Hindawi Publishing Corporation Applied Bionics and Biomechanics Volume 2015, Article ID 151686, 9 pages http://dx.doi.org/10.1155/2015/151686

Research Article Terrain Adaptability Mechanism of Large ’ Feet on the Kinematics View

Qun Zhang,1 Xilun Ding,1 and Kun Xu1,2

1 Space Robot Laboratory, School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China 2State Key Laboratory of Robotics and System, HIT, Harbin 150001, China

Correspondence should be addressed to Xilun Ding; [email protected]

Received 7 July 2015; Revised 2 November 2015; Accepted 22 November 2015

Academic Editor: Young-Hui Chang

Copyright © 2015 Qun Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Ruminants live in various parts of land. Similar cloven assist ruminants in adapting to different ground environment during locomotion. This paper analyzes the general terrain adaptability of the feet of ruminants using kinematics of the equivalent mechanism model based on screw theory. Cloven hooves could adjust attitude by changing relative positions between two digits in swing phase. This function helps to choose better landing orientation. “Grasping” or “holding” a rock or other object on the ground passively provides extra adhesion force in stance phase. Ruminants could adjust the position of the metacarpophalangeal joint or metatarsophalangeal joint (MTP or MCP) with no relative motion between the tip of feet and the ground, which ensures the adhesion and dexterity in stance phase. These functions are derived from an example from chamois’ feet and several assumptions, which are believed to demonstrate the foundation of adaptation of ruminants and ensure a stable and continuous movement.

1. Introduction “grasp” the rock to avoid slip [10, 11]. Previous studies [12–15] take the foot as a whole to discuss the moving characteristics Animalshaveevolvedthefeetofvariousshapesandfunctions and no theoretical analysis of kinematics of the foot is to adjust to complex terrains. Insects’ feet have small spines elucidated. Indeed, the foot is very complicated, constituted and hooks that help them to climb [1], while geckos have by skeleton, multiple joints, ligament, muscles, subcutis, and the ability to climb vertical surfaces like walls and even some skin modifications [16]. We believe that the similar ceilings, using five hundred thousand keratinous hairs [2, structure forms the basis of adaptability of ruminants during 3]. have less diversity than insects and reptiles. locomotion. The theoretical analysis of kinematics of the foot Ruminants (Artiodactyl) that have larger cloven than reveals moving ability and dexterity in general and makes most insects and geckos, including , , , and us understand more about the terrain adaptability of large , have a wide distribution on the earth. Despite the ruminants. various living environments, ruminants have a similar foot This paper aims to figure out the general functions of the structure [4, 5], covered by neither setae nor hook, which feet as follows: how the feet could adjust the attitude in swing is quite different from that of insects and reptiles. With the phase and how the feet could keep the dexterity and adhesion assistance of simple, reliable, and strong feet, they adjust to in stance phase. An equivalent mechanism model of the foot the terrain on which they feed, mate, and avoid predators very is built based on the skeleton and joints of the foot. Using the well. Buffalo and cattle can go through mud easily [6]. mechanism kinematics and screw theory, we could determine have special and soft feet to cross deserts [7]. and blue the workspace of the mechanism as a dexterity indicator. can climb up and down cliffs and ledges steadily and Finally, we employ the length of digits’ bones of chamois on fleetly to feed on any grass, shrub, or trees [8, 9]. Their cloven the theoretical model to discuss the terrain adaptability of the hooves can spread apart when contacting the ground and ruminants. 2 Applied Bionics and Biomechanics

Antebrachium

Carpal bones Carpal joint Proximal row Distal row Hinge joint, 1 DOF

3rd metacarpal bone

4th metacarpal bone

Fetlock joint Hinge joint, 1 DOF Proximal phalanx Pastern joint Middle phalanx Saddle joint, 2 DOF Coffin jo int Distal phalanx Saddle joint, 2 DOF

Figure 1: The skeleton and articulation of the ox’s manus (schematic), which can be equivalent to an articulated mechanism.

