Rigid-Body Nano-Manipulator Design Criteria for … Nanotechnology and Molecular Engineering Workshop, Caltech, Pasadena, January 5-17, 2004 1 Andrés Jaramillo-Botero Pontificia Universidad

Computational Nanotechnology and Molecular Engineering Computational Nanotechnology and Molecular Engineering Workshop, Caltech, Pasadena, January 5Workshop, Caltech, Pasadena, January 5--17, 200417, 2004

11

Andrés Jaramillo-BoteroPontificia Universidad JaverianaCali, Colombia

Rigid-Body Nano-Manipulator Design Criteria for Reduced Constrained Dynamics


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ObjectiveObjective

Design a 3-6DOF nano-scale rigid-body manipulator with implicit simplified dynamic complexity, to:

1. Reduce real-time control requirements. Increased ∆t, decoupled and/or invariant Articulated Body Inertia (ABI), Single Input Single Output (SISO) control, and/or,

2. Reduce molecular modeling and simulation requirements (rigid-body model, scalable algorithms)

Other criteria: speed, dexterity, repeatability, payload, compliance motion, reachable workspace.PREMISE: Macroscopic counterparts are not good a starting pointPREMISE: Macroscopic counterparts are not good a starting point


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Motion ControlMotion Control

SimulationSimulation

Robotics IntroRobotics Intro

Inverse Kinematics

Forward Kinematics

Element and Joint Element and Joint ParametersParameters

Position and Orientation of End-Effector

Joint Angles and Displacements

Inverse Dynamics

Element and Joint Element and Joint Parameters (mass)Parameters (mass)

Kinematics Parameters (trajectory)

Effective Forces

Kinematics Dynamics

End-Effector

Joints

Elements

Base

5 Degree of 5 Degree of Freedom Freedom Spatial Robot Spatial Robot Manipulator, Manipulator, Mitsubishi Mitsubishi RVRV--M1M1

Forward Dynamics (EOM)


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Examples: Puma 560Examples: Puma 560Coupled, configuration dependent

Jaramillo-Botero, A. “Design Criteria for Simplified Robot Dynamics” Epiciclos Magazine, 2002

θ1

θ2

θ3

θ4,5,6

Puma Robot, Unimation Inc.


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Examples: Stewart PlatformExamples: Stewart Platform

fineMotion970116.pdb

Highly Coupled, low dexterity, configuration dependent, reduced workspace, actuators, good payload

Molecular Robot: Stewart Platform (2,596 atoms). Eric Drexler.Macroscopic Stewart Platform Robot


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Constrained RigidConstrained Rigid--Body Molecular DynamicsBody Molecular Dynamics

nn Simplified description and Simplified description and analysis analysis –– implicit constraints implicit constraints

nn Study conformations and Study conformations and rigidrigid--body dynamics (system body dynamics (system partitioned in rigidpartitioned in rigid--bodies bodies connected by hinges)connected by hinges)

nn EOM in internal coordinates EOM in internal coordinates nn Dense Mass OperatorDense Mass Operatornn IncreasedIncreased timetime--stepstep

Pi,i+1


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Equations of Motion Equations of Motion –– EOM EOM (Internal Coordinates)(Internal Coordinates)

MicrocanonicalMicrocanonical EnsembleEnsemble (NVE)(NVE)

Where, from NewtonWhere, from Newton--Euler dynamics using spatial operators,Euler dynamics using spatial operators,

( ) ( ),T M Cq q q q q= +&& & &

( ) ( )1 ,M T Cq q q q q− − = & & &&

T TM H I Hr r=( )T T T TC H I H I H Hr r r r = + + & &&

Mathiowetz, Jain, Karasawa, Goddard, NEIMO, [94]Fijany, Cagin, Jaramillo-Botero, Goddard, MCFA , [96]

TextJ F+

In contact with environment


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EOM …EOM …

Equivalently, from LagrangeEquivalently, from Lagrange--Euler, the velocity dependent terms Euler, the velocity dependent terms are expressed as a function of are expressed as a function of MM

