XXIV ICTAM, 21-26 August 2016, Montreal, Canada
PERISTALTIC WAVE LOCOMOTION WITH A MILLIPEDE INSPIRED MECHANISM
Javad S. Fattahi1 and Davide Spinello∗2
2
Department of Mechanical Engineering, University of Ottawa, Ottawa, Ontario, Canada
1
SUNLAB, University of Ottawa, Ottawa, Ontario, Canada
Summary We propose the model of a flexible mechanism that adapts to unstructured terrains. The locomotion mechanics is inspired by the
one of multi-legged organisms such as millipedes and millipedes, whose efficiency in exploring unstructured environments is due to high
redundancy and robustness of their locomotion system based on multiple contacts with the terrain. Leg to ground contact kinematics can be
modeled as a peristaltic displacement wave [1], with periodic contact that provides redundancy and robustness for locomotion. The body of
the mechanism is modeled as a Timoshenko beam, whose kinematics captures the basic morphology of millipedes and centipedes bodies.
With respect to our work in [2] we propose a different locomotion model that is based on the mechanics of legged peristaltic locomotion
[3]. A Lagrangian approach is used to derive the governing equations of the system that couple locomotion and shape morphing, that are
illustrated by simulation results.
RELATED WORK
An important part of current robotic research has been and is inspired by features characterizing animal organisms, due
to their inherent robustness and adaptability to uncertainties in the interaction with unstructured environments [4]. In order
to achieve robustness and system adaptability with respect to different environments, mobile multibody robots have been
developed from classical manipulators by considering rigid body chains interacting with the environment in a variety of
modes. In the framework of flexible members, several researches have recently proposed finite dimensional (with large
number of degrees of freedom) and distributed systems capable of high deformability and large displacement of the end
effector [5].
DESCRIPTION OF THE MODEL AND RESULTS
Body
Legs
Contact
(a)
(b)
Figure 1: (a) Details of the dorsal structure of selected millipedes. (By Animalparty (Own work) [CC-BY-SA-3.0 (http://
creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons; (b) schematic snapshot of the profile
of the legs’ extremities.
Millipedes have a slender body comprised of an exoskeleton which can be schematized as a chain of coaxial rings with two
legs each, which possesses an internal cartilaginous spine with spinal elements connected by three degrees of freedom joints,
see Fig. 1(a). The locomotion can be characterized by two mechanisms, namely the spinal locomotion and the concertina
locomotion, and it is schematized by the two stages kinematics illustrated in Fig. 1(b), comprised of a stage in which the leg is
in contact with the ground to create the thrust associated to friction, and a recovery stage in which the leg is not in contact. In
the works [3, 1] the overall pattern has been identified as analogous to a peristaltic wave, that typically propagate in the bodies
of warm-like organisms. This mechanism is inherently redundant and robust as many contact points with the ground ensure
persistent thrust for forward locomotion, and allows shape morphing with respect to nonzero curvature of the substrate, with
filtering of local asperities through the coupling between the legs and the body. Fig. 1(b) schematically shows a snapshot of
the profile of the legs’ extremities; by considering a continuous distribution of legs, we approximate the contact profile by the
traveling wave
2πN
x + ct , x ∈ [0, `],
r sin
`
∗ Corresponding
author. Email: [email protected]
where r is the length of the limbs (assumed to be the same), c is the speed of propagation, x is the abscissa along the body of
length `, and the integer N represents the number of legs in contact with the substrate at a given time t. Note that the sign of
the time term in the traveling wave profile is chosen so that contact points travel left to right with respect to the body oriented
as in Fig. 1(b).
To describe the motion of the system we consider the composition of a rigid body placement with a small deformation
about the rigid body configuration. The rigid body placement is described by the change of coordinates x(X, t) = d(t) +
R(θ(t)) (X − δ`E1 ) that maps X from the reference configuration to x in the current rigid body configuration. The rigid
change of coordinates is as usual composed of a rigid body displacement d with respect to the origin of a fixed reference
frame, and by the action of the rotation tensor R(θ). The small deformation about the rigid body placement is described by a
map χ that takes points x and maps them to the point χ(x, t) in the current deformed configuration as χ(x, t) = x + U(x, t),
where U is a small deformation, that consistently with the the linearized planar Timoshenko beam theory is given by U(x, t) =
(u(x1 , t) − X2 ψ(x1 , t))e1 + w(x1 , t)e2 , where ei are rotated orthonormal basis vectors (floating reference frame) that are
used to describe the rigid body placement, and u, ψ and w are respectively the axial displacement, the rotation of the cross
section, and the traversal displacement all with respect to the rigid body placement. The velocity of a point χ in P is obtained
as its material time derivative, that is therefore given by
χ̇ = ḋ + θ̇W(x − d) + U̇
(1)
where θ̇W is the skew-symmetric tensor ṘRT . The system is driven by the peristaltic wave pattern on the legs described
above, which is either constitutively related to a distributed action on the body, in which case the action is equivalent to an
external force, or it is geometrically related to kinematic descriptors in (1), in which case it acts as a prescribed kinematic
term. The evolution equations are then obtained by by minimizing the action functional, and the corresponding weak form of
the system of governing partial differential equations is reduced to a system of ordinary differential equations by projecting
the spatial parts of the deformation fields along the Timoshenko beam basis functions [2].
Simulation results in Fig. 2 are obtained by setting a nondimensional peristaltic wave propagation speed c = 0.1, and
by letting the system track a profile composed as the sequence of parabolic and straight portions, with non-smooth curvature
transition between the two parts. Figure 2(a) shows eight snapshots of the system, with the state across the point with change
of curvature occurring around time 100 sec. The zoom of this snapshot is shown in Fig. 2(b). Results illustrate how the system
driven by legs’ peristaltic locomotion patterns result in path tracking and shape morphing, and it delineates a theoretical and
simulation framework to design a class of autonomous mobile robotic devices.
10
8
0.06
6
0.04
4
0.02
2
0
0.00
0
2
4
6
8
10
12
14
9.2
9.4
9.6
x
(a)
9.8
10.0
10.2
x
(b)
Figure 2: (a) Snapshots of the system tracking a profile (terrain) and shape morphing while driven by a peristaltic wave; and
(b) zoom of the snapshot at the critical point corresponding to non-smooth change of curvature of the terrain.
References
[1] Shigeru Kuroda, Itsuki Kunita, Yoshimi Tanaka, Akio Ishiguro, Ryo Kobayashi, and Toshiyuki Nakagaki. Common mechanics of mode switching in
locomotion of limbless and legged animals. Journal of The Royal Society Interface, 11(95):20140205, 2014.
[2] J. Fattahi and D. Spinello. Path Following and Shape Morphing With a Continuous Slender Mechanism. ASME Journal Dynamic Systems, Measurements, Control, 2015.
[3] Yoshimi Tanaka, Kentaro Ito, Toshiyuki Nakagaki, and Ryo Kobayashi. Mechanics of peristaltic locomotion and role of anchoring. Journal of The
Royal Society Interface, 9(67):222–233, 2011.
[4] M. Sitti, A. Menciassi, A. Jan Ijspeert, K. H. Low, and S. Kim. Survey and Introduction to the Focused Section on Bio-Inspired Mechatronics.
IEEE/ASME Transactions on Mechatronics, 18(2):409–418, Apr 2013.
[5] Frédéric Boyer, Shaukat Ali, and Mathieu Porez. Macrocontinuous dynamics for hyperredundant robots: Application to kinematic locomotion bioinspired by elongated body animals. Robotics, IEEE Transactions on, 28(2):303–317, 2012.