inaryID is

moredom,actua-

s, and

Vivek A. Sujan

Steven Dubowsky

Department of Mechanical Engineering,Massachusetts Institute of Technology,

Cambridge, MA 02139

Design of a LightweightHyper-Redundant DeployableBinary ManipulatorThis paper presents the design of a new lightweight, hyper-redundant, deployable BRobotic Articulated Intelligent Device (BRAID), for space robotic systems. The BRAintended to meet the challenges of future space robotic systems that need to performcomplex tasks than are currently feasible. It is lightweight, has a high degree of freeand has a large workspace. The device is based on embedded muscle type binarytors and flexure linkages. Such a system may be used for a wide range of taskrequires minimal control computation and power resources.@DOI: 10.1115/1.1637647#

o

se.e

s

tt

l

sr

nnncic

vg

of

ical

d tos de-eersnta-s ae

n therays,

res.rmsns.

alsoh asxcep-

, andh allelre-peri-moryly-clearetsm.

emByeryms.llel

theitsgen-

ar-ts—verst

egs

1 IntroductionIn recent years, significant efforts have been made by the

botics research community to simplify and improve the perfmance and control of manipulators. An interesting example isnary manipulators@1–6#. In this concept, a manipulator icontrolled by energizing actuators that can assume only ontwo states~‘‘on’’ or ‘‘off’’ !. The joint level control is very simpleBy simply triggering the given actuator in the system a discrchange in state is obtained. Often, the control does not reqfeedback sensors. The two states are the extreme positions oactuator. Examples include pistons, solenoids or motors. Tform of control has been classified as sensor-less manipula@2,3,7#. As the number of binary actuators in the system increathe capabilities of the device approach that of a conventional ctinuous manipulator. This is analogous to the revolution ofdigital computer replacing the analog computer. However,leads to mechanisms with complex system kinematics. Studiethe kinematics and control of such ‘‘hyper-redundant’’ maniputors, both with and without binary actuation have been perform@1,8–16#.

Future space robotic missions will require robots to perfocomplex tasks such as scouting, mining, conducting scienceperiments, and constructing facilities for human explorers andtlers @4,17#. To accomplish these objectives, future planetarybotic systems will need to work faster, travel larger distancperform highly complex operations, and work with a higher dgree of autonomy than today’s technology will permit. New apractical design concepts are required to meet these challeFor example, robots will require devices for deploying commucation antennas, rapidly positioning scientific sensors and fatating assembly@17#. Such devices may also have commercmedical applications where simple, robust and accurate menisms for positioning surgical tools are needed.

A single, yet lightweight, robust and simple device that couperform a number of these tasks would be highly desirablewould need to have fine motion resolution, a large motion wospace, multiple degrees of freedom, control simplicity, and havsmall stowed volume. Designing such a system posses secomplex challenges such as keeping the system lightweisimple, accurate and choosing the number of degrees of freeneeded. To address some of these challenges, we need to exmethods to reduce mechanical complexity introduced by convtional bulky components and large number of power wires. Tuse of non-conventional components such as binary muscle-actuators and elastic bistable joints is appealing. However, th

Contributed by the Mechanisms and Robotics Committee for publication inJOURNAL OF MECHANICAL DESIGN. Manuscript received July 2002; revised Jun2003. Associate Editor: D. C. H. Yang.

Copyright © 2Journal of Mechanical Design

ro-r-bi-

of

teuiref thehistiones,on-hehiss ofa-ed

rmex-et-o-es,e-dges.i-ili-alha-

ld. Itrk-e aeralht,

domploreen-hetypeese

introduce new limitations. Elastic joints have a finite rangemotion and short fatigue lives. Shape Memory Alloy~SMA!muscle wires have stringent thermal, geometric and mechanproperties and consume a lot of power.

This paper presents the design of an element that is intendemeet these requirements and overcome the challenges. Thivice, called a Binary Robotic Articulated Intelligent Devic~BRAID!, consists of compliant mechanisms with large numbof embedded actuators and is a step toward practical implemetion of binary devices for space robotic systems. Figure 1 showpotential application concept for the BRAID element. In somways it resembles deployable systems that have been used ipast for space applications as: deployable booms, solar arantennas, articulated masts and others@18–21#. These devices canbe deployed from a small package into relatively large structuHowever, in general these structures are not controllable in teof being able to assume different commanded configuratioOnce deployed, they are usually not retractable. They areusually constructed from heavy and complex components, sucgears, motors, cables, etc., although there are some notable etions @4,22,23#.

This paper addresses the design issues, system kinematicspractical implementation concerns that go into developing sucsystem. The BRAID element is made of a serial chain of parastages~see Fig. 2~a!!. Each three DOF stage has three flexubased legs, each with muscle type binary actuators. In the exmental system described in this paper these are shape mealloy ~SMA! actuators. As discussed later, more promising pomer actuators are now being implemented in this study. Musactuation allows binary operation of each leg. The flexuressimple and light weight. The experimental BRAID built consisof five parallel stages, yielding 15 binary degrees of freedoThus it has 215 (32,768) discrete configurations. Thus, the systcan approximate a continuous system in dexterity and utility.its polymer construction and binary actuation the design is vlightweight and simple, appropriate for space exploration systeThis paper is divided as follows: Section 2 describes the paralink stage design of the BRAID element. Section 3 explainskinematics of the BRAID. Section 4 presents a discussion ofactuators and the actuator control. Section 5 presents the firsteration in experimental implementation of the BRAID.

