The modelling and control of flexible space structures is a topic of much current research. Some examples are the NASA Remote Manipulator System (RMS) on the Space Shuttle and appendages of satellites (e.g., solar panels). Typically, these structures are required to slew about a fixed or hinged location and are subsequently made rigid and massive to prohibit bending or torsion along any of their axes. In space-borne structures, size and mass are a significant factor wherein a less massive structure results in a lighter, faster mechanism and more economial operation and cost-effectiveness. The combination of low mass and long lengths results in a highly flexible structure with low natural frequencies.
If the vibrations from a lightweight, flexible, slewing link are chaotic, then precise prediction of the long term time history is impossible, while the near term history may be quite good. Small uncertainties in the initial conditions lead to diverging responses. A statistical measure of the chaotic dynamics can be obtained by using the probability density functions [1]. Measurement of the density functions (probability, PDF, and cumulative, CDF) can be used as a diagnostic tool especially in periodically forced systems. Probabilistic theory is capable of distinguishing different degrees of complexity of motions, and presents a further step in bridging the gap between simple systems and complicated dynamics.
This note will focus on the physically relevant case of a flexible link (a simple model of a robot manipulator arm) undergoing periodic slewing motion (as normally occurs in repetitious tasks). The equations of motion take into account Coriolis and centripetal forces as well as other non-linear terms. This system is known to undergo chaotic motion for certain parameters (e.g., initial deflection and amplitude of the forcing function) [2]. The amplitude PDF and CDF for the position and velocity, P(y) and P(y˙), of the system will be used as an identification technique for chaos in a probabilistic setting. The system is investigated for multi-maxima peaks typical of chaotic distributions. The chaotic distributions of the model are compared to the standard normal distribution. both the chi-squared and Kolmogorov-Smirnov goodness of fit tests are used.