DSCC MANEUVERABILITY AND HEADING CONTROL OF A QUADRUPED ROBOT UTILIZING TAIL DYNAMICS

Transcription

1 Proceedings of the ASE 07 Dynaic Systes and Control Conference DSCC07 October -3, 07, Tysons, Virginia, USA DSCC ANEUVERABILITY AND HEADING CONTROL OF A QUADRUPED ROBOT UTILIZING TAIL DYNAICS Wael Saab Robotics and echatronics Lab echanical Engineering Dept. Virginia Tech Blacksburg, VA, USA Pinhas Ben-Tzvi Robotics and echatronics Lab echanical Engineering Dept. Virginia Tech Blacksburg, VA, USA bentzvi@vt.edu ABSTRACT This paper presents odeling and analysis of a quadruped robot that utilizes tail dynaics to control its heading angle. The tail is envisioned to assist locootion as a eans separate fro its legs to generate forces and oents to iprove perforance in ters aneuverability. Tail otion is analyzed for both low and high-speed tail actuation to derive sufficient conditions to aintain equilibriu and forulate aneuverability relations that result in rotation and translation of the robotic syste. Sensitivity analysis is presented to select optial tail ass and length ratios to axiize the change of the heading angle. A heading controller is then proposed and siulated to achieve a desired heading angle utilizing tail dynaics. Results of this research will assist in the design, odeling, and analysis of robotic systes sharing siilar topologies to the proposed odel, such as obile robots with wheeled, tracked, ulti-legged, or articulated-body based locootion with swinging extreities such as tails, torsos, and libs. bodies [6, 7]. Siilarly, cheetahs have been observed to swing their tails in rapid otions while chasing prey to aintain the roll of their bodies [8]. Lizards have also been observed to use their tails during aerial aneuvers to adjust their pitch for a sooth landing [9, 0]. By observing nature s abundant store of solutions, engineers can gain a source of inspiration in designing robots to address the challenges persisting in the field of legged robot design and control []. These challenges involve designing legged robots with high aneuverability and liited size, weight, and cost, and developing algoriths to control foot trajectories to achieve forward walking and turning gaits. The ajority of research in the field of legged robotics has focused on the ipleentation of steady state trotting, galloping, and bounding [-4]. We hypothesize that the burden on the legs to siultaneously stabilize and aneuver the robot can be reduced by utilizing an inertial appendage in the for of an attached tail. This can lead to siplifications of leg designs and control algoriths. Various ethods have been investigated to enhance aneuverability of obile robots using an attached inertial appendage. A lizard-inspired tail has been successful eployed on a wheeled robot to dynaically self-right and control pitch to soothly transition between various ground slopes, using the assuption of zero net angular oentu exchange [5, 6]. A siilar approach was presented for a iniature juping robot and a bio-inspired kangaroo robot with an active tail utilized for pitch control between juping instances [7, 8]. A Newton-Euler approach was used to odel a cheetah robot with a two degree-of-freedo (DOF) attached tail during idair flight. The authors developed and siulated an attitude controller and deonstrated preliinary experiental results of disturbance rejection to aintain stability [8]. A oth inspired robot successfully deonstrated self-righting and id-air turning using an inertial appendage [9]. Patel et al. derived a INTRODUCTION Biologists have long observed the uses and benefits that tails provide anials, which range fro fat storage to counication []. Perhaps the ost useful tail functionalities applicable to the field of obile legged robotics are enhanced stabilization and aneuverability capabilities [, 3]. Hickan [4] provided an extensive review of tail usage in aals. For instance, kangaroos use their tails as a counterbalance while standing on their hind legs and as a eans of energy storage to assist in hopping [5]. Kangaroo rats have been observed to rapidly swing their tails between hops to rotate their bodies in idair and continue hopping in a different direction. Dinosaurs, such as the Tyrannosaurus, are believed to have used their tails as an active counter balance and as a eans of stabilization when walking forward, to aintain the yaw orientation of their Copyright 07 ASE

