RFT-Based Analysis of Mobility Performance of Hopping Rover On

RFT-based Analysis of Mobility Performance of Hopping Rover on Soft Soil for Planetary Exploration Kosuke Sakamoto1, Masatsugu Otsuki2, Takashi Kubota2, Yoshiki Morino1 1Waseda University, 3-4-1 Okubo Sinjuku Tokyo, Japan, E-mail: [email protected], [email protected] 2ISAS,JAXA, 3-3-1 Yoshinodai Chuo Sagamihara Kanagawa, Japan, E-mail: [email protected], [email protected]

ABSTRACT Future planetary exploration requires rovers to ex- hibit advanced surface mobility. Various remote observations reveal that a number of scientifically significant sites are located on rough terrain such as cliff, steep slopes, or moon hole. Instead of improving wheel ’s performance, this paper studies the performance of a contact-based hopping mechanism on soft soil terrain. In order to design the concept of a jumping mechanism for challenging planetary environments, the movement of rovers on loose regolith is studied based on Resistive Force Theory (RFT). Particularly, this paper analyzes the effect of stepper Figure 1: MER Spirit wheels got stuck in soft angles against the ground plane through RFT-based soil (Image by NASA/JPL-Caltech) simulation and experiments. Vertical jumping tests were performed on silica sand. The trend of simulation and experiment is consistent, while small numerical errors can be observed. The results show stuck in soft soil. Due to this adversity, it is difficult that RFT is effective to get trend of mobility perfor- for Spirit to continue mobile exploration. In addi- mance on soft soil. tion, to explore small solar system bodies that have low or ultra-low gravity is interesting for many countries recently. However, these bodies are hard to ex- 1 INTRODUCTION plore for conventional wheeled rover because of their A lot of planetary exploration missions have carried low-gravity level. There are two approaches to solve out by a lot of countries so far. In the late 1960s and these challenges: improving wheel’s performance, the early 1970s, the Soviet Union’s Lunokhod 1 and and developing different locomotive systems. As 2 [1] had reached the moon for the first time in his- an example of the former approach, NASA/JPL- tory, and the U.S.’s Lunar Roving Vehicle (LRV) [2] Caltech developed two wheeled tether rover, called of the Apollo missions made contribution to the suc- Axel [8], that was developed to provide access and cess of the human explore missions. As more recent get samples of extreme terrain. Axel’s wheels are example, China’s Jade Rabbit [3] had also reached large diameter and have grousers. Rovers which the moon in 2013. As other examples of rovers, have large diameter grouser wheels can get over ob- NASA’s Sojouner [4] of Mars Pathfinder, Mars Ex- stacles as large as wheel’s diameter without suspen- ploration Rovers (Spirit and Opportunity) [5] and sion system like a rocker-bogie. Mars Science Laboratory (Curiosity) [6] have ex- On the other hand, non-wheeled rovers are also de- plored the red martian surface. These rover’s loco- veloped in a lot of countries or research institutions. motion mechanism is wheeled platforms due to their The PrOP-F Phobos probe [9] was a hopping rover simplicity, energy efficiency, and reliability. that was used for to jump on low or ultra-low gravity However, various remote observations discovered bodies. DLR’s MASCOT lander [10] was also able that a number of scientific significant terrains, such to hop using reaction wheels. The Hedgehog [11] as cliffs, steep slopes, and moon hole [7], are thought that was developed by NASA/JPL-Caltech can per- hard to traversed by conventional wheeled rovers. form pivoting, slipping and hopping by using three For example, Figure. 