2 BRAID DesignFigure 2~b! shows one stage of the BRAID element. Each p

allel link stage has three legs. Each leg has three flexure jointwo one DOF joints and one three DOF joint. This results in fiaxes per leg: three in parallel, the fourth orthogonal to the fithree and the fifth orthogonal to the fourth. Coupling the three ltogether~symmetrically 120° apart! gives the parallel link stage

thee

004 by ASME JANUARY 2004, Vol. 126 Õ 29

o

p

ma-

hereeyest

nd

-

ility,

wo-neethestic

tents

three DOF mobility ~vertical translation, pitch, and yaw! con-trolled by actuators placed on each leg. However, in the physimplementation of the design~see Fig. 2~b!! the fifth DOF in eachleg was removed, since this motion is small and can be accmodated by elastic deflections. This is described in greater dein Section 3.

The concept of using flexures to replace hinges and bearingnot new. Short plastic hinges can commonly be seen in commcial applications such as cabinet doors, tool box lids, shambottle caps, etc. Their design requires careful attention to thstructure mechanics. In the BRAID application large rangesmotion and low stiffnesses in the axes of rotation are desirwhile maintaining high stiffnesses in all other axes. Repeabending of a flexure can cause fatigue failure. The relationsbetween performance and fatigue life can be estimated toorder by considering a simple beam of thicknesst, with Young’smodulusE, bent elastically to a radius ofR. The surface strain andmaximum elastic stress is given by:

«5t

2Rand s5E

t

2R(1)

Fig. 1 Potential BRAID applications—coring rock samples †4‡

30 Õ Vol. 126, JANUARY 2004

ical

m-tail

s iser-ooeirof

ed,tedhipfirst

This stress must not exceed the fatigue yield strength of theterial, s f . Hence, the minimum bend radius is given by:

R>t

2 S E

s fD5

t

2 S 1

M D (2)

Materials that can be bent to the smallest radius or the ones wM ~defined in Eq.~2!! is maximized are desirable because thgive the largest range of motion. Literature suggests the bchoices are polymeric materials and elastomers withM equal to331022 @24#. Materials such as polyethylene, polypropylene anylon fall into this category. For comparison, for spring steelMequals 0.531022 ~which would be appropriate when high stiffness and small range of motion is desired!. An ultra high molecu-lar weight polyethylene is chosen here, based on its machinabfatigue life, stiffness, weight, and cost.

Actuation of each leg is accomplished using a muscle-type tstate~or binary! actuator such as an SMA. If each leg has only oactuator~an SMA wire in tension! the restoration force to changthe binary states is provided by the elastic flexure joints. Onother hand, if a pair of antagonistic actuators is used, the elarestoration force is not needed. As proposed in the past, de

Fig. 3 Detent based binary joint †4,6‡

Fig. 2 BRAID design concept, „a… Assembled structure; „b… Single parallel link stage final design

Transactions of the ASME

Jou

Fig. 4 i th parallel link stage, „a… Physical parallel link stage; „b… diagrammatic representation

rh

ah

a

(

d

the

help lock each binary leg into a discrete state~see Fig. 3! andprovide more accurate and repeatable positioning@4,6#. They alsoeliminate the need for power while the BRAID is stationary.

3 Kinematics

3.1 Forward Kinematics. Binary devices present a numbeof challenges not found in conventional continuous systems.example, a continuous robot can reach an infinite numbepoints within its workspace while a binary robot can only reacfinite number of points. A binary robot’s workspace is thus a dcrete point cloud. A BRAID based system adds further challengdue to the complexity of its kinematics. A typical four by fouhomogeneous transformation matrix is formulated as a combtion of a rotation matrix and a translation vector of one coordinframe with respect to another. The kinematic variables are tthree rotational and three translational variables~six DOF!. In asingle parallel link stage of the BRAID system, the three legspositioned about the vertices of two equilateral triangles~see Fig.4~b!!. Additionally, based on the joint configuration of each lethe single stage has only three degrees of freedom—pitchux)and yaw (uy) rotations and a vertical~z! translation~couplingeffects lead to non-independent motions in thex andy directionsas well!. In general, given the four by four transformation matrA i 21,i , of the i th coordinate frame with respect to thei 21thcoordinate frame~see Fig. 4~b!!, one can derive the forward kinematics of the entiren-staged system.A0n defines the forward ki-nematics from base to end-effector of the entire system ansimply given by:

A0n5A0,1A1,2A2,3¯An21,n5)i 51

n

A i 21,j (3)

whereA i 21,i is given by:

A i 21,i5g01i•g12

i•g23

i 5F Rxi 0

0331T 1

G •F Ryi 0

0331T 1

G •F xe

I333 ye

ze

0331T 1

G5F xi

R333 yi

zi

0331T 1

G (4)

rnal of Mechanical Design

rFor

ofa

is-es,rina-teen

re

g,

ix

-

is

(I333 is the 333 identity matrix! and the 333 rotation matricesRx

i andRyi are given by:

Rxi 5F 1 0 0

0 cosuxi 2sinux

i

0 sinuxi cosux

iG and Ry

i 5F cosuyi 0 2sinuy

i

0 1 0

sinuyi 0 cosuy

iG

(5)

In this formulation the leg lengths,l ji , ~see Fig. 4! are the control

variables. The relationship between these leg lengths andpitch, yaw, and vertical translation of thei th coordinate framewith respect to thei 21th coordinate frame~see Fig. 4! can beformulated. From Fig. 4:

g1i 5uy

i g2i 5g2x

i 1g2y

i g3i 5g3x

i 1g3y

i (6)

where

g2x

i 5sin~p/6!uxi (7)

g3x

i 52sin~p/6!uxi (8)

d3i sing3y

i

cos~p/3!5r cos~p/6!2r cos~p/6!cos~uy

i !2d1i sin~g1

i ! (9)

d2i sing2y

i

cos~p/3!5r cos~p/6!2r cos~p/6!cos~uy

i !1d1i sin~g1

i !

