Optimal Gaits and Design for Locomoting Systems Intellectual Merit


Oct 30, 2013 (4 years and 8 months ago)


Optimal Gaits and Design for Locomoting Systems
Intellectual Merit
Locomotion is everywhere.The capabilities that animals exhibit in converting internal joint motions into
displacements inspires us to to ask the question:“Howcan we imbue artificial systems with the same ease of
motion?” We propose to design,control,and plan motions for a broad class of locomoting systems.These
systems maneuver both on land and in the water,and experience dynamics and nonholonomic constraints
which may change dynamically.The true goal of this program is not to build a specific system,but rather
to address the scientific underpinnings of locomotion.Specifically,the proposed work seeks to:(1) develop
tools to design and optimize gaits,i.e.cyclic internal motions that result in a desired net motion and (2) use
these tools to design optimal morphologies for locomoting mechanical systems.Experiments will validate
and refine the ideas in the proposed work on systems familiar to the PIs.
The proposed work will draw upon fundamentals of differential geometry to develop techniques to effi-
ciently design gaits.This is a significant improvement over the state-of-the-art,which is limited to evaluating
specific gaits.These evaluation techniques are based on the reconstruction equation,which relates the in-
ternal velocities of the degrees of freedom of a mechanical system to its external velocities,and our recent
development of optimal coordinates,which allow for more efficient integration of the reconstruction equa-
tion over gait cycles.The proposed work will develop tools to analyze and manipulate the reconstruction
equation so that it can be used for gait design,as well as evaluation.With these new efficient tools,we can
design optimal gaits for locomoting systems,starting with kinematic systems and moving on to dynamic
systems,both with nonholonomic constraints.This new framework also enables us to define new metrics
to evaluate optimal agility in locomoting systems.The proposed work will then use these gait development
tools to design optimal robot morphologies.
Broader Impacts
An improved understanding of locomotion is vital for mechanisms that can navigate in challenging
terrains,e.g.,for applications like urban search and rescue,designing medical devices with the capability to
self-propel within the body,or searching for improvised explosive devices (IEDs) in hard to reach locations,
such as the nooks and crannies prevalent in the ports of major cities.The proposed work will not provide a
specific system for these tasks,but rather the theory developed in this work will define metrics to quantify
and improve robotic mobility,so that such applications can be achieved and the scientific understanding of
locomotion is advanced.
The proposed work will also have considerable impact on education and outreach.Funding will primar-
ily be used to support graduate students and the proposed program will offer a number of graduate course
design projects.To coordinate the collaboration and foster an interdisciplinary atmosphere,graduate stu-
dents funded through this program will participate in a summer exchange program in which CMU students
will come to MIT and vice-versa.Undergraduates will be involved through undergraduate research support
programs and through the undergraduate Robotics Clubs at Carnegie Mellon and MIT.An exchange pro-
gramwill allowtwo club members each year to visit the other university to share ideas.Finally,our labs will
host open houses for student groups and Take Your Child to Work Day.Both PI’s have a strong history of
fostering undergraduate involvement in research and bringing robotics to K-12 students through programs
such as FIRST and FETCH!,a PBS science/reality show that reaches approximately 3.5 million viewers
each week.
Key Words:robotic locomotion,locomotion,swimming,dynamic systems,motion planning
1 Introduction 1
2 Background 2
2.1 The Reconstruction Equation..................................2
2.2 Gait and Stroke Design.....................................3
2.2.1 Connection Vector Fields and Height Functions....................4
2.2.2 Optimized Coordinates and Height Functions for Translation.............5
2.3 Morphology...........................................6
3 Proposed Work 7
3.1 Gait Design I:Exploiting the Optimized Coordinates.....................7
3.1.1 Height Function Based Optimization.........................7
3.1.2 “Height functions” in Higher Dimensions.......................8
3.2 Gait Design II:New Physical Domains.............................8
3.2.1 Swimming Systems...................................9
3.2.2 Dynamics........................................9
3.2.3 Switching Constraints.................................10
3.3 Maneuverability and Agility..................................10
3.4 MechanismDesign:Optimizing Morphology.........................11
3.4.1 Optimization Framework................................12
3.4.2 Morphology in Hydrodynamical Systems.......................12
3.5 Experiments...........................................13
4 Broader Intellectual Impact:Education and Outreach 14
5 Management Plan and Collaboration Strategy 15
6 Results fromPrior NSF Support 15
1 Introduction
Locomotion is everywhere.Snakes crawl,fish swim,birds fly,and all manner of creatures walk.The
facility with which animals use internal joint motions to move through their environments far exceeds that
which has been achieved in artificial systems.Consequently there is a growing interest in raising the lo-
comotion capabilities of such systems to match – or even surpass – those of their biological counterparts.
Fundamentally,animal locomotion is primarily composed of gaits,i.e.,cyclic shape changes which trans-
port the animal.Examples of such gaits (often referred to as “strokes” in swimming organisms) include a
horse’s walking,trotting,and galloping,a fish’s translation and turning strokes,and a snake’s slithering and
sidewinding.The efficacy of these motions in natural systems,along with the abstraction that they allow
from shape to position changes,suggests that gaits should play an equally important role in artificial loco-
motion.Therefore,the proposed work will establish a geometric understanding as to why gaits work,and
use this understanding to:
1.Produce tools for designing and optimizing gaits for mechanical systems,and
2.Use these tools to design optimal morphologies for locomoting mechanical devices.
These tools will apply to conventional wheeled mobile robots,and more importantly,to multi-component
systems that rely on cyclic internal motions (changes in shape) to produce a net displacement,say in the
way of a snake,fish or microorganism.
Much of the prior work in gait design has taken the approach of choosing parameterized basis functions
to represent the gaits,simulating the resulting motion of the systemwhile executing the gaits,and optimizing
the input parameters – which represent the kinematics of the organism or device – to find gaits which meet
the design requirements.Such optimization with forward simulation bears a high computational cost and
suffers from the presence of local minima.By taking recourse to fundamentals in differential geometry,
computationally efficient techniques to evaluate gaits have been developed,but such techniques,including
our own,are limited in that they do not consider second order dynamic effects,such as drift,nor allow
for varying constraints.Moreover,prior techniques produce only “small,” inefficient motions and in some
cases,do not even provide desired displacement information because they are restricted to analysis in the
body frame.
The proposed work will overcome limitations of prior work and contribute to the science and under-
standing of locomotion.Specifically,this effort will (1) create a new framework with a rigorous foundation
in differential geometry and motion planning that includes dynamics (e.g.,inertial effects),(2) design new
gait optimization techniques that efficiently exploit the geometric structure of this framework,(3) apply this
structure,in conjunction with a newmetric for maneuverability,to measure optimal agility of these systems,
and contrast that with locomotion efficiency,and (4) design mechanisms that exhibit the optimal agility and
locomotive capability derived in the previous points.While the proposed work subsumes prior work for
single rigid-body wheeled systems,it is especially relevant to systems with indirect means of propulsion,
nonholonomic constraints,or both,for which conventional approaches are of limited use.Such systems
include (but are not limited to) fish,snakes,and their respective robotic equivalents.As such,the proposed
gait and mechanism design framework applies to both kinematic,drift-free systems,and those which have
drift dynamics and hence can “coast.” Additionally,it allows for discrete changes in system constraints,
such as those that occur when a snake lifts a segment of its body,a wheel breaks loose and begins to skid,
or a fish sheds a vortex fromits tail.
The core of the proposed work centers on the reconstruction equation,which provides a functional
relationship between the velocities of the internal and external degrees of freedom of a mechanical system.
We propose to develop analytic tools to efficiently manipulate and analyze the reconstruction equation so as
to more effectively design gaits.The tools will be based on recent results fromChoset’s group using optimal
choices of coordinates to efficiently approximate the displacement obtained by integrating the reconstruction
equation over one cycle.This type of efficient evaluation is vital in the design of optimal locomoting
morphologies – such as the lowReynolds number morphologies investigated by Hosoi – as the effectiveness
of each morphology (in a vast parameter space of possibilities) must be evaluated based on the best possible
gait that particular formcan achieve.To implement and evaluate the framework,we write the reconstruction
equation for a number of systems,including those that undulate on the ground or in a fluid,with both
constant and varying constraints.By posing the locomotion problemin this manner,we can put gait design
for a variety of systems into a common framework.While both PIs are expected to contribute to all aspects
of the proposed research program,Choset will lead the framework development effort and Hosoi will direct
phases related to design of optimal morphologies.Concomitant experiments will be used to validate and
refine the developing theory.
2 Background
The proposed work builds on the body of locomotion literature which uses geometric mechanics to sepa-
rate internal shape changes fromthe external motions they produce.The application of geometric mechanics
to locomotion,pioneered by Shapere and Wilczek [1] and further developed by Murray and Sastry [2],Kr-
ishnaprasad and Tsakiris [3],and Kelly and Murray [4],provides a powerful mathematical framework for
analyzing locomotion.A key product of this work is the reconstruction equation for locomoting systems,
which relates internal shape changes to body velocity for locomoting systems.
Past researchers have derived the reconstruction equation for a diverse set of systems,including those
that locomote across land [4],swimin a variety of fluid regimes [5],or float freely in space [6].Additionally,
a number of algorithms using the reconstruction equation to plan motion trajectories for these systems have
been proposed,with emphases on either implementing asymptotically stable controllers,or designing gaits
and strokes,i.e.,cyclic changes in shape,which produce desired net changes in position and/or orientation.
The bulk of the literature is geared towards gait design for mechanisms with a fixed morphology,while a
smaller set of work has looked at optimizing the morphology itself.
2.1 The Reconstruction Equation
At the heart of the proposed work is the reconstruction equation [7],which relates changes in the internal
shape of a locomoting system to its motion through the world.The configuration q 2 Q of such systems
separates naturally into q = (g;),where g 2 Gis the position and orientation of the systemin the ambient
space,and  2 M is the shape of the system,i.e.,the relative positions of the component bodies.The
reconstruction equation relates the shape and position velocities as
 = A() _ +()p;(1)
where  is the body velocity,i.e.,the velocity in the forward,lateral,and rotational directions,A() is a
matrix termed the local connection and () distributes the generalized momentump into the body velocity.
This momentumis directed along the unconstrained position directions of the system[7],and evolves as
_p = p