1 shows MER spirit wheel got reaction wheels and spikes attached its cubical body. The Comet Hopper (CHopper) [12] was developed for NASA Discovery Program mission, which could hop multiple times on the surface during descents. JAXA’s MINERVA [13] that was deployed from Hayabusa had two wheels to induce hopping force and determine hopping direction. The Highland Ter- rain Hopper (HOPTER)[14] had a originality locomotive system to explore small solar system bodies that are lower gravity level than the Earth. Research and development of locomotive mechanism is one of the most important points for planetary exploration rovers that have to survive the harsh environment differed from Earth, such as strong cosmic radia- Figure 2: Moving of micro-plate in granular tion, thermal cycling and thermal extremes due to a media (original figure in [18]) thin or lack of an atmosphere and gravity levels. At the point of moving efficiency, hopping is more useful than wheeled platform in lower gravity levels. called stepper, is given by: The contribution of this paper is analysis of hop- ∫ ping performance on soft soil. Some parts of plan- FRFT = ζ α ( β, γ) z dAS (1) etary terrain are soft soil, such as regolith of the S Moon. The experiment that is supposed lower or where α( β, γ) denotes the resistive stress per unit microgravity level planets is difficult to test under depth, dAS denotes the area of stepper, and z de- the Earth environment due to difference of grav- notes the sinkage depth from the soil surface. α ity levels. As such, microgravity or lower gravity denotes the function of β, attack angle of stepper, γ, testing becomes confined to few key experiments intrusion angle of stepper, and ζ denotes the SF that under well-defined conditions. Therefore, models indicates soil feature. In this paper, as a stepper, flat construction and simulations of mobility platforms plate was employed. in different environments of kind of planets help us to develop better devices for planetary exploration. Figure. 3 shows the schematic of hopping rover. Pa- rameters are defined as follows: z0 denotes an initial sinkage depth, d denotes an initial sinkage of spring, 2 DYNAMICS MODEL φ denotes a slope angle of stepper, Mb denotes a There are few conventional mobility platform mod- mass of body, mj denotes a mass of stepper, and ks, els on soft soil which field of study is called ter- cs denote spring and damping constants. The area ramechanics. The classical terramechanics was de- of stepper, As, is given by: veloped by M. G. Bekker [15], later extended by J. Y. Wong [16] or A. R. Reece [17]. As = h0w (2) Recently, the new terramechanical method is proposed by Li et al [18]. The proposed approach where h0 denotes height of stepper, w denotes width stands for Resistive Force Theory (RFT)[19]. The of stepper. proposed method is also based on empirical models The schematic of vertical hopping experiment is from experimental results, where they measure the shown in Figure. 4. In this paper, hopping direc- Resistive Force (RF) of small plate that is moving tion is only vertical. Hence, only vertical height freely in granular media. And, there is only single h is measured and simulated. On the other hand, parameter, called “scaling factor (SF)” that we need motions and rolling of horizontal direction are not to measure on each terrain type. The new model considered. As a result, the equations of motion is is more effective to simulate the dynamic motion of expressed by: hopping rover (hopper), because their model dose ( ) not rely on conventional terramechanical models. M (z¨ (t) + g) + c z˙ (t) − z˙ (t) b b ( s b ) j According to [15], or [17], Only vertical pressure of +k z (t) − z (t) = 0 (3) horizontal plate is determined by the conventional ( s b) (j ) models. In addition, the conventional method is ap- mj z¨j (t) + g − cs z˙b (t) − z˙j (t) plied to only wheel model. According to [18], the ( ) − − = reaction or resistive force (RF) exerted leg elements, ks zb (t) zj (t) FRFT (4) Table 1: Parameters for simulation and experiment

