Scaling Hard Vertical Surfaces with CompliantMicrospine ArraysAlan T. Asbeck, Sangbae Kim,and M.R. CutkoskySchool of EngineeringStanford UniversityStanford, California 94305–2232Email: aasbeck, sangbae, email@example.comWilliam R. ProvancherDepartment of Mechanical EngineeringUniversity of UtahSalt Lake City, Utah 84112-9208Email: firstname.lastname@example.orgMichele LanzettaUniversity of PisaPisa, ItalyEmail: email@example.comAbstract— A new approach for climbing hard vertical surfaceshas been developed that allows a robot to scale concrete, stucco,brick and masonry walls without using suction or adhesives.The approach is inspired by the mechanisms observed in someclimbing insects and spiders and involves arrays of microspinesthat catch on surface asperities. The arrays are located on thetoes of the robot and consist of a tuned, multi-link compliantsuspension. In this paper we discuss the fundamental issues ofspine allometric scaling versus surface roughness and modelthe interaction between spines and surfaces. We also discusstoe suspension properties needed to maximize the probabilitythat each spine will find a useable surface irregularity and todistribute climbing loads among many spines. The principles aredemonstrated with a new climbing robot, SpinybotII, that canscale a wide range of flat exterior walls, carry a payload equalto its own weight, and cling without consuming power. We alsodiscuss how toe parameters scale with robot mass and how spineshave also been used successfully on the larger RiSE robot.I. INTRODUCTIONIn recent years, there has been considerable progress insmall, legged robots that can run rapidly and stably overrough terrain , , , . Climbing and maneuveringon vertical surfaces presents a more difficult challenge, onethat robots are just beginning to address. Small robots thatcan climb a variety of hard and soft surfaces unobtrusivelyand cling for extended periods of time without high powerconsumption would be beneficial for applications such assurveillance or the inspection of hard-to-reach locations.Previously developed climbing robots have generally employedsuction cups , , , magnets ,  oradhesives , ,  to cling to smooth vertical surfacessuch as windows and interior walls. None of these approachesis suitable for porous or dusty exterior surfaces such as brick,concrete, stucco or stone. Other climbing robots employ handand foot holds in the manner of a human climber , .A recent innovation employing a controlled vortex  tocreate negative aerodynamic lift and force the robot againsta surface has been demonstrated on brick and concrete wallswith considerable success. However, this approach consumesThis work is an extension of two papers presented at conferences, see ,.significant power (including when the robot is stationary),unavoidably generates noise and is more difficult to adapt tonon-flat surfaces such as window ledges and corners.When we look at animals that exhibit agility on verticalsurfaces, we find a variety of methods employed . Largeranimals such as cats and raccoons have strong claws that canpenetrate wood and bark surfaces. Tree frogs and many insectsemploy sticky pads , . Geckos and some spiders utilizelarge numbers of very fine hairs that achieve adhesion viavan der Waals forces on almost any kind of surface , ,. Other insects, arthropods and reptiles employ small spinesthat catch on fine asperities . All of these approachesare worthy of examination for climbing robots. However, dryadhesives and spines are particularly attractive for hard, dusty,exterior surfaces.Several researchers are currently working on creating syntheticversions of the setae found on geckos or the scopulaeseen on spiders , , . The early results are intriguingbut current synthetic adhesives are not able to sustain thetensile loads needed at the forelimbs of a climbing robot.Moreover, they are fragile and lack the self-cleaning propertythat allows geckos to climb dusty walls.In nature, spines are found on a variety of animals. Insectsand arthropods that climb well on vertical surfaces often havelegs equipped with large numbers of small, sharp spines. At alarger scale, geckos that frequent rock surfaces such as cliffsand caves have small claws on each toe in addition to theirdry adhesive structures . Unlike the larger claws of a cat,spines do not need to penetrate surfaces. Instead, they engagesmall surface asperities. Several studies in the biology literaturehave considered the problem of spine/surface interaction.Dai et al.  present a planar model of spine/asperity contactand compute the maximum load per spine as a function ofspine strength, relative size of the spine tip versus that of anasperity, and coefficient of friction. As expected, for roughsurfaces the mechanical strengths of the spine and asperitybecome the limiting factors; for smoother surfaces friction ismore important, and the ability to pull along the surface ismuch reduced.In the following sections, we examine the issues involvedwith using spines to adhere to surfaces. First, in Section II
Fig. 1. Profile photograph of typical spine, with shaft dia., d = 270µm andtip radius r s = 10µm, engaging a profile of a rough concrete surface.we analyze how spines interact with surfaces. We address thefollowing questions: Where can spines of a certain size perchon a given surface? How does the number of perching-spotsvary with spine size? In Section III we examine the strength ofthe spine/asperity contacts. The analysis in Sections II and IIIlends insight into the design considerations of climbing withspines: In order to attach a robot of a given mass to a givensurface, how small do the spines need to be and how many ofthose spines are needed?In Section IV we discuss foot and leg design considerationsfor making spines attach to surfaces. In Section V we presentthe design of a climbing robot, SpinybotII, that uses spines toclimb to a variety of wall surfaces. In particular, we discuss thefeatures of SpinybotII that facilitate load sharing among manyspines. Finally, in Section VI we discuss how toe complianceparameters scale with robot mass and how spines have beenadapted to the much heavier RiSE  robot.II. SPINE/SURFACE MODELINGNumerical simulations were performed to model the behaviorof spines interacting with surfaces. In general, thesimulations fitted 2-D spine cross-sections to 2-D surfaceprofiles and determined the locations where the simulatedspine could perch on the profile. This analysis gives insightinto what size spines are needed to climb various surfacesand the properties of a surface that make it climbable. Thisis a geometric analysis, indicating trends in the number ofasperities (perchable regions) per unit length of surface. Spineand surface strength are also important considerations and areaddressed in Section III and Appendix II.Two-dimensional profiles of several stone and sandpapersurfaces were obtained using a profilometer (Taylor HobsonTalysurf Series 2), with a conical stylus (15 ◦ cone angle)terminating in a spherical tip of radius 2 µm. Profiles ofrougher concrete surfaces were measured using a laser micrometer(Omron Z4M-N30V), since the profilometer couldnot measure very rough surfaces. The measured profiles were5 cm long. The profilometer had a vertical resolution 0.426 µmFig. 2. Diagram of the spine/surface interaction model. The spine is modeledby a circle of radius r s approaching in a direction indicated by the approachvector, which creates a swept volume. The intersection of the swept volumeand the original profile is found and the center of the spine tip circle isrecorded to create a traced surface. Contiguous shelf-like regions, startingwhen θ, the normal vector to the traced surface, is larger than some criticalangle θ min , are available for sustaining loads.and samples were taken every 1 µm; the laser micrometer hada spot size 64 µm, a vertical resolution of 0.977 µm, andsamples were taken every 2.4 µm.In analyzing these profiles we consider a typical spine asshown in the photograph in Figure 1. The spine is a gentlycurved beam with a circular cross section that tapers to arounded tip of radius r s . The spine is attached to a robot limbthat drives it toward the surface and subsequently strokes itdownward along the surface, hoping to engage an asperity. Inthe simulation, we model the spine tip as a circle of radius r sapproaching the surface at an angle θ a .In Figure 2, a diagram of our model, a circle of radiusr s has approached the surface in a direction indicated by theapproach vector. The surface shown in the figure is an actualprofile of a rough-cut granite surface.