2 atheatical odel of a wheeled robot with a single DOF tail capable of roll otion that was assued to be rotating about a grounded pivot point using the Euler-Lagrange forulation. The authors developed a control algorith and deonstrated with siulations that the robot could perfor a high-speed roll faster than a tail-less robot [0]. The ajority of research on robotic tails has focused on attitude control during id-air flight using the zero-net transfer of angular oentu, an approach that has been extensively proposed in space robots and satellites for attitude control [-3]. There has also been research conducted to analyze the dynaics of obile robots with ground contact friction. Kohut et al. investigated dynaic turning of a hexapod robot using yaw otion of a tail [4, 5]. In this analysis, it was assued that the tail torque occurs as soon as the tail is actuated that overwhels static friction; therefore, dynaic friction between the feet and ground induces a oent about an effective radius during tail actuation. However, this odel did not consider the effects of translation caused by inertial forces. Casarez et al. developed an analytical odel of a obile robot with a reaction wheel that transferred a pure oent, neglecting effects of inertial forces, to the robotic body to deterine the effects of otor voltage, friction, and wheel inertia had on heading angle [6]. Chernous Ko odeled ulti-body echanical systes consisting of several links which can perfor snake-like locootion s along a horizontal plane in the presence of dry friction. Periodic otions consisting of slow and fast phases were used to create controllable longitudinal, latitudinal translational and rotational otions using a sequence of eleentary otions [7, 8]. This paper adapts the fraework presented in [5, 7, 8] to odel and analyze a robotic syste consisting of a quadruped robot and swinging inertial appendage in the for of a tail. This paper is part of ongoing research that investigates the use of inertial tails on legged robots, which are envisioned to provide a eans of external loading separate fro the leg echaniss, to generate forces and oents to aid aneuverability and stabilization [9-38]. The ain contributions of this paper are: () odeling of the quadruped and swinging inertial appendage in the presence of friction to analyzes the effects of both low-speed and high-speed tail actuations on the robotic syste in ters of aneuverability, defined as the resultant rotation and translation of the syste, caused by both inertial forces and oents, () sensitivity analysis that utilizes the developed odels to copute optial tail ass and length ratios to axiize heading angle rotation of the legged robot, and (3) design and siulate a heading angle controller that utilizes tail dynaics. This paper is organized as follows. Sections and 3 present odeling and kineatic analysis of the robotic syste. Section 4 presents dynaic analysis of low and high-speed tail actuations, to derive sufficient conditions to aintain equilibriu and the aneuverability relations to produce translation and rotation of the robotic syste. Section 5 presents sensitivity analysis to copute optial tail length and ass ratios to axiize heading angle change. Section 6 presents siulation results of a heading angle controller. Concluding discussions are then presented in section 7. ROBOTIC SYSTE ODELING This section presents the robotic syste under study and the odel that is used for analysis throughout this paper. Figure shows a scheatic diagra (side view) of robotic syste that consists of a quadruped robot with an attached tail developed by the authors [37, 39]. The quadruped consists of four robotic odular legs that are designed to be low-inertia, two DOF echaniss capable of perforing planar walking gaits that enable planar forward locootion. It is envisioned that rotation of the tail will enable to quadruped to adjust its heading angle for left and right turning as will be deonstrated in section 6. The quadruped s ass is concentrated in the body region due to the locations of its otors. It is assued that during a walking gait the legs do not significantly change the ass properties of the robot. The tail is coposed