Abstract
In this study, to analyze the influence of the cross-sectional shape of a rigid foot on its sinking performance in soft soil, a numerical model of the subsidence of a single leg was built using the arbitrary Lagrangian–Eulerian method in ABAQUS to describe the interaction between the foot and soil. After performing a simulation of circular, annular, X-shaped, and improved X-shaped feet, all with the same cross-sectional area, the end resistance, side resistance, and distribution of contact pressure along the radial direction were analyzed. The simulation results showed that the soil arching area caused by the hole at the bottom of a rigid foot can effectively increase the end resistance and that the side resistance increases with an increase in the side perimeter. The end resistance and side resistance of the improved X-shaped robot foot were higher than those of the X-shaped robot foot during the sinking process. The effects of the improved geometric parameters of the X-shaped robot foot on its sinkage were analyzed through orthogonal experiments. The simulation results can provide a reference for the future structural design and optimization of robot feet working in soft soil environments.
Keywords
Introduction
Soft ground has strong water permeability, poor compression resistance, and low shear resistance. When terrain machines travel on soft terrain, they can easily suffer from sinkage issue. The Opportunity and Curiosity Mars rovers have a difficult time moving normally because the soft terrain on Mars causes a large amount of subsidence. 1 Developing anti-sinking technology for ground machinery walking on soft terrain is an urgent necessity, particularly for machines working on oil exploitation in desert areas, the development of the wind and tidal energy in coastal tidal flat areas, and the exploitation and protection of marsh wetland resources.
Anti-sinkage has become a topic of considerable interest in the field of terrain vehicle research. Research related to anti-sinkage methods for wheel terrain machines has primarily focused on changing the structural design parameters, such as increasing the wheel surface, track width, and tire size, 2 using deformable wheels, such as arched tires and concave elastic screen wheels, 3 and developing a bionic rigid wheel by studying the traveling mechanism of ostriches and camels walking on sand.4–6 However, the shortcomings of grounding pressure, slipping sinkage, and other inherent defects make wheels unsuitable for use in irregular and uneven terrain. In contrast, leg–foot traveling mechanisms are highly adaptable to terrain, with more freedoms and better stability. Thus, legged robots can move smoothly to perform fieldwork. 7 Researchers have studied the design of robot legs and gait optimization to improve the walking stability of robots and the adaptability of robots to the terrain. However, the anti-sinkage ability of robot feet has primarily been improved by increasing the contact area. Javadi and Spoor 8 increased the number of legs and changed the layout of legs to improve the bearing capacity of the robot and reduce its footprint depth, thereby avoiding the compression or damage of deep soil. Zhang et al. 9 increased the contact area to reduce sinkage in sand in their investigations of bionic walking wheels. However, increasing number of wheels and legs or increasing the contact area by designing a foot mechanism increases the complexity of the mechanical structure, motion control system, and control arithmetic. Furthermore, a larger robot foot will also be heavier, which increases its resistance when walking on soft terrain. Therefore, the area of a robot foot cannot be infinitely increased and has an upper limit. The bearing capacity of a robot foot is affected by not only the contact area but also the side friction of the foot and the yield flow of the soil underneath the foot. In addition, these factors are closely related to the structure of the robot foot. Few studies have investigated the effect of foot shape on the yield flow of the soft soil under foot and the bearing capacity to improve the anti-sinkage ability. B Yeomans and CM Saaj 10 established the single-leg test bed to compare sinking characteristics of four different types of robot feet, but the general law of the influence of the geometry shape of a rigid foot on its sinking performance was not reported, and the sinking mechanism related to aspects of soil stress and yield flow was not further elaborated.
We have performed explorative theoretical research on the feasibility of improving the anti-sinking ability of a robot foot by changing its shape, which can cause the transference of soil stress. 11 This article investigates the influence of the geometry shape of a rigid foot on its sinking performance, and ABAQUS is used to establish a three-dimensional simulation model of single-leg subsidence. The simulations of the interaction between rigid feet of different cross-sectional shapes and soft soil are running, which can provide a reference for optimizing the foot shape of legged robots working in soft soil conditions.