12

TC M Mqq q q= −& & & &

Canonical EnsembleCanonical Ensemble (NVT)(NVT)Introduce normal experimental conditions:Introduce normal experimental conditions:

i)i) Interaction with a heat bathInteraction with a heat bathii)ii) Hamiltonian chosen for proper canonical distribution in momentumHamiltonian chosen for proper canonical distribution in momentum

and configuration spaceand configuration space


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Extended EOM Extended EOM –– Canonical NewtonCanonical Newton--HooverHoover

Where,Where,

EOM same form as EOM same form as microcanonicalmicrocanonical, except friction term and , except friction term and nonbondnonbond force gradient,force gradient,

( ) ( ) ( ),H HT M Cq q q q q= +&& &

( )12

NBTH XC M MMqq x q qq q Φ= − + +& & & &&

21 1s B

TTx t = −

&

NoséNosé:: introduces an additional coordinate variable introduces an additional coordinate variable (virtual time(virtual time--scalingscaling, s) and conj., s) and conj.HooverHoover: real: real--time explicit by transforming time explicit by transforming Nosé’sNosé’s EOMEOM

Friction


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Underlying AssumptionUnderlying Assumption

nn It is known from the structure of It is known from the structure of MM that it only depends on the that it only depends on the kinematics of the manipulator arm and mass properties of its kinematics of the manipulator arm and mass properties of its individual links.individual links.

nn It then follows, from the established EOM for both the It then follows, from the established EOM for both the microcanonicalmicrocanonical and canonical forms that a decoupled (i.e. and canonical forms that a decoupled (i.e. diagonal) form of diagonal) form of MM and/or configuration invariant form of and/or configuration invariant form of MMwould greatly simplify the complexity of the systems dynamicswould greatly simplify the complexity of the systems dynamics

T TM H I Hr r=


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Structure of Structure of MM: Symmetric Positive Definite (SPD) : Symmetric Positive Definite (SPD) for serial chains (direct or remote actuation)for serial chains (direct or remote actuation)Method Method [Jaramillo[Jaramillo--BoteroBotero, 2002], 2002]: : Decompose ABI in diagonal/offDecompose ABI in diagonal/off--diagonal terms to determine decoupling and invariance conditionsdiagonal terms to determine decoupling and invariance conditions..

Diagonal TermsDiagonal Terms (dependence on principle moments)(dependence on principle moments)

OffOff--diagonal Termsdiagonal Terms (decoupling requires elimination, invariance constancy)(decoupling requires elimination, invariance constancy)

,1

n TT k ki i i i i i ik

k iM H I P I P H

= +

= + ∑

,1

n TT i k ki j i j j i j jk

k jM H P I P I P H i j

= +

= + > ∑


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Decoupling: proof of existence and criteriaDecoupling: proof of existence and criteria

MM defines a metric tensor on the configuration manifold (serial sdefines a metric tensor on the configuration manifold (serial spatial patial manipulators) [manipulators) [SpongSpong, 1992]. Then, if Euclidian, the system is said to be , 1992]. Then, if Euclidian, the system is said to be diagonalizable. This involves an analytic (symbolic) solution tdiagonalizable. This involves an analytic (symbolic) solution to the Ricci tensor o the Ricci tensor to determine conditions under which it vanishes [Jaramilloto determine conditions under which it vanishes [Jaramillo--BoteroBotero, 2002]. , 2002].

{ }[ ] { }[ ]

222 212

, , 0

ij hjhk ikhijk

i j i jh k k hl lij ik

l

MMM MR

hk l hj lq q q q q q q q

∂∂∂ ∂ = + − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + − = ∑


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Invariance: proof of existence and criteriaInvariance: proof of existence and criteria

Involves calculating theInvolves calculating the MMqq (partial derivative of the spatial articulated body (partial derivative of the spatial articulated body inertia with respect to joint states) to determine the sensibiliinertia with respect to joint states) to determine the sensibility of ty of MM with respect with respect to manipulator pose [Jaramilloto manipulator pose [Jaramillo--BoteroBotero, 2002]. , 2002].

