The “Heat Flow Property Package Instrument” (HP3) is part of NASA’s current Mars mission “InSight”, which was launched in 2018 and currently operates on the surface of Mars. The instrument needs to remain at its initial position and orientation during operation. Although the landing site can have significant tilt and can be covered with low cohesion soil, any mechanical excitation might make the instrument slip. Therefore, the instrument is using a tailored feet design, which can withstand lateral loads. Future instruments might require higher resistance against slip. This can be due to stronger tilted landing sites or due to higher shocks emitted from stronger penetration probes. This paper introduces a novel design for those instruments based on the idea of the “spaced-link track” of Bekker to further minimize slippage. This design concept is originally used on tracks of heavy machinery. It is presented how the major design feature can be incorporated into the current design. A newly developed analytical-numerical model is utilized to estimate the track force of the new design. The paper closes with a design study at which the new design and the current design are compared to each other for different sized feet.
The “Heat Flow Property Package Instrument” (HP3) [
Left: artist’s illustration of the InSight lander on the surface of Mars (Image Credit: NASA/JPL [
Tracks are used on heavy machinery to ensure their mobility on soft or muddy terrain. They provide enough resistance against sinkage due to their large contact area. Their blades penetrating the soil ensure forward traction. Research on the shape of those tracks is done since the beginning of the last century. In the 60s, Bekker published his idea of a differently designed track called the “spaced-link track” [
Comparison between a conventional track (a) and Bekker’s spaced-link track (b). Figures taken from [
Bekker derived analytical equations for a spaced-link track based on the logarithmic spiral method in combination with a virtual wall. He assumed that the horizontal track force (
Description of the geometrical parameters, virtual wall, and the rupture line of a single cleat.
Furthermore, the total sinkage of the track
The number of cleats on a track (
The total length of the rupture area (
Assume slippage Calculate Assume one value for Calculate Calculate new Compare the assumed and calculated values for
Bekker’s analytical expressions for the force generated by a track and the spacing between the cleats were shown in the previous chapter. Though the analytical equations are more convenient to be used, they use empirical factors derived for large scale applications under earth gravity. Hence, a new numerical model based on the FEM is introduced, which can be scaled to different gravities and dimensions. Since the time Bekker derived these equations, numerical methods were developed extensively. Therefore, it became a usual practice to use those to investigate even more complex problems. One of those methods is the Finite Element Method in combination with an elasto-plastic material model. Many researchers [
Material elasticity is based on the isotropic material parameter of the soil. Plasticity is incorporated by including a failure criterion to define a limit for the elasticity behavior, a flow rule, which gives the stress-strain relationship in a plastic state and a consistency criterion to avoid that stress is exceeding the yield limit. Failure criteria can be generalized as a closed surface in three-dimensional principal stress space (see [
The parameters A and B are defined in relation to the Mohr-Coulomb failure criterion. In literature, they are defined such that the cone of the Drucker Prager failure criterion inscribes, middle-circumscribes, or circumscribes the hexagonal-shaped cone of the Mohr-Coulomb failure criterion [
Parameters A and B depending on the type of surface surrounding the Mohr-Coulomb yield surface.
Type of Mohr-Coulomb conversion | Parameter A | Parameter B |
---|---|---|
Circumscribes | ||
Middle-circumscribes | ||
Incribes |
[DP_Parameters].
For the numerical analysis, the inscribing configuration of the Drucker-Prager parameter conversion is chosen, as it provides the best fit to the measurements.
The numerical modelling is performed in ANSYS 16.0. The analysis approach is used by [
Overview of the FEM model with boundary conditions, dimensions, and global interface to rigid body.
Soil properties and references used for the finite element model.
Soil parameter | Size | Unit | Reference |
---|---|---|---|
E-modulus | 50 000 | [MPa] | Derived from [ |
Poisson ratio | 0.2 | [-] | Derived from [ |
Density | 1540 | [ | [ |
Cohesion | 1034 | [Pa] | [ |
Friction angle | 29.3 | [deg] | [ |
Detailed view of the modelling of a single cleat in soil.
Contact properties used for finite element model.
Property | Size | Unit |
---|---|---|
Soil-metal friction coefficient | 0.3 | [-] |
Penalty stiffness | 1e10 | [-] |
Tangential stiffness | 1e10 | [-] |
[contactproperties].
If the spaced link track is used for tracks, it is of major importance to choose the spacing between the cleats, such that the soil failure in front of each cleat does not affect the next cleat. If not, the spaced link track design would become a conventional one. Figure Assume Derive Derive mass per cleat using Build and solve FEM model Read-out Compare the assumed and calculated values for Read-out force per cleat and derive total force
By using this approach the spacing of the track and the number of the cleats is automatically adjusted to the parameter of the global track.