Numerical model of single-leg subsidence
The sinking performances of feet with different cross sections are comparatively analyzed by numerical simulation. Based on an analysis of the sinking performance and optimization of the foot’s geometric shape, it is reasonable and time-saving to establish an appropriate scheme of the foot sample. In contrast, it is difficult to obtain practical results directly by testing in a soil bin without the necessary theoretical research or numerical simulations. Compared with the method of a soil bin test, the numerical simulation method makes it easy to obtain the soil stress and strain, and the results of simulations can also be helpful for studying the soil stress transfer mechanism in the sinking process of a foot.
Figure 1 shows the single-leg subsidence model. The robot foot penetrates into soft soil vertically by assigning displacement or applying force. The equilibrium relationship in the vertical direction is
where FLoad is the applied load, FN is the end resistance, and Ff is the side resistance.
Model of single-leg subsidence.
Due to the serious soil disturbance in the sinking process of the robot foot, the large deformation of the soil leads to the distortion of elements near the contact surface between the foot and soil, which may cause the simulation to terminate. The arbitrary Lagrangian–Eulerian (ALE) method in ABAQUS/Explicit is used in the simulation to solve dynamic problems involving large deformation in the sinking process. Combining the advantages of Lagrangian analysis and Eulerian analysis, the ALE method can describe the mesh flow and material motion independently and avoid the distortion of the computational mesh. 12
To reduce the computation time, the study ignores the internal force and deformation of the rigid foot during the sinking process. In this article, the three-dimensional model of the robot foot is simplified: the robot foot is defined as a rigid body, and a fillet of the same radius is set at the edge of the sole. Figure 2 illustrates the finite element model of the soil, which is described by the modified Drucker–Prager (D–P) cap model. The main parameters of the soil are shown in Table 1, and the parameters of the modified D–P/cap model in ABAQUS are referenced from the literature. 11
The dimensions of foot–soil interface model (unit: mm): (a) front view and (b) vertical view.
Main parameters of soil.
The soil near the bottom and side of the robot foot, where the ALE adaptive mesh is used, exhibits large deformation. Other areas of the soil are defined by Lagrangian meshes. The contact surface between robot foot and soil is defined as hard contact, where the two contact surfaces are separated, and the contact constraints on the corresponding nodes are removed when the contact pressure is 0 or negative. Furthermore, the penalty function method is used to describe the friction behavior in the tangential direction. The vertical displacement of the bottom soil and the horizontal displacement of the side soil are set to 0. The displacement boundary condition is applied by making the foot move into the soil 6 cm in 1 s.
The phenomenon of the soil arching effect caused by a hole in the soil has been observed in previous two-dimensional research, where the stress transfer occurs at the same time. In this article, the influence of the hole and section parameters of the rigid foot on the resistance in the sinking process is studied using the numerical simulation method. In geotechnical engineering, the X-shaped cast-in-place concrete pile in composite foundations has a large unit material volume and specific surface area, which can effectively improve the bearing capacity of a single pile. 13 Inspired by this result, this article presents an X-shaped foot with the hole in the cross section and conducts comparative simulation experiments with circular, annular, and X-shaped feet having equal cross-sectional areas. Figure 3 describes three types of horizontal section shapes of robot feet used in this simulation. The height of all rigid feet is 100 mm.
Three types of horizontal section shape of robot feet (unit: mm): (a) circular, (b) annular, and (c) X-shaped.
The section parameters of the circular, annular, and X-shaped feet are shown in Table 2. For a given cross-sectional area, the perimeter of the X-shaped section is the largest, and the perimeter of the circular section is the smallest.
Geometric parameters of circular, annular, and X-shaped foot with equal cross-sectional area.