We create a spine swept volume from the spine tip circle,bounded at the upper edge by the angle θ a and at the loweredge by a ray perpendicular to the surface face. A tracedsurface is formed by moving the spine swept volume over theprofile and recording the position of the center of the spinetip circle. We assume that the details of the spine, includinga conical spine end and the spine curvature, fall within theswept volume. Thus, the spine swept volume accounts for thefact that an actual spine is not be able to reach all parts of thesurface (e.g. the bottoms of deep pits) if it moves towards thesurface at an angle or if it is oriented at an angle as it slidesover the surface.To find regions of the surface that a spine could perch on(i.e., asperities), we proceed along the traced surface frombottom to top in Figure 2. We search for locations in whichthe angle θ of the normal vector to the traced surface is above aminimum useable angle, θ min , which depends on the angle atwhich the spines are loaded, θ load , and coefficient of friction,µ, as seen in Equation 1:TABLE ITABLE OF SURFACES AND ROUGHNESS PARAMETERSSurf. Surface Average RMSNo. Roughness R a Roughness R q(µm)(µm)1 cobblestone 56.9 78.12 machined granite 6.6 10.33 rough cut granite 42.7 56.14 polished granite 13.2 21.05 paving stone 73.0 92.26 concrete cinderblock 93.0 131.97 concrete 2 70.1 88.48 Al-oxide 80 42.1 57.09 Al-oxide 100 35.8 49.910 Al-oxide 120 20.3 26.011 Al-oxide 150 21.7 27.812 Painter’s 100 30.5 38.9θ min = θ load + arccot(µ). (1)We consider regions between these locations and the subsequentminimum in the traced surface to be “useable asperities,”as shown in Figure 2. In essence, we find “shelves” thatare terminated by an angle θ min near regions of maximumprojection from the wall surface. We assume that the spineslides quasi-statically along the surface from the top down. Ifthe spine tip falls into any of the “useable asperity” regions,it will continue to move along the surface until it reachesthe point at which the normal vector to the traced surfaceis at the angle θ min , and at that location the spine will stopand catch on the surface. The spine can then be loaded awayfrom the wall (i.e., used to generate adhesion) at an angle upto θ load before it begins to slip off the asperity. For a givenRoot-Mean-Square (RMS) surface roughness, R q , the useableasperity regions can vary greatly, depending on the details ofthe profile. This purely geometric analysis does not accountfor spine bouncing or other dynamic effects; however, asdiscussed in the next section, it provides useful predictions ofthe comparative ease with which surfaces of varying roughnessand asperity sharpness can be climbed.We use the number of asperities per centimeter as a metricrather than other measures such as the effective length ofthe asperities because the former is a better indication of theprobability that a spine will encounter an asperity during astroke of a given length. (For example, a single long asperitywould be equivalent to many short asperities in terms ofeffective asperity length.) A more accurate assessment wouldinclude complete information about the probability distributionof asperities and asperity lengths on the wall.A. Simulation resultsProfiles were taken for several stone surfaces and severaltypes of sandpaper, listed in Table I. The table also showsthe linear and RMS roughness measures R a and R q (seeFig. 3. Number of asperities per centimeter versus spine tip radius for varioussurfaces, with θ a = 45 ◦ and θ min = 75 ◦ . The numbers in the legendcorrespond to the surfaces in Table I. Solid lines indicate stone and concretesurfaces, and dashed lines indicate sandpaper surfaces. A band correspondingto the 1/r trend that would be expected for a perfect fractal surface has beenadded for reference. The surfaces not plotted had curves very similar to thoseshown.Appendix III) for these surfaces. Figure 3 shows typical resultsfor the number of asperities per centimeter for selected stoneand sandpaper surfaces, with constant values of θ min and θ a .For the stone surfaces (solid lines) the number of asperitiesper centimeter decreases at a rate close to 1/r for smallertip radii. This is to be expected, since many surfaces areapproximately fractal , , , so the surface propertiesshould be similar at different length scales. Above a certaintip radius, however, the number of available asperities dropsrapidly, implying that the surfaces do not behave fractally atlarge length scales. This is an effect of the cutting, polishing
Fig. 4. Number of asperities per centimeter versus spine tip radius for severalvalues of θ min , for a stone surface (values of θ min are marked on the graph).θ a is held constant at 65 ◦ . A band corresponding to the 1/r curve expectedfor a perfect fractal surface is shown for comparison.Fig. 5. Number of asperities per centimeter versus θ min for various surfaces,for a constant spine tip radius of 30µm. θ a is held constant at 45 ◦ . Thenumbers in the legend correspond to the surfaces in Table I. The numberof asperities/cm can be seen to drop drastically as θ min approaches 90 ◦ ,showing that there are many fewer ledge-like regions that are perpendicularto the surface face than sloped regions more parallel to the surface face.and wearing processes that have partially smoothed the stonesurfaces so that large asperities are rare.The distribution of lengths between asperities is approximatelydescribed by an exponential random variable, withprobability density functionf X (x; λ) = λ exp(−λx), x ≥ 0, (2)where x is the distance between asperities and λ is the numberof asperities per centimeter. The mean distance betweenasperities is 1/λ and the variance is 1/λ 2 . The asperity lengthsare also approximately exponential random variables.In Figure 3, the curves for the sandpaper surfaces have ashallow slope of less than 1/r for small tip radii. This occursbecause the sandpaper surfaces have a relatively uniformparticle size, so the number of asperities is closer to beinga constant for small tip radii.Figure 4 shows the number of asperities per centimeter versusspine tip radius for several values of θ min and constant θ a .As θ min increases, corresponding to the need for asperities thatare more shelf-like, the simulation is less likely to find useableasperities, as expected. This corresponds to a downward shiftof the lines in the figure. All of the lines can be seen to followthe expected 1/r trend.Figure 5 shows the number of asperities per centimeterversus θ min and a constant spine tip radius. There are manyasperities for low values of θ min , which corresponds to regionsof the surface that are nearly vertical. Making use of theseasperities would require extremely high friction between thespines and surface as well as the spines being loaded nearlyparallel to the surface. There are relatively few asperities forhigh values of θ min , which correspond to shelf-like regions.If θ a is varied while holding θ min constant, the number ofasperities/cm changes relatively little, decreasing significantlyonly for large approach angles where the spine is nearlyparallel to the wall. For small approach angles, the spine’sability to reach the regions of the surface it hooks on to isonly slightly affected. In our analysis, we examined θ a =0 ◦ , 45 ◦ , 65 ◦ , and 80 ◦ . There was essentially no