-prismatic planar robot arm in Section 4. Concluding remarks are given in Section 5.

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2. Tracking control design procedure 2.1. Characteristics of mechanical systems Consider the motion of an n degrees of freedom mechanical system given by ’ þ gðqÞ ¼ Q þ td ; MðqÞq. þ Cðq; qÞ

ð2:1Þ

where q 2 Rn are the generalized coordinates, MðqÞ 2 Rnn is the symmetric positive’ definite inertia matrix, Cðq; qÞrepresents the Coriolis and centripetal terms, gðqÞrepresents the potential terms, Q is a control law and td represents bounded disturbances. The system in Eq. (2.1) has the following characteristics: (i) The inertia matrix MðqÞis bounded: lm I4MðqÞ4lM ;

lm ; lM 2 R þ :

ð2:2Þ

(ii) The Coriolis and centrifugal terms C ðq; q’Þ satisfy * ’ ’ 2; kCðq; qÞk4v b ðqÞkqk

ð2:3Þ

where vb* ðqÞ is a known scalar function. % qÞsuch ’ (iii) It is always possible to find Cðq; that
 T 1 ’ % % qÞ ’ ¼ Cðq; qÞ ’ q; ’ ’ z¼0 Cðq; qÞ z MðqÞ Cðq; 2

8z 2 Rn :

ð2:4Þ

(iv) The gravity vector gðqÞsatisfies kgðqÞk4g0* ðqÞ; (v)

ð2:5Þ

where g0* ðqÞ is a known function. The system dynamics in Eq. (2.1) can always be written as ’ qÞr . ¼ Q; Yðq; q;

ð2:6Þ

’ qÞ . 2 Rnr and r 2 Rr is a vector of system parameters. where Yðq; q; (vi) Its assumed that the disturbances are bounded as follows: ktd k4Y:

ð2:7Þ

2.2. Tracking control of mechanical systems Consider a mechanical system described by Eq. (2.1) having characteristics given by Eqs. (2.2)–(2.7). The objective is to design a control Q such that the state z ¼ ½q; q’ T of the system follows the state of a model system z ¼ ½qm ; q’m T in the presence of parameter uncertainties and bounded disturbances. It is assumed that

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qm ; q’m 2 Rn satisfy the equation Mm ðqm Þq.m þ Cm ðqm ; q’m Þ þ gðqm Þ ¼ vc

ð2:8Þ

provided that kqm k4wm1

and

kq’m k4wm2 :

ð2:9Þ

Proposition 1. Consider the system in Eq. (2.1) with the following control law: Q ¼ QL þ Qr ;

ð2:10Þ

QL ¼ Kd B; 8 < sðBÞ a * B kBk2 Qr ¼ : se

ð2:11Þ if kBk2 l5e;

e 2 Rþ ;

e4rB ;

ð2:12Þ

if kBk2 5e;

with definitions e q qm ; B e’ þ LfðeÞ;    @f    ’ 22 þ g0 þ Y þ ZlM kBk2 ; a * ¼ lM q.m L e’ þvb kqk @e 2
T
T sðBÞ s1 ðBÞ . . . sn ðBÞ ; fðeÞ ¼ f1 ðe1 Þ . . . fn ðen Þ ;

ð2:13Þ ð2:14Þ ð2:15Þ

where se is an arbitrary function, fðeÞ is a diffeomorphism, Kd and L are constant diagonal positive definite matrices, Z > 0 is a constant, and the following conditions are satisfied: Bi si ðBÞ50;

i ¼ 1; . . . ; n;

fi ð0Þ ¼ 0;

dfi ðei Þ > 0; dei

ð2:16Þ i ¼ 1; . . . ; n:

ð2:17Þ

Assume that the initial conditions satisfy kqð0Þk2 4 w1 ; þ ’ kqð0Þk 2 4rv þ wm2 4

lmin ðKd Þ ; vb

ð2:18Þ

where w1 > 0 is constant, lmin ðKd Þ is the minimum eigenvalue of Kd and rþ v is a positive constant that depends on design parameters and initial conditions. Assume also that condition (2.9) is satisfied. Then the closed-loop system has the following properties: (i) The system converges exponentially to a manifold: ’ kBðeðtÞ; eðtÞÞk 2 4rB þ gB exp ð ZtÞ; where gB and rB are positive constants that depend on design parameters and initial conditions.

