tailieunhanh - Mobile Robots -Towards New Applications 2008 Part 3

Tham khảo tài liệu 'mobile robots -towards new applications 2008 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Modelling of Bipedal Robots Using Coupled Nonlinear Oscillators 71 Fig. 21. Trajectory in the phase space limit cycle . Fig. 22. Phase portrait. Coupled Oscillators System Oscillators are said coupled if they allow themselves to interact in some way one with the other as for example a neuron that can send a signal for another one in regular intervals. Mathematically speaking the differential equations of the oscillators have coupling terms that represent as each oscillator interacts with the others. According to Kozlowski et al. 1995 since the types of oscillators the type and topology of coupling and the external disturbances can be different exist a great variety of couplings. In relation to the type of coupling considering a set of n oscillators exists three possible basic schemes Low Reinhall 2001 1 coupling of each oscillator to the closest neighbours forming a ring with the n-th oscillator coupled to the first one i 1 n i 1 i j i 1 i -1 i -1 5 1 i 1 i n 72 Mobile Robots Towards New Applications 2 coupling of each oscillator to the closest neighbours forming a chain with the n-th oscillator not coupled to the first one i 1 i 1 i j ii 1 -1 i -1 i n 6 3 coupling of each one of the oscillators to all others from there the term mutually coupled i j 1 .n j i 7 This last configuration of coupling will be used in the analyses since it desires that each one of the oscillators have influence on the others. Figure 23 presents the three basic schemes of coupling. Fig. 23. Basic schemes of coupling in ring a in chain b and mutually coupled c . Coupled Oscillators with the Same Frequency From the equation 3 considering a net of n-coupled Rayleigh oscillators and adding a coupling term that relates the velocities of the oscillators we have e -S 1 -qii i2 di Q2 e -e o - j -6j 0 i 1 2 . n 8 j 1 where Si qi Qi and Ci j are the parameters of this system. For small values of parameters determining the model nonlinearity we will assume that the .