Elsevier

Journal of Terramechanics

Volume 62, December 2015, Pages 63-73
Journal of Terramechanics

Measurement and modeling for two-dimensional normal stress distribution of wheel on loose soil

https://doi.org/10.1016/j.jterra.2015.04.001Get rights and content

Highlights

  • We developed a method to measure two-dimensional stress distribution beneath a wheel.

  • The stress distribution shows a mountain-shape in the wheel’s width direction.

  • The distribution area moves to a forward part along with the increase of slip ratio.

  • We constructed a model of the stress distribution based on the experimental data.

  • We reconstructed entire normal stress distribution from two points’ data.

Abstract

On the Moon or Mars, typical target environments for exploration rovers are covered with fine sand, so their wheels easily slip on such weak ground. When wheel slippage occurs, it is hard for the rover to follow its desired route. In the worst case, the rover gets stuck in loose soil and cannot move anymore. To reduce the risk of the rover getting stuck, analysis of the contact mechanics between the soil and wheel is important. Various normal stress distribution models for under the wheel surface have been proposed so far. However, classical models assume a uniform stress distribution in the wheel’s width direction. In this study, we measured the two-dimensional normal stress distribution of a wheel in experiments. The results clarified that the stress distribution in the wheel’s width direction is a mountain-shape curve with a peak located at the center of the wheel. Based on the results, we constructed a stress distribution model for the wheel’s width direction. In this paper, we report our measurements for the two-dimensional stress distribution of a wheel on loose soil and introduce our stress distribution model for the wheel’s width direction based on our experimental results.

Introduction

In recent years, planetary exploration using lunar/planetary exploration mobile robots (rovers) have been planned and conducted by major developed countries. Typical target environments for such exploration rovers are covered with fine sand, so their wheels easily slip on such weak ground. When wheel slippage occurs, it is hard for the rover to follow its desired route. In the worst case, the rover gets stuck in loose soil and cannot move anymore. For instance, NASA/JPL’s Mars Exploration Rover (Spirit), which landed on Mars in 2004, was buried in loose Martian soil in 2009 (Kerr, 2009). In 2010, NASA/JPL gave up trying to free the rover and converted it to a static observation station. Thus, wheel slippage is a serious problem for lunar/planetary surface exploration missions. To avoid such situations, understanding of wheel–soil interaction is essential.

Bekker developed the fundamental terramechanics model of a wheel in the 1960s (Bekker, 1956, Bekker, 1960, Bekker, 1969), and Wong, Reece, and others improved upon Bekker’s work in their drawbar pull models (Reece, 1965, Onafeko and Reece, 1967, Wong, 2008). These drawbar pull models are based on normal stress distributions generated beneath a wheel and shear stress distributions generated over the same area as the normal stress. Several studies have been conducted on stress distributions that are generated beneath the wheels of lunar/planetary rovers on loose soil.

Hegedus (1963) mounted three pressure sensors on a wheel surface and measured the stress distribution on flat and soft soil in the circumferential direction. He assumed that the normal force acting on a wheel edge was zero and deduced the distribution based on the quadratic approximation of three points: the wheel center, near the left edge of the wheel, and the left edge of the wheel; and the wheel center, near the right edge of the wheel, and the right edge of the wheel. He performed experiments using the wheels and demonstrated that the shape of the normal stress distribution in the wheel circumferential direction corresponds to the slip ratio.

In a previous study, we mounted four thin pressure sensors on the rover’s wheel surface and measured the normal stress distributions (Nagatani et al., 2009). In our experiments, we confirmed that the stress distribution in the width direction of the wheel is not uniform. However, the mean of the sensors’ output was used because of the low resolution. We reported that (1) the slip ratio increases with the wheel slippage, (2) a normal stress distribution is generated at the front portion of the wheel, and (3) the generation region moves forward as the wheel slippage increases.

In other research related to the measurement of the normal stress distribution generated beneath the wheel, Krick, 1969, Senatore and Iagnemma, 2014 measured normal and tangential stresses using strain gages. Oida et al. (1991) measured three-dimensional stress distributions using a three-axial force transducer on tire surface. In addition, Iizuka and Kubota, 2010, Narita et al., 2011 measured the stress distributions for an elastic wheel.

Several researchers have observed the sand particles beneath a wheel directly to analyze the normal stress distribution. In typical cases, the sand particles were observed with camera devices (Fukami et al., 2006, Moreland et al., 2012, Vlahinic et al., 2012). A half cut model of the wheel would be developed, and the cut surface would be placed on a clear glass or plastic plate to observe the movement of the sand particles while the wheel rotated. In contrast, Kinugasa et al. successfully tracked the three-dimensional movement of soil particles beneath a wheel by using a radioactive isotope tracking system (Kinugasa et al., 2013).

