As displayed in Fig. 1 and Supplementary Movie 1, the control torque τ, applied to the lander at rest, actuates it to pivot over the leading edge, and then the bottom side is dragged downwards in contact with the granular regolith, yielding a reaction force on to the lander (Allen et al. 2013). Given sufficient acceleration impulsive moment, the lander will leave the surface, hopping forwards in a ballistic trajectory. Hockman et al. (2017) stated that the key mechanism of this internally actuated mobility is the transmission of rotational energy to translation energy, indicating that the acceleration duration T of the control torque could determine the hopping outcomes. To elucidate the fundamental dynamics that governs surface locomotion on asteroids, we adopt 12 different acceleration durations ranging from 20 to 800 $\rm {ms}$ in this section, in which μ is set to 0.5, β is set to 1.0, and c is set to 0 Pa. Two typical dynamics data, corresponding to 20 and 600 $\rm {ms}$, are displayed in the insets of Figs 2(a) and (b), which shows instantaneous velocity $v$ and angle θ (relative to horizon), versus time t. Generally, during the hopping process, the acceleration first shows a gradual increase, and then decreases rapidly at the end of control torque, during which the reaction force is highly fluctuating due to intermittent transmissions of acoustic energy along force-chain-like networks (Clark, Kondic & Behringer 2012; Cheng et al. 2018), as illustrated in Fig 3. In this collisional scenario, the acceleration torque excites strong force chains which exert sporadic impacts on the bottom of the lander, while after the end of acceleration, the force networks become sparse and fragmented immediately, leading to the rapid decrease of the reaction force as evidenced by Fig. 2(c). The work done by the lander to impact with regolith surface generates the contact force, and then increase its velocity and height before the lander loses contact with the regolith at time tesc. After that, the lander behaves as free projectile motion under only gravity, which corresponds to a gradual decrease of the vertical velocity, forming a peak value in the velocity curve and a turning point in the angle curve. Hereafter, we define these two values as hopping velocity V and hopping angle ϑ, respectively. Note that the lowercase $v$ and θ represent the instantaneous velocity and angle, while the uppercase V and ϑ represent hopping outcomes. The hopping distance can be deduced from V and ϑ, given as 2V2cos (ϑ)sin (ϑ)/g. Thus in the case of T = 600 ms, the lander hops about 57 m under the gravity of 1.0 × 10−4$\rm {m\, s^{-2}}$, corresponding to a large surface coverage on small asteroids. And in a lower acceleration duration like T = 40 ms, the lander only moves about 45 cm, which gives a much more precise form of surface relocation. The results presented here provide strong evidence on the feasibility of the hopping landers for the exploration of low-gravity bodies. Additionally, our simulations show the lander takes about 1–17 min to complete the hop, which is far less than the battery life of MASCOT (16 h, Ho et al. 2017). This information is crucial for the design of landers equipped without solar batteries.
Citation: Cheng et al. 2018
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