Sinusoidal Commutation

Six-step commutation jumps the field vector between only six positions. The natural fix is to stop jumping and instead drive the current in each of the three phases as a smooth sine wave, each shifted 120° from the other two. Instead of hopping the "push" direction six times per revolution, a sinusoidally-commutated motor can point it in any direction at any instant, gliding continuously around the motor.

Source: Wikimedia Commons

$$I_a = I_{peak}\sin (\theta_e) \qquad I_b = I_{peak}\sin (\theta_e - 120°) \qquad I_c = I_{peak}\sin (\theta_e + 120°)$$

Add the magnetic effect of all three phases together at any instant, and the result is a single, smoothly-rotating magnetic field vector, no jumps, no steps. This is smoother, quieter, and more efficient than six-step commutation, but it requires much finer rotor position resolution (a real encoder, not just six Hall states) so the controller always knows exactly which point on the sine wave each phase should be at.

Source: Wikimedia Commons

Sinusoidal commutation is a big improvement, but on its own it's still an open-loop shape, the controller is just tracing out a predetermined sine wave locked to rotor angle. It has no direct handle on how much torque that produces, or how to efficiently correct it in real time. That's the problem Field-Oriented Control solves.