(10)

c1i 50 c2

i 5c2x

i 1c2y

i c3i 5c3x

i 1c3y

i (11)

where

c2x

i 52p

6sin~ux

i ! (12)

c3x

i 5p

6sin~ux

i ! (13)

c2y

i 5p

6sin~uy

i ! (14)

c3y

i 5p

6sin~uy

i ! (15)

The deflection parameters (d i ,g i ,c i) give us the coupledxi andyi

translation of thei th stage:

JANUARY 2004, Vol. 126 Õ 31

a

er,y bele-

en

,the

l-

xi52d1i sin~g1

i ! (16)

yi52S r

22

d3i sing3x

i

cosp/62

r

2cosux

i D (17)

Next, the vertical projections (b22 and b3

2) of the leg lengths,l 2i

and l 3i , are found from the following:

~ l 2i !25a2

21b225S d2

i sing2x

i

cosp/6D 2

1S d2i sing2y

i

cosp/3D 2

1b22 (18)

~ l 3i !25a3

21b325S d3

i sing3x

i

cosp/6D 2

1S d3i sing3y

i

cosp/3D 2

1b32 (19)

Figures 5 and 6 shows the projections of sections ABCDEFGH defined in Fig. 4. Using these projections, the relationsbetween the desired unknowns,zi , ux

i , uyi and the known link

lengthsl 1i , l 2

i , l 3i can be established.

First from Fig. 5,h1i can be found as:

h1i sinu25

b21b3

2(20)

A relationship between the anglesu1 , u2 , u3 , u4 , andux , canalso be established:

u35a cosS ~h1i !21S 3

2r D 2

2b2

2~h1i !S 3

2r D D where

b25S 3

2r 2 l i

i cosu1D 2

1~ l 1i sinu1!2 (21)

Fig. 5 Projection of section ABCD from Figure 4

Fig. 6 Projection of section EFGH from Figure 4 „b…

32 Õ Vol. 126, JANUARY 2004

ndhip

u25u281u29 where u285a sinS l 1i sinu1

b D and

u295a sinS 3/2r sinu3

b D (22)

uxi 5p2u32u2 (23)

Also from Fig. 5 we have:

h1i sinu22 l 1

i sinu153

2r sinux (24)

Using Eqs.~18–23! and plugging into Eq.~24! we get one equa-tion in two unknowns (ux and uy). A similar equation can bederived using the geometry in Fig. 6.

From Fig. 6,h2i andh3

i are found as follows:

h2i sina25

2

3b21

1

3l 1i sinu1 (25)

h3i sina15

2

3b31

1

3l 1i sinu1 (26)

Once again, a relationship between the anglesa1 , a2 , a3 , a4 ,anduy , can be established:

a35cosS ~h2i !21~2r tanp/6!22b2

2~h2i !~2r tanp/6!

D where

b25~2r tanp/62h3i cosa1!21~h2

i sina1!2 (27)

a25a281a29 where a285a sinS h3i sina1

b D and

a295a sinS 2r tanp/6 sina3

b D (28)

uyi 5p2a32a2 (29)

Also from Fig. 6 we have:

h2i sina22h3

i sina152

)r sinuy (30)

Using Eqs.~18,19–29! and plugging into Eq.~30!, we get oneequation in two unknowns (ux anduy). Equations~24! and ~30!now give two independent equations in two unknowns. Howevboth are highly non-linear transcedental equations and can onlsolved numerically. A Newton-Raphson algorithm was impmented to solve for the unknowns,ux anduy . Finally, the verticaltranslation can be solved using solutions forux and uy and Eq.~31!:

zi5h2i sina22

1

)r sinuy5h1

i sinu221

2r sinux (31)

This is the general solution for the BRAID system for the givleg lengths. Hence,Ai 21,i is only a function of the variable leglengths of thei th stage (l 1,2,3

i ). However, for a binary systemsince only two leg lengths need to be considered, given one oftwo lengths~i.e., the maximum or open leg length! the second oneis then a well defined function of the first. This is found as folows. The minimum value of the leg lengthl 1

i , given the maxi-mum value ofh1

i ~from Fig. 5! is found by solving:

r 5~r 1 l 1i !cosu15~r 1 l 1

i !•S ~3r /2!21~3r /21 l 1i !21~h1

i !2

3r ~3r /21 l 1i !

D(32)

The minimum value of the leg lengthh1i given the maximum

value of l 1i ~from Fig. 5! is found by solving:

Transactions of the ASME

u

o

n

s

n

hip.ayid-

ac-

frac-

em.a-

ipu-n-rseion.for

, thed-theore

u-the

thmThebe-

ill

-re

-

es,lly-ma-oneply

andolu-ndalss

gen-ne

ceday

r

25S r

21h1

i D cosu25S r

21h1

i D •S ~3r /2!21~3r /21h1i !22~ l 1

i !2

3r ~3r /21h1i !

