Lift-off performance of a jumping robot on hard and soft ground

We study lift-off during jumping on hard ground and granular media in a simple robot composed of a linear actuator in series with a spring. On hard ground, the robot jumps from a metallic base. On granular media (GM) composed of $0.3$ mm glass particles, a circular foot is attached to the spring and an air-fluidized bed sets the initial volume fraction, $\phi$. The actuator frequency and phase are systematically varied to find optimal performance. On both substrates, optimal jump height does not occur at the robot's resonant frequency $f_0$. Two distinct jumping modes emerge: a simple jump which is optimal above $f_0$ is achievable with a squat maneuver, and a ``stutter" jump which is optimal below $f_0$ is generated with a counter-movement. For hard ground, both modes exhibit similar performance. On closely packed GM ($\phi=0.62$), the simple jump becomes the favored mode. On loosely packed GM ($\phi=0.58$), jump height performance is significantly reduced due to greater yielding in the material. A dynamical model reveals how optimal lift-off results from non-resonant transient dynamics.