Abstract: For providing the theory basis for wheel configuration design and motion control, the rolling and steering mechanical models were developed to analyze the effect of wheel parameters on motion performance of lunar rover. And then the curves which illustrated the dependence of wheel motion performance on the wheel parameters, were deduced. The analysis results indicated that increasing the wheel diameter was more effective than increasing wheel width in enhancing the drawbar pull. In contrast, steering maneuverability was significantly affected by the wheel width. Furthermore, the dimensionless evaluation indexes were carried out to study the effect of wheel parameters on its motion performance. Five samples with different wheel width-diameter ration, namely 0.4, 0.45, 0.5, 0.55 and 0.6, were taken as the analysis object to compute the performance values such as traction performance, steering performance and overall performance index. The results revealed that the samples exhibited different superiority in single evaluation index. However, the value of comprehensive evaluation index decreases with the increasing width-diameter ratio and its maximum value appeared at about 20% slip.
INTRODUCTION
Designing a lunar rover implies that the device should be constructed to accomplish many challenging tasks by traveling long distance. The successful degree of a newly designed lunar rover locomotion systems depends on how well it can operate in the rough terrains (Pandey and Ojha, 1978). Wheels play an important role in generating traction and providing locomotion for rover. However, the rover’s traction performance and steering ability are dependent on the wheel parameters, namely the wheel diameter and width. When the unknown terrain is so severe that researcher can not determine which motion performance should be enhanced, it is necessary to ensure the comprehensive motion performance for lunar rover.
The traction force, produced from the interaction between the wheel and the sand, determines the rover’s ability to accelerate, climb slope and cross over obstacles (Seidi et al., 2009). In terramechanics, the drawbar pull of a wheel is calculated by the normal and shear stresses under the wheel (Bekker, 1969; Wong, 2001). Its value can be increased by changing the wheel parameters (Liu et al., 2010; Ding et al., 2011; Ray et al., 2009), although it is restricted due to constraints such as vehicle dimensions and the power consumption. The previous researches (Liu et al., 2008, 2011) measured the traction force and steering resistance moment by changing the grouser spacing, height, thickness and circumferential angle in soil bin with loose sand. The results revealed that improvement degree of motion performance is highly dependent on the grouser height and slip. Irani et al. (2011) expanded on the traditional terramechanic models of Bekker and Wong to develop a new analytical model that captures the dynamic effects caused by grousers. Recently, simulation of the wheel-soil interaction was employed to investigate the wheel motion performance (Fang et al., 2011; Xia, 2011; Nakashima et al., 2011). Results indicated that the FEM and DEM model used for this study might be applicable to determine terramechanical interactions under lunar surface.
On the study of steering characteristic of vehicles, terramechanics was mainly applied to investigate the steering dynamics of mobile robot as its motion direction kept a certain angle with longitude symmetry plane of wheel (Raheman and Singh, 2003; Ishigami et al., 2007). Gee-Clough and Sommer (1981) adopted a dimensional analysis to study the steering forces on undriven angled wheels and revealed that the side force coefficient was related to slip angle by an exponential relationship. Furthermore, the investigation on the characteristics of wheel-sand interaction can provide the theory basis for the wheel robot to establish dynamic analysis (Peng et al., 2009; Bigdeli et al., 2008), path planning (Kuppan-Chetty et al., 2011), motion control (Yu et al., 2008; Zerigui et al., 2008) and motion evaluation (Hegazy and Dhaliwal, 2011).
The main objectives of this study are to establish the rolling and steering theoretical model for lunar rover and study the effect of wheel parameters on traction ability and steering maneuverability. Based on the analysis results, the dimensionless performance evaluation indexes and criterion for the wheel parameters is proposed in order to investigate the relationship between wheel parameters and motion performance indexes which can be beneficial to determine the wheel design parameters.
EFFECT OF WHEEL PARAMETERS ON TRACTION PERFORMANCE
Rolling mechanical model: When the driven wheel rolls on sand, the maximum radial stress occurs in front of bottom-dead-centre and shifts forward on rim, whose position is marked by angle θm, as shown in Fig. 1.
In Fig. 1, D and B are the wheel diameter and width, respectively. W is the payload acting on the wheel. Tr is the driving torque. PD is the drawbar pull. θf, θr and h are the entry angle, exit angle and sinkage of the wheel, respectively. σr(θ) and τr(θ) are the radial stress and shear stress acting on the intercepted cylinder.
In Fig. 1, θm is a function of θf and wheel slip, i which can be expressed as (Wong, 2001):
| (1) |
where, c1 and c2 are constant coefficients. i is the wheel slip which can be described as:
| (2) |
where, ω and v is the rotating angular velocity and the travel velocity of the wheel, respectively.
