Simplified Motion Modeling for Snake Robots

Transcription

1 Smplfed Moton Modelng for Snake Robots Floran Enner, Davd Rollnson and Howe Choset Abstract We present a general method of estmatng a snake robot s moton over flat ground usng only knowledge of the robot s shape changes over tme. Estmatng world moton of snake robots s often dffcult because of the complex way a robot s cyclc shape changes (gats) nteract wth the surroundng envronment. By usng the vrtual chasss to separate the robot s nternal shape changes from ts external motons through the world, we are able to construct a moton model based on the dfferental moton of the robot s modules between tme steps. In ths way, we effectvely treat the snake robot lke a wheeled robot where the bottom-most modules propel the robot n much the way the bottom of the wheels would propel the chasss of a car. Expermental results usng a 16-DOF snake robot are presented to demonstrate the effectveness of ths method for a varety of gats that have been desgned to traverse flat ground. I. INTRODUCTION Snake robots are a class of hyper-redundant mechansms [1] consstng of knematcally constraned lnks chaned together n seres. Ther many degrees of freedom gven them the potental to navgate a wde range of envronments. Our group has developed modular snake robots that rely solely on ther nternal shape changes to locomote through ther envronment [2]. To smplfy control of the robot s many degrees of freedom, we have developed cyclc motons, known as gats, that undulate the robot s jonts accordng to parameterzed sne waves [3]. We have developed and mplemented gats that can traverse a varety of terrans, ncludng flat ground, slopes, and ppes. In partcular, for flat ground there are a number of gats that are used to move the snake forward, sdeways, or turn n place as shown n Fg. 1. Despte the successful mplementaton of these gats, representng and estmatng external moton of these gats through the world has proven to be dffcult. The nternal shape changes that a snake robot uses to locomote nvolve the entre body and ntroduce two sgnfcant problems. Frst, no coordnate frame that s statc wth respect to a sngle pont on the robot ntutvely represents the pose of the entre robot at all tmes. Second, the segments of the robot that are nteractng wth ground (and thus nducng moton n the world) are constantly changng. To address the frst problem, the we have developed the vrtual chasss [4] that reles on the overall shape of the robot to defne a unque body frame at every pont n tme. In partcular, ths body Floran Enner s a graduate student n the Salzburg Unversty of Appled Scences, Austra fenner.ts-m21@fh-salzburg.ac.at Davd Rollnson s a graduate student n the Robotcs Insttute, Carnege Mellon Unversty, Pttsburgh, PA, drollns@cs.cmu.edu Howe Choset s an assocate professor n the Robotcs Insttute, Carnege Mellon Unversty, Pttsburgh, PA, choset@cs.cmu.edu Fg. 1: Examples of a snake robot executng the gats dscussed n ths paper. Movng across from top left to bottom rght: rollng, sdewndng, sltherng, and turn-n-place. frame has the property that t s constantly algned wth the prncple components of the robot s overall shape. It provdes a formally defned body frame whch closely matches the ntutve way n whch an operator thnks about the poston and orentaton of the robot. The second problem s addressed n ths work by explotng the observaton that gats consstently nteract wth the ground parallel to the flattest sde of the robot s shape (or more formally, the plane normal to the thrd prncpal axs of nerta). By defnng the bottom sde of the robot n ths way we are able to approxmate the effects that the relevant modules have on the overall moton of the robot. By averagng the translatonal motons nduced by the bottom modules, together wth the rotatonal motons of these modules around the geometrc center of the robot, we are able to estmate the net translaton and rotaton of the robot through arbtrary shape changes. Ths smplfed model estmates the moton of a snake robot n much the same way that a smple no-slp assumpton s frequently (and effectvely) used to estmate the moton of a wheeled robot. It s our hope that ths technque wll begn to allow many plannng and estmaton tools from the wheeled robot communty to be used on hyper-redundant and artculated locomotng robots. Anmatons of ths moton model are presented n an accompanyng vdeo. II. PRIOR WORK There s a sgnfcant amount pror work n the study of the moton of bologcal snakes [5], [6] and snake robots [1], [7].

