Abstract:
A biologically inspired climbing device with toes and spines is provided. The device has toes capable of moving independently from each other. Each toe distinguishes a compliant linkage, which has non-compliant parts and compliant parts. Spines are distributed over each of the toes. The spines have a diameter that is less than or equal to a diameter of an asperity of a surface. The toes with spines could be arranged into one or more feet. The feet could then be organized in legs enabling the device to climb a surface in a gait pattern. The device uses low power and is quiet, leaves no traces or tracks behind, works well on smooth, rough, uneven, porous and dirty surfaces, and is able to carry and support its own body weight.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
       [0001]    This application is cross-referenced to and claims benefit from U.S. Provisional Application 60/635,160 filed Dec. 9, 2004, which is hereby incorporated by reference. 
     
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
       [0002]    This invention was supported in part by the Defense Advanced Research Projects Agency (DARPA) under Grant Number N66001-03-C8045. The U.S. Government may have certain rights in the invention. 
     
    
     FIELD OF THE INVENTION 
       [0003]    This invention relates generally to devices and methods to climb over surfaces. More particularly, it relates to a devices and methods for climbing or clinging without using suction or vacuum methods and without using sticky materials such as glue. 
       BACKGROUND OF THE INVENTION 
       [0004]    In recent years, there has been considerable progress in small, legged robots that can run rapidly and stably over rough terrain. Climbing and maneuvering on a vertical surface presents a more difficult challenge, which robots are just beginning to address. For applications such as surveillance or the inspection of hard-to-reach locations, one would like to have small robots that can climb a variety of hard and soft surfaces unobtrusively and cling for extended periods of time without high power consumption. 
         [0005]    Previously developed climbing robots have generally employed either suction cups, magnets, or sticky adhesives to cling to smooth vertical surfaces such as windows and interior walls. None of these approaches is suitable for porous and typically dusty exterior surface such as brick, concrete, stucco or stone. A recent development employing a controlled vortex to create negative aerodynamic lift has been demonstrated on brick and concrete walls (www.vortexhc.com). However, this approach consumes significant amount of power, whether the robot is stationary or moving, unavoidably generates noise, and is difficult to adapt to non-smooth surfaces such as window ledges, corners, and corrugated surfaces. Still other robots employ hand and foot holds in the manner of a human climber. 
         [0006]    Observing animals that exhibit scansorial (vertical surface) agility, one can find a variety of methods employed. Larger animals such as cats and raccoons employ strong claws that penetrate wood and bark surfaces. Tree frogs and many insects employ sticky pads. Geckos and some spiders employ large number of hairs that achieve adhesion via van der Waals forces on almost any kind of surface. Other insects, arthropods and reptiles employ small spines that catch on fine asperities. Accordingly, it would be considered an advance in the art to design biologically inspired climbing devices and mechanism. 
       SUMMARY OF THE INVENTION 
       [0007]    The present invention is a biologically inspired device that has toes and spines for climbing on a surface or clinging onto a surface. The device includes a plurality of toes that are capable of moving independently from each other. Each toe distinguishes a compliant linkage, which has a plurality of non-compliant parts and one or more compliant parts. In one example, the compliant linkage includes a plurality of compliant parts. The linkage further distinguishes an un-deflected position and a deflected position. Each toe could include, either as an intrinsic (passive) property of the design or explicitly added to the design, an overload mechanism and/or a disengagement mechanism. 
         [0008]    A plurality of spines is distributed over each of the toes and attached to at least one of the non-compliant parts of a toe. In one example the spines are attached to a plurality of non-compliant parts of a toe. The spines have a diameter that is less than or equal to a diameter of an asperity of a surface. In case the spines are tapered they would have a tip diameter less than or equal to the diameter of the asperity of the surface. In some cases, spines are capable of moving independent from each other within a toe. The movements of the toes and spines would increase the probability of spines finding an asperity to latch or cling onto. The device could further include an under-actuated mechanism or a tendon capable of engaging and disengaging the spine(s), toe(s) or any combination thereof. 
