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

Contents

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

1

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

T

pp

p +p

T

p _

_ + _

T

__

_;(2)

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)

2

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.

(a)

(b)

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 (

1

;

2

) 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

3

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,

i

=

~

A

i

() _ = k

~

A

i

()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

~

A

i

() and _,we can easily visually

determine the sign of

i

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

~

A

i

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

~

A

y

ﬁeld is null at all points in the

shape space,as the middle wheelset constrains

y

to zero under all conditions.Due to singularities in the vector ﬁelds at the lines

1

= ,

2

= ,and

1

=

2

,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

4

identical in world and body coordinates,i.e.,

_

=

,and so fromthe application of Stokes’s theoremto the

~

A

vector ﬁeld,the net rotation over a gait is

=

Z

T

0

() d =

Z

~

A

() d =

ZZ

a

curl

~

A

i

() d;(5)

where

a

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.

(a)

(b)

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

1

=

2

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

x

and

y

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

~

A

x

and

~

A

y

;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

approaches.

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

(

x

;

y

).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%.

5

(a)

(b)

(c)

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

~

A

,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

~

A

between the choices must thus be the gradient of this function.

Third,that this gradient-relationship between expressions of

~

A

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

~

A

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

6

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.

1

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

1

The proposed work will also consider which of the available spline deﬁnitions best works as a gait parameterization.

7

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 (

1

;

2

) 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

2

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

2

i.e.,the displacement over a gait is a combination of drift-free,kinematic effects and rate-dependent,dynamic drift.

8

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

~

A

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

~

A

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

components?

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

~

A

) 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.,

@p

@

/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

9

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

~

A,and

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 =

p

det JJ

T

,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

10

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

p

det AA

T

,and the agility to be

1

I

p

det AA

T

,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

velocities.

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

11

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

p

det AA

T

+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

1

and R

2

.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

1

and R

2

) rather than performing a

functional optimization (as with

1

(t) and

2

(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,`

cr

,is reached.At this critical

length,there is a transition:organisms smaller than`

cr

tend to have one ﬂagellum;organisms larger than

`

cr

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

microswimmers.

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-

12

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

data.

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.

(a)

(b)

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

radii.

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

13

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

14

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].

15

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