differencebetween any of the approach angles for 65 ◦ or less, whilethe 80 ◦ case showed a reduction in the number of asperities.B. Correlation with climbing robot performanceIn this section we compare the results of the foregoinganalysis to the empirical results obtained with our climbingrobot. The parameters θ min and θ a used by our climbingrobot, SpinybotII (see Figure 6), were measured and used inthe simulation. Based on the geometry of SpinybotII, the feetare loaded at angles 3.5 ◦ < θ load < 8 ◦ degrees from the wall.The coefficient of friction between stainless steel spine tips androck is generally between 0.15 and 0.25, which correspondsto θ min between 86.5 ◦ and 81 ◦ for an average θ load of 5 ◦using Equation 1. The approach angle, θ a , is from 45 ◦ -65 ◦ ,based on the angle of the spines themselves and the motionof the tip over the surface. The tip radii, r s , of the spines are10-15 µm for new spines and 25-35 µm for spines dulled dueto heavy use.The results using these values qualitatively correspond tothe actual performance of SpinybotII on the correspondingsurfaces. SpinybotII can adhere extremely well to the Aloxide100 and 80 grit sandpapers, indicating that there is ahigh density of asperities with ledge angles of at least 81-86.5 ◦ . SpinybotII adhere less well to the Al-oxide 150 and120 grit sandpapers, with the feet slipping down these surfaces
for a longer distance before solidly engaging, indicating alower asperity density. It adheres poorly to the painter’s 100grit sandpaper, only rarely finding asperities. On the naturalsurfaces, SpinybotII adheres very well to the rough concretes,moderately well (i.e., barely well enough to climb) to therough cut granite surface, and not at all to the ground andpolished surfaces. In the simulation, if θ min is 82 ◦ -85 ◦ andθ a is 45 ◦ or 65 ◦ , at radii of 10-40 µm the ordering of thesurfaces in terms of asperities/cm corresponds to the orderingof the same surfaces in terms of how well SpinybotII canadhere to them.As discussed in the next section, SpinybotII’s spines donot scrape quasi-statically over surfaces. However, the effectsof spine bouncing and reduced dynamic friction are mainlya reduction in the apparent asperity density. Also, asperityfailure was not frequently observed for light loads on thesesurfaces. Thus, the relative ordering of how well SpinybotII’sfeet perform on different surfaces generally matches the predictedordering in terms of asperities per unit length in thesimulation.The concrete profiles show somewhat fewer asperities thanwould be expected for small tip radii and very large valuesof θ min . This is because they were recorded using a lasermicrometer with a 64 µm spot size that produced some lowpassfiltering. Consequently the concrete profiles appear tohave few asperities in the simulation using these parameters,although in actuality they have many asperities. Also, all of thesurfaces show many fewer asperities than would be expected ifθ min is increased above around 82 ◦ . This discrepancy is likelydue to the reduced ability of the measurement instruments toaccurately record abrupt changes in surface height.C. DiscussionDai et al.  present a planar model of spine/asperitycontact that includes the relative size of the spine tip versusthat of an asperity and coefficient of friction. Our analysisis similar, except that we use actual surface contours andcompute the corresponding traced surface for the swept volumeproduced by a spine. Dai et al. state that the angleof spine/surface contact is key to obtaining traction betweenbeetle claws and surfaces. However, they also discuss theirresults in terms of surface roughness, a different parameterentirely (though it is usually correlated with the spine/surfacecontact angles). Our analysis indicates that the normal angle,θ, is the critical parameter, and that the linear and RMSroughness measures R a and R q are not always correlated withactual spine performance. As an example, the R a and R q of thepainter’s 100 grit sandpaper are larger than the correspondingvalues for 120 and 150 grit Al-oxide, but it has many feweruseable asperities for large assumed values of θ min , due to itsmore rounded features.Our present analysis is limited in that it cannot detect thepresence of angles more than 90 degrees (upward-leaningledges). Even at angles close to 90 degrees it is less accuratedue to the non-zero cone angle of the profilometer stylus.Fig. 6. Picture of upper section of SpinybotII on concrete wall and detailedview of several spines independently engaging asperities on the concretesurface.For some surfaces, SpinybotII’s spines may attach to manyasperities with angles greater than 90 degrees.The observed behavior of spines interacting with surfacesalso differs slightly from that assumed by the model. As afoot is brought into contact with a surface and begins itsdownward stroke, we observe that many spines briefly catch on“pseudo-asperities” and then break away as the load is rampedup. There are several reasons why the spines probably detachfrom these “pseudo-asperities.” The spines can deflect slightlyor undergo rotations, which will lead to their slipping. Thiscan be due to non-idealities in the toe linkage design as wellas spine bending. Another possibility is that these “pseudoasperities”are actually valid asperities, but are quite weak andbreak off the wall once the load increases past a certain point(see Appendix II for analysis). Indeed, on some surfaces smallparticles can be observed breaking off the wall as a foot slidesover it.One final possibility is that the “pseudo-asperities” areregions where there is a ledge with a very shallow angle. Inthese regions, initially a spine can get stuck if there is a smallbut positive normal force being applied to the spine and asmall shear force. However, as the shear force is increased(and normal force possibly decreased), eventually the frictionforce between the spine and the ledge is not large enough to
overcome the applied load, and the spine slips. These “pseudoasperities”would be useable if the friction force between thespine and surface was higher or the loading angle was keptmore towards the wall (or into the wall).The spines also tend to skip over the surface (i.e., to becomebriefly airborne) after slipping off the “pseudo-asperities”and they undergo alternating regimes of static and dynamicfriction. The effective coefficient of friction while this occursis probably quite low. Finally, there is a chance that as aspine on a compliant suspension drags down the wall itmay tend to follow a local groove or valley and thereby besteered away from protruding asperities. Conversely, negativeasperities (pits) will tend to steer the spine into a favorablelocation for obtaining a grip. Hence, actual spine/surfacedynamics vary depending on the surface type as well as howmuch the spine is able to move in the direction perpendicularto its travel.It was mentioned earlier that the model does not takeasperity strength into account. However, the trend of thenumber of asperities/cm varying as 1/r still should hold true,even if weak asperities are excluded. The strength of theasperities will depend on the material properties as well assurface geometry (see Appendix II).III. SPINE/ASPERITY CONTACT STRENGTHWhile smaller spines are more effective at engaging asperitieson smooth surfaces, they also carry