Locomotion with Continuum Limbs - CiteSeerX

2012 IEEE/RSJ International Conference on Intelligent Robots and Systems October 7-12, 2012. Vilamoura, Algarve, Portugal

Locomotion with Continuum Limbs Isuru S. Godage♯ , Thrishantha Nanayakkara⋆ , and Darwin G. Caldwell♯

locomotion in unstructured environments. In contrast to rigid limbs, continuum limbs open up a wide range of potential applications in legged locomotion in cluttered environments, since omni-directional continuous smooth bending (Fig. 2a) and inherent, uniformly distributed compliance can be used for versatile purposes like whole arm grasping and as support appendages to surmount barriers [6]. Moreover, this new class of limbs can easily adapt to irregular environments through different types of movement patterns like crawling and walking, since their broad actuation space as shown in Fig. 2b enables easy real-time generation of different limb shapes and gait patterns. Soft appendages also mean lower collision forces given a speed. This feature can be very useful in many direct encounters with humans such as mobile robots in crowded places. Moreover, the softness of the legs will be an advantage to deform the whole body to squeeze through narrow gaps in many search and rescue, defense, and industrial applications. Given the state of the art of continuum robotic arm designs, the idea of continuum limbs is not without its limitations. The main challenge lies with the load bearing capacity in comparison with rigid legged robots [6]. Serial link rigid legs are relatively straightforward to design, accurately controllable, and have superior performance in highly structured environments. However, they exhibit less applicability and controllability for reliable operation in unstructured environments due to the lack of passive dexterity and appendages to counter the unanticipated force perturbations and impulses of ground contacts [5]. Moreover due to their rigid construction, they only demonstrate limited variation in types of movements. The field of continuum arms is increasingly drawing the attention of researchers due to its challenging and opportune outcome in many practical applications. Number of interesting continuum robotic arms have been proposed over the years such as [7], [8], [9]. OctArm by Walker et al. is a good example of a multisection continuum arm built with pneumatic muscle actuators (PMA) and demonstrates potential for different applications [10]. Jones et al. describe the

Abstract— This paper presents the kinematics, dynamics, and experimental results for a novel quadruped robot using continuum limbs. We propose soft continuum limbs as a new paradigm for robotic locomotion in unstructured environments due to their potential to generate a wide array of locomotion behaviors ranging from walking, trotting, crawling, and propelling to whole arm grasping as a means of negotiating difficult obstacles. A straightforward method to derive the kinematics and dynamics for the proposed quadruped has been demonstrated through numerical simulations. Initial experiments on a prototype continuum quadruped demonstrate the ability to stand up from a flat-belly stance, absorb external disturbances such as maintaining stability after dropping from a height and after being perturbed by a collision, and crawling on flat and cluttered environments. Experiment results provide evidence that locomotion with soft continuum limbs are feasible and usable in unstructured environments for variety of applications.

I. I NTRODUCTION Muscular hydrostats such as squids and octopus have been known to demonstrate a remarkable ability to use their continuum soft arms as articulated limbs for walking as shown in Fig. 1. Compared to other terrestrial locomotion methods, it requires a higher level of neuromuscular coordination with an exceptional level of dexterity, planing and control [1]. However, the particular muscular arrangement in the soft tentacles provide infinite degrees of freedom allowing it to compose a rich array of movement types i.e., reaching, bending, and whole arm grasping [2]. All modes of legged locomotion is generally characterized by punctuated contact force perturbations at each leg collision on the ground [3]. Since impedance parameters of the ground vary across contact points, the perturbation forces at each collision vary across ground contact points giving rise to variability in steady state dynamics. In order to be efficient and stable in the presence of such variability of dynamics, many biological walkers with soft limbs adopt different strategies such as real-time adaptation of shape, mass distribution, and stiffness distribution of limbs during stance and swing phases, in order to tune their body frequencies to optimize walking [4]. In this context, compliant limbs can be useful to find a compromise between efficiency and robustness of walking in unstructured terrains with variable dynamics across ground contact points [5]. Therefore, legged locomotion using soft continuum limbs provides a fascinating novel paradigm for robotic legged ♯ Isuru S. Godage and Darwin G. Caldwell are with the Istituto Italiano di Tecnologia, Department of Advanced Robotics, Via Morego 30, 16163 Genoa, Italy. {isuru.godage; darwin.caldwell}@iit.it ⋆ Thrishantha Nanayakkara is with Division of Engineering, King’s College, University of London Strand, London WC2R 2LS. [email protected]