of an actuation unit that houses a geared otor assebly and a rigid rod with an attached tip ass. Tail Actuation Unit Qaudruped Tip ass Body Isoetric View Legs Figure. Scheatic diagra of the robotic syste coposed of a quadruped and tail. Figure depicts the free body diagra (top view) of the robotic syste. A body attached frae of reference (O', b, b ) is fixed to the quadruped at O' = (x 0, y 0 ). The bodies are odeled as point asses 0 (actuation unit), (quadruped body), and (tail tip ass). In the figure, the quadruped and tail are disconnected at the tail revolute joint. The tail gearbox and actuator assebly is located at point O'. The bodies can translate in the inertial e e -plane and the quadruped and tail can rotate relative to one another, about the e 3 -axis. The heading angle of the quadruped and relative yaw angle of the tail are defined as θ and β, respectively. Therefore, the syste is odeled with four DOFs. The following variables will be used to represent the physical paraeters of the syste in this analysis: i represents luped asses, v i is the ass velocities, and l i represents effective radius, defined as half the diagonal distance between legs. The index i = {0,, } refer to the actuation unit, quadruped, and tail bodies, respectively. Copyright 07 ASE

3 An input torque of agnitude applied onto the tail produces yaw rotation about its revolute joint. A torque of equal agnitude and opposite direction acts on the quadruped body during tail actuation. H (sin( ) ( 0 0) cos( ) ( O l y x l x0 y0) l[cos( ) x( ) sin( )( y( ) x ) cos( ) y ] y x x y ) e (3) O e Tail F r e F t l b O' 0 F r β θ F t l b Quadruped Figure. Free body diagra (top view) of the robotic syste consisting of a quadruped robot and tail Coulob friction is used to odel friction forces that act on the syste at points of contact between the quadruped s feet and ground. During instances of sliding contact otion, the suation of friction forces is equal to μg and opposes the direction of sliding velocity Here, μ represents the friction coefficient (static and dynaic friction coefficients are assued to be equivalent), is the total ass of the syste = 0 + +, and g is the gravitational acceleration. 3 KINEATIC ANALYSIS This section presents kineatic analysis of the robotic syste, and derivations of the center of ass location and angular oentu of the robotic syste. With reference to Fig., the CO of the syste is coputed as follows x0 l cos( ) lcos( ) xc e () y0 l sin( ) lsin( ) yc e By defining the angular velocities of the quadruped and tail to be e3 and e 3, the total angular oentu of the robotic syste about point O is given by Eq. () 3 H R v () O i i i i0 Substituting position and velocity relations into the above equation and evaluating the cross products yields the total angular oentu of the syste: 4 DYNAIC ANALYSIS This section analyses low and high-speed tail actuation to deterine the conditions required to aintain equilibriu and derive the relations of aneuverability that result in translation and rotation of the robotic syste. Analysis of this low-speed tail otion will study the forces and oents produced by the tail to deterine the conditions required to aintain equilibriu. Low-speed tail actuation involves rotating the tail to desired initial conditions while aintaining equilibriu. Equilibriu refers to steady state conditions where the syste reains stationary due to resistive friction forces. To aneuver, a high-speed tail actuation will be perfored to generate the necessary forces and oents to overcoe friction. Let the agnitude of the oent induced by friction forces be denoted by f. During this type of actuation, it is assued that the input torque agnitude is larger than the friction induced oent > f ; therefore, the external frictional forces between the quadruped s feet and ground can be neglected; thus, satisfying the conservation laws of linear and angular oentu. Since the syste is initially at equilibriu, after fast tail otions, x c and y c are constants of otion and H 0 = 0. These conservation laws will then be used to analyze high-speed tail actuation, to forulate relations of aneuverability in ters of variations of translation and rotation of the robotic syste. 