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Towards terrain interaction prediction for bioinspired planetary exploration rovers
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Bioinspir. Biomim. 9 016009 (http://iopscience.iop.org/1748-3190/9/1/016009) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 130.64.11.153 This content was downloaded on 19/06/2017 at 00:03 Please note that terms and conditions apply.
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Bioinspiration & Biomimetics Bioinspir. Biomim. 9 (2014) 016009 (15pp)
doi:10.1088/1748-3182/9/1/016009
Towards terrain interaction prediction for bioinspired planetary exploration rovers Brian Yeomans 1 and Chakravathini M Saaj Surrey Space Centre, Department of Electronic Engineering, University of Surrey, Guildford, UK E-mail: [email protected] and [email protected] Received 16 October 2013, revised 3 December 2013 Accepted for publication 9 December 2013 Published 16 January 2014 Abstract
Deployment of a small legged vehicle to extend the reach of future planetary exploration missions is an attractive possibility but little is known about the behaviour of a walking rover on deformable planetary terrain. This paper applies ideas from the developing study of granular materials together with a detailed characterization of the sinkage process to propose and validate a combined model of terrain interaction based on an understanding of the physics and micro mechanics at the granular level. Whilst the model reflects the complexity of interactions expected from a walking rover, common themes emerge which enable the model to be streamlined to the extent that a simple mathematical representation is possible without resorting to numerical methods. Bespoke testing and analysis tools are described which reveal some unexpected conclusions and point the way towards intelligent control and foot geometry techniques to improve thrust generation. Keywords: bioinspired, terrain, model (Some figures may appear in colour only in the online journal)
respect to limiting the mass of the vehicle, with an expected target of the order of 20 kg in total. The surfaces of both the Moon and Mars are mainly covered with dry, granular regolith material [2, 3] and therefore any vehicle operating in those environments must both offer acceptable performance across such material and minimize the risk of becoming immobilised. Consequently the behaviour of wheeled vehicles on granular soils has been quite well studied [4], although terrain interaction prediction for planetary exploration missions so far has mainly relied— it should be noted with reasonable success—on the semiempirical methods developed by Bekker [5] and Wong [6]. These were however developed in the context of the behaviour of large, heavy vehicles; whilst it is perhaps helpful that each generation of planetary rover is becoming more massive [7], it is by no means clear that these methods remain applicable to the analysis of the behaviour of a much smaller, lighter, much more complex walking vehicle. Whilst much work has been done on the interaction of granular materials with simple intruders [8, 9], there remain gaps in the knowledge of how granular materials interact with a more complex intruder such as the leg/foot assembly of
1. Introduction The remote, unstructured environments of the Moon and rocky planets such as Mars present an extreme challenge to the effective mobility of the vehicles sent to explore their surface. So far, only wheeled vehicles have been deployed for this task, and whilst this approach has seen much success, it also brings considerable limitations; in particular, steep, rocky terrain for the time being will remain out of bounds, even though these high relief, exposed strata regions may be of the highest scientific interest. A strategy that offers considerable potential as a means to improve the science return was described in [1]. This comprises a mother rover/scout combination where the scout is a bioinspired, highly agile walking design intended for relatively short trips through high relief terrain; the objective would be to despatch the scout, perhaps to deploy instruments or gather a sample, returning to the mother rover with the results and to recharge batteries ready for the next trip. The design criteria for the scout are stringent, particularly with 1
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Bioinspir. Biomim. 9 (2014) 016009
B Yeomans and C M Saaj
Figure 1. DFKI crater explorer (CREX).
a walking rover. However, some general principles can be established.
of simple cross-section was examined experimentally in [20]. The rod served as an example lower leg section of a possible walking rover of typical hexapod design, such as that shown in figure 1 [21]; testing was undertaken with a variety of loose, dry, granular materials; loose material was deliberately selected as that was expected to be the most challenging type of material, from a mobility perspective, for a lightweight walking rover. The first category of model was based on ‘wedge theory’ [22, 23]—this is a quasi static terramechanics analysis technique which seeks to predict the geometry of failure and stress at failure of the granular material; it is based ultimately on the work of Coulomb [24] and treats the granular material as a continuum. This was contrasted with the approach developed by Albert et al [8, 15, 16] and Hill [17] which is based on stochastic analysis of behaviour at the individual grain level, applying the work of Coppersmith et al [25]; this approach seeks to describe the dynamic behaviour of the material as it flows around the rod, and the processes of force chain formation, collapse and reformation. Clearly whilst both methods are seeking to describe the same phenomena, the theoretical basis for each could not be more different. The work in [20] found that the quasi static analysis wedge theory techniques were an unreliable predictor of leg/soil force, whereas the dynamic model of grain level behaviour not only gave consistently accurate predictions, but was capable of simple mathematical expression without any need to resort to numerical techniques. In its general form the relationship found was:
(i) Granular materials exhibit behaviours similar to a solid, or behaviours similar to a fluid, depending on the stresses and displacements they are subjected to [9]. The fluid behaviour is not that of a normal, Newtonian fluid, however; the material does not obey the Navier–Stokes equations, but rather acts as a ‘frictional fluid’ [8], within which interactions are dominated by the frictional behaviour of the grain to grain contacts. (ii) Granular materials transmit forces through networks of force chains. These may be spatially extensive and relatively sparse, leaving much of the material only lightly loaded [10]. (iii) Unlike a simple scenario such as flow down a chute, where the material can be analysed as a continuum [11, 12], the feet of a walking rover represent relatively small, discrete intruder objects interacting in a highly localized manner with the granular material.This can be expected to be complex and time varying. To date, work on modelling walking rover leg/soil interaction has focussed largely on terramechanics models, by extension of the substantial work on wheeled vehicles [13], or on models such as that described in [14] which apply the granular physics work of Albert et al, Hill and others [8, 15–17], although other approaches exist. In [18], a modelling technique applicable to walking rovers is described based on a particle system called plastic terramechanics particle (PTP); PTP is intended to avoid the computational disadvantages of approaches such as discrete element modelling and so to be real time capable. In [19], a machine learning approach to leg/soil behaviour using a genetic algorithm is described. To begin to develop a more reliable quantitative model of walking rover terrain interaction, the prediction accuracy of two alternative methods for modelling the drag on a rod
dF = K dAh1.5
(1)
where the force dF is that arising on a small cross-sectional area of an intruder dA, K is a constant which is characteristic of the granular material and h is the depth of immersion in the granular material of the small area dA. The total force on the intruder can then be derived by integration of this equation across the cross-section of the intruder, and 2
Bioinspir. Biomim. 9 (2014) 016009
B Yeomans and C M Saaj
Figure 2. Hexapod kinematics illustration.
Figure 2 illustrates the kinematics of a typical hexapod with three actuated joints per leg, and shows the different phases of ground contact. It can be seen how the foot rotates in the soil as the robot moves forward, how the joints adjust to maintain the lateral distance between the robot body and the foot, and how the joints lower and raise the foot at each end of the ground phase. The conventional hexapod design was chosen to form the basis of this study because the rock and slope climbing ability of a vehicle like CREX provides a significant extension in the types of location reachable by the mission, whilst complying with strict mass constraints and utilizing the hexapod as a well established, thoroughly tested design form able to achieve the required agility and mobility on steep and rocky surfaces. A hexapod with three degree of freedom legs was chosen as the baseline vehicle type given it offers static stability in multiple gait formations (between three and five feet in ground contact at any time), whilst utilizing the minimum number of actuators per leg consistent with good agility. Unlike some other robot designs, it can be seen from figure 2 that the lower leg sections spend much of their time normal or close to normal to the terrain surface, which somewhat simplifies the analysis. In summary, the approach adopted was as follows.
this was verified experimentally as applicable for the simple cylindrical shapes used in the experiments. These experiments also confirmed a number of aspects of the behaviour seen as being consistent with dense granular flow [8] including the characteristic frequency signature of force chain breakage and reformation,a lack of sensitivity to the velocity of the intruder, and a dependence on the cross-sectional area of the intruder but not on its precise geometry [26]. Chen Li et al in [14] developed a model of leg/granular material force prediction for arbitrary shaped legs. Although founded on similar principles, the approach was extended from a 2D to 3D analysis using the ‘resistive force theory’ originally developed in [27] to demonstrate that the force on a leg of complex shape could be predicted by summing the forces on infinitesimal leg elements, taking account of both the angle of attack and angle of intrusion of the leg element. The accuracy, repeatability and relative simplicity of these models of leg/soil interaction forces thus offers the possibility of a fast, accurate predictor of legged rover performance and paves the way for the work described in this paper. The objective of this paper is therefore to draw on the evolving field of granular material physics, to derive lessons from what is now known about the behaviour of these ubiquitous but challenging materials, and to use this evidence to shape an investigation aimed to progress the development of a usable model of granular terrain interaction for a lightweight walking rover by developing the basic drag model into a comprehensive predictor of performance applicable to a hypothetical scout rover of a type similar to the DFKI ‘crater explorer (CREX)’ vehicle shown in figure 1. This is a rather different machine to that used in [14], which was very small and light and had highly curved ‘C’ shaped legs, and thus had significantly different geometry from a more conventional walking hexapod.
(i) In view of the complexity of events at the leg/soil interface, it was decided to study one single leg of a hexapod rather than the entire vehicle. However, measurement techniques where the leg interacts with the soil material in a manner that differs from that in the actual machine (for example, measurement of forces on a leg section held in a manipulator) were considered to have major drawbacks; for example, it is not possible to accurately reproduce partial slip conditions in such a configuration. Therefore it was decided to develop a bespoke kinematically accurate test facility equipped to directly measure forces at the leg 3
Bioinspir. Biomim. 9 (2014) 016009
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Figure 3. SLTB facility.
(ii)
(iii)
(iv)
(v)
assembly/soil interface and thus verify the accuracy of the model predictions in a realistic environment. To seek to identify effects deriving from changes in the properties of the granular material itself, tests were conducted using three dry granular materials of quite widely differing physical properties. In [14] the foot/soil shear force, which is the primary generator of thrust for a wheeled vehicle, is modelled as part of the empirically measured gravity asymmetric dependence on the intrusion angle γ and attack angle β. Use of a conventional hexapod allows the leg to be positioned approximately normal to the soil surface and the leg/soil shear force measured directly. To facilitate the comparison between predicted and actual results, our preference was to develop an explicit predictor for shear force. Sinkage will primarily be determined by vehicle mass, gait, and foot geometry. It has been established in [14, 20] that the drag on the robot leg depends on leg sinkage in a highly nonlinear manner (sinkage2.5 for a simple cylinder), and thus to accurately predict the leg/soil interaction forces, an accurate predictor of sinkage will also be required. It was suspected that modifications to foot geometry could be a useful tool in controlling sinkage, and thus improve the control of the overall thrust available from the leg/soil interface. Therefore the behaviour of a variety of foot designs in addition to a simple flat foot was investigated. To assist with the analysis of results, and to provide a platform upon which to base ‘what if’ type investigations using the model, a parametric tool was developed to facilitate further desktop investigation using the model algorithm.
