Simple Trajectory Generation


The world is full of PID-loops, thermostats, and PLLs. These are all feedback loops where we control a certain output variable through an input variable, with a more or less known physical process (sometimes called "plant") between input and output. The input or "set-point" is the desired output where we'd like the feedback system to keep our output variable.

Let's say we want to change the set-point. Now what's a reasonable way to do that? If our controller can act infinitely fast we could just jump to the new set-point and hope for the best. In real life the controller, plant, and feedback sensor all have limited bandwidth, and it's unreasonable to ask any feedback system to respond to a sudden change in set-point. We need a Trajectory - a smooth (more or less) time-series of set-point values that will take us from the current set-point to the new desired value.

Here are two simple trajectory planners. The first is called 1st order continuous, since the plot of position is continuous. But note that the velocity plot has sudden jumps, and the acceleration & jerk plots have sharp spikes. In a feedback system where acceleration or jerk corresponds to a physical variable with finite bandwidth this will not work well.


We get a smoother trajectory if we also limit the maximum allowable acceleration. This is a 2nd order trajectory since both position and velocity are continuous. The acceleration still has jumps, and the jerk plot shows (smaller) spikes as before.


Here is the python code that generates these plots:

See also:

I'd like to extend this example, so if anyone has simple math+code for a third-order or fourth-order trajectory planner available, please publish it and comment below!

EMC2 tpRunCycle revisited

I started this EMC2 wiki page in 2006 when trying to understand how trajectory control is done in EMC2. Improving the trajectory controller is a topic that comes up on the EMC2 discussion list every now and then. The problem is just that almost nobody actually takes the time and effort to understand how the trajectory planner works and documents it...

A recent post on the dev-list has asked why the math I wrote down in 2006 isn't what's in the code, so here we go:

I will use the same shorthand symbols as used on the wiki page. We are at coordinate P ("progress") we want to get to T ("target") we are currently travelling at vc ("current velocity"), the next velocity suggestion we want to calculate is vs ("suggested velocity), maximum allowed acceleration is am ("max accel") and the move takes tm ("move time") to complete. The cycle-time is ts ("sampling time"). The new addition compared to my 2006 notes is that now the current velocity vc as well as the cycle time ts is taken into account.

As before, the area under the velocity curve is the distance we will travel, and that needs to be equal to the distance we have left, i.e. (T-P). (now trying new latex-plugin for math:) (EQ1)

T-P = {1 \over 2} t_s v_c + {1 \over 2} t_m v_s

Note how the first term is the area of the red triangle and the second therm is the area of the green triangle. Now we want to calculate a new suggested velocity vs so that using the maximum deceleration our move will come to a halt at time tm, so(EQ2):

 v_s = a_m (t_m - t_s)

Inserting this into the first equation gives (EQ3):

T-P = {1 \over 2} t_s v_c + {1 \over 2} t_m a_m (t_m - t_s)

this leads to a quadratic eqation in tm (EQ4):

{1 \over 2} a_m t_m^2 - {1 \over 2} a_m t_s t_m + {1 \over 2} t_s v_c - (T-P) = 0

with the solution (we obviously want the plus sign)(EQ5)

 t_m = {1 \over 2}t_s + \sqrt{ {t_s^2 \over 4} - {t_s v_c - 2(T-P) \over a_m} }

which we can insert back into EQ2 to get (EQ6) (this is the new suggested max velocity. we obviously apply the accel and velocity clamps to this as noted before)

 v_s = a_m (t_m - t_s) = -{1 \over 2 }a_m t_s + a_m \sqrt{ {t_s^2 \over 4} - {t_s v_c - 2(T-P) \over a_m} }

It is left as an (easy) exercise for the reader to show that this is equivalent to the code below (note how those silly programmers save memory by first using the variable discr for a distance with units of length and then on the next line using it for something else which has units of time squared):

over and out.