Folding Cartons with Fixtures: A Motion Planning Approach Liang Lu - - PDF document

Folding Cartons with Fixtures: A Motion Planning Approach

Liang Lu Srinivas Akella Beckman Institute for Advanced Science and Technology University of Illinois at Urbana-Champaign

Industrial Need

Flexible systems that:

• automatically plan how to fold a

given carton

• enable designers to evaluate

carton foldability properties Benefits:

• Reduce time and cost to

manufacture new products

• Enable virtual prototyping and

computer-aided design

Minimalist Approach

Develop simple and flexible systems to accomplish task Robust systems with simple hardware elements Complexity is in analysis and planning software Characterize the capabilities of system Examples: Erdmann(1993); Canny and Goldberg(1994); Bicchi(1995); Chirikjian(1995); Lynch(1996)

Manipulation of Articulated Objects

How many robot degrees of freedom are needed to control object degrees of freedom? What planning algorithms enable desired manipulation of objects? Can robot system easily adapt to new

• bjects?

Related Work

Carton folding machines: Ward (1981); Capdeboscq (1985); McBride and Lile (1986); Marschke (1989); Dorell Equipment (1990s) Sheet metal bending: Inui et al. (1987); Wang and Sturges (1993); Radin, Shiptalni, and Hartman (1997); Gupta et al. (1998) Motion planning: Lozano-Perez (1983, 1987); Barraquand and Latombe (1991); Gupta and Guo (1995); Kavraki et al. (1996); Kavraki, Lamiraux, and Holleman (1998); Amato et al. (1998) Assembly planning: Nevins and Whitney (1978); Krishnan and Sanderson (1991); Wilson and Latombe (1994); Goldberg and Moradi (1996) Computational Geometry and Origami: Lubiw and O’Rourke (1996); Biedl et al. (1998); Demaine, Demaine, and Mitchell (1999) Minimalism: McGeer (1990); Erdmann(1993); Canny and Goldberg(1994); Bicchi(1995); Chirikjian(1995); Lynch(1996)

Carton Folding

Folding cartons to package two-way radios Need automated solution:

• eliminate repetitive stress injuries
• adaptable to different models

Carton Ready For Loading Carton Blank folding

Example Folding Sequence

Sequence of folds to transform carton blank to folded carton

Carton Blank θ ( ) θ θ ( )

5 7

θ ( ) θ ( ) θ θ ( )

6 2 3 1 4

θ θ θ θ θ θ θ

1 2 3 4 5 6 7

Folding with Fixtures

Design fixture shape and select cardboard motions so blank is folded into carton

θ ( )

2

θ ( )

3

F1 Carton Blank θ θ θ θ θ θ θ θ ( )

1 4

θ θ ( )

5 7 1

θ

2 3 4 5 6 7

Fold Sequencing

Fold sequencing as motion planning for an articulated robot Given: Cardboard robot with n rotary joints. Find: Sequence of (possibly simultaneous) collision-free joint motions from initial state to folded state

θ θ θ θ θ θ θ

1 2 3 4 5 6 7

Configuration Space Representation

Recursive slice projection (Lozano-Perez, 1987) Modifications: Nonserial manipulator Generate all feasible folding sequences Exploit symmetry

θ θ θ θ θ θ θ

6 4 3 2 1 7 5

Modeling Assumptions

Carton joints rotate due to contact with the fixture A joint that begins rotating continues to rotate till it reaches its goal orientation All joints rotate with the same angular velocity Multiple joints can move together; they may begin moving simultaneously or with delays Generate folding sequences by identifying valid unfolding sequences ω

Motion Planning Model

Generate folding sequences by identifying valid unfolding sequences

• 1. Carton joints rotate due to contact

with the fixture

• 2. Joints rotate monotonically to goal
• rientation at angular velocity
• 3. Multiple joints can move together;

they may start simultaneously or with delays ω

Motion Constraints

Folding with fixtures constrains possible carton motions For the kth joint: Motion constraints: tk

start

δ j ω

• j

1 = k 1 –

∑

= tk

end

highik lowik – ω

• tk

start

+ = θik t ( ) ωt = lowik + tk

start

t tk

end

≤ ≤ θi2 t ( ) θi1 t ( ) δ1 – = θin t ( ) θin

1 –

t ( ) δn

1 –

– = θik t ( ) θik

1 –

t ( ) δk

1 –

– = t2

start

t t1

end

≤ ≤ tk

start

t tk

1 – end

≤ ≤ tn

start

t tn

1 – end

≤ ≤

Motion Constraints (cont’d)

Initial constraints: Goal constraints: t tk

start

< k , 1 … n , , = θik t ( ) lowik = θik t ( ) highik = t tk

end k

, > 1 … n , , =

2 goal

q

init

q

1 2 1 5 4 3 2 1 3 1 2

θ θ δ δ l l l θ P P l l

Search Algorithm

Generate n! joint sequences For each joint sequence, find all feasible discretized values of for each joint Worst-case time complexity: Maximum number of line segments in a path is 2n-1 Pruning tricks: Eliminate all paths that share a line segment that intersects an obstacle Identify subset of joints that can be first and second joints δ O n!sn ( )

Folding Sequence

5 7 4 6 3 1 2 unfolded carton folded carton

Folding Sequence

5 7 4 6 3 1 2 unfolded carton folded carton

Multiple Angular Velocity Formulation

Permit different carton joints to rotate at different angular velocities For the kth joint: tk

start

δ j γi jω

• j

1 = k 1 –

∑

= tk

end

highik lowik – γikω

• tk

start

+ = θik t ( ) γikωt = lowik + tk

start

t tk

end

≤ ≤ γik 1 ≥

Motion Constraints

Initial constraints: Goal constraints: t tk

start

< k , 1 … n , , = θik t ( ) lowik = θik t ( ) highik = t tk

end k

, > 1 … n , , =

1 2 5 1 2 goal

2 1 3

init

q q

3 4

θ θ

δ δ

θ

l l l l l

Implemented Folding Sequence

5 7 4 6 3 1 2 unfolded carton folded carton

Implementation

AdeptOne robot moves carton blank through the fixture

• Holding Plate

Flat Carton Blank Fixture Robot Actuator Actuator

Carton Styles

Radio carton HP carton Slope carton

Exploiting Carton Symmetry

HP carton robot with 7 joints HP carton robot with 5 joints

θ θ θ θ θ θ

1 6 7 3 4 2

θ

5

θ θ θ θ

1 2 3 4

θ

5

Folding Sequence: HP Carton

4 1 5 3 7 2 6 unfolded carton folded carton

Folding Sequence and Fixture for HP Carton

θ

1

θ

5

θ

7 )

( θ

2

θ

6

( ) θ

3

( )

3 1 2 4 5 6 7

θ θ θ θ θ θ θ Fixture for the HP Carton

Summary

• Flexible method to fold cartons

using interchangeable fixtures

• Model cartons as robots and use

motion planning to generate folding sequences

• Design fixture for selected

folding sequence

• Implementation of carton folding

system

Future Work

• Develop planner to automatically

design carton folding fixtures

• Generate folding sequences and

fixtures considering robot degrees of freedom

• Design cartons that are easy to

fold

• Create 3-D MEMS structures

from 2-D elements