In simplest terms, the variable frequency drive (VFD) starts at zero hertz (Hz) and can run there indefinitely. I can make such a bold statement because at that frequency, and with the motor stationary, there is only magnetizing current being taken through the semiconductors. The variable frequency drive normal cooling system can disperse the heat that would be generated in this situation.
The motor in this case is likely one of two types; Induction or Permanent Magnet (PM). We also need to be clear when discussing 'frequency' and 'speed' as they pertain to motors. While they are related, they are not necessarily the same.
In the case of the Induction motor, at zero frequency or even up to 0.2 or 0.3 Hz there is little or no induction of the rotor bars taking place so only magnetizing current and low level active (torque producing) current is being taken from the variable frequency drive, even at zero speed.
A typical high efficiency induction motor needs approximately 0.75 to 1.0 Hz of rotor bar frequency to develop full torque. That is to say, with a locked rotor condition (zero speed) the motor shaft would be developing rated torque when the rated (slip) frequency and rated stator magnetic field strength is circulating around the rotor. This relationship of frequency and speed is consistent throughout the rated speed range of the induction motor.
In the PM motor, the rotor flux is established by means of the permanent magnets rather than induction. Therefore, with the proper the stator flux the motor shaft would be developing rated torque almost immediately as the stator frequency beings to rise. This also means that the PM motor, with proper stator magnetic field strength, would tend to hold its rotor position relative to the stator field poles, even at zero Hz.
If position accuracy is the objective, an encoder feedback and a field oriented control algorithm are likely to give the best results.
Whether it is an induction motor or PM motor, the important factor is the proper management of the stator field flux and orientation so that the needs of the rotor are fulfilled by the variable frequency drive. Attention must also be given to the cooling needs of the motor at low speeds.
In either case, the properly sized variable frequency drive is not likely to have an issue with low frequency operation.