2. Methods mechanismconstitutedbythetwobranches.Thatis,wefocus on the functions of two digits, which are essentially the same 2.1. Structure of Ruminants’ Feet. The skeletons and joints in the manus and tarsus. In the following part, we define of ruminants’ manus and tarsus are similar, but they vary phalanxes and corresponding joints as foot (as shown in in size, such as bovine [17], camel [18], and sheep [19]. The Figure 2). structure of ruminants’ manus is shown in Figure 1 [16]. The skeleton of the manus consists of carpal bones, metacarpal 2.2. Observations of Goat’s Feet. The feet is observed from bones, and phalanges. In ruminants, there are two digits left. three adult female goats (Capra hircus, breed), ranging in Each digit has three phalanges. The carpal joint (MCP in body mass from 23 to 28 kg. All goats were from Beijing forelimb) is composite articulations. The carpal joint acts as Badaling Safari World and apparently healthy. Under the a hinge joint. Due to the complex anatomy of the carpal approval of Beijing Badaling Safari World and the profes- skeleton complemented by numerous ligaments of the carpus, sional guidance and assistance of the staffs, the goats were the primary movements of the carpal joint are flexion and made to lie on their side. The metacarpal bones or metatarsal extension [16]. In ruminants, the remaining third and fourth bones were grabbed by two of the staff to provide a closer metacarpal bones are fused and no movement is possible observation. Thus, unaffected by the movement of the upper [4]. The two fetlock joints are hinge joints that can only flex limb, the movement of feet was scrutinized. The goats were and extend. The pastern joints are saddle joints due to the examined and released very fast to avoid stress and injury. concavo-convex shape of the joint surfaces. They act mainly as hinge joints [20]. However, served as saddle joints, pastern joints are biaxial joint, allowing flexion and extension and 2.3. Kinematics in Swing Phase. The mechanism in Figure 2 a limited range of lateral movements [16]. The coffin joints is open chain mechanism, composed of a fixed platform at are similar to the pastern joints. Tarsal joint (MTP in the the top connected two branches, which can move freely in hindlimb) is also composite joints. The tarsal bones and joints the swing phase. Both branch I and branch II consist of three are different from the corresponding ones in the forelimb [4], serially connected joints: one revolute joint and two universal while the bones and joints of the metatarsus and digits are joints. The universal joints are modeled as two revolute joints 𝜉 similar. intersecting at one point. In the joint notation 𝑖𝑗 (also screw 𝑖 Due to the anatomy in Figure 1 (modified from [21]), notation), the first subscript denotes the branch number and 𝑗 wecandrawdiagramofamechanism.Therevolutejoint thesecondsubscript denotes the joint number within the 𝜃 ∈𝑅 𝑖 = 1,2,...,5 𝑗=1,2 connects the two same branches; each one consists of one branch. 𝑖𝑗 ( ; ) denotes the amount 3×1 revolute joint and two universal joints. Since the two branches of the joint 𝜉𝑖𝑗 rotation, and 𝐿𝑖𝑗 ∈𝑅 (𝑖 = 1, 2,; 3 𝑗=1,2) contact the ground in stance phase, we just discuss the denotes the link of the branch. Applied Bionics and Biomechanics 3

z z Branch I y S Branch II y S 𝜉 𝜉11 ξ 11 𝜉 O x 21 O x 21 l l l1 l1 l1 l1

𝜉12 𝜉 𝜉 22 12 𝜉22 𝜉13 𝜉 𝜉 l 23 l 13 𝜉23 2 2 l2 l2

𝜉 𝜉25 𝜉 15 𝜉15 25 𝜉24 𝜉14 𝜉 𝜉24 l3 14 l3 l z 3 z l3 y z y y L=l O x O x O x T1 T2 T

Figure 2: The mechanism of foot in swing phase (reference config- Figure 3: The mechanism of grasping rock passively (reference con- uration). figuration).

