Mathematical Modeling and Dynamic Simulation of Frog Jumping for Bio-Inspired Robotics

The biomechanics of frog jumping has been a subject of significant interest in both biology and engineering, driven by the high efficiency of their movement. This study presents the dynamic simulation of a frog’s complete jump cycle, from take-off to landing and re-stabilization, to advance the development of bio-inspired jumping robots for irregular terrains. As a primary contribution, and unlike previous studies that focus exclusively on the propulsion phase, this work addresses all stages, using direct servomotor actuation without mechanical energy storage. Biological joint kinematics were mathematically characterized using Cubic Smoothing Splines. By empirically tuning the smoothing parameter ( p ), the trajectories achieved the continuous differentiability required for electromechanical actuation. These curves were implemented into a 3D multibody simulation (Altair Inspire), where a PID-based tracking framework managed the mechanically nonlinear multibody dynamics governing the jump (arising from contact forces, impacts, and time-varying inertial effects) to ensure stabilization during the complex landing phase. Validating the model against previous studies, the simulation successfully achieved a maximum horizontal jump distance of 24.12 cm (4.02 body lengths) and a peak velocity of 1.45 m/s. The kinematic fidelity of the model was mathematically validated, yielding a maximum Normalized Root Mean Square Error (NRMSE) of 4.121% relative to biological reference trajectories. Furthermore, the robustness of the landing and re-stabilization phases was demonstrated through a continuous double jump covering a total distance of 45.83 cm. Finally, a dynamic scaling analysis was performed to evaluate the feasibility of implementing real motors. Ultimately, this study establishes a mathematically robust framework for replicating frog-inspired jumping dynamics, contributing a transferable methodology for the design and control of articulated bio-inspired robotic systems.