smaller loads. Whensteel spines catch on asperities, the contact typically fails inone of three ways:• the spine fails plastically at its base due to tensile stressfrom bending,• the spine tip rotates elastically such that it slips off theasperity,• the asperity fails, typically as a particle becomes unbondedfrom the surrounding matrix.In each of these cases, if we take a dimension such as thespine tip radius, r s , as a characteristic length and scale everythinguniformly, then the maximum load of the spine/asperitycontact increases as r 2 s (see Appendix II for details).Figure 7 shows graphically how the maximum load ofthe spine/asperity contact increases as r 2 , while the expectednumber of asperities per unit area decreases as 1/r 2 . As spinesbecome smaller we can ascend smoother surfaces because thedensity of useable spine/asperity contacts increases rapidly.However, to support a given load we need proportionally largernumbers of spines because each contact can sustain less force.Thus, in order to make use of large numbers of spines, the twokey design principles are:• ensure that as many spines as possible will independentlyattach to asperities,• ensure the total load is distributed among the spines asuniformly as possible.Figure 7 also shows data that supports the r 2 and 1/r 2trends for spine/asperity contact strength and asperity densityper unit area, respectively. Measurements of contact strengthFig. 7. Log/log plot showing the expected r 2 trend of spine/asperity contactstrength versus the expected 1/r 2 trend for asperities per unit area of thewall. The number of asperities per unit area for surfaces of rough cut graniteand concrete are plotted for comparison with the expected asperity densitytrend. Individual tests of spine/asperity failures are plotted for concrete andsandpaper surfaces, showing the force required to break the spine/asperitycontact. Failures occurred either through asperity brittle failure or spinebending (either elastically or plastically).were done using spines of various sizes on concrete and sandpapersamples attached to a load cell. We note that the sandpaperconsisted primarily of male features (which had a smallbonding cross-section). Therefore, asperity failure tended tooccur before spine failure on that surface. In contrast, the castconcrete primarily consisted of female features that were muchstronger; so spine failure, either by excessive end rotationor plastic bending, tended to dominate. For other surfaces,all three failure modes tended to occur simultaneously. Thefigure also shows the asperity density data for a concrete andmachined granite surface as compared to the expected 1/r 2 forfractal surfaces. At the bottom of the figure, the representativeasperity length scales for a few different surfaces are indicated.These indicate, for example, that rough concrete surfaces willpresent useable asperities of up to 300 µm in radius whereassmooth concrete or stone panels will present asperities of upto 20 µm. These values essentially impose an upper limit tothe spine size that can be used with these surfaces.For our first climbing robot, SpinybotI, we employed 4spines per foot, each with a tip radius of approximately 30-50 µm. This machine was able to climb stucco and roughconcrete reliably. The spine/asperity contacts could sustainloads of several Newtons (N), usually limited by brittle failureof the asperity rather than of the spine. However, for surfacessuch as smooth concrete and dressed stone, the probabilityof a spine encountering a useful asperity during a verticalstroke length of approximately 2 cm was too low for reliableclimbing. SpinybotII employs two rows of spines on each foot,
Fig. 8. Photograph and equivalent elastic linkages for one toe of SpinybotII.Linkage at left shows the deflected position for a 40g load, superimposed onthe undeflected position (shown in dotted lines). Key to labels: 1. 200 µmshaft diameter spines (inside dotted circles), 2. tendon for applying loads, 3.soft urethane flexure permitting travel in y direction, 4. buckling flexures withlow stiffness in the −x direction under compression and higher stiffness undertension, 5. primarily rotational flexure for the proximal spine, 6. buckling/liftoffflexure for proximal spine, 7. hard member used to attach toe to tendonassembly.each spine having a tip radius of approximately 15-20 µm.The maximum force per spine/asperity contact is 1-2 N, andthe probability of finding useable asperities per centimeter ofstroke length is high. On most surfaces that SpinybotII canclimb, 30-40% of the spines engage.The design of feet that enable spines to effectively attachto asperities and share the load is described in Section IV. Inaddition, as with any climbing robot, it is important to keepthe center of gravity as close to the wall as possible and toavoid imposing any forces or moments at the feet that couldlead to premature detachment. The features of SpinybotII thatachieve these effects are described in Section V.IV. TOE AND FOOT DESIGN: PROMOTING ATTACHMENTAND LOAD SHARINGThe feet on SpinybotII represent the sixth generation ofa compliant, spined design. Each foot contains a set of tenidentical planar mechanisms, or “toes,” as can be seen inFigure 8.In discussing the toe and foot design, we assume that thefoot undergoes a general motion to engage the spines on thewall. In particular, we assume the foot is presented to the wallin some orientation and is pressed against the wall lightly. Thefoot is then pulled downwards along the wall. In this manner,the spines undergo a motion similar to that assumed for anindividual spine in the analysis in Section II. The spines attachto the wall at some point while the foot is being dragged downthe wall. The normal force must then decrease and eventuallybecome negative for the spines to generate adhesion. Duringthis process, the foot maintains its original orientation withrespect to the wall.Each of the toes is a compliant multi-bar linkage, independentof its neighbors and able to stretch parallel to the wallunder a load. This feature of stretching along the wall undera load is key to enabling multiple spines per foot to attach: ifa toe catches an asperity, neighboring toes are not preventedfrom catching their own asperities because they will continueto slide down the wall as the caught toe stretches. Figure 6shows a side view of the robot on a concrete wall and a detailof a single foot, showing several of the planar toe mechanisms,each of which bears two spines (several of which are visiblyengaged). The manner in which the spines stretch along thewall independently can also be seen in Figure 9 in a top viewof a foot.The mechanisms are created using a rapid prototyping process,Shape Deposition Manufacturing ,  that permitshard and soft materials to be combined into a single structure.In the present case, the white and grey materials are hardand soft urethanes, of 75 Shore-D and 20 Shore-A hardness,respectively (Innovative Polymers Inc.). The resulting structurecan be approximated as an elastic multi-link mechanism, asshown in Fig. 8.The spines are approximately 1.5 mm long with a 200 µmshaft diameter and 10-35 µm tip radius. They are embeddeddirectly into the hard white links during the SDM process. Thesoft urethane flexures provide both elasticity and viscoelasticdamping. They permit greater extensions without failure thanminiature steel springs (as were used on some of the earlierfoot designs).For small deflections, the linear and rotational stiffness ofeach spine in the (x, y) plane can be modeled using a 3x3stiffness matrix, K, taken with respect to a coordinate systemembedded in the spine (see Figure 8 for a diagram of thecoordinate system):⎡⎣ k ⎤xx k xy k xθk xy k yy k yθ⎦k xθ k yθ k θθFor a spine to appropriately move along the wall andeffectively engage asperities, this K matrix should have certainproperties. At initial contact, k xx should be very small fordisplacements in the −x direction, so that a large numberof toes can conform to uneven surfaces without requiring asignificant normal force. In SpinybotII’s toes, this is accomplishedthrough the flexures at the end of each toe (labeled