Section 2 describes the apparatus and facilities used, section 3 the experimental results and analysis, section 4 the model validation and presentation of the effect of foot geometry changes, and section 5 discussion and conclusions. These lead to a new, comprehensive model of leg/terrain interaction applicable to lightweight walking rovers of conventional design. 2. Test facilities 2.1. The single leg test bed (SLTB)
The single leg test bed (SLTB) shown in figure 3 was developed recognizing the dynamic nature of the terrain interactions involved. It was considered essential to reproduce as far as possible the actual kinematics and dynamics of walking rover operation, for example to ensure that dynamic slip effects all the way from 0% to 100% could be reproduced. The SLTB comprises a number of integrated subsystems. (i) The mechanical system comprises a carriage constrained so it can move horizontally along a linear rail assembly. A system of cords and pulleys balance the carriage, enabling most of the residual friction to be compensated for and so that the carriage can be braked or accelerated as required. (ii) Drawbar pull and position sensing—in series with the cord system is a strain gauge, amplifier and digitizing microcontroller so that drawbar pull on the carriage can be directly measured and recorded. Carriage position is sensed and digitized using a linear encoder and associated microcontroller. (iii) Leg mechanical System—a complete three degree of freedom leg is attached to the carriage, comprising commercially sourced actuators [28], off the shelf and custom made brackets and links. The actuator axes mimic those of a three degree of freedom hexapod leg; thus the axis of the innermost joint is vertical relative to the SLTB, whereas the axes of the two outer joints are horizontal.
The following sections of this paper summarize the experiments and analysis undertaken, using granular materials with mean particle sizes ranging from 0.045 to 0.4 mm. 4
Bioinspir. Biomim. 9 (2014) 016009
B Yeomans and C M Saaj
Figure 4. Rotation sinkage measurement.
A wide variety of legged rover systems can be tested by (vi) Integration of SLTB control and data generation takes place in a LabVIEW environment [30]. LabVIEW installation of an actual leg from the proposed design. coordinates the various sub systems using its multiActuation and feedback of joint position, velocity and threading facilities and generates an integrated comma torque is provided using the capabilities of the commercial separated value (.csv) file of results, using the off the shelf equipment. microcontroller clocks to timestamp each data collection (iv) Soil force sensing—this is achieved using a combination cycle. of a miniature six axis force/torque sensor forming the foot of the single leg, and two force sensing resistors (vii) Foot positioning is also recorded using digital video data of the foot movement. The foot position is tracked using (FSRs) wrapped around the lower leg section, giving eight coloured markers (a blue disk on the surface of the lower channels of force data in all. To achieve and maintain leg opposite the carriage, where the red tape can be seen realistic loadings of the foot/soil interface, enabling a in figure 3). The marker position in the video frame is range of potential vehicle designs and masses to be extracted using computer vision techniques based on the emulated, real time feedback of force sensor data is OpenCV libraries [31], and the position data synchronized utilized to achieve proportional integral derivative control with other data sources in the LabVIEW environment. of the leg actuators; the feedback is conditioned and sensor noise reduced using averaging of sensor results with a The result is a comprehensive data set covering the whole dedicated 32 bit microcontroller board [29], which carries of the leg stepping cycle. This is then post processed and out a number of tasks in real time including initialization further analysed using MATLAB scripting. of the sensors, noise reduction via a 40 tap moving average calculation, and generation of 32 bit floating point force 2.2. Rotation sinkage testing and torque readings.Force values from the six axis sensor and strain gauge are accurate to within 0.2N, and torque A feature of conventional hexapod design is that the foot rotates to within 0.03N; force readings from the FSRs have a around an axis aligned with the length of the lower leg during maximum expected error of 0.9N, but these are used the leg cycle [32]. This rotation creates sinkage, as described primarily to cross check data from the six axis sensor in section 3.2.3. To characterize this behaviour, the apparatus shown in figure 4 was developed. and strain gauge rather than on a stand alone basis. Feet of differing sizes/geometries can be attached to a (v) Certain of the testing requires accurate maintenance of a known, constant load, for example, when investigating motor driven rotating tube with a vertical axis; drive to the tube shear force prediction (section 3.1) or lateral sinkage is provided by a DC motor and geartrain, and can be varied (section 3.2.2). In these cases, rather than rely on the to control torque and tube angular velocity. Foot loading is feedback mechanism, constant loading is achieved by varied by adding mass to the platform at the top of the tube. attaching a mass carrier and masses directly to the top Tube rotation and sinkage is measured using computer vision analysis of a video recording of the tube motion; markers on face of the outermost actuator. 5
Bioinspir. Biomim. 9 (2014) 016009
B Yeomans and C M Saaj
Figure 5. Shear force measurement. Table 1. Soil simulant parameters.
2.3.3. Surrey space centre planetary simulant three (SSC-3).
This is a wind sorted beach sand, sub angular, with a very consistent grain size. The material was collected, dried, and sieved to generate material within a narrow particle size distribution of 150 to 212 μm, and thus falling between that of the other two materials.
Parameter
ES-3
SSC-2
SSC-3
Mineralogy PSD Mean particle size Particle shape Bulk density (kg m−3 ) Particle density (kg m−3 ) Void ratio Friction angle (◦ )
Quartz 0.2–0.9 mm 0.4 mm Sub rounded 1.46 × 103
Crushed garnet 0.02–0.09 mm 0.045 mm Angular - sharp 2.22 × 103
Quartz 0.15–0.21mm 0.18mm Sub angular 1.48 × 103
2.6 × 103
4.1 × 103
2.63 × 103
3. Results
0.78 35.76
0.85 43.34
0.78 33.68
Here the results from a number of procedures are presented. Although the images in figure 6 feature a hemispherical foot for general illustrative purposes, all of the numerical data, plots and comparisons in this section are based on feet of planar circular geometry. Section 4.1 presents work on the effect of variation to foot geometry based on a variety of alternatives to the planar foot.
the tube at 90◦ intervals enable the rotation of the tube to be measured, and a marker band around the tube tracks sinkage. 2.3. Materials
The physical properties of the granular materials used are summarized in table 1.
3.1. Shear force prediction
For level, flat foot geometries, the foot/soil shear force arising can be measured directly in the SLTB. Figure 5 plots the results obtained using the SLTB of shear force against position for three foot loadings, using a 25 mm diameter flat foot. All soil materials show qualitatively similar results, this plot utilizes SSC-2 material. It can be seen that the shear force rises to a maximum quite gradually as displacement increases. This is expected behaviour for loose sands, as described in [36]. It was found that shear force could be predicted reliably by a two stage process.
This comprises a coarse quartz sand based on the commercially available Leighton Buzzard DA 30 material. It serves as an analogue of the coarse Martian regolith found on dune slopes [33]. The particle size distribution is 0.2 to 0.9 mm. A detailed summary of the properties and physical characteristics is set out in [34]. 2.3.1. Engineering soil simulant-3 (ES-3).
2.3.2. Surrey space centre planetary simulant two (SSC-2).
This comprises a fine silty sand manufactured from crushed almandine garnet. Although originally intended as an analogue for fine grained Martian regolith, the particle size distribution and angular particle shape of the material also make it suitable as a lunar regolith simulant [35]. The particle size distribution is 20 to 90 μm. A detailed description of the material properties can be found in [13].
(i) The maximum shear force is based on an application of the Mohr Coulomb relationship, as shown in equation (2): τmax = σnormal tan φeffective 6
(2)
Bioinspir. Biomim. 9 (2014) 016009
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Figure 6. Sinkage behaviour.
where the shear force τmax is proportional to the normal stress on the foot σnormal . Cohesion is ignored as dry granular materials are used. Figure 5 shows the predicted maximum shear force for each load level using an angle of internal friction φeffective of 25◦ . For each of the materials, this value of φ is somewhat lower than those derived using conventional testing with direct shear test equipment [37], which range between 33◦ and 43◦ . (ii) The ‘rise’ behaviour was predicted using equation (3) based on [36] τ = τmax (1 − e−slip/Kshear )
(iii) Frame 3 shows the leg at the end of the subsequent lateral sinkage phase. Particularly of note is the fact that the leg initially slips, as can be seen from the impression on the soil, but stops slipping as the sinkage created by the slip increases drag on the leg assembly. It can also be seen that sinkage in this phase is quite substantial. (iv) Frame 4 shows the leg at the end of the subsequent rotation sinkage phase. The leg no longer slips but remains centred in the soil—however the rotation of the foot is associated with more sinkage, which is also substantial. (v) Finally, Frame 5 shows the exit as the leg lifts; the additional slip is a consequence of the leg unloading whilst it is still moving.
(3)
where Kshear is an empirical shear displacement factor. A value of Kshear = 30 mm results in the predicted response shown by the purple curve; this can be compared with the actual load = 23.05N results, shown in green. Interestingly, this is quite a slow rate of rise with respect to displacement, compared with other observations, for example in [38].
It can be seen that sinkage comprises up to three distinct phases. (i) Initial pressure sinkage as the leg touches down—this will always occur. (ii) Should the force applied to the leg/foot assembly exceed the drag arising from the leg/soil interface, lateral slip occurs following touchdown. This will in turn create lateral sinkage, further increasing the drag from the soil. Whether lateral sinkage initially arises and then stops, or continues, will depend on the relative levels of the force applied to the leg and the resistance force arising through drag. Thus the extent and duration of lateral sinkage will depend critically on the level of actuator torque applied. (iii) If the forces from applied torque continue to exceed the leg/soil resistive forces, lateral sinkage will continue as the dominant behaviour through the remainder of the leg cycle. However if this is not the case and lateral sinkage ceases, the third sinkage type—rotation sinkage—takes
3.2. Sinkage prediction
Qualitative examination of the behaviour of the leg in the SLTB reveals that sinkage behaviour comprises not one single type but a sequence of distinguishable sinkage phenomena. Figure 6 shows a series of images derived from the SLTB. Qualitatively the behaviour does not change with the design of the foot, here one of hemispherical design is shown. (i) Frame 1 shows the leg at the moment of touchdown. (ii) Frame 2 shows the leg at the end of the initial, pressure sinkage phase. 7
Bioinspir. Biomim. 9 (2014) 016009
B Yeomans and C M Saaj
Figure 7. Pressure sinkage response.
over. Additional sinkage arises related to the rotation of the loaded foot in the soil, and continues until the leg cycle completes.