2 A survey of a wde varety of snake robots and snake robot locomoton s presented n [8]. More recent research on both bologcal snakes [9], [1] and robotc snakes [11], [12] has focused on a snake s nteracton wth ts envronment durng locomoton. Our method dffers from [11] n that we model the robot as beng purely knematc. Whle [1] also presents a knematc model for hyper-redundant systems, the model reles on havng a contnuous backbone curve n order to understand and control the desred macroscopc shape of the robot. The assumptons made by our model are n much the same sprt as prevous work that has been done for sdewndng n both bologcal snakes [13] and robotc snakes [14], [15]. Our model dffers n that t uses the shape of the robot drectly and s agnostc to the underlyng functons used for moton control. Rather than fully account for the true forces actng on the robot or the true shape of terran wth whch we nteract, our goal s to present a smplfed model that s stll a reasonable approxmaton of the robot s moton. Furthermore we would lke to pursue a model that s computatonally nexpensve and can be easly be mplemented for systems that lack the necessary tactle, force, torque, or pose sensors needed to nform a full dynamc moton model. These trats make ths method potentally desrable to be used as the underlyng moton model of varous state estmaton technques such as partcle flters or kalman flters that are wdely used n the wheeled robot communty [16], [17]. III. MOTION MODEL In ths secton, we descrbe our mathematcal model for the robot s external moton due to ts nternal shape changes. We use the vrtual chasss [4] at each tmestep t as the body frame from whch to observe the moton of the robot s modules. Ths body frame has the property that t s constantly algned wth the snake robot s overall shape, and effectvely dentfes the flattest dmenson of the snake s shape (the dmenson correspondng to the thrd prncpal moment of nerta). It should be assumed that any use of the term body frame n ths paper means the body frame defned by the robot s vrtual chasss. Our model assumes that the axs correspondng to the flattest prncpal component s algned wth the ground surface normal, and ths axs s assgned to the z axs of the vrtual chasss. Ths makes our model smlar to that of most flat ground wheeled vehcle models, where we assume that translatons occur only n the x-y plane and that rotatons occur about only the z axs. In a way ths moton model can be thought of as a numerc dervaton of the common no-slp model for wheeled robots. It would be analogous to observng the dfferental moton of the bottom of a vehcle s wheels n the body frame at each tme step and usng that moton to predct the moton of the vehcle n the world. A. Gats and Robot Knematcs To smplfy control of the 16 degrees of freedom used to locomote our robot, we rely on gats that are pre-defned Fg. 2: Examples of the vrtual chasss body frame for the four gats n ths paper. Movng across from top left to bottom rght: rollng, sdewndng, sltherng, and turn n place. cyclc undulatons that are passed through the length of the snake. All of the gats presented n ths paper use parameterzed sne waves that are smlar to Hrose s serpenod curve [7], and ts 3D extensons [18]. The serpenod curve descrbes the curvature of a backbone as a functon of tme, poston on the backbone, and other gat-specfc parameters. To provde 3D moblty and manpulaton, the robot s jonts are alternately orented n the lateral and dorsal planes of the robot. Because of ths desgn, our framework for gats conssts of separate parameterzed sne waves that propagate through the lateral (even-numbered) and dorsal (odd-numbered) jonts. We refer to ths framework as the compound serpenod curve, α(n, t) = { βodd + A odd sn(ξ odd ) odd β even + A even sn(ξ even + η) even ξ odd = ψ odd n + ν odd t ξ even = ψ even n + ν even t. In (1) β, A and η are respectvely the angular offset, ampltude, and phase shft between the lateral and dorsal jont waves. In (2) the parameter ψ descrbes the spatal frequency of the macroscopc shape of the robot wth respect to module number, n. The temporal component ν determnes the frequency of the actuator cycles wth respect to tme, t. Usng equatons (1) and (2), the parameters descrbng a gven gat, and the robot s mechancal desgn parameters (confguraton of jont axes and the dstance between jonts), we can generate the 3-dmensonal shape of the snake robot from the forward knematcs of the system. At each tme step, the vrtual chasss body frame for the robot s shape s calculated as descrbed n [4]. The vrtual chasss for the gats presented n ths paper s shown n Fgure 2. B. Module Moton The means n whch a module s moton can cause external movement are twofold: a translaton of a module s center n the body frame and a rotaton about a module s center that causes a translaton at the surface of the module touchng the ground (Fg. 3). The effect of rotatonal module moton s mostly seen n gats lke rollng (top-left n Fgs. 1 and (1) (2)