4. Low-Speed Tail Actuation In this section we forulate the conditions to aintain equilibriu during low-speed tail actuation. Assuing that the syste is initially at equilibriu, we first estiate the upper bound of generated forces fro tail actuation and use the to analyze the resultant force balance on the quadruped, to deterine the axiu threshold of tail otion such that friction induced forces cancel those caused by the actuator torque and loading generated by the tail. For low-speed tail actuation, the axiu generated forces resulting fro tail actuation that aintain syste equilibriu occur at a certain upper, axiu threshold of tail otion paraeters defined by ε and Ω, which represent the agnitude of angular velocity and acceleration, respectively. Using this notation and considering the tail as the syste of study, the axiu agnitudes of F t and F r, tangential and radial force agnitude coponents exerted on the tail by the quadruped, and can be estiated by Fr l Ft l l (4) 3 Copyright 07 ASE

4 We now study the effects of these axiu forces and oent on the quadruped consisting of asses 0 and. Rotational equilibriu will be aintained if the agnitude of the axiu induced oent, coposed of the input torque and the oent resulting fro axiu generated forces F t and F r, about the cobined CO does not exceed the friction induced oent agnitude f. Translational equilibriu will be aintained if the agnitude of the axiu cobined generated forces, F, does not exceed the friction force agnitude F f. This eans that the generated loading due to the low speed-tail actuation will not cause the quadruped to aneuver, since it is opposed by the friction induced forces and oents. Therefore, the conditions to aintain equilibriu are defined as F F f A balance of forces and oents about the cobined CO with respect to the body attached frae yields the following equation for F and F ( Fr cos Ft sin ) b( Fr sin Ft cos ) b [ x ( F cos F sin ) y ( F sin F cos )] b (6) c t r c t r 3 Substituting Eq. 4 into Eq. 6 and using the Cauchy- Schwarz inequality, relations for F and can be defined as F l l l 4 f l l ( ) For ulti legged robotic systes, F f can be estiated as a force proportional to the total weight of the robotic syste g and μ. Siilarly, f is proportional to F f and the effective radius l. Substituting these relations and Eq. 7 back into Eq. 5 yields the necessary condition to aintain equilibriu: l l l l gl ( ) l 4 g If the above inequalities are satisfied, the syste will aintain equilibriu and reain stationary, enabling the robotic syste to position the tail during low-speed tail actuation prior to aneuvering using high-speed tail actuation. (5) (7) (8) 4. High-Speed Tail Actuation In this section, the laws of conservation of linear and angular oentu, discussed in Section 4, are applied to derive the relations of aneuverability during high-speed tail actuation. Using the relation xc yc 0 fro conservation laws, the tie derivative of Eq. is coputed to derive the velocity of point ass 0 : x y l sin l ( )sin( ) l cos l ( )cos( ) 0 e 0 e Substituting Eq. 9 into the relation for total angular oentu, Eq. 3, where H 0 = 0 due to conservation laws, yields an expression for the heading angle rate of change l ( 0 ) d 0 0 l ( ) l ( ) d (9) (0) Where d ll cos. It can be observed fro Eq. 0 that θ is inversely related to β. This relation is intuitive since any action of the tail will have an equal and opposite reaction on the quadruped. Nuerical integration of Eq. 0 yield the relation of heading angle variation defined by f ( ) d () f i where β i and β f represent the initial and final values of β. By integrating Eq. 9 over the initial and final values of tail otion, the translation of the syste is coputed to be x y acos a cos( ) asin a sin( ) 0 e 0 e () Equations () and () represent aneuverability relations of the legged robot due to high-speed tail actuations. 5 HEADING ANGLE SENSITIVITY ANALYSIS The aneuverability relations derived in this section are applicable to a wide range of obile robots that utilize dynaics of a tail. This section analyses the sensitivity of the tail ass and length ratios, defined as σ = /( 0 + ), λ = l /l, on the variation of heading angle. For a quadruped of ass = 3 kg, 0 = kg and effective radius l = 0.5, Eq. () was evaluated for / rad. Figure 3 shows the plot of heading angle vs. ass ratio for various length ratio values λ = {0.5,,, 3, 4}. 4 Copyright 07 ASE