D(33)

The minimum leg lengthh2i given maxh3

i ~from Fig. 6! is foundby solving:

r

)5S r

)1h2

i D cosu35S r

)1h2

i D•S ~2r /) !21~2r /)1h2

i !22~h3i !2

2~2r /) !•~2r /)1h2i !

D (34)

Since every leg in the system can be in only one of two sta~binary!, each leg length can have only one of two values. Heneach term ofAi 21,i can have only 8 different discrete values coresponding to the 8 possible states of a single BRAID stage~seeFig. 7!. An interesting aspect of binary robotic devices is thforward kinematics computations, described above, can be dwithout repeated use of trigonometric computations during rtime. Whereas continuous robots have infinite solution spacesquiring complex geometric calculations be performed during rutime, binary-actuated robot geometries can be computed offlahead of time since there are only a finite number of statesvolved. The solution of the module kinematics need onlysolved once, possibly on a different computer than the one beused for real-time control. During run-time, forward kinematcomputations are reduced to simple matrix multiplication, bason values stored in memory. No computationally costly maematics is required at run-time and can be implemented withmost basic processing capabilities.

Figure 8 shows the workspace generated for a 5 stage BRAIDelement. The workspace for this example consists of 215 uniquestates. However, the workspace is clearly non-uniform in its dtribution. Optimization of this workspace in terms of densityreachable states is an area of continuing research.

The forward kinematic calculations can be further extendedformulate the BRAID system Jacobian matrix,JBRAID . Currently,no closed form solution toJBRAID is known. However, the aboveforward kinematics formulation may be directly applied for oline numerical computation ofJBRAID . Well known applicationsof this matrix include relating the end-point Cartesian motions (p)to the individual joint motions (q) and end-point forces/torque~F! to the individual joint level forces~t!, independent of binaryactuation. This is summarized in Equation~35!:

p5JBRAIDq

t5JBRAIDT F (35)

In particular, the latter allows for an efficient end-point force cotrol formulation architecture. In this architecture, the required e

Fig. 7 One stage of the BRAID, showing its eight binaryconfigurations.

Journal of Mechanical Design

tesce,r-

atonen-re-n-inein-

beingiced

th-the

is-f

to

-

n-d-

point forces would be generated using the above relationsHowever, due to the hyper-redundancy of the BRAID, we mconcurrently establish desired end-point positions. This hybrhyper-redundant manipulator control architecture with binarytuator inputs is currently under development.

3.2 Inverse Kinematics. Once the forward kinematics othe system have been developed, the problem of executing ptical tasks requires the solution to the inverse kinematics problWhile the hardware costs and control complexity of a binary mnipulator are lower than those of a continuously-actuated manlator, there is a tradeoff in the complexity of the trajectory planing and inverse kinematics software. As expected, the invekinematics problem cannot be expressed in a closed form solutBrute force or exhaustive search methods may prove appealingsystems with few stages~less than 5!, but become impractical forlarger systems. As the number of degrees of freedom increasecomplexity of the workspace grows exponentially. For every aditional stage there is about an order of magnitude increase innumber of states in the workspace. Hence, for large systems mefficient search methods are required to find ‘‘optimum’’ soltions. This section presents two possible search methods forinverse kinematics problem. The first is a genetic search algoriand the second is a combinatorial heuristic search algorithm.search metric for both algorithms is to minimize the distancetween the end-effector and desired position. Both algorithms wbe briefly described and their results will be presented.

3.2.1 Genetic Search Algorithm.Here a classical genetic algorithm approach for finding an optimum solution is used, wheeach generation consists ofm-bit binary words describing the manipulator state~wherem is the number of actuators!. The geneticalgorithm begins with a set of random manipulator solution statcalled a generation. The genetic algorithm uses simple physicabased rules, or tests, to produce a fitness value for a givennipulator configuration. The fitness value is used to comparemanipulator state to another. For the task at hand, fitness is simdefined as the error between the current end-effector positionthe desired position. To generate successive generations of stions, the algorithm allows for reproduction, genetic crossover agenetic mutation. The probability of reproduction is proportionto the fitness value of a solution. Solutions with better fitnevalues have a better chance of being reproduced in the nexteration. The algorithm then combines some attributes from osolution with those of another, selected from the set of reprodusolutions. This process is called crossover. The algorithm m

Fig. 8 Position workspace of 5 stage BRAID element „BRAIDelement base center Äorigin …

JANUARY 2004, Vol. 126 Õ 33

oao

f

t

ar

ctly

ing aran-etshlyn at isularfromper-

icsus-12

ge-lier,

rowna-uire

reas aialsithhe

fordis-theese

archgesedases

us,in-

y tontly.ullythe

also add new characteristics that were not present in the prevgeneration. This process is called mutation. Using the techniqof crossover and mutation, a new generation of manipulator cfigurations, is evolved. Appropriately designed genetic searchgorithms will converge to a locally optimum solution. This algrithm design involves selection of a fitness function, crossovermutation probabilities, and the population size. A full descriptiof genetic algorithms can be found in@25#.

