To obtain the polynomial profile, the displacement is assumed to be a polynomial function. This function is differentiated to obtain the velocity, acceleration, and jerk or pulse functions. The equation for the displacement is:
Jerk or Pulse:
The curves of a polynomial depend on the constraints the user places on the equations. The constraints will in turn determine the values of the coefficients A0 through An. The two polynomial curves, shown in the figures below, use different constraints. In figure 9 the displacement, velocity and acceleration are constrained to zero at the beginning. At the other end, the displacement is constrained to a value of h; while, the velocity and acceleration are zero. In figure 10, two additional constraints are added at the ends; namely, the jerk is constrained to zero.
Polynomials can be used to force values in the kinematics that the other profiles cannot attain. There is some danger in this however; sometimes the constraints will either cause incompatibilities in the curves or be impossible.
The properties for the polynomial are whatever the user wants to make. This is the most flexible profile there is. They are virtually unlimited in flexibility.