p +p

p _
_ + _

in which the s are scaling matrices defined by the physical parameters of the system.
For driftless locomoting systems [5,8,9],the generalized momentum vanishes from the reconstruction
equation.These systems have either as many constraints as position directions,or the property that the
generalized momentum starts at zero and is unchanging.For both classes of systems,the reconstruction
equation is thus further reduced to the kinematic reconstruction equation,
 = A() _;(3)
where we view the local connection as acting as a body-frame Jacobian,mapping from velocities in the
shape space to the corresponding body velocity.
Much early interest in applying the reconstruction equation to locomotion focused on understanding
the motion of nonholonomically constrained systems,and started with simple examples,such as the rolling
disk and differential drive or Ackerman-steered cars.These analyses were soon extended to more com-
plex nonholonomic systems,such as serial link mechanisms with active joints and passive wheelsets.Key
contributions to the development of this field include the works of Krishnaprasad and Tsakiris [3],Kelly
and Murray [4],and Ostrowski and Burdick [10,11],who advanced the modeling techniques to consider
systems like the snakeboard,which exhibit complex kinodynamic behavior.In our prior work [8,12],we
simplified the expression of the kinodynamic effects through the introduction of a scaled momentum term.
The reconstruction equation has also been investigated for a range of swimming systems,including one
of the foundational works on the topic,by Shapere and Wilczek [1].Most recently,Kelly [9],Melli et
al.[5],and Avron and Raz [13] have developed driftless models for swimming in the limits of both low and
high Reynolds numbers (i.e.,in the viscously dominated and inertially dominated regimes,respectively).
More complex models,which allow for drift,include those of McIsaac and Ostrowski [14],Mason and
Burdick [15],Xiong and Kelly’s [16]and Morgansen et al.[17].
2.2 Gait and Stroke Design
In nature,animals typically locomote by repeating gaits or strokes,i.e.,cyclic patterns of shape changes.
When planning motions for robots,it is convenient to adopt this cyclic motion,and design gaits which
produce desired net motions,rather than trying at each instant to find the differential motion which best
contributes to the net displacement.To aid in our formal discussion of gaits,we define three important terms
illustrated in Fig.1:shape changes,gaits,and image-families.Ashape change 2 is a trajectory in the
shape space M of the robot over an interval [0;T],and a gait  2   is a cyclic shape change.Each
gait has a defined start point (0),and two gaits whose images in M are the same closed curve,but with
different start points,are distinct.The image-family of a gait is the set of all gaits which share an image (i.e.,
trace the same closed curve) in M with that gait,but may have different starting points or speeds.
Figure 1:(a) The kinematic snake (studied by Choset’s group) and three-link swimmer (designed and constructed in Hosoi’s lab)
are each parameterized by a position (x;y;) and shape (
) They respectively locomote by pushing against nonholonomic
constraints or the surrounding fluid.(b) A shape change is a trajectory in the shape space,and a gait is a cyclic shape change.
Given a gait ,it is relatively straightforward to find the displacement resulting from executing the
gait,by passing its derivative
 through the reconstruction equation in (1) to find the body velocity during
the gait,and then integrating to find position change.The inverse problem,designing gaits to produce
specified position change,is more difficult.There have been two broad approaches to designing gaits in the
geometric mechanics framework.One approach,e.g.[2,11,17],seeks to find sets of small oscillations that
differentially move systems in all possible directions,while the second approach seeks out gaits that produce
larger “steps,” trading finer resolution in exchange for greater efficiency.A straightforward approach to
finding such gaits is to identify a domain of possible gaits with a parameterized function,and then to optimize
those parameters by forward integration of the reconstruction equation or equations of motion over the gaits.
This technique is in widespread use in a broad range of contexts;examples of especial relevance to the
proposed work include the results of Ostrowski et al.[18] Tsakiris et al.[19],who considered the motion of
snake-like robots on land,and our prior work [20,21] on low Reynolds number swimming systems.
2.2.1 Connection Vector Fields and Height Functions
Optimization of gaits via forward simulation,the chief technique used in previous work,is computa-
tionally expensive.Further,it is sensitive to local minima,the gait function used,and the initial choice of
parameters.A number of groups,including our own,have addressed this problemby identifying qualitative
geometric features of the reconstruction equation which highlight how the system translates and rotates in
response to changes in shape [8,5,12,13,22,23].These features in turn facilitate the initial selection of
gait forms and parameters.
In our prior work,we introduced the concept of connection vector fields [22] and connection height
functions [8,12,23],the latter of which were independently explored in [5,13],as tools for visualizing the
reconstruction equation and designing gaits for kinematic systems.Each row of the local connection A()
in the kinematic reconstruction equation (3) can be considered as defining a vector field whose dot product
with the shape velocity produces the corresponding component of the body velocity,


()  _ = k

()kk _k cos ;(4)
where  is the angle between the vectors and,for convenience,we wrap the negative sign into the vector
field definition.As the cos  term is a measure of the alignment of

() and _,we can easily visually
determine the sign of 
resulting froma given choice of _,by observing whether this alignment is positive,
negative,or zero.Similarly,a segment of a shape change that follows a flow line of

produces positive
motion in that body direction,while a shape change that cuts across the flow lines produces no motion in
that component.A sample set of connection vector fields is shown in Figure 2.
Figure 2:Connection vector fields for the three-link kinematic snake shown in Fig.1(a).The

field is null at all points in the
shape space,as the middle wheelset constrains 
to zero under all conditions.Due to singularities in the vector fields at the lines

= ,
= ,and 
= 
,the magnitudes of the vector fields have been scaled to their arctangents for readability.The
input shape vectors at a and b will produce pure forward translation and pure negative rotation,respectively.
Connection height functions,which illustrate the curl of the connection vector fields,illustrate the net
motion over gaits.For a planar system with position and orientation g = (x;y;),the rotational velocity is
identical in world and body coordinates,i.e.,
 = 

,and so fromthe application of Stokes’s theoremto the

vector field,the net rotation over a gait is
 =

() d =


() d =


() d;(5)
where 
is the area of a surface on M bounded by the gait.By plotting the curl of the connection vector
field as a height function of H

() on the shape space,as in Figure 3,we can easily identify by inspection
image-families of gaits which produce various net rotations.Rules for designing curves which produce
desired values of the integral in (5) are given in [8] and illustrated in Figure 3.
Figure 3:The connection height function for rotation,H

for the kinematic snake,along with two gait image-families.In (a),
the loop encircles equal positive and negative areas of the height function,so any gait fromthis family produces zero net rotation.
The two loops of the image-family in (b) have opposite orientations and encircle oppositely-signed regions of H

,so gaits from
this family produce non-zero net rotation.To accommodate the singularity along the 
= 
line,the height function is scaled
to its arctangent for display.
2.2.2 Optimized Coordinates and Height Functions for Translation
The height function approach described in Section 2.2.1 is not directly applicable to the translational
components of g.The correspondence of _x and _y to 
and 
varies as  changes and rather than measuring
the displacements over gaits,the height functions measure the net “forwards minus backwards” motion
experienced by the systems,which we term the body velocity integral,or BVI [24].Melli et al.achieved
some success in generating displacement height functions for these components by incorporating Lie bracket
information into the height functions corresponding to


;while these height functions are only
valid for relatively small gaits and do not provide the exact results of the  height function,they do provide
some indication of the resulting translation,and serve as a graphical reinforcement of the small-amplitude
In our recent work [24],we have approached this problem from a new angle,and shown that through
an appropriate choice of coordinates,we can generate translational height functions that provide a close
approximation of the displacement for macroscopic gaits.At the heart of this approach is the recognition
that some choices of body frame for a given system rotate less than others in response to joint motion
(though with the same net rotation over a gait cycle),and thus produce less discrepancy between ( _x;_y) and
).For example,in [24] we redefined the orientation of the kinematic snake to be the mean orientation
of the three links,rather than the orientation of the center link.As the center link has a tendency to rotate
in the opposite direction from the outer links,the mean orientation line moves very little in response to
shape changes.With this choice of orientation,the translational BVI approximated the net translation of the
kinematic snake over a representative gait with an error of less than 10%.
Figure 4:Quality of the approximation using the optimized coordinate choice.(a) The approximated and exact displacements
corresponding to a representative image-family of gaits.For the optimized measure of orientation,they are represented respec-
tively by the cross above the x-axis and a very short arc section at the same location.The approximated and exact displacements
using the mean-orientation coordinate system from [24] and the center-link coordinate system from [8] are included for refer-
ence.(b) Maximum radius of a circular gait image-family centered at a given point on the shape space that produces less than
10%error.(c) Visual representation of gait image-families that produce less than 10%error.The heavy line is the representative
image-family used to produce the figure in (a).
More recently,we have developed techniques to automatically optimize the choice of coordinates for a
given system without relying on physical intuition about the system (and thus generalizing the approach).
These optimization techniques rest on three key observations.First,that minimizing the rotation of the body
frame for a given is achieved by minimizing the magnitude of