it can be seen that there is an inbuilt dependence, via the b factor, on the dimensions of the contact patch. In practice it was found that this method did not give an accurate predictor of sinkage, probably because the parameters are derived as an average over a range which includes much higher stress levels than those applied in these tests. In addition, in these conditions of relatively low stress, the dependence of the sinkage response on b is not seen at all. This can be seen in figure 8 which shows the sinkage response in ES-3 material using a range of foot diameters, each of which follow the same trend line. It was found that the behaviour more closely resembles that seen in [39], which showed a generic linear relationship between stress and sinkage for an object vertically penetrating a granular material, as shown in equation (5):
Thus sinkage can be seen to be progressive, dynamic, and with respect to lateral sinkage, dependent on the balance between the ability of the leg/soil interface to resist deformation, and the force applied. Figure 7 shows the pressure sinkage response. These results comprise linear fits to averages of actual results comprising three or more test runs per material, with sinkage measured at a series of discrete normal stress values; incremental loads were added to the apparatus and the sinkage measured at each load. The reason for the averaging is that in a pressure sinkage scenario, the particles of the material are in a quasi static equilibrium which is very far indeed from a true equilibrium. It was found experimentally that for each load value, there is quite a wide range of possible sinkage values—this is not surprising as the load bearing ability for each test will vary about a mean, depending on precisely how the grains are organized. By carrying out more tests and averaging the results, the mean value can be derived. For a simple, flat foot, these mean values of the sinkage results shows an approximately linear response to the stress applied to the foot. The conventional terramechanics approach to predicting pressure sinkage uses the Bernstein—Bekker Pressure sinkage equation as described in [5] and [6], shown in equation (4): kc + kφ zn . (4) p= b Here kc , kφ and n are empirically derived parameters typical of the granular material, p is stress, z sinkage and b the radius of the contact patch. The parameters are derived by a curve fitting approach over a wide range of stress levels, and 3.2.1. Pressure sinkage prediction.
dF = αμ(ρbulk gz) dA
(5)
where μ is an internal friction coefficient, ρbulk gz is the gravitational loading pressure based on the bulk density of the material ρbulk , dF is the force normal to an incremental area dA and α is a factor indicating the penetration or shear resistance of the material. What is particularly interesting in these results is the size and range of the dimensionless factor α; from 239 to 612. The size of the factor indicates that the forces resisting sinkage are very much higher than those arising from the self-weight under gravity of the displaced soil material, as so the ‘self weight’ factor must be acting as an analogue for the length and thus the frictional resistance of the force chains formed. The wide range seen in the factor is also interesting and suggests that characteristics in addition to the bulk properties of the material are relevant; a likely candidate may be the effect the shape of the material grains themselves has on force chain stability, as it would be expected that angular grains would create more durable force chains than those formed of rounded or subrounded grains [40]. 8
Bioinspir. Biomim. 9 (2014) 016009
B Yeomans and C M Saaj
Figure 8. Pressure Sinkage—effect of change in contact patch diameter.
Figure 9. Lateral sinkage response.
(iii) Variation in material properties has only a small effect on the response, with the results varying between ±20% of the average. (iv) At the typical foot velocities of a walking rover (1–10 cm s−1 ), little or no dependence on velocity is seen.
Figure 9 plots sinkage against slip in the lateral sinkage (slipping) phase. There are several interesting features of this behaviour.
3.2.2. Lateral sinkage prediction.
(i) The response has an asymptotic nature, trending toward a maximum at high slip. (ii) Whilst at very low stress levels, there is a separation in the response, at the stress levels typical of a lightweight walking rover, the response compresses to the extent that there is negligible dependence on either applied stress or intruder geometry—note the bunching of curves at other than very low stress.
The experimental work found that lateral sinkage behaviour could be described reasonably well by the following equation: hlateral = A(1 − eB∗slip ). 9
(6)
Bioinspir. Biomim. 9 (2014) 016009
B Yeomans and C M Saaj
Figure 10. Rotation sinkage—effect of changes to foot dimension.
The relevant model prediction is shown as the purple curve labelled ‘Model’ in figure 9. Typical values, where both slip and lateral sinkage hlateral are expressed in mm, are −0.04 for B, found to be approximately constant irrespective of material type, and between 35.0 and 42.0, depending on the material type, for A (A is effectively the maximum sinkage at large slip levels). Therefore it can be concluded that there is much less sensitivity to material type when compared with the pressure sinkage case.
subject to a 22.27N load. It can be seen that there is a reasonable correlation between the predicted and actual results. It is quite surprising that sinkage depends on both stress and the foot radius2 . The reasons for this are not currently understood; possibly the increased volume of soil under the foot, within which the particles are mobilized by the foot rotation, or the increased linear velocity of the foot rim, both of which are related to increased foot area, might play a part. Certainly this behaviour merits further study. K was found to range between 6.8359 × 10−04 and 8.8867 × 10−04 for the materials tested, and therefore it can be concluded that soil properties also have only a moderate effect on the rotation sinkage component, when compared with the pressure sinkage case.
Figure 10 shows the response in the rotation sinkage phase. Colours identify the diameter of the flat foot used, whereas the load applied is identified by the line style. This phase shows the most complex set of characteristics.
3.2.3. Rotation sinkage prediction.
(i) Each response follows a similar asymptotic profile. (ii) For a given foot diameter, increasing load and thus increasing stress results in an increase in sinkage. However, increasing foot diameter also results in increasing sinkage, with sinkage dependent on the foot radius2 . The result is that the response curves appear to form ‘families’ (shown by the double headed arrows) which appear to be associated with a given load level as the product Stress × foot radius2 depends on Load. (iii) The responses form smooth curves.
4. Model validation and effect of foot geometry variation
The experimental work found that rotation sinkage behaviour could be described by the following equation:
(i) Geometry: 25 mm diameter leg section, and 25 mm diameter planar, circular foot. (ii) Soil material: SSC-2 (iii) Load: 23.09N. (iv) Wave gait kinematics
sinkage = K ∗ Stress ∗ footradius2 (1 − eBθ )
The SLTB enables the overall leg/soil forces to be measured by directly measuring the force exerted on the SLTB carriage; this enables an overall check of the validity of the modelling approach to be made. Figure 11 shows a typical plot of predicted and actual forces against position, derived from the SLTB. The parameters for this scenario are as follows.
(7)
where θ is rotation in radians, stress is in Pa and foot radius in m. Figure 10 illustrates two examples of the model predictions, shown by purple curves labelled as follows; ‘Model Fit 1’, which is for a 25 mm diameter foot subjected to a 32.08N load, and ‘Model Fit 2’, which is for a 50 mm diameter foot
The force predictions are based on the equations previously described above. Whilst a comparison can also be made here between actual (in red dotted) and predicted 10
Bioinspir. Biomim. 9 (2014) 016009
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Figure 11. SLTB verification.
(in purple dash dotted) shear force, in the same way as figure 5, this plot also uses the drag, shear force and sinkage prediction algorithms described earlier in this paper to compute the predicted total force on the carriage, which is plotted in cyan. This can be compared directly with the measured force, plotted in blue. It can be seen that there is a close correlation between the predicted and actual results, in particular with the prediction faithfully capturing the dynamic changes in force as the leg assembly moves. This correlation provides confidence that the combined force prediction and sinkage algorithms can in fact reliably predict walking rover leg/soil forces. 4.1. Foot geometry changes
It became apparent during SLTB testing that a conventional hexapod design equipped with a flat foot has essentially pre-determined sinkage behaviour. This is because the soil interaction is determined largely by the load on each foot and the cross-section rather than the precise shape of the intruder interacting with the soil, as was found in [26]. The load per foot can be varied significantly via gait induced foot loading changes; for example, the load per foot for a 20 kg rover will be 39.2N in a wave gait configuration, and 65.4N using a tripod gait. However it may not be possible in practice to switch gait patterns without serious adverse consequences; for example, a wave gait configuration may be required to maximize stability if climbing a slope. Therefore a number of alternative foot geometries were tested with a view to determining if geometry modification might be a way of improving control over leg thrust. Figure 12 shows those design types investigated in addition to the flat feet tested. The rationale for choosing each of these designs is as follows.
Figure 12. Foot designs.
means it presents little resistance to movement in the plane of the blade and so can easily penetrate the soil material, yet presents a large cross-sectional area normal to the direction of thrust in the soil material. This method of propulsion for a robot was studied in [41], which describes a hatchling sea turtle-inspired robot, FlipperBot. A potential issue with this design is that the lack of rotational symmetry will mean the foot will strongly resist being rotated in the soil. (ii) Hemisphere—this design avoids this issue as it is rotational symmetric. The small initial diameter presented by the shape should increase initial pressure sinkage.
(i) Blade/Flipper type—this design comprises a thin blade whose plane is normal to the base of the foot. This shape 11
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Figure 13. Foot geometry comparison.
(iii) Cone—this is a more extreme version of the hemisphere, similar considerations should apply as it is rotationally symmetric and should increase initial pressure sinkage. Here a stepped cone shape is used.
prevents further rotation of the foot. This creates so much drag on the rover that it is effectively incapacitated. This problem should be soluble without undue complexity however through incorporation of a passive ankle joint that would allow the foot to rotate. Such a mechanism, probably deployable when needed and then retractable, could substantially aid thrust on granular materials. It can be seen that the hemisphere shape gives a modest increase in drag whereas the increase with the cone shape is at higher sinkages similar to that of the flipper. The modest effect with the hemisphere is not surprising; its radius is only 12.5 mm and so makes relatively little difference to the sinkage behaviour at higher levels of sinkage; nevertheless, it generates a useful increase. The large increase for the cone shape came as a surprise; although it was anticipated that the cone design would increase sinkage in the initial, pressure sinkage phase, there was expected to be a trade off between increased sinkage and reduced drag (at a given sinkage level). This is because the cone cross-sectional area is, for a simple cone, half that of the equivalent cylinder, with the ‘missing’ area at greater depth where the drag force per incremental area is higher. However, in practice, this trade off was found not to materialize, and the drag experienced for the cone shape for a given sinkage level was in fact similar to that of a cylinder with equivalent root diameter. Further investigation of this behaviour is required, which was found consistently in all soil materials and which suggests that at least in this scenario, the resistive force theory does not result in an accurate drag prediction.
Figure 13 shows a comparison of the drag generated by four different foot geometries as the leg/foot assembly moves through the soil material. The scenarios in each of these cases are identical other than with respect to the choice of foot. Details are as follows.