         [0009]    In the example of a climbing robot, the toes with spines could be arranged into one or more feet. The feet could then be organized in legs enabling the robot to climb a surface in a gait pattern. The under-actuated mechanism or a tendon could then also be attached to each foot. 
         [0010]    Unlike existing wall-climbing robots that rely on noisy suction or vacuum methods, devices of this invention require low power and are quiet. Unlike conventional wall-climbing robots that use sticky substances such as glues, devices of this invention usually leave no traces or tracks behind. More importantly, devices of this invention work well on porous and dirty surfaces, overcoming a significant limitation of prior approaches. In general, a robot according to this invention has the ability to climb on (or cling onto) smooth, rough or uneven surfaces such as concrete, stucco, sandstone, brick, and similar building walls, interior walls, or drywalls. Furthermore, a robot utilizing the mechanisms of the present invention is able to carry and support its own body weight. 
         [0011]    Robots utilizing the present invention would be highly desirable in many practical applications such as surveillance robots, toys, military applications, etc. Moreover, the design and mechanisms could be used as climbing aids for humans and to other applications that require high adhesion on hard, slightly rough surfaces such as concrete, stucco, or cut stone and even smoother surfaces. Furthermore, the design and mechanisms could also be used to position or move devices on walls, angled or vertical surfaces. For example, the feet and/or toe mechanisms could be used to position and/or move a billboard, cling to the outside of a spacecraft, make a clamping mechanism that can adhere to these surfaces and can support other objects, include feet for a robot to enable it to climb impenetrable surfaces as well as penetrable surfaces. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0012]    The objectives and advantages of the invention will be understood by reading the following detailed description in conjunction with the drawings, in which: 
           [0013]      FIG. 1  shows examples of spines according to the present invention. Example  110  shows a spine with diameter d and a tip diameter d s . Example  120  shows a spine with diameter d and a tip diameter d s  relative to a surface with asperity d a . 
           [0014]      FIG. 2  shows an example of a compliant linkage toe design according to the present invention. It is noted that in photograph  220  spines  230  are shown inside the dotted circles. 
           [0015]      FIG. 3  shows an example of a robot climbing whereby several spines are engaging asperities on a concrete surface  310  according to the present invention. 
           [0016]      FIG. 4  shows a plan  410  view and side  420  view of one robotic foot containing ten toes, each toe is like the toe shown in  FIG. 2 . The figure shows that the toes can deflect independently of each other. In addition, the entire foot can displace in the distal (y) direction due to an unactuated prismatic joint. The attachment/detachment trajectory of the foot as illustrated could be an upward (+y) motion, followed by lift-off motion (−x), touchdown (+x), and a downward pull (−y). It is noted that the upward (+y) and liftoff (−x) motions are part of the disengagement motion to remove the foot from the wall. The attachment trajectory just includes the touchdown (+x) and downward pull (−y). The sequence of motions could be accomplished using an under-actuated mechanism consisting of a single rotary RC servo-motor and an elastic band that is initially loose and becomes taut as the leg moves upward. A hard stop functions to remove the leg from the wall when the servo rotates to the upward position (the end of the arm moves in the (+y) direction). When the servo rotates to the upward position, the elastic band becomes taut and pushes the leg against the hard stop. When the servo rotates to the downward position ((−y) direction), the elastic band becomes loose and the leg rotates freely about the joint where it is connected to the servo arm. 
           [0017]      FIG. 5  shows a plan view  510  and a photograph  520  of a climbing robot according to the present invention. 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 could help to reduce the pitching moment. The center of mass (COM) is ideally within the polygon of contacts, to minimize yawing rotations in the plane of the wall. 
           [0018]      FIGS. 6-8  show different examples of disengagement and/or overload protection mechanisms according to the present invention. 
           [0019]      FIGS. 9-12  show different examples of mechanisms capable of engaging and disengaging spine(s), toe(s) or combinations thereof according to the present invention. 
       
