By Nicholas J Giordano; Hisao Nakanishi

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**Extra info for Computational physics**

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The theory of wavelets is widely used in identification problems, where a studied system evolution is deduced from the wavelet transform response. Wavelets of the Daubechies, Morlet, and Gauss families are used to detect faults in gears and gearboxes as well as to estimate the location of cracks [21, 25, 30]. The wavelet-type analysis is applied to lumped mechanical systems governed by ordinary differential equations [12, 17]. The proposed procedure can be viewed as a logarithmic transform applied to a fitted component of the original response.

10) whereby these speed thresholds only depend on N and ˛. Remark 2. , mi D m; their actual value is not required to find these speed thresholds for switching the number of active spikes. Now, if the slope is known (measured), it is then possible to choose adequate gaits, with respect to low spike forces. Remark 3. The estimate of the actuator forces (see (8)) gives identical speed thresholds, as it differs from the spike force estimate only by the addition of a weight term of one mass; this term does not influence the intersection of curves in Fig.

Since all system parameters are assumed to be unknown or, more precisely, uncertain, an adaptive controller is used to achieve the tracking of certain gaits. After finding optimal gaits, a procedure for automatic gait shift based on the measurement of certain restricted system variables is implemented. 1 Modeling The starting point is the chain of mass points in a common straight line as shown in Fig. 1. t / (i D 0; : : : ; n) are the current coordinates of the segments. t /. , they restrict velocities from being negative [10].