Improving Limitations of Rover Missions in the Moon and Planets by Unifying Vehicle–Terrain Interaction Models

Abstract

This paper extends the analysis of empirical methods for describing vehicle–terrain interactions in lunar terrain. Given analytical formulation to predict mobility performance in extraterrestrial environments requires reliable in situ testing campaigns, this imposes fundamental restrictions to conceive a more consolidated theory, and further any possibility to use improved empirical methods to design better space hardware. Hence, we propose an analytical approach to extrapolate data taken in parabolic flights to model vehicle performance in multiple gravity regimes. The extrapolation technique and respective reported uncertainties can be used, therefore, to tune fitting parameters of a set of general formulas in the domain of Terramechanics, allowing to have empirical estimates of the wheel mobility in cases where testing is inherently very complex. Finally, by the analysis of previous Moon and Mars rover missions with the fitted equations, we report an empirical design criterion that allows to generate first estimates of the optimal wheel dimensions, taking into account the eventual longitudinal slip of the rover and including the desired mass of the payload. The introduced rules for geometry optimization can be important for future space rover missions in remote soils.

Appendices

Appendix A: Set of Time-Domain Equations for $z^\mathrm{I}$

1.1 A.1 Defining Extrapolation Hypothesis for Sinkage Functions

There are three forms do define functions of the type $z^{\mathrm{I}}$: (i) as parallel lines to h(m) sharing the same slope but different intersect, (ii) sharing the same intersect but having different slopes, and (iii) intersecting h(m) in a point but having distinct slopes and intersects. Only the first two can be quantified, from which the parallel line hypothesis was tested in Sect. 3. For the second method, if the slope of $z^{\mathrm{I}}(m,g_{\mathrm{ex}})$ differs from h(m) by $\alpha$:

$$\begin{aligned} z^{\mathrm{I}}(m,g_{4})=\left( a'+\alpha '\,(m_4,g_{\mathrm{ex}})\right) \,m+b' \pm \epsilon _{hf}. \end{aligned}$$

Term $\alpha$ will be defined as

$$\begin{aligned} \alpha \,(m_4,g_{\mathrm{ex}})={\left\{ \begin{array}{ll} \frac{f(g_{\mathrm{ex}})-b'}{m_4},\, \text {if } g_{\mathrm{ex}}\ne g_4 \\ 0,\, \text {if } g_{\mathrm{ex}}=g_4. \end{array}\right. } \end{aligned}$$

Notice the error $\epsilon _{hf}$ will be defined similarly as in Sect. 3.

1.2 A.2 Statistical Error for $z^I$ and $z^{II}$ Functions

In the case of h(m) and $f(g_{\mathrm{ex}})$ functions, the standard errors for the estimate will serve to address the propagation of error in $z^{\mathrm{I}}$ and $z^{\mathrm{II}}$ formulae. The $\epsilon _h$ within 90% of confidence at specific masses in OGE (Fig. 4), we can have in millimeters:

