Studying slippage on pushing applications with snake robots

Transcription

1 DOI /s RESEARCH Open Aess Studying slippage on pushing appliations with snake robots Fabian Reyes and Shugen Ma * Abstr In this paper, a framework for analyzing the motion resulting from the interion between a snake robot and an et is shown. Metris are derived to study the motion of the et and robot, showing that the addition of passive wheels to the snake robot helps to minimize slippage. However, the passive wheels do not have a signifiant imp on the fore exerted onto the et. This puts snake robots in a similar framework as roboti arms, while onsidering speial properties exlusive to snake robots (e.g., lak of a fixed-base, interion with the environment through frition. It is also shown that the onfiguration (shape of the snake robot, parameterized with the polar oordinates of the robot s COM, plays an important role in the interion with the et. Two examples, a snake robot with two joints and another with three joints, are studied to show the appliability of the model. Keywords: Snake robot, Cont, Modeling Bakground Robots that are apable of loomotion in unstrutured onditions are neessary for realisti appliations. However, loomotion alone may not be suffiient when more dexterous interion with the environment is needed. Therefore, roboti systems with apability to loomote and also inter dexterously with their surroundings are desirable, and indeed a natural extension of robotis researh. Snake robots have shown promise regarding loomotion [1]. Loomotion in planar environments has been probably the main topi of researh for snake robots [2 4] and has been extended to motion in planar slopes [5, 6], motion in 3D-spae [7, 8], and more broad studies on loomotion [9]. An interesting idea that ombines loomotion and interion with the environment, alled obstale-aided loomotion (OAL, has been proposed in [10] where obstales in the environment are used as auxiliary soures for propulsion or to avoid jamming. Although snake robots ould exel in loomotion, it is not lear if they an be used to inter with the environment (or an et dexterously. Its struture resembles *Correspondene: shugen@se.ritsumei.a.jp Department of Robotis, Ritsumeikan University, Kusatsu, Shiga , Japan a roboti manipulator, but there are key differenes that have not been fully addressed in previous researh (.f. Fig. 1. The lak of a fixed-base makes it diffiult for a snake robot to manipulate an et as dexterously as a roboti arm. Another differene is that a snake robot has ont with the environment through frition at several points of its body. Additionally, mass beomes a very important parameter to study. Unlike researh regarding roboti arms where it is assumed that the arm an lift the et and it is a matter of hoosing an optimal input, snake robots may not be able to move the et due to its inertial properties. Beause the kinemati struture of a snake robot resembles a roboti arm, papers that deal with similar (but not exly the same situations an be found in existing literature. In [11] a hyper-redundant serial robot was onsidered and both loomotion and manipulation of an et were onsidered. However, the analysis was purely kinemati while assuming a fixed-base roboti system. In other words, there was not fore analysis showing the onditions for feasibility of the problem. In [12], the duality between loomotion and manipulation of a snake robot was onsidered under the assumption that the snake robot an be treated similarly to a roboti arm with a fixed-base when manipulating an et. This The Author(s This artile is distributed under the terms of the Creative Commons Attribution 4.0 International Liense ( whih permits unrestrited use, distribution, and reprodution in any medium, provided you give appropriate redit to the original author(s and the soure, provide a link to the Creative Commons liense, and indiate if hanges were made.

