Problem in design?

[ccurtis]

The simpler circuit will need base resistors on the PNP transistors to keep them turned off as the bases are floating at the moment.
The 10uH choke will not be needed as it will not have any effect in this situation. The 25R (27R) resistor will reduce the back emf from the relay coil.

The series inductor-resistor combo is the relay coil equivalent. At first glance, large value resistors from the PNP bases to 5V would seem necessary to keep them turned off, but not much of anything can happen until deliberate turn-on of one of the NPNs occurs, after which niether base is floating.

 

[Roff]

Either way a total drop of 0.15 v for the bridge is very optimistic.

No doubt. LTspice, with the PN2222 and PN2907 models I have, showed about 4.66V. Still pretty optimistic.

 

[colin55]

After the relay has been pulsed in one direction (either direction) the base of each PNP transistor is connected in such a way that any leakage from them will turn them on more. This is basically called "floating."