3.2.2 Combinatorial Heuristic Search Algorithm.This algo-rithm was first described in@3#. To avoid exponential growth othe search space as the number of actuators grow, the invkinematics are solved by changing the state of only a few actors at any given time. This is perceived as ak-bit change to thegiven system state, where any system state is defined by anm-bitword ~m is the number of actuators!. At any state all possiblechanges~up tok-bits, wherek<m) are evaluated to determine thone that optimizes the search metric~i.e., reduces the error between the end-effector position and the desired position!. Thisoptimal change forms the new state of the system and the seprocedure repeats until convergence. This search method redthe computational complexity fromO(2m) to O(mk) or from ex-ponential computational time to polynomial computational timThis is seen through the following observation. For the binrobot being considered withm actuators, moving towards its taget by changing up tok-bits requires a search through (m/0)1(m/1)1 . . . 1(m/k) possible states, where (m/k)5m!/k!(m

Fig. 9 Inverse kinematics solution times for variousalgorithms.

34 Õ Vol. 126, JANUARY 2004

iousueson-al--ndn

erseua-

e-

archuces

e.ry

-

2k)!. Expanding these terms, the highest power ofm is found tobe k. An exhaustive search of same system would require exa2m search states.

3.2.3 Search Algorithm Simulation Results.Performance ofthe two search methods is quantified on a stochastic basis usMonte Carlo method. One hundred target points are selecteddomly within the volume of a binary workspace cloud. The targare chosen from within a spherical region, whose radius is roug90% of the radius of the actual point cloud. Each target is giverandom orientation. The inverse kinematics for each target pointhen solved for and the solution times, displacement and angerrors are computed and recorded. For comparison, resultsexhaustive searching are also presented. Simulations wereformed on a 750 MHz Pentium III computer.

Figure 9 shows the times for solving the inverse kinematproblem using the two algorithms described above. The exhative search proves to be quite fast for systems with less thanDOF ~4 parallel link stages!, the combinatorial algorithm is thefastest for systems having between 12 and 40 DOF, and thenetic algorithm fastest for larger systems. As mentioned earwith increasing complexity of the workspace~i.e., increasingDOF!, solution times associated with exhaustive searching gexponentially. However, both the genetic algorithm and combitorial search methods appear to converge to solutions that reqpolynomial computational times.

Errors in position and orientation for the various algorithms aalso quantified. Figure 10 shows how the average errors dropfunction of the number of BRAID stages for the combinatorand genetic search algorithms. Note that after about 10 stage~30DOF!, there are only marginal improvements in performance wincreasing number of DOF. An example of the distribution of terrors is shown in Fig. 11 for the case of a 15 stage~45 DOF!BRAID. The shapes of the error distributions are very similareach of the algorithms, and most closely resemble a gammatribution. The outliers are generally near the boundaries ofworkspace, and in practice tasks could be planned to avoid thregions.

Figure 12 shows an example of convergence of the two sealgorithms to reach a single random point as the number of sta~n! of the BRAID changes. The actuated leg length is normalizto one inch. Figure 12 shows that, the steady state error decrewith increasingn. Note that the absolute error difference~afterconvergence! for 12 stages and 17 stages is 0.015 inches. Thalthough the number of reachable points and the workspacecreases significantly with the number of stages, the accuracreach a particular point may not necessarily increase significa

These errors should not be viewed as the results of a foptimized system. The same stage geometry is used for all

Fig. 10 Average errors vs. number of DOF for different algorithms „100 samples per DOF …, „a… displacement error;„b… angular error

Transactions of the ASME

Journ

Fig. 11 Error distribution for a 15-stage BRAID „100 samples …, „a… displacement error, „b… angular error

znt

t

v

tlr

‘

ofmlyti-into

erateden ofgo-s-t forinases. Anorrrorhed in

d abe

systems analyzed, regardless of DOF. In reality, an optimistage geometry would be different for a 9-DOF system tha90-DOF system. Consequently, numerical optimization of allgeometric parameters would need to be carried out to determmore accurate limits on the errors of such systems.

3.2.4 Curve Fitting. For several practical tasks, in additioto endpoint positioning, the BRAID system may be required toa curve. For example, in the placement of a sensor while avoidobstacles, the BRAID mechanism would have to appropriabend around the obstacles~see Fig. 13!. Such curves may beformed as splines, polynomials or other approximations of curaround obstacles defined using way-points~or sometimes calledvia-points!. In such scenarios the hyper-redundancy of the sysmust be exploited. Conceptually, this inverse kinematics probis equivalent to the endpoint-positioning problem, as it requithe sequential iteration of the inverse kinematics solution on eintermediate stage. Each stage has an associated desiredpoint’’ position that lies on the curve—or as close as possiblethe curve. The genetic search algorithm is modified and appliethis problem. Fitness of a given system state is defined asr.m.s. error formed by the individual displacement errors froeach stage of the BRAID with respect to the desired curve.though several possible fitness functions or search metrics maemployed, this works well. The ability to plan such a task isprerequisite for demonstrating feasibility and utility of the binarobotic concept in real-world scenarios.

al of Mechanical Design

eda

heine

nfitingely

es

ememesach‘end-to

d tothemAl-y bea

ry

Simulations are carried out to demonstrate the feasibilityplanning actuator sequences. Cubic spline curves are randogenerated within the BRAID mechanism’s workspace, by idenfying a set of way-points. An example of such a curve is seenFig. 14 using 5 way-points. In Fig. 14, the curve is projected onthe XZ plane and the units arenormalizedwith respect to themaximum displacement of a single BRAID stage.