2 Page 2 of 12 a b Fig. 1 Thought experiment. General senario of a snake robot onting an et. a The snake robot onts an et while it may also be onting the environment either with its belly (frition or pushing against a wall, for example. b The snake robot may be able to move the et. The et may be very heavy and the snake robot will move around the et was ahieved by making the first link of the snake robot behave similarly to fixed-base (due to its shape and mass, but the results annot be extended to the ase of a general snake robot. The problem of analyzing and ontrolling a snake robot under these onditions has been reported in [13], where it is shown that several assumptions made in previous published literature are not enough to guarantee aurate ontrol of a planar snake robot with fritional onts with the ground. The main etive of this paper is to study the resulting interion between a snake robot and an et, when the task is to push the et. We onsider this to be a prelude to more interesting interions like grasping or dexterous manipulation. However, it is important to understand the basis of the interion first. This is an extension of previously published work [14, 15] where a more omplete mathematial modeling of the problem has been presented. In [15], the optimal onfigurations of the snake robot to maximize the fore exerted onto the et have been presented. This paper fouses on the motion of the system, rather than fores. The main motivation for this study is that, as presented in [13], alulating an optimal input instantaneously (i.e., at one instant of time is not enough to aurately ontrol the system. We onjeture that understanding how the system will behave over time is also important. In other words, if the task is to push an et, then the motion of the et must be maximized, while the motion of the snake robot minimized. Throughout this paper, there are several assumptions that have to be made beause of the omplexity of the problem: 1. All bodies in the system are rigid. 2. Conts between the snake robot and ets or the environment (exept the ground are onsidered fritionless point onts. 3. The snake robot has passive wheels or any other mehanial means to ahieve anisotropi frition between the robot s belly and the ground. 4. There is only one onstraint per link of the snake robot. 5. The operational spae is a plane (embedded in a full 3D spae. 6. The snake robot has only one ont point with the et to be manipulated. Assumption 1 allows to have a lear mathematial model of the problem without making assumptions about the ompliane of the bodies whih may be unrealisti to know a priori in real-life situations. Although this assumption may be relaxed by onsidering some sort of virtual ompliane at the onts (e.g., [16, 17], it does not neessarily imply more realisti or orret results. In partiular, it may lead to a stiff system of differential equations and several other problems. As presented in [14], a snake robot may have too many onts with the environment leading to a statially indeterminate system [18]. In order to ensure Assumption 4, we assume that the onstraints from passive wheels are removed when that link is onting an et or a wall, for example. This an be done by lifting the links [19] or with retrable passive wheels, for example. Assumptions 1 through 4 allow to onsider that all onstraint fores are linearly independent and a unique solution an be found. Assumption 5 is made in order to limit the number of parameters and get lear and meaningful results whih may be diffiult for 3-dimensional spae, sine the problem presented in this paper is still very broad and has

3 Page 3 of 12 not been learly defined as it has been disussed in this setion. However, the models presented in this paper are based on spatial vetors [20] whih are trivial to extend from 2D to 3D, so in the future more results an be obtained for more speifi tasks. Assumption 6 is made beause it is not the intent of this paper to study any type of grasp losure or dexterous manipulation, but to understand the interion itself first. The paper is organized as follows. In Mathematial bakground setion, the neessary mathematial bakground to understand this paper is presented along referenes neessary to develop the onepts further. Motion of the system due to the interion setion is the main body of the paper; the modeling of the system is presented, and metris and quantities mentioned in this setion are derived. In Results setion, a speifi example is studied to show the appliation of the proposed metris. Disussion and future appliations setion inludes several omments regarding the sope and limitations of the results presented in this paper. The paper onludes with some remarks in Conlusion setion. Mathematial bakground In this setion, we give a very brief introdution to the mathematial topis neessary to understand this paper. We reommend [20 22] for a more detailed treatment. In partiular, the foundations of the model used in this paper have been presented in [15]; readers are enouraged to read this referene for a more detailed treatment of snake robots in the framework of artiulated-bodies. As stressed in previous researh, it is important to guarantee invariane of metris in order for the results to be meaningful. Not only to avoid inonsisteny of units, but also for the metri to be invariant to a hange of oordinates. To derivate the analysis of the system to lead to meaningful metris, we employ dual vetors [23] and basi differential geometry [21]. Differential geometry: twists, wrenhes, and metris A twist (onatenation of linear and angular veloity υ M n an be expressed w.r.t. a ovariant basis e =[ e 1,..., e n ] T as υ = e T υ. The element υ R n an be interpreted as the (vetor of ontravariant omponents of υ. A wrenh (onatenation of linear fore and torque f F n an be expressed w.r.t. a ontravariant basis e =[ e 1,..., e n ]T as f = e T f and f R n an be interpreted as the ovariant omponents of f. It is important to notie that both bases e and e may not be orthogonal, so the ommon definition of inner produt (e.g., υ υ = υ T υ would give inorret results. Let us denote the metri tensor of the ovariant basis as I = ee T and its inverse by I 1. The (squared length of a twist and wrenh is an invariant quantity and an be obtained using the salar produt { } while taking into aount the metri tensor as υ 2 = υ υ = υ T ee T υ = υ T Iυ, f 2 = f f = f T e e T f = f T I 1 f. Unonstrained model of the snake robot The snake robot an be modeled as a series of rigid links onneted by revolute joints. All joints have their axes parallel to eah other; therefore, the snake robot is onstrained to move on a plane (but is unonstrained in any other way. The kinemati model of a snake robot is similar to an open-hain roboti manipulator (.f. Fig. 2. The model has been previously studied in [13, 15, 24, 25]. The snake robot has a total of n s N degrees of freedom (DOFs, and its generalized oordinates are enapsulated in the vetor q s (t R n s. The snake robot has n l = n a + 1 links eah with mass m i. The Jaobian for the ith link is a mapping from the vetor of generalized veloities q s to the twist υ i R 3 of the link and is denoted as J i R 3 n s υ i = J i q s. (3 The equations of motion of the snake robot an be presented in the anonial form [ e T Ms q s + h ] s = e T [ Bτ + τ ] ext Fig. 2 Kinemati model. The generalized oordinates of the snake robot and its COM are shown. Also, the interion between the snake robot and et with its orresponding ont fore an be seen f (1 (2 (4

