The sp_scurve code is based on S-curve (7-segment) trajectory planning mathematics, which uses jerk-limited motion profiles derived from the kinematics equations of motion.
Mathematical Foundation
The core math is based on third-order polynomial motion equations with controlled jerk (rate of change of acceleration):
Position: P(t) = P₀ + V₀·t + ½·A₀·t² + ⅙·J·t³
Velocity: V(t) = V₀ + A₀·t + ½·J·t²
Acceleration: A(t) = A₀ + J·t
These equations appear explicitly throughout the code (e.g., lines sp_scurve.c:187-197, 842-844, 1089-1091).
Key Mathematical Components
7-Segment S-Curve Profile - The trajectory is divided into 7 phases (S0-S6/n0-n6):
S0: Increasing jerk (acceleration ramps up)
S1: Constant acceleration
S2: Decreasing jerk (acceleration ramps down to 0)
S3: Constant velocity (cruise)
S4: Decreasing jerk (deceleration begins)
S5: Constant deceleration
S6: Increasing jerk (deceleration ramps to 0)
Cubic Equation Solving - The solve_cubic() function at lines 80-184 solves cubic equations using Cardano's formula and trigonometric method for finding roots.
There is a Newton-Raphson Method - The solute() function at lines 54-68 uses Newton-Raphson iteration to solve polynomial equations, but it is not in use, Cardano's formula is computationally superior.
Stopping Distance Calculation - Based on the relationship Amax² = V·J + 0.5·a² (line 949), derived from integrating the jerk-limited motion equations.
Attribution
The code credits go to 杨阳 (Yang Yang) as the original author, he also referenced a Chinese technical document at doc88.com/p-9119146814737.html for the S-curve formulation.
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