0T i T i TM H H M M H Hq d dr r r r = − = % %


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Arm Design for Decoupled ABI Arm Design for Decoupled ABI (case: 2 DOF direct drive)(case: 2 DOF direct drive)Method: Immobilizing all joints, but one, then generalizeMethod: Immobilizing all joints, but one, then generalize

Structure:Structure: for arms with open kinematic chain structure, the ABI cannot be decoupled unless the joint axes are orthogonal to each other (2 DOF).Mass properties:Mass properties: and 2 z xzm Lr I= 0yzI =

0zr =0L =

Two options for mass properties to be validTwo options for mass properties to be valid

z

b2b1

z

b1

b2-rz

2DOF Arm2DOF Arm

y

z

x

m2

m1

L

-rz

b1

b2Joint 1

Joint 2


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Arm Design for Invariant ABI Arm Design for Invariant ABI (case 1: 2 DOF direct drive)(case 1: 2 DOF direct drive)Method: Immobilizing all joints, but one, then generalizeMethod: Immobilizing all joints, but one, then generalize

Structure:Structure: for arms with open kinematic chain structure, the ABI can only be invariant, iff, one of the following conditions is satisfied:

10

0

z

x y

bb bL

== == 2

00

x y

xy yz xz

xx yy

r rI I II ML I

= == = =+ =

Type: IType: I--11 Type: IType: I--22

z

xL

b2

b1

b2

b1


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Arm Design for Invariant ABI Arm Design for Invariant ABI (case 1: 2 DOF direct drive)(case 1: 2 DOF direct drive)Method: Immobilizing all joints, but one, then generalizeMethod: Immobilizing all joints, but one, then generalize

2

2

0

0

y

xx yy

xy yz

z xz

rI I MLI Im Lr I

== += =

=

00

0xx yy

xy yz xz

LI II I I

== == = =

Type: IType: I--33 Type: IType: I--44y

z

x

b2

b1

b2

y

z

x

m2

L

-rz


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Arm Design for Decoupled and Invariant ABIArm Design for Decoupled and Invariant ABI(direct drive)(direct drive)Method: Immobilizing all joints, but one, then generalizeMethod: Immobilizing all joints, but one, then generalize

The necessary and sufficient conditions for an open kinematic chain structure to possess a decoupled and invariant ABI matrix are given by:

Structure:Structure: 2 DOF arms with orthogonal axes.

Mass properties:Mass properties: I-3 or I-4 in Theorem 2Theorem 2. I-3=DI-1 and I-4=DI-2


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Arm Design for Invariant ABI Arm Design for Invariant ABI (case 2: 3 DOF direct drive)(case 2: 3 DOF direct drive)Method: Adding configurations with previous conditions metMethod: Adding configurations with previous conditions met

2,3

2 2,2 3 2,3

3 3 3

3 3

00

0xy yz xz

xx yy

Lm q m qI I II I

=+ =

= = ==


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Arm Design for Invariant ABI Arm Design for Invariant ABI (case 3: 3(case 3: 3-- 6 DOF direct drive)6 DOF direct drive)Method: Making the Systems Energy Position IndependentMethod: Making the Systems Energy Position Independent

( )( )[ ]( ) ( ) ( )[ ]

( ) ( )( )

22 3 2 2

23 3

22 2 2 21 1 2 2 3 2 3 3 2 2

2 2 22 2 1

2 2 22 3 2 3 1

2 22 2 3 1 3 2 2 3

223 2 3

3

2 3

3

3 2 3 3 2 3

12

1 cos sin21 cos sin2

cos cos cos

zz xx

zz xx

xx yy yyK I I I M a

M a p

I M a I

I M p I

M a p M a p

q q q q q

q q q

q q q q qq q q q q q q q

q q

= + + + + + +

+ + + ++ + + +

+ +

+

+

& & & & &

&

&

& & & &

& &

( ) ( ) ( )2 2 3 2 3 32 2 3sin sinM p a MP M pg gq q q++= +

Design yellow terms, so that,2

3 2 3 2 2

3 3 2 2 3 2

0;; 0zz xx

zz xx

p I M a II I M p M a

= + == + =

( ) [ ]22 2 2 2 21 1 2 2 3 2 3 3 2 2 2 3 1

12 xx yy yy xx xxK I I I M a I Iq q q q q q = + + + + + +

& & & & & &0P =

Kinetic and Potential Energy expressions become: Yang and Tzeng, 1986


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Simple ValidationSimple Validation ((MatlabMatlab))

Jaramillo-Boter, Daza, Diaz, 2002


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Direct Drive vs. Remote ActuationDirect Drive vs. Remote Actuation

nn One way to relax the One way to relax the orthogonalityorthogonality condition for condition for decoupling is to relocate the decoupling is to relocate the actuators to reduce the actuators to reduce the reaction torques exerted by reaction torques exerted by one actuator and which do one actuator and which do not necessarily act on the not necessarily act on the adjacent link [Asada et al. adjacent link [Asada et al. 86].86].