TABLE IIyztip trajectoryyx[ Outer Spine:16 −8 0.30−8 73 −0.280.30 −0.28 90SPINYBOTII TOE STIFFNESS MATRICES] [ Inner Spine:14 −2 0.25−2 16 −0.010.25 −0.01 58][ Units:N/m N/m N/radN/m N/m N/radN/rad N/rad Nm/rad]] [ K Matrix:kxx k xy k xθk xθ k yθ k θθk xy k yy k yθTABLE IIISTIFFNESSES AND DAMPING PARAMETERS FOR TOE LINKAGE2.3.1.Fig. 9. Side and plan view of one foot containing ten toes. The toes candeflect independently of each other. In addition, the entire foot can displacein the distal (y) direction due to an un-actuated prismatic joint (1.). Thetrajectory of the foot consists of detachment, an upward (+y) motion followedby a lift-off motion (−x), and attachment, consisting of touchdown (+x) anda downward pull (−y). The sequence of motions is accomplished using anunder-actuated mechanism consisting of a single rotary RC servo motor (2.)and an elastic band (3.) that is taut while the foot is disengaged and becomesloose as the foot engages. See Figure 10 for further details about the footactuation and trajectory.4. in Fig. 8), which are designed to buckle so that they havea very low stiffness for −x deflections. It is best to apply assmall of a normal force as possible while engaging the spinesbecause this will minimize the number of spines that attach toundesirable “pseudo-asperities”, as discussed in Section II.If there are small tensile loads on the foot (in the +xdirection), some toes will remain compressed from the foot’sengaging motions; k xx should again be small so these compressedtoes do not push the foot away from the wall. Thisconcern presents a design compromise since a larger k xxwill bring the spines back to the wall more quickly if theyshould slip and and bounce as described earlier. Finally, formoderate tensile loads such as are encountered when the footis disengaging, k xx should be large so the toes can disengageeffectively. If k xx is too small, spines can occasionally jamin deep pits on a surface as the foot moves away from thespine during disengagement. Jamming is prevented by pullingparallel to the spine axis with moderate tension. On SpinybotIIthis is also accomplished with the flexures at the end of thetoe.In the orthogonal direction, k yy should be moderate, asit represents a different trade-off. A softer k yy allows eachLocation(numberedlabel,Fig. 8)Parameter in kinematic modelk = linear stiffness elementc = linear damping elementk t = rotational stiffness element3. k = 60 N/mc = 0.1 Ns/mk t =0.005 Nm4. k = 90 N/m in tensionk = 5 N/m in compressionc = 0.02 Ns/m5. k =100 N/mc = 0.001 Ns/mk t = 0.001 Nm6. k = 60 N/mc = 0.1 Ns/mtoe to stretch more in the longitudinal direction to increasethe probability that each spine will catch an asperity duringthe downward stroke of the foot; but if k yy is too soft, themechanism will require an excessive stroke length to supporta given load. In essence, these factors determine the “asperitysearch length” for the downward stroke. At the same time,k xy should be small so that stretching in the y directiondoes not cause the spines to retract. The k xθ and k yθ termsshould also be small and, preferably, slightly negative so thatdisplacements in the x or y direction are not accompanied byanticlockwise rotations in the (x, y) plane that would lead topremature disengagement.The toes on SpinybotII were designed to conform to theserequirements for effective spine attachment. Their actual behaviorwas confirmed by measurements. First, the stiffnessmatrix K was measured on a toe, for both the outer and innerspines, relative to the hard member of the toe that attachedto the rest of the foot (7. in Figure 8). The stiffnesses weremeasured around an operation point of (-0.13 cm, 0.13 cm)in the (x,y) directions. The results are shown in Table II, andthey can be seen to generally correspond to the desired valuesas discussed previously.The toe mechanism shown in Fig. 8 was also modeledusing Working Model T M software (MSC Inc.), and the variouslinear and rotational stiffness elements were adjusted to matchbench-top test results of SpinybotII toes. The results are
ContactyxFully Engaged(Free Hang)DisengagementTension onElastic bandNo tensionFree pivotJoint limitFig. 10. Diagram of how the feet engage and disengage on SpinybotII. Duringengagement (left), the elastic band at the back of the leg unit is in tension.Since the foot is attached to the servo arm through a pin joint (labeled “freepivot” in the middle figure), the tension on the elastic band presses the footagainst the wall as the servo rotates downward. The elastic band graduallyloses its tension based on the geometry of the leg, such that when the foot isfully engaged there is no tension on the elastic band (middle). This allows therobot to hang freely from the foot and prevents forces that would push therobot body away from the wall. In disengagement (right), the servo reversesits direction of motion, reversing the trajectory of the foot. Eventually a hardstop on the foot hits the servo arm, causing the foot to lift off the wall.summarized in Table III. The mechanism is designed so thatinitial contact at the inner, or proximal, spine actually forcesthe distal spine slightly outward (+x direction) to increase theprobability that it will also contact an asperity.Once one or both spines have contacted the wall, the toe canapply a force that is mainly vertical, with a small inward (+x)component to help the robot climb. Fig. 8 shows the effect ofa typical 40 gram load sustained by one toe in climbing.V. BODY DESIGN: PROMOTING LOAD SHARING ANDSTABILITYAn important observation of agile scansorial animals likegeckos is that they employ multi-level conformability (e.g.lamellae, toes, and limbs) and redundancy (multiple pads pertoe, multiple toes per foot and multiple feet in contact) forreliable climbing. The same principles have been found necessaryfor SpinybotII. Accordingly, the entire foot mechanismis mounted on a prismatic joint with an elastic suspension thatallows it to move up to 1 cm in the distal (+y) direction (1. inFig. 9). In addition, the entire foot assembly is spring loadedby a second elastic element behind the pivot (3. in Fig. 9),where it is connected to a rotary RC servo motor (2. in Fig. 9).The result is an under-actuated R-R-P serial kinematic chainthat traces a loop trajectory, as shown in Fig. 9, when the servoFig. 