978-1-4673-1736-8/12/S31.00 ©2012 IEEE

Articulated Legs

Articulated Legs

(a) (b) Fig. 1: Octopus using articulated legs for locomotion (a) bipedal walking (b) multipedal walking.

293

z 0.15

z

Tip

z (m) 0.05

2

y

0.1

Base

0 −0.1 −0.05

y o −0.050 x

0.05

0 x (m)

o

4

x

Z

1

Robot body

Y Leg numbers X 3 O Fig. 3: Prototype robot with floating base representation.

y (m)

(a) (b) Fig. 2: (a) Continuum arm bending (b) 3D kinematic workspace of a continuum arm.

12

kinematics of such continuum arms but the use of curvature introduces complicated resulting expressions and singularity problems [11]. Godage et al. proposed a new kinematic and dynamic models based on mode shape functions (MSF) to eliminate those limitations [9], [12]. The models are based on the actual kinematics of the arm structure and solutions are obtained directly in the joint space. This work allowed to simulate continuum arm movements and generate inverse kinematic solutions efficiently without singularity problems. In the area of locomotion, the six legged robot presented in [13] was the first to use flexible legs. A study on continuum limbs for grasping in climbing situations appeared in [14] for a tripedal robot. Also, passive walking with fixed microrubber legs has been explored in [15]. This paper presents a new quadruped using continuum limbs with preliminary experimental results. The kinematic and dynamic models proposed in [9], [12] are extended to develop the kinematics, dynamics and gait generation. With the inherent compliance and rich limb actuation space, the robot poses interesting research problems in the areas of stability and compliant control for locomotion in cluttered non-engineered terrains. Proposed work seeks to find symbiosis between skeletal locomotion and continuum structures to harness the potential to develop a new breed of hybrid locomotion. This is believed to be the first demonstration of walking using soft limbs with variable stiffness. The structure of this paper is as follows. Section II introduces the proposed continuum legged robot, its construction, features and the experimental setup. The complete kinematic model and walking trajectory generation is presented in section III. The dynamic model and simulation results are given in section IV. The Experimental results are presented in section V. A discussion is presented in section VI followed by the conclusion in section VII.

Pressure source

Pressure regulators

12

Fig. 4: System block diagram.

source, pressure control commands, and digital pressure controllers. The robot consists of a light weight body frame (12.5 cm X 25 cm X 4 cm) made of 1.5 mm PVC sheets with four single section continuum limbs attached. Each continuum limb, as illustrated in Fig. 5, uses three variable length PMAs in extending mode [16] (unactuated length 11 cm and mean maximum extension 3.5 cm) supported by a rigid frames at each end at a radius 9 mm and 120◦ apart. The PMAs were custom made using Silicone tubes (∅ 4 mm) and braided Nylon sheaths and are operated within 0−5 bars pressure range. PMAs of these limbs are mechanically constrained to extend in parallel to their neutral axis by bundling together with plastic wire ties at regular intervals and polyethylene spiral bindings. Therefore, they uniformly extend upon actuation and bend for asymmetric PMA length changes [17]. Because of the lightweight flexible materials used in fabrication, the proposed continuum quadruped robot has special features that its rigid counterpart does not such as minimal leg-ground impacts. The prototype is designed to analyze the concept of continuum legged robots with gait generation, stability and dynamic walking. The input pressure is supplied by a constant 6 bar pressure source while it is regulated by 12 digital pressure regulators (Pneumax 171E2N.T.D.0009S). The controlled pressure is then carried to respective PMAs via long tubes (∅ 4 mm). Pressure controlling commands are given via a 4800bps RS232 channel to each pressure regulator at 40 Hz from a computer that computes the pressure variation. Three USBRS232 converters (four RS232 interfaces each) are used to interface to all regulators. A user interface in the computer executes appropriate commands to control the movements.