Although some researchers have examined the normal stress distribution of wheels on weak soil, in almost all cases the normal stress distribution in the width direction of the wheel was assumed to be uniform, as shown in Fig. 1. However, the stress distribution shape generated beneath a wheel is actually a mountain-shaped curve. Therefore, we concluded that the uniform stress distribution models are inadequate for analysis of the contact mechanics for the interaction between the wheel and loose soil. In order to obtain the normal stress distribution precisely, an array sensor of the stress, in the width direction of the wheel, is needed. However, this approach is currently difficult because commercially available array sensors are expensive. Moreover, the space and force resolutions of these array sensors are too low to measure the normal stress distribution precisely. Therefore, we measured the two-dimensional (2D) normal stress distribution by using a six-axis force/torque (F/T) sensor. This sensor cannot measure the normal stress distribution in all areas simultaneously but can measure the normal stress distribution at a specific point in the wheel width direction precisely. Therefore, we can repeat the measurement of the normal stress distribution in the wheel circumferential direction while changing the measurement point in the wheel width direction. We then superimpose all of the normal stress distribution data. With this, we can obtain a result similar to that using the array sensor.

In an actual field, online estimation of the normal stress distribution of the wheel is needed in order to predict the risk for rover immobilization. Knowing the normal stress distribution allows the forces acting on the wheel (e.g., drawbar pull) to be estimated. To do so, precise measurement of the entire normal stress distribution of the wheel is essential. However, in the actual field, changing the measurement point in the wheel width direction of the sensor is impossible. Hence, we can measure the 2D normal stress distribution of the wheel using a six-axis F/T sensor a priori and model the normal stress distribution in the wheel width direction. With this, we can estimate the normal stress distribution of the wheel precisely by substituting the normal stress distribution data in the wheel circumferential direction at several points into the model.

In this study, our aim was to reconstruct the 2D normal stress distribution beneath a wheel based on a normal stress distribution model in the wheel width direction and a few pressure sensors mounted on the wheel. To achieve this goal, understanding and constructing a model of the 2D normal stress distribution beneath the target wheel are necessary. Therefore, we first measured the 2D stress distribution beneath the wheel in detail. Next, we approximated the stress distribution along the wheel’s width direction to construct a stress distribution model in the wheel width direction. Finally, we reconstructed the wheel’s 2D normal stress distribution using two points’ experimental data.

This paper is organized as follows. Section 2 presents the strategy of our approach. In Section 3, we report our measurement method for the 2D normal stress distribution of a wheel and our experimental results. In Section 4, we discuss the validity of the results. In Section 5, we propose a stress distribution model for the width direction of the wheel based on the measured results presented in Section 3. Finally, we present our conclusions on our research in Section 6.

Section snippets

Strategy in this work

When a wheel comes in contact with weak ground, it sinks into the loose soil. During this time, normal stress is generated between the wheel and soil. In addition, shear stress is generated in the same area as the normal stress when the wheel rotates. Thus, the shear stress distribution can be estimated from the normal stress distribution. The drawbar pull can be estimated from the normal and shear stress distributions. Based on the above, obtaining the normal stress distribution beneath the

Measurement experiment of 2D stress distribution

To construct a 2D normal stress distribution model, understanding the actual normal stress distribution generated beneath the wheel is essential. Therefore, we conduct measurement experiments with a special wheel that is equipped with an F/T sensor to measure the normal stress. In this section, we introduce our special wheel and report the measurements for the 2D normal stress distribution on weak soil.

Evaluation of measured 2D stress distribution

In order to evaluate the validity of the measured 2D normal stress distribution, we examined whether the weight of the two-wheeled test bed is balanced by summing the vertical components of the normal stress σ(θ) and shear stress τ(θ) generated beneath the wheel. The vertical force of the wheel, Fz, can be calculated by the following equation:Fz=r-b/2b/2θrθf{τ(θ)sinθ+σ(θ)cosθ}dθdy,where r is the wheel radius, b is the wheel width, θf is the entry angle into the soil, θr is the departure angle

Modeling 2D stress distribution of wheel

The measured 2D stress distributions showed a mountain-shaped curve along the width direction of the wheel. However, classical stress distribution models treat the distribution in the width direction of the wheel as uniform. Thus, if the wheel’s whole stress distribution cannot be measured, the overall stress distribution cannot be obtained accurately using classical models.

In order to measure the overall normal stress distribution of the wheel, a pressure array sensor that covers the entire

Conclusion

In this paper, we presented the following:

  • 1.

    We reported the measured 2D stress distributions generated beneath a wheel for both flat loose soil and loose soil slopes. The results were as follows:

    • A.

      When a wheel travels on flat loose soil, the 2D stress distribution becomes a mountain-shape curve along the width direction of the wheel.

    • B.

      When the robot climbs up a loose soil slope, the normal stress distribution shape also becomes a mountain-shaped curve, and its peak decreases as the inclination angle

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