    
    
     DETAILED DESCRIPTION 
     A. Spines in Nature 
       [0020]    Insects and arthropods that climb well on man-made and natural surfaces often use legs equipped with large numbers of small, sharp spines. Even geckos that frequently rock surfaces such as cliffs and caves have small claws on each toe in addition to their dry adhesive structures. Unlike the claws of a cat, the small spines or claws do not need to penetrate the surface. Instead, they exploit small asperities (bumps or pits) on the surface. Several studies in the biology literature have considered the problem of spine/surface interaction. For example, Dai et al. (2002) presented a planar model of spine/asperity contact and computed the maximum load per spine as a function of spine strength, relative size of the spine tip versus that of an asperity, and coefficient of friction (see Dai et al. (2002) in a paper entitled “Roughness-dependent friction force of the tarsal claw system in the beetle  Pachnoda marginata  (Coleoptera, Scarabaeidae)” and published in  The Journal of Experimental Biology  205:2479:2488). As expected, for rough surfaces the mechanical strengths of the spine and asperity become the limiting factors; for smoother surfaces friction is more important and the ability to pull in toward the surface is much reduced. 
       B. Spine Scaling for a Climbing Robot 
       [0021]    Given the observed relationship between spine or claw size and animal size, one is led to ask: For a climbing robot of a given size, how large should the spines be? If one considers a robot that weighs approximately 0.5 Kg, one might expect spines or claws similar to those seen in squirrels or large climbing lizards. However, this argument ignores the point that spines made of hardened steel are much stronger and stiffer than natural spines and can therefore be smaller while supporting a comparable load. 
         [0022]    Indeed, if the strength of the spine/asperity contact were not a constraint, one should make the spines as small as possible. The reason behind this argument is that many natural surfaces, and some man-made surfaces such as concrete and stucco, have an approximately fractal surface topography so that characteristic surface features (asperities) can be found over a wide range of length scales. As shown in  FIG. 1 , for spines of a certain tip diameter, d s , and for the purpose of this invention, we are interested in asperities of average diameter d a ≧d s  to obtain effective interlocking (d a  is the diameter of an asperity). Given the self-similar nature of fractal surfaces, one can expect the density of such asperities to grow at least as 1/d a   2  per unit area of the wall. This invention can also be used on very non-fractal surfaces, so long as the spines have tip diameters less than the diameters of asperities on the surface. 
         [0023]    In practice, there is a lower limit to the useful spine dimensions. We have found that when steel spines catch on asperities on concrete or stucco, the contact typically fails in one of three ways:
       (1) Plastic failure of the base of the spine in bending,   (2) Excessive elastic rotation of the spine tip causing it to slip off the asperity, or   (3) Brittle failure of the asperity itself.       
 
         [0027]    In each of these cases, if we take a dimension such as the spine tip diameter, d s , as a characteristic dimension and scale everything uniformly, then the maximum load of the spine/asperity contact increases as d s   2 . 
         [0028]    The spine/asperity contacts have three primary failure modes. The first mode of failure is due to the tensile stress at the base of the spine, causing the spine to plastically deform. Maximum stress on a cylindrical cantilever beam is: 
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         [0029]    The second mode of failure is excessive spine tip rotation. Deflection angle at the tip of cantilever beam is: 
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         [0030]    The third mode of failure is that the asperity itself may break off or fail in shear. Shear stress failure is: 
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         [0031]    The details of the asperity failure will depend on whether the material is brittle and whether cracks or defects are present. However, the strength of the asperity is generally expected to increase as the square of asperity diameter. 
         [0032]    As a person of average skill in the art would appreciate, as spines become smaller the climbing robot can ascend smoother surfaces because the density of useable spine/asperity contacts increases rapidly. However, one would need a larger number of spines because each contact can sustain less force. To make use of a larger number of spines, the first two design principles behind climbing with spines are therefore:
       (1) Ensure that as many spines as possible will independently find asperities to attach to, and   (2) Ensure that the total load is distributed among the spines as uniformly as possible.       
 