5 Fro the plot, it can be observed that increasing the values of β and λ does increase θ, since the variation of heading angle changes with design paraeters and reaches near steady state conditions. For exaple, past a certain ass ratio threshold of σ ~ 0.05, the addition of tail ass does not significantly cause large variations of heading angle, since the slope of θ nearly approaches zero. This trend physically akes sense because tail ass does increase the inertial forces transferred to the quadruped, but also increases the overall weight of the syste, resulting in a larger friction induced oent that ipedes otion of the syste. It can be observed that heading angle is ore sensitive to the tail length ratio. This sensitivity can be seen in Fig. 3 by observing the large differences between steady state θ values as λ increases. However as λ is increased beyond a threshold λ ~, the effectiveness of heading angle variation decreases. This trend can be physically explained due to the quadratic increase of tail inertia with respect to tail length; however, inertial forces in the tangential and radial directions also affect θ, as seen in Fig., and are expected to liit the variation of heading angle for large values of λ. Therefore, the ost effective eans to obtain large variations of heading angle is to axiize tail length while satisfying diension constraints. Results of this analysis can be used to effectively size a robotic tail for a given legged robot of known ass and diensions. approxiation u l and put into state space for with the state vector defined as [, ]. The gains of the LQR controller were anually tuned to produce acceptable rise tie and settling tie for a desired heading angle θ des, and input saturation u sat was incorporated to better approxiate actuator perforance. Siulation paraeters of the physical syste, Fig., were obtained fro CAD software as: 0 = kg, = 3 kg, = 0.3 kg, l = 0.5, l =. Quadruped ass paraeters were extracted fro coputer aided design software. The tail ass and length ratios were chosen based off the analysis in section 5 such that σ = 0.0 and λ =. Figure 4 shows siulation results of heading angle and heading angular velocity for desired heading angle set point values θ d = {π/8, π/4, π/} rad and u sat = ± 5 N. Figure 5 shows the siulation results of the resultant tail angle for the three desired heading angle set points. Fro Figs. 4 and 5 is can be observed that: ) the LQR controller rise tie is approxiately equivalent to 3 s for all three case scenarios although θ d is increasing, this results in a higher tail angular velocity to achieve the desired set-points, ) β increases inversely proportional with respect to θ d as expected fro Eq. 0. These results can be used to design the physical syste for experientation. For exaple, the high requireents of agnitudes ust be practically achieved by the chosen geared otor based on specifications and the tail ust be designed to avoid interference with the quadruped body for large desired heading angle values based on siulated results of the resultant β angular displaceents or the control algorith should be odified to account for a liited tail range of otion. Using the resultant tail angular profiles in Fig. 5, the expected translation of the syste can be coputed using Eq.. Coputed results of translation of the robotic syste are presented in Table. Figure 3. Plot of heading angle vs. ass ratio for various length ratio values. 6 HEADING CONTROLLER The robotic syste is odeled as four DOFs with one active DOF in the for of torque input to the tail. Therefore, the syste is under actuated. At this stage, the heading angle of the quadruped will be deeed the ost iportant and a controller will be designed to close the loop on this variable; hence, the reaining DOF s will be uncontrolled. As a preliinary approach to designing a heading angle controller, we choose to ipleent a linear quadratic regulator (LQR). For ease of siulation, the tie derivative of Eq. 0 was coputed and linearized using a sall angle Figure 4. Siulation results of heading angle and velocity for various θ d set point values. 5 Copyright 07 ASE