The result of the numerical analysis is a specific force per unit-width (
Sandy loam is used as material. This is mainly driven by drag measurements of the spaced-link track performed by Bekker in this soil. Bekker published cohesion and the friction angle of the soil together with the results of his measurements. The E-Modulus and poisson ratio needed for the numerical analysis were not published. Sandy loam consists of clay, sand, and silts in a variable fraction. The individual numbers of E-modulus and poisson ratio for the materials were shown in [
Bekker performed field testing of the spaced-link track using a small track vehicle called the “groundhog”. This vehicle has two tracks, each with a width of 36 in. and a total length of 108 in. The cleats have a dimensions of
With the given dimensions, the numerical model is used to derive the results of the total track force for the same loads as specified in [
Comparison of the total drag between measurement, analytical results (both reprinted from [
The HP3 instrument uses four, rigid feet as a contact interface to the soil. A bottom view of one single foot is shown in Figure
Bottom view of one HP3 flight-model foot.
The total mass at the shear failure plane (
The bulldozing force generated by one foot is determined specifically for a unit-width blade. The specific force is then integrated in track direction at the outer edge of the foot. The specific force can be estimated by equations presented by [
Integrated between
The HP3design shall be replaced by a design based on Bekker’s spaced-link track to generate more force against slip. The given volume and outer shape shall remain the same, as those are the result of the available volume on the lander deck. The available space is circular with a diameter of about 80 mm at available depths of 10 mm. Hence, the linear cleat design of Bekker’s track needs to be transformed into a circular design feasible with the given volume. First, the baseline design was transformed by modifying the brim of the current foot design such that it resembles a cleat. It was investigated secondly, how the remaining inner diameter can be used for further cleats: At a cleat ratio of 1 : 1 (depths to height), the available diameter becomes 60 mm. The required spacing between the cleats is between 16 and 19 mm at a contact area of 2200 mm2 (see Table
Results of the numerical analysis of single cleats with unit-width of Bekkers spaced-link track for Earth and Mars gravity.
Contact area | Force (Earth) | Force (Mars) | ||
---|---|---|---|---|
[N/mm] | [mm] | [N/mm] | [mm] | |
250 | 0.117 | 41.5 | 0.0865 | 27.5 |
750 | 0.089 | 26.5 | 0.071 | 19.5 |
1500 | 0.079 | 21.5 | 0.066 | 17.25 |
2500 | 0.075 | 18.5 | 0.06 | 15.5 |
Bottom view of a foot equipped with two cleats.
Schematic top view of the new proposed design.
Same as the original design, this foot design uses a cylindrical strut as an interface to the structure of the instrument located at the center of the foot. The length of the strut can be adjusted to the needed space between the bottom part of the instrument and the soil. The material thickness of the side walls is kept as thin as possible to ensure a sufficient sinkage of the instrument into the soil. The performance is estimated with a similar approach as for the current design: The specific forces per unit-width are estimated numerically with the model described in Section
Integrated between
Figure
Results of the design study.
The generated force increases with the available foot diameter. This result can be explained by the larger circumference of the foot, which increases the area used for bulldozing (HP3) and for the spaced-link designed edge. This effect is further explained by the significant larger available area for cohesion. The generated force under Earth gravity is higher than the force under Mars gravity. Gravity contributes linearly to the Mohr-Coulomb terms of the used equations, and it also contributes linearly to the force generated by bulldozing in front of the foot. Hence, a change in gravity also has a significant influence on the total generated force. The graphs clearly indicate that the currently used design has better efficiency than the new proposed design in terms of slippage resistance. The generated force by the currently used design is up to 50% larger than the force generated by a foot based on the spaced-link track. This effect is present for both studied gravities. The specific bulldozing force can be estimated by equation (
The paper studied a new design idea to increase the slippage resistance of successors of the Mars instrument HP3. The design idea is based on Bekker’s spaced-link track. The analytical equations to determine the force generated by a track are recapitulated. The analytical equations are based on many empirical parameters, and the accuracy is not known for other gravities and small scales. Therefore, a numerical model based on the FEM is introduced and verified based on experimental results published by Bekker. A new foot design based on the spaced-link track is proposed. The analytical estimation to determine their slippage resistance is presented and used to determine their efficiency for different foot diameters. It turned out that the efficiency of the new design is very low, and it can therefore not compete with current used design. This discards the design from further studies of foot designs for this instrument. But it shall be pointed out here that this is mainly driven by the available little space. Applying the design idea to future instruments with more space available is reasonable as more cleats can be used. This might make the design competitive with other.
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The manuscript has not been published and is not under consideration for publication in any other journal.
We have no conflicts of interest to disclose.