The schematic is OK.
The problem is to implement an algorithm to get the motor to run with reasonable torque.
I assume the motor is a brushless, sensorless DC motor.

Such motors typically require a predefined starting point along with back emf sensing.
At any point in time only two of the three windings are energised in a predefined sequence.
Initially the motor is brought into a known position by energising only two predetermined windings.
This is followed by the next pair of windings being energised so that the rotor starts to move ( there are six valid combinations of energising windings for a full 360 degree turn of the shaft.)
Back emf builds up with shaft speed and depending on how well the BEMF sensing circuitry works the motor may be switched from a slow ... very slow acceleration regime untill BEMF sensing works ok to full synchronised switching , synchronised to motor rotor speed.

Starting and direction changes are a problem without hall sensors.
as MGP pointed out.. a hard thing to tune and get to run well .