3 -z a t d_ 2 a t Ω t d_ 2 b t Fg. 3: A smplfed 2-D dagram showng the effects of translaton and rotaton of a module on the pont that s assumed to be n contact wth the ground. Weght Poston n Normalzed Threshold Fg. 4: Plot of our weghtng functon over dfferent curvature parameters. 2) where module postons stay fxed n the body frame, but rotate n place. More complcated motons such as sdewndng and sltherng (top-rght and bottom-left n Fgs. 1 and 2) are drven prmarly by the translatonal module moton where the postons of the modules themselves move wth respect to the body frame. To calculate a module s translatonal moton at each pont n tme, we subtract the module s poston n the body frame at the prevous tmestep from ts current poston n the body frame. The poston of the th module of the robot at tme t s a t = x y. (3) z Thus the dfferental moton of the th module due to ts translaton n the body frame at tme t s t a t = a t a t 1. (4) To calculate the effects of a module s rotatonal moton, each module can be pctured as a wheel. Modules are modeled as spheres, where we are nterested n the pont on the sphere that ponts down (n the -z drecton of the body frame) at each pont n tme. We can defne ths drecton n the frame of each module by r t = (R t) 1 d/2. (5) In (5) d s the dameter of the module, and R t s the rotaton matrx that descrbes the orentaton of the th module n the body frame at tme t. We can descrbe the dfferental rotaton of the th module (n that module s frame) at tme t by Ω t = (R t 1) 1 R t. (6) We can apply ths dfferental rotaton both forwards and backwards to the pont r t, average the dsplacements due to the rotatons, and rotate the result back nto the body frame b t = R t Ω tr t (Ω t) 1 r t. (7) 2 Fnally, we can sum the moton component due to a module s translaton, a t, and a module s rotaton, b t, p t = a t + b t (8) Fg. 5: Dagram qualtatvely llustratng a possble threshold range for sdewndng. n order to capture the full moton that a module would mpart on the body frame of the robot, f t were contactng the ground n the -z drecton n the body frame. C. Ground Contact Model The next step of the moton model s to determne whch modules are assumed to be n contact wth the ground. Snce for most cases the flat axs of the vrtual chasss s relatvely normal to the ground, we assume that modules wthn a threshold of the lowest poston n the z-axs of the body frame (axs closest to pontng upwards n the real world) are havng ground contact. For reference, each module of the robot s 5 cm n dameter. The average module range of moton n the z-axs of the vrtual chasss body frame s shown n table I. To avod dscontnutes n the model as modules pass n an out of the ground contact threshold, we ntroduce a normalzed exponental smoothng functon that weghts the lowest modules more strongly and gradually decreases the weght to zero through the range of the threshold, γt = 1 z t z mn, f zt z mn < τ τ (9), otherwse w t = 1 e δγ t 1 e δ (1) ŵ t = wt n. (11) =1 w t In (9), τ s the threshold parameter that controls the range of postons for whch modules are consdered to be n contact wth the ground, zt corresponds to the z poston TABLE I: Gat Z-Axs Range of Moton Gat Range (cm) Rollng.4 Sdewnd 4.5 Slther 4.7 Turn-n-place 4.8

4 of the th module and z mn s the mnmum z poston of all modules at tme t. In (1), δ s a curvature parameter that controls the shape of the weghtng functon through the threshold. Fgure 4 shows the effect of δ on the shape of the weghtng functon. Ths smplfcaton ntroduces errors whch depend partly on the angle between z-axs of the vrtual chasss and the vector normal to the actual ground as well the number of modules that mght be contactng the ground smultaneously, due to the rregular shape of the modules. Snce these trats dffer for every gat, we explored optmzng τ and δ on a gat-by-gat bass. In secton IV we present results for motons that are optmzed for translatng gats, as well as results for a general set of parameters that have been optmzed across a range of motons. D. Body Frame Translaton Usng the estmated dfferental moton for each module, and the weghts that represent assumed ground contact, we are able to estmate the overall moton of the robot. The predcted translatonal moton of the robot s calculated by the weghted average of module motons n the body frame n m t = ŵt p t. (12) =1 Whle the predcted translaton wll contan components n the x, y and z drectons, the component of moton n the z drecton s small compared to x-y and wll be neglected due to the constrants of the planar moton model. It s mportant to note the negatve sgn on the sum n (12), snce our model assumes the moton of the robot s due to the reacton of the dsplacements of modules n the body frame. E. Body Frame Rotaton The rotatonal moton of the robot can also be