Nonlinear Model Predictive Control for mechanical applications is often used to perform the tracking of time-varying reference trajectories, and is typically implemented using quadratic penalty functions. Controllers for mechanical systems, however, are often required to handle large deviations from the reference trajectory. In such cases, it has been observed that NMPC schemes based on quadratic penalties can have undesirably aggressive behaviours. Heuristics can be developed to tackle these issues, but they require intricate and non-systematic tuning procedures. This paper proposes an NMPC scheme based on a specific class of penalty functions to handle large deviations of quadrature states from their reference, offering an intuitive and easy-to-tune alternative. The behaviour of the proposed NMPC scheme is analysed, and the conditions for its nominal stability are established. The control scheme is illustrated on a simulated quadcopter.