II. P ROTOTYPE C ONTINUUM Q UADRUPED An introduction to the proposed continuum quadruped along with its features is given in this section. Fig. 3 shows the prototype robot and details the coordinate frames that are being used in subsequent development of kinematics and dynamics. The body coordinate frame (ob xb yb zb ) of the floating base coordinate space (xb ) lies in the geometric center of the robot body. The overall system block diagram is illustrated in Fig. 4. There are four main parts to the experimental setup: the robot, input pressure

Robot body

Constraining wire ties

Pneumatic muscle actuators

Fig. 5: A prototype continuum limb.

294

III. K INEMATICS OF C ONTINUUM Q UADRUPED A. Introduction: Continuum Limb Kinematics Due to the deforming nature of the continuum limb, it has a continuous homogeneous transformation matrix (HTM) to define position and orientation along the neutral axis [12]. Let the joint space vector (i.e., the length changes of PMAs) for any ith continuum limb attached to the quadruped T be q Li = li1 (t) li2 (t) li3 (t) where i ∈ {1, 2, 3, 4}. Using MSF based approach, the HTM for the limb is given by (1) where ΦR and φp are rotation matrix and position vector respectively. The scalar ξi ∈ [0, 1] defines the points along the neutral axis of the continuum limb.   ΦR (ξi , q Li ) φp (ξi , q Li ) TΦ (ξi , q Li ) = (1) 0 1

Fig. 7: Velocity/gravity forces of a continuum limb slice and robot body.

2) Via Inverse Velocity Kinematics: If the vector of leg constraints is pc ∈ Rk and the robot is sufficiently constrained, required actuator joint velocities are calculated by (4) where I12 is the identity matrix, [ ]+ is the right pseudo-inverse, JC ∈ Rk×18 is the constraint Jacobian matrix, J ∈ R12−k×18 is the Jacobian matrix of the unconstrained velocities, and p˙ d ∈ R12−k×1 is the desired limb tip velocity vector. Then using (3), the robot body frame velocity is calculated by (5).       JC + 0k×1 (4) q˙ r = I12×12 012×6 p˙ d J h i+ r b x˙ b = − J∂x J∂q (5) C C q˙ r

B. Floating Base Representation T  Define the floating base joint space as q = q Tr xTb where q r ∈ R12×1 is the composite actuator joint space vector T  defined as q r = q TL1 q TL2 q TL3 q TL4 and xb ∈ R6×1 is the position and orientation of the body with respect to the inertial frame OXY Z (Fig. 3). Using the result in (1), the HTM for the ith continuum limb attached to the body relative to the inertial frame is given by (2) where Tb is HTM of the body frame with respect inertial frame, Ti (constant) is HTM of the limb base with reference to the body coordinate frame (oB xB yB zB ). Ri and pi are the rotation matrix and position vector along the neutral axis of the limb relative to the inertial frame. Ti (ξi , q) = Tb (xb ) Ti TΦ (ξi , q r )   Ri (ξi , q) pi (ξi , q) = 0 1

Discontinuous gait walking and trotting shown in Fig. 6a and Fig. 6b are generated by employing (4) and (5). Accompanying video shows the simulation of these kinematic walking patterns. IV. DYNAMICS OF C ONTINUUM Q UADRUPED

Utilizing Lagrangian formulation, authors previous works presented dynamic models for continuum arms using hydraulic muscle actuators that are intended for underwater operations [9], [18]. The continuum arm is considered to made up of thin slices and kinetic and potential energies are found for a general slice. Then the total energies are found by integrating slice energies to derive the equations of motion. This approach is extended in this section to formulate the system dynamics of the continuum quadruped. Consider any ith continuum limb attached to the robot body shown in Fig. 7. Using the HTM given in (2), the linear and angular velocity vector with respect to the body coordinate frame Oib Xib Yib Zib for a slice at ξi in floating base representation (Vib ) is given by (6) where Jbi is the body Jacobian. The dependency variables are dropped for brevity. h  ∨ iT ˙i Vib (ξi , q, q) ˙ = RTi p˙ i = Jbi (ξi , q) q˙ (6) RTi R