4 Page 4 of 12 where M s (q s R n s n s is the inertia matrix of the snake robot (a symmetri positive definite (PD matrix, h s (q s, q s R n s 1 ontains Coriolis and entripetal effets, and τ ext (q, q Rn s 1 is a vetor of torques produed by external fores (e.g., kineti frition. The matrix B R n s n a defined as [ ] 03 na B :=, 1 (5 na n a is a matrix that projets the vetor of input fores τ into the spae of generalized fores. The matrix 1 denotes the identity matrix of appropriate dimensions. Unonstrained model of an et A rigid body is able to move in its operational spae with dimensions n op, and its equations of motion an be omply written as e T[ ] I a + p = e T[ ] f, (6 where a, p, and f Rn op denote the aeleration, veloity-produed terms, and total wrenh ing on the body, respetively. If the body is onstrained to move in a plane (but unonstrained in any other way, it will have three DOFs (i.e., n op = 3. I R n op n op denotes the inertia tensor of the rigid body. The mass of the et m will be denoted as a multiple of the mass of a link of the snake robot as m = κm i. In other words, κ is a proportionality oeffiient relating the masses of interest. If all links of the snake robot have the same mass m, then the inertia matrix of the snake robot an be fored as M s := m M, and the inertia matrix of an et as I = m Ī, where the new inertia matries M and Ī orrespond to inertia matries with unitary mass. Summary of onstraints The interion between a snake robot and an external et reates a set of fores between them that avoid penetration (also alled kinemati onstraints or nonpenetrability onstraints [21, 26, 27]. Additionally, the (stati frition fores between the belly of the robot and the ground an also be modeled as onstraint fores (bounded by their frition limit. Assuming there are n N onstraint fores in total, the onstraint fores f Rn span the onstrained subspae C ={f : f = Tλ }, (7 where the matrix T n opn n is a matrix spanning the onstraint fores on the operational spae [21], and λ R n ontains the magnitude of the onstraint fores (in the ontext of optimization this vetor is usually alled the Lagrangian multipliers [21, 26]. To failitate the oupling between the snake robot and environment/et(s, it is useful to put together all the onstraints in vetor/matrix form. All the onstraints an be put together into the following form ] qs A[ 0 υ (8 where the A R n s n is alled the onstraint matrix and takes the following form A = [ J s G T ], where J s R n n s is alled the robot Jaobian (also alled hand Jaobian [26, 27] whih projets the vetor of generalized veloities of the snake robot onto the onstrained subspae T T 1 0 J s = T T n where T k spans the onstrained spae for the kth onstraint and J denotes the Jaobian orresponding to the link under that onstraint, without any speifi ordering. The matrix G R n op n is usually referred to as Grasp Matrix and its transpose is a mapping from the motion spae of the et to the onstrained subspae; it an be onstruted in a similar manner to the robot Jaobian. The onstraint fores projeted bak onto the snake robot and et are [ ] [ ] τ J T f = s λ (11 G = A T λ, where τ Rn s is simply the projetion of the onstraint reion fore f onto the spae of generalized fores of the snake robot. Motion of the system due to the interion Now that it is assumed that the snake robot is touhing at least one et, the new oupled equations of motion an be written as along the onstraints J.. J, e T [ Ia + p ] = e T [ f + A T λ ], [ ] [ ] Ms 0 qs I = a = 0 I a [ h p ] [ = s Bτ p f ] = f Aa + Ȧυ 0, (9 (10 (12 (13