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Decoupling by Actuator RelocationDecoupling by Actuator Relocation

nn 4 DOF multi4 DOF multi--bar spatial bar spatial mechanism: partiallymechanism: partially--decoupled decoupled and Invariant ABIand Invariant ABI

nn Decoupling handled under Decoupling handled under special pose, control dependent special pose, control dependent (O(1) SISO controller possible)(O(1) SISO controller possible)

nn The rotation of any link is due to The rotation of any link is due to 1 and only 1 actuator1 and only 1 actuator

8

6

5

4 1

3

2

7

9

1q

2q

3q4q

{ }{ }{ }

1 4 6

52 8

3 7 9

234

S S SS S SS S S

= = == = == = =


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Computed Torque Control LawComputed Torque Control Law

Where Where ee, , KKvv and and KKpp are constant gain matricesare constant gain matrices..

The error dynamics:The error dynamics:

FF:FF: provides necessary drive to move along nominal path. provides necessary drive to move along nominal path. FB:FB: provides correction torques to trajectoryprovides correction torques to trajectory

( ) ( ) ( )( ), v pdT M C M K e K eq q q q q q= + + − −&& & & &

de q q= −

0te →∞ →

( )( ) ( ),v pdT M K e K e Cq q q q q= − − +&& & &&

( )( ) 0v pM e K e K eq + + =&& &

FF: Feedforward Component FB: Feedback Component

Introduce actual error in trajectory


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Thermal Noise and Positional Thermal Noise and Positional Variance (uncertainty)Variance (uncertainty)

( )( )[ ]( )[ ]

exp /

exp /B

x

B

V x k Tf xV x k T dx

∞

−∞

−=−∫

( )[ ][ ]

( )

( )

2

2

2

2 2

exp /2

exp /2

exp /22

1 exp /22

s Bx

s B

s B sB

classclass

k x k Tf xk x k T dx

k x k T kk T

xp

sps

∞

−∞

−=−

−=

= −

∫

( ) 212 sV x k x=

From Classical Statistical Mechanics

Potential Energy Harmonic Oscillator

PDF for position coordinate x


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Thermal Noise and Positional Thermal Noise and Positional Variance (uncertainty)Variance (uncertainty)

2 BS

k Tks =

s : RMS positional errorkS: Transverse stiffnesskb: Boltzmann’s constantT: temperature

s 2: 0.02 nm (0.2 Å)KS: 10 N/mkb: 1.38 x 10-23 J/KT: 300 K

Thermal noise can be controlled by decreasing the temperature T or by increasing the restoring force fR (measured by the stiffness).

Less than an Atomic diameterPositional VariancePositional Variance

R Sf k d= − Diamond parametersChosen for example


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Thermal Noise and Positional Thermal Noise and Positional Variance (the Quantum version)Variance (the Quantum version)

s : RMS positional errorh: Planck’s constantkb: Boltzmann’s constantks: Stiffnessw: frequencyT: temperature

2 1 1 1 12 2exp 1 exp 1s s

B B

ww wk mk

k T k Ts

= + = + − −

h hh h

PDF for position and momentum

Drexler, K.E., Nanosystems: molecular machinery, manufacturing, and computation, Wiley and Sons, 1992.