calculated n a smlar manner as translaton. Because the moton model s constraned to representng translatons and rotatons n the x-y plane, we compute a weghted average of the rotatons that the modules nduce about the geometrc center of the robot (the orgn of the body frame). For ths, we frst compute moment vectors from the cross product of the poston of each module n the body frame wth the z-axs and normalze the resultng vector to be unt length d t = 1 d t = a t (13) d t d t (14) We then calculate the weghted sums of the dot product of each module s dsplacement wth the module s correspondng moment vector dvded by the dstance of that module to the geometrc center of the robot θ t = n =1 ŵ t ( p t d t) a. (15) t Fg. 6: Dagram showng the estmated translaton and rotaton of the robot by averagng the weghted moton of all modules. As wth the robot s translaton, the sgn of the summed rotaton s flpped because fact that the robot s rotaton s caused by the reacton of the modules dsplacements about the geometrc center of the robot as vew n the body frame. F. Full Body Frame Moton The total translatonal and rotatonal moton at tme t can be combned nto a homogeneous transform that can be appled to the body frame of the robot at each pont n tme T t = cos( θ t ) sn( θ t ) m x t sn( θ t ) cos( θ t ) m y t 1 1. (16) In (16), m x t and m y t are respectvely the x and y components of the robot s translatonal moton, m t, from (12). IV. EXPERIMENT To verfy our model, we ran experments usng the four gats mentoned prevously: rollng, sdewnd, slther, and turn-n-place. To optmze τ and δ we performed test runs of the dfferent gats on our lab floors. Four trals of turn-nplace and sx trals each of rollng, sdewnd, and slther were conducted. The runs were started at dfferent phases wthn ther respectve gat cycles, moved over varous dstances, and run n forward and reverse n order to gve a wde samplng of external moton wthn the real world. For each tral we recorded the feedback jont angles from the robot and manually measured the change n poston and orentaton of the robot n the world. After tunng the model parameters, 24 new trals were run to test the model s accuracy The model was also tested on prevously logged data of the robot movng n a moton capture lab. A. Model Optmzaton To represent the poston and orentaton of the robot we chose to use polar coordnates nstead of cartesan coordnates, snce t more ntutvely and stably represents motons where the movements are prmarly along a sngle axs ɛ r = r meas r pred r meas (17) ɛ φ = φ meas φ pred π (18)

5 TABLE II: Table of Parameters.6 Total Error.1 Threshold (cm) 8 2 Curvature Fg. 7: Averaged total errors for all of the tranng trals, plotted for a range of thresholds and curvatures. ɛ θ = θ meas θ pred π 2 (19) In our parameterzaton r s the dstance the robot traveled from the orgn, φ s the angle of the robot s fnal poston, and θ s the angle of the robot s orentaton. These errors were normalzed to be percent error for the dstances (17) and by π radans for the angles (18) (19). The total error functon s a weghted summed squares ɛ total = λ r (ɛ r ) 2 + λ φ (ɛ φ ) 2 + λ θ (ɛ θ ) 2 (2) where λ vares the relatve weghtng of the dfferent errors. For the gats that are used for translatng the snake (sdewnd, rollng and slther) we used the weghts: λ r =.2, λ φ = 1, λ θ = 1. For the turn-n-place gat where φ s not meanngful and θ s more mportant, we used the weghts: λ r =.5, λ φ =, λ θ = 1. Fgure 7 shows the total error for a range of thresholds and curvatures, averaged for the 22 tranng trals of the varous gats. It llustrates the farly wde regon of usable parameters for the model. The optmal parameters for predctng moton vared accordng to the type of gat, and are shown n Table II. Notably, the predcted moton for the turn-n-place gat became more accurate wth a much hgher threshold and steeper curvature. Snce the other gats are less senstve to varyng the model parameters, the overall best parameters based on the tranng data were very smlar to the ones for turn-n-place. B. Results To test the accuracy of the model, 6 new trals of each gat were conducted for a total of 24 trals. As wth the prevous trals, gats were started at varous ponts n ther respectve cycles and run n dfferent drectons n order sample a large range of moton. Tables III and IV present the means and standard devatons for both the specfc and general model parameters for each gat. Note that the results for the turn-n-place gat are presented dfferently from the other gats. Due to the fact that the goal of the turn-n-place gat s to rotate the snake, the yaw motons are presented n terms of percentage error, rather than degrees. Smlarly, the translaton of the robot s so small that they are presented n Gat Type τ (cm) δ Rollng / Sdewnd / Slther Turn-n-place Overall percentage of robot length nstead of percentage of dstance traveled. The moton model was