6 ACKNOWLEDGENTS This aterial is based upon work supported by the National Science Foundation under Grant No Figure 5. Siulation results of tail angle and velocity for various θ d set point values. Table. Coputed translation of the robotic syste due to tail rotation. θ d {β i,β f } x 0 (c) y 0 (c) (rad) (rad) π/8 {0, -0.96} π/4 {0, -.9} π/ {0, -3.8} CONCLUSION In this paper, we analyzed a robotic syste coposed of a quadruped robot and an attached tail that is used to provide a eans separate fro its legs to generate external forces and oents to aneuver and control the heading angle. odeling and analysis of the syste was perfored, where sufficient conditions to aintain equilibriu and produce aneuverability using low and high-speed tail actuations were derived. Using the derived expression of heading angle variation, sensitivity analysis of heading angle dependent on design paraeters was presented and an LQR controller was ipleented to control heading angle. Results of this analysis can be used to choose tail length and ass ratios to axiize the variation of heading angle utilizing tail dynaics. Siulation results of the tail angular displaceent and velocities will be used in designing a physical prototype of the robotic syste and will aid the process of actuators selection. Future work will involve expanding the kineatic and dynaic odels to analyze the effects of articulated tail structures as opposed to the single bodied rigid pendulu odel used in this paper. A nonlinear controller will be developed to ore accurately estiate the response of the syste. REFERENCES [] Aleksiuk,., 968, "Scent-ound Counication, Territoriality, and Population Regulation in Beaver " Journal of aalogy, 49(4), pp [] Jindrich, D. L., and Qiao,., 009, "aneuvers During Legged Locootion," Chaos: An Interdisciplinary Journal of Nonlinear Science, 9(), p [3] Webb, P. W., 004, "aneuverability-general Issues," IEEE Journal of Oceanic Engineering, 9(3), pp [4] Hickan, G. C., 979, "The aalian Tail: A Review of Functions," aal review, 9(4), pp [5] Proske, U., 980, "Energy Conservation by Elastic Storage in Kangaroos," Endeavour, 4(4), pp [6] Howell, A. B., 944, "Speed in Anials, Their Specialization for Running and Leaping," Aerican Journal of Physical Anthropology, 3(). [7] Benton,. J., 00, "Studying Function and Behavior in the Fossil Record," PLoS biology, 8(3). [8] Briggs, R., Lee, J., Haberland,., and Ki, S., 0, "Tails in Bioietic Design: Analysis, Siulation, and Experient," Proc. International Conference on Intelligent Robots and Systes IEEE/RSJ, Vilaoura, Algarve Portugal, pp [9] Jusufi, A., Kawano, D., Libby, T., and Full, R., 00, "Righting and Turning in id-air Using Appendage Inertia: Reptile Tails, Analytical odels and Bio-Inspired Robots," Bioinspiration and Bioietics, 5(4), p [0] Libby, T., oore, T. Y., Chang-Siu, E., Li, D., Cohen, D. J., Jusufi, A., and Full, R. J., 0, "Tail-Assisted Pitch Control in Lizards, Robots and Dinosaurs," Nature, 48(7380), pp [] Bowling, A., 0, "Ipact Forces and Agility in Legged Robot Locootion," Journal of Vibration and Control, 7(3), pp [] Kazei, H., ajd, V. J., and oghadda,.., 03, "odeling and Robust Backstepping Control of an Underactuated Quadruped Robot in Bounding otion," Robotica, 3(03), pp [3] Krasny, D. P., and Orin, D. E., 03, "Evolution of Dynaic aneuvers in a 3d Galloping Quadruped Robot," Proc. International Conference on Robotics and Autoation, IEEE, Orlando, FL, USA, pp [4] Wang, X., Li,., Wang, P., and Sun, L., 0, "Running and Turning Control of a Quadruped Robot with Copliant Legs in Bounding Gait," Proc. IEEE International Conference on Robotics and Autoation IEEE, Shanghai, China, pp [5] Chang-Siu, E., Libby, T., Toizuka,., and Full, R. J., 0, "A Lizard-Inspired Active Tail Enables Rapid aneuvers and Dynaic Stabilization in a Terrestrial Robot," Proc. International Conference on Intelligent 6 Copyright 07 ASE

