How to PID tune a four wheel robot?

Question

Four wheel mecanum rover with a mass of approximately 50 Kg. Each wheel has a 20:1 gearbox with encoder at the wheel (rather than the motor).

If this was something like a robot arm I would PID tune each DOF starting with the distal joints and work my way back to the base. This is a case of four motors collaborating to accelerate and control the total mass. What's the best approach to doing this?

One thought I had was to setup a tuning jig for single motor+gearbox+wheel, load it with 1/4 of the mass and tune them individually in this fashion. This would likely work just fine. Perhaps not the most elegant but it might do OK.

The other one is to run the system open loop, issue a step change command, observe and record the response and use this to iterate a solution. In many ways this is similar to the Twiddle algorithm, which is nothing more than sequential approximation and optimization.

I'll have the hardware built next week so I can't test at the moment. I am leaning towards tuning each motor+gearbox+wheel using the 1/4 mass jig first. This could be followed by a fine tuning process using fine-step successive approximation to get it right with all four motors driving the entire mass.

One of the questions I have has to do with what happens to this tuning in the context of a holonomic drive, which can move the mass in multiple directions and rotate about its axis (as well as rotate about any point on the xy plane).

Twiddle algorithm: https://www.youtube.com/watch?v=2uQ2BSzDvXs

Answer 1

I can think of three ways to control this robot:

1. Do as you described and have one separate controller for each motor, tuned independently using a test fixture;
2. Use a MIMO PID controller to simultaneously control the four motors;
3. Use two PID controllers, one for linear speed, another for angular.

The last option would take advantage of the fact that, according to the kinematic equations for a four-wheeled drive, the front and rear left wheels $W_{LF}$ and $W_{LR}$ can be combined into a single "virtual" wheel $W_L$ such that:

$$ \omega W_L = \frac{\omega W_{LF} + \omega W_{LR}}{2} $$

Where $\omega W_L$, $\omega W_{LF}$ and $\omega W_{LR}$ are respectively the angular speeds of the virtual left, left front and left rear wheels. A virtual right wheel $W_R$ can be defined analogously. This reduces the system definition to a differential drive, for which linear and angular speeds can be computed as:

$$ v = \frac{r (\omega W_L + \omega W_R)}{2} $$

$$ \omega = \frac{\omega W_R - \omega W_L}{2l} $$

Where $r$ is the wheel radius and $l$ the lateral distance between wheels. From the formulas above you get:

$$ \omega W_L = \frac{v}{r} - \omega l $$

$$ \omega W_R = \frac{v}{r} + \omega l $$

Allowing you to use linear and angular speeds as references to control the virtual wheels.