According to screw theory [22, 23], the twist coordinates toed mammals like cannot seize a stone by their odd 6×1 3×1 of kinematic pair are 𝜉(v, 𝜔)∈𝑅 ,where𝜔 ∈𝑅 is the digit. There are interdigital ligaments (cruciate ligament) 3×1 axis of rotation and v =−𝜔 × q (q ∈𝑅 is a point on the between the two digits (the space between two claws is called axis) if the joint is a revolute joint. interdigital cleft), which are not found in the digit of horse The cross product by 𝜔 is a linear operator, and 𝜔 × q can [4]. Distal interdigital ligament bridges middle phalanx and be represented using a matrix: distal phalanx of two digits. When the rock is embedded in the interdigital cleft, the foot will splay out. Since the 0−𝜔𝜔 3 2 interdigital ligaments limit this movement [16], the two digits [ ] 𝜔 × q = 𝜔̂q = [ 𝜔3 0−𝜔1] q. (1) will tend to close. This is like “grasping” or “holding” the rock except that it is passive. When ruminants firmly “grasp” a −𝜔 𝜔 0 [ 2 1 ] rock in stance phase, the relative position between two digits remains unchanged. This situation resembles “grasping” an × 𝜉̂ The 4 4matrix given in (2) is the generalization of the object using two manipulators. The mechanism transforms 𝜔̂ ∈ 𝑠𝑜(3) skew-symmetric matrix into the single loop mechanism with grasping constraint 𝜔̂ v (Figure 3). The mechanism is composed of a moving platform ̂ 𝜉=[ ]. (2) at the bottom connected to a fixed platform at the top by two 00 branches. 𝜃 = 0 We let correspond to the fully extended configuration 2.4.1. DOF in the Reference Configuration andattachbaseandtoolframesasshowninFigure2.Due to the product of exponentials formula for the manipulator {S }={𝜉 , 𝜉 Branch I. The branch motion-screw system b1 11 12, forward kinematic, the transformation between tool and base 𝜉13, 𝜉14, 𝜉15} is described by its basis frames in branch I is given by 𝑇 { 𝜉11 = (0, 0, 0, 1, 0, 0) } ̂ ̂ ̂ ̂ ̂ { } 𝑏1 𝜉11𝜃11 𝜉12𝜃12 𝜉13𝜃13 𝜉14𝜃14 𝜉15𝜃15 𝑏1 { 𝑇 } 𝑔𝑠𝑡 (𝜃) =𝑒 𝑒 𝑒 𝑒 𝑒 𝑔𝑠𝑡 (0) , (3) { 𝜉 =(0,−𝑙,0,1,0,0) } { 12 1 } { } 𝑔𝑏1(0) { 𝑙 𝑇 } where 𝑠𝑡 refers to the transformation between tool and {S }= 𝜉 =(𝑙,0,− ,0,1,0) . 𝜃 = 0 𝑏1 { 13 1 2 } (5) base frames at . { } { 𝑙 𝑇} The transformation between tool and base frames in {𝜉 =(𝑙+𝑙 ,0,− ,0,1,0) } { 14 1 2 } branch II is similar { 2 } 𝑇 ̂ ̂ ̂ ̂ ̂ 𝑏2 𝜉21𝜃21 𝜉22𝜃22 𝜉23𝜃23 𝜉24𝜃24 𝜉25𝜃25 𝑏2 { 𝜉15 =(0,−𝑙1 −𝑙2,0,1,0,0) } 𝑔𝑠𝑡 (𝜃) =𝑒 𝑒 𝑒 𝑒 𝑒 𝑔𝑠𝑡 (0) . (4) Constraint-screw F(f, 𝜏) and its motion-screw 𝜉(v, 𝜔) are 2.4. Kinematics in Stance Phase. After regulating the attitude related by of the two digits in swing phase, ruminants can embed F ⋅ 𝜉 =0, (6) the protrusion of rock between their digits in stance phase f v to increase the adhesion and remain stable. However, odd where F =[𝜏 ], 𝜉 = [ 𝜔 ]. 4 Applied Bionics and Biomechanics

𝑟 According to (6), a basis for branch S𝑏1 constraint-screw 2.4.2. Inverse Kinematics. The 𝑋𝑌𝑍 Euler angles (𝛼,𝛽,𝛾) system can be calculated are available for describing the orientation of the moving platformrelativetothebaseand[𝑥𝑦𝑧] refer to origin 𝑇 { 2 } coordinates of moving platform. Through constraint-screw 𝑟 F11 = (0, 0, ,0,1,0) 𝑟 𝑟 {S }= 𝑙 . systems {S𝑏1} and {S𝑏2},movingplatformcannotrotateabout 𝑏1 { } (7) 𝑇 𝑧-axis in the reference configuration. So, the angles (𝛼,) 𝛽 { F12 = (0, 0, 0, 0, 0, 1) } can describe the orientation of the moving platform of the mechanism. We define that 𝛼 represents the roll angle, 𝛽 𝛾 Branch II. The branch motion-screw system {S𝑏2}={𝜉21, 𝜉22, refers to the pitch angle, and refers to the yaw angle. [𝑥𝑦𝑧] 𝜉23, 𝜉24, 𝜉25} is described by its basis refer to origin coordinates of moving platform

