A Systematic Approach to Programming an Analog Computer to Generate a Large Class of Trajectories

The negative gradient method is extended to the stable analog computer programming of a class of nonlinear time-varying trajectory problems defined by ¿(x, t) = 0, f(u, x, t) = 0, u = x = dx/dt, where x and u are n-dimensional vectors, ¿ is an m-dimensional position vector function (m≪n) and f is an (n-m)-dimensional velocity vector function. This class of problems includes the amplitude stabilized harmonic oscillator, two-or three-dimensional contour tracing, planetary motion simulation and certain vibration problems in mechanics involving Lagrange´s equations and positional constraints. An augmented velocity vector function (¿) is defined by ¿ = f + d¿/dt which provides n independent equations ¿ = 0. For a fixed x, these equations can be solved for u in the computer ``reset´´ mode. In the ``compute´´ mode u is connected to the integrator developing x. The computer program is shown to be easily stabilized by time scaling.