11. Photograph of SpinybotII wall and diagram of climbing mechanism.Each set of three legs is attached to a mechanism that allows the robot to“ratchet” its way up the wall with an alternating tripod gait. A long tail helpsto reduce the pitching moment. The center of mass (COM) is within thepolygon of contacts, to minimize yawing rotations in the plane of the wall.motor rotates back and forth. Further explanation of the legmotion can be seen in Figure 10. After some experimentation,the best elastic elements were found to be 6.4mm diameterelastic bands commonly used for dental braces.This leg construction effectively moves the foot over thewall in a desirable manner similar to the “general motion”described earlier. The foot maintains its orientation parallel tothe wall during the entire time it is touching the wall, and thepositive normal force provided by the rear elastic element (3.in Fig. 9) enables the spines to engage early in the stroke downthe wall. The normal compliance of the foot provided by thiselastic element also enables the foot to move over bumps orirregularities on the wall while maintaining contact. When theleg has completed its downward motion, the elastic elementgoes slack, allowing the loading angle of the foot to be asclose to the wall as possible, since the robot is then hangingfreely from the foot by a pivot on the servo arm (see Figure10).The robot utilizes an alternating tripod gait, similar tothat found in climbing insects (see Figure 11). At any time,the robot is nominally clinging by three feet. Like manyclimbing animals, the robot also has a tail which reduces theforces required at the front limbs to overcome body pitchbackfrom the wall. This pitch-back moment is producedby gravity acting at the center of mass, which is locatedapproximately 2 cm outward from the wall. The weight of therobot, including lithium polymer batteries, wireless camera,and PIC microprocessor is 0.4 Kg. It can carry an additionalpayload of 0.4 Kg while climbing.
Fig. 12. Photograph of two toe designs that employ overload stops to preventtoe damage. A pin 1. attached to the robot’s foot provides overload protectionin conjunction with the hole in each toe surrounding the pin. This ideallycauses the toe to disengage upon excessive extension, or in the worst caseprevents further extension if the toe does not disengage.will usually vary as r s . Usually the required number of toes nshould be chosen based on the spine size, leading to n ∝ 1/r sfor a constant robot mass.One situation that requires attention is if a robot climbs asurface with a low density of useable asperities, or a foot failsto attach properly to the wall. In these cases, very few spineswill attach to the wall, causing them to have abnormally-highloads. To prevent spine or toe damage, one possible solutionis to employ overload stops for the toes such that the spinesautomatically disengage if the overload condition is reached.Examples of toes that employ overload stops can be seenin Figure 12. These toes have been used on another robot,the RiSE platform, a 3.2 Kg, 6-legged, 2-DOF/leg robot thatemploys many of the same features as SpinybotII, includinga tail, pull-in motions, and a sprawled posture with the centerof mass close to the wall . Although the RiSE platformis approximately 7.5 times heavier than SpinybotII, it hasbeen able to climb multi-story stucco and rough concretewalls reliably due to its use of active force sensing (to avoidoverloading any of the feet) and active steering to navigatearound smooth patches. It presently cannot climb some of thesurfaces SpinybotII can, but work is ongoing to equip it withsmaller and more numerous spines.A. ConclusionsVII. CONCLUSIONS AND FUTURE WORKSpinybotII climbs reliably on a wide variety of hard, outdoorsurfaces including concrete, stucco, brick, and dressedsandstone with average asperity radii >25 µm. The essentialprinciples behind its operation include using many miniaturespines with a compliant suspension that ensures that the loadis shared uniformly among them. The same principles can alsobe applied to larger robot platforms. Desired spine tip radiusis a function of average asperity size for the surfaces to beclimbed and not of robot size.A more challenging problem is to tackle rough or corrugatedsurfaces or, in general, surfaces that have roughnesscomparable to spine length. Either the feet and toes must haveenough “suspension travel” to accommodate the contours ofthe surface or they must have an additional active degree offreedom, like the toes of geckos or the tendon-actuated tarsusof insect legs. On contoured surfaces it should be possibleto exploit internal “grasp” forces, in a manner similar to thatused by robots that climb with hand-holds and foot-holds ,, for additional security. The compliant suspension of thespines will become an increasingly difficult design challengeas spines are made smaller: smaller spines necessarily havea smaller clearance, possibly preventing them from reachingdeep holes in the surface while maintaining a favorable loadingangle.Another challenging problem is to climb surfaces withmuch lower roughness than concrete or sandstone, such aspolished stone or interior wall panels. The scaling argumentsin Sections II and III should still apply. However, for smoothpanels the average asperity radius may be on the order of afew micrometers, requiring spine tip radii of perhaps 1 µm.These extremely small spines will be over 100 times weakerthan the spines on SpinybotII and a large number of themwill be required, unless the overall mass of the robot can bereduced correspondingly. Going still smaller, we approach thedimensions of the hairs that are being investigated for syntheticdry adhesives , , , . At the smallest scales, hairsutilizing adhesion have two advantages over microspines ornanospines: they are less sensitive to the local surface normaldistribution and they are loaded primarily in tension, ratherthan in bending.For a given surface, at a small enough length scale, thesurface will appear fractal. For spine dimensions smaller thanthis, the number of asperities per unit area will grow as 1/r 2 .Since the spine strength grows as r 2 , we hypothesize that thetotal weight that can be sustained per unit of surface areausing spines is approximately constant. This weight per unitarea number will depend on the distribution of surface normalangles, which is related to the surface’s fractal dimension.An interesting question is whether some combination ofspines and adhesive hairs will ultimately prove most effectivefor scaling a variety of hard vertical surfaces. Differentsurfaces have different distributions of surface normal angles.Spines perform best on surfaces with normal angles close to90 ◦ , while dry adhesives do best on smoother surfaces withnormal angles closer to 0 ◦ . To be able to climb the widestvariety of surfaces, both spines and dry adhesives could beused.B. Future workFuture work in modeling spine/surface contacts could benefitfrom 3-D surface measurement, which would enablemore accurate asperity location predictions. For very smoothsurfaces, an atomic force microscope or similar instrumentmay be required to accurately measure the surface heights. Forrough surfaces, using a laser interferometer with the surfacetilted at an angle could permit measurement of surface normalangles in excess of 90 ◦ . The model could also be extended toinclude asperity strength. For the robot feet, making smallerspines and better suspension systems is the correspondingnext step. It is still unknown what the minimum scale is forfabricating and using spines effectively.