𝜉 = (0, 0, 0, 1, 0, 0)𝑇 𝑥 { 21 } [ ] { } 𝑅𝑝 [𝑅 (𝛼) 𝑅 (𝛽) 𝑦] { 𝜉 =(0,−𝑙,0,1,0,0)𝑇 } [ 𝑥 𝑦 ] { 22 1 } 𝑔𝑑 =[ ]=[ ] , (16) { } 01 [ 𝑧] { 𝑙 𝑇 } {S }= 𝜉 =(𝑙,0, ,0,1,0) . 01 𝑏2 { 23 1 2 } (8) [ ] { } { 𝑙 𝑇} {𝜉 =(𝑙+𝑙 ,0, ,0,1,0) } where 𝑔𝑑 ∈𝑆𝐸(3)is the desired configuration of the tool { 24 1 2 } { 2 } frame. 𝑇 {𝜉25 =(0,−𝑙1 −𝑙2,0,1,0,0) } The forward kinematics is described in exponential coor- dinates as S𝑟 Abasisforbranch 𝑏2 constraint-screw system can be ̂ ̂ ̂ ̂ ̂ 𝜉11𝜃11 𝜉12𝜃12 𝜉13𝜃13 𝜉14𝜃14 𝜉15𝜃15 calculated 𝑔𝑠𝑡 (𝜃) =𝑒 𝑒 𝑒 𝑒 𝑒 𝑔𝑠𝑡 (0) (17) 𝑇 ̂ ̂ ̂ ̂ ̂ 2 𝜉21𝜃21 𝜉22𝜃22 𝜉23𝜃23 𝜉24𝜃24 𝜉25𝜃25 { } =𝑒 𝑒 𝑒 𝑒 𝑒 𝑔𝑠𝑡 (0) . 𝑟 F21 = (0, 0, − ,0,1,0) {S𝑏2}={ 𝑙 } . (9) { 𝑇 } 𝑔 { F22 = (0, 0, 0, 0, 0, 1) } Given a desired configuration 𝑑, 𝑔 (𝜃) =𝑔 . Dai et al. proposed a generalized Kutzbach-Grubler¨ mobility 𝑠𝑡 𝑑 (18) criterion [24] to calculate the degrees of freedom (DOFs) 𝑀 −1 For branch I, postmultiplying this equation by 𝑔𝑠𝑡 (0) isolates for 𝑛 bodies connected by 𝑔 joints, each with 𝑓𝑖 degrees of freedom the exponential maps: ̂ ̂ ̂ ̂ ̂ 𝑔 𝜉11𝜃11 𝜉12𝜃12 𝜉13𝜃13 𝜉14𝜃14 𝜉15𝜃15 −1 𝑒 𝑒 𝑒 𝑒 𝑒 =𝑔𝑑𝑔𝑠𝑡 (0) =𝑔1. (19) 𝑀=𝑑(𝑛−𝑔−1)+∑𝑓𝑖 + V −𝜁, (10) 𝑖=1 3 Apply both sides of (19) to a point 𝑝3 ∈𝑅which is the where 𝑑 is the order of mechanism, V represents redundant common point of intersection for the universal joint axes 𝜉𝜃̂ constraint, and 𝜁 is passive DOF. For single loop mechanism, (𝜉14, 𝜉15). Since 𝑒 𝑝=𝑝if 𝑝 is on the axis of 𝜉, this yields V is equal to 0 [25] ̂ ̂ ̂ 𝜉11𝜃11 𝜉12𝜃12 𝜉13𝜃13 𝑒 𝑒 𝑒 𝑝13 =𝑔1𝑝13, (20) 𝑑=6−𝜆, (11) 𝜉̂ 𝜃 −𝜉̂ 𝜃 −𝜉̂ 𝜃 𝑒 13 13 𝑝 =𝑒 11 11 𝑒 12 12 𝑔 𝑝 . (21) where 𝜆 is common constraint. 13 1 13 Thus, at the reference configuration in Figure 3, Projecting both sides of (21) to the 𝑥-axis, 𝑟 𝑟 𝜆= ({S }∩{S }) = 1. ̂ dim 𝑏1 𝑏2 (12) 𝜉13𝜃13 [1000]𝑒 𝑝13 (22) The degrees of freedom of the mechanism are drawn ̂ ̂ −𝜉11𝜃11 −𝜉12𝜃12 =[1000]𝑒 𝑒 𝑔1𝑝13. 𝑀=5×(6−6−1) + (1+1+2+2+2+2) =5. (13) 𝜃11 and 𝜃12 are eliminated, and we can determine 𝜃13 as When the mechanism is not at the reference configuration, follows: generally, 𝑥 − (𝑙 𝛼) /2 + 𝛼𝑙 +𝑙/2 𝜃 = 0 cos sin 3 . 𝑟 𝑟 sin 13 (23) −𝑙2 𝜆=dim ({S𝑏1} ∩ {S𝑏2}) =0. (14)

Since 𝜃13 is known, (20) becomes Thus, the degrees of freedom of the mechanism are drawn 𝜉̂ 𝜃 𝜉̂ 𝜃 𝜉̂ 𝜃 𝑒 11 11 𝑒 12 12 (𝑒 13 13 𝑝 )=𝑔𝑝 . 𝑀=6×(6−6−1) + (1+1+2+2+2+2) =4. (15) 13 1 13 (24) Applied Bionics and Biomechanics 5

Table 1: Average length of osseous structures of the digit in chamois (unit: mm).