Results of the numerical simulations
The influence of soil’s height on simulation result
In the real condition, the size of soil that robot is walking on is infinite, but the size of soil used in the simulation is limited because of the limited capacity of computer’s memory and computation. However, the size of soil has an influence on the simulation result. In order to reduce the error caused by the restriction on soil’s size in the radius direction as far as possible, non-reflecting boundary conditions are applied in the vertical boundary of soil. However, the height of soil used in the simulation is directly related to the sinkage of robot foot. So three soil meshes with the total height of 400, 600, and 800 mm are used to simulate the sinking process of the circular foot, and the results of load–sinkage curves are shown in Figure 4 to compare the influence of soil’s height on the simulation result. There is little difference between the three curves, which illustrates that with the soil height we used the edge effect can be controlled in an appropriate value.
The load–sinkage curves of difference soil’s height.
Comparative analysis of circular and annular feet
The load–sinkage curves for the circular and annular feet are shown in Figure 5. Overall, sinkage increases with load, and compared to the circular foot, the annular foot requires a larger load when penetrating the same depth, which illustrates that the annular foot has a higher vertical bearing capacity. The end resistances of the circular and annular feet having the same cross-sectional area are compared in Figure 6 to illustrate the influence of the hole on the bottom of a rigid foot with the same cross-sectional area on the end resistance. For a given sinkage, the end resistance of the annular foot is larger, which indicates that the annular section can effectively increase the end resistance. Combined with the normal contact pressure of each element on the sole of the foot and the distribution of the major principal stress vectors, the reasons leading to differences in behavior between circular and annular feet are further analyzed.
The load–sinkage curves of circular and annular foot.
The end resistance–sinkage curves of circular and annular foot.
Because the shape and size of the mesh are not strictly symmetric, there is a certain numerical error in the simulation results, and the contact pressure of nodes with the same radius is not strictly uniform. Figure 7 shows the distribution of nodes with respect to the output contact pressure. A number of equidistant nodes are selected in the radial direction of the bottom surface, and the contact pressure of 24 circumferential nodes with the same radius is averaged to reduce the numerical error caused by mesh dividing.
The distribution of unit nodes to output contact pressure: (a) the sole of circular foot and (b) the sole of annular foot.
The radial distribution of the contact pressure of nodes in the sole of the circular foot at different degrees of sinkage is shown in Figure 8(a). The distributions at different sinkage levels exhibit similar trends: the contact pressure of the nodes increases slowly along the radial direction and reaches the maximum near the outer edge of the circular foot. Due to the fillet radius, the soil around the edge tends to flow along the tangential direction of the fillet, and therefore, the contact pressure decreases at the outer edge of the sole. The radial distribution of the contact pressure of the nodes in the annular foot’s sole at different sinkage levels is shown in Figure 8(b) to investigate the effects of the hole in the bottom of a rigid foot. The maximum contact pressure is concentrated at the area near the edge of the sole, and the maximum is slightly larger than the contact pressure of the circular foot. The distribution of contact pressure in the middle area of the sole, except the area near the edge, is notably larger than that for the circular foot. Based on the previous result that the end resistance of the annular foot is larger, the larger end resistance is attributed to the increased contact pressure in the middle area of the sole because the contact pressure of the nodes near the edge is smaller than that of the circular foot, which ultimately reduces the end resistance. In other words, the hole in the bottom of the rigid foot can increase the end resistance. This phenomenon is due to the soil arching effect caused by the relative displacement of soil during the sinking process.
The distribution of contact pressure of unit nodes in radial direction: (a) circular foot and (b) annular foot.
