What methods and equipments can I used to measure and record the current variation of CNC servo motor while machining.
There are two features of Motor control worth sampling. The first is the current you mention. There are two parts of the current path that are going to be interesting, and you probably need to measure them separately. The first half is the rise current, the part of the commutation when the motor is connected to power. The natural induction of the motor windings will cause the current to rise gradually, until the control loop shuts it off, either because it is using current limiting (chopper drive), or some other part of the control loop is turning the motor off. In the case of a normal H-Bridge, this is the only current path. In the off state, no current can flow in or out of the motor because it is completely disconnected. More elaborate circuits may also use regeneration, in which the motor is connected to another part of the circuit to assist in deceleration, either a dump resistor or a battery recharging circuit.
For small currents (about 1A or less), the usual way to do this is with a low value current sense resistor, connected in series with the motor windings. The current passing through the resistor causes a voltage difference on the opposite sides of the resistor, which can be measured quite easily.
Larger currents require fancier tricks, but a good option is a hall effect current sensor. The result is quite similar, an output voltage in proportion to the instantaneous current. This type of sensor has a slight propagation delay ( a few microseconds ) because of the multiple stages of amplification needed with the hall effect sensors.
The other feature of motor control worth looking at is Back EMF. This can be useful in some types of (really simple) servo control to determine motor velocity, since back EMF is proportional to motor speed. This feature can only be measured on windings that are 'off', since the voltage of the power supply will dominate the higher impedance windings. In the case of a PMDC motor, Back emf would be measured during the off periods of motor control. The back emf at the moment of switch-off will be equal to the supply voltage, but will decay, due to the natural inductance of the windings until it reaches the level determined by its speed.
One thing you can do to get an approximate measure, that isn't too complicated, is to measure the DC current going into the switching bridge. If you know the current and voltage of the DC supply, you get at least an upper limit to what the output power can be. This doesn't account for the efficiency of the switching network nor of the motor itself, but has the advantage that it's simpler to implement, and in some cases is sufficient.
Otherwise, you're going to have to measure the phase currents into the motor. First you need to know how many current sensors you will need, and how you determine that will depend on the number of phases and their connection. That done, you then need to deal with what amounts to AC current with a boatload of harmonics on it, and possibly under conditions of changing frequency due to changes in the mechanical load on the motor, so this is definitely a more complicated way to go.