,which encodes the local gradient of 
with respect to the shape.Second,that for any pair of valid coordinate choices under which the reconstruc-
tion equation can be derived,the difference in  between the two choices must be a function only of the
robot’s shape,and that the difference in

between the choices must thus be the gradient of this function.
Third,that this gradient-relationship between expressions of

under different choices of coordinates
means that we can find the optimal choice by applying Hodge-Helmholtz decomposition to separate out the
gradient component of

and then integrating this gradient to find the corresponding optimal choice of .
We have applied this coordinate optimization approach to the kinematic snake robot,with the results
illustrated in Fig.4.As shown in Fig.4(a),the BVI and displacement for gaits in a representative image
family are almost identical,which is a significant improvement over both the mean-orientation coordinate
system from [24] and the center-link coordinate system from [8].Figures 4(b) and 4(c) illustrate the do-
main of gaits over which the approximation error is less than 10%.Note that this approach is not about
designing gaits that have minimal intermediate rotations,but instead finds the coordinate system in which
the intermediate rotations over any gait produce the least error in the approximation.
2.3 Morphology
While significant effort has been directed towards optimizing gaits,studies on optimal systemmorphol-
ogy are limited,owing to the challenges inherent in optimizing kinematics and geometry simultaneously.A
few studies exist that investigate optimal morphology for a given gait – e.g.,[25] reports optimal geome-
tries for a low Reynolds number three-link swimmer executing the Purcell stroke in which the “arms” move
in series through prescribed angles,and [26] investigate optimal geometries for swimming using gaits that
mimic the stroke patterns observed in live fish – however there is a danger that by using arbitrary,potentially
suboptimal gaits these types of analyses may result in suboptimal designs.
Hosoi’s group has achieved a number of successes in the past integrating both geometry and kinematics
in the optimization of extremely idealized systems (e.g.the three-link swimmer [20] and mammalian sperm
morphologies [21]).These systems were analyzed using slender body theory,without the benefit of the
geometric mechanics tools developed by Choset and others.While slender body theory sufficed for these
simple geometries,it is computationally infeasible to go beyond these initial steps without the ability to
cheaply and efficiently identify optimal gaits.Hence one of our goals in the proposed research is to develop
and analyze the appropriate reconstruction equation for low Reynolds number swimmers with complex
morphologies and to use this systemas a testbed for our new framework.
3 Proposed Work
The proposed work focuses on optimization for locomoting dynamic systems,both from a controls
and a design perspective.By taking recourse to our previous work centered on the connection and recon-
struction equation,we will develop new methods to prescribe optimal gaits for a broad class of systems,
including nonholonomically constrained locomotors and swimming systems,and those that feature higher-
dimensional shape spaces,varying constraints,and dynamics.We then derive a new general calculation
that allows one to measure the efficacy of these gaits for the aforementioned systems.However,the work
becomes more provocative when these ideas are extended and applied to the design of such systems.So in-
stead of optimizing parameters over a gait space,the proposed work also develops the algebraic machinery
to efficiently optimize over the design parameters.
3.1 Gait Design I:Exploiting the Optimized Coordinates
In the optimized coordinates,the height functions are raised froma visualization tool for understanding
the net displacement over gaits into a computational tool that accurately approximates this net motion.To
more efficiently design gaits,the proposed work will develop new gait design algorithms that take full
advantage of the height functions.These algorithms will address two key weaknesses in typical functional
optimization approaches to gait design:First,the relationship between the functional parameters and the net
displacement must be calculated numerically as a path integral,which provides no tractable representation
of the gradient of the net displacement with respect to the parameters.Second,the computational cost of
the optimization grows exponentially with the number of gait parameters,compromising the completeness
of the search through the gait-function space.
3.1.1 Height Function Based Optimization
The basic idea underlying the algorithms to be developed in the proposed work is that the height func-
tions directly encode the gradient of the net displacement with respect to variations in the gait functions:
expanding a gait to enclose more positive volume under a height function increases the net displacement in
the corresponding direction.By itself,this gradient relationship is only marginally useful for guiding gait
optimization,as changing even a single gait parameter (say,the scale or eccentricity of an elliptical trajec-
tory through the shape space) moves all points on the gait curve,making it difficult to precisely change the
enclosed area to follow a gradient.In the proposed work,however,we will start by defining gaits not by pa-
rameterized curves,but by sets of spline control-points.
This definition decouples the individual sections of
the curves,thus allowing for targeted changes to the enclosed area and facilitating the definition of gaits with
an effectively larger number of parameters.The proposed work will combine this decoupled representation
with the gradient-property of the height functions and investigate algorithms by which to “flow” the spline
through the shape space to enclose appropriate areas under the height functions.For example,“outward”
and “inward” steps for each control point can be simply computed from its position relative to neighboring
control points,and these steps can then be collected together as primitives for adjusting the spline position.
Further research in the proposed work will explore an alternate gait optimization approach that steps
The proposed work will also consider which of the available spline definitions best works as a gait parameterization.
back from directly considering the path of the gait through the shape space,and instead constructs an area
on the shape space over which the height function integrals have the desired values,then generates the
gait itself from the boundary of the area during post-processing.Such an approach would take advantage
of similar expansion and flow rules to the spline algorithm,and would also be especially conducive to
including higher-level guiding principles to gait construction,such as the gait design rules in [8],which
utilize symmetries in the height functions.
3.1.2 “Height functions” in Higher Dimensions
Previous works involving height functions [5,6,8,24] have been limited to systems that have two active
joints and translate and rotate in the plane.The limitation on the number of joints is tied to the use of
height functions as visualization tools (it is difficult to display trajectories through a shape space greater
than three dimensions,and for more than two dimensions the curl integral in (5) becomes significantly more
complicated).In comparison,the limitation on the position space stems from the previous applicability of
the height functions to only calculating net planar rotation (making larger position spaces mostly irrelevant).
With the new height function optimization approaches described above,however,we have the opportunity
to expand the height function approach to these higher-dimensional spaces.
For example,the proposed work will first address higher dimensions by considering the case of a four-
link system with three active joints.To make this jump from two- to three-dimensional shape spaces,it is
helpful to recognize that the curl height functions are actually the magnitudes of a curl vector field normal
to the (
) plane,and that integrating the area under the height function over a region bounded by a
gait is equivalent to integrating the flux of this curl field through that region.The connection vector fields
for a systemwith three joints are three-dimensional,and their curls are themselves three-dimensional vector
fields.The equivalent to integrating under the height function is integrating the flux of the curl field through
a surface bounded by the gait (any such surface,as by Stokes’s theoremall choices of surface will yield the
same result).While this formulation pushes the bounds of tractability for a human gait designer,it appears
ripe for the application of a surface-growing algorithm like that described in Section 3.1.1.Consequently,
the proposed work will seek to develop the additional structure to handle the increased dimensionality of
extra joints in the system.
Similarly,working with automated gait design algorithms makes it feasible to consider systems with
more position degrees of freedom than a human designer can reasonably keep track of.One obvious target
for such developments is lowReynolds number swimmers with multiple flagella.These complex morpholo-
gies are commonly observed in nature and admit full spatial motion,rather than being confined to the plane.
We are also interested in unconventional position spaces,such as the double-SE(2) position space of a pair
of planar swimmers operating together.These types of systems have generated considerable recent inter-
est as they provide a means of breaking symmetry,enabling swimmers to move in groups in environments
where solo swimming is physically prohibited [27].The key difficulty when working with these larger po-
sition spaces is finding gaits that move the system by the desired amount in each direction,so the proposed
work will investigate means of simultaneously optimizing over multiple objective functions,and incorporate
theminto the algorithms described in Section 3.1.1.
3.2 Gait Design II:New Physical Domains
Many interesting systems exhibit kinodynamic behavior
as encoded in the full reconstruction equa-
tion (1).Additionally,hybrid systems experience discrete changes in their constraints,such as a wheel
transitioning from rolling to skidding,a snake lifting a portion of its body from the ground,or a fish shed-
ding vortices from its tail.None of these effects is captured directly by the connection vector fields,and
i.e.,the displacement over a gait is a combination of drift-free,kinematic effects and rate-dependent,dynamic drift.
therefore,this phase of the proposed work will expand our “height function”-based gait design approach to
handle these added complexities.
3.2.1 Swimming Systems
So far,our investigations of optimal coordinates have focused on systems with pure nonholonomic
constraints,such as the kinematic snake.The principles of this optimization,however,are applicable to
a broad range of systems,and one aspect of the proposed work will be to explore the space of system
models from other physical domains that we can incorporate into our framework.In this effort,we are
especially interested in the mechanics of swimming systems and will start with two such models:the low
Reynolds number swimmer Hosoi has previously examined without benefit of the geometric mechanics
framework [28],and the high Reynolds number swimmer studied by Kanso et al.[29] and Melli et al.[5].
In our preliminary work,we have observed that because these systems do not have hard singularities in their
constraints,the curls of their

