Bipolar H-bridge chips are common, relatively cheap and easy to use, but have far higher losses than MOSFET H-bridges so will require heatsinking. The trouble with MOSFET H-bridges is finding one! Integrated ICs tend to be intended for low voltage low power stuff like pager/cellphone vibration motor control, and are invariably in packages that are evil or impossible to hand solder. You can easily find H-bridge driver ICs but must then add external MOSFETS.
The schematic you posted looks OK. However the bridge drivers *MUST*be proper MOSFET drivers, not some crappy 4000 series logic chip that cant supply enough gate current. There are also a lot of possible issues like dead-time and shoot-through that make a dedicated H-Bridge driver advisable, and prototyping it is going to be a beast as you don't have the experience and probably don't have the test equipment to develop something like this without filling a couple of 1lb jars with blown-up MOSFETs and burnt-out controller chips. This can be alleviated to some extent if the H-Bridge driver chip manufacturer has a development kit available that you can copy the PCB layout from, but if all you've got is loose components and datasheets, much trouble and smoke is certain.
Unless you are going to production or need to minimise size/weight or power consumption and losses, I'd go with bipolar H-bridge chips on a large heatsink. It will be cheaper for a one-off and far easier to get going.
Anyway, on the assumption that all the coils are of similar resistance value, what would differ? I plan to make them closer in size.
If each coil has a significantly different resistance, as previously stated, you either need:
* a PWM capable H-bridge for each with a sense resistor and a feedback loop to a PWM controller to set the current - which is moderately complex + you need to repeat the circuit 20 times,
or you need to:
* control the supply voltage to each individual H-bridge to set the current for its coil, so 20 H-briges + an adjustable switching regulator for each.
If the resistance values are similar, you can supply all the H-bridges with the same voltage and the current will be determined by I=V/R, so if all R falls within a 10% range, so will all I. This is a great simplification as you now only need one switching regulator, albeit a heavy duty one.