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(ii) The tracking error is input-to-state stable:   keðtÞk2 4bðkeð0Þk2 ; tÞ þ g sup kBðtÞk2 ; 04t4t

where g is class K(strictly increasing and gð0Þ ¼ 0) and b is class KL (for each fixed t, it belongs to class K with respect to keð0Þk2 and, for each fixed keð0Þk2 , is decreasing with respect to t and bðkeð0Þk2 ; tÞ ! 0 as t ! 1). Proof. Choose the following function: E ¼ 12 BT MB:

ð2:19Þ

Differentiation with respect to time, substitution of Eqs. (2.1), (2.10) and (2.11) in definition (2.19) results in @f ’ E’ ¼ BT ð C% q’ g Kd B þ Qr þ td Þ BT M q.m þ BT ML e’ þ 12 BT MB: @e ð2:20Þ Eq. (2.4) we get

yields

1 T ’ 2 B M ðqÞB

% qÞB, ’ ¼ BT Cðq;

and

denoting

f ¼ M q.m þ C% q’ þ g

  @f % E’ ¼ BT Qr f þ ML e’ þ td BT ðKd CÞB: @e

ð2:21Þ

We have that % q’ > 0 ) q’T Kd q’ > q’T C% q’ q’T ðKd CÞ ’ 22 4q’T Kd q4l ’ 22 ’ lmin ðKd Þkqk max ðKd Þkqk

and because

BT Kd C% B > 0 if lmin ðKd Þ ’ : kqðtÞk 25 vb

ð2:22Þ inequality (2.3) implies that ð2:23Þ

This inequality is satisfied provided condition (2.18) is satisfied. Using Eq. (2.21) we get   @f T ’ E4B Qr f þ ML e’ þ td : ð2:24Þ @e Property (i). Define a state x ½e

e’ T and apply inequality (2.2) to Eq. (2.19):

2 2 1 1 T 1 2 lm kBk2 42 B MB42 lM kBk2 :

Condition (5.2) is satisfied for V E, with o1 ¼ condition (5.3), we need ’ 2ZðE E þ Þ; E4

ð2:25Þ 1 2 lm

and o2 ¼

1 2 lM .

To satisfy

where E þ is a design prameter. From inequalities (2.24) and (2.26), we get   @f T B Qr f þ ML e’ þ td 42ZðE E þ Þ: @e

ð2:26Þ

ð2:27Þ

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Consider the control Q given in (2.10)–(2.12), then rearranging inequality (2.27) yields     @f  2   ’ 2 þ g0 þ Y þ ZlM kBk2 kBk2 þ BT sðBÞ e’ þ vb kqk lM q.m L @e     @f T % . ’ ’ ð2:28Þ 5 B M qm L e þ C q þ g td þ ZBT MB 2ZE þ : @e From properties (2.2), (2.3) and (2.5) we get      @f  @f T   kBk2 lM q.m L e’ 5 B M q.m L e’ ; @e  @e ’ kBk2 vb jjq’jj22 5 BT C% q; kBk2 ðg0 þ YÞ5 BT ðg td Þ; ZlM kBk22 5ZBT MB:

&

ð2:29Þ

Hence, using condition (2.16) we have exponential convergence to the manifold B ¼ 0, according to Theorem 1 in Appendix A, with sffiffiffiffiffiffiffi sffiffiffiffiffiffiffi Eþ Eþ rB ¼ ¼ ; ð2:30Þ o1 lm 8 > 0 if 12 lM kB0 k22 4E þ ; > < sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ ð2:31Þ gB ¼ lM 2 lM kB0 k2 E > > if 12 lM kB0 k22 > E þ kB0 k22 r2B ¼ : 1 lm 2 lm and parameter e satisfying sffiffiffiffiffiffiffiffiffiffi 2E þ e4rB ) e4 : lm

ð2:32Þ

Property (ii). Consider the system ’ B ) e’ ¼ B LfðeÞ: e’ þ LfðeÞ ¼ uðe; eÞ

ð2:33Þ

Choose the function V ¼ 12 fT ðeÞfðeÞ:

ð2:34Þ

Then, the first condition of the input-to-state theorem (see [25]) is satisfied using condition (2.17). Differentiating Eq. (2.34) and applying Eq. (2.33) yields n  X dfi 2 dfi ’ V¼ Lii fi þ fi Bi : ð2:35Þ dei dei i¼1 Using inequality (2.17) on the second condition of the input-to-state theorem (see [25]), yields dfi df 5fi i Bi ) Lii f2i 5fi Bi for kek2 5rðkBk2 Þ > 0: ð2:36Þ Lii f2i dei dei

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And after some algebra (see [26]) we get vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n  uX 1 kBk  2 2 kek2 5t fi

rðkBk2 Þ L ii i¼1

359

ð2:37Þ

satisfying the second condition in the input-to-state theorem and thus proving Property (ii). In the special case of linear manifold we have the following result. Corollary 1. Consider the controller given by Eqs. (2.10)–(2.15) and satisfying conditions (2.16)–(2.18). For B ¼ e’ þ Le

ð2:38Þ

Z > lmax ðLÞ lmin ðLÞ;

ð2:39Þ

and

where lmin ðLÞ and lmax ðLÞ are the minimum and maximum eigenvalues of L, we have (i) keðtÞk2 4re þ ge exp ð Ze tÞ; ’ (ii) keðtÞk 2 4rv þ gv exp ð Zv tÞ; where re ; ge ; Ze ; rv ; gv ; and Zv are positive constants that depend on design parameters and initial conditions. Proof. See [26].

&

Notes. (i) The design parameters are the radius of convergence rB and e. (ii) Possible choices for sðBÞ in definition (2.12) are sðBÞ ¼ ½0 2 Rn ;

si ðBi Þ ¼ Ki* Bi ;

i ¼ 1; . . . ; n;

Ki* > 0:

(iii) For kBðxÞk5e the control is arbitrary. A possible choice is B for kBðxÞk2 5e Qr ¼ se sðBÞ ae e

ð2:40Þ

ð2:41Þ

thus, satisfying continuity at kBðxÞk2 ¼ e. Moreover, for kBðxÞk2 5e it constitutes a proportional-derivative (PD) control logic with and additional dissipating term. From Eq. (2.6) it can be shown that every element of the inertia matrix M, denoted mij , can be expressed as h i ð2:42Þ mij ¼ y# ij ðrÞ þ Dij ðrÞ mij* ðqÞ; where mij* ðqÞ is a scalar function of the generalized coordinates, y# ij ðrÞ is the nominal function of the physical parameters of the system and Dij ðrÞ is a parameter perturbation. Define Dij ðrÞ D% ij ðrÞ ; y# ij ðrÞ

m# ij ðqÞ ¼ y# ij ðrÞmij* ðqÞ;

ð2:43Þ

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n
o % ij ðrÞ D% M 5 maxi; j¼1; ...; n supr2Dq Rr D then after some algebra it can be shown that 
 # lM ¼ nð1 þ D% M Þ supq2D Rn lmax MðqÞ 5 supq2D q

ð2:44Þ

n q R

flmax ½MðqÞ g;

ð2:45Þ

where lmax ½ denotes the maximum eigenvalue and lM is defined in inequality (2.2).