For the SpinybotII robot body, obvious improvements areto increase the climbing speed and to provide additionalarticulation so that the robot can negotiate vertical/horizontaltransitions such as window sills. These features are beingpresently added to the RiSE robot, which will enable it tohave a wide range of behaviors. Adding degrees of freedomis straightforward, except that the center of mass must remainclose to the wall and the additional degrees of freedom mustnot interfere with the compliant design principles of the toes,feet and legs as described in this paper.APPENDIX I: INDEX TO MULTIMEDIA EXTENSIONSThe multimedia extensions to this article can be foundonline by following the hyperlinks from www.ijrr.org.Extension Media type Description1 Video A video of SpinybotII climbing variousbuildings around the Stanford campus andsome close shots of its feet and toesengaging asperities.APPENDIX II: SPINE FAILURE MODESWe have observed that the spine/asperity contacts have threeprimary failure modes.1) The spine fails plastically at its base due to tensilestress from bending.2) The spine deflects elastically such that it slips off theasperity.3) The asperity fails, typically as a particle becomesunbonded from the surrounding matrix.Fig. 13.Curved beam with variables used in spine failure mode analysis.Figure 13 shows a curved beam used in spine failure modeanalysis. Shown in the figure are the following variables,which are used in the subsequent calculations:R = Radius of curvature of the spine,d = diameter of cross section of spine,β = angle from the y-axis to the tip of the spine,α = angle the tip of the spine rotates (about the z-axis) inresponse toF = force exerted on tip of the spine.The first mode of failure is due to the tensile stress at thebase of the spine. For a long curved spine, the maximum stressis essentially the same as it would be for a straight cantileverbeam :σ max = Mc =Iwhere32F ldπd 4 ∝ 1 d 2 (ifld = const).M = applied moment at end of beamI = moment of inertia for circular beam cross-sectionc = distance from the neutral axis of the beam to an edgel = equivalent beam length.The second mode of failure is excessive tip rotation. Herewe can apply Castigliano’s Theorem to solve for the tipdeflections and rotations for a curved spine . Applyinga dummy end moment, M, and solving for the end rotation,α (see Figure 13), we obtain:α = ∂U/∂M= R22EI [−2F y + (2F x + F y (π + 2β))cos(β)+(−2F y + F x π + 2F x β)sin(β)] ∝ 1 d 2R(ifd = const. at given β, F x and F y ) (3)whereU = strain energy in the beamE = Young’s modulus for the beam material.The third mode of failure is that the asperity itself may breakoff. The literature on surface failure or erosion (e.g. , )for cementitious materials such as concrete, or rock with hardcrystals in a weaker matrix, generally starts with the Hertzstress distribution at the contact . The maximum pressureis at the center of the contact patch:wherep max = 3f/2πa 2 = (6fE 2 /π 3 R 2 ) 1/3a = (3fR/4E) 1/3E = (1 − ν 2 s )/E s + (1 − ν 2 a)/E a(1/R = 1/r s + 1/r a )and the subscripts s and a refer to the spine tip and asperity,respectively, andν = Poisson’s ratio for each materiala = radius of the contact patchf = normal force applied to the spine/asperity contact.The variable r s is the same variable discussed in SectionsII and III. This will be less than half of the beam diameter(r s < d/2) in Figure 13 if a spine tapers to its point as shownin Figure 1.The worst case tensile stress is at the periphery of thecontact patch:
σ T = ((1 − 2ν a )p max )/3The actual failure will depend on the local stress state, numberof cracks and fracture toughness of the material. However, itwill be a function of the maximum tensile stress. Thereforewe can write thatf max = [(πσ max /(1 − 2ν a )) 3 (9/2E 2 )]R 2The quantity in square brackets is a constant depending on thematerials so that, in the end, the maximum sustainable load isexpected to vary as the square of the radii of curvature of thespine tip and asperity.APPENDIX III: R a AND R q CALCULATIONThe linear roughness R a was calculated byR a = 1 N∑abs(x[i] − ¯x)Ni=1and RMS roughness R q was calculated byR q = √ 1 N∑(x[i] − ¯x)N2 ,i=1where x[i] is the height of point i in the profile, ¯x is theaverage height of the profile, and N is the number of pointsin the profile.ACKNOWLEDGMENTThis work was supported in part by the DARPA BiodynoticsProgram under Contract NC66001-03-C-8045. Additional supportwas provided by a DCI fellowship for W. Provancherand a NSF fellowship for A. Asbeck. The authors thank V.Mattoli for his development of the PIC processor program forcontrolling the RC servos. Thanks are also due to J. Lee forher help in designing and fabricating SpinybotI.REFERENCES E. Arzt, S. Gorb, and R. Spolenak. From micro to nano contacts inbiological attachment devices. In Proceedings of National Academy ofSciences, volume 100, pages 10603–10606, 2003. A.T. Asbeck, S. Kim, W.R. Provancher, M.R. Cutkosky, andM. Lanzetta. Scaling hard vertical surfaces with compliant microspinearrays. In Proceedings, Robotics: Science and Systems Conference.Boston, MA, June 2005. K. Autumn and A. Peattie. Mechanisms of adhesion in geckos.Integrative and Comparative Biology, 42(6):1081–1090, 2002. C. Balaguer, A. Gimenez, J.M. Pastor, V.M. Padron, and C. Abderrahim.A climbing autonomous robot for inspection applications in 3d complexenvironments. Robotica, 18:287–297, 2000. D. Bevly, S. Dubowsky, and C. Mavroidis. A simplified cartesiancomputedtorque controller for highly geared systems and its applicationto an experimental climbing robot. Transactions of the ASME. Journalof Dynamic Systems, Measurement and Control, 122(1):27–32, 2000. M. Binnard and