4.1.1. Foot design changes—results.
(i) The leg diameter is 25 mm. (ii) The root dimension of the rotationally symmetric shapes is 25mm. (iii) The flipper blade is 25 mm wide and 50 mm long. (iv) The soil material is ES-3 and the normal stress on the foot is 26.98 kPa. The ‘position’ axis in the plot effectively represents slip; the experiment ensures the assembly is forced to slip to confirm the soil is failing and so the thrust the soil can provide at failure, rather than a potentially lower value, is derived. Some dynamic effects can be seen here, particularly with the cone foot—on occasion the foot even rebounds slightly before proceeding. The flipper type foot was found to be extremely effective in increasing leg thrust, as it penetrated the material easily and presented a broad, high drag profile to the material. It can be seen that after the initial penetration, the flipper adds over 20N of thrust when compared with the flat foot without the flipper. The flipper is, however, not suitable for a conventional hexapod rover with a 3 degree of freedom leg design, unless that design is modified. This is because whilst this manner of construction simplifies the leg design, the kinematics of operation force the foot to rotate in the soil material. The flipper is so effective that the flipper jams in the soil material and
5. Discussion and conclusions The work presented in this paper confirms that analysing the behaviour of a walking rover when interacting with granular material is a complex and challenging process. However, 12
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Figure 14. Parametric tool—main screen.
aggregate leg/soil force. Based on the entered parameters, the algorithm analyses the ground contact phase dividing it into 100 timesteps. The algorithm loops through the timesteps sequentially, predicting the slip, sinkage and thrust at each step and generating a results array which is plotted graphically. Any change to any of the parameters results in an immediate re-computation and re-plot of the profile of predicted thrust covering the whole leg cycle. Permanent data such as soil parameter information is not hard coded but contained in a database read by the Tool; this enables data to be updated easily, or what if? scenarios easily run based on hypothetical parameters. The tool has a graphical user interface which enables parameters such as rover and foot geometry, mass, gait pattern etc to be set via spinboxes; all parameters can be changed at will with the results of the change presented immediately as both a graph and a saveable dataset. In addition to the spinboxes, two key parameters can be set or modified by sliders; these are the magnitude of acceptable lateral slip, and the magnitude of acceptable sinkage. This treatment was chosen as these parameters were found to be both critical to the results, and ones which might be adjusted dynamically, for example if the underlying algorithm were implemented in the vehicle controller. Usage of the tool has highlighted the following interesting conclusions.
despite the complexity, some clear themes emerge which enable behaviours to be categorized and predicted. Firstly, although on the face of it the behaviour of the granular material as it flows round an intruder is quite different from the quasi static pressure sinkage behaviour under the foot, the actual differences may be less than expected. The model of each behaviour type is hydrostatic, with force predictions scaling approximately with immersion depth. The principal difference between the models of each scenario is only the magnitude of the factor α; high α is associated with a quasi static environment, particularly where angular materials are employed, and lower α, with less sensitivity to material properties, where the material is in motion. The apparent distinction between the two behaviour types may therefore be only one of identifying whether the local environment will tend to produce either persistent (quasi static) or constantly reforming force chains. Secondly, despite the complexity, all of the behaviours seen here can be described by simple mathematics, with no need to resort to numerical methods; this is very encouraging in the context of the possible development of real time terrain interaction prediction models. To further investigate the dynamics of the force prediction/sinkage prediction algorithms described here, a parametric tool was developed as part of this study; this enables many possible scenarios of rover geometry, soil type and behaviour to be investigated, leading to some surprising and counter intuitive conclusions. The user interface with the tool can be seen in figure 14. The tool implements an algorithm which incorporates the equations described earlier in this paper in order to predict
(i) The thrust profile is clearly highly dynamic in nature, and so to take full advantage of the thrust available will require intelligent control of leg actuators designed to replicate this profile. 13
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Figure 15. Parametric tool—effect of change in soil type.
(ii) The initial slip of the foot post touchdown generates substantial sinkage; control of the amount of slip, for example with an intelligent torque controller, has a major impact on the thrust available. Thus acceptance of some degree of initial slip is probably essential in order to achieve acceptable thrust levels. (iii) Contrary to expectations, the tool shows that ‘weaker’ materials can generate more thrust, provided overall sinkage levels remain acceptable and the vehicle does not become immobilized. Figure 15 compares thrust profiles for ‘weaker’ ES-3 and ‘stronger’ SSC-2 material, showing that thrust using ES-3 is greater. The reason for this is that sinkage (primarily pressure sinkage) is much greater with ES-3, and whilst the drag experienced at a given sinkage level in the stronger material is greater, this is more than outweighed by the drag resulting from the increased aggregate sinkage. Clearly care will be needed in practice, given the risk that increased sinkage will erode the benefits in other ways, for example by generating drag between the soil and rover body, or by being so large that the legs are no longer able to lift themselves clear of the soil when stepping forwards. For this reason the tool enables a safe sinkage limit to be preset, so that an impractical recommendation is not generated. This plot also confirms that an accurate thrust prediction requires a degree of knowledge of the soil material physical parameters, something that remains a challenge in a planetary exploration context where soil parameter information may be sketchy or non-existent.
slip using an intelligent controller, or through modifications or deployable additions to foot geometry. The prediction model has only so far been validated with an example single leg rather than the full vehicle, and so future work will include implementation of the algorithm that has been developed in a rigid body dynamics simulation of the full rover to verify its accuracy, and ultimately development of a walking vehicle with intelligent actuator control based on this model. Acknowledgments This work was partially funded by the European Space Agency under NPI contract 4000102014/NL/11/PA. Useful discussions with Michel van Winnendael of the European Space Agency and Dr Marcus Matthews of the Department of Civil and Environmental Engineering, University of Surrey, UK, are also gratefully acknowledged. References [1] Colombano S, Kirchner F, Spenneberg D and Hanratty J 2004 Exploration of planetary terrains with a legged robot as a scout adjunct to a rover Space 2004 Conf. on American Institute of Aeronautics and Astronautics (San Diego, California) [2] McKay D S, Heiken G, Basu A, Blanford G, Simon S, Reedy R, French B M and Papike J 1995 The lunar regolith Lunar Sourcebook: A User’s Guide to the Moon ed G Heiken, D Vaniman and B M French (Cambridge: Cambridge University Press) pp 285–356 [3] Bell J 2008 The Martian Surface: Composition, Mineralogy, and Physical Properties (Cambridge: Cambridge University Press) [4] Patel N, Slade R and Clemmet J 2010 The ExoMars rover locomotion subsystem J. Terramech. 47 227–42 [5] Bekker M G 1959 Introduction to Terrain Vehicle Systems, Part 1 - The Terrain and Part 2 - The Vehicle (Ann Arbor, MI: University of Michigan Press)
This paper demonstrates progress towards a reliable terrain interaction prediction model for a walking planetary exploration rover of conventional hexapod design, and has highlighted a number of open problems and unexpected conclusions from the work. It suggests that the interaction might be more susceptible to control than might at first sight appear to be the case, either through active management of 14
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[6] Wong J Y 2008 Theory of Ground Vehicles (New York: Wiley-Interscience) [7] Showstack R 2011 Curiosity on the way to Mars Eos Trans. Am. Geophys. Union 92 455 [8] Albert R, Pfeifer M A, Barab´asi A L and Schiffer P 1999 Slow drag in a granular medium Phys. Rev. Lett. 82 205–8 [9] Schiffer P E 2010 Fun in the sand: some experiments in granular physics http://nanohub.org/resources/8491 [10] Behringer R P, Howell D, Kondic L, Tennakoon S and Veje C 1999 Predictability and granular materials Physica D 133 1–17 [11] Jop P, Forterre Y and Pouliquen O 2006 A constitutive law for dense granular flows Nature 441 727–30 [12] Chehata D, Zenit R and Wassgren C R 2003 Dense granular flow around an immersed cylinder Phys. Fluids 15 1622–30 [13] Scott G P and Saaj C M 2012 The development of a soil trafficability model for legged vehicles on granular soils J. Terramech. 49 133–46 [14] Li C, Zhang T and Goldman D I 2013 A terradynamics of legged locomotion on granular media Science 339 1408–12 [15] Albert I, Tegzes P, Albert R, Sample J G, Barab´asi A L, Vicsek T, Kahng B and Schiffer P 2001 Stick-slip fluctuations in granular drag Phys. Rev. E 64 1–9 [16] Albert I, Tegzes P, Kahng B, Albert R, Sample J G, Pfeifer M, Barab´asi A L, Vicsek T and Schiffer P 2000 Jamming and fluctuations in granular drag Phys. Rev. Lett. 84 5122–5 [17] Hill G, Yeung S and Koehler S A 2005 Scaling vertical drag forces in granular media Europhys. Lett. 72 137 [18] Yoo Y-H and Quack L 2012 Towards a real-time capable realistic soil contact model using a particle system IMSD-12: 2nd Joint Int. Conf. on Multibody System Dynamics [19] Roemmermann M, Ahmed M, Quack L and Kassahun Y 2011 Modeling of leg soil interaction using genetic algorithms ISTVS’11: 17th Int. Conf. of the Int. Society for Terrain-Vehicle Systems [20] Yeomans B, Saaj C M and van Winnendael M 2013 Walking planetary rovers—experimental analysis and modelling of leg thrust in loose granular soils J. Terramech. 50 107–20 [21] Cordes F, Roehr T M and Kirchner F 2012 RIMRES: a modular reconfigurable heterogeneous multi-robot exploration system iSAIRAS’12: Proc. 11th Int. Symp. on Artificial Intelligence, Robotics and Automation in Space [22] Terzaghi K 1943 Theoretical Soil Mechanics 3rd edn (London: Wiley) [23] McKyes E 1985 Soil Cutting and Tillage (Amsterdam: Elsevier) [24] Coulomb C A 1776 Essai sur une application des r`egles de maximis & minimis a` quelques probl`emes de statique, relatifs a` l’architecture (Paris: De l’Imprimerie Royale)
[25] Coppersmith S N, Liu C, Majumdar S, Narayan O and Witten T A 1996 Model for force fluctuations in bead packs Phys. Rev. E 53 4673–85 [26] Albert I, Sample J G, Morss A J, Rajagopalan S, Barab´asi A-L and Schiffer P 2001 Granular drag on a discrete object: shape effects on jamming Phys. Rev. E 64 061303 [27] Gray J and Hancock G J 1955 The propulsion of sea-urchin spermatozoa J. Exp. Biol. 32 802–14 [28] Robotis Co. Ltd 2013 Robotis dynamixel range (www.robotis.com/xe/dynamixel_en) [29] Leaflabs LLC. Maple Board, Leaflabs 2012 Robotis dynamixel range (http://leaflabs.com/devices/maple/) [30] Johnson G W and Jennings R 2001 LabVIEW Graphical Programming: Practical Applications in Instrumentation and Control (New York: McGraw-Hill) [31] Bradski G 2000 The OpenCV Library Dr. Dobb’s Journal of Software Tools [32] Yeomans B, Saaj C M and van Winnedael M 2013 Modelling leg/terrain interaction for a legged planetary micro-rover ASTRA’13: 13th ESA Workshop on Advanced Space Technologies for Robotics and Automation [33] Gouache T P, Patel N, Brunskill C, Scott G P, Saaj C M, Matthews M and Cui L 2011 Soil simulant sourcing for the ExoMars rover testbed Planet. Space Sci. 59 779–87 [34] Brunskill C, Patel N, Gouache T P, Scott G P, Saaj C M, Matthews M and Cui L 2011 Characterisation of Martian Soil Simulants for the ExoMars rover testbed J. Terramech. 48 419–38 [35] Carrier W D, Olhoeft G R and Mendell W 1995 Properties of the lunar surface: section 9.1.1—particle size distribution Lunar Sourcebook: A User’s Guide to the Moon ed G Heiken, D Vaniman and B M French (Cambridge: Cambridge University Press) pp 477–8 [36] Janosi Z and Hanamoto B 1961 The analytical determination of drawbar pull as a function of slip for tracked vehicles in deformable soils ISTVS 1st Int. Conf. on Mechanics of Soil-Vehicle Systems (Turin, Italy) [37] ASTM Standard 2004 Standard test method for direct shear test of soils under consolidated drained conditions ASTM Standard D3080-04 [38] Godbole R and Alcock R 1995 A device for the in situ determination of soil deformation modulus J. Terramech. 32 199–204 [39] Brzinski T A, Mayor P and Durian D J 2013 Depth-dependent resistance of granular media to vertical penetration Phys. Rev. Lett. 111 168002 [40] Ting J M, Meachum L and Rowell J D 1995 Effect of particle shape on the strength and deformation mechanisms of ellipse-shaped granular assemblages Eng. Comput. 12 99–108 [41] Mazouchova N, Umbanhowar P B and Goldman D I 2013 Flipper-driven terrestrial locomotion of a sea turtle-inspired robot Bioinspir. Biomim. 8 026007
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Towards terrain interaction prediction for bioinspired planetary exploration rovers
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Bioinspir. Biomim. 9 016009 (http://iopscience.iop.org/1748-3190/9/1/016009) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 130.64.11.153 This content was downloaded on 19/06/2017 at 00:03 Please note that terms and conditions apply.
