# Third order point-to-point motion profile

## Dynamics & Control

#### Introduction

In many mechatronic applications where a movement from A to B needs to be performed, a third order point to point motion profile is used.
To enable early insight in the relevant parameters of a motion profile it is useful to calculate and visualize the relevant parameters (position, speed, acceleration and jerk).

This sheet provides analytical formulas to calculate the quickest motion between point A to point B based on given maximum levels speed, acceleration and jerk.

#### Approach

The typical time plot of the parameters of a third order profile is depicted on the right.
The difficulty in defining the motion trajectory is that the shape is not always the same. For example, there are cases where the maximum speed or acceleration level is not achieved, because there is not enough time to build up to the maximum before slowing down again. These different cases (I … VI) are captured by the following criteria table: Then the following formula’s define the motion parameters over time:

jerkaccelerationvelocityposition
t0…t1jmaxjmax ∙ (t - t0)1/2 ∙ jmax ∙ (t - t0)21/6 ∙ jmax ∙ (t - t0)3
t1…t20a1 = a2v1 + a1 ∙ (t - t1)p1 + v1 ∙ (t - t1) + 1/2 ∙ a1 ∙ (t - t1)2
t2…t3-jmaxa2 - jmax ∙ (t - t2)v2 + a2 ∙ (t - t2) + 1/2 ∙ -jmax ∙ (t - t2)2p2 + v2 ∙ (t - t2) + 1/2 ∙ a2 ∙ (t - t2)2 + 1/6 ∙ - jmax ∙ (t - t2)3
t3…t400v3 = v4p3 + v3 ∙ (t - t3)
t4…t5-jmax-jmax ∙ (t - t4)v4 + 1/2 ∙ - jmax ∙ (t - t4)2p4 + v4 ∙ (t - t4) + 1/6 ∙ - jmax ∙ (t - t4)3
t5…t60a5 = a6v5 + a5 ∙ (t - t5)p5 + v5 ∙ (t - t5) + 1/2 ∙ a5 ∙ (t - t5)2
t6…t7jmaxa6 + jmax ∙ (t - t6)v6 + a6 ∙ (t - t6) + 1/2 ∙ jmax ∙ (t - t6)2p6 + v6 ∙ (t - t6) + 1/2 ∙ a6 ∙ (t - t6)2 + 1/6 ∙ jmax ∙ (t - t6)3
t1 = tjt2 = tat3 = ta + tjt4 = tvt5 = tv + tjt6 = tv + tat7 = tv + tj + ta