I'd like build tracking system for under-actuated robot. The optimal reference paths has been calculated by an optimization code for path planning. My question is it necessary to linearize the system in each sample time or I must linearize the model at one point (for example the end desired end point) and use this model
You don't need to linearize in anyway. You can design tracking control using integrator backstepping method. Check our paper if your system can be transformed into strict feedback form (or can be feedback linearized).
Article Adaptive Controller Design for Tracking and Disturbance Atte...
No need to linearize. The result you will get will not be an optimal if you linearize.
For understanding I always first linearize and observe the solution. Later on remove the linearity constraint and move with exact problem.
Linearization is not needed, however you can do linearization about certain operating points at the beginning to comprehend the behaviour then omit.
In EFK, you need to linearize around the mean sample by taking Jacobian around that point.
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