22 Smks = h


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Transverse stiffness of Transverse stiffness of a cylindrical a cylindrical rodrod of radius r and length Lof radius r and length L

E: Young’s modulusk: transverse stiffnessr: radiusL: lengthI: Moment of inertia cross section

43 3

3 34S

EI r Ek L Lp= =

E: 1012 N/m2

k: 10 N/mr: 8 nmL: 50 nm

s 2 =0.007 nm

2 BS

k Tks = 3

24

43

BL k Tr Es p=

Diamond parametersChosen for example


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1

00 0 0 1

i i i i i i i

i i i i i i iii

i i i

cos cos sin sin sin a cossin cos cos sin cos a sin

Asin cos d

θ α θ α θ θθ α θ α θ θ

α α−

− ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ = ,

04

0 0 0 1

xn n n nx x x

n n n ny y y y

n n n nz z z z

x y z px y z p

Tx y z p

= .

n n n nx y y x

n n n nx y y x

n n n nx y y xn

i nx

n y

nz

x p x p

y p y p

z p z pdTJd x

x

x

θ

− +

− +

− +⇒ =

Kinematic Propagation of Kinematic Propagation of RMS Positional Variance:RMS Positional Variance:RMS Workspace Error VolumeRMS Workspace Error Volume

nn θθ:: Joint angleJoint anglenn αα:: Angle between Angle between

adjacent joint axisadjacent joint axisnn d, a:d, a: Distances Distances

between adjacent jointsbetween adjacent joints

0 1 2 3 01 2 3 4 4A A A A T=

0θ400

0θ3500

0θ2500

50θ10ππ/2/2

diNm

θiRad

ainm

αirad

dV dJdt dtq=


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Molecular Building BlocksMolecular Building Blocks

DNA packagingDNA packagingMotor (Purdue),Motor (Purdue),ATPase (UCB),ATPase (UCB),Viral PeptideViral PeptideVPL (Rutgers).VPL (Rutgers).

Electric Electric Motors, Motors, Pneumatics, Pneumatics, HydraulicsHydraulics

ActuatorsActuators

Proteins, DNA, Proteins, DNA, NanojointsNanojointsRotational, prismatic, spherical …Rotational, prismatic, spherical …JointsJoints

Metal, Plastic, WoodMetal, Plastic, WoodStructural ElementsStructural Elements

DNA CNB

M. G. Rossmann et al. 2003. Bacteriophage f29 scaffolding protein gp7 before and after prohead

assembly. Nat. Struct.Biol. 10:572-576

Motor that powers the DNA packaging system in a virus

Hongyun Wang, George Oster, Nature396, 279-282 (1998)).

Mavroidis, C., and Dubey, A., 2003, "From Pulses to Motors",

Nature Materials, 9 2(9):573-574


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Molecular Building BlocksMolecular Building Blocks

Magnetic force sensing Magnetic force sensing Force sensors, position Force sensors, position sensors, pressure, sensors, pressure, temperaturetemperature

SensorsSensors

DNA double DNA double crosscross--over (Rutgers),over (Rutgers),Carbon Carbon nanotubesnanotubes

Gears, Belts, ChainsGears, Belts, ChainsTransmission ElementsTransmission Elements

DendrimersDendrimers(NASA Ames Research Center)

WiresWiresInterconnectsInterconnects


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Mol

ecul

ar

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arts

Parts

FullereneFullerene BasedBased NanogearsNanogearsD.H. D.H. RobertsonRobertson, B.I. , B.I. DunlapDunlap*, & C.T. *, & C.T. WhiteWhite*, *Naval *, *Naval ResearchResearch LaboratoryLaboratory, Washington DC., Washington DC.

0 to 0.1 revs/ps in 10ps 0 to 0.5 revs/ps in 50ps 0 to 0.5 revs/ps in 10ps

0.5 rev/ps=5x10 11rev/s=500GHz=1/2 trillion revs/sec


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Carbon Carbon NanotubeNanotube Based Based NanogearsNanogearsJieJie Han, Al Han, Al GlobusGlobus, Richard Jaffe, Glenn , Richard Jaffe, Glenn DeardorffDeardorff, NASA Ames Research Center, NASA Ames Research Center

Rack and pinion : rack follows Rack and pinion: pinion follows

Gear rotation in vacuum (50/70/100 rot/ns)

HydrogenBencene Molecule

Gear rotation at room temperature (50/70/100 rot/ns)

>100 rot/ns


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Planetary Gear (version 1) (Version 2)

Kinetic T / Rotation step Kinetic T / Time

Differential Gear

Planetary GearsPlanetary GearsCagin, T., Jaramillo-Botero, A., Gao, G., W. A. Goddard, III, Nanotechnology 9 (3), 143-152 (1998).Eric Drexler, Institute of Molecular Manufacturing