fields are significantly smaller than was the case for the kinematic snake.
This suggests that the gradient-based separation may work even better for these systems than it did for the
kinematic snake,producing a smaller

field and thus an even better displacement approximation.The
coordinate optimization and height function gait generation tools should thus be directly applicable to these
swimming systems,providing both geometrically motivated explanations for the results of previous works,
and the benefits of new optimization routines and analyses described below.
3.2.2 Dynamics
Removing,for example,one of the constraints on the kinematic snake gives it the ability to “coast”
even if the joints are locked.To include such dynamic effects,i.e.,the momentum distribution function
 and the generalized momentum p from (1),into our gait design approach,the proposed work considers
two questions.First,does the small-rotation condition on which we base our use of the BVI still hold
in the presence of dynamics?Second,what tools can we introduce to design gaits with desired dynamic
Initially,we will drawupon our prior work [12] to design gaits over which the momentumis always zero,
and for which the resulting motion is dictated by the local connection.While these kinematic gaits share
the time scalability and simple design rules of gaits for kinematic systems,there are efficiency benefits to be
gained by storing and managing momentum.Therefore,the proposed work will also investigate means of
finding choices of coordinates that keep 

,the contribution of p to 

,small.Gaits in these regions will be
able to accumulate momentum in translational directions,while keeping their rotations small.Alternately,
we will also consider gait design algorithms that balance kinematically induced rotations (from a non-zero

