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““Automatic Synthesis of Automatic Synthesis of a Planar Linkage Mechanisma Planar Linkage Mechanism””

Yoon Young Kim

Seoul National University

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Our Goal: Automatic Mechanism Synthesis?

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Research MotivationResearch Motivation…………

More Use of Robots or artificial life-forms Motion-generating mechanisms become also important.

Optimization Methodology Can Be a Critical Design Tool.

Space Robot at MIT Binary Manipulatorat Johns Hopkins

Arrow Model at Brandeis Univ. Binary Manipulator

at SNU

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Issues in Mechanism Design?

1θ

2θ

3θ

( , , )x y z

[1] Forward Kinematics Analysis

Input iθ Output motion?

[2] Inverse Kinematics Analysis

Output( , , )x y z Input motion?

[3] Mechanism Synthesis [Given Input & Desired Output Motion Mechanism?]

The Subject of This Talk

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Related Research

Design variables: linkage shape, joint positionsObjective: minimize maximum forces required by cylinders

Optimal Shaping and Sizing of Mechanism Design

Fujita et al. (2000)

Example 1 : Design problem of hydraulic shovel mechanism

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Related Research

Design variables: linkage lengths, anglesObjective: maximum force outputConstraint: cover human finger’s stochastic workspace

Workspace analysis of human index finger

Example 2 : Design problem of Fingertip Haptic Display

FHD (U. of Washington, 2004)

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What if no good initial design is available?

Develop a new design method not requiring baseline designs, “Automatic Design Synthesis Method”

Many trials and errors

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Our Goal: Automatic Mechanism Synthesis

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Research Direction ?

0. Synthesize a desired mechanism without an Initial Layout.

?Rigid Link Revolute

Joint

1. To begin with, we will work with 2-D link mechanisms consisting of rigid links and revolute joints only.

2. We will set up Mechanism Synthesis as Optimization Problem, but how?

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Optimization Setup?

( )0

2Prescribed Actual Minimize max ( ) ( )f

P Pt t tf t t

≤ ≤• = −r r

P(t)

1) Kinematic equations are highly nonlinear.

2) The whole motion path should be traced.

3) Gradient-based Optimizer is always preferred for efficiency.

Difficulties?

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Design Domain Discretization Approach?

Workspace discretization

Link

Joint1. Revolute, 2. Rigid, 3. No joint

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Simply consider all possible cases?

Revolute Joint Rigid Joint

(Rigid Connection)

Candidate Mechanism 1

. . . . . .

. . . . . . . . . . . . . . . . . .

Candidate Mechanism 2

Candidate Mechanism i

Candidate Mechanism N

By GA? not so efficient, difficult for complex mechanisms

Better to use gradient-based optimizers if possible

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KEY: Need a Single Unified Model Representing “All Possible Kinematic Configurations” by Real-valued Variables.

Vary Link Density? does not make sense in rigid-body links

How about using the modeling technique used for

Topology Optimization?

Rigid or No Link

“Revolute J.”, “Rigid Joint” or Disconnected

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Model joints by means of springs with varying stiffness

Our Idea?

“Joint”

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Discretized Mechanism Configuration

Translational Spring KT

Rotational Spring KR

• Design Variables min MaxT T Tk k k≤ ≤

min maxR R Rk k k≤ ≤

Price: Kinematic analysis Kinematic analysis and static force equilibrium analysis (or equations of motion)

• Our Proposition!!

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This spring idea really works?

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1. Check Open Link Case1. Check Open Link Case

?Input

Motion

Desired Output Motion

Springs with varying stiffness Design

Variables

Our Model

t=0 sec t=1 sec t=2 sec

t=0 sec t=1 sec t=2 sec

Flying links with weak springs

At the 1st Optimization Iteration

At the final Optimization Iteration

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2. Check Closed Link Case

Known Solution (Mechanism)

Springs

Our Model?

Input Motion

Output Motion

PROBLEM: The Ranges of Relative stiffness Rates of rotational springs to translational springs are hard to adjust. Then what?

Animation for Optimization

History

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New Modeling Ideas:

“Replace Links with Rigid Blocks”

Links are Now Modeled by Rigid Blocks, and No Rotational Springs are used.

1ijK 2ijK

Better not to Use Rotational Springs, but Only Translational Springs. But How?

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This idea Can Be Useful?

i.e., all possible link combinations (Revolute Joint, or Rigid Joint, or disconnected Joint ) by adjusting stiffness values?

min max 1ijk k kε = ≤ ≤ =

Varying Translational Stiffness K

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1) Revolute Joint (at Location 1)

maxijk k= min

ijk k=

1 2

F

(Overlapping does not interfere with actual motion.)

