Sinkage

A possible drawback to a grouser wheel is that it could potentially sink deeply into soft soils as may be found on the Moon or Mars, greatly affecting the rover’s ability to maneuver. Here, we develop a model to estimate the depth of grouser wheel sinkage in loose sand based on Bekker’s equations for deformable terrain [21]. We then compare the model to sinkage data taken in the Mini Mars Yard at JPL. The data show that sinkage is not a debilitating issue with this design1_.

Wheel sinkage can be divided into two types: static and dynamic sinkages. The static sinkage is a result of the vertical load of the wheel, and the dynamic sinkage is caused by the wheel’s rotation. We assumeAxel movesquasi-statically, meaning that at the slow speeds typical of the rover, we can approximate its motion with static analysis. As a result, we focus our analysis on the static sinkage of the grouser wheel.

Bekker’s pressure-sinkage equation relates the depth to which a thin vertical plate penetrates the soil to characteristics of the soil and the pressure on the plate [21, 37]. It is calculated as

p(h) =

_kc

b +kφ

hn , (3.13)

wherehis the depth of the blade penetration into the ground, bis the width of the plate, andpis 1_{This section describes joint work done with Sandeep Chinchali.}

Figure 3.5: Submerged flat plate at arbitrary angleαwith key parameters labeled. The plate models a grouser.

the pressure on the plate. kc and kφ are pressure-sinkage modules, andnis the sinkage exponent, which is based on the soil type [21]. Previous works have conducted experiments to quantify these parameters for a variety of terrains [57, 69, 67]. Shibly [57] found that over a range of terrain parameters reasonable for a planetary rover on deformable soil, the sinkage exponentn≈1, allowing simplification of Equation 3.13. Dry sand, for example, which was typically used forAxel’s mobility experiments, has a sinkage exponentn= 1.1.

We incorporate this equation into our sinkage model by consideringAxel’s grousers as submerged plates at some arbitrary angle, as in Figure 3.5. Integrating the pressure equation along the direction

l and approximating the dry sand sinkage exponent n = 1.1 ≈ 1 in order to realize an analytic expression, we find the total vertical force on one grouser to be approximately

Fgrouser= bh 2 2 cosα kc b +kφ , (3.14)

whereαis the angle between plate and the vertical.

Axel 1’s wheels feature two slightly different grousers, one larger and one smaller, mounted in an alternating fashion around its rim. While the grouser wheels rotate,Axel switches between having one and two grousers in contact with the ground. Thus, the total vertical force between the ground and the wheels is a function ofβleave, the angle at which one grouser loses contact with the ground as the wheel drives forward. From the diagram in Figure 3.6, this angle is calculated as

βleave=α−θsep=cos−1

_r+d

r+lp,small

−θsep , (3.15)

whereris the radius of the wheel,lp,smallis the length of the small plate, andθsep is defined as the separation angle between the grousers (36.8◦inAxel 1). The force on each grouser can be calculated as a function of the single variabled, the distance from the bottom of the wheel rim to the ground. In static equilibrium, the force from the grousers will balance with the weight of the rover, and this

(a) Wheel with two grousers submerged in the ground.

(b) Wheel just as one grouser loses contact with the ground.

Figure 3.6: Diagrams showingAxel 1’s wheel as it rotates. The wheel’s grousers alternate between large and small sizes. Note that the diagram is not to scale and shows only two grousers for clarity.

analysis will estimate the depth to which the grousers sink into the soil.

Two sets of measurements of the wheel height, d, were carried out in the JPL Mini Mars Yard for various values ofβ, the angle between the larger grouser and the vertical. The rover was initially placed on a flat patch of sand with both grousers at approximately equal angles to the vertical. The wheel motor was then driven to rotate the wheel, and measurements ofdwere taken until the grouser passed the vertical and the rover tipped to the other side. The data is plotted alongside the theoretical model in Figure 3.7, where the wheel height is calculated as a function of the large grouser angle from the vertical, β. The data from the two tests fit the model reasonably well and deviate mostly in the extremes where the large grouser is almost vertical or close to leaving the ground. It may be necessary to develop a different model for these scenarios in order to more closely approximate the observed results.

The model and our experiments suggest the benefit in rock climbing ability gained by the intro- duction of the grouser does not lead to a wheel design which might sink deeply into soft soils and therefore become bogged down.