5 Page 5 of 12 where equality holds for onstraints imposed by frition. Equation (13 is the derivative of (8. As disussed in [21], both holonomi and non-holonomi onstraints an be obtained in this uniform manner at the aeleration level. The hange from the ontravariant basis e to the basis of the onstrained spae e an be obtained by using the projetor e e : R n R n [15, 21] defined as e e := (AI 1 A T 1 AI 1 = G 1 AI 1. (14 The positive semidefinite (PSD matrix G := (AI 1 A T represents the metri tensor for the basis e and has full rank if all onstraints are linearly independent. (Additional omments regarding the rank of this metri tensor are loated in Disussion and future appliations setion. The mapping (14 an be interpreted as the left pseudo inverse of the matrix A T as A T := (AI 1 A T 1 AI 1 A T A T = 1. (15 The onstraint fores λ = e T λ an be obtained by projeting the equations of motion (12 onto the onstrained subspaes using the projetor (14 and taking into aount the onstraint (13 as e T [ ] λ e T A T f + (AI 1 A T 1 (AI 1 p Ȧυ. (16 The right-hand side (RHS of (16 has two terms. The first term depends purely on the set of fores exerted onto the system, either by the uators of the snake robot or an external wrenh exerted onto the et. The seond term inludes terms produed by veloity and will vanish if the system starts from an equilibrium onfiguration. This (affine system of equations is usually interpreted as a fore ellipsoid [27 29], and it maps a quadrati region in the input spae f to an ellipsoid in the output spae of onstraint fores λ, while the veloity-produed terms will shift the origin of suh ellipsoid. In this paper, it is assumed the system starts from equilibrium so that the following linear mapping an be defined λ = e e f. The obtained ontrained fores (due to the inputs in the system an be substituted bak onto the equations of motion (12. The resulting aeleration of the system is e T a = e T [ I 1 ( e e f ], where the new projetor e e : R n R n defined as e e := 1 A T A T (17 (18 (19 is a projetor from input wrenhes e T f to the spae orthogonal to the onstrained spae C, but with oordinates expressed with respet to the original basis e. The extra inverse inertia I 1 transforms to oordinates in e basis (i.e., transforms from wrenhes to twists. By solving for q s and a, the motion of the snake robot and et an be obtained as ( qs q s = τ τ ( qs + f f (20, a = ( ( a τ τ + a f f, where the auxiliary mappings q s τ : R n a R n s, q s f : R n op R n s, a τ : R n a R n op, a f : R n op R n op an be defined as q s τ := 1 m q s f := 1 κm and The (squared length of the aelerations of the snake robot q s 2 or et a 2 an be obtained in an invariant way by onsidering its metri tensors, where the total expression an be divided into three terms as We will onentrate on the ontributions of the inputs of the snake robot. It an be verified that the auxiliary mappings τ and τ, after some manipulation, an be defined as τ := 1 (28 m BT( 1 Ĵ T TM ( 1 s s 1 Ĵ T s B where the auxiliary term ( ( 1 M s 1 J T s ( 1 M s J T s G 1 G 1 a τ := 1 ( 1 κmī G G 1 a f := 1 1 (1 1κ ( κmī G q s 2 = τ T ( τ + τ T a 2 = τ T ( τ + f T τ f ( τ + τ T ( G T Ī 1 J s M 1 s f τ + f T τ f G 1 τ := 1 κm BT M 1 s J T s G 1 G T Ī 1 Ĵ T s := J T s G 1 J s M 1 s J s M 1 s B B G T Ī 1 ( f f ( f f f. GG 1 (21 (22 (23 (24 (25 (26 (27 J s M 1 s B (29

