As locomotion decisions must be taken by considering the future, most existing quadruped controllers are based on a model predictive controller (MPC) with a reduced model of the dynamics to generate the motion, followed by a second whole-body controller to follow the movement. Yet the choice of the considered reduction in the MPC is often ad-hoc or decided by intuition. In this article, we focus on particular MPCs and analyze the effect of the reduced models on the robot behavior. Based on existing formulations, we offer additional controllers to better understand the influence of the reductions in the controller capabilities. Finally, we propose a robust predictive controller capable of optimizing the foot placements, gait period, center-of-mass trajectory and corresponding ground reaction forces. The behavior of these controllers is statistically evaluated in simulation. This empirical study is a basis for understanding the relative importance of the components of the optimal control problem (variables, costs, dynamics), that are sometimes arbitrarily emphasized or neglected. We also provide a qualitative study in simulation and on the real robot Solo.