$$\begin{aligned} \begin{bmatrix} \epsilon _h\,(m_1,s_1)\\ \epsilon _h\,(m_1,s_2)\\ \epsilon _h\,(m_1,s_3)\\ \epsilon _h\,(m_1,s_4)\\ \epsilon _h\,(m_1,s_5)\\ \epsilon _h\,(m_1,s_6)\\ \epsilon _h\,(m_1,s_7) \end{bmatrix}=&\begin{bmatrix} 1.9\\ 1.29\\ 3.07\\ 3.54\\ 5.79\\ \mathrm{N/A}\\ \mathrm{N/A} \end{bmatrix}; \begin{bmatrix} \epsilon _h\,(m_2,s_1)\\ \epsilon _h\,(m_2,s_2)\\ \epsilon _h\,(m_2,s_3)\\ \epsilon _h\,(m_2,s_4)\\ \epsilon _h\,(m_2,s_5)\\ \epsilon _h\,(m_2,s_6)\\ \epsilon _h\,(m_2,s_7) \end{bmatrix}\\ =&\begin{bmatrix} 1.42\\ 0.99\\ 2.36\\ 2.72\\ 4.46\\ \mathrm{N/A}\\ \mathrm{N/A} \end{bmatrix}; \begin{bmatrix} \epsilon _h\,(m_3,s_1)\\ \epsilon _h\,(m_3,s_2)\\ \epsilon _h\,(m_3,s_3)\\ \epsilon _h\,(m_3,s_4)\\ \epsilon _h\,(m_3,s_5)\\ \epsilon _h\,(m_3,s_6)\\ \epsilon _h\,(m_3,s_7) \end{bmatrix}= \begin{bmatrix} 1.2\\ 0.86\\ 2.06\\ 2.37\\ 3.88\\ \mathrm{N/A}\\ \mathrm{N/A} \end{bmatrix}\\ \begin{bmatrix} \epsilon _h\,(m_4,s_1)\\ \epsilon _h\,(m_4,s_2)\\ \epsilon _h\,(m_4,s_3)\\ \epsilon _h\,(m_4,s_4)\\ \epsilon _h\,(m_4,s_5)\\ \epsilon _h\,(m_4,s_6)\\ \epsilon _h\,(m_4,s_7) \end{bmatrix}=&\begin{bmatrix} 1.18\\ 0.86\\ 2.04\\ 2.36\\ 3.86\\ \mathrm{N/A}\\ \mathrm{N/A} \end{bmatrix}; \begin{bmatrix} \epsilon _h\,(m_5,s_1)\\ \epsilon _h\,(m_5,s_2)\\ \epsilon _h\,(m_5,s_3)\\ \epsilon _h\,(m_5,s_4)\\ \epsilon _h\,(m_5,s_5)\\ \epsilon _h\,(m_5,s_6)\\ \epsilon _h\,(m_5,s_7) \end{bmatrix}= \begin{bmatrix} 2.53\\ 1.74\\ 4.13\\ 4.77\\ 7.08\\ \mathrm{N/A}\\ \mathrm{N/A} \end{bmatrix}, \end{aligned}$$

where in $s_6$ and $s_7$, uncertainties cannot be defined with accuracy due to the only few data available. With the purpose to apply Eq. 12 and find the propagated error for $z^{\mathrm{I}}$, thus $\epsilon _f\,(g_{\mathrm{ex}},s_i)$ shall be defined. Errors for $f(g_{\mathrm{ex}})$ predictions in the Moon and Earth will be

$$\begin{aligned} \begin{bmatrix} \epsilon _f\,(g_1,s_1)\\ \epsilon _f\,(g_1,s_2)\\ \epsilon _f\,(g_1,s_3)\\ \epsilon _f\,(g_1,s_4)\\ \epsilon _f\,(g_1,s_5)\\ \epsilon _f\,(g_1,s_6)\\ \epsilon _f\,(g_1,s_7) \end{bmatrix}= \begin{bmatrix} 1.28\\ 1.17\\ 0.81\\ 1.5\\ 4.39\\ 3.02\\ 6.91 \end{bmatrix};\begin{bmatrix} \epsilon _f\,(g_4,s_1)\\ \epsilon _f\,(g_4,s_2)\\ \epsilon _f\,(g_4,s_3)\\ \epsilon _f\,(g_4,s_4)\\ \epsilon _f\,(g_4,s_5)\\ \epsilon _f\,(g_4,s_6)\\ \epsilon _f\,(g_4,s_7) \end{bmatrix}= \begin{bmatrix} 0.86\\ 0.78\\ 0.54\\ 1.02\\ 2.36\\ 1.46\\ 4.83. \end{bmatrix} \end{aligned}$$

