Trapezoidal motion profile

Time to move a distance with velocity and acceleration limits, with the velocity curve.

Inputs

Distance

Maximum velocity

Maximum acceleration

Result

Total time

1.1

Peak velocity: 2
Time accelerating: 0.5
Time cruising: 0.1

Velocity over the move

Data table

Time (s)	Velocity (m/s)
0	0
0.088	0.352
0.176	0.704
0.264	1.056
0.352	1.408
0.44	1.76
0.528	2
0.616	1.936
0.704	1.584
0.792	1.232
0.88	0.88
0.968	0.528
1.056	0.176
1.1	0

How it’s calculated

Formula

t_a = \frac{v_\text{max}}{a_\text{max}}, \qquad d_a = \frac{v_\text{max}^2}{2 a_\text{max}}

t = 2 t_a + \frac{d - 2 d_a}{v_\text{max}} \quad \text{if } 2 d_a < d, \qquad \text{else } t = 2\sqrt{d / a_\text{max}}

With your numbers (converted to SI units)

t_a = \frac{2}{4} = 0.5\,\text{s}, \quad d_a = \frac{2^2}{2 \times 4} = 0.5\,\text{m}

t = 2 \times 0.5 + \frac{1.2 - 2 \times 0.5}{2} = 1.1\,\text{s}

What it means

A trapezoidal profile accelerates at the limit, cruises at the maximum velocity, then decelerates. If the move is too short to reach cruise speed it becomes a triangle, and the peak speed is lower than the limit. Raising acceleration helps short moves most; raising velocity helps long ones.

Assumptions

Equal acceleration and deceleration, starting and ending at rest.
Infinite jerk (instant changes in acceleration).

Learn the concepts behind this on resources.simbotics.org · All formulas