Chaotic Phenomena and Fractional-Order Dynamics in the Trajectory Control of Redundant Manipulators

1、Nonlinear Dynamics 29: 315342, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.Chaotic Phenomena and Fractional-Order Dynamics in the Trajectory Control of Redundant ManipulatorsFERNANDO B. M. DUARTEDepartment of Mathematics, School of Technology, Polytechnic Institute of Vis

2、eu, Campus Politécnico, P-3504-510 Viseu, PortugalJ. A. TENREIRO MACHADODepartment of Electrical Engineering, Institute of Engineering, Polytechnic Institute of Porto, R. Dr. António Bernardino Almeida, P-4200-072 Porto, Portugal(Received: 18 June 2001; accepted: 7 December 2001)Abstract.

3、Redundant manipulators have some advantages when compared with classical arms because they allow the trajectory optimization, both on the free space and on the presence of obstacles, and the resolution of singularities. For this type of arms the proposed kinematic control algorithms adopt generalize

4、d inverse matrices but, in general, the corresponding trajectory planning schemes show important limitations. Motivated by these problems this paper studies the chaos revealed by the pseudoinverse-based trajectory planning algorithms, using the theory of fractional calculus.Keywords: Planar manipula

5、tors, redundant manipulators, chaos, fractional calculus.1.IntroductionThis paper discusses a fractional calculus perspective in the study of the trajectory control of redundant manipulators and establishes a connection between the theory of fractional-order dynamical systems, chaotic phenomena, fra

6、ctals and robotics.Fractional calculus goes back to the beginning of the theory of differential calculus but its inherent complexity postponed the application of the associated concepts. Nevertheless, in the last decade the progress in the areas of chaos and fractals revealed subtle relationships wi

7、th the fractional calculus leading to an increasing interest in the development of the new paradigm. In the area of automatic control some work has been carried out but the proposed algorithms are still in a preliminary phase of establishment.On the other hand, the area of robotics has been develope

8、d since the seventies and re-searchers have recognized that the addition of extra Degrees of Freedom (DOF) to form a redundant robot overcomes the functional limitations of conventional nonredundant manipu-lators. However, the kinematic-based redundancy approaches that have been proposed cannot prot

9、ect against chaotic-like joint motions and high transients.Having these ideas in mind, the paper is organized as follows. Section 2 develops the formalisms for the fractional calculus and matrix generalized inverses. Section 3 introduces the fundamental issues for the modeling of redundant manipulat

10、ors. Section 4 analyses the resulting chaotic phenomena revealed by the trajectory planning algorithms. Finally, Section 5 presents the main conclusions.316 F. B. M. Duarte and J. A. Tenreiro Machado2.Fundamental AspectsThis section introduces the fundamental mathematical aspects of the theories of

11、fractional calculus and matrix generalized inverses.2.1. FRACTIONAL CALCULUSFractional calculus is a natural extension of the classical mathematics. In fact, since the beginning of the theory of differential and integral calculus, mathematicians such as Euler and Liouville investigated their ideas o

12、n the calculation of noninteger-order derivatives and integrals. Nevertheless, in spite of the work that has been done in the area, the application of Fractional Derivatives and Integrals (FDIs) has been scarce until recently. In the last years, the advances in the theory of chaos revealed profound

13、relations with FDIs, motivating a renewed interest in this field.The basic aspects of the fractional calculus theory, the study of its properties and research results are addressed in 113. In what concerns the application of FDI concepts we can mention a large volume of research about viscoelasticit

14、y/damping 1429 and chaos/fractals 3032. However, other scientific areas are currently paying attention to the new concepts and we can refer the adoption of FDIs in biology 33, electronics 34, signal processing 35, 36, system identification 3739, diffusion and wave propagation 4042, percolation 43, m

15、odelling and identification 44, 45, chemistry 46, 47 and automatic control 4854. This work is still giving its first steps and, consequently, many aspects remain to be investigated.Since the foundation of the differential calculus the generalization of the concept of deriv-ative and integral to a no

16、ninteger-order has been the subject of several approaches. Due to this reason there are various definitions of FDIs (Table 1) which are proved to be equivalent.Nevertheless, from the control point of view some definitions seem more attractive, namely when thinking in a real-time calculation. The Lap

17、laceFourier definition for a derivative of order C is a direct generalization of the classical integer-order scheme with the mul-tiplication of the signal transform by the s/j operator. In what concerns automatic control theory this means that frequency-based analysis methods have a straightforward