θm divides region from θr to θf into two regions, that is from θm to θf and from θr to θm. Using the Reece formulation and the relationship hθ = (cosθ-cosθf)r into these two regions, the radial stress distribution σr(θ) can be described as (Wong, 2001):
| (3) |
| (4) |
| Fig. 1: | Forced diagram of a driven wheel on loose sand |
| (5) |
| (6) |
where, kc and kφ are the pressure sinkage modulus, n is the sinkage exponent, c is the soil cohesion, φ is the internal friction angle, K is the shear deformation modulus, j is the shear deformation distance which can be written as:
| (7) |
Based on the force balance in the Fig. 1, the equations of W, Tr and DP can be derived by the following equations:
| (8) |
| (9) |
| (10) |
where, RB is the bulldozing resistance.
According to the passive earth pressure theory, bulldozing resistance acting on the area with unit width and sinkage h will be:
| (11) |
where, γ is the unit weight of sand, Kpc= (Nc-tanδ) cos2δKpc, Kpγ = (2Nγ/tanδ+1)cos2δ, Nc and Nγ are the Terzaghi’s bearing factors and δ = arctan (2tanφ/3).
Therefore, RB can be expressed as:
| (12) |
When W, i and soil parameters are known, Eq. 1-12 describe the dependence of wheel traction performance on wheel parameters.
Theoretical effect of D and B on PD and Tr: Defining W = 49N, i = 0.3 and the terrain parameter values described in Table 1 (Liu et al., 2010) in Eq. 1-12, then it can be derived the theoretical effect of D and B on PD and Tr (as shown in Fig. 2).
It can be observed from Fig. 2a that PD increases with the increasing D and B. However, the dependent degree of PD on D and B is different. Increasing the wheel diameter is more effective than increasing wheel width in improving the wheel drawbar pull. This is can be explained by Eq. 9 that the larger wheel diameter is more beneficial to decrease the wheel sinkage and the bulldozing resistance will also descends with the decreasing sinkage. Meanwhile, the increasing D and B lead to the increase of driving torque as shown in Fig. 2b. The stronger traction ability is favorable to enhance the adaptability of lunar rover in strict terrain. However, the increasing driving torque implies that the driving motor will consumes much energy and this is disadvantageous for the lunar rover with limited energy. Therefore, the reasonable design of wheel parameters should comprehensively considerate the balance relation between traction performance and power consumption.
EFFECT OF WHEEL PARAMETERS ON STEERING MANEUVERABILITY
Steering mechanical model: Lunar rovers are expected to travel long distances to fulfill challenging mission by means of executing the remote control command. Due to the effect of unknown terrain and the time delay of information transmission between lunar rover and control center, lunar rover must steer frequently at static position in order to avoid the obstacles or change the path. For getting a rover with compact structure, researchers usually put the steering equipment above the wheel and keep the axis of steering motor passing the center of wheel. Therefore, the steering mechanical model in this paper is established for such steering motor assignation.
Figure 3 illustrates the groove and forced diagram of a steering wheel on loose sand.
In Fig. 3, the shadowed region is the horizontal projection of contact area between steering wheel and sand. Ts is the output torque of steering motor, l and ρ are the distance between the center of micro-units area dxxdy on the wheel rim and area dxxdz on the side surface of wheel and the axis O(O'), respectively.
| Fig. 2: | Dependence of diameter and width of wheel on the PD and Tr respectively, when W=49, I=0.3 |
| Table 1: | Mechanical parameters of sand used in the experiments |
The steering wheel will stop moving when the rover needs to change the motion direction after moving to a certain position at a slip value, i. To the closed wheel, the seven contact forces (Fig. 3). Groove and forced diagram of a steering wheel on loose sand between wheel and sand can be deduced as shown in Fig. 3. RSBL, RSBR and RSFL, RSFR are the lateral bulldozing resistance and friction force acting on both sides of wheel, respectively. Rτ is the shear force acting on the wheel rim. RLBF and RLBF is the longitudinal bulldozing resistance acting on the wheel rim.
| Fig. 3(a-b): | (a) Groove of steering wheel and (b) Force acting on steering wheel |
To calculate the moments of all of force about axis O'(O'), the resultant Steering Resistance Moment (SRM) can be derived finally.
