I'm guessing the outputs of each of these pedal things is a analog voltage? Are the pedals pots that put out a voltage as a function of pedal position?
If so, then the question is basically asking for a crossbar switch of 20 analog outputs to 20 analog inputs. I'll assume this is what you want.
This could be done with relays, but you would need a great many of them. If each input must be arbitrarily switchable to any pedal, then you need 400 separate switches. This would be physically large and fairly expensive. Solid state relays are not appropriate since they will have voltage drops accross them. They are meant for switching power where dropping a Volt or so doesn't matter.
There are such things as analog switches. These are ICs with several analog paths that are either open or closed to a few 10s of Ωs and are digitally controlled. You could put 20 channels of these switches in front of every input and have firmware in a microcontroller turn on one of them per input. This is basically the same as the relay concept, except that the analog switches will be much smaller and cheaper. Ultimately there are still 400 individual switches. You might want to buffer the output of each individual mux (collection of 20 switches with their outputs tied together) with a opamp. This avoids issues due to the on-state impedance of the individual switches.
However, there might be a better way altogether than a brute force crossbar with 400 independent switches. You could read each of the 20 pedal output voltages into A/Ds in a microcontroller, do the selecting of which inputs are used for each of the outputs, then write the selected values back out to D/As, once for each input. Instead of 400 switches you have 20 A/D input channels and 20 D/A output channels. Since the pedals change their outputs in human time, you've got lots and lots of time to scan the A/D inputs in a microcontroller. There are micros that have 20 or more A/D inputs, so all you would need is little R-C filtering and input protection, then right into 20 separate pins of the micro.
In the digital approach, the muxed pedal signals don't need to be real D/As. They could be low pass filtered PWM outputs from the micro. No micro has 20 PWM output channels that I know of, but again your signals are so slow that the PWM should be doable in firmware with a interrupt per PWM slice. For example, a 100 kHz interrupt inside the micrco that updates each PWM output would give you 1 part in 1000 resolution (much better than you can hold your foot in a given position) updated at 100 Hz rate. That would require some low pass filtering, but the output should be able to keep up with how fast a human can move a foot pedal. Each low pass filtered result should be buffered with a opamp to make the impedance low.
I'd seriously look into this last approach. The electronics will be simpler, smaller, and cheaper. The foreground code in the micro can be interpreting commands received via a UART to indicate which pedal should drive each output. It can also store the last configuration in non-volatile memory.