Now, the associated uncertainties reported in Table 1 for $z_{\mathrm{Bekk}}$ involve several sources of error: propagated by the $z^{\mathrm{I}}$ functions with regard to the measured data, and the Bekker’s formula itself that fits $z^{\mathrm{I}}$ with $R^2 >97\%$. Considering only the first, the static sinkage prediction for, e.g., the Yutu rover using the values set initially is

$$\begin{aligned} \epsilon _{hf}^{I}&=\sqrt{\epsilon ^{2}_{h}\left( 22.4 \,\text { kg},s_1\right) +\epsilon ^{2}_{f}({g_1,s_1)+\epsilon ^{2}_{h}(m_4,s_1)}}\\&=\sqrt{(2.99)^2+(1.28)^2+(1.18)^2} =3.45. \end{aligned}$$

Now, in the high slip regime a similar procedure can be established:

$$\begin{aligned} \epsilon _{hf}^{I}&=\sqrt{\epsilon ^{2}_{h}\left( 22.4 \,\text { kg}, s_3\right) +\epsilon ^{2}_{f}({g_1,s_3)+\epsilon ^{2}_{h}(m_4,s_3)}}\\&=\sqrt{(4.85)^2+(0.81)^2+(2.04)^2}=5.23. \end{aligned}$$

The process is repeated to obtain Luna and LRV errors, knowing $\epsilon _{h}\left( 94.5 \text { kg},s_1\right) =17.54$, $\epsilon _{h}\left( 94.5 \text { kg},s_3\right) =27.79$, $\epsilon _{h}\left( 177 \text { kg},s_1\right) =34.43$, $\epsilon _{h}\left( 177 \text { kg},s_1\right) =54.45$. Although a $90\%$ confidence interval certainly makes prediction errors to be very large, it is important to see this uncertainty is more tolerable at lower masses.

1.3 A.3 Functions $z^\mathrm{I}$

The following equations relate the $z^{\mathrm{I}}$ and $z^{\mathrm{II}}$ functions from the previous analysis, as function of slip (or time).

$$\begin{aligned} \begin{bmatrix} z^{\mathrm{I}}(m,g_{\mathrm{ex}},s_1)\\ z^{\mathrm{I}}(m,g_{\mathrm{ex}},s_2)\\ z^{\mathrm{I}}(m,g_{\mathrm{ex}},s_3)\\ z^{\mathrm{I}}(m,g_{\mathrm{ex}},s_4)\\ z^{\mathrm{I}}(m,g_{\mathrm{ex}},s_5)\\ z^{\mathrm{I}}(m,g_{\mathrm{ex}},s_6)\,\\ z^{\mathrm{I}}(m,g_{\mathrm{ex}},s_7) \end{bmatrix}&= \begin{bmatrix} -1.856\\ -4.618\\ -3.167\\ -0.343\\ 5.656\\ 2.228\\ 2.1 \end{bmatrix}+ \begin{bmatrix} 0.231\\ 0.145\\ 0.018\\ -0.057\\ -0.362\\ 0.128\\ 0.244 \end{bmatrix}\,g_{\mathrm{ex}}\nonumber \\&\quad + \begin{bmatrix} 0.436\\ 1.343\\ 1.768\\ 1.834\\ 1.937\\ 1.91\\ 1.92 \end{bmatrix}\,m \end{aligned}$$

By the model proposed in Eq. 16, sinkage functions can be used to define a set of $n_o$ and $n_1$ that minimize the error expressed by the following formula:

$$\begin{aligned} z^{I}(D,m,g_{\mathrm{ex}},t_i)=\left[ \frac{3\,m\,g_{\mathrm{ex}}}{D^{0.5+(n_o+n_1\,m\,g_{\mathrm{ex}})} \,b\,k\,(3-n_o)}\right] ^\frac{1}{n_o+\frac{1}{2}}. \end{aligned}$$