(2)


The tip velocity p˙ i ∈ R3×1 relative to the inertial frame is derived in (3) where Ji ∈ R3×18 is the modal Jacobian matrix for the ith continuum limb.  r p˙ i (ξi , q) = Ji (ξi , q) q˙ = J∂q i

b J∂x i



q˙ Tr

x˙ Tb

T

(3)

C. Walking Trajectory Generation 1) Via Inverse Position Kinematics: In this method, modal position kinematic formulation given by (2) and (3) is used to generate individual leg trajectories for walking. Firstly, the stance and swing phase of walking is defined in the inertial frame and then using iterative inverse kinematic (IK) Newton-Raphson algorithm or constrained optimization techniques, joint angle trajectories are determined [12]. However, this method requires the user to know each leg trajectory prior to generating actuator solutions while the next method, based on constrained velocity kinematics avoids this problem.

th The kinetic energy for a slice is then given by  in i limb (7) where δM = mδξi · diag 1, 1, 1, 14 r2 , 41 r2 , 12 r2 is the slice inertia matrix and m is the limb mass. Similarly, the slice gravitational potential energy (GPE) is derived in (8) where g is the gravity vector in the inertial frame. Similar to the single continuum section case in [9], [18], slice energies are integrated along the length of the limb to compute the limb kinetic and potential energies.

295

1

1

2

5

4

3 (a)

6

2

5 6 4 3 (b) Fig. 6: Snapshots of walking (a) Kinematic simulation of discontinuous gait walking with center of gravity projection onto the support triangle (b) Trotting simulated with floating base coordinate system.

 1 T  bT Ji δMJbi q˙ q˙ 2 δPg = (mδξ) pTi g

δKi =

Param mL Kb mB

(7) (8)

1

The total kinetic energy of the system is given by (9). The first term is the sum of kinetic energies of the four continuum limbs. The second term is the kinetic energy of the robot body where MB ∈ R6×6 is its inertia matrix and JbB ∈ R6×18 is the Jacobian of the robot body velocity with respect to OB XB YB ZB . " 4 Z # 1   1 T X bT b bT b K = q˙ Ji δMJi dξi +JB MB JB q˙ (9) 2 0 i=1

3

4

A. Control Using the Computed Torque Approach Using the pseudo inverse on (12) while the joint constraints remain valid, computed torques on the robot can be ¨ d is the desired joint accelerations. calculated by (13) where q The input joint torques, τ are then computed in (14) by adding proportional velocity and positional feedback to the computed torques via inverse dynamics where KP and KD are position and velocity feedback error gains.
+ τ ID = Su QT ST Su QT (D¨ q d + h) (13) τ = τ ID + KP S (q d − q) + KD S (q˙ d − q) ˙ (14) B. Simulation Results Initial dynamic simulations are carried out using the system dynamics described by (12). Fig. 8 shows instances of the dynamic simulation done on robot standing up from lying-flat position (zero joint torque) and then executing a discontinuous gait walking and accompanying video shows the full simulation. Since the contact points are constrained ground reaction forces do not appear in (12) [19]. Input joint torques are calculated by (14) utilizing walking trajectories generated in section III-C. Parameter values based on approximated measured values given in Table I were used in the simulations and as a first approximation only linear kinetic energies of continuum limbs were considered.

Using the Lagrangian L = K − P, the dynamic representation of the full quadruped is given by (11) where D ∈ R18×18 is the generalized inertia matrix, τ ∈ R18×1 is the input force vector in the joint space, h ∈ R18×1 stands for the centrifugal and Coriolis force, conservative force, and damping force matrices. External  forces due to contacts are λ and the S = In×n 0n×6 is the actuated joint selection matrix [19]. (11)

Employing  the  QR decomposition technique T = Q R 0 ) on a sufficiently constrained system, (11) can be decomposed into (12) where Su = 0(18−k)×k I(18−k)×(18−k) is used to isolate the lower part of (11) [19].