R.M.S. errors in position of the BRAID with respect to thdesired curve are presented. 100 cubic spline curves are geneand tested for each of then-staged BRAIDs. Figure 15 shows threlation of the average r.m.s. displacement errors as a functiothe number of stages in the BRAID for the genetic search alrithm. Note that for practical applications high DOF BRAID sytems would be necessary. Thus the simulation was carried oua minimum of 10 stage BRAIDs. This effect is also reflectedFig. 15 where the average r.m.s. displacement errors decresignificantly after 10 stages, and tapers off at about 40 stagesexample of the distribution of the errors is shown in Fig. 16 fthe case of a 50 staged BRAID. Again, the shapes of the edistributions most closely resemble a gamma distribution. Toutliers are generally near the boundaries of the workspace, anpractice tasks could be planned to avoid these regions.

4 Actuator ControlFor some applications, a hyper-redundant BRAID would nee

large number of actuators. In the future, the actuators would

Fig. 12 Search algorithm convergence for a single target point for n -staged BRAIDs, „a… Genetic searchconvergence; „b… Combinatorial search convergence

JANUARY 2004, Vol. 126 Õ 35

lloysrn totureio ofn 10go-

irege ofon-CCDago-ver,

rig-cep-ed-aketor

o-htIDing

malla

alformar-woofb-

er.ise.Thebee

ally

ili-esis-the

ntere.

ance

psin

di-can

the

5:1

he

anthodffersingac-

as

expected to be polymer-based smart materials, such as condupolymers and electrostrictive polymers@26,27#. However, whilethese materials are anticipated to meet the needs of the concethe future, they have not yet reached a sufficient state of devement to perform practical experimental demonstrations in devitoday. In the near term, shape memory alloys~SMAs! are being

Fig. 13 Endpoint positioning and avoiding obstacles †4‡

Fig. 14 Spline curve to match „see text for description …

Fig. 15 Average r.m.s. displacement error vs. number ofBRAID stages „100 samples …

36 Õ Vol. 126, JANUARY 2004

cting

pt inlop-ces

used as surrogate muscle actuators. SMAs are a class of awhich are able to remember their shape and are able to retuthat shape after being deformed, given a certain temperachange@28#. These alloys can be used as actuators, as the ratthe deformation stress to the recovery stress can be higher thato 1. It is proposed that each link be actuated by a pair of antanistic SMA wires, which open and close the link. Appropriate wlength and thickness are easily determined based on the ranmotion and the force output desired. For this laboratory demstration, the payload is assumed to be a sensor such as acamera weighing less than 50 grams. The stiffness of the antnistic pair and the flexures must also be accounted for. Howethe design of the SMA actuators is not a serious challenge.

To actuate the BRAID element, the actuators need to be tgered selectively, as required by the inverse kinematics. Contually, such a binary control is simple requiring no sensory feback. However, the large number of actuators can rapidly mthe physical realization of such a system difficult, if each actuarequires unique power supply lines. A multitude of wiring intrduces possibility for error and would result in additional weigand volume, large external forces, and complexity. The BRAconcept uses a more compact and efficient form of supplypower and control~see Fig. 17!. A common power line andground are provided to all the actuators. Each actuator has a s‘‘decoder’’ chip that can be triggered into either binary state bycarrier signal ‘‘piggybacked’’ on the power line. The carrier signis a sequence of pulses that identifies a unique address in theof a binary word for the actuator that requires toggling. Thischitecture reduces the wiring of the entire system to only twires ~see Fig. 17!. This can easily be implemented in the formconducting paint/tape~on the non-conducting polyethylene sustrate!. This bus architecture is currently being implemented.

The signal~consisting of a sequence of pulses! is extractedfrom the main power line by a simple Thevenin voltage dividThis is then driven through a de-bouncer circuit to remove noA counter then adds the number of pulses in the pulse train.output of the counter forms the binary address of the SMA totriggered~on or off!. A buffer between the counter output and thlatching circuit prevents intermediate count values to accidenttrigger the wrong SMA.

The buffer can be implemented in several ways. Two possibties are described here. First, an RC delay can be used. The rtor ~R! and capacitor~C! values can be changed so as to adjustrise time (t50.63R C), to allow sufficient time for the entiresignal to be processed by the counter before TTL~or CMOS!voltage thresholds are reached. Hence, the output of the couaffects the latching circuit only after a delay equal to the rise timTo be practical~allowing for variations in signal transfer time!,this may require large values for the resistance and capacitterms, making the circuit bulky.

A second option for the buffer is to use a series of flip-flotriggered by an end-of-pulse-train flag, added to the signal linethe form of a voltage spike. By introducing another voltagevider between the signal line and the buffer, all address pulsesbe ignored as they would lie below the TTL~or CMOS! voltagethreshold level. Hence, the buffer would only be triggered byend-of-pulse-train flag. For example, for TTL thresholds~1.3V!,the signal line with address pulses could peak at 2V. A 2.voltage divider would force the buffer to see a 0.8V~,TTLthresh.! signal. The buffer trigger pulse would peak at 5V on tsignal line. This would force the buffer to see a 2V~.TTLthresh.! signal, thus triggering it. This end-of-pulse-train flag calso be used to reset the counter. In testing, the second meproves to be more reliable and was implemented. After the buacquires the signal, a latching circuit decodes the address ustandard combinatorial logic, and latches it to the appropriatetuator trigger line using sequential logic~see Fig. 18!. For sim-plicity in fabrication each decoder chip can be identical, having

Transactions of the ASME

Fig. 16 Error distributions for a 50 staged BRAID „100 samples …

Fig. 17 SMA power and control bus, „a… Overview of actuator control electronics; „b… Power Õcontrol bus decoder architecture

Journal of Mechanical Design JANUARY 2004, Vol. 126 Õ 37

38 Õ Vol. 126, JANUA

Fig. 18 SMA power bus address decoding and latching electronics

i

.apfnetb

p

o

iu

nas,re-

allm-to aure-only

F.

canh atrol

cor-histhef itsse.nd a

beThese-

llowityooderturethendnts

many output lines as there are actuators. However, for anydecoder chip only one line is connected, thus providing uniqaddressing.