Shear resistance moment: To take the micro-unit area dxxdy on the wheel rim as the research object as shown in Fig. 3, the distance:
The steering wheel yaws around the axis O(O'), the shear resistance acting on the unit are dxxdy can be expressed as:
| (13) |
where, js is the shear displacement of unit area dxxdy as the wheel yaws angle β which can be given as follows:
| (14) |
In Fig. 3, the contact area between steering wheel and sand is less than that between rolling wheel and sand. However, the sinkage is assumed to be constant, then the stress σs(θ) acting on the steering wheel rim can be defined as σs(θ)≈σr(θ)/ξ, ξ is the ratio of contact area between steering wheel to that between rolling wheel and sand. Then, the σs(θ) can be described as:
| (15) |
Substituting Eq. 15 and 14 into Eq. 13, shear resistance acting on the rim can be derived as follows:
Thus, the shear resistance moment produced by Rτ can be written as:
| (16) |
In Eq. 13, the value of dry sand cohesion c is smaller so that it almost can be ignored, then τs(θ) is approximately proportional to σs(θ). Therefore, it can be deduced from Eq. 16 that Mτ is independent of ξ.
Longitudinal resistance moment: According to the passive earth pressure theory, bulldozing resistance acting on the rim area with unit width and h singkage will be:
| (17) |
Therefore, the RLBF and RLBR can be expressed as:
where, hr, hr = R(1-cosθr), is the sinkage of rim at the angle θr.
Thus, the longitudinal bulldozing resistance moment acting on the wheel rim can be written as follows:
| (18) |
Lateral resistance moment: When the wheel begins to steer, the far-rear side and the near-front side of wheel from the axis O(O') will impact the sand outside and inside, respectively. The impact forces include the lateral bulldozing resistance and friction as shown in Fig. 3.
Here, taking the micro-unit area dxxdz on the contact area between wheel side and sand as the research object, the distance, ρ, can be described as:
From Eq. 17, it can be deduced that the passive earth pressure acting on unit area dxxdz to be:
| (19) |
Based on the geometry relationship in Fig. 3, forces RSBL and RSBR are equal and can be computed as:
| (20) |
Therefore, the bulldozing resistance moments MSB generated by force RSBL and RSBR can be described as:
| (21) |
Due to the impacting force and relative motion between wheel side and sand, the unit friction force acting on the area dxxdz can be expressed as following equations by employing the J. Janosi shear theory:
| (22) |
where, jlateral is the shear displacement of area dxxdz as the wheel yaws angle β which can be written as:
| (23) |
Then, the resistance moment MSF produced by force RSFL and RSFR can be computed as:
| (24) |
Therefore, the resistance moment Mlateral acting on the wheel side of wheel can be expressed as:
| (25) |
Resultant steering resistance moment: Based on the above analysis, the resultant SRM, M, can be described by the equation:
| (26) |
Using the Eq. 26, the SRM can be evaluated, whose value can be used to predict the steering ability and guide the selection of steering motor.
Theoretical effect of D and B on SRM: Figure 4 shows the dependence of SRM on D and B when W = 49N, i = 0.3, β = 25°.
It can be seen from the Fig. 4 that SRM slightly decrease with the increasing of D when the B is a certain value. The increasing wheel diameter will induce the reducing of wheel sinkage and the contact area between wheel and sand will increase as well as so that the unit radial stress descends at the same time. Therefore, it is difficult to intuitively judge the changing trend of SRM affected by the wheel diameter. According to the calculation from Eq. 16, 18 and 25, the longitudinal and lateral resistance moment descend slightly after raising the value of wheel diameter and holding the value of wheel width.
| Fig. 4: | Dependence of SRM on wheel width and diameter |
In contrast, the improvement range of shear resistance moment is much less so that the resultant steering resistance moment, M, basically remain the same despite the increasing of wheel diameter.
On the contrary, SRM are significantly improved with the increase of wheel width when the wheel diameter is constant as shown in Fig. 3. The increasing wheel width will reduce the wheel sinkage. However, the theoretical calculation reveals that the longitudinal bulldozing resistance increases sharply to the larger wheel width, meanwhile, the contact area between wheel and sand rise as well as despite of the reduced wheel sinkage. Therefore, the shear resistance moment are raised with the increasing wheel width.
On the basis of the theoretical investigation on wheel steering ability, it can be deduced that the favorable way to improve the steering maneuverability will be to decrease the wheel width. However, the effect of wheel width on wheel traction ability and steering performance is different. Therefore, defining the wheel parameters in designing wheel stage should to balance the motion performance of lunar rover wheel in order to obtaining the optimized wheel configuration parameters.
MOTION PERFORMANCE EVALUATION OF LUNAR ROVER WHEEL
Dimensionless analysis approach: The above analysis reveal that enhancing the wheel motion ability need the opposite changing trend of wheel parameters. A dimensionless analysis is carried out to study the effect of individual wheel parameters on the traction performance and steering ability of lunar rover wheel. Table 2 shows the dimensionless evaluation indexes of motion performance.
| Table 2: | Dimensionless evaluation indexes of motion performance |
In Table 2, the larger value of ηs illuminates that steering maneuverability is more excellent. The traction performance index, Ψ, is the product of tractive efficiency and drawbar pull under unit weight. The larger value of Ψ indicates that the larger work output under the efficiently tractive efficiency can be generated by the driven wheel.