) with dynamically induced rotations (froma non-zero 

),and thus maintain a steady orientation.
This idea of using momentum to counteract kinematic rotations leads to the second question about
dynamics that the proposed work will address,regarding what tools we need to design gaits with specific
dynamic properties.While our prior work [12] explored tools for designing gaits whose momenta are
sign-definite,it did not explore how to control the momentum with any finer granularity.To address this
shortcoming,the proposed work will develop tools based not only on the curve a gait makes in the shape
space,but also the rate at which the curve is followed.By reparameterizing the momentum evolution
equation (2),we see that the rate of change of the momentum with respect to shape is proportional to the
rate of the shape change,i.e.,
/k _k,from which it is clear that the momentum will change more over
a gait segment that is executed quickly than over the same segment executed at a slower pace.By varying
the speed of the gait at key segments,it should therefore be feasible to design a gait which maximizes the
net change in momentum,or one that starts and ends at the same momentum,but with a large intermediate
momentum.The proposed work will investigate means of presenting this relationship in a visually intuitive
manner,which can be used to directly design the pacing of gaits by inspection,and will also support the
development of efficient algorithms for optimizing the dynamic contribution of the gait,in the manner of
Section 3.1.1.
3.2.3 Switching Constraints
Hybrid systems discretely switch among sets of constraints or dynamical states.For example,a hybrid
extension of the kinematic snake could remove or reposition one or more of its constraint wheelsets,or a
fish could gain a discrete change in momentum by shedding a vortex.This gives them increased freedom
of movement,but at the expense of requiring more complex motion planning.The proposed work will
build upon the motion planning tools described above to account for these added complexities.Specifically,
we will address what changes connection vector fields and height functions require in order to work with
switching constraints,and we will generate gaits for such systems.
On first appearance,the basic application of connection vector fields to hybrid systems appears relatively
straightforward – assign one set of fields for each constraint set,and have gaits switch fields as they make
their way through the shape space.However,this does not consider transient effects incurred when the robot
switches state while in motion.Additionally,one must consider the growth in computational complexity as
the number of available fields increases.The height functions,on the other hand,will require more attention,
as they must generated fromthe curls of smooth connection vector fields.The proposed work will seek to to
build composite connection vector fields that smoothly interpolate the
As taken from individual constraint
sets in different regions of the shape space,and generate height functions fromthese vector fields.For gaits
which exist in regions of M where
A is mostly undistorted by the interpolation,the area integral under
the height function will closely approximate the actual BVI (and thus the displacement) for the gait with
switching constraints.The proposed work will investigate methods for constructing these composite fields,
with a specific focus on finding rules for the blending interpolation that minimize the distortion of
for limiting the computational complexity of switching among a sizable number of constraint sets.
3.3 Maneuverability and Agility
The fundamental agility and maneuverability of locomoting systems has received little attention in the
context of geometric mechanics.We feel that both of these measures,however,bear a strong resemblance
to the concept of manipulability which has been developed for fixed base manipulators.At their most basic
levels,maneuverability and manipulability measure the freedom a system has to relocate its body or end
effector,respectively.This freedomis often examined through the systemJacobian,which maps input joint
velocities into body or end effector workspace velocities.
The most widely used measure of manipulability is Yoshikawa’s kinematic manipulability w [30],de-
fined as w =
det JJ
,where J is the Jacobian of the system relating workspace velocity v to the joint
configuration velocity _q,i.e.,v = J _q.This quantity is the volume of the manipulability ellipse – i.e.,the
workspace velocities reachable with unit-norm control inputs.Other metrics based around the kinematic
manipulability ellipse have also been considered,such as Klein and Blaho’s discussion of joint range avail-
ability,the Jacobian’s condition number,or the minimum singular value of the manipulability ellipse [31].
Another closely-related but distinct metric is the dynamic manipulability [32],which calculates a manipu-
lability ellipse taking the dynamics and torques of the mechanism into account.Other refinements to the
manipulability have taken into account non-Euclidean spaces or the presence of joint limits and singulari-
ties [33,34,35,36,37,38].
The study of locomotor maneuverability [39,40,41,42] has received significantly less attention than has
manipulability,and has focused on measures specialized to specific systems.The proposed work will initially
investigate the maneuverability and agility of our locomoting systems with the manipulability framework
developed by prior researchers,identifying the two concepts with kinematic and dynamic manipulability,
respectively.The key innovation of this portion of the proposed work will be the recognition that the local
connection fromthe reconstruction equation is a Jacobian fromshape to position velocities.We thus propose
that the maneuverability to be
det AA
,and the agility to be
det AA
,taking into account the inertia
of the system.For kinodynamic systems,the the proposed work will also consider how the maneuverability
and agility depend on the momentum of the system,which will warp the ellipsoids of possible position
While the manipulability-inspired measures of maneuverability and agility give a good first order mea-
sure of the freedom of movement,as a strictly local measure there are several qualities which it does not
capture.First,the motion of undulatory systems,such as the three-link snake robot in Figure 1(a) is very
much restricted by hard joint limits,such as the need to avoid collisions between the links.These limits
clearly reduce the system’s freedom to move – both at the limit itself and in the vicinity – but are only
reflected in the manipulability by the cutoff to zero at the limit.To address this concern,the proposed work
will draw upon the weighted manipulability concepts introduced in [36,37,38],which scale the manipu-
lability by the distance to the nearest joint limit.The prior work in this area has not,however,rigorously
considered the formthis weighting function should take.The proposed work will carry out this investigation
to produce an improved measure of maneuverability.
An equally important limitation of applying basic manipulability theory to locomoting systems is that
their maneuverability ellipsoids will often be lower-dimensional than their position space.For instance,the
three-link snake can move in x,y,and ,but with just two input shape variables it can only move indepen-
dently in two position directions (which two directions depends on the present shape).The maneuverability
“ellipsoids” for this systemwill thus be two-dimensional ellipses,with zero volume in the three-dimensional
position space.A simple solution to this problem might be to take the area of this ellipse as the maneuver-
ability;the proposed work will investigate whether this is indeed the right measure,or if a better one exists.
These new measures of maneuverability are particularly exciting as they allow us to tackle both biolog-
ical and mechanical design questions related to morphology.It is widely believed that the morphology of
many biological organisms represents a compromise between stability and maneuverability [43].While op-
timal efficiency,speed,and stability have been relatively well-studied in many biological systems,studies on
maneuverability and agility lag behind partially due to the lack of a good metric to quantify these concepts.
With the ability to quickly and efficiently select optimal gaits and a well-defined objective function that
quantifies maneuverability (both of which will be addressed in the proposed work),we have the potential to
address a whole host of open questions:Is there a trade-off between agility and stability?Can we predict
