Simple Trajectory Generation

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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.

1st_order_trajectory

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.

2nd_order_trajectory

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!

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