5 Experimental SystemThe experimental system constructed is shown in Fig. 19

consists of five parallel link stages coupled serially, giving aDOF end-effector. This device has yet to have its bistable joinstalled. With binary control this structure has 2335 ~or 32768!possible states giving the device suitable freedom for a numbeapplications. For other applications this could be extended toor 20 stages giving 23320 ~approximately 1018) possible statesWhile this closely approximates a continuous workspace, it leto some interesting inverse kinematic problems due to the hyredundancy of the system,~see Section 3!. A second generation othe BRAID system is currently in progress. This system will icorporate changes in the hinge/flexure design to increase thtigue life and improve the stiffness of the joints in each leg ofBRAID. Additionally, bistable elements, such as those descriin section 2, will be incorporated. This will reduce power cosumption needed to drive the BRAID. Finally, new binary tyactuators are being evaluated to overcome thermal, mechanand geometric limitations of SMA actuators. Polymer actuat~such as Electrostrictive Polymer Artificial Muscle! may allow forreduction in size, weight and power usage, with increased rangmotion, higher forces and longer fatigue life.

6 ConclusionsA new lightweight hyper-redundant binary device with potent

application to space robotics systems has been presented. S

RY 2004

oneue

. It5

nts

r of10

dser-

-fa-

heed

n-eical,rs

e of

alch a

system may be used for the deployment of booms and antenrapid and accurate positioning of sensors, and manipulationquiring simple and highly articulated onboard manipulators,while maintaining a small stowed volume. This device is coprised of a serial chain of parallel stages and can collapsesmall stowed volume. Each stage is composed of three flexbased elements, each with muscle type actuators, permittingbinary operation of each link. Each parallel link stage has 3 DOBy coupling these parallel stages in series, a 5 DOFend-effectorpose is obtained. With a large number of actuators, this systemapproximate a continuous system in dexterity and utility. In sucsystem, only a binary word command signal is needed to conthe motion of the entire system. Controller feedback and theresponding computational load is completely eliminated. Tscheme eliminates all but one power and one ground line insystem. The system does not have a closed form solution oforward kinematics. Numerical solutions to the BRAID inverkinematics problem~or trajectory planning!, are also presentedThese search techniques are based on a genetic algorithm acombinatorial heuristic algorithm. Both methods prove tohighly efficient as compared to exhaustive search methods.workspace of such manipulators and the actuator triggeringquence are areas of current study. Such optimizations would afor better distributed workspaces, allowing good dexterthroughout a region. Preliminary experimental results show gpromise@4,6#. Further, to eliminate the need for individual powwires to the large array of actuators, a system power bus struchas been developed. Current work continues to exploreflexure/hinge design of the BRAID to increase the fatique life aimprove the stiffness of the structure. Further, bistable eleme

Fig. 19 Experimental platform of BRAID

Transactions of the ASME

se

D

fE

rs

o

r

r

’’

e

red

o-In-

co-er-

d-In-

he,

ly-nal

hal-T-

ys-

and

y-nst.

nn-

le.,

ande,

-

a-

ing

c-

will be incorporated at each hinge. Alternative actuation schemsuch as Electrostrictive Polymer Artificial Muscles~EPAM!, arebeing explored.

AcknowledgmentsThe authors would like to acknowledge the NASA Institute f

Advanced Concepts~NIAC! for their support in this work.Also the technical contributions of Matthew Lichter~MIT ! andGuillermo Oropeza~MIT ! is acknowledged.

References@1# Ebert-Uphoff, I., and Chirikjian, G. S., 1996, ‘‘Inverse Kinematics of Di

cretely Actuated Hyper-redundant Manipulators Using Workspace DensitiProceedings of the IEEE International Conference on Robotics and Autotion, 1996,1, pp. 139–145.

@2# Erdmann, M. A., and Mason, M. T., 1988, ‘‘Exploration of Sensor-less Mnipulation,’’ IEEE J. Rob. Autom.,4, pp. 369–379.

@3# Lees, D. S., and Chirikjian, G. S., ‘‘A Combinatorial Approach to TrajectoPlanning for Binary Manipulators,’’ Proceedings of the IEEE InternationConference on Robotics and Automation, Minneapolis, Minnesota, April 19

@4# Lichter, M. D., Sujan, V. A., and Dubowsky, S., 2000, ‘‘Experimental Demostration of a New Design Paradigm in Space Robotics,’’ Proceedings ofSeventh International Symposium on Experimental Robotics, ISER 00.10–13, 2000, Honolulu, Hawaii.

@5# Oropeza, G., 1999, ‘‘The Design of Lightweight Deployable StructuresSpace Applications,’’ Thesis for the Bachelors of Science in Mechanicalgineering, Massachusetts Institute of Technology, May 1999.