         [0035]    The climbing device of this invention has feet (hands would be equivalent) and toes (fingers would be equivalent) that embody these design principles. In addition, as with any climbing mechanism, it is important to keep the center of gravity as close to the wall as possible and to avoid imposing any forces or moments at the feet that could lead to premature detachment. The features of a climbing device according to the present invention that achieve these effects are also described. 
       C. Foot and Toe Design: Promoting Attachment and Load Sharing 
       [0036]      FIG. 2  shows a schematic  210  and a photograph  220  of a compliant suspension linkage for one toe of the climbing device. In this example, the compliant linkage utilizes a combination of hard materials (non-compliant) and soft (compliant) materials. More specifically, the toe distinguishes three hard materials  240 ,  242 , and  244  that are connected with soft materials  250 ,  252 ,  254  and  256 . As shown herein the number of non-compliant and compliant parts as well as the connections between the two parts could be varied and depends on the toe design and requirements for the climbing robot or other applications. It is noted that instead of having a toe design made of different hard and soft materials, one could also make a toe out of one type of material. In the latter case, one would distinguish non-compliant and compliant parts (or portions of the same toe) that vary in dimensions (e.g. thickness). It is also possible to combine these two approaches. 
         [0037]    Each spine  230  or spine group per non-compliant element is compliantly supported with respect to its neighbors so that every spine (or spine group) could find a useful asperity (or asperities) onto which to latch as shown in  FIG. 3 . In the example in  FIG. 2 , if the spine attached to  240  grabs onto an asperity, then the spine attached to  242  is forced towards the wall reaching toward an asperity. However, the spines on the neighboring toes are not necessarily affected. If the spine attached to  242  grabs onto an asperity, the spine on  240  is not necessarily forced towards the wall. In general, a spine or spine group on any toe could force spines on other toes, e.g. neighboring toes, towards the wall. All these movements of the toes are important to maximize the probability that many spines on each foot will find asperities of surface  310  and share the total load. 
         [0038]    Each robotic foot contains a set of compliant toes with spines. In the example of  FIG. 4 , the number of toes is  10 . In general the number of toes could be in the range of two to millions or more per foot. The number of spines per toe could range from one to more than 100, and is predominantly dependent on the spine characteristics, total number of toes for the climbing mechanism/device/robot, type of asperity of the surface, and weight and size of the climbing mechanism/device/robot. The spines are preferably positioned at the non-compliant parts, such as  240  and  242 , and particularly at the distal end of each these linkages as shown in  FIG. 2 . A cable or a tendon  280  can be employed to actively engage or disengage the spines. Such as tendon could be attached to the foot or to each toe.  FIGS. 9-12  shows variations to these engagement and disengagement mechanisms. In the example of  FIG. 2 , non-compliant element  244  connects to a supporting structure for the toes. 
         [0039]    Once a toe makes contact with a surface or an asperity a load is applied to its toe linkage. As a result the toe linkage changes position from an un-deflected position, which is shown in dotted lines and superimposed in  210 , to a deflected position as shown in  FIG. 2 . The positional change depends e.g. on the stiffness characteristics of the compliant materials and mechanical interactions with the connected non-compliant materials as well as the applied load. For example, compliant material  250  has preferably low stiffness in the −x direction when the linkage is under compression and a higher stiffness when the linkage is under tension. Compliant material  250  also preferably provides damping of the motion of non-compliant element  242 . Compliant material  252  has preferably similar properties to compliant material  250 . Compliant material  254  primarily allows for rotational flexure of the proximal spine (bottom  230 ). Compliant material  256  permits travel in y direction. 
         [0040]    For small deflections, the linear and rotational stiffness of each spine in the (x,y) plane can be modeled using a 3×3 stiffness matrix, K, taken with respect to a coordinate system embedded in the spine (see  FIG. 2 ): 
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         [0041]    At initial contact, we require that k xx  be very small for displacements in the −x direction, so that there is little force that tends to push other toes back from the wall and cause them to disengage. Compliant part  250  at the end of the toe are designed to buckle so that they have a very small stiffness for −x deflections and a higher stiffness for tensile loads (+x direction), once the climbing robot starts to transfer part of its weight to the toe. This higher stiffness for tensile loads is preferably used to support the load of the robot pulling in the (−x) direction, and also to disengage the spine(s) from the asperity (asperities) during a later disengagement motion. At the same time, k yy  should be moderate and, more importantly, k xy  should be preferably small and non-negative so that stretching in the y direction does not cause any retraction of the spine in the x direction. Small negative values of k xy  are acceptable, however. Finally, the k xθ  and k yθ  terms should be small and, preferably, negative so that displacements in the x or y direction are not accompanied by anticlockwise rotations in the (x, y) plane that would lead to premature disengagement. Small positive values of the k xθ  and k yθ  terms are also acceptable, however. 
         [0042]    The toe linkage mechanism shown in  FIG. 2  was modeled in the Working Model™ software (MSC Inc.) and the various linear and rotational stiffness elements were adjusted until the model matched deflections obtained when applying small loads and measuring the corresponding displacements in bench-top tests. The results are summarized in Table I. 
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                 TABLE I 
               