also tested on prevously logged moton capture data of a smlar robot and envronment. The robot n the moton capture data had a dfferent head, tal, and tether from the confguraton used n the other trals. It was was also movng on the concrete floor of the moton capture studo, nstead of the tle floor n the lab. In Fg. 8 we show the estmated moton from the logged jont angles, compared to the moton capture data of the robot s actual moton. V. CONCLUSIONS We have presented a general method for estmatng the moton of a hgh DOF snake robot on flat ground. Our method makes use of the robot s vrtual chasss body frame to separate the robot s nternal shape changes from external moton n the world as well as to dentfy the parts of the robot that are n contact wth the ground. The utlty of ths model les n ts smplcty. As long as one knows the knematc confguraton at each tmestep, ths method can be appled to a robot that lacks force feedback, tactle sensng, or even orentaton sensng (beyond what s needed to properly ntalze the vrtual chasss). Even though the assumptons of our model neglect dynamc effects lke nerta, frctonal forces, and ntermttent ground contact, t provdes a relatvely accurate estmate of moton at much less computatonal expense compared to full 3D dynamc smulatons or more complex dynamc models. TABLE III: Moton Model Errors - Rollng / Sdewnd / Slther Parameters r (%) φ (deg) θ (deg) Mean Dev. Mean Dev. Mean Dev. Rollng 9 % 2 % Sdewnd 5 % 2 % Slther 6 % 3 % r (%) φ (deg) θ (%) Mean Dev. Mean Dev. Turn-n-place 12 % 4 % 7 % 2 % TABLE IV: Moton Model Errors - General Parameters r (%) φ (deg) θ (deg) Mean Dev. Mean Dev. Mean Dev. Rollng 9 % 2 % Sdewnd 18 % 2 % Slther 6 % 3 % r (%) φ (deg) θ (%) Mean Dev. Mean Dev. Turn-n-place 9 % 7 % 34 % 11 %

6 x (cm) Actual Predcted 5 5 y (cm) 1 15 Fg. 8: Comparson between moton capture data and estmated moton for sdewndng at a speed of approxmately 1 cm/s. Although the parameters of the model can be optmzed for a gven gat or moton, a general set of model parameters have been demonstrated to be effectve for a wde range of gats that are frequently used for traversng flat ground. Further trals wll be performed wth addtonal motons and on a varety of terrans (carpet, grass, asphalt, etc.) to see how effectvely ths model extends to other operatng condtons. VI. FUTURE WORK A. Gat Desgn and Moton Plannng One of the ongong challenges wth controllng snake robots s desgnng effcent and useful motons that take nto account nteracton wth the world. Most of the gats developed by our lab have been the result of hand-tuned functons followed by ether expermental refnement or some form of offlne optmzaton [15], [19]. One lne of future work could nvolve usng ths model as part of more nteractve tools for gat development, or serve as the model for onlne optmzaton of gat transtons or other non-cyclc motons. B. State Estmaton / SLAM Perhaps the most relevant extenson of ths work would be ncorporate t nto state estmaton or use as a moton model for SLAM algorthms. In many ways the approxmaton of ths model mrrors the no-slp assumpton often made n the moton model of many wheeled robotc systems. Gven the success that wheeled robots have had n usng a smplfed and often poor assumpton of ground nteracton, we feel that moton model framework presented n ths work may fulfll such a role for snake robots and artculated locomotng systems. C. Extendng to Other Terrans The most sgnfcant lmtatons of the current model s the assumpton of flat ground. However, ths assumpton s used prmarly as a stand-n for a lack of true ground contact sensng. If the robot s modules were equpped wth tactle sensors or some other method of determnng ground contact was avalable ths model could be adapted to handle full 3D moton n a farly straght-forward manner. In the meantme, other structured envronments for whch to adapt ths model are the nsdes and outsdes of ppes. REFERENCES [1] G. Chrkjan and J. Burdck, The knematcs of hyper-redundant robot locomoton, IEEE Transactons on Robotcs and Automaton, vol. 11, no. 6, pp , [2] C. Wrght, A. Johnson, A. Peck, Z. McCord, A. Naaktgeboren, P. Ganforton, M. Gonzalez-Rvero, R. L. Hatton, and H. Choset, Desgn of a Modular Snake Robot, Proceedngs of the IEEE Internatonal Conference on Intellgent Robots and Systems, Oct. 27. [3] M. Tesch, K. Lpkn, I. Brown, R. L. Hatton, A. Peck, J. Rembsz, and H. Choset, Parameterzed and Scrpted Gats for Modular Snake Robots, Advanced Robotcs, vol. 23, pp , June 29. [4] D. Rollnson and H. 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Choset, Usng Response Surfaces and Expected Improvement to Optmze Snake Robot Gat Parameters (211), n Internatonal Conference on Intellgent Robots and Systems, 211. VII. ACKNOWLEDGEMENTS The authors would lke to thank the members of the Borobotcs Lab and Robotcs Insttute, partcularly Ross Hatton, Matt Tesch, Austn Buchan and Justn MacEy.