Species Proximal phalanx Middle phalanx Distal phalanx Environment Chamois 38.0 23.3 25.9 Alpine

Applying Paden-Kahan subproblem-rotation about two non- Table 2: Parameter of the mechanism (unit: mm). 𝜃 𝜃 intersecting axes [23], we solve for 11, 12. 𝑙 𝑙 𝑙 𝑙𝐷 The remaining kinematics can be written as Parameters 1 2 3 Chamois 38 23.3 25.9 17.525 14 ̂ ̂ ̂ ̂ ̂ 𝜉14𝜃14 𝜉15𝜃15 −𝜉11𝜃11 −𝜉12𝜃12 −𝜉13𝜃13 𝑒 𝑒 =𝑒 𝑒 𝑒 𝑔1. (25) Table 3: The angle range of joints (unit: rad). Apply both sides of (25) to any point 𝑝 which is not at the Joints 𝜃11, 𝜃12, 𝜃15, 𝜃21, 𝜃22, 𝜃25 𝜃13, 𝜃14, 𝜃23, 𝜃24 intersection of the universal joint axes (𝜉14, 𝜉15) as follows: Angle range −𝜋/6∼𝜋/6 −𝜋/10∼𝜋/10 ̂ ̂ ̂ ̂ ̂ 𝜉14𝜃14 𝜉15𝜃15 −𝜉11𝜃11 −𝜉12𝜃12 −𝜉13𝜃13 𝑒 𝑒 𝑝=𝑒 𝑒 𝑒 𝑔1𝑝. (26) 𝜌 is increased until the point is out of the workspace). The Applying Paden-Kahan subproblem-rotation about two sub- volume of the reachable workspace is determined by sequent axes, 𝜃14 and 𝜃15 are found. So, all 𝜃11 through 𝜃15 are 1 determined on branch I. The inverse kinematics of branch II 𝑉= ∑ ∑𝜌2Δ𝜑Δ𝑧. 2 𝑗 (28) canbesolvedsimilarly. Slices 𝑗

2.4.3. The Workspace of the Mechanism. While solving 2.5. Parameter Determination. In this paper, we focus on inverse kinematics, there could be multiple solutions. We the digits of chamois, which lives in the alpine zone [27]. need to determine whether the solutions satisfy the constraint Alpine terrain has snowy mountains, rough terrain, and conditions. Workspace is considered as a useful measure of alpine meadow [28]. This is very complex terrains for large the range of the mechanism given the orientation. There animals. To live under high alpine conditions, animals have are two types of kinematic constraints affecting the available evolved various adaptions [29]. Thus, the feet of chamois may workspace of the mechanism: joint angle limitations and link be a complex whole of adaptability. The method above can interference [26]. The joints of animals cannot rotate 360 also apply to other ruminants with similar structure of feet. degrees; thus, the motion is restricted by physical construc- The average length of the digits from both the manus and tion. Since the bones of animals have physical dimensions, thetarsusisshowninTable1[30].Basedonthedata,we interference might happen when the mechanism moves. can get the parameters in the mechanism (Table 2), where Since links have geometry shapes and physical dimensions, the distance between two digits (𝑙)isanestimate.Duetothe link interference may appear during moving. To keep things lack of concrete data and analysis, we assume that the shortest simple, assume that every link is cylindrical with the same distance between two adjacent links should be greater than diameter. The shortest distance between two adjacent links 𝐷 𝐷 14 mm, which is greater than the width of proximal phalanx should be greater than the diameter .Let 𝑖 be the (13.6mm).Previousmeasurementsofjointangleofgoat’sfoot minimum distance between the centerlines of two adjacent 𝐷 indicate that the average joint angle of MTP and MCP during links. Since 𝑖 is the minimum distance between two line stance phase (during level, uphill, and downhill) is 17.6 to ∘ segments, it may not be equal to the common perpendicular 28.6 which is related to the configuration in Figure 3 and Δ ∘ segment of the two links ( 𝑖). There are the intersection the max joint angular excursion of them is 26.1 [13]. While points of two links with their common normal 𝑛𝑖. 𝐷𝑖 is equal Δ ruminants walk on flat ground, the joints probably will not to 𝑖 only if both intersection points are on the links. If one reach the maximum angle during swing phase and stance of the intersection points or both are not on the links (i.e., on 𝐷 phase. The angular range is larger than the measurements the extension line), 𝑖 is either the distance of an endpoint of during stance phase in case that ruminants go through rugged one link to the other link or the distance of the endpoints of terrain or other harsh environments. Though we lack the two links. The detailed method is discussed in [26]. amplitude of lateral angle range of digit joint, we truly know Thus, the inverse solutions of kinematics are subject to the that the range is small. So, we can assume the angular range following constraints: from reference configuration (𝜃 = 0)inTable3.