optimal morphologies for agile motion?Do biological organisms that live in complex environments exhibit
“optimally agile” geometries?
3.4 MechanismDesign:Optimizing Morphology
So far we have primarily discussed selecting gaits in systems for which the morphology is given.Ulti-
mately,in our proposed work we will move beyond prescribed geometries and consider design of locomoting
mechanical systems and rationalization of morphologies observed in biology.These challenges will be ad-
dressed in the latter stages of the research since they will make use of the gait selection tools developed in
the earlier phases of the proposed work.To illustrate that morphological design and gait selection are inex-
tricably linked,consider the following example.Suppose we are given two organisms with vastly different
morphologies (e.g.,a giraffe and a rhinoceros) and we are asked to select which morphology is better suited
for speed.If the optimal fast gait (galloping) is known,this is a simple comparison.However,if the optimal
gait is unknown,using arbitrarily chosen gaits (e.g.,racing a hopping giraffe against a skipping rhinoceros)
can clearly lead to erroneous conclusions about morphology.Hence the question of optimal gait selection is
embedded within the larger question of geometric design in locomoting systems.
3.4.1 Optimization Framework
In this phase of the research,we will investigate swimming and crawling morphologies that are opti-
mized for speed,efficiency,and/or maneuverability.Optimal speed and efficiency have both been studied
extensively for a number of modes of locomotion and specific morphologies hence our primary new contri-
butions will be:(1) simultaneous optimization of both morphology and kinematics/dynamics and (2) design
of optimally agile morphologies that arise from new measures of maneuverability descibed below.Initial
work will consider an objective function of the form
det AA
+B where Ais the local connection,B
is e.g.,a drag term,and Aand B are functions of both robot configuration and morphology.For example,
consider the three-link snake depicted in Figure 1(a).The morphology of such a crawler can be system-
atically varied by changing the ratio of the length of the links.For each morphology,we can construct a
connection vector field,find the optimal gait and visualize howthe optimal gait evolves as a function of link
length ratios,R
and R
.Note that although this this adds a two newoptimization parameters,it is not likely
to be prohibitively expensive as we are finding optimal scalar values (R
and R
) rather than performing a
functional optimization (as with 
(t) and 
(t)).The proposed work will develop the optimization proce-
dures which will allowus to efficiently search the space defined by the objective function.Such an objective
function will inform a designer of the trade-offs made,e.g.,between efficiency and agility,and allow us to
investigate how optimal morphologies change as the weighting of the objective functions is varied.
3.4.2 Morphology in Hydrodynamical Systems
With the development of optimization algorithms for higher dimension shape spaces,we will be able to
address a number of hydrodynamical locomotion questions that were previously intractable.Two topics of
particular interest to us are:(1) optimal morphologies in multi-flagellar low Reynolds number swimmers
and (2) optimal morphologies for maneuverability in high Reynolds number systems.It has been well-
established that a reconstruction equation can be formulated for the three-link geometry in both the low
[1,13] and high Reynolds number limits [5,9].For questions regarding optimal morphologies,previously
derived equations will be modified slightly to incorporate a larger number of internal degrees of freedom;for
questions regarding maneuverability,existing forms of the reconstruction equations can be used and applied
to our new maneuverability metrics.
The first topic,optimal morphologies in multi-flagellar low Reynolds number swimmers,arises from
an earlier study of flagellar propulsion in which Hosoi’s group observed a curious trend in the biological
data.Very small eukaryotic organisms tend to swim with a single flagella.As the size of the organism
increases,the length of the flagella increases until a critical length scale,`
,is reached.At this critical
length,there is a transition:organisms smaller than`
tend to have one flagellum;organisms larger than
tend to have two.At even larger length scales there are additional transitions to increasing numbers of
flagella.While Hosoi’s group was able to find optimal kinematics and morphologies for organisms with one
or two flagella,more complex morphologies proved to be computationally prohibitively expensive.Using
our new higher dimensional “height functions” which allows for efficient identification of optimal gaits,
we plan to investigate these morphological transitions and ideally predict the optimal number of tails for
The second topic we will investigate is maneuverability of high Reynolds number swimmers.In previous
work,Melli et.al.[5] investigated “efficient turning gaits” for three-link swimmers in potential flow.In this
study,they do not explicitly define a metric for maneuverability and instead examine the efficiency of gaits
that reorient the swimmer.Curiously,their study leads to the conclusion that slender fish are better at turning.
In contrast,one of the most famously maneuverable fish is the boxfish [44],named for it’s decidedly non-
slender shape.To decipher this discrepancy we will apply our new metrics for maneuverability and agility
to optimize agility of swimmers in potential flow.These results will be compared with existing biological
3.5 Experiments
While the primary goal of the proposed effort is to advance the science of locomotion,we will validate
our results by implementing them on physical systems.In our prior research,we have constructed physical
instantiations of several idealized locomotors,including swimmers [45] and the three-link snake robot from
Figure 1(a).In the proposed work,we will expand the capabilities of these systems,e.g.,by providing
means to alter the morphology,or to change constraints as described in Section 3.2.3.For the experiments
themselves,in which we will evaluate the systems’ agility and performance while executing gaits,we will
make extensive use of the robotics and fluid mechanics infrastructures extant in Choset’s and Hosoi’s labs,
respectively,examples of which are shown in Fig.5.
Figure 5:Example experimental platforms.(a) Kinematic snake with optical mice for odometry.The passive wheel constraints
can be shifted or removed to alter the systemdynamics,and extra links added to increase the dimensionality of the shape space.
(b) Robotic swimmer fromHosoi’s lab.The body houses a geared DC motor;rotation was converted into an angular oscillation
using a Scotch yoke and a lever.To control the kinematics,the base angle of the tail – a stainless steel wire – can be varied
sinusoidally at oscillation frequencies between 5 and 0.4 rad/s and amplitudes between 0.814 and 0.435 rad,with the resulting
motion tracked by a motion capture system.Morphologies can be easily varied by interchanging tails of different lengths and
Ideally,we would like to compare the efficiency and completeness of our new techniques with those
of previous locomotion algorithms,but to the best of our knowledge such data does not exist for robotic
locomoting systems.One possible area of research for a future REU student may be cataloging such data
that can be used by us and the community at large.Naturally,the proposed work would then develop means
by which measurements of performance among the different optimization techniques would be standardized
to allow for an apples-to-apples comparison.
A second way to substantiate the efficacy of the gaits and morphologies in the proposed work is to
compare themwith existing solutions found in nature.While we make no claims that biology finds the global
optimum solution,we do believe that it provides good local optima for comparison.In comparing optimal
morphologies with swimming biological organisms,we will make use of existing data in the literature.The
most rudimentary search reveals extensive data on fish morphology.Guided by our optimization results for
agility,we will identify correlations between fish shape (characterized by body length,height,and width) and
environmental parameters (e.g.,“mean free path” or characteristic distance between obstacles normalized by
fish size).Although there is high variability across biological organisms,there is a plethora of data available
for fish morphologies so statistical approaches are feasible.These methods have been successfully applied
by Hosoi and Tamin the past to find correlations between optimal flagella length and observed morphologies