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Bioinspiration & Biomimetics Bioinspir. Biomim. 9 (2014) 016009 (15pp)
doi:10.1088/1748-3182/9/1/016009
Towards terrain interaction prediction for bioinspired planetary exploration rovers Brian Yeomans 1 and Chakravathini M Saaj Surrey Space Centre, Department of Electronic Engineering, University of Surrey, Guildford, UK E-mail: [email protected] and [email protected] Received 16 October 2013, revised 3 December 2013 Accepted for publication 9 December 2013 Published 16 January 2014 Abstract
Deployment of a small legged vehicle to extend the reach of future planetary exploration missions is an attractive possibility but little is known about the behaviour of a walking rover on deformable planetary terrain. This paper applies ideas from the developing study of granular materials together with a detailed characterization of the sinkage process to propose and validate a combined model of terrain interaction based on an understanding of the physics and micro mechanics at the granular level. Whilst the model reflects the complexity of interactions expected from a walking rover, common themes emerge which enable the model to be streamlined to the extent that a simple mathematical representation is possible without resorting to numerical methods. Bespoke testing and analysis tools are described which reveal some unexpected conclusions and point the way towards intelligent control and foot geometry techniques to improve thrust generation. Keywords: bioinspired, terrain, model (Some figures may appear in colour only in the online journal)
respect to limiting the mass of the vehicle, with an expected target of the order of 20 kg in total. The surfaces of both the Moon and Mars are mainly covered with dry, granular regolith material [2, 3] and therefore any vehicle operating in those environments must both offer acceptable performance across such material and minimize the risk of becoming immobilised. Consequently the behaviour of wheeled vehicles on granular soils has been quite well studied [4], although terrain interaction prediction for planetary exploration missions so far has mainly relied— it should be noted with reasonable success—on the semiempirical methods developed by Bekker [5] and Wong [6]. These were however developed in the context of the behaviour of large, heavy vehicles; whilst it is perhaps helpful that each generation of planetary rover is becoming more massive [7], it is by no means clear that these methods remain applicable to the analysis of the behaviour of a much smaller, lighter, much more complex walking vehicle. Whilst much work has been done on the interaction of granular materials with simple intruders [8, 9], there remain gaps in the knowledge of how granular materials interact with a more complex intruder such as the leg/foot assembly of
1. Introduction The remote, unstructured environments of the Moon and rocky planets such as Mars present an extreme challenge to the effective mobility of the vehicles sent to explore their surface. So far, only wheeled vehicles have been deployed for this task, and whilst this approach has seen much success, it also brings considerable limitations; in particular, steep, rocky terrain for the time being will remain out of bounds, even though these high relief, exposed strata regions may be of the highest scientific interest. A strategy that offers considerable potential as a means to improve the science return was described in [1]. This comprises a mother rover/scout combination where the scout is a bioinspired, highly agile walking design intended for relatively short trips through high relief terrain; the objective would be to despatch the scout, perhaps to deploy instruments or gather a sample, returning to the mother rover with the results and to recharge batteries ready for the next trip. The design criteria for the scout are stringent, particularly with 1
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Bioinspir. Biomim. 9 (2014) 016009
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Figure 1. DFKI crater explorer (CREX).
a walking rover. However, some general principles can be established.
of simple cross-section was examined experimentally in [20]. The rod served as an example lower leg section of a possible walking rover of typical hexapod design, such as that shown in figure 1 [21]; testing was undertaken with a variety of loose, dry, granular materials; loose material was deliberately selected as that was expected to be the most challenging type of material, from a mobility perspective, for a lightweight walking rover. The first category of model was based on ‘wedge theory’ [22, 23]—this is a quasi static terramechanics analysis technique which seeks to predict the geometry of failure and stress at failure of the granular material; it is based ultimately on the work of Coulomb [24] and treats the granular material as a continuum. This was contrasted with the approach developed by Albert et al [8, 15, 16] and Hill [17] which is based on stochastic analysis of behaviour at the individual grain level, applying the work of Coppersmith et al [25]; this approach seeks to describe the dynamic behaviour of the material as it flows around the rod, and the processes of force chain formation, collapse and reformation. Clearly whilst both methods are seeking to describe the same phenomena, the theoretical basis for each could not be more different. The work in [20] found that the quasi static analysis wedge theory techniques were an unreliable predictor of leg/soil force, whereas the dynamic model of grain level behaviour not only gave consistently accurate predictions, but was capable of simple mathematical expression without any need to resort to numerical techniques. In its general form the relationship found was:
(i) Granular materials exhibit behaviours similar to a solid, or behaviours similar to a fluid, depending on the stresses and displacements they are subjected to [9]. The fluid behaviour is not that of a normal, Newtonian fluid, however; the material does not obey the Navier–Stokes equations, but rather acts as a ‘frictional fluid’ [8], within which interactions are dominated by the frictional behaviour of the grain to grain contacts. (ii) Granular materials transmit forces through networks of force chains. These may be spatially extensive and relatively sparse, leaving much of the material only lightly loaded [10]. (iii) Unlike a simple scenario such as flow down a chute, where the material can be analysed as a continuum [11, 12], the feet of a walking rover represent relatively small, discrete intruder objects interacting in a highly localized manner with the granular material.This can be expected to be complex and time varying. To date, work on modelling walking rover leg/soil interaction has focussed largely on terramechanics models, by extension of the substantial work on wheeled vehicles [13], or on models such as that described in [14] which apply the granular physics work of Albert et al, Hill and others [8, 15–17], although other approaches exist. In [18], a modelling technique applicable to walking rovers is described based on a particle system called plastic terramechanics particle (PTP); PTP is intended to avoid the computational disadvantages of approaches such as discrete element modelling and so to be real time capable. In [19], a machine learning approach to leg/soil behaviour using a genetic algorithm is described. To begin to develop a more reliable quantitative model of walking rover terrain interaction, the prediction accuracy of two alternative methods for modelling the drag on a rod
dF = K dAh1.5
(1)
where the force dF is that arising on a small cross-sectional area of an intruder dA, K is a constant which is characteristic of the granular material and h is the depth of immersion in the granular material of the small area dA. The total force on the intruder can then be derived by integration of this equation across the cross-section of the intruder, and 2
Bioinspir. Biomim. 9 (2014) 016009
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Figure 2. Hexapod kinematics illustration.
Figure 2 illustrates the kinematics of a typical hexapod with three actuated joints per leg, and shows the different phases of ground contact. It can be seen how the foot rotates in the soil as the robot moves forward, how the joints adjust to maintain the lateral distance between the robot body and the foot, and how the joints lower and raise the foot at each end of the ground phase. The conventional hexapod design was chosen to form the basis of this study because the rock and slope climbing ability of a vehicle like CREX provides a significant extension in the types of location reachable by the mission, whilst complying with strict mass constraints and utilizing the hexapod as a well established, thoroughly tested design form able to achieve the required agility and mobility on steep and rocky surfaces. A hexapod with three degree of freedom legs was chosen as the baseline vehicle type given it offers static stability in multiple gait formations (between three and five feet in ground contact at any time), whilst utilizing the minimum number of actuators per leg consistent with good agility. Unlike some other robot designs, it can be seen from figure 2 that the lower leg sections spend much of their time normal or close to normal to the terrain surface, which somewhat simplifies the analysis. In summary, the approach adopted was as follows.
this was verified experimentally as applicable for the simple cylindrical shapes used in the experiments. These experiments also confirmed a number of aspects of the behaviour seen as being consistent with dense granular flow [8] including the characteristic frequency signature of force chain breakage and reformation,a lack of sensitivity to the velocity of the intruder, and a dependence on the cross-sectional area of the intruder but not on its precise geometry [26]. Chen Li et al in [14] developed a model of leg/granular material force prediction for arbitrary shaped legs. Although founded on similar principles, the approach was extended from a 2D to 3D analysis using the ‘resistive force theory’ originally developed in [27] to demonstrate that the force on a leg of complex shape could be predicted by summing the forces on infinitesimal leg elements, taking account of both the angle of attack and angle of intrusion of the leg element. The accuracy, repeatability and relative simplicity of these models of leg/soil interaction forces thus offers the possibility of a fast, accurate predictor of legged rover performance and paves the way for the work described in this paper. The objective of this paper is therefore to draw on the evolving field of granular material physics, to derive lessons from what is now known about the behaviour of these ubiquitous but challenging materials, and to use this evidence to shape an investigation aimed to progress the development of a usable model of granular terrain interaction for a lightweight walking rover by developing the basic drag model into a comprehensive predictor of performance applicable to a hypothetical scout rover of a type similar to the DFKI ‘crater explorer (CREX)’ vehicle shown in figure 1. This is a rather different machine to that used in [14], which was very small and light and had highly curved ‘C’ shaped legs, and thus had significantly different geometry from a more conventional walking hexapod.
(i) In view of the complexity of events at the leg/soil interface, it was decided to study one single leg of a hexapod rather than the entire vehicle. However, measurement techniques where the leg interacts with the soil material in a manner that differs from that in the actual machine (for example, measurement of forces on a leg section held in a manipulator) were considered to have major drawbacks; for example, it is not possible to accurately reproduce partial slip conditions in such a configuration. Therefore it was decided to develop a bespoke kinematically accurate test facility equipped to directly measure forces at the leg 3
Bioinspir. Biomim. 9 (2014) 016009
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Figure 3. SLTB facility.
(ii)
(iii)
(iv)
(v)
assembly/soil interface and thus verify the accuracy of the model predictions in a realistic environment. To seek to identify effects deriving from changes in the properties of the granular material itself, tests were conducted using three dry granular materials of quite widely differing physical properties. In [14] the foot/soil shear force, which is the primary generator of thrust for a wheeled vehicle, is modelled as part of the empirically measured gravity asymmetric dependence on the intrusion angle γ and attack angle β. Use of a conventional hexapod allows the leg to be positioned approximately normal to the soil surface and the leg/soil shear force measured directly. To facilitate the comparison between predicted and actual results, our preference was to develop an explicit predictor for shear force. Sinkage will primarily be determined by vehicle mass, gait, and foot geometry. It has been established in [14, 20] that the drag on the robot leg depends on leg sinkage in a highly nonlinear manner (sinkage2.5 for a simple cylinder), and thus to accurately predict the leg/soil interaction forces, an accurate predictor of sinkage will also be required. It was suspected that modifications to foot geometry could be a useful tool in controlling sinkage, and thus improve the control of the overall thrust available from the leg/soil interface. Therefore the behaviour of a variety of foot designs in addition to a simple flat foot was investigated. To assist with the analysis of results, and to provide a platform upon which to base ‘what if’ type investigations using the model, a parametric tool was developed to facilitate further desktop investigation using the model algorithm.
