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This work presents a predictive-control approach to active mechanical filtering of complex, periodic motions of organs induced by respiration or heart beating in robotized surgery. Two different predictive-control schemes are proposed for the compensation of respiratory motions or cardiac motions. For respiratory motions, the periodic property of the disturbance has been included into the input-output model of the controlled system so as to have the robotic system learn and anticipate perturbation motions. A new cost function is proposed for the unconstrained generalized predictive controller (GPC), where reference tracking is decoupled from the rejection of predictable periodic motions. Cardiac motions are more complex, since they are the combination of two periodic nonharmonic components. An adaptive disturbance predictor is proposed which outputs future predicted disturbance values. These predicted values are used to anticipate the disturbance by using the predictive feature of a regular GPC. Experimental results are presented on a laboratory testbed and in vivo on pigs. They demonstrate the effectiveness of the two proposed methods to compensate complex physiological motion.