Synthesis of universal motion curves in generalized model

The universal motion curve is synthesized in a generalized model. The acceleration curve is divided into seven intervals of which the curve shapes on intervals can be prescribed by arbitrary functions. The characteristic of the motion curve are controlled completely by four shape functions of acceleration and seven time increments. To simplify the synthesis derivation, normalized function and symmetric function are defined to represent these shape functions. The displacement, velocity, acceleration, and jerk curves are derived in piecewise function forms and the peak of velocity and acceleration are also expressed in explicit parametric forms, The designers can quickly construct the motion curves according to the synthesis formulas and the step-by-step procedures. The helpfulness and convenience of these formulas are demonstrated by three examples in symbolic or numerical solution.