             
             
               
                   
               
               
                 STIFFNESS AND DAMPING PARAMETERS FOR TOE LINKAGE 
               
             
          
           
               
                   
                 Parameter in kinematic model 
               
               
                 Location 
                 k = linear stiffness element 
               
               
                 numbered in 
                 c = linear damping element 
               
               
                 FIG. 2 
                 k I  = rotational stiffness element 
               
               
                   
               
               
                 252 
                 k = 60 N/m 
               
               
                   
                 c = 0.1 Ns/m 
               
               
                 256 
                 k = 60 N/m 
               
               
                   
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                 k I  = 0.005 Nm 
               
               
                 250 
                 k = 90 N/m in tension 
               
               
                   
                 k = 0.005 N/m in compression 
               
               
                   
                 c = 0.02 Ns/m 
               
               
                 254 
                 k = 100 N/m 
               
               
                   
                 c = 0.001 Ns/m 
               
               
                   
                 k I  = 0.001 Nm 
               
               
                   
               
             
          
         
       
     
         [0043]    The toe linkage mechanism of  FIG. 2  is designed so that initial contact at the inner, or proximal, spine actually forces the distal spine slightly outward (+x direction) to increase the probability that it will also contact an asperity. Once one or both spines have contacted the wall, the toe can apply a force that is mainly vertical, with a small inward (+x) component to help the robot climb. 
         [0044]    An important observation of agile scansorial animals like geckos is that they employ multi-level conformability (e.g. lamellae, toes, and limbs) and redundancy (multiple pads per toe, multiple toes per foot, multiple feet in contact) for reliable climbing. Similar principles have been found necessary for the robotic climbing mechanism of this invention. To employ multi-level conformability, the entire foot mechanism can be mounted on a prismatic joint with an elastic suspension (e.g. an elastic band) that allows it to move up to  1  cm in the distal (+y) direction as shown in  FIG. 4 . In addition, the entire foot assembly is spring loaded by a second elastic element behind the pivot, where it is connected to a rotary RC servomotor. The result is an under-actuated R-R-P serial kinematic chain that traces a loop trajectory, as shown in  FIG. 4 , when the servomotor rotates back and forth. In one example of a climbing robot designed by the inventors, the best elastic elements were found to be about 6.4 mm diameter elastic bands commonly used for dental braces. 
       D. Body Design: Promoting Load Sharing and Stability 
       [0045]    Moving from the foot to the body as a whole,  510  in  FIG. 5  shows that the robot could utilize an alternating tripod gait, as found in climbing insects. The climbing robot could have two body halves each having three feet with multiple toes as discussed herein (see  FIG. 2-4 ). In the example of  FIG. 5 , the climbing robot has two body halves, each having three feet,  530 A and  530 B, respectively. At any time, the climbing robot in the example of  FIG. 5  could be clinging to a surface by three feet. As a person of average skill in the art would readily appreciate, the number of legs, feet and toes could be varied and is not limited to the examples provided herein. Furthermore, the type of gait or climbing techniques could also be varied and is not dependent on a tripod gait. 
         [0046]    While one of the concerns for vertical climbing is to avoid pitching back from the plane of the wall, it is also important to maintain rotational stability in the plane of the wall. As shown by  520  in  FIG. 5  the center of mass (COM) of the climbing robot preferably lies within a polygon of contacts at all times. Also, as observed in climbing insects and reptiles, the legs have a slight inward pull, toward the centerline of the robot. This arrangement reduces the upsetting moments (in the plane of the wall) about the center of mass, should one of the legs momentarily lose its grip. 
       E. Disengagement and Overload Mechanisms 
       [0047]    In some cases, especially when adapting the design and mechanisms of the present invention to heavier and larger robots, it would be desired to include a disengagement and/or overload mechanism in the toe design. The spine  610  of a toe would preferably disengage when the toe reaches a certain stretch position as is illustrated in  FIG. 6 . The disengagement mechanism could be accomplished by having an inextensible linkage  620  that attached to the top part of the toe. In (1), the toe approaches surface  630 . In (2), spine  620  has hooked onto a bump on the surface, and a small load F is being applied to the toe, shown by the arrow. In (3), the load F has increased to the point where the top member  620  has fully elongated. In (4), an additional load is applied again, further increasing F, causing the front part  640  of the toe to rotate backwards as indicated by the curved arrow. In (5), the spine has disengaged from the wall. This could either occur by the spine rotating back enough that the angle change is enough to make it disengage, or by the back part of the member  650  pushing down on the surface and resisting further rotation, so that the spine is forcibly lifted off the surface vertically. 
         [0048]    In another embodiment shown in  FIG. 7 , one could also have a disengagement mechanism that pulls the spine  710  off the wall  720  perpendicularly (i.e. vertically in  FIG. 7 ) instead of rotating it.  FIG. 7  shows a toe mechanism with a wedge-shaped part of the outer member  730  that slides between the inner member  740  and the wall (compare (1) and (2)), prying it vertically in the figure as shown up and off the bump, thereby disengaging the spine. 
         [0049]    The disengagement/overload solutions implemented in  FIGS. 6-7  employ overload stops for elastic toes with a passive end-of-travel mechanism such that the spine ideally automatically disengages from the surface if the overload condition is reached.  FIG. 8  shows an alternate design of a disengagement/overload mechanism for toe  810 . In this example, the toe can extend forward (see right-hand example and arrow) until non-compliant part  820  with the opening/hole in it hits pin  830 , which is attached to the foot. Such a construction would cause the toe to disengage upon excessive extension. Essentially, the toe stretches until pin  830  through the hole in the middle of the toe hits the back of the hole (back of the enclosure), which causes a large disengaging moment so the toe rotates backwards about the spine/wall contact until it lifts off the wall. The pin can also serve to merely restrict further forward motion of the toe, to prevent damage to the compliant parts  840 ,  850  even if it does not cause the toe to disengage from the surface. 
       F. Parameters 
       [0050]    The following is a list of parameters that could be used in the toe design, foot design and/or robot design:
       Number of legs: 2-20, or more.   Number of feet: 2-20, or more.   Number of toes per foot: 2-1000, or more.   Number of Spines per toe: 1-100, or more.   Length of spine: 10 cm or larger down to 20 micrometers or less.   Diameter of spine: 1 cm or more down to 0.5 micrometers or less.   Diameter of tip of spine: 1 cm or more down to 0.5 micrometers or less.   Material of spine metal, plastic, ceramic or any type of hard or non-compliant material.   Spine shape: Any shape, but in particular straight, curved, which could be tapered or not tapered.       
 