Notice the fitting parameters will depend in the case of $z^{\mathrm{I}}$ functions, in a fixed $g_{\mathrm{ex}}$ and $t_i$. Thus, for Earth, we have

$$\begin{aligned} \begin{bmatrix} n_o (z^{\mathrm{I}},g_4,s_1)\\ n_o(z^{\mathrm{I}},g_4,s_2)\\ n_o(z^{\mathrm{I}},g_4,s_3)\\ n_o(z^{\mathrm{I}},g_4,s_4)\\ n_o(z^{\mathrm{I}},g_4,s_5)\\ n_o(z^{\mathrm{I}},g_4,s_6)\,\\ n_o(z^{\mathrm{I}},g_4,s_7) \end{bmatrix}=&\begin{bmatrix} 0.26\\ 0.48\\ 0.55\\ 0.565\\ 0.59\\ 0.59\\ 0.59\\ \end{bmatrix}; \begin{bmatrix} n_1 (z^{\mathrm{I}},g_4,s_1)\\ n_1(z^{\mathrm{I}},g_4,s_2)\\ n_1(z^{\mathrm{I}},g_4,s_3)\\ n_1(z^{\mathrm{I}},g_4,s_4)\\ n_1(z^{\mathrm{I}},g_4,s_5)\\ n_1(z^{\mathrm{I}},g_4,s_6)\,\\ n_1(z^{\mathrm{I}},g_4,s_7) \end{bmatrix}\\ =&\begin{bmatrix} 0.95\cdot 10^{-5}\\ 1\cdot 10^{-5}\\ 1.5\cdot 10^{-5}\\ 1.3\cdot 10^{-5}\\ 1.3\cdot 10^{-5}\\ 1.3\cdot 10^{-5}\\ 1.3\cdot 10^{-5}\\ \end{bmatrix} \end{aligned}$$

For the Moon ($g_1=\frac{1}{6}G$):

$$\begin{aligned} \begin{bmatrix} n_o (z^{\mathrm{I}},g_1,s_1)\\ n_o(z^{\mathrm{I}},g_1,s_2)\\ n_o(z^{\mathrm{I}},g_1,s_3)\\ n_o(z^{\mathrm{I}},g_1,s_4)\\ n_o(z^{\mathrm{I}},g_1,s_5)\\ n_o(z^{\mathrm{I}},g_1,s_6)\,\\ n_o(z^{\mathrm{I}},g_1,s_7) \end{bmatrix}=&\begin{bmatrix} 0.61\\ 0.8\\ 0.85\\ 0.88\\ 0.9\\ 0.9\\ 0.9\\ \end{bmatrix}; \begin{bmatrix} n_1 (z^{\mathrm{I}},g_1,s_1)\\ n_1(z^{\mathrm{I}},g_1,s_2)\\ n_1(z^{\mathrm{I}},g_1,s_3)\\ n_1(z^{\mathrm{I}},g_1,s_4)\\ n_1(z^{\mathrm{I}},g_1,s_5)\\ n_1(z^{\mathrm{I}},g_1,s_6)\,\\ n_1(z^{\mathrm{I}},g_1,s_7) \end{bmatrix}\\ =&\begin{bmatrix} 0.155\cdot 10^{-4}\\ 1.1\cdot 10^{-3}\\ 1.5\cdot 10^{-3}\\ 1.5\cdot 10^{-3}\\ 1.5\cdot 10^{-3}\\ 1.5\cdot 10^{-3}\\ 1.5\cdot 10^{-3}\\ \end{bmatrix}. \end{aligned}$$

For Mars ($g_{\mathrm{ex}}\approx \frac{3}{8}G$):

