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In my understanding the integrator in the block diagrams is simply a placeholder for a controller with integral action (I or PI controller). But the diagram doesn't tell about the kind of involved signals. You can e.g. think of incremental encoders, then phase comparator and integrator would be represented by some logic and an up/down counter.
I don't know if somebody asked you about how you control the motors. Is it through potentiometer or joystick or something else. If you are controlling through pot or joystick. You will get this problem of sync. The only option with this is manual adjustment of either the pot, joystick or adjustment in software if you are controlling through PWM. The other problem for non-sync is your weight of robo on both sides. If one side is heavier then that side will definately move slow that means imporper CG. So a balance of weight is requied.
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