(JTC

Su QT (D¨ q + h) = Su QT ST τ

2

Fig. 8: Dynamic simulation: robot standing up from the flat-belly stance and then engage in discontinuous gait walking.

Similarly, the  system potential energy is derived in (10) where Ke = diag {Ke11 , Ke12 , · · · , Ke43 } :∈ R12×12 is the stiffness coefficient matrix of the joint actuators, Kb and φi are bending stiffness coefficients and bending angles of continuum limbs [9], and PB = MB pTB g is the potential energy of the robot body.  4 Z 1 X
1 1 mpTi g dξi + Kb φ2i + q Tr Ke q r +PB (10) P= 8 2 0 i=1

D¨ q + h = ST τ + JTC λ

Value Param Value 0.017 kg Ke 700 N m−1 0.1 N m rad−1 g 9.81 ms−2 0.15 kg lmax 0.035m TABLE I: Parametric values

V. E XPERIMENTAL R ESULTS Two prototype continuum quadrupeds were built to explore the feasibility and carry out preliminary experiments with compliant behavior and crawling locomotion. Firstly, a prototype continuum limb demonstrates walking trajectory (swing and stance paces). Then a continuum quadruped

(12) 296

2

1

1

2

1

2

2

4

3

4

4

3

Fig. 9: Illustrative leg trajectory for discontinuous gait walking.

1

3

3

(a)

4

Fig. 10: Robot stand up from the flat-belly stance to full upright pose.

(b) Fig. 11: Compliance demonstration via (a) heavy load impact rejection and (b) ground contact impact rejection.

standing from flat-belly stance and impact rejection capability of the compliant limbs are presented. Finally crawling locomotion is demonstrated on flat and cluttered grounds without quantitative analysis to assert the concept of continuum locomotion. Future works intend to present in-depth analysis of like robots with continuum limbs. Readers are referred to the accompanying video for complete demonstrations. A. Walking Trajectory of a Single Continuum Limb

1

2

3

4

5

6

Fig. 12: Continuum quadruped crawling on an even ground.

A cyclic walking pattern was demonstrated by employing a continuum limb. Fig. 9 depicts the instances of the leg during the stance pace. The pressure input for PMAs were controlled to obtain the desired limb tip trajectory that is calculated using (4). Results show its feasibility and capability for realizing walking trajectories.

in which the robot traversed without any problem. Due to the light weight body, sometimes the force generated by the legs are in excess and causes the robot to jump in air as the accompanied video shows. Locomotion experiment carried out on an uneven terrain (created by throwing irregularly shaped paper clutter under a cloth) where the obstacles are of around 45% of the limb length is illustrated in Fig. 13. The cloth provided a slippery ground but the robot was able to negotiate the difficulties without many problems. The impact incidents from the limbground contacts on the soft terrain are hardly noticeable due to the lightweight and compliant limbs of the robot.

B. Continuum Quadruped: Standing Up from Flat-belly Pose Fig 10 shows the robot standing up from lying-flat pose in two stages. First, the co-lateral limbs in one side are actuated followed by the limbs on the other side to attain full standing posture. Then it is actuated back to starting pose. This experiment demonstrates the continuum limb’s ability to transform from a complete soft straight structure to a load bearing stiff structure that can function to support a body.

VI. D ISCUSSION

C. Impact Tolerance via Inherent Compliance

The proposed method of modeling kinematics and dynamics using of continuum limbs uses a modal approach and reduces the computational time [12]. Therefore it can serve in real-time trajectory generation to circumvent difficulties during walking in non-engineered terrains. It has been experimentally shown that the continuum leg can stiffen from the soft unactuated state to achieve an upright posture, and support crawling locomotion. This can be useful in a number of dexterous locomotion strategies like going under a barrier,