$$\begin{aligned} \begin{bmatrix} n_o (z^{\mathrm{I}}, g_{\mathrm{ex}},s_1)\\ n_o(z^{\mathrm{I}}, g_{\mathrm{ex}},s_2)\\ n_o(z^{\mathrm{I}}, g_{\mathrm{ex}},s_3)\\ n_o(z^{\mathrm{I}}, g_{\mathrm{ex}},s_4)\\ n_o(z^{\mathrm{I}}, g_{\mathrm{ex}},s_5)\\ n_o(z^{\mathrm{I}}, g_{\mathrm{ex}},s_6)\,\\ n_o(z^{\mathrm{I}}, g_{\mathrm{ex}},s_7) \end{bmatrix}=&\begin{bmatrix} 0.46\\ 0.68\\ 0.75\\ 0.76\\ 0.77\\ 0.77\\ 0.77\\ \end{bmatrix}; \begin{bmatrix} n_1 (z^{\mathrm{I}},g_{\mathrm{ex}},s_1)\\ n_1(z^{\mathrm{I}},g_{\mathrm{ex}},s_2)\\ n_1(z^{\mathrm{I}},g_{\mathrm{ex}},s_3)\\ n_1(z^{\mathrm{I}},g_{\mathrm{ex}},s_4)\\ n_1(z^{\mathrm{I}},g_{\mathrm{ex}},s_5)\\ n_1(z^{\mathrm{I}},g_{\mathrm{ex}},s_6)\,\\ n_1(z^{\mathrm{I}},g_{\mathrm{ex}},s_7) \end{bmatrix}\\ =&\begin{bmatrix} 9.5\cdot 10^{-6}\\ 2.1\cdot 10^{-4}\\ 2.7\cdot 10^{-4}\\ 3.1\cdot 10^{-4}\\ 3.2\cdot 10^{-4}\\ 3.2\cdot 10^{-4}\\ 3.2\cdot 10^{-4}\\ \end{bmatrix}. \end{aligned}$$

Appendix B

The functions $z^{\mathrm{II}} (m,g_{\mathrm{ex}})$ can be defined as parallel lines to $z^{\mathrm{II}} (m_4,g_{\mathrm{ex}})$ (as shown in Fig. 1). The master equation would be

$$\begin{aligned} z^{\mathrm{II}}(m,g_{\mathrm{ex}})=f(g_{\mathrm{ex}})+ p(m,g_4) \pm \epsilon _{hf}^{\mathrm{II}}, \end{aligned}$$

(18a)

where $p\,(m,g_4)$ would be the vertical displacement from $f(g_{\mathrm{ex}})$ (assume $z_{i,j}^{\mathrm{I}}=z_{i,j}^{\mathrm{II}}$ as before in the analysis of Sect. 3):

$$\begin{aligned} z^{\mathrm{II}}(m,g_{\mathrm{ex}}) = f(g_{\mathrm{ex}}) + p\,(m,g_{4}) \end{aligned}$$

with $p\,(m,g_{4})$ defined as previously for q, but using intersecting points at a constant gravity $g=g_4$ :

$$\begin{aligned} p\,(m,g_{4})={\left\{ \begin{array}{ll} h(m)-f(g_4)\quad \text {if } m\ne m_4 \\ 0\qquad \text {if } m=m_4. \end{array}\right. } \end{aligned}$$

The $z^{\mathrm{II}}$ functions resulting from the previous equations are the following:

$$\begin{aligned} \begin{bmatrix} z^{\mathrm{II}}(m,g_{\mathrm{ex}},s_1)\\ z^{\mathrm{II}}(m,g_{\mathrm{ex}},s_2)\\ z^{\mathrm{II}}(m,g_{\mathrm{ex}},s_3)\\ z^{\mathrm{II}}(m,g_{\mathrm{ex}},s_4)\\ z^{\mathrm{II}}(m,g_{\mathrm{ex}},s_5)\\ z^{\mathrm{II}}(m,g_{\mathrm{ex}},s_6)\,\\ z^{\mathrm{II}}(m,g_{\mathrm{ex}},s_7) \end{bmatrix}= \begin{bmatrix} -0.94\\ -3.04\\ -3.13\\ -0.31\\ 2.95\\ -0.05\\ -0.7 \end{bmatrix}+ \begin{bmatrix} 0.231\\ 0.145\\ 0.018\\ -0.057\\ -0.362\\ 0.128\\ 0.244 \end{bmatrix}\,g_{\mathrm{ex}}+ \begin{bmatrix} 0.436\\ 1.343\\ 1.768\\ 1.834\\ 1.937\\ 1.91\\ 1.92 \end{bmatrix}\,m. \end{aligned}$$

Because $z^{\mathrm{II}}$ functions physically represent the sinkage with variation of gravity at constant mass, they do not establish a useful scenario for analysis in the context of this paper. Thus, no fitting parameters for the aim of a more general formulation were found.