High tractive efficiency is an object pursued in driving the lunar rover. However, the tractive efficiency does not incorporate the work output of a driven wheel. The possibility exists that in some conditions a device may have high efficiency yet do little work, whereas in other conditions it may do considerable work but have poor efficiency (Pandey and Ojha, 1978). Thus, the work output and tractive efficiency are both required to be combined into one dimensionless term, defined traction performance index, Ψ, as shown in Table 2.
To unmanned lunar rover, it should synchronously possess the excellent traction ability and steering capacity. Therefore, the comprehensive dimensionless evaluation criterion is defined as following:
| (27) |
where, A1 and A2 are the weight coefficient of Ψ and ηs, respectively, whose value can be obtained by means of fuzzy evaluation theory or expert evaluation method.
RESULTS AND DISCUSSION
Here, the evaluation criterions are employed to investigate the reasonable match between wheel diameter and width. In the calculation of evaluation index, the load W is 32.7 N and the sand parameters are assigned as listed in Table 2.
The concept, named width-diameter ratio ζ, is introduced into the motion performance evaluation of lunar rover wheel, whose definition is the ratio of wheel width to wheel diameter. The width-diameter ration is regard as the design index of wheel so as to reflect the reasonable match between wheel width and diameter.
Five samples are selected by combining the different wheel parameters, as shown in Table 3.
| Table 3: | Analysis samples |
| Fig. 5: | Dependence of evaluation indexes on slip, respectively |
Substituting the parameters of samples (listed in Table 3) in to the Eq. 9, 10, 26 and the evaluation indexes (listed in Table 2), it can derive the dependence of evaluation indexes value on wheel slip as shown in Fig. 5.
To the single evaluation index, it can be observed from the Fig. 5 that the sample exhibits different superiority. However, the results can guide the wheel design in order to enhance a certain item of lunar rover under the specific tasks or a certain terrain conditions.
The manned lunar rover is expected to accomplish many challenging tasks by traveling long distance. When the unknown terrain is so severe that designer can not determine which motion performance item should be enhanced, the comprehensive motion performance of lunar rover should be the research focus so as to ensure the lunar rover can cope with the tough terrain.
| Fig. 6: | Dependence of comprehensive evaluation of motion performance for samples at various slip. (—□—) D = 0.3 m, B = 0.12 m, ζ = 0.4; (—●—) D = 0.33 m, B = 0.15 m, ζ = 0.45; (—Δ—) D = 0.3 m, B = 0.15 m, ζ = 0.5; (—▼—) D = 0.27 m, B = 0.15 m, ζ = 0.55; (—ο—) D = 0.3 m, B = 0.18, ζ = 0.6 |
Therefore, Eq. 27 is adopted to systematically evaluate the traction ability and steering capacity, where A1 and A2 are 0.7 and 0.3, respectively derived by the fuzzy evaluation theory. The Fig. 6 illustrates the dependence of comprehensive evaluation of driving performance for wheel samples at various slip.
The Fig. 6 indicates that the value of χ decreases with the increasing width-diameter ratio, ζ. The maximum value of χ appears at about 20% slip. Therefore, the lower ζ value is advisable when design the wheel in order to improve its comprehensive motion performance.
CONCLUSION
Based on the wheel-sand interaction theory, traction force, driving torque, shear force beneath the steering wheel, bulldozing resistance acting on steering wheel rim and side surface, respectively are investigated. The quantitative relation between these items and wheel parameters is established. And then the theoretical analysis on the effect of wheel parameters on traction performance and steering ability indicates that increasing the wheel diameter is more effective than increasing wheel width in improving the wheel drawbar pull, whereas it is at the cost of increasing the power consumption. In contrast, steering ability is sensitive to rim width. Through the dimensionless evaluation analysis established to study the effect of wheel parameters on the traction performance and steering maneuverability, the results reveal that the samples with different width-diameter ratio exhibit various superiority in single evaluation index which can be utilized to guide the wheel design in order to enhance a certain item of lunar rover under the specific tasks or certain terrain conditions. However, the value of comprehensive evaluation index decreases with the increasing width-diameter ratio, and its maximum value appear at about 20% slip. Therefore, the research results are beneficial to design the wheel in order to enhance lunar rover’s comprehensive motion performance on unknown tough terrain.
ACKNOWLEDGMENT
This research was supported by National Natural Science Foundation of China (Grant No. 61005073), Creative Research Fund of Shanghai University, Creative Research Fund for Graduate Student of Shanghai University and the Special Fund of Shanghai University Scientific Selection and Cultivation for Outstanding Young Teachers.