3. Tracking fuzzy logic controllers The fuzzy control system is shown in Fig. 1. The fuzzy control law can be implemented as follows. 3.1. Structure of proposed fuzzy logic controllers In this subsection, relevant terminology and results are presented. For a more detailed discussion see [9].

Fig. 1. Closed-loop fuzzy tracking control system (closed_loop_tracking_system.wmf ).

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361

The fuzzy membership functions are chosen to satisfy the normality requirement mAk ðxÞ41 8 k and the centers of the membership functions satisfy xl 5xlþ1 . The % % fuzzy logic controller consists of the following modules: Fuzzy rule base: The canonical fuzzy IF–THEN rules are chosen to be Ruleð1Þ: IF x is Al THEN y is Bl

ð3:1Þ

where Al and Bl are fuzzy sets in U  R and V  R, respectively, x 2 U and y 2 V are the input and output (linguistic) variables of the fuzzy system, respectively. It is required that the fuzzy sets satisfy consistency ðif mAk ðxi Þ ¼ 1 ) mAj ðxi Þ ¼ 0 8 j 6¼ kÞ and completeness ð8x 2 Ux 9 mAk ðxÞ > 0Þ. Fuzzy inference engine: Product inference engine was chosen, using individual rule based inference with union combination, Mamdani’s product implication, algebraic product of all the T-norm operators and max for all the S-norm operators: " !# n Y M mB ðyÞ ¼ maxl¼1 supx2U mA ðxÞ mAli ðxi ÞmBl ðyÞ ð3:2Þ i¼1

Fuzzifier: Singleton fuzzifier was used: ( 1 if x ¼ x * ; mA0 ðxÞ ¼ 0 otherwise: Defuzzifier: Center average defuzzifier was used: PM l y% wl * y ¼ Pl¼1 ; M l¼1 wl

ð3:3Þ

ð3:4Þ

where y%l is the center of the lth fuzzy set and wl is its height. It can be shown (see [9]) that if the fuzzy set Bl in (3.1) is normal with center y%l , and we have fuzzy rule base (3.1), product inference engine (3.2), singleton fuzzifier (3.3), and center average defuzzifier (3.4) then $ PM l #Qn % m ð x Þ y l i l¼1 i¼1 Ai $; f ðxÞ ¼ P #Q ð3:5Þ M n l¼1 i¼1 mAl ðxi Þ i

n

where x 2 U  R is the input to the fuzzy system, and f ðxÞ 2 V  R is the output of the fuzzy system. 3.2. Fuzzy manifold-boundary robust controller Here we implement QL as a conventional controller (non-fuzzy) and Qr having a fuzzy component.

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3.2.1. Design of Qr For kBk2 > e, Qr is implemented using equation (2.12) with sðBÞ ¼ 0. For kBk2 4e, Qr is given by B Qr ¼ se ¼ a * Kf ; e

ð3:6Þ

where Kf is the crisp output of a fuzzy logic controller described as follows. Step 1 } Choice of membership functions: Define se ¼

kBk2 : e

ð3:7Þ

Choose N fuzzy membership functions for se with centers satisfying sle 50;

sle 41;

sle 4slþ1 e :

ð3:8Þ

Step 2 } Formulation of fuzzy rules: Formulate the IF–THEN rules of the form IF se is Alse THEN Kf is BlKf ;

l ¼ 1; 2; . . . ; N:

ð3:9Þ

The centers of the THEN parts of the fuzzy sets BlKf , denoted by clKf , are 04se clKf 41

;

clKf 5clþ1 Kf :

ð3:10Þ

Step 3 } Obtaining crisp values: The output is given by Pl¼N l l¼1 mAl ðse ÞcKf : Kf ¼ P l¼N l¼1 mAl ðse Þ