M.R. Cutkosky. A design by composition approachfor layered manufacturing. ASME Transactions, Journal of MechanicalDesign, 122(1):91–101, 2000. T. Bretl, S. Rock, and J.C. Latombe. Motion planning for a threelimbedclimbing robot in vertical natural terrain. In Proceedings of theIEEE International Conference on Robotics and Automation, volume205, pages 2946 – 2953. Piscataway, NJ, USA : IEEE, 2003. M. Catmill. Climbing. In M. Hildebrand, D.M. Bramble, K.F. Liem,and D.B. Wake, editors, Functional Vertebrate Morphology, pages 73–88. The Belknap Press, Cambridge, 1985. J. G. Cham, S. A. Bailey, J. E. Clark, R. J. Full, and M. R. Cutkosky.Fast and robust: Hexapedal robots via shape deposition manufacturing.The International Journal of Robotics Research, 21(10), 2002. M. A. Costa and M. R. Cutkosky. Roughness perception of hapticallydisplayed fractal surfaces. In Proceedings of the ASME Dynamic Systemsand Control Division, volume 69-2, pages 1073–1079, 2000. Z.D. Dai, S.N. Gorb, and U. Schwarz. Roughness-dependentfriction force of the tarsal claw system in the beetle pachnodamarginata (coleoptera, scarabaeidae). Journal Of Experimental Biology,205(16):2479–2488, 2002. K.A. Daltorio, S. Gorb, A. Peressadko, A.D. Horchler, R.E. Ritzmann,and R.D. Quinn. A robot that climbs walls using micro-structuredpolymer feet. In International Conference on Climbing and WalkingRobots (CLAWAR), pages 131–138, 2005. K.A. Daltorio, A.D. Horchler, S. Gorb, R.E. Ritzmann, and R.D. Quinn.A small wall-walking robot with compliant, adhesive feet. In Int. Conf.on Intelligent Robots and Systems (IROS), pages 4018–4023, 2005. S.B. Emerson and D. Diehl. Toe pad morphology and mechanisms ofsticking in frogs. Biological Journal of the Linnean Society, 13(3):199–216, 1980. W. Federle, M. Riehle, A.S.G. Curtis, and R.J. Full. An integrativestudy of insect adhesion: mechanics and wet adhesion of pretarsal padsin ants. Integrative and Comparative Biology, 42(6):1100–1106, 2002. Y. Fukuoka, H. Kimura, and A.H. Cohen. Adaptive dynamic walkingof a quadruped robot on irregular terrain based on biological concepts.Int. Journal of Robotics Research, 22(3-4):187–202, 2003. A. K. Geim, S. V. Dubonos, I. V. Grigorieva, K. S. Novoselov, A. A.Zhukov, and S. Yu. Shapoval. Microfabricated adhesive mimickinggecko foot-hair. Nature Materials, 2:461–463, 2003. J. A. Greenwood. Contact of rough surfaces. In I. L. Singer and H. M.Pollock, editors, Fundamental of Friction: Macroscopic and MicroscopicProcesses, pages 37–56. Kluwer Academic Publishers, Dordrecht, TheNetherlands, 1992. J. A. Greenwood. Problem with surface roughness. In I. L. Singerand H. M. Pollock, editors, Fundamental of Friction: Macroscopicand Microscopic Processes, page 5776. Kluwer Academic Publishers,Dordrecht, The Netherlands, 1992. K.L. Johnson. In Contact Mechanics, pages 72–94. Cambridge UniversityPress, 1985. A. B. Kesel, A. Martin, and T. Seidl. Adhesion measurements on theattachment devices of the jumping spider evarcha arcuata. The Journalof Experimental Biology, 206:2733–2738, 2003. S. Kim, A.T. Asbeck, M.R. Cutkosky, and W.R. Provancher. Spinybot:Climbing hard walls with compliant microspines. In Proceedings ofthe IEEE International Conference on Robotics and Automation, pages601–606. Seattle, WA, USA : IEEE, July 2005. G. La Rosa, M. Messina, G. Muscato, and R. Sinatra. A lowcostlightweight climbing robot for the inspection of vertical surfaces.Mechatronics, 12(1):71–96, 2002. R. Lal Tummala, R. Mukherjee, N. Xi, D. Aslam, H. Dulimarta, JizhongXiao, M. Minor, and G. Dang. Climbing the walls [robots]. IEEERobotics and Automation Magazine, 9(4):10–19, 2002. C. Menon, M. Murphy, and M. Sitti. Gecko inspired surface climbingrobots. Proceedings of the IEEE International Conference on Roboticsand Biomimetics (ROBIO), 2004. P. Menzel and F. D’Aluisio. Biobots. Discover, 21(9), 2000. R. Merz, F.B. Prinz, K. Ramaswami, M. Terk, and L. Weiss. Shapedeposition manufacturing. In Proceedings of the Solid Freeform FabricationSymposium, volume 1-8, August 1994. A. W. Momber. Damage to rocks and cementitious materials from solidimpact. Rock Mechanics and Rock Engineering, 37(1):57–82, 2004. R. D. Quinn, G.M. Nelson, R.J. Bachmann, D.A. Kingsley, J. Offi,T.J. Allen, and R. E. Ritzmann. Parallel complementary strategies forimplementing biological principles into mobile robots. Int. Journal ofRobotics Research, 22(3):169–186, 2003. U. Saranli, M. Buehler, and D.E. Koditschek. Rhex: A simple and highlymobile hexapod robot. Int. Journal of Robotics Research, 20:616–631,2001. A. Saunders, D.I. Goldman, R.J. Full, and M. Buehler. The RiSEclimbing robot: Body and leg design. Proc. SPIE Int. Soc. Opt. Eng.,6230:623017, May 2006. G.L. Sheldon and I. Finnie. The mechanism of material removal in theerosive cutting of brittle materials. ASME Journal of Engineering forIndustry, 88B:393–400, 1966. C.R. Shigley, J.E.; Mischke. In Standard Handbook of Machine Design(2nd Edition). McGraw-Hill, 1996.
 M. Sitti and R.S. Fearing. Synthetic gecko foot-hair micro/nanostructuresfor future wall-climbing robots. In Proceedings of the IEEEInternational Conference on Robotics and Automation, volume 1, pages1164 – 1170. Piscataway, NJ, USA : IEEE, 2003. Vortex HC, LLC. VRAM Mobile Robot Platform. [Online],http://www.vortexhc.com/vmrp.html, 2003. Z.L. Xu and P.S. Ma. A wall-climbing robot for labelling scale of oiltank’s volume. Robotica, 20:209–212, 2002. P.A. Zani. The comparative evolution of lizard claw and toe morphologyand clinging performance. Journal of Evolutionary Biology, 13:316–325,2000. J. Zhu, D. Sun, and S.K. Tso. Development of a tracked climbing robot.Journal Of Intelligent And Robotic Systems, 35(4):427–444, 2002.