Optimal excitation for identification of a cam set-up B. Demeulenaere, V. Lampaert, J. Swevers, J. De Schutter Department of Mechanical Engineering, Division PMA, K.U.Leuven, Belgium e-mail: bram.demeulenaere@mech.kuleuven.ac.be Abstract This paper describes the design of optimal excitations for the identification of a cam set-up. The goal of the identification is obtaining an accurate machine model such that it can be integrated in the design of the cam that drives the set-up. The first part of the paper discusses the identification of the cam set-up dynamics for a given excitation. The identification is done in the time domain, using a maximum likelihood estimator, and explicitly takes advantage of the periodicity of the measurements. The second part describes a methodology to optimize the excitation, which can be done in two ways: (i) optimizing the drive speed of the drive motor and (ii) optimizing the cam profile. A critical part of the optimization algorithm is the calculation of the steady state response (to the applied excitation) of the cam set-up. This is done in an iterative way in the frequency domain and requires a frequency domain description of the non-linear cam-follower mechanism, which is based on an analogy with phase modulation. 1 Introduction Cam-follower mechanisms are often used for realiz- ing fast, periodical motions, as they occur in looms. The fierce competition in the textile industry forces the loom manufacturers to produce increasingly faster machines, which causes problems such as inaccuracy, vibrations and wear. A major cause of these prob- lems is the fluctuation of the drive speed of the cam- follower mechanism. Generally speaking, the cam is designed for a constant drive speed, which is a reas- onable assumption in slow machines, where the drive speed fluctuation is limited. In fast machines how- ever, these fluctuations are substantial, which results in a more pronounced excitation of the machine res- onances than in the case of an almost constant drive speed. The traditional solution to this problem is a fly- wheel, which keeps the drive speed as constant as pos- sible. However, a better - mechatronic - solution is to calculate the drive speed fluctuations in advance and to take them into account when designing the cam. This approach towards cam design is significantly dif- ferent from the traditional one and provides a way to develop machines with fewer problems at high drive speeds. A primary requirement to implement this ap- proach is an accurate dynamic machine model. This model can be obtained through system identification, an experimental technique which can be formally de- scribed as [4]: The selection of a model for a system, using a lim- ited number of measurements of the input and outputs, which may be disturbed by noise, and a priori system knowledge. Generally, a system identification procedure con- sists of three steps [4]. The first step is experiment design: designing an experiment with the purpose of collecting useful data. Theoretically, this boils down to designing a persistent excitation of the system, which means that the inputs should be sufficiently rich such that all modes are excited and observable in the output sequence 1 . However, appropriate ex- periment design goes beyond satisfying the condi- tion of persistency of excitation: another possibility is designing excitations that are optimal according to some optimization criterion. This is discussed in sec- tion 3. The second step is the design of an identification model, which is an equation that allows the calcula- tion of the system dynamics on the basis of the meas- urements. In this case, the identification model has the following form: (1) in which is a column vector containing the unknown parameters that describe the system dynamics. This regression equation is explained in detail in section 2.3. 1 As a rule of thumb, an input signal should contain at least different sinusoids in order to identify an -th order system.