Section 2 describes the apparatus and facilities used, section 3 the experimental results and analysis, section 4 the model validation and presentation of the effect of foot geometry changes, and section 5 discussion and conclusions. These lead to a new, comprehensive model of leg/terrain interaction applicable to lightweight walking rovers of conventional design. 2. Test facilities 2.1. The single leg test bed (SLTB)
The single leg test bed (SLTB) shown in figure 3 was developed recognizing the dynamic nature of the terrain interactions involved. It was considered essential to reproduce as far as possible the actual kinematics and dynamics of walking rover operation, for example to ensure that dynamic slip effects all the way from 0% to 100% could be reproduced. The SLTB comprises a number of integrated subsystems. (i) The mechanical system comprises a carriage constrained so it can move horizontally along a linear rail assembly. A system of cords and pulleys balance the carriage, enabling most of the residual friction to be compensated for and so that the carriage can be braked or accelerated as required. (ii) Drawbar pull and position sensing—in series with the cord system is a strain gauge, amplifier and digitizing microcontroller so that drawbar pull on the carriage can be directly measured and recorded. Carriage position is sensed and digitized using a linear encoder and associated microcontroller. (iii) Leg mechanical System—a complete three degree of freedom leg is attached to the carriage, comprising commercially sourced actuators [28], off the shelf and custom made brackets and links. The actuator axes mimic those of a three degree of freedom hexapod leg; thus the axis of the innermost joint is vertical relative to the SLTB, whereas the axes of the two outer joints are horizontal.
The following sections of this paper summarize the experiments and analysis undertaken, using granular materials with mean particle sizes ranging from 0.045 to 0.4 mm. 4
Bioinspir. Biomim. 9 (2014) 016009
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Figure 4. Rotation sinkage measurement.
A wide variety of legged rover systems can be tested by (vi) Integration of SLTB control and data generation takes place in a LabVIEW environment [30]. LabVIEW installation of an actual leg from the proposed design. coordinates the various sub systems using its multiActuation and feedback of joint position, velocity and threading facilities and generates an integrated comma torque is provided using the capabilities of the commercial separated value (.csv) file of results, using the off the shelf equipment. microcontroller clocks to timestamp each data collection (iv) Soil force sensing—this is achieved using a combination cycle. of a miniature six axis force/torque sensor forming the foot of the single leg, and two force sensing resistors (vii) Foot positioning is also recorded using digital video data of the foot movement. The foot position is tracked using (FSRs) wrapped around the lower leg section, giving eight coloured markers (a blue disk on the surface of the lower channels of force data in all. To achieve and maintain leg opposite the carriage, where the red tape can be seen realistic loadings of the foot/soil interface, enabling a in figure 3). The marker position in the video frame is range of potential vehicle designs and masses to be extracted using computer vision techniques based on the emulated, real time feedback of force sensor data is OpenCV libraries [31], and the position data synchronized utilized to achieve proportional integral derivative control with other data sources in the LabVIEW environment. of the leg actuators; the feedback is conditioned and sensor noise reduced using averaging of sensor results with a The result is a comprehensive data set covering the whole dedicated 32 bit microcontroller board [29], which carries of the leg stepping cycle. This is then post processed and out a number of tasks in real time including initialization further analysed using MATLAB scripting. of the sensors, noise reduction via a 40 tap moving average calculation, and generation of 32 bit floating point force 2.2. Rotation sinkage testing and torque readings.Force values from the six axis sensor and strain gauge are accurate to within 0.2N, and torque A feature of conventional hexapod design is that the foot rotates to within 0.03N; force readings from the FSRs have a around an axis aligned with the length of the lower leg during maximum expected error of 0.9N, but these are used the leg cycle [32]. This rotation creates sinkage, as described primarily to cross check data from the six axis sensor in section 3.2.3. To characterize this behaviour, the apparatus shown in figure 4 was developed. and strain gauge rather than on a stand alone basis. Feet of differing sizes/geometries can be attached to a (v) Certain of the testing requires accurate maintenance of a known, constant load, for example, when investigating motor driven rotating tube with a vertical axis; drive to the tube shear force prediction (section 3.1) or lateral sinkage is provided by a DC motor and geartrain, and can be varied (section 3.2.2). In these cases, rather than rely on the to control torque and tube angular velocity. Foot loading is feedback mechanism, constant loading is achieved by varied by adding mass to the platform at the top of the tube. attaching a mass carrier and masses directly to the top Tube rotation and sinkage is measured using computer vision analysis of a video recording of the tube motion; markers on face of the outermost actuator. 5
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Figure 5. Shear force measurement. Table 1. Soil simulant parameters.
2.3.3. Surrey space centre planetary simulant three (SSC-3).
This is a wind sorted beach sand, sub angular, with a very consistent grain size. The material was collected, dried, and sieved to generate material within a narrow particle size distribution of 150 to 212 μm, and thus falling between that of the other two materials.
Parameter
ES-3
SSC-2
SSC-3
Mineralogy PSD Mean particle size Particle shape Bulk density (kg m−3 ) Particle density (kg m−3 ) Void ratio Friction angle (◦ )
Quartz 0.2–0.9 mm 0.4 mm Sub rounded 1.46 × 103
Crushed garnet 0.02–0.09 mm 0.045 mm Angular - sharp 2.22 × 103
Quartz 0.15–0.21mm 0.18mm Sub angular 1.48 × 103
2.6 × 103
4.1 × 103
2.63 × 103
3. Results
0.78 35.76
0.85 43.34
0.78 33.68
Here the results from a number of procedures are presented. Although the images in figure 6 feature a hemispherical foot for general illustrative purposes, all of the numerical data, plots and comparisons in this section are based on feet of planar circular geometry. Section 4.1 presents work on the effect of variation to foot geometry based on a variety of alternatives to the planar foot.
the tube at 90◦ intervals enable the rotation of the tube to be measured, and a marker band around the tube tracks sinkage. 2.3. Materials
The physical properties of the granular materials used are summarized in table 1.
3.1. Shear force prediction
For level, flat foot geometries, the foot/soil shear force arising can be measured directly in the SLTB. Figure 5 plots the results obtained using the SLTB of shear force against position for three foot loadings, using a 25 mm diameter flat foot. All soil materials show qualitatively similar results, this plot utilizes SSC-2 material. It can be seen that the shear force rises to a maximum quite gradually as displacement increases. This is expected behaviour for loose sands, as described in [36]. It was found that shear force could be predicted reliably by a two stage process.
This comprises a coarse quartz sand based on the commercially available Leighton Buzzard DA 30 material. It serves as an analogue of the coarse Martian regolith found on dune slopes [33]. The particle size distribution is 0.2 to 0.9 mm. A detailed summary of the properties and physical characteristics is set out in [34]. 2.3.1. Engineering soil simulant-3 (ES-3).
2.3.2. Surrey space centre planetary simulant two (SSC-2).
This comprises a fine silty sand manufactured from crushed almandine garnet. Although originally intended as an analogue for fine grained Martian regolith, the particle size distribution and angular particle shape of the material also make it suitable as a lunar regolith simulant [35]. The particle size distribution is 20 to 90 μm. A detailed description of the material properties can be found in [13].
(i) The maximum shear force is based on an application of the Mohr Coulomb relationship, as shown in equation (2): τmax = σnormal tan φeffective 6
(2)
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Figure 6. Sinkage behaviour.
where the shear force τmax is proportional to the normal stress on the foot σnormal . Cohesion is ignored as dry granular materials are used. Figure 5 shows the predicted maximum shear force for each load level using an angle of internal friction φeffective of 25◦ . For each of the materials, this value of φ is somewhat lower than those derived using conventional testing with direct shear test equipment [37], which range between 33◦ and 43◦ . (ii) The ‘rise’ behaviour was predicted using equation (3) based on [36] τ = τmax (1 − e−slip/Kshear )
(iii) Frame 3 shows the leg at the end of the subsequent lateral sinkage phase. Particularly of note is the fact that the leg initially slips, as can be seen from the impression on the soil, but stops slipping as the sinkage created by the slip increases drag on the leg assembly. It can also be seen that sinkage in this phase is quite substantial. (iv) Frame 4 shows the leg at the end of the subsequent rotation sinkage phase. The leg no longer slips but remains centred in the soil—however the rotation of the foot is associated with more sinkage, which is also substantial. (v) Finally, Frame 5 shows the exit as the leg lifts; the additional slip is a consequence of the leg unloading whilst it is still moving.
(3)
where Kshear is an empirical shear displacement factor. A value of Kshear = 30 mm results in the predicted response shown by the purple curve; this can be compared with the actual load = 23.05N results, shown in green. Interestingly, this is quite a slow rate of rise with respect to displacement, compared with other observations, for example in [38].
It can be seen that sinkage comprises up to three distinct phases. (i) Initial pressure sinkage as the leg touches down—this will always occur. (ii) Should the force applied to the leg/foot assembly exceed the drag arising from the leg/soil interface, lateral slip occurs following touchdown. This will in turn create lateral sinkage, further increasing the drag from the soil. Whether lateral sinkage initially arises and then stops, or continues, will depend on the relative levels of the force applied to the leg and the resistance force arising through drag. Thus the extent and duration of lateral sinkage will depend critically on the level of actuator torque applied. (iii) If the forces from applied torque continue to exceed the leg/soil resistive forces, lateral sinkage will continue as the dominant behaviour through the remainder of the leg cycle. However if this is not the case and lateral sinkage ceases, the third sinkage type—rotation sinkage—takes
3.2. Sinkage prediction
Qualitative examination of the behaviour of the leg in the SLTB reveals that sinkage behaviour comprises not one single type but a sequence of distinguishable sinkage phenomena. Figure 6 shows a series of images derived from the SLTB. Qualitatively the behaviour does not change with the design of the foot, here one of hemispherical design is shown. (i) Frame 1 shows the leg at the moment of touchdown. (ii) Frame 2 shows the leg at the end of the initial, pressure sinkage phase. 7
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Figure 7. Pressure sinkage response.
over. Additional sinkage arises related to the rotation of the loaded foot in the soil, and continues until the leg cycle completes.