𝜃min ≤ 𝜃 ≤ 𝜃max, (27) 3. Result 𝐷𝑖 ≥𝐷. Figure 4 shows that the forefoot of the goat can spread out and The workspace is divided into slices of thickness Δ𝑧 parallel close freely. Both forefoot and hindfoot of three goats were to the 𝑋𝑌 plane. As to each slice, the boundary is determined examined, demonstrating similar ability. by polar coordinates search method [26] (from a point within We could plug the value of parameters and angle the workspace, the angle 𝜑 is increased by Δ𝜑 and the radius limitation into (3) and (4) using Monte Carlo method. 6 Applied Bionics and Biomechanics

rock” at 𝛽=0shows the largest workspace volume and the configuration of holding “large rock” at 𝛽=0has the smallest workspace volume.

4. Discussion The goal of this study is to investigate possible functions of feet in large ruminants based on similar structure. We give Spread a method that can be used to investigate the functions of all Close apart largeruminants.Inaddition,themountainhabitatofchamois has gradients of aspect, vegetation, altitude, valleys, ridges, edges, and streams [27], which contains various terrains. The foot of chamois somehow has representative among large ruminants. Thus, the result from chamois could give us some general functions of the foot. Due to the anatomy of the foot (Figure 1), the primary flexion and extension of the foot act like a small limb with parallel hinge joints; the reverse lateral movements (the 3rd Figure 4: The forefoot of the goat which can close and spread out. and4thtoebonesmoveatthereversedirection)causethe claws to splay out and close (Figure 4). Figure 5 shows the movement scope of the tip of the foot in swing phase. Given the limitation of the joints, the foot is able to adjust distance The workspace of two branches can be drawn in Figure 5. between two hooves and the rotation angle (lateral or front- Using the method in [31], the workspace volume of branch back) before the foot touches the ground. While the volume 4 3 Iis1.4 × 10 mm . of the workspace of the horse is only half, the cloven-hoofed Figure 5 shows the graphical representation of the animals have more flexible movement to choose the posture workspace of two branches when the two digits are during of the foot by changing the relative position between two swing phase. The set of points defines the available workspace digits. References to earlier papers have described how the that the end-effector of the two branches can reach under cattle or goat goes through the soft terrain [8, 32]; they touch joint angle limitations. Each digit could achieve flexion- the ground with claws splayed out. Soil or small stones are extension and lateral movements individually. With two end- embedded in interdigital cleft and clamped to provide more points of the digits chosen in the corresponding workspace contact area, friction angle, and adhesion, produce more during swing phase, the attitude of the foot is determined propulsion, and reduce subsidence of the foot [7]. When when these two digits step on the ground. encountering the rock ground, a cloven-claw foot can also The single loop mechanism (Figure 3) depicts the foot grasp the sharp edge of the rock passively. This is similar to holding the rock passively duringstancephase.Thedistance how a human grips a stick using two fingers, only passive. between two endpoints of the digits is equal to that between Goatsandbluesheeptendtosplayouttheirclawswhen two fetlock joints. Ruminants could also hold other sizes of walkingdownhilltoincreasethecontactareaandavoidthe rock passively, bigger or smaller. Under this definition, one slip.Thefootisabletoadjustdistancebetweentwohooves possible configuration is shown in Figure 6. Based onthe to adapt to different sizes of small stones or rock ledges. method given above, the workspace of the mechanism could Moreover, though the lateral movements of these feet are be determined similarly. limited, the foot can tilt laterally by manipulating two digits to All three mechanisms in Figure 3 and Figure 6 are reach different heights. This function prevents the ruminants symmetric around 𝑦-axis and 𝑧-axis, so the corresponding from overturn and improves stability on the cross slope. Only workspacehasthesameshapeandsizeifgiventhesame having two digits which can splay out and close can let the absolute values of the orientation (𝛼,). 𝛽 We choose the values ruminants accomplish this adaptability. of (𝛼,) 𝛽 of the first quadrant. After holding a rock or some other bulges passively, When the foot holds different rock size (let “large rock” two digits in stance phase cannot move as dexterously and represent the condition 𝐿 = 𝑙+6,let“bigrock”be𝐿= freelyastheonesinswingphase.Theworkspaceinstance 𝑙+3,let“normalrock”be𝐿=𝑙,andlet“smallrock”be phase is much smaller than the one in swing phase (less 4 3 𝐿=𝑙−3), the corresponding workspace will change as shown than 1000 versus 1.4 × 10 mm ). Though the movement in Figure 7. The workspace volume is much larger at 𝛽=0 of the mechanism is restricted within the workspace in than at 𝛽=𝜋/36at different distance between two digits. stance phase, the DOF of the mechanism remain 4 or 5. The increase of the roll angle 𝛼 has little effect on the volume The workspace relative to the base (the top plate of the of the workspace (decrease the volume). The workspace of mechanism) is important when planning a task for the foot. holding “small rock” at 𝛽=0isnearlythesameastheoneof Let 𝑔𝑡𝑠 betheconfigurationofthebaseframerelatedtothe holding “big rock” at 𝛽=0, while the workspace of holding tool frame (Figure 3) as follows: “small rock” at 𝛽=𝜋/36is larger than the one of holding 𝑔 =𝑔−1 ∈𝑆𝐸(3) . “big rock” at 𝛽=𝜋/36. The configuration of holding “normal 𝑡𝑠 𝑠𝑡 (29) Applied Bionics and Biomechanics 7

