Movement & arriving using forces

Question

3D. Given a point mass m, inital position p0 initial velocity v0, and a desired location d, how do I apply forces (with magnitude no greater then fMax) to move m to d and stop. I know how to apply forces, it's the strategy for determining what force to apply each frame.

You would think this would be easy - but it's turned out to be difficult.

A solution that only converges at the limit is not acceptable (i.e. having the mass oscillate back/forth over the destination by smaller and smaller distance)

Edit: This is not the same as this question. Finding the stopping distance is trivial, but is (maybe) only part of the solution.

Edit: Also not the same as this question, because the highest answer is incorrect (as stated by my constraints)

Answer 1

What you're describing is a classical controls problem. An old subject and widely studied by most engineering students.

There are two approaches you can take, one that involves tweaking (PID controller) and one that's based on math & control theory.

1. PID controller: proportional-integral-derivative controller (more on Wikipedia) relies on the difference in your position and the destination (proportional), your total error distance (integral), and the speed (derivative). This video explains it very well. It works by multiplying each input value by a constant factor (called the gains) to obtain a force to apply. So you'd have a function, getControl(), which provides how much force to apply.

   def getControl(gains, dist2dest, totalError, speed):
       P, I, D = gains
       control = P*dist2dest + I*totalError + D*speed
   

Picking values for the gains is called 'tuning' and can be done interactively with your physics model. Once you're satisfied with the behavior, save the values.

Info on how you move (i.e. what is the position as a function of force?) is required to solve this with maths.

2. Root Locus (more on Wikipedia). Once you have a transfer function, G (function that describes your system dynamics --> pos/force = G(s), in Laplace-space), you can plot the root locus (matlab fxn 'rlocus') to visualize the dynamics and add controller zeros/poles automatically (matlab fxn 'place') or by hand to place the closed-loop poles where you want in order to obtain desired behavior (no overshoots, short transient time, etc.)

Force can be constrained by simply setting any force greater than fMax to fMax.