ð3:11Þ

3.3. Fuzzy manifold robust controller The error manifold B ¼ 0 is formulated using fuzzy logic, and Qr has the fuzzy component described for the fuzzy-manifold boundary robust controller. Choose a manifold that minimizes time to reach the origin for a system of unit masses in the absence of friction (see [27]): e.i ¼ ui ;

ð3:12Þ

jui j41:

The resulting error manifold chosen is B ¼ e’ þ LfðeÞ;

fðeÞ ¼ diag½f1 ðe1 Þ    fn ðen Þ T ;

fi ðei Þ ¼ signðei Þ

pffiffiffiffiffiffiffiffiffi 2jei j: ð3:13Þ

To improve performance further, we use the fuzzy se described previously for kBk2 4e (see Eq. (3.6)). The resulting controller, denoted by fuzzy manifold robust controller, is described in the following. Step 1 } Choice of membership functions: Choose N fuzzy membership functions for each ei . Their centers satisfy

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eli 5elþ1 i ;

363

ð3:14Þ

where l ¼ 1; 2; . . . ; N and i ¼ 1; 2. Step 2 } Formulation of fuzzy rules: Formulate the IF–THEN rules of the form: IF ei is Alei THEN f is Blfi :

ð3:15Þ

The centers of the THEN parts of the fuzzy sets Blfi , denoted by clfi , satisfy: # $rffiffiffiffiffiffiffi  l l cfi ¼ sign eli ð3:16Þ ei ; where eli is the center of membership function Alei . Step 3 } Obtaining crisp values: The output is given by PN l l¼1 mAl ðei Þcfi f i ¼ PN l¼1 mAl ðei Þ

ð3:17Þ

which is continuously differentiable if mAl ðei Þ is continuously differentiable. It can be seen that f0i ðei Þ given by Eq. (3.13) is discontinuous at ei ¼ 0. This problem is alleviated by the fuzzy logic implementation.

4. Simulation Consider a two-link planar robotic arm shown in Fig. 2. The vector of generalized coordinates is q ¼ ½x y T where x is the second joint displacement and y is the first joint angle. The corresponding generalized forces vector is Q ¼ ½F t T . The equations of motion are given in the form of (2.1) with % qÞ ’ q’ þ gðqÞ ¼ Q; MðqÞq. þ Cðq; ð4:1Þ " MðqÞ ¼

% qÞ ’ ¼ Cðq; " gðqÞ ¼

# 0 ; m1 x2 þ 43 m2 L2c

m1 0 "

0

m1 xy’

m1 xy’

m1 xx’

ð4:2Þ

# ð4:3Þ

;

m1 g cosðyÞ ðm1 x þ m2 Lc Þg sinðyÞ

# ;

Lc ¼

L 2

ð4:4Þ

The physical properties of the links used for the control systems are given in Table 1. The gain matrices were " # " # 10 0 40 0 L¼ ; Kd ¼ : ð4:5Þ 0 10 0 40

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Fig. 2. Two degree of freedom system (pr_robot_arm.wmf ).

Table 1 Parameters of the RP robot arm

Mass of prismatic link (kg) Mass of rotational link (kg) Length of rotational link (m)

True

Minimum

Nominal

Maximum

5.0 2.0 1.0

4.0 1.0 0.99

4.8 2.9 1.0

10.0 3.0 1.1

In all simulation cases, the system was required to follow a model system given by the dynamics equation (2.1) with the following parameters: " Mm ðqÞ ¼

m1m a21m

0

0

m2m a22m

m1m ¼ 1:0;

# ;

m2m ¼ 1:5;

" ’ ¼ Cm ðq; qÞ

# 9:72y’ 1 ; 7:90y’ 2

a1m ¼ 3:0;

The model’s reference input, Qm , was 8 > ð t 8Þ : 4 1 þ cos 5

" gm ðqÞ ¼

a2m ¼ 2:0;

for

t48 s;

for

t > 8 s:

# m1m ga1m y1 ; m2m ga2m y2

g ¼ 9:81:

ð4:6Þ

ð4:7Þ

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365

The initial conditions of the model and plant were qm0 ¼ ½0:35 0:0 T ;

q’m0 ¼ ½0:0

0:0 T ;

q0 ¼ ½0:5

0:0 T ;

q’0 ¼ ½0:0

0:0 T : ð4:8Þ

Resulting bounds on model rates and accelerations were supðkq.m k2 Þ ¼ 0:4483;

supðkq’m k2 Þ ¼ 0:3465:

ð4:9Þ

In all the simulations, the following design parameters were chosen (see Eqs. (2.12) and (2.14)): e ¼ 5:910 6 ;

Z ¼ 100:

ð4:10Þ

The control laws proposed in the previous section were compared via MATLAB/ Simulink. The non-fuzzy implementation of the proposed control law, named ‘‘new robust controller’’ in all figures, had sðBÞ ¼ 0 and Qr given by Eq. (2.41) for kBk5e. In all simulations, the mass of the prismatic link of the manipulator was 8 for 04t420 s > < 5 kg for 205t440 s ð4:11Þ m1 ðtÞ ¼ 10 kg > : 4 kg for 405t460 s Fig. 3 presents a comparison of the control efforts of the new robust controller, fuzzy manifold-boundary controller (see Section 3.2) and fuzzy manifold robust controller (see Section 3.3). It can be observed that the control efforts were similar and smooth for all the proposed controllers. Figs. 4 and 5 show the tracking errors and rates for all the proposed controllers. Fig. 4 shows that the smallest tracking errors and fastest convergence are obtained for the fuzzy manifold robust controller. In Fig. 5 it can be seen that the tracking error rates obtained for the fuzzy manifold-boundary and fuzzy manifold controllers were 50% in magnitude of the error rates obtained for the non-fuzzy robust controller.

Appendix A. Exponential convergence to a manifold [26] Definition 1 (Uniform exponential convergence to a manifold). The system x’ ¼ f ½t; xðtÞ is said to be uniformly exponentially convergent to within a radius rB of the manifold BðxÞ ¼ 0 iff there exists aB > 0 with the property that, for any initial conditions t0 2 R and x0 2 D  Rn , there exists cB ðx0 Þ50 such that, if xðÞ is a solution of x’ ¼ f ½t; xðtÞ with xðt0 Þ ¼ x0 , then kBðxðtÞÞk4rB þ cB ðx0 Þ exp½ aB ðt t0 Þ

ðA:1Þ

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Fig. 3. Comparison of control efforts (torques_franklin.eps).

for all t5t0 . Moreover, if D ¼ Rn then the system is globally uniformly exponentially convergent to within a radius rB of the manifold BðxÞ ¼ 0. Theorem 1. Suppose that for a system x’ ¼ f ½t; xðtÞ x 2 Rn ; t 2 Rþ and a continuously differentiable function BðxÞ : Rn ! Rn , there exists a continuously differentiable function V ðxÞ : Rn ! R with the following properties: (i) There are scalars o1 ; o2 > 0 such that o1 kBðxÞk2 4V ðxÞ4o2 kBðxÞk2

8x 2 Rn :

ðA:2Þ

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367

Fig. 4. Comparison of tracking errors (errors_franklin.eps).

(ii) There are scalars V * 50; aB 50 such that dV ðxÞ 4 2aB ½V ðxÞ V * 8x s:t: V ðxÞ > V * : ðA:3Þ dt Then the system is uniformly exponentially convergent to within a radius rB of the manifold BðxÞ ¼ 0 with rate aB , where 8t 2 R :

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Fig. 5. Comparison of tracking rates errors (rateserrors_franklin.eps).

rffiffiffiffiffiffiffiffiffi V* rB ¼ ; o1

8 >
: o1 4

if

V ðx0 Þ4V * ;

if

V ðx0 Þ > V * :

ðA:4Þ

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369

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