0 0 Branch II Branch II −20 1 Branch I −20 1 Branch I 3 −40 3 −40 3 (mm)

3 (mm) 2 2 z z 5 −60 2 5 −60 2 5 5 4 −80 y −80 4 4 z y x y x 50 z 50 z x 20 x 20 y 0 10 y 0 10 (mm) 0 (mm) 0 −50 −10 −50 −10 −20 x (mm) −20 x (mm) (a) (b)

Figure 5: The available workspace of two branches with angle limitation during swing phase. The set of black points in (a) and (b) describes the workspace of branch I, while the red ones in (b) show that of branch II.

z z y S y S 𝜉11 𝜉 𝜉11 𝜉 O x 21 O x 21 l l l1 l1 l1 l1

𝜉 𝜉12 𝜉22 12 𝜉22 𝜉13 𝜉13 𝜉23 𝜉23 l l2 l2 2 l2 𝜉 𝜉 25 𝜉 15 𝜉15 25 𝜉 𝜉 𝜉 𝜉14 24 14 24 l l 3 z 3 z l3 y l3 y L L 𝜉 𝜉 26 16 𝜉26 𝜉16 O x O x T T (a) (b)

Figure 6: The mechanism of foot while grasping the groundsively. pas The distance between two endpoints of the digits 𝐿>𝑙,while𝜃13 >0 (relative to the configuration in Figure 3), 𝜃23 =−𝜃13, 𝜃𝑖𝑗 (𝑖 =1,3,𝑗̸ =3)=0̸ in (a); 𝐿<𝑙, 𝜃13 <0(relative to the configuration in Figure 3), 𝜃23 =−𝜃13, 𝜃𝑖𝑗 (𝑖 =1,3,𝑗̸ =3)=0̸ in (b).

Because of the rigid body transformation, the workspace of the influence of the volume of the workspace at different thetopplaterelativetothegroundisthesameastheone relative orientation when the hooves grasp different size of which is calculated above. The parallel mechanism could theobjectpassivelyandfirmlywithoutrelativemotion.The change the position of the carpal joint (Figure 1) when the roll angle 𝛼 represents the movement of flexion and extension endpoints of two branches of the parallel mechanism are fixed of the foot and the pitch angle 𝛽 refers to the lateral movement with the ground. It depicts the ability of the foot to adjust of the foot. If there are no lateral movements, the workspace thepositionoftheMCPandMTP(thebaseframesatthe is rather large; that is, the foot shows great dexterity under the toplinkinFigure3)atthegivenrelativeorientationwhen primary movement of the foot (flexion and extension). Due the tip of the digit is fixed with the ground. The volume of to the limited lateral movements of the digits, once the lateral the workspace can be used as a measure of the foot dexterity. movements occurs, the volume of the workspace decreases a For ruminants, appropriate foothold can be chosen by the lot; that is, the dexterity of foot is weakened at any roll angle adjustment within the workspace of the foot to regulate the 𝛼. The foot has to sacrifice the dexterity to attain the lateral orientation of the limb and the body, even though the tip of movements. the foot is fixed. This function of the foot can help ruminants Holding different size of rocks passively has an influence to adapt to rough terrain and increase stability. Figure 7 shows on the dexterity of the foot. The excessive distance between 8 Applied Bionics and Biomechanics