6 Page 6 of 12 has been introdued for a more omp notation and any further simplifiation has been omitted for simpliity s sake. However, the linear relationship w.r.t. the masses of the system beomes evident. Slippage ratio As stated in Bakground setion, it is an important problem to predit motion and not only fores, in order to understand the interion between the snake robot and et and try to aomplish a task. If the task is to manipulate an et, then it is desirable to maximize the motion of the et a 2 while minimizing the slippage of the snake robot q s 2. On the other hand, a snake robot ould loomote using the environment as a soure of propulsive fores or as a support, similar to the idea of limbing [30 32]. This ase resembles more a walking robot where the ont with the environment is neessary for the robot to move. To the best of our knowledge, this distintion has not been studied with snake robots. To analyze this, we propose the ratio of aelerations sr := a 2 a 2 + q s 2, (30 and all it slippage ratio whih is a dimensionless salar quantity bounded as sr [0, 1]. Using this ratio, we an analyze the following three general situations: sr 1 whih implies that the aeleration of the snake robot is minimal ( q s 2 a 2 or q s 2 0. sr 0.5 whih implies a similar magnitude of aeleration for the two subsystems ( q s 2 a 2. sr 0 whih implies that the magnitude of the aeleration of the et is minimal ( q s 2 a 2 or a 2 0. Results To study the interion between snake robot and et, we an apply the framework proposed in this paper while hanging the number and type of onstraints and studying the resulting aelerations of the system. In general, we propose three different senarios depending on the type of onstraints present on the system as follows: Senario 1: The snake robot is in ont with an et but unonstrained in any other way. The frition between the snake robot and ground is negligible. Senario 2: The snake robot onts one et and has passive wheels in all other links. The frition between the passive wheels and ground is bounded by its limit surfae. Senario 3: The snake robot is onting one et and has passive wheels in all other links. The passive wheels impose (unbounded and bilateral non-holonomi onstraints. In other words, we will hange the properties of the interion between the snake robot and the environment (ground and then analyze the resulting aeleration of the et a 2, of the snake robot q s 2 and the slippage ratio sr as a result. Senario 1 allows us to onsider only the inertial properties of the system. Senario 2, on the other hand, allows us to study the effet that passive wheels have on the system, but with a bounded frition oeffiient µ s. Senario 3 onsiders ideal passive wheels and ould be onsidered as the extreme ase when µ s. This is the most ommon model used for studying loomotion of snake robots. This quantity an be seen as the ratio between a desired output and the total output. By analyzing the slippage ratio sr, given a onfiguration and input, we an understand better the behavior of the system. Polar oordinates of the COM of the snake robot In order to ompare snake robots with different number of joints, it is neessary to parameterize the onfiguration of the robot with a set of parameters in ommon. The idea of using the polar oordinates of the COM of the snake robot w.r.t. the ont point with the et has been introdued in [15] and further explored in [24]. The (unsigned distane from the COM of the snake robot to the ont point is denoted by COM, and the angle between this vetor and the link onting the et is denoted as COM. These quantities an be seen in Fig. 3. a b Fig. 3 Three senarios. Snake robot with 2 joints onting an et. a The frition between the snake robot and the ground is negligible. b The snake robot has passive wheels (represented by the blak retangles with a limit surfae for the frition fore. Ideal passive wheels are assumed