Figure 11a shows how the intrinsic compliance of the continuum legs remain stable after being disturbed by dropping a heavy object (a large stapler weighing 250 g) from a height around 20 cm. Fig. 11b shows how the compliance properties help to self stabilize after being dropped from around 20 cm height. These two passive capabilities suggest promising advantages to survive in unstructured and hazardous environments with minimum planning and active control. D. Continuum Quadruped Crawling In this experiment, continuum limbs are actuated in a cyclic manner in open loop where leg no. 1,4 and 2,3 are in phase to obtain crawling motion. The control pressure inputs for the PMAs of a limb are phase shifted sinusoidal functions of the form Pj (t) = a0 + a1 cos (2πt + ϕj ) where j = {1, 2, 3} is the PMA number, a0 = 2.5 bar, a1 = 2.5 bar 2π is the phase shift. This pressure and ϕj = 0, 2π 3 ,− 3 input generates circular arm tip motions and Fig. 12 shows snapshots of experimental results of crawling on flat terrain

1

2

3

4

5

6

Fig. 13: Continuum quadruped overcoming uneven terrain.

297

Uncleared area

Pressure source

Mine detection Equipment

Pressure commands to limbs

Pressure control

Cleared area

experimental evidence suggests that soft continuum limbs are feasible and a potentially viable mode of locomotion to overcome a range of challenges faced in legged locomotion on unstructured environments to serve many useful applications. Detailed studies on variable stiffness and interaction forces of continuum limbs will be presented in future publications.

Mines Mines

Fig. 14: Demining application with sample equipment set up.

R EFERENCES swimming, and full body grasping to surmount barriers, etc. Moreover, the light-weight structure of a soft continuum robot makes it suitable for other weight sensitive applications such as planetary exploration, floating and swimming, etc. The major limitation of the concept of continuum robots is the low payload due to the limited stiffness that PMAs can obtain. However, one can substitute PMAs with hydraulic muscle actuators [9]. They are similar in construction and actuating mechanism but able to tolerate high pressures resulting higher stiffness values. Therefore increased payload are possible at the cost of compliance of PMAs. This modification also facilitates underwater walking robots (without the buoyancy of PMAs underwater) as exemplified by the biological counterparts in Fig. 1. Continuum robots can also have potential field applications in the area reduction phase of humanitarian demining where a suspected area with landmines have to be surveyed without exerting potentially hazardous forces on the ground contaminated with landmines [20]. Fig. 14 shows typical setup for a demining robot using continuum limbs where as the heavy components of the system such as pressure source and the pressure regulators are isolated from the robot and located in cleared areas while only pressure carrying tubes are connected to the robot in uncleared area. This drastically reduces the mass of the robot to support only the mine detecting equipment (≃ 1 kg) to greatly reduce the danger of setting off mines. This robot can also be modified to a search and rescue operations by scaling down the design to fit a camera and other sensing devices. However, there are open challenges such as the ability to control the softness to meet contradictory requirements of load bearing capacity, predictability, and accuracy of state estimation. Problems related to stability and balance with the low control accuracy can be avoided by using 6 or more legged robots and this also increase the payload at the cost of more control hardware such as pressure regulators.