1000 952.3 in other ruminants. Based on these common functions, 900 878.7 idioadaptive evolution of different ruminants will be detected

) 800 using similar methods and adding the ligament limitation in 3 future works. Different species have different length 700 of digits, also the age, the gender, and the different digit will 600 affect the parameters of osseous structures. The difference in 500 length of digits and rotation range of joints may be one of 400 the reasons why ruminants are able to adapt to the different terrains. The terrain adaptability of large ruminants’ foot may 300 224.5 223.7 help to inform the foot design of highly adaptable robot. 200 207.1 206.4 Volume of workspace (mm workspace of Volume 113.5104.7 100 69.764.3 Conflict of Interests 00 0 8.4 7.7 4.6 4.2 l−3 l l+3 l+6 The authors declare that there is no conflict of interests Distance between two digits regarding the publication of this paper. (𝛼 = 0, 𝛽 = 0) (𝛼 = 0, 𝛽 = 𝜋/36) (𝛼 = 𝜋/36, 𝛽 =0) (𝛼 = 𝜋/36, 𝛽 = 𝜋/36) Acknowledgments Figure 7: The volume of workspace versus the distance between two The financial support from the National Basic Research digits at four different orientations of the moving platform. Program of China (973 Program), the National Science Foundation for Distinguished Young Scholar, China (Grant no. 51125020), the National Natural Science Foundation of two digits (large size rock, 𝐿=𝑙+6)wouldincurtheloss China (Grant no. 51305009), and State Key Laboratory of of the dexterity at 𝛽=0.Itisbecauseholdinglargesizerock Robotics and System (HIT) are gratefully acknowledged. The passivelyneedstoopenthedigits,whichalreadycauseslateral authors also would like to thank the staffs of Beijing Badaling Safari World for the help of taking photos of goats. movements; that is, the initial joint angle of 𝜃13,𝜃23 is not zero ∘ ∘ (𝜃13 = 3.5 , 𝜃23 = −3.5 ). This occupies part of the angular excursion to retain the configuration. This disadvantage of References occupying angular excursion also contributes to the loss of the dexterity when holding small size rock passively (𝐿 = 𝑙−3) [1] S. Gorb, Attachment Devices of Insect Cuticle, Springer, 2001. or big size rock (𝐿=𝑙+3) at the orientation of 𝛽=0.In [2] K. Autumn, Y. A. Liang, S. T. Hsieh et al., “Adhesive force of a fact,itcausesalmostthesamelossoftheworkspacevolume. single gecko foot-hair,” Nature,vol.405,no.6787,pp.681–685, 2000. However, compared with the reference distance (𝐿=𝑙), the larger distance (𝐿=𝑙+6) provides the largest dexterity at [3] K. Autumn, M. Sitti, Y. A. Liang et al., “Evidence for van der Waalsadhesioningeckosetae,”Proceedings of the National 𝛽=𝜋/36. Actually, with the increased distance, the dexterity 𝐿>𝑙 Academy of Sciences of the United States of America,vol.99,no. augments ( ). So, when the lateral movements are needed, 19, pp. 12252–12256, 2002. holding the larger rock passively would be the better choice [4] S. Sisson, The Anatomy of the Domestic Animals,W.B.Saunders, to keep both stability and dexterity. 1921. [5] S. Qiusheng, Anatomy of the Metacarpus, Phalanges and 5. Conclusion Sesamoids of Bactrian Camel, Yak, Cattle, Yak-Cattle Hybrid and Buffalo, Lanzhou University, 2008. Therefore, we can summarize several functions of the foot to [6] Y. Chen, J. Tong, and B. Chen, “Inverse kinetics analysis adapt to the terrain. The foot could change relative position ofthegaitofthecattlebostauruswalkingonsoftground,” between two digits to splay out or tilt to adapt to the slope Transactions of the Chinese Society for Agricultural Machinery, in swing phase. The orientation of the foot is prepared for vol.38,no.10,article043,pp.165–169,2007. the movement function, that is, grasping a rock passively in [7] W.Zhihao, Q. Xiding, and J. 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