(eng): This paper deals with a novel nonlinear design of the discrete model predictive control represented by two algorithms based on the features of linear methods for the numerical solution of ordinary differential equations. The design algorithms allow more accurate motion control of robotic or mechatronic systems that are usually modelled by nonlinear differential equations up to the second order. The proposed ways apply nonlinear models directly to the construction of equations of predictions employed in predictive control design. These equations are composed using principles of explicit linear multi-step methods leading to straightforward and unambiguous construction of the predictions. Examples of the noticeably improved behaviour of proposed ways in comparison with conventional linear predictive control are demonstrated by comparative simulations with the nonlinear model of six-axis articulated robot.