         [0060]    The number of hard (non-compliant) parts depends on the number of spines per toe. For example, if you have 100 spines per toe, there could be several hundreds of non-compliant parts in the toe. The number of soft (compliant) parts also depends on the number of spines per toe. 
         [0061]    The type of materials could be:
       For non-compliant parts plastic, metal, ceramic, or any material that does not deform very much. In a particular embodiment, a hard urethane of 75 Shore-D hardness could be used.   For compliant parts any material that is compliant either by material characteristics or dimensions. For example, rubber, polyurethane, plastic, metal (if you make it thin enough), or the like. In a particular embodiment, a soft urethane of 20 Shore-D hardness could be used. Soft urethane flexures provide both elasticity and viscoelastic damping. The inventors of this invention have found that soft urethane flexures permit greater extensions without failure than miniature steel springs.       
 
         [0064]    It is noted that D a  as referred to herein represents an approximate maximum value of D a . Since the surfaces are all approximately fractal, all of them will have asperities of size smaller than these values, down to the smallest useable asperity sizes (&lt;0.5 micrometer). The quoted values below are approximately the largest reasonable values for these surfaces. If a range is stated, it is intended to mean that there are different types of that material, and the different types have different maximum asperity sizes. For example, tree bark can be quite rough like an oak tree (D a  of about 1 cm) or can be very shiny and smooth (D a &lt;10 micrometer). The following are some examples of surface with asperities:
       Weathered painted exterior surfaces could have a D a  about 1-5 micrometer and smaller.   Tree bark could have a D a  of about ˜1 cm to 10 micrometers.   Stucco could have D a  of about 3 mm to 0.1 mm.   Concrete (including cinder block, cast concrete, etc.) could have a D a  of about 2 mm to 20 micrometers.   Cut or machined stone could have a D a  of about 1 mm to 10 micrometers.   Brick could have a D a  of about 2 mm to 10 micrometers.   Painted interior drywall could have a D a  of about 10 micrometers to 0.5 micrometer.       
 