7 Page 7 of 12 The norms (26, (27 and slippage ratio (30 have to be studied over regions of the input spae on a speifi kth onfiguration. The inputs are restrited to the quadrati region of the input ( τ ={τ : τ T M 1 s τ 1} (31 whih will be, in general, an ellipsoid and not a unitary sphere as is usually onsidered (i.e., τ 2 1 is not the same as τ T τ 1. Case study 1: Snake robot with two joints In order to show more speifi and qualitative results, we apply the mappings and study a snake robot with two joints (.f. Fig. 3. The small number of joints allows us to show graphially the magnitude of the studied norms as a funtion of the joint torques. First, we study a snake robot with the parameters desribed in Table 1. The snake robot is onting an et with its tail (first link and the ont ours at the middle of the link (.f. Fig. 3. The angles of the joints are varied in the range [ 135, 135 ] every 10 (784 onfigurations in total, and the metris (26, (27, and (30 are alulated within the quadrati region (31. One example onfiguration an be seen in Fig. 4, where it is assumed that the et has a hundred times the mass of a link of the robot (i.e., κ = 100. The first, seond, and third olumns represent the three senarios depited in Fig. 3, respetively. A lighter olor represents a higher value of the depited norm. Figure 4a shows the magnitude of the aeleration of the et a 2. It an be seen that it barely hanges regardless of the senario (i.e., independently of the f that the snake robot has or has not passive wheels, the et will aelerate the same given the same input. Figure 4b shows the magnitude of the aeleration of snake robot q s 2. This shows learly that, even if the et s aeleration is similar for all three senarios, the behavior of the snake robot hanges. The addition of passive wheels (seond and third olumns inreases the area where the snake robot s slippage Table 1 Parameters of the simulation for ase study 1 Symbol Value Unit Desription n 5 Number of DOFs of the system n a 2 Number of uated joints m i 1 (kg Mass of link i, i = 1,..., n l l i 0.15 (m Length of link i, 1 = 1,..., n l I om,i (kg m 2 Rotational inertia for the ith link τ =[τ a1, τ a2 ] T (N m Input joint torques µ s 0.1 Coeffiient of (stati frition used for Senario 2 is minimal. Without passive wheels, the snake robot will slip in almost any diretion of the input spae. The slippage ratio (30 gives quantitative information about the movement of the system and an be studied in the same way as the previous norms. Figure 4 shows the value of the slippage ratio for all three senarios with several values for κ for one onfiguration. It an be seen that sr 0 in the region where there is no ont with the et (i.e., the snake robot an move freely and therefore a 2 0. It is interesting to see that in all three senarios it is always possible to make the et move. However, Senario 2 (non-ideal passive wheels has a limited region where the slippage ratio is high, ompared to Senario 3, where a whole region seems to give high values of slippage ratio. These regions in the input spae are highlighted in Fig. 4b,. Regions where the slippage of the snake robot are minimized tend to have higher slippage ratio. Case study 2: Snake robot with three joints The proposed framework and metris an be applied to a snake robot with any number of joints. In this setion, a snake robot with three joints is studied. However, studying the three-dimensional input spae ould be umbersome. Instead, the norms λ 2, a 2 and slippage ratio (30 are studied as a funtion of the polar oordinates of the COM of the snake robot ( COM, COM w.r.t. the ont point, as disussed in previous setions. Figure 5a reports the result for the norm of onstraint fores λ 2. The polar plots show the results for senario 1, 2, and 3, respetively. The higher the value, the bigger the onstraint fores. It an be seen that in senario 1 (negligible frition there is a lear trend for onfigurations with the COM of the snake robot at angles 90 and 90 to have a higher imp on the wrenh applied to the et. Although senarios 2 and 3 report a higher norm of the onstraint fores, this is due to the addition of passive wheels. From this figure alone, it is not possible to asertain the imp on the et. Figure 5b reports the result for the norm of the et s aeleration a 2. The polar plots report the results for senarios 1, 2, and 3, respetively. It an be seen that the addition of passive wheels (even ideal ones have little imp on the aeleration of the et. However, the onfiguration of the snake robot, parametrized with the polar oordinates of its COM, has a lear and meaningful imp on the aeleration of the et. Although basi intuition would tell that the addition of onstraints (i.e., passive wheels should have an imp on the fore applied to the et (through λ, and onsequently on its aeleration a, this study shows that is not the ase (at least, not that simply. An important addition of this paper w.r.t. [15, 24] is the study of the slippage ratio sr. By studying the

8 Page 8 of 12 a b Fig. 4 Norms of motion of the system. The first, seond, and third olumn represent senario 1, senario 2, and senario 3, respetively. A higher value represents more power transmitted to the respetive motion. The onfiguration of the robot is q s ={0, 0, 0, 135, 135 }. a Aeleration of the et. b Aeleration of the snake robot. Slippage ratio. Several values of κ are shown

9 Page 9 of 12 a b Fig. 5 Norms studied over all onfigurations of the snake robot. a Constraint fore (from left to right: senarios 1, 2, and 3. b Aeleration of the et (from left to right: senarios 1, 2, and 3. Slippage ratio (from left to right: senarios 1 and 2 relationship between motions of both systems (snake robot and et, we an understand how the additional onstraints have an imp on the system. A snake robot without passive wheels will slip as it pushes the et. Therefore, minimizing this motion while keeping a steady fore on the et (and therefore produing

10 Reyes and Ma Robot. Biomim. (2017 4:9 an aeleration is desirable. Figure 5 shows the result of the slippage ratio (30. It an be seen that in Senario 1 (without passive wheels the same trend as with the et s aeleration appears. However, passive wheels (even non-ideal ones have a big imp on the slippage ratio (take notie of the hange of sale. To show more learly the imp of the onfiguration of the snake robot on the aeleration and slippage of Page 10 of 12 the system, Fig. 6 shows the best and worst onfigurations for the aeleration of the et Fig. 6a and for slippage ratio Fig. 6b. The results are summarized in Table 2. It an be seen that passive wheels (Senario 2 and 3 have little imp on the aeleration, but a signifiant one on dereasing the slippage of the snake robot (sr 1. Fig. 6 Representative onfigurations hosen among the best and worst onfigurations of the snake robot. a Aeleration of the et. b Slippage ratio