[1] M. Buehler, “Dynamic locomotion with one, four and six-legged robots,” Journal of the Robotics Society of Japan, vol. 20, no. 3, pp. 15–20, 2002. [2] W. M. Kier and K. Smith, “Tongues, tentacles and trunks: the biomechanics of movement in muscular-hydrostats,” Zoological Journal of the Linnean Society, vol. 83, pp. 307–324, 1985. [3] B. Ahlborn, R. Blake, and W. Megill, “Frequency tuning in animal locomotion,” Zoology, vol. 109, no. 1, pp. 43–53, 2006. [4] C. Moritz and C. Farley, “Passive dynamics change leg mechanics for an unexpected surface during human hopping,” Journal of Applied Physiology, vol. 97, no. 4, pp. 1313–1322, 2004. [5] J. Rummel, Y. Blum, and A. Seyfarth, “Robust and efficient walking with spring-like legs,” Bioinspiration & Biomimetics, vol. 5, no. 4, p. 046004, 2010. [6] I. Walker, D. M. Dawson, T. Flash, F. Grasso, R. Hanlon, B. Hochner, W. M. Kier, C. Pagano, C. D. Rahn, and Q. Zhang, “Continuum robot arms inspired by cephalopods,” in Proc. 2005 SPIE Conf. Unmanned Ground Vehicle Technology IV, pp. 303–314. [7] G. Immega, K. Antonelli, K. Inc, and B. Vancouver, “The KSI tentacle manipulator,” in IEEE Int. Conf. on Robotics and Automation, 1995, pp. 3149–3154. [8] W. McMahan, B. Jones, and I. Walker, “Design and implementation of a multi-section continuum robot: Air-octor,” in IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, 2005, pp. 2578–2585. [9] I. S. Godage, D. T. Branson, E. Guglielmino, G. A. Medrano-Cerda, and D. G. Caldwell, “Shape function-based kinematics and dynamics for variable length continuum robotic arms,” in IEEE Int. Conf. on Robotics and Automation, 2011, pp. 452–457. [10] W. McMahan, V. Chitrakaran, M. Csencsits, D. Dawson, I. Walker, B. Jones, M. Pritts, D. Dienno, M. Grissom, and C. Rahn, “Field trials and testing of the octarm continuum manipulator,” in IEEE Int. Conf. on Robotics and Automation, may 2006, pp. 2336–2341. [11] B. Jones and I. Walker, “Kinematics for multisection continuum robots,” IEEE Tran. on Robotics, vol. 22, no. 1, pp. 43–55, Feb. 2006. [12] I. S. Godage, E. Guglielmino, D. T. Branson, G. A. MedranoCerda, and D. G. Caldwell, “Novel modal approach for kinematics of multisection continuum arms,” in IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, 2011, pp. 1093–1098. [13] K. Suzumori, F. Kondo, and H. Tanaka, “Applications of flexible microactuators to walking robots,” in Video proc. of IEEE Int. Conf. on Robotics and Automation, 2003. [14] I. Walker, R. Mattfeld, A. Mutlu, A. Bartow, and N. Giri, “A novel approach to robotic climbing using continuum appendages in in-situ exploration,” in IEEE Aerospace Conference, 2012, pp. 1–9. [15] D. Zhou and K. Low, “Combined use of ground learning model and active compliance to the motion control of walking robotic legs,” in IEEE Int. Conf. on Robotics and Automation, 2001, pp. 3159–3164. [16] D. G. Caldwell, G. A. Medrano-Cerda, and M. Goodwin, “Control of pneumatic muscle actuators,” IEEE Control Systems Magazine, vol. 15, no. 1, pp. 40–48, 1995. [17] I. S. Godage, D. T. Branson, E. Guglielmino, and D. G. Caldwell, “Pneumatic muscle actuated continuum arms: Modeling and experimental assessment (accepted),” in IEEE Int. Conf. on Robotics and Automation, 2012. [18] I. S. Godage, D. T. Branson, E. Guglielmino, G. A. Medrano-Cerda, and D. G. Caldwell, “Dynamics for Biomimetic Continuum Arms: A Modal Approach,” in IEEE Int. Conf. on Robotics and Biomimetics, 2011, pp. 104–109. [19] M. Mistry, J. Buchli, and S. Schaal, “Inverse dynamics control of floating base systems using orthogonal decomposition,” in IEEE Int. Conf. on Robotics and Automation, 2010, pp. 3406–3412. [20] T. Nanayakkara, T. Dissanayake, P. Mahipala, and K. Sanjaya, “A human-animal-robot cooperative system for anti-personal mine detection,” Humanitarian Demining: Innovative Solutions and the Challenges of Technology, I-Tech Education and Publishing, pp. 347–361, 2008.

VII. C ONCLUSIONS This paper presents the analytical and experimental results of legged locomotion with soft continuum limbs. This work proposes a continuum quadruped and formulates its kinematic and dynamics in floating base representation utilizing MSF based approach. Gait generation is discussed and kinematic simulations for discontinuous gait walking and trotting is presented along with an initial dynamic simulation. Experiments were carried out to demonstrate walking cycle, stiffening to reach a stand up posture, and passive inherent compliance of continuum limbs. Experiments on crawling locomotion were successfully performed in both flat and unstructured terrains. The theoretical and preliminary 298