@6# Sujan, V. A., Lichter, M. D., and Dubowsky, S., 2001, ‘‘Lightweight Hyperedundant Binary Elements for Planetary Exploration Robots,’’ Proceedingthe IEEE/ASME Conference on Advanced Intelligent Mechatronics~AIM’01!, July, 2001. Como, Italy.

@7# Goldberg, K., 1992, ‘‘Orienting Polygonal Parts without Sensors,’’ CompStruct., 1992, Special Robotics Issue.

@8# Chirikjian, G. S., and Burdick, J. W., 1995, ‘‘The Kinematics of Hyperedundant Robot Locomotion,’’ IEEE J. Rob. Autom.,11~6!, pp. 781–793.

@9# Huang, M. Z., and Shou-Hung Ling, 1994, ‘‘Kinematics of a Class of HybRobotic Mechanisms with Parallel and Series Modules,’’ Proceedings of1994 IEEE International Conference on Robotics and Automation,1.3, pp.2180–2185.

@10# Hughes, P. C., 1991, ‘‘Trussarm-A Variable Geometry Truss Manipulator,Intell. Mater. Syst. Struct.,2, pp. 148–160.

@11# Kwon, S., and Youngil, Youm, ‘‘General Algorithm for Automatic Generatioof the Workspace forn-link Redundant Manipulators,’’ Proceedings of th

Journal of Mechanical Design

es,

or

-s,’’

ma-

a-

ryal96.n-theec

orn-

-of

s.

-

idthe

J.

n

International Conference Advanced Robotics, 1991. ‘Robots in UnstructuEnvironments,’1.2 pp. 1722–1725.

@12# Chirikjian, G. S., and Burdick, J. W., 1990, ‘‘An Obstacle Avoidance Algrithm for Hyper-redundant Manipulators,’’ Proceedings of the 1990 IEEEternational Conference on Robotics and Automation. 13–18 May.1, pp. 625–631.

@13# Umetani, Y., and Hirose S., 1973, ‘‘Biomechanical Study of Serpentine Lomotion,’’ Proceedings of the 1st RoManSy Symp. 1973, Udine, Italy, SpringVerlag, pp. 171–184.

@14# Umetani, Y., and Hirose, S., 1976, ‘‘Biomechanical Study of Active CorMechanism with Tactile Sensors,’’ Proceedings of the 6th Int. Symp. ondustrial Robots, 1976, Nottingham, pp. c1-1-c1-10.

@15# Hirose, S., and Umetani, Y., 1978, ‘‘The Development of Soft Gripper for tVersatile Robot Hand,’’ Mechanism and Machine Theory, Pergamon Press13,pp. 351–359.

@16# Gravagne, I. A., and Walker, I. D., 2000, ‘‘On the Kinematics of Remoteactuated Continuum Robots,’’ Proceedings of the 2000 IEEE InternatioConference Robotics and Automation3, pp. 2544–2550.

@17# Huntsberger, T. L., Rodriguez, G., and Schenker, P. S., 2000, ‘‘Robotics: Clenges for Robotic and Human Mars Exploration,’’ Proceedings of ROBOICS2000, Albuquerque, NM, Mar 2000.

@18# Dotson R. D., 1995, ‘‘Spacecraft Deployable Structure Testing,’’ Space Stems Design and Development Testing~AGARD-CP-561!. AGARD. 1995, pp.6/1–12. Neuilly Sur Seine, France.

@19# Gantes, C., Connor, J., and Logcher, R. D., 1989, ‘‘Structural AnalysisDesign of Deployable Structures,’’ Comput. Struct.,32~3/4!, pp. 661–669.

@20# Meguro, A., Mitsugi, J., and Ando, K., 1993, ‘‘A Modular Cable-mesh Deploable Structure for Large Scale Satellite Communication Antennas,’’ Trans. IElectron., Inf. Commun. Eng. B-II,J76B-II ,~5!, pp. 476–84. Japan.

@21# Syromiatnikov, V. S., 1992, ‘‘Manipulator System for Module Redocking othe Mir Orbital Complex,’’ Proceedings of the 1992 IEEE International Coference on Robotics and Automation,1 pp. 913–918.

@22# Darby, A. P., and Pellegrino, S., 1999, ‘‘Modeling and Control of a FlexibStructure Incorporating Inertial Stick-slip Actuators,’’ J. Guid. Control Dyn22, pp. 36–43.

@23# Pellegrino, S., and Guest, S. D., 1998, ‘‘Deployable Structures: TheoryApplications,’’ Proceedings of IUTAM-IASS Symposium held in CambridgSeptember 1998, Kluwer Academic Publishers, Dordrecht.

@24# Ashby, M., 1992, ‘‘Material Selection in Mechanical Design,’’ ButterworthHeinemann, Oxford.

@25# Goldberg, D., 1989, ‘‘Genetic Algorithms in Search, Optimization, and Mchine Learning,’’ Addison-Wesley, Reading, MA.

@26# Madden, J. D., Cush, R. A., Kanigan, T. S., et al., 2000, ‘‘Fast-contractPolypyrrole Actuators,’’ Synth. Met.,113, pp. 185–193.

@27# Pelrine, R., Kornbluh, R., Pei, Q., et al., 2000, ‘‘High-speed Electrically Atuated Elastomers with Over 100% Strain,’’ Science,287~5454!, pp. 836–839.

@28# Gilbertson, R., 1994, Muscle Wires. San Alselmo, CA.

JANUARY 2004, Vol. 126 Õ 39