Key Engineering Takeaways
- •A path is purely geometric (sequence of poses); a trajectory adds time parameterization (velocities, accelerations, jerks).
- •Cubic polynomials (3rd order) only guarantee continuous velocity; quintic polynomials (5th order) ensure continuous acceleration and bounded jerk.
- •Zero initial and final accelerations prevent mechanical backlash shock on gearboxes.
Prerequisites
- • Polynomial calculus
- • Linear systems of equations
Deriving the 5th-Order Quintic Polynomial System
A quintic trajectory has the form:
$$s(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3 + a_4 t^4 + a_5 t^5$$
With 6 boundary conditions (initial/final position, velocity, and acceleration), solving a $6 \times 6$ linear system yields the coefficients.
Tags:#Trajectory Generation#Quintic Splines#Jerk Reduction#Motion Planning#Interpolation