7 Robots and Systes, IEEE/RSJ, San Francisco, CA, USA, pp [6] Johnson, A.., Libby, T., Chang-Siu, E., Toizuka,., Full, R. J., and Koditschek, D. E., 0, "Tail Assisted Dynaic Self Righting," Proc. International Conference on Clibing and Walking Robots and the Support Technologies for obile achines, Springer, Baltiore, D, USA. [7] Liu, G.-H., Lin, H.-Y., Lin, H.-Y., Chen, S.-T., and Lin, P.- C., 04, "A Bio-Inspired Hopping Kangaroo Robot with an Active Tail," Journal of Bionic Engineering, (4), pp [8] Zhao, J., Zhao, T., Xi, N., Cintrón, F. J., utka,. W., and Xiao, L., 03, "Controlling Aerial aneuvering of a iniature Juping Robot Using Its Tail," Proc. International Conference on Intelligent Robots and Systes IEEE/RSJ, Tokyo, Japan, pp [9] Deir, A., ANKARALI,.., Dyhr, J., organsen, K., Daniel, T., and Cowan, N., 0, "Inertial Redirection of Thrust Forces for Flight Stabilization," Proc. International Conference on Clibing and Walking Robots and Support Technologies for obile achines, Springer, Baltiore, D, USA, pp [0] Patel, A., and Braae,., 03, "Rapid Turning at High- Speed: Inspirations fro the Cheetah's Tail," Proc. International Conference on Intelligent Robots and SystesTokyo, Japan, pp [] Fernandes, C., Gurvits, L., and Li, Z., 993, "Attitude Control of Space Platfor/anipulator Syste Using Internal otion," Space Robotics: Dynaics and Control, Springer, pp [] Papadopoulos, E., and Dubowsky, S., 99, "On the Nature of Control Algoriths for Free-Floating Space anipulators," IEEE Transactions on Robotics and Autoation, 7(6), pp [3] Yoshida, K., and Uetani, Y., 990, "Control of Space Free-Flying Robot," Proc. Conference on Decision and Control, IEEE, Honolulu, HI, USA, pp [4] Kohut, N., Haldane, D., Zarrouk, D., and Fearing, R., 0, "Effect of Inertial Tail on Yaw Rate of 45 Gra Legged Robot," Proc. International Conference on Clibing and Walking Robots and the Support Technologies for obile achines, Springer, Baltiore, D, USA, pp [5] Kohut, N. J., Pullin, A. O., Haldane, D. W., Zarrouk, D., and Fearing, R. S., 03, "Precise Dynaic Turning of a 0 C Legged Robot on a Low Friction Surface Using a Tail," Proc. IEEE International Conference on Robotics and Autoation, IEEE, Karlsruhe, Gerany, pp [6] Casarez, C., Penskiy, I., and Bergbreiter, S., 03, "Using an Inertial Tail for Rapid Turns on a iniature Legged Robot," Proc. International Conference on Robotics and Autoation IEEE, Karlsruhe, Gerany, pp [7] Chernous' ko, F., 005, "odelling of Snake-Like Locootions," odeling, Siulation and Optiization of Coplex Processes, pp [8] Chernous' ko, F., 00, "Controllable otions of a Two- Link echanis Along a Horizontal Plane," Journal of Applied atheatics and echanics, 65(4), pp [9] Rone, W. S., and Ben-Tzvi, P., 0, "Continuu anipulator Statics Based on the Principle of Virtual Work," Proc. International echanical Engineering Congress and Exposition, ASE, Houston, TX, USA, pp [30] Rone, W. S., and Ben-Tzvi, P., 03, "ulti-segent Continuu Robot Shape Estiation Using Passive Cable Displaceent," Proc. International Syposiu on Robotic and Sensors Environents, IEEE, Washington, DC, USA, pp [3] Rone, W. S., and Ben-Tzvi, P., 04, "Continuu Robotic Tail Loading Analysis for obile Robot Stabilization and aneuvering," Proc. International Design Engineering Technical Conferences & Coputers and Inforation in Engineering Conference, ASE, Buffalo, NY, USA, pp. V05AT08A009-V005AT008A009. [3] Rone, W. S., and Ben-Tzvi, P., 04, "echanics odeling of ultisegent Rod-Driven Continuu Robots," ASE Journal of echaniss and Robotics, 6(4), p [33] Rone, W. S., and Ben-Tzvi, P., 04, "Continuu Robot Dynaics Utilizing the Principle of Virtual Power," IEEE Transactions on Robotics, 30(), pp [34] Rone, W. S., and Ben-Tzvi, P., 05, "Static odeling of a ulti-segent Serpentine Robotic Tail," Proc. International Design Engineering Technical Conferences and Coputers and Inforation in Engineering Conference, ASE, Boston, A, USA, pp. V05AT08A054-V005AT008A054. [35] Rone, W. S., and Ben-Tzvi, P., 06, "Dynaic obile Robot aneuvering Utilizing Tail-Induced Inertial Loading," ASE Journal of Dynaic Systes easureent and Control [36] Saab, W., and Ben-Tzvi, P., 06, "Design and Analysis of a Discrete odular Serpentine Tail " Proc. International Design Engineering Technical Conferences & Coputers and Inforation in Engineering Conference, ASE, Charlotte, NC, USA, p. 8. [37] Saab, W., and Ben-Tzvi, P., 06, "Design and Analysis of a Robotic odular Leg," Proc. International Design Engineering Technical Conferences & Coputers and Inforation in Engineering Conference, ASE, Charlotte, NC, USA, p. 8. [38] Saab, W., Rone, W., and Ben-Tzvi, P., 07, "Discrete odular Serpentine Robotic Tail: Design, Analysis and Experientation," Robotica. [39] Saab, W., Rone, W., and Ben-Tzvi, P., 06, "Robotic odular Leg: Design, Analysis and Experientation," Journal of echaniss and Robotics. 7 Copyright 07 ASE