it can be seen that there is an inbuilt dependence, via the b factor, on the dimensions of the contact patch. In practice it was found that this method did not give an accurate predictor of sinkage, probably because the parameters are derived as an average over a range which includes much higher stress levels than those applied in these tests. In addition, in these conditions of relatively low stress, the dependence of the sinkage response on b is not seen at all. This can be seen in figure 8 which shows the sinkage response in ES-3 material using a range of foot diameters, each of which follow the same trend line. It was found that the behaviour more closely resembles that seen in [39], which showed a generic linear relationship between stress and sinkage for an object vertically penetrating a granular material, as shown in equation (5):
Thus sinkage can be seen to be progressive, dynamic, and with respect to lateral sinkage, dependent on the balance between the ability of the leg/soil interface to resist deformation, and the force applied. Figure 7 shows the pressure sinkage response. These results comprise linear fits to averages of actual results comprising three or more test runs per material, with sinkage measured at a series of discrete normal stress values; incremental loads were added to the apparatus and the sinkage measured at each load. The reason for the averaging is that in a pressure sinkage scenario, the particles of the material are in a quasi static equilibrium which is very far indeed from a true equilibrium. It was found experimentally that for each load value, there is quite a wide range of possible sinkage values—this is not surprising as the load bearing ability for each test will vary about a mean, depending on precisely how the grains are organized. By carrying out more tests and averaging the results, the mean value can be derived. For a simple, flat foot, these mean values of the sinkage results shows an approximately linear response to the stress applied to the foot. The conventional terramechanics approach to predicting pressure sinkage uses the Bernstein—Bekker Pressure sinkage equation as described in [5] and [6], shown in equation (4): kc + kφ zn . (4) p= b Here kc , kφ and n are empirically derived parameters typical of the granular material, p is stress, z sinkage and b the radius of the contact patch. The parameters are derived by a curve fitting approach over a wide range of stress levels, and 3.2.1. Pressure sinkage prediction.
dF = αμ(ρbulk gz) dA
(5)
where μ is an internal friction coefficient, ρbulk gz is the gravitational loading pressure based on the bulk density of the material ρbulk , dF is the force normal to an incremental area dA and α is a factor indicating the penetration or shear resistance of the material. What is particularly interesting in these results is the size and range of the dimensionless factor α; from 239 to 612. The size of the factor indicates that the forces resisting sinkage are very much higher than those arising from the self-weight under gravity of the displaced soil material, as so the ‘self weight’ factor must be acting as an analogue for the length and thus the frictional resistance of the force chains formed. The wide range seen in the factor is also interesting and suggests that characteristics in addition to the bulk properties of the material are relevant; a likely candidate may be the effect the shape of the material grains themselves has on force chain stability, as it would be expected that angular grains would create more durable force chains than those formed of rounded or subrounded grains [40]. 8
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Figure 8. Pressure Sinkage—effect of change in contact patch diameter.
Figure 9. Lateral sinkage response.
(iii) Variation in material properties has only a small effect on the response, with the results varying between ±20% of the average. (iv) At the typical foot velocities of a walking rover (1–10 cm s−1 ), little or no dependence on velocity is seen.
Figure 9 plots sinkage against slip in the lateral sinkage (slipping) phase. There are several interesting features of this behaviour.
3.2.2. Lateral sinkage prediction.
(i) The response has an asymptotic nature, trending toward a maximum at high slip. (ii) Whilst at very low stress levels, there is a separation in the response, at the stress levels typical of a lightweight walking rover, the response compresses to the extent that there is negligible dependence on either applied stress or intruder geometry—note the bunching of curves at other than very low stress.
The experimental work found that lateral sinkage behaviour could be described reasonably well by the following equation: hlateral = A(1 − eB∗slip ). 9
(6)
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Figure 10. Rotation sinkage—effect of changes to foot dimension.
The relevant model prediction is shown as the purple curve labelled ‘Model’ in figure 9. Typical values, where both slip and lateral sinkage hlateral are expressed in mm, are −0.04 for B, found to be approximately constant irrespective of material type, and between 35.0 and 42.0, depending on the material type, for A (A is effectively the maximum sinkage at large slip levels). Therefore it can be concluded that there is much less sensitivity to material type when compared with the pressure sinkage case.
subject to a 22.27N load. It can be seen that there is a reasonable correlation between the predicted and actual results. It is quite surprising that sinkage depends on both stress and the foot radius2 . The reasons for this are not currently understood; possibly the increased volume of soil under the foot, within which the particles are mobilized by the foot rotation, or the increased linear velocity of the foot rim, both of which are related to increased foot area, might play a part. Certainly this behaviour merits further study. K was found to range between 6.8359 × 10−04 and 8.8867 × 10−04 for the materials tested, and therefore it can be concluded that soil properties also have only a moderate effect on the rotation sinkage component, when compared with the pressure sinkage case.
Figure 10 shows the response in the rotation sinkage phase. Colours identify the diameter of the flat foot used, whereas the load applied is identified by the line style. This phase shows the most complex set of characteristics.
3.2.3. Rotation sinkage prediction.
(i) Each response follows a similar asymptotic profile. (ii) For a given foot diameter, increasing load and thus increasing stress results in an increase in sinkage. However, increasing foot diameter also results in increasing sinkage, with sinkage dependent on the foot radius2 . The result is that the response curves appear to form ‘families’ (shown by the double headed arrows) which appear to be associated with a given load level as the product Stress × foot radius2 depends on Load. (iii) The responses form smooth curves.
4. Model validation and effect of foot geometry variation
The experimental work found that rotation sinkage behaviour could be described by the following equation:
(i) Geometry: 25 mm diameter leg section, and 25 mm diameter planar, circular foot. (ii) Soil material: SSC-2 (iii) Load: 23.09N. (iv) Wave gait kinematics
sinkage = K ∗ Stress ∗ footradius2 (1 − eBθ )
The SLTB enables the overall leg/soil forces to be measured by directly measuring the force exerted on the SLTB carriage; this enables an overall check of the validity of the modelling approach to be made. Figure 11 shows a typical plot of predicted and actual forces against position, derived from the SLTB. The parameters for this scenario are as follows.
(7)
where θ is rotation in radians, stress is in Pa and foot radius in m. Figure 10 illustrates two examples of the model predictions, shown by purple curves labelled as follows; ‘Model Fit 1’, which is for a 25 mm diameter foot subjected to a 32.08N load, and ‘Model Fit 2’, which is for a 50 mm diameter foot
The force predictions are based on the equations previously described above. Whilst a comparison can also be made here between actual (in red dotted) and predicted 10
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Figure 11. SLTB verification.
(in purple dash dotted) shear force, in the same way as figure 5, this plot also uses the drag, shear force and sinkage prediction algorithms described earlier in this paper to compute the predicted total force on the carriage, which is plotted in cyan. This can be compared directly with the measured force, plotted in blue. It can be seen that there is a close correlation between the predicted and actual results, in particular with the prediction faithfully capturing the dynamic changes in force as the leg assembly moves. This correlation provides confidence that the combined force prediction and sinkage algorithms can in fact reliably predict walking rover leg/soil forces. 4.1. Foot geometry changes
It became apparent during SLTB testing that a conventional hexapod design equipped with a flat foot has essentially pre-determined sinkage behaviour. This is because the soil interaction is determined largely by the load on each foot and the cross-section rather than the precise shape of the intruder interacting with the soil, as was found in [26]. The load per foot can be varied significantly via gait induced foot loading changes; for example, the load per foot for a 20 kg rover will be 39.2N in a wave gait configuration, and 65.4N using a tripod gait. However it may not be possible in practice to switch gait patterns without serious adverse consequences; for example, a wave gait configuration may be required to maximize stability if climbing a slope. Therefore a number of alternative foot geometries were tested with a view to determining if geometry modification might be a way of improving control over leg thrust. Figure 12 shows those design types investigated in addition to the flat feet tested. The rationale for choosing each of these designs is as follows.
Figure 12. Foot designs.
means it presents little resistance to movement in the plane of the blade and so can easily penetrate the soil material, yet presents a large cross-sectional area normal to the direction of thrust in the soil material. This method of propulsion for a robot was studied in [41], which describes a hatchling sea turtle-inspired robot, FlipperBot. A potential issue with this design is that the lack of rotational symmetry will mean the foot will strongly resist being rotated in the soil. (ii) Hemisphere—this design avoids this issue as it is rotational symmetric. The small initial diameter presented by the shape should increase initial pressure sinkage.
(i) Blade/Flipper type—this design comprises a thin blade whose plane is normal to the base of the foot. This shape 11
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Figure 13. Foot geometry comparison.
(iii) Cone—this is a more extreme version of the hemisphere, similar considerations should apply as it is rotationally symmetric and should increase initial pressure sinkage. Here a stepped cone shape is used.
prevents further rotation of the foot. This creates so much drag on the rover that it is effectively incapacitated. This problem should be soluble without undue complexity however through incorporation of a passive ankle joint that would allow the foot to rotate. Such a mechanism, probably deployable when needed and then retractable, could substantially aid thrust on granular materials. It can be seen that the hemisphere shape gives a modest increase in drag whereas the increase with the cone shape is at higher sinkages similar to that of the flipper. The modest effect with the hemisphere is not surprising; its radius is only 12.5 mm and so makes relatively little difference to the sinkage behaviour at higher levels of sinkage; nevertheless, it generates a useful increase. The large increase for the cone shape came as a surprise; although it was anticipated that the cone design would increase sinkage in the initial, pressure sinkage phase, there was expected to be a trade off between increased sinkage and reduced drag (at a given sinkage level). This is because the cone cross-sectional area is, for a simple cone, half that of the equivalent cylinder, with the ‘missing’ area at greater depth where the drag force per incremental area is higher. However, in practice, this trade off was found not to materialize, and the drag experienced for the cone shape for a given sinkage level was in fact similar to that of a cylinder with equivalent root diameter. Further investigation of this behaviour is required, which was found consistently in all soil materials and which suggests that at least in this scenario, the resistive force theory does not result in an accurate drag prediction.
Figure 13 shows a comparison of the drag generated by four different foot geometries as the leg/foot assembly moves through the soil material. The scenarios in each of these cases are identical other than with respect to the choice of foot. Details are as follows.
4.1.1. Foot design changes—results.
(i) The leg diameter is 25 mm. (ii) The root dimension of the rotationally symmetric shapes is 25mm. (iii) The flipper blade is 25 mm wide and 50 mm long. (iv) The soil material is ES-3 and the normal stress on the foot is 26.98 kPa. The ‘position’ axis in the plot effectively represents slip; the experiment ensures the assembly is forced to slip to confirm the soil is failing and so the thrust the soil can provide at failure, rather than a potentially lower value, is derived. Some dynamic effects can be seen here, particularly with the cone foot—on occasion the foot even rebounds slightly before proceeding. The flipper type foot was found to be extremely effective in increasing leg thrust, as it penetrated the material easily and presented a broad, high drag profile to the material. It can be seen that after the initial penetration, the flipper adds over 20N of thrust when compared with the flat foot without the flipper. The flipper is, however, not suitable for a conventional hexapod rover with a 3 degree of freedom leg design, unless that design is modified. This is because whilst this manner of construction simplifies the leg design, the kinematics of operation force the foot to rotate in the soil material. The flipper is so effective that the flipper jams in the soil material and
5. Discussion and conclusions The work presented in this paper confirms that analysing the behaviour of a walking rover when interacting with granular material is a complex and challenging process. However, 12
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Figure 14. Parametric tool—main screen.