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NanoactuatorNanoactuator: : NeonNeon AtomAtom BombBombCagin, T., JaramilloCagin, T., Jaramillo--Botero, A., Gao, G., W. A. Goddard, IIIBotero, A., Gao, G., W. A. Goddard, IIIEric Eric DrexlerDrexler, , InstituteInstitute ofof Molecular Molecular ManufacturingManufacturing

LaserLaser ActuatedActuated NanomotorNanomotorDr. Don W. Dr. Don W. NoidNoid, Dr. , Dr. BobbyBobby G. G. SumpterSumpter, , RobertRobertTuzunTuzun, , OakOak RidgeRidge NationalNational LaboratoryLaboratory (ORNL)(ORNL)

Positive Charge

Negative Charge

Mol

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arts

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•Fixe

d Molec

ular mach

ines can

be act

uated e

lectrica

lly.

•Floa

ting Molec

ular mach

ines (in

liquid) c

an be ac

tuated

via

chemical

energy

, light o

r acous

tic ener

gy

•Molec

ular mach

ines ins

ide the

body ca

n be act

uated v

ia

glucos

e -oxyge

n combus

tion cel

ls


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arts

OrganicOrganic ActuatorsActuatorsLet the RecBCD helicase enzyme adhere to the ADN strand by its loose end. As the RecBCDuncoils the AND strand, the fluorescent dye is removed (the ADN string seems to shorten). 1000 base pairs/sec. RedBCD is composed of 3 proteins, 2 of which are molecular motors!

Dillingham, M. S., Spies, M. and Kowalczykowski, S.C. (2003) RecBCD enzyme is a bipolar DNA helicase. Nature, in press. Universidad de California en Davis.

DNA Throttle Controls Molecular Machine DNA Throttle Controls Molecular Machine

Kinesin is a power generating enzyme, a protein motor that converts free energy

from the phosphate-gamma bond of ATP into mechanical work. This work is used to transport intracellular organisms along

the axonal microtubules.


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ConclusionsConclusionsnn NanomanipulatorNanomanipulator cannot be treated as Macroscopic counterpartscannot be treated as Macroscopic counterpartsnn Open Open kinematickinematic chain DD arms cannot have > 2 decoupled DOF,chain DD arms cannot have > 2 decoupled DOF,nn Decoupling can be achieved for > 2 DOF arms by actuator Decoupling can be achieved for > 2 DOF arms by actuator

relocation (closed relocation (closed kinematickinematic chains),chains),nn Configuration invariance possible for more than 3DOF by Configuration invariance possible for more than 3DOF by

elimination of nonlinear terms due to velocity (elimination of nonlinear terms due to velocity (microcanonicalmicrocanonical),),nn Linearization improves highLinearization improves high--speed performance (speed performance (microcanonicalmicrocanonical))nn Full decoupling (3DOF): SISO control O(1) parallel dynamic Full decoupling (3DOF): SISO control O(1) parallel dynamic

((microcanonicalmicrocanonical),),nn Simplified dynamics larger integration timeSimplified dynamics larger integration time--steps in rigidsteps in rigid--body MD,body MD,nn Full linearization not possible for canonical due to friction anFull linearization not possible for canonical due to friction and d

nonbondednonbonded interactions, but improved performance achieved,interactions, but improved performance achieved,nn Decoupling/Invariance: reduction in computational steps (/1,000 Decoupling/Invariance: reduction in computational steps (/1,000

without reduction of friction terms in canonical)without reduction of friction terms in canonical)


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AcknowledgmentsAcknowledgments

nn Institute of Pure and Applied Mathematics (IPAM), University Institute of Pure and Applied Mathematics (IPAM), University of California Los Angeles (UCLA).of California Los Angeles (UCLA).

nn National Science Foundation (National Science Foundation (NanoscaleNanoscale Science and Science and Engineering Program)Engineering Program)

Special thanks Special thanks to:to:nn Dr. Mario Blanco, MSC, CaltechDr. Mario Blanco, MSC, Caltech

Documents

Rigid-Body Nano-Manipulator Design Criteria for … Nanotechnology and Molecular Engineering Workshop, Caltech, Pasadena, January 5-17, 2004 1 Andrés Jaramillo-Botero Pontificia Universidad