Terms Symbol Value Unit Gravitational acceleration g 9.81 m/s2 Mass of body Mb 0.074 kg Mass of stepper mj 0.016 kg Spring coeﬃcient ks 1025 N/m Damping coeﬃcient cs 0.87 Ns/m Initial sinkage of spring d 0.037 m Initial position z0 0.001 m Width of stepper w 0.08 m Height of stepper h0 0.06 m Thickness of stepper b 0.004 m SF ζ 2.0 -

cated as following equations. Figure 3: Schematic of hopper π β = − φ (7) 2 π γ = (8) 2

Note that Eq. (5) is only completed in zj ≤ 0, because FRFT is the resistive force that is generated only when objects move in granular media. Therefore, if zj ≥ 0, Eq. (5) is rewritten as:

FRFT = 0 (9) In addition, after the spring length becomes natural length, the spring is ﬁxed. Therefore, only the gravity acts to hopper body and stepper. As a result, the equations of motion when the hopper and stepper Figure 4: Schematic of vertical hopping exper- are in air, are expressed by: iment M (z¨ (t) + g) = 0 (10) b ( b ) mj z¨j (t) + g = 0 (11) where each zb and zj denote coordinate axises along the vertical direction of hopper body and stepper, and the RF, FRFT, is rewritten as: 3 NUMERICAL SIMULATION ∫ ( ) The numerical simulation result of the time histories FRFT = ζ α( β, γ) −zj (t) sin φdAS (5) of hopping height is shown in Figure. 5. All the pa- S rameters that were used in this numerical simulation and experiment are indicated in Table. 1. To calcu- α β, γ where ( ) is given by following equation. late Eq.5 and Eq. (4), the Runge-Kutta methods was used. Figure. 5 shows that the slope angle of the [ ( ) ∑1 ∑1 φ = φ m β nγ highest point is 90 [deg]. And, as the is re- α( β, γ) = A , cos 2π + m n π 2π duced, the height is reduced proportionately. Also, m=−1 n=0 ( )] the rate of decrease of maximum hopping height is m β nγ not constant related to φ. The reason is considered +Bm,n sin 2π + (6) π 2π as follows: FRFT is proportional to sin φ. Hence, if FRFT is not large enough, stepper sink into the sand where Am,n and Bm,n are parameters appeared in until forces are balanced. As a result, it is consid- [18], and in vertical hopping case, β and γ is indi- ered that the initial potential energy is dissipated in φ = 15 1 0.6 φ = 30 φ = 45 0.9 φ = 60 0.5 φ = 75 ks=1400 φ = 90 0.8 0.4 ks=1200 0.7 ks=1000 0.3 0.6 ks=800

0.5 0.2 ks=600 Hopping height [m]

Maximum Hopping height [m] 0.4 0.1

0.3 0 0.2 30 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 φ [deg] time [s] Figure 5: Time history of hopping height of the Figure 7: The comparison of maximum hopping hopper height with the value of ks

×10-3 φ = 25 5 φ = 30 φ = 45 0 φ = 60 φ = 75 φ = 90 -5

-10

-15 Hopping height [m] -20

-25

0 2 4 6 8 time [s] ×10-3 Figure 6: Time history of the sinkage of stepper Figure 8: Stepper proportion to sinkage depth. per. Springs are taken up by the motor, then re- leased. Using springs forces and stepper, the hopper φ In terms of the sinkage of stepper, as the is in- hops vertically. To measure the hopping height, mo- creased, the sinkage depth is decreased, as shown in tion capture cameras, as shown in Figure. 10, and Figure. 6. If the sinkage is increased, the dissipa- optical motion capture software were used. The tion of energy is also increased. If the dissipation software is “Motive: Tracker” of OptiTrack and of energy is proportion to ks, hmax is also propor- these cameras are “OptiTrack S250:E”. The hop- tion to ks. However, Figure. 7 shows that hmax is per body is a commercial item that is a “Parrot not proportion to ks. The result indicates that the MINIDRONES-Jumping sumo”. As shown in Fig- dissipation of energy has some sort of non-linear ure. 8, the steppers were modiﬁed by using 3D- parameters. printer with PLA resin, and prepared seven forms; 4 HOPPING EXPERIMENTS φ = 90, 75, 60, 45, 30, 25, 20. The testbed soil is sil- In this section, Vertical hopping experiments and ica sand whose SF is two. Each parameters that are these experimental results are shown to verify the used in this experiments are shown in Table. 1. numerical simulations results of section 3. After The results are shown in Table. 2. The values of hop- describing those, the simulated results and experi- ping height are indicated in mean s.d. These ex- mental results are compared and considered. periments were tested ten times each φ. The results indicate that, in φ = 90, 75, 60, 45, 30, 25[deg], the 4.1 Vertical hopping experiments hopping height is increased with increase in φ. How- The hopper is shown in Figure. 9. The developed ever, at the φ = 20[deg], the hopping height is higher hopper is attached two springs, a motor, and a step- than the hopping height at the φ = 25, 30[deg]. The Table 2: Vertical hopping height with varying stepper angles

Stepper angle φ [deg] Hopping height h [cm] 90 63.5 1.09 75 62.2 1.02 60 59.7 0.274 45 53.1 0.790 30 39.6 0.400 25 30.4 0.642 20 46.6 0.927