18、adaptation to FDIs.Consider the elemental control system represented in Figure 1 (with 1 < < 2) with transfer function G(s) = Ks in the forward path. The open-loop Bode diagrams (Figure 2) of amplitude and phase have a slope of 20 dB/dec and a constant phase of /2 rad, respectively. Therefore,

19、 the closed-loop system has a constant phase margin of (1/2) rad, that is, independent of the system gain K. Likewise, this important property is also revealed through the root-locus depicted in Figure 3. For example, when 1 < < 2 the root-locus follows the relation / = cos1 , where is the dam

20、ping ratio, independently of the system gain K.The implementation of FDIs based on the LaplaceFourier definition adopts the frequency domain and requires an infinite number of poles and zeros obeying a recursive relationship 48, 49. Nevertheless, this approach has several drawbacks. In a real approx

21、imation the finite number of poles and zeros yields a ripple in the frequency response and a limited bandwidth. Moreover, the digital conversion of the scheme requires further steps and additional approx-imations making it difficult to analyze the final algorithm. The method is restricted to cases w

22、here a frequency response is well known and, in other circumstances, problems occur for its implementation. An alternative approach is based on the concept of fractional differentialChaotic Phenomena and Fractional-Order Dynamics 317Table 1. Definitions of FDIs.1xLiouville(Ic)(x) =(t)dt, < x <

23、 + ()(x t)1x(x t) dt, < x < +(Dcf )(x) = (1 ) dx1df (t)1x(t)RiemannLiouville(I+ )(x) =dt, a < x ()(xt)1ax(D+ f )(x) = (1 ) dx(x t) dt, a < xa1df (t)1xHadamard(I+)(x) =(t)dt, x > 0, a > 0 ()tln(t/x)10(D+ f )(x) =1 (1)xf (x) f (t) dt, ln(x/t)1 t+01+xa/ h (+j )GrünwalsLetnikov(I+

24、 )(x) =h lim0 h+(x j h) ()= (j1)01tf(1) m d, m 1 < < m(m)=m0d ) + CaputoDf (t) (m )(tdtm f (t), m = x+= (1 )(x t)1+Marchaud(D f )(x)f (x) f (t)dtFourierF I±) = F /(±j ), 0 < Re() < 1F D± = (±j )F , Re() > 0LaplaceLI0+ = L/s, Re() > 0LD0+ = sL, Re() 0Figure 1. Bl

25、ock diagram for an elemental feedback control system of fractional-order .318 F. B. M. Duarte and J. A. Tenreiro MachadoFigure 2. Open-loop Bode diagrams of amplitude and phase for a system of fractional-order 1 < < 2.Figure 3. Root locus for a feedback control system of fractional-order 1 <

26、; < 2.of order . The GrünwaldLetnikov definition of a derivative of fractional-order of the signal x(t), Dx(t), motivated an approximation based on an n-term truncated series in the discrete-time domain, that in z-transform is given by 5052 T k=0k! ( k + 1)=TruncnTZDx(t)1n(1)k ( + 1) zkX(z)1

27、 z1X(z). (1)An important property revealed by the GrünwaldLetnikov definition and aproximation(1) is that while an integer-order derivative implies simply a finite series, the fractional-order derivative requires an infinite number of terms. This means that integer derivatives are local operato

28、rs in opposition with fractional derivatives that have, implicitly, a memory of all past events.2.2. GENERALIZED INVERSESThis subsection addresses the generalization of the concept on matrix inversion.For Am×n and Xn×m, the following relations are used to define a generalizedinverse A, a r

29、eflexive generalized inverse Ar and a pseudoinverse A#:AXA = A,(2)XAX = X,(3)(AX)T = AX,(4)(XA)T = XA.(5)Chaotic Phenomena and Fractional-Order Dynamics 319Conditions (2) through (5) are called the Penrose conditions. A generalized inverse of matrix A m×n is a matrix X = A n×m satisfying c

30、ondition (2). On the other hand, a reflexive generalized inverse of matrix A m×n is a matrix X = A r n×m satisfying both conditions (2) and (3). Finally, a pseudoinverse of a matrix A m×n (the so-called MoorePenrose inverse) is a matrix X = A# n×m satisfying conditions (2) throug

31、h (5) 5558.The generalized inverse is not unique and, in general, if A is a particular matrix satisfying (2), then all the generalized inverse of matrix A are given by (6) where Y varies overall possible n × m matrices:A + Y AAYAA.(6)Suppose that A has rank r and that its rows and columns have been permuted to make the leading r × r submatrix nonsingular. Therefore, to compute a generalized inverse of A, we must apply row operations to the augmented matrix Aa = A, Im (assumi