Lift-Off Dynamics in a Simple Jumping Robot

We study vertical jumping in a simple robot comprising an actuated mass-spring arrangement. The actuator frequency and phase are systematically varied to find optimal performance. Optimal jumps occur above and below (but not 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 peculiar stutter jump, which is optimal below $f_0$, is generated with a countermovement. A simple dynamical model reveals how optimal lift-off results from nonresonant transient dynamics.

Figure 1

Jumping robot (a) schematic diagram. (b), (c) Actuator position $x_a$ and video tracked position of the thrust rod $x_p$ (dotted trajectory), and the sensor voltage for $A=0.19\mathrm{mm}$ in which the robot does not lift off and $A=0.30\mathrm{mm}$ in which the robot lifts off.

Figure 2

Jump height and jumping modes. (a) Experimental jump height $h$ as a function of $f$ and $\varphi$ with illustrated actuator trajectories (top) at different $\varphi$ with $A=4\mathrm{mm}$. White solid lines (derived from model) separate different jumping modes, with ST indicating stutter jump, S for single jump, and MJ for multijump. (b) The inset shows the model [Eq. (1)] with the same parameters. Panels (c) and (d) illustrate trajectories for the stutter and single jumps. The robot is airborne (white robots) when the rod position, $x_p>0$. The global actuator position $x_A$ is not to scale with rod length.

Figure 3

Experimental (solid) and simulation (dashed) results of (a) maximum jump height at each phase, (b) the corresponding optimal frequency for each phase, (c) time to lift-off, and (d) deformation power at optimal frequency; initial transients are omitted in this calculation.

Figure 4

Simulated time trajectories of absolute actuator velocity (black) and power input by external forces (red, lighter color) for $f/f_0=0.54$, 0.72 (optimal), and 1, at $\varphi =\pi /2$. The vertical dotted line indicates when the actuator stops. Light gray areas: aerial state ($x_p>0$), dark gray: negative force ground state ($mg/k<x_p\le 0$), white: positive force ground state [$x_p\le (mg/k)$].