How to make a robot move using Arduino other than timing to predefined locations?

Without sensors or encoders, there is no way to know where you are. Without timing, there is no way to guess how far you may have travelled.

Without sensors, encoders, or timing, it is impossible to do what you are asking.

Cartesians Robot use sensors. All robot need sensors. Sometime they are external (camera looking at the robot), sometime they are on the robot (IMU).

Basically what you're asking is :

• How can an animal without any way to sense the world move from an exact point to another ?

Well, it can't.

Why do you wish not to use sensors ? There are very cheap ones out there that could help you with your project. You just need to define this project a little better so you can find the sensor that works for you.

Btw, going to predefined location without sensor like encoders, is not really feasible either. Or a pain. You'll need to calculate an approximate speed from the voltage (and, as someone pointed out, knowing the voltage would need a sensor) you input in the motor and, trust me, it's not going to be very accurate. The drift is going to kill you.