Electronic – arduino – Stepper Motor Big Easy Driver Undervoltage

The rearranged version of your schematic looks like

^{simulate this circuit – Schematic created using CircuitLab}

The problems have already been mentioned by jippie, and rearranging the schematic shows that the lower four transistors (Q8,Q9,Q6,Q11) are inverted (collector<->emitter).
Also note that even if the level of the bridge supply was the same as the control level you would need to use one resistor for each base because a common one would make both upper/lower transistors conduct simultaneously.

One solution you can use to control the bridge from two logic level control signals is shown in A simple H-Bridge design

Assuming that the outputs of your brushed drivers are a voltage or current proportional to the commanded speed, you can certainly drive a bipolar stepper with two brush drivers. Let's assume that the output of a driver set for 100 gives exactly the drive current needed by a stepper, and that -100 gives exactly the current needed, but with opposite polarity. Then the two driver channels can be driven like this:

^{simulate this circuit – Schematic created using CircuitLab}

The position of the stepper motor is given by the relative amplitudes of the two phases. In this case, the requirement is that they be 90 degrees apart. In the figure, reading from left to right gives rotation in one direction, while reading from right to left gives rotation in the opposite direction. Always keep track of what step you are at, and only change the drives so as to move one step to the right or left.