in mammalian sperm[21].
4 Broader Intellectual Impact:Education and Outreach
The bulk of the funds will go towards the support and interdisciplinary training of two PhD students.
The PhD students will serve as a link between Choset’s robotics group at Carnegie Mellon and Hosoi’s
fluids group at MIT,and,in addition to biweekly teleconferences and other correspondence,will visit each
other once a year.This arrangement fits in well with the increasingly interdisciplinary climate that is being
fostered by the administration in the Robotics Institute at Carnegie Mellon and in the School of Engineering
at MIT,both of which encourage faculty to collaborate across traditional disciplinary boundaries.
The PIs have a track record of pushing research results into graduate education.Choset is the lead
author of the motion planning textbook “Principles of Robot Motion,” which has been widely adopted.
Hosoi has co-organized several workshops aimed at graduate students including “A Day of Locomotion.”
The workshop was designed to bring together graduate students and researchers from various disciplines
and was described by one student participant as:“it was as if the best parts of several conferences were
distilled and mixed together.”
In addition,Choset and Hosoi have a strong history of including undergraduates in their research pro-
grams – simply look at the PIs’ web sites for a long list of undergraduate involvement.This semester,Choset
is advising 10 undergraduates,which include women and minorities,who are developing snake robots and
Hosoi is supervising eight undergraduate researchers,five of which are women (as a reference,the pro-
portion of undergraduate women in engineering at MIT is approximately 33%).One particular component
of the proposed program that lends itself to undergraduate participation is the initial phases of “optimiza-
tion” in which,once the numerical codes are in place,will sample various parameter regimes to find the
most promising regions in phase space for “optimal” (e.g.speediest,most agile,etc.) locomotion.We have
found in the past that undergraduates are often particularly adept at “nosing out” interesting phenomena in
these types of exploratory – yet bounded – projects.As the topic of this proposal is in an emerging inter-
disciplinary field of both scientific and technological relevance,it is likely to appeal to our undergraduate
students,particularly as they will be able to see explicitly how their contribution relates to the project as a
whole.Undergraduates will also be involved in maintaining the robot systems at Carnegie Mellon,as well
as performing experiments on them.The PIs will apply for additional REU support for undergraduates.
Moreover,these undergraduate research topics (and possibly others) will be supported as SURG (Small
Undergraduate Research Grants) at Carnegie Mellon and UROP (Undergraduate Research Opportunities
Program) projects at MIT,in addition to senior thesis topics.
Although the proposed work is of a fundamental theoretical nature,it is a goal of the PIs to help raise
general awareness of scientific progress and results.Previous NSF-sponsored projects by the PIs have been
of interest to the popular media,and have been given significant coverage on Slashdot,the New York Times
science page,and other popular outlets.Research topics proposed herein have a similar flavor in that certain
aspects are accessible to a non-scientific audience.
Finally the PIs are involved in numerous outreach activities that will be enhanced through this collabo-
ration.Choset has delivered many talks to local junior high and high school students,given demonstrations
at the Carnegie Museum of Science,and served as a judge for the US First Junior competition.Hosoi has
also served as a judge in the FIRST Robotics Competition annually since 2004 and has appeared twice as a
“science expert” on FETCH!a PBS reality/science show aimed at K-12.According to the Center for Ad-
vancement of Informal Science Education (CAISE) it is estimated that FETCH!reaches “approximately 3.5
million viewers each week.” An equal number of girls and boys watch the show,and 43%of the viewers are
African American or Hispanic.She also serves as the faculty advisor for Discover Mechanical Engineering,
a program that provides freshmen with a hands-on introduction to mechanical engineering culminating in
the popular soccer-bot competition at the Boston Museum of Science;and she is a regular speaker at WTP,
MIT’s Women’s Technology Program,a summer programthat brings high school women to MIT.
5 Management Plan and Collaboration Strategy
Specific Roles.The PIs have a long history working with colleagues in interdisciplinary groups and have
already been working together planning the desired outcomes of the proposed work,if supported.We plan
to investigate all aspects of the proposal together but clearly each PI will have his or her own “center of
expertise.” These topics include control,mechanics,design,biomechanics and fluids dynamics.As the two
PIs bring complementary skill sets to the table – both of which are needed for the successful execution of
the proposed work – we envision a tight connection between the two PIs research efforts.
The particular areas that the two PIs will manage are as follows.Choset will draw upon his prior
contribution to robotic locomotion to lead the development of the gait controllers,specifically defining them
for higher dimensional shape and position spaces,and systems with dynamics and varying constraints;he
will also lead the effort on developing measures of maneuverability.Hosoi will lead the effort on both
low and high Reynolds number swimmers including gait (or “stroke”) design,incorporation of long range
viscous effects in the connection,and interpretation of maneuverability results in the context of biological
data.Finally,both Choset and Hosoi will supervise students running experiments on the actual robots.
Collaboration Strategy.Both of the senior personnel are dedicated to fostering a collaborative community
among their graduate students,one advised by Choset at CMU and one by Hosoi at MIT.Choset and Hosoi
will encourage the graduate students to work with one another and and make visits to each others’ univer-
sities;already Choset’s graduate student Hatton has visited Hosoi and her group.Formal communications
will take the form of maintaining a Wiki to exchange and archive documents,having monthly telecons (via
Skype),and writing joint conference papers,journal papers and annual progress reports to the NSF.We will
also have a student exchange programs where graduate students fromCMUwill spend the summers at MIT,
and vice versa.Once a year,the two research groups will get together to present new results.
One of the unique aspects of our collaboration is that we will apply the a common platform to multiple
systems reported in the literature.Choset will apply for supplemental REUfunds to support undergraduates
who will help the graduate students maintain the experimental apparatus.
6 Results fromPrior NSF Support
Choset:Tying Together Low-level and High-level Planners with Cellular Decompositions.Grant:IIS-
0308097,Project Period 9/1/03 – 8/31/06,Funding:$286,000.The purpose of this work was to use a
high-level planner to divide a non-Euclidean configuration space into smaller contractible regions inside of
which local planners can be easily defined.We specifically focused on path planning for a two-body robot.
We also addressed related topics for a mobile manipulation task.This work has resulted in several papers,
including [46,47,48] and supported two different graduate students at various stages of their careers.
Hosoi:Collaborative Research-ITR:Higher Order Partial Differential Equations:Theory,Computational
Tools,and Applications in Image Processing,Computer Graphics,Biology,and Fluids.Grant:ACI-
0323672,Project Period:9/15/03 – 8/31/08.,Funding:$350,000.ACI-0323672 supported research related
to developing reduced models for PDEs,in particular those arising in the context of thin films,with topogra-
phy.Thin films arise in a number of settings fromindustrial coating flowapplications such as spin coating in
microcircuit fabrication to biological systems such as the liquid lining in the lung and gastropod locomotion.
We have investigated and developed mathematical models for free surface films of a variety of materials in-
cluding Newtonian fluids [49,50],non-Newtonian fluids [51,52,53] and particle-laden flows [54,55,56];
in various geometries such as inclined planes [55],corner geometries [57,58],rotating cylinders [54];and
subject to a variety of different forcings [59].
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