F

1 2

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m inijk k=

1 2

m axijk k=

Interpretation: Rigid Block with Springs Link with Revolute Joints

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2) Rigid Joint

Interpretation

maxijk k=

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3) No Connection (or Disconnected Joint)

minijk k=

Interpretation

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Now Back to Optimization Formulation for Now Back to Optimization Formulation for Mechanism SynthesisMechanism Synthesis

P(t)1ijK 2ijK

min maxwith , 1,2, ,ik k k i N≤ ≤ =

( )0

2Prescribed Actual Minimize max ( ) ( )N f

P Pt t tf t t

≤ ≤∈• = −

kr r

• Subject to the field equations

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Equation of Motion or Force Equilibrium Equation

(1 )TZ+ Φ =M Y λ Q

Since velocities and accelerations are zero for static equilibrium, the equation yields equilibrium equations as

(1')TΦ − =Zλ Q 0 (2 )Φ = 0

Incremental equations for (1’) and (2) can be written as

( )⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −ΦΔ

Δ

ΦΦΦ R

λZ

0Qλ

Z

ZZZZTT

QλR Z −= TΦwhere,

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Mechanism Synthesis : Case Study

Input Motion

?oMotion 2:-20 Time×

oMotion 1:30 Time×

Objective: Minimize

Number of Design Variables: 19(open), 8(closed)( )

0

2Prescribed Actualmax ( ) ( )f

P Pt t tt t

≤ ≤−r r

Output Motion

Input Motion ?

oMotion :30 Time×

Output Motion

Open linkage

Closed linkage

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Need for 2nd-Generation Rigid Block Model

•We may need “anchored” revolute joints inside the configuration region to expand the design space. Then, what should we do ?

Input motion

Desired output motion ?

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m

“Connecting”Springs

Let’s Introduce “Anchoring” Springs

“Anchoring”Springs

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Check effectiveness of Anchoring Springs

anchoringk

connectingk

m

Case 1 41,10piling

connecting piling

kk k= = Case 2 41, 10connecting piling connectingk k k= =

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Now, Application……

Input Motion

?oMotion :30 Time×

( )0

2Prescribed ActualMinimize max ( ) ( )f

P Pt t tt t

≤ ≤−r r

Problem Definition

P(t)

Output Motion (click!) Connecting

Springs:

Anchoring Springs:

Discretized Design Model

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Convergence History

Max

imum

of

Dis

tanc

e

Stiff

ness

Objective Function Design variables

Convergence History

Synthesized Mechanism

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Alternative anchoring spring model: model 2

“Connecting”Springs

“anchoring” Springs

m

Model 1

Model 2

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Linear motion mechanism

Number of Design Variables: 59

Problem Definition

Input Motion

( )0

2Target ActualMinimize max ( ) ( )f

P Pt t tt t

≤ ≤−r r

?oMotion :30 Time×

Output Motion (click)

Discretized Design Model

Our Modeling

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Objective Function Design variables

Convergence History

Synthesized Mechanism

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Door-opening mechanism design test

Problem Definition

Output Motion (click)

0

Minimize max ( ) ( )ft t t

y y x xt t

x x y y≤ ≤

⎛ ⎞− −+⎜ ⎟

⎜ ⎟− −⎝ ⎠

* *

* *

* *

A BA BAA BB

A BA B

r r

Connecting Springs:

anchoring Springs:

Discretized Design Model

?oMotion :30 Time×

Input Motion

Output Motion (click)

*AA

*B

B

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Objective Function Design variables

Convergence History

Synthesized Mechanism

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ConclusionsConclusions

• Automatic Synthesis of Rigid-link and Revolute Joint Mechanism Was Investigated.

• A Synthesis Model Consisting of Rigid Blocks and Springs Is Proposed.

• Some Successful Test Results Were Obtained.

Remark: still a long way to go……, it is just a beginning.

1ijK 2ijK

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Based on : Y.Y.Kim et al, “Automatic Synthesis of a Planar Linkage Mechanism with Revolute Joints by Using Spring-Connected Rigid Block Models,”ASME J. Mech. Design, Accepted (2006)

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Mechanism synthesis using ADAMS Mechanism synthesis using ADAMS

Step 1: Discretize by Block links and dummy links

Green links: dummy links Dummy

motion

Point with desired motion

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1ijK 2ijK

Connecting Rigid-body block by springs? Use “Bushing Elements”

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Block motion

Point with desired motion

Connecting Blocks by bushing element

Bushing ElementsBushing Elements…………

1ijK 2ijK

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Bushing stiffness varies as a function of a design variable

Create stiffness variables

Create auxiliary design variables

Make bushing stiffness vary as a ftn. of a design variable

Step 2:

1ijK 2ijK

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Mechanism synthesis using ADAMS Mechanism synthesis using ADAMS Step 3: Define objective function

Marker number of output point of Block link

Marker number of output point of dummy link

Objective function name

* DM: Distance between output point of block link and dummy link

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Mechanism synthesis using ADAMS Mechanism synthesis using ADAMS

Step 4: Set and solve optimization problem