The automatic generation of animal-like motions in quadruped robots remains challenging, especially in achieving natural joint movements through reinforcement learning (RL). Although RL-based approaches simplify policy optimization, maintaining smooth motions still requires extensive hyperparameter adjustments. We address this challenge by developing: (1) An incremental foot position controller that regulates displacement of foot position increments instead of absolute joint positions, intrinsically reducing abrupt joint movements through action space regularization. (2) A phase-synchronized reward function that explicitly promotes inter-leg coordination patterns. Experimental results in both the simulated and real world demonstrate that our approach generates more coherent gait cycles compared to direct joint position control methods, with the quadruped robot exhibiting consistently smooth and coordinated locomotion transitions. Supplementary video evidence is available at https://vsislab.github.io/ifpc/ .