Figure 9: Experimental setup of hopper

0.8 Simulation Experiment 0.7

0.6

0.5

0.4

Hopping height [m] 0.3

0.2

0.1 10 20 30 40 50 60 70 80 90 φ [deg] Figure 10: Motion capture cameras Figure 11: Comparison of vertical hopping height between experiment and sim- comparison of results of vertical hopping experi- ulation ments and numerical simulations is shown in Fig- ure. 11. The simulated results are shown as blue line, and the experimental results are shown as red the sand makes experimental values of FRFT bigger square with error bar. In 25 ≤ φ ≤ 90[deg], the than simulated value, because FRFT is proportional same trend of experimental values and simulated to sinkage depth. Secondly, as shown in Figure. 13, values are confirmed from these results. This result if the slope angle becomes smaller than an angle of shows that RFT is useful to research the trend of repose of sand, a part of stepper is above the surface motion on soft soil. of sand. This part of stepper makes the experimental value of F smaller than the simulated value. On the other hand, the error between experimental RFT Figure. 14 shows that conditions of stepper are on values and simulated values are also confirmed from sand each angles. The red circle points indicate same results. As the trend of error, it is confirmed non-modeled parts. In the using model, these parts that the smaller the values of φ become, the larger were not modeled. Hence, the errors between ex- the values of error. periments and simulations were caused. 4.2 Discussions In these two reasons, the first one has a great effect The two reasons of errors and trend of error are on these errors, because almost all experimental val- considered. Firstly, as shown in Figure. 12, if the ues are larger than the simulated values. Especially, φ = φ is small enough, i.e., smaller than specific value, at the 20[deg], large error is observed. The a part of stepper sinks in the sand. Therefore, this most of the errors are caused by the part of stepper part of stepper is not on surface of sand, the values that is in sand. As a result, the experimental FRFT is much bigger than the simulated one. of FRFT are different between experiment and simulation. Especially, a part of stepper that is under Figure. 15 and Figure. 16 are the comparisons of (a) Horizontally (b) A little oblique

1 Experiment 0.9 d = 0.001 d = 0.002 0.8 d = 0.003 d = 0.004 0.7 d = 0.005

0.6

0.5

0.4

0.3 Maximum Hopping height [m] 0.2

0.1 20 30 40 50 60 70 80 90 φ [deg] Figure 15: The comparison of maximum hop- Figure 13: Separation of sand and stepper ping height with the value of d maximum hopping height when the initial sinkage all the experimental values were bigger than the sim- of spring d, and scaling factor ζ are changed. In- ulated values. Although the errors were confirmed, creasing the initial sinkage of spring, the difference RFT is useful to study the trend of motion on soft of trend between the values of simulations and ex- soil. periments are increased, as shown in Figure. 15. The References difference of trend is also confirmed from Figure. 16 [1] Carrier D (1922) Soviet rover systems. In: Pro- by increasing ζ. These results indicate that the error ceedings of 1922 AIAA Space Programs and was caused by effects of surface. Technologies Conference. 5 CONCLUSION [2] Asnani V, Delap D and Creager C (2009) The The mobility performance of hopping rover is an- development of wheels for the Lunar Roving alyzed by using RFT that is a new method of ter- Vehicle. Journal of Terramechanics, 46 (3): ramechanics. The numerical simulation and vertical 89-103. hopping experiments are conducted. As the simulation, the maximum hopping height is increased [3] Lakdawalla E (2014) China lands on the moon. while the stepper angle φ is increased. The same Nature Geoscience, 7 (81): 81. trend was also observed by vertical hopping exper- [4] Wilcox B, Nguyen T (1998) Sojourner on iments. However, the errors between the simulated mars and lessons learned for the future plane- values and the experimental values were also con- tary rovers. SAE Technical Paper, Tech. Rep. firmed. These errors seem to be caused by the im- 981695. proper placement of the stepper on loose sand. In this experiments, it is considered that main part of [5] Squyres SW, Arvidson RE et al (2003) Athena the errors were the former reason’s effect, so almost mars rover science investigation. Journal of [12] Clark BC et al (2008) Comet Hopper: A Mis- 0.7 Experiment ζ = 2.1 sion Concept for Exploring the Heterogeneity ζ = 2.2 0.6 ζ = 2.3 of Comets. LPI Contributions, 1405: 8131. ζ = 2.4 ζ = 2.5 0.5 [13] Yoshimitsu T (2004) Development of au- tonomous rover for asteroid surface explo- 0.4 ration. In: Proceeding of 2004 IEEE Interna- tional Conference on Robotics and Automation 0.3 2004, Louisiana, USA, pp. 2529-2534.

Maximum Hopping height [m] 0.2 [14] Me`ge D et al (2016) The Highland Terrain

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