aggregate leg/soil force. Based on the entered parameters, the algorithm analyses the ground contact phase dividing it into 100 timesteps. The algorithm loops through the timesteps sequentially, predicting the slip, sinkage and thrust at each step and generating a results array which is plotted graphically. Any change to any of the parameters results in an immediate re-computation and re-plot of the profile of predicted thrust covering the whole leg cycle. Permanent data such as soil parameter information is not hard coded but contained in a database read by the Tool; this enables data to be updated easily, or what if? scenarios easily run based on hypothetical parameters. The tool has a graphical user interface which enables parameters such as rover and foot geometry, mass, gait pattern etc to be set via spinboxes; all parameters can be changed at will with the results of the change presented immediately as both a graph and a saveable dataset. In addition to the spinboxes, two key parameters can be set or modified by sliders; these are the magnitude of acceptable lateral slip, and the magnitude of acceptable sinkage. This treatment was chosen as these parameters were found to be both critical to the results, and ones which might be adjusted dynamically, for example if the underlying algorithm were implemented in the vehicle controller. Usage of the tool has highlighted the following interesting conclusions.
despite the complexity, some clear themes emerge which enable behaviours to be categorized and predicted. Firstly, although on the face of it the behaviour of the granular material as it flows round an intruder is quite different from the quasi static pressure sinkage behaviour under the foot, the actual differences may be less than expected. The model of each behaviour type is hydrostatic, with force predictions scaling approximately with immersion depth. The principal difference between the models of each scenario is only the magnitude of the factor α; high α is associated with a quasi static environment, particularly where angular materials are employed, and lower α, with less sensitivity to material properties, where the material is in motion. The apparent distinction between the two behaviour types may therefore be only one of identifying whether the local environment will tend to produce either persistent (quasi static) or constantly reforming force chains. Secondly, despite the complexity, all of the behaviours seen here can be described by simple mathematics, with no need to resort to numerical methods; this is very encouraging in the context of the possible development of real time terrain interaction prediction models. To further investigate the dynamics of the force prediction/sinkage prediction algorithms described here, a parametric tool was developed as part of this study; this enables many possible scenarios of rover geometry, soil type and behaviour to be investigated, leading to some surprising and counter intuitive conclusions. The user interface with the tool can be seen in figure 14. The tool implements an algorithm which incorporates the equations described earlier in this paper in order to predict
(i) The thrust profile is clearly highly dynamic in nature, and so to take full advantage of the thrust available will require intelligent control of leg actuators designed to replicate this profile. 13
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Figure 15. Parametric tool—effect of change in soil type.
(ii) The initial slip of the foot post touchdown generates substantial sinkage; control of the amount of slip, for example with an intelligent torque controller, has a major impact on the thrust available. Thus acceptance of some degree of initial slip is probably essential in order to achieve acceptable thrust levels. (iii) Contrary to expectations, the tool shows that ‘weaker’ materials can generate more thrust, provided overall sinkage levels remain acceptable and the vehicle does not become immobilized. Figure 15 compares thrust profiles for ‘weaker’ ES-3 and ‘stronger’ SSC-2 material, showing that thrust using ES-3 is greater. The reason for this is that sinkage (primarily pressure sinkage) is much greater with ES-3, and whilst the drag experienced at a given sinkage level in the stronger material is greater, this is more than outweighed by the drag resulting from the increased aggregate sinkage. Clearly care will be needed in practice, given the risk that increased sinkage will erode the benefits in other ways, for example by generating drag between the soil and rover body, or by being so large that the legs are no longer able to lift themselves clear of the soil when stepping forwards. For this reason the tool enables a safe sinkage limit to be preset, so that an impractical recommendation is not generated. This plot also confirms that an accurate thrust prediction requires a degree of knowledge of the soil material physical parameters, something that remains a challenge in a planetary exploration context where soil parameter information may be sketchy or non-existent.
slip using an intelligent controller, or through modifications or deployable additions to foot geometry. The prediction model has only so far been validated with an example single leg rather than the full vehicle, and so future work will include implementation of the algorithm that has been developed in a rigid body dynamics simulation of the full rover to verify its accuracy, and ultimately development of a walking vehicle with intelligent actuator control based on this model. Acknowledgments This work was partially funded by the European Space Agency under NPI contract 4000102014/NL/11/PA. Useful discussions with Michel van Winnendael of the European Space Agency and Dr Marcus Matthews of the Department of Civil and Environmental Engineering, University of Surrey, UK, are also gratefully acknowledged. References [1] Colombano S, Kirchner F, Spenneberg D and Hanratty J 2004 Exploration of planetary terrains with a legged robot as a scout adjunct to a rover Space 2004 Conf. on American Institute of Aeronautics and Astronautics (San Diego, California) [2] McKay D S, Heiken G, Basu A, Blanford G, Simon S, Reedy R, French B M and Papike J 1995 The lunar regolith Lunar Sourcebook: A User’s Guide to the Moon ed G Heiken, D Vaniman and B M French (Cambridge: Cambridge University Press) pp 285–356 [3] Bell J 2008 The Martian Surface: Composition, Mineralogy, and Physical Properties (Cambridge: Cambridge University Press) [4] Patel N, Slade R and Clemmet J 2010 The ExoMars rover locomotion subsystem J. Terramech. 47 227–42 [5] Bekker M G 1959 Introduction to Terrain Vehicle Systems, Part 1 - The Terrain and Part 2 - The Vehicle (Ann Arbor, MI: University of Michigan Press)
This paper demonstrates progress towards a reliable terrain interaction prediction model for a walking planetary exploration rover of conventional hexapod design, and has highlighted a number of open problems and unexpected conclusions from the work. It suggests that the interaction might be more susceptible to control than might at first sight appear to be the case, either through active management of 14
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[6] Wong J Y 2008 Theory of Ground Vehicles (New York: Wiley-Interscience) [7] Showstack R 2011 Curiosity on the way to Mars Eos Trans. Am. Geophys. Union 92 455 [8] Albert R, Pfeifer M A, Barab´asi A L and Schiffer P 1999 Slow drag in a granular medium Phys. Rev. Lett. 82 205–8 [9] Schiffer P E 2010 Fun in the sand: some experiments in granular physics http://nanohub.org/resources/8491 [10] Behringer R P, Howell D, Kondic L, Tennakoon S and Veje C 1999 Predictability and granular materials Physica D 133 1–17 [11] Jop P, Forterre Y and Pouliquen O 2006 A constitutive law for dense granular flows Nature 441 727–30 [12] Chehata D, Zenit R and Wassgren C R 2003 Dense granular flow around an immersed cylinder Phys. Fluids 15 1622–30 [13] Scott G P and Saaj C M 2012 The development of a soil trafficability model for legged vehicles on granular soils J. Terramech. 49 133–46 [14] Li C, Zhang T and Goldman D I 2013 A terradynamics of legged locomotion on granular media Science 339 1408–12 [15] Albert I, Tegzes P, Albert R, Sample J G, Barab´asi A L, Vicsek T, Kahng B and Schiffer P 2001 Stick-slip fluctuations in granular drag Phys. Rev. E 64 1–9 [16] Albert I, Tegzes P, Kahng B, Albert R, Sample J G, Pfeifer M, Barab´asi A L, Vicsek T and Schiffer P 2000 Jamming and fluctuations in granular drag Phys. Rev. Lett. 84 5122–5 [17] Hill G, Yeung S and Koehler S A 2005 Scaling vertical drag forces in granular media Europhys. Lett. 72 137 [18] Yoo Y-H and Quack L 2012 Towards a real-time capable realistic soil contact model using a particle system IMSD-12: 2nd Joint Int. Conf. on Multibody System Dynamics [19] Roemmermann M, Ahmed M, Quack L and Kassahun Y 2011 Modeling of leg soil interaction using genetic algorithms ISTVS’11: 17th Int. Conf. of the Int. Society for Terrain-Vehicle Systems [20] Yeomans B, Saaj C M and van Winnendael M 2013 Walking planetary rovers—experimental analysis and modelling of leg thrust in loose granular soils J. Terramech. 50 107–20 [21] Cordes F, Roehr T M and Kirchner F 2012 RIMRES: a modular reconfigurable heterogeneous multi-robot exploration system iSAIRAS’12: Proc. 11th Int. Symp. on Artificial Intelligence, Robotics and Automation in Space [22] Terzaghi K 1943 Theoretical Soil Mechanics 3rd edn (London: Wiley) [23] McKyes E 1985 Soil Cutting and Tillage (Amsterdam: Elsevier) [24] Coulomb C A 1776 Essai sur une application des r`egles de maximis & minimis a` quelques probl`emes de statique, relatifs a` l’architecture (Paris: De l’Imprimerie Royale)
[25] Coppersmith S N, Liu C, Majumdar S, Narayan O and Witten T A 1996 Model for force fluctuations in bead packs Phys. Rev. E 53 4673–85 [26] Albert I, Sample J G, Morss A J, Rajagopalan S, Barab´asi A-L and Schiffer P 2001 Granular drag on a discrete object: shape effects on jamming Phys. Rev. E 64 061303 [27] Gray J and Hancock G J 1955 The propulsion of sea-urchin spermatozoa J. Exp. Biol. 32 802–14 [28] Robotis Co. Ltd 2013 Robotis dynamixel range (www.robotis.com/xe/dynamixel_en) [29] Leaflabs LLC. Maple Board, Leaflabs 2012 Robotis dynamixel range (http://leaflabs.com/devices/maple/) [30] Johnson G W and Jennings R 2001 LabVIEW Graphical Programming: Practical Applications in Instrumentation and Control (New York: McGraw-Hill) [31] Bradski G 2000 The OpenCV Library Dr. Dobb’s Journal of Software Tools [32] Yeomans B, Saaj C M and van Winnedael M 2013 Modelling leg/terrain interaction for a legged planetary micro-rover ASTRA’13: 13th ESA Workshop on Advanced Space Technologies for Robotics and Automation [33] Gouache T P, Patel N, Brunskill C, Scott G P, Saaj C M, Matthews M and Cui L 2011 Soil simulant sourcing for the ExoMars rover testbed Planet. Space Sci. 59 779–87 [34] Brunskill C, Patel N, Gouache T P, Scott G P, Saaj C M, Matthews M and Cui L 2011 Characterisation of Martian Soil Simulants for the ExoMars rover testbed J. Terramech. 48 419–38 [35] Carrier W D, Olhoeft G R and Mendell W 1995 Properties of the lunar surface: section 9.1.1—particle size distribution Lunar Sourcebook: A User’s Guide to the Moon ed G Heiken, D Vaniman and B M French (Cambridge: Cambridge University Press) pp 477–8 [36] Janosi Z and Hanamoto B 1961 The analytical determination of drawbar pull as a function of slip for tracked vehicles in deformable soils ISTVS 1st Int. Conf. on Mechanics of Soil-Vehicle Systems (Turin, Italy) [37] ASTM Standard 2004 Standard test method for direct shear test of soils under consolidated drained conditions ASTM Standard D3080-04 [38] Godbole R and Alcock R 1995 A device for the in situ determination of soil deformation modulus J. Terramech. 32 199–204 [39] Brzinski T A, Mayor P and Durian D J 2013 Depth-dependent resistance of granular media to vertical penetration Phys. Rev. Lett. 111 168002 [40] Ting J M, Meachum L and Rowell J D 1995 Effect of particle shape on the strength and deformation mechanisms of ellipse-shaped granular assemblages Eng. Comput. 12 99–108 [41] Mazouchova N, Umbanhowar P B and Goldman D I 2013 Flipper-driven terrestrial locomotion of a sea turtle-inspired robot Bioinspir. Biomim. 8 026007
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