Next, the major principal stress of the surrounding soil elements is analyzed to explore the stress transfer mechanism that occurs during the sinking process of the annular foot. The major principal stresses of the soil elements near the bottom of the annular foot in the medial surface are shown in Figure 9 for sinkage z = 30 mm. The outline of the foot is plotted and marked on the figure. The direction of the major principal stress vectors is not entirely vertical, and the path of the major principal stress vectors forms an inverted arch under the sole of the annular foot that is marked by a black dotted line. The penetration of the foot leads to deformation and condensation of the soil due to the squeezing effect. Due to the non-uniform force acting on the soil, uneven displacement occurs and produces a certain wedge effect between the soil particles. This causes the stress in the soil to reach the yield state and transfer to the surrounding soil, and the major principal stress vectors deflect following the development of the soil arching effect. As shown in Figure 9, in the arched area denoted by a black dotted line, the vertical compressive stress of the soil under the arch foot gradually deflects to the central axis, which causes the vertical compressive stress to transform into radial compressive stress. The annular foot’s symmetric structure causes the radial compressive stress at the apex of the arch to equal 0, and a reaction force is generated on the sole of the annular foot, which increases the end resistance. In addition, the major principal stress vectors in the area marked by the red dotted line are vertical, which indicates that the soil under the hole bears the vertical load and is essentially equivalent to increasing the bearing range of the soil compared to the circular foot.
The distribution of the major principal stress vectors.
Comparative analysis of annular and X-shaped feet
The X-shaped foot with a middle hole has a larger side perimeter than the annular foot with the same cross-sectional area. The end resistance–sinkage curves and side resistance–sinkage curves for the annular and X-shaped feet are shown in Figures 10 and 11, respectively. Figure 10 shows that the annular foot has only a slightly larger end resistance than the X-shaped foot due to the smaller area of the fillet on the edge of the annular foot’s sole. This finding indicates that the side shape is not the key factor affecting the end resistance for rigid feet with the same cross-sectional area. Figure 11 and Table 2 illustrate that the X-shaped foot has a larger side resistance than that of the annular foot with the same cross-sectional area due to the larger perimeter of the X-shaped foot. For the first segment OA of the side resistance–sinkage curves, the shear stress of the soil gradually reaches the shear strength during the side flow of the soil, and a balance is achieved at a certain moment between points A and B. With an increase in the applied load, the steady state is broken after point B and the side resistance continues to increase. The sum of the end resistance and side resistance in the sinking process of the X-shaped foot is larger than those of circular and annular feet with the same cross-sectional area, and the bearing capacity of the X-shaped foot is higher.
The end resistance–sinkage curves of annular and X-shaped foot.
The side resistance–sinkage curves of annular and X-shaped foot.
As shown in Figure 12, based on the structure of the X-shaped foot, an improved X-shaped foot with the same cross-sectional area can be obtained by adding two holes in the directions of the X-axis and Y-axis. The sum of the areas of the five identical holes in the improved X-shaped section is equal to the area of the hole in the X-shaped section. The distribution of the major principal stress vectors of the soil under different cross sections at z = 30 mm is shown in Figure 13. The soil arching effect caused by the major principal stress under the vertical section is observed around each hole, which indicates that with the same cross-sectional area, the range of soil influenced by the soil arching effect under the sole is considerably larger for the improved X-shaped foot than for the X-shaped foot. Figures 14 and 15 present the end resistance–sinkage curves and side resistance–sinkage curves for the X-shaped and improved X-shaped feet, respectively. The results indicate that the end resistance and side resistance of the improved X-shaped foot are larger than those of the X-shaped foot, and the difference increases with an increase in the sinkage level.
The improved X-shaped foot (unit: mm).
The distribution of the major principal stress vectors under different cross sections: (a) A-A cross section and (b) B-B cross section.
The end resistance–sinkage curves of X-shaped foot and improved X-shaped foot.
The side resistance–sinkage curves of X-shaped foot and improved X-shaped foot.
Orthogonal experiment and optimization
Orthogonal simulation experiment
To analyze the influence of the geometric parameters on the sinking performance of the improved X-shaped robot foot with the same cross-sectional area in soft soil, the radius R1 of the hole in the cross section (factor A), the side length “a” of the cased square section (factor B), and the radius R2 of the distribution circle (factor C) are selected as the three factors of the orthogonal experiment to explore the effect regularity on the sinkage of the robot foot under identical loads. The three factors are shown in Figure 16.
Three factors in the orthogonal simulation experiment.