11 Page 11 of 12 Table 2 Results of the norms Conept Senario 1 Senario 2 Senario 3 Worst a Best a Worst sr Best sr Disussion and future appliations Several assumptions have been made in this line of researh, espeially the number of onts onsidered between ets, and their rigidity. This is beause we are interested in giving a solution that is mathematially rigorous while guaranteeing uniqueness of solution. To inlude more ont points means to loose this in favor of robustness. For example, a penalty method (aka. virtual springs or barrier funtions may be onsidered, whih is ommon for whole-arm body manipulation (WAM tasks. Although our assumptions are restritive, it allows us to give a solid foundation for the researh. Other models or onsiderations an be used for more realisti senarios, but the rigid-body assumption used here allows to have a lear basis for omparison. Considering the gaps in knowledge regarding snake robots (as highlighted in Bakground setion, we onsider the model and results presented to be useful for moving researh forward. The holonomi onstraints (e.g., onstraints due to joints of the snake robot are already enoded in the kinemati model of the snake robot presented in Mathematial bakground setion. These holonomi onstraints are desribed by the use of the robot s Geometri Jaobians. Further distintion between holonomi and nonholonomi onstraints is not neessary, as both an be expressed in the same unified manner (13, as mentioned in [21, 33]. The metris and general framework presented an be used to analyze more omplex tasks involving snake robots (or similar roboti systems intering with an et or the environment. This gives an opportunity to study both snake robots and roboti arms in the same framework, sine the analysis is similar to the often-used fore/manipulability ellipsoids used to study roboti arms or hands [34]. However, the metris presented in previous researh do not onsider the motion of the robot itself, sine a fixedbase was always assumed. The framework presented in this paper an be applied to other mobile systems in a more omplete manner than reported in the literature [28, 29]. More speifially, the analysis presented extends the onept of fore or manipulability ellipsoids [26, 34], from the ase of a fixed-base robot with end-effetor, to a mobile robot without an end-effetor. For a given task and onfiguration, an analysis an be arried out to find the optimal input (vetor of joint torques to minimize or maximize slippage of the system. A few onlusions an be drawn from analyzing norms (26 and (27 on the input spae (.f. Fig. 4. In all senarios, τ 2 whih is the joint furthest away from the et has almost no effet on the aeleration of the et. However, the addition of passive wheels helps to anhor the snake robot and ouples the effet of τ 2 on the system. Conlusion In this paper, a modeling and analysis framework for snake robots in ont with an external body has been presented. Results show that the addition of passive wheels has little effet on the wrenh applied to the et and therefore, little hange in its aeleration. However, the passive wheels do have an effet on the motion of the robot itself. In other words, under ertain onditions the slippage of the robot an be minimized while pushing the et. This ould be benefiial for pushing or manipulation tasks. To the best of our knowledge, this problem has not been fully studied with snake robots. Authors ontributions FR onduted mathematial analysis, programming to perform the simulations, and wrote the manusript. SM supervised the researh. Both authors read and approved the final manusript. Aknowledgements This study was in part supported by Strategi Researh Foundation Grantaided Projet for Private Universities ( from Ministry of Eduation, Culture, Sports, Siene and Tehnology, Japan, and R-GIRO (Ritsumeikan Global Innovation Researh Organization. We aknowledge and thank their support. Competing interests The authors delare that they have no ompeting interests. Funding This researh did not reeive funding from any partiular organization. Publisher s Note Springer Nature remains neutral with regard to jurisditional laims in published maps and institutional affiliations. Reeived: 8 September 2017 Aepted: 24 Otober 2017 Referenes 1. Liljebäk P, Pettersen KY, Stavdahl Ø, Gravdahl JT. A review on modelling, implementation, and ontrol of snake robots. Robot Auton Syst. 2012;60(1: Ma S. Analysis of reeping loomotion of a snake-like robot. Adv Robot. 2001;15(2: Saito M, Fukaya M, Iwasaki T. Serpentine loomotion with roboti snakes. IEEE Control Syst Mag. 2002;22(1: Liljebäk P, Pettersen KY, Stavdahl Ø, Gravdahl JT. A simplified model of planar snake robot loomotion. In: Proeedings of IEEE/RSJ international

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