Capacitor Powered Vehicle Project
First of all, you're premise is, well, wrong.  Ultracapacitors are not meant to be used as bulk energy storage.  That's not what they're for.  They give non-useful results and offer no benefit over traditional energy storage methods.  Ultracapacitors have high power density - NOT energy density.  The entire point of ultracapacitors is that they are used IN TANDEM with a rechargeable battery of some sort.  The capacitors handle things when the motor demands a ton of current, but this doesn't last long, and this removes the necessity for a battery to be forced to supply this high peak current draw. 

You say you want to prove they are viable for electric vehicle energy storage.  But they're not.  You're not proving anything.  No one doubts that they have sufficient power density to power a vehicle - in fact, they are used in a lot of vehicles, busses in particular, in combination with batteries and for high energy recovery from regenerative breaking.  The power is not the problem and is not the reason capacitors are not suitable for electric vehicles.  

The entire reason they aren't useable for electric vehicles is they DON'T have the energy storage needed.  The problem facing EVs has always been range.  It has never been lack of power.  By making this car, you will be doing nothing but proving what we already know - that they have much much to little energy storage capacity to be practical.  When or if new capacitors based on graphene are commercialized, then if you can build a 1/10th scale car that has equal or better range, or if it has less, somehow justifies it with some other advantage, than a battery based car.

A much better demonstration would be using capacitors correctly and comparing a car with a battery alone and a car with a battery and capacitors.  Without even doing much of anything, you'll see the capacitors add range to a battery powered vehicle if added.

Also, this is the one thing I don't understand.  Everyone wants to do capacitor this, capacitor that.  Why? What is the advantage to using capacitors? Batteries have flat discharge curves, can be recharged with >99% efficiency, and are becoming cheaper every day. 

Before you say the rapid charging and the nearly unlimited cycles, which I think is probably the most common reason behind wanting to use capacitors, this is not a reasonable expectation.  If you do not understand how capacitors work, then it may seem like they would be revolutionary, but unfortunately, the physical realities are not so sunny.

Graphene capacitors do nothing to address the single biggest issue that dwarfs even energy density: it's electrolytic.  Electrolytic capacitors dry out.  And faster the hotter they get.  Really fast.  Most have ratings of 1000-2000 hours at 85°C.  Capacitors drying out are why most electronics fail.  And, interestingly enough, if you had a smartphone powered by a graphene super capacitor, it would need to be replaced several times before the battery needed to be replaced once, because batteries don't dry out like electrolytic capacitors do.

If you're thinking, "we could just seal the capacitor"...we can't.  Electrolytes, by the property that makes them useful and conductive in the first place, all produce hydrogen gas as a by-product of current conduction.  We must provide a way for this gas to escape from the capacitor, or it will explode.  

The capacitor you are using has a rated life of 1500 hours.  It only has the cycle life and '10 year DC life' if kept at 25°C over that entire operational life.  That is, well, ridiculous.  Automotive applications require much higher temperatures, and beyond that, using the capacitors heats them up.  So even if they were magic graphene capacitors with the same energy storage as a battery, a battery will still outlast them, which begs the question: why? Why capacitors? What is the point?

As for the proper use of ultracaps, it works like this: batteries are happier and will provide more joules of energy if they are discharged steadily and the lower the current, the better.  So, by combining ultracapacitors to handle high power demands with a battery to handle the energy storage (which is using both for the thing they are designed for), you get a far more effective vehicle that not only can manage high acceleration or efficient regenerative breaking (by dumping the energy into the capacitors rather than the battery - as they can absorb as quickly as they can dispense), but actually will extract more useful energy out of the same sized battery.

Lets ignore the fact that you're using ultracapcitors in the worst possible way for seemingly no reason at all.  You're right, the buck-boost converter is needed, and bidirectional since ultracapacitors are not meant to be used like this and don't have a flat discharge curve at all (which is solved by using them correctly, with a battery).  But boost converters are less efficient the lower the input voltage, and 2.85V is effectively too low to start with, and will only get worse.  You would do much better with several capacitors in series, because that voltage is simply too low to be practical.

Finally, you can't limit the current of a brushless motor and have it operate correctly.  They need the current they demand, period.  It is more than likely that the peak stall current will be much to large for a buck-boost converter to provide, regardless of if it is needed for only a millisecond or not.  You should have a direct connection from the capacitors/batteries to the motor controller.  Limiting the current will at best use the same energy less efficiently to do the same thing.  It takes so many joules to get the vehicle moving to a certain speed from a dead stop, and limiting the current does nothing to change that.  You'll discharge that much from the capacitors no matter what, but the losses in a buck-boost converter coupled with the losses from not giving the motor what it wants will make it take far more joules to do anything you could do by simply hooking the bank directly to the speed controller.