The orthogonal experiment of L9 (34) is performed on the numerical simulation of single-leg subsidence in the same soil models. The optimum ranges of each factor are determined according to the actual situation, where if the side length “a” of the cased square section is overly large or small, each foot would take up more space and the small side would limit the size of the hole in the cross section. The radius R1 of the hole and the radius R2 of the distribution circle should not be overly large or small due to the need to reserve certain installation sizes for legs and feet; R1 values of 20, 25, and 30 mm, R2 values of 75, 80, and 85 mm, and “a” values of 245, 250, and 255 mm are selected as the levels of the experimental factors. During the bulk of the time spent walking, the feet of the conventional quadruped or hexapod robots are typically normal or close to normal to the soft soil surface; simulations that apply a vertical load of 1200 N on rigid feet with cross-sectional areas of 23562 mm2 at a uniform rate of 1 s simplify the analysis. The results of the simulated experiments are shown in Table 3.
Orthogonal experiment and data.
Range analysis
According to a range analysis of the statistical results, the radius R1 of the hole in the cross section has the largest influence on the sinkage range (R = 8.5 mm). The sinkage is reported to decrease with an increase in R1 and exhibits a larger decline when R1 changes from 20 to 25 mm. The sinkage is not affected by the side length a of the cased square section or the radius R2 of the distribution circle. With an increase in a or R2, the sinkage initially decreases and then increases, but the overall change is small. The results of the orthogonal experiment show the primacy sequence of the experiment factors: the radius R1 of the hole in the cross section, the side length a of the cased square section, the radius R2 of the distribution circle, and the combination of the minimal sinkage is A3B2C2.
Variance analysis
To determine the significance of the influence of each factor on the results, orthogonal experimentation data are analyzed using an F-test to assess the distinctiveness of the factors. The results of the variance analysis are shown in Table 4. The influence of the radius R1 of the hole on sinkage is significant, whereas the effects of the side length a of the cased square section and the radius R2 of distribution circle are not significant. The results can be further discussed according to the range analysis and variance analysis while ensuring that a certain assembly space is reserved for the fixed connecting piece. The side length a of the cased square section and the radius R2 of the distribution circle have only a slight influence on sinkage and thus do not need to be considered. A greater range of soil arching area is produced due to an increase in R1; with the same cross-sectional area, the soil arching area, which forms deflection of the principal stress vectors, accounts for a larger proportion of the end surface and has a positive effect on the bearing capacity. However, an increase in R1 will increase the space occupied by the robot foot, which should be limited in the design process.
Variance analysis of sinkage.
Denotes the factor to be significant when F0.01(2,2) ≥ Ffactor > F0.05(2,2).
Conclusion
ABAQUS/Explicit was adopted to analyze the indentation of a rigid foot in soft soil. Three-dimensional numerical simulations were performed using the ALE method to obtain the force–sinkage relationship.
The applied load and end resistance were compared between circular and annular feet with identical cross-sectional areas, and the results indicated that the soil arching area caused by the hole in the bottom of the annular foot can effectively increase the end resistance. The end resistance and side resistance were compared for annular and X-shaped feet with identical cross-sectional areas, and the results indicated that a change in the cross-sectional side shape has only a slight influence on the end resistance and that increasing the side perimeter can increase the side resistance.
Several soil arching areas formed under the improved X-shaped foot, and the number of soil arching areas was consistent with the number of holes in the cross section. The improved X-shaped foot had a higher vertical bearing capacity than that of the X-shaped foot due to the increase in the end resistance and side resistance in the sinking process.
The influences of the geometric parameters of the improved X-shaped robot foot on the sinkage were analyzed based on the results of the orthogonal experiment method. The sequence of factors affecting the sinkage is the radius R1 of the hole in the cross section, the side length a of the cased square section, and the radius R2 of the distribution circle. Minimal sinkage was observed when levels of the experimental factors were as follows: R1 of 30 mm, a of 250 mm, and R2 of 80 mm. These results can provide a useful reference for optimizing the feet of legged robots working in soft soil conditions.
Footnotes
Acknowledgements
The authors wish to thank the reviewers for their valuable comments.
Academic Editor: Yangmin Li
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (nos 51375141 and 51375140).