         [0072]    All of these surfaces can also be painted or unpainted. If a surface is painted, the paint may fill in many of the smaller asperities, so there may only be asperities approximately larger than 50 micrometers or so and less than approximately 5 micrometers. In other words, there may be very few asperities between 50 micrometers and 5 micrometers. Unpainted surfaces will generally have a continuous range of asperity sizes. 
         [0073]    The characteristic stiffness k yy , i.e. the force required to move a spine relative to its forward displacement (e.g. in the +y direction in  FIG. 2 ), could vary depending on the desired load per spine, the number of spines, the desired forward motion of the toe or spine (which can be related to the asperity density on the wall), and possibly other factors. As such, there can be quite a wide range of desired characteristic stiffnesses k yy . One example was discussed herein with reference to Table I, which was suited for the characteristics of the climbing robot in that example. Similarly, the characteristic stiffness k xx , i.e., the force required to move a spine relative to its displacement in the direction normal to the surface (the −x direction in  FIG. 2 ), depends on the number of spines per foot, the roughness of the surface, and possibly other factors. As such, there can also be quite a range of desired stiffnesses for k xx . 
       G. Variations 
       [0074]    The present invention has now been described in accordance with several exemplary embodiments, which are intended to be illustrative in all aspects, rather than restrictive. Thus, the present invention is capable of many variations in detailed implementation, which may be derived from the description contained herein by a person of ordinary skill in the art. For example, the number of hard and soft structures in a compliant linkage could be varied to less or more than the example of  FIG. 2 . Also there are many possible mechanisms by which the spines could be made to move in the desired manner, including, but not limited to, (i) using non-compliant parts that are connected by rotary compliant joints that are not extensible, (ii) mechanisms that use a fluid- or air-filled cavity to provide the desired compliances in either or both directions, (iii) or any other mechanism or combinations of mechanisms. In another variation, compliance could be adapted in normal and tangential direction for higher load and different stroke lengths. Furthermore, the foot width could be increased and/or length could be decreased to avoid interference. Yet another variation is to have a heterogeneous spine/toe population for wide range of surfaces. Furthermore, the spines could be distributed over the foot in any type of pattern. Still another variation is to have a tail, which reduces the “pull-in” forces needed at the front limbs to overcome the pitching moment produced by gravity acting at the center of mass. 
         [0075]    Still another variation pertains to actively sweeping the spines over the surface via a tendon or action while keeping the foot stationary on the surface (or the foot could move, but it could remain stationary), instead of e.g. moving the foot down the wall and allowing the spines to sweep over the surface that way (see  FIGS. 9-12 ).  FIGS. 9-11  show an example of toe  900  with a tendon  910  pulling non-compliant part  920  whereby tendon  910  passes through an opening  932  in non-compliant  930  ( 940  is a compliant part of toe  900 ;  950  is spine of toe  900 ). The tendon  910  is used to engage or disengage the spine(s) on toe  910 .  FIG. 9  shows an un-deflected toe  900 .  FIG. 10  shows toe  900  deflected downward (+x) and backward (−y) in a direction indicated by arrow  1020  when a force  1010  is applied to tendon  910 . The motion of spine  950  has caused it to become hooked onto an asperity on the surface  1030 . Non-compliant part  930  can be held in a fixed position with respect to surface  1030 , can be moved along surface  1030 , or can be moved along surface  1030  during spine engagement.  FIG. 11  shows tendon  910  stretched caused by force  1110  because spine  950  is hooked onto an asperity and is not able to move further.  FIG. 12  shows an example where each toe  900  is connected to tendon  910 , and tendons  910  for each toe are connected (via a connection  1210 ) together to form another foot-level tendon  1220 . This foot-level tendon  1220  can be used to engage or disengage spines  950  (not indicated in  FIG. 12 , but see  FIG. 9 ). Tendon  1220  could be inextensible. Tendons  910  could be compliant. 
         [0076]    All such variations are considered to be within the scope and spirit of the present invention as defined by the following claims and their legal equivalents.