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 objects?

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 folding Carton Blank Carton Ready For Loading Need automated solution: • eliminate repetitive stress injuries • adaptable to different models

Example Folding Sequence Sequence of folds to transform carton blank to folded carton θ 7 θ θ θ 6 3 4 θ 5 ( ) θ θ θ 2 5 7 θ 1 Carton Blank ( ) θ θ 4 1 ( ) θ 3 ( ) θ 2 ( ) θ 6

Folding with Fixtures Design fixture shape and select cardboard motions so blank is folded into carton θ 7 θ θ θ 6 3 4 θ 5 θ 2 θ 1 Carton Blank ( ) θ θ 7 5 ( ) θ θ 4 1 ( ) θ 3 ( ) θ 2 F1

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 7 θ 6 θ θ θ 3 5 4 θ 2 θ 1

Configuration Space Representation Recursive slice projection (Lozano-Perez, 1987) θ 1 θ 2 θ 3 θ 4 θ 6 θ 7 θ 5 Modifications: Nonserial manipulator Generate all feasible folding sequences Exploit symmetry

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 ω velocity Multiple joints can move together; they may begin moving simultaneously or with delays Generate folding sequences by identifying valid unfolding sequences

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 ω orientation at angular velocity 3. Multiple joints can move together; they may start simultaneously or with delays

Motion Constraints Folding with fixtures constrains possible carton motions For the k th joint: = + θ ik t ( ) ω t low ik start end ≤ ≤ t k t t k – 1 k δ j ∑ start = - - - - - t k ω = 1 j – high ik low ik end = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + start t k t k ω Motion constraints: = – θ i 2 t ( ) θ i 1 t ( ) δ 1 start end ≤ ≤ t 2 t t 1 = – θ ik t ( ) θ ik ( ) δ k start end ≤ ≤ t – 1 t k t t k – 1 – 1 = – θ in t ( ) θ in ( ) δ n start end ≤ ≤ t t n t t n – 1 – 1 – 1

Motion Constraints (cont’d) Initial constraints: = θ ik t ( ) start = 1 … n < , , , low ik t t k k Goal constraints: end k = = 1 … n θ ik t ( ) > , , , high ik t t k θ 3 P P q 2 1 goal q l θ δ 5 init 2 2 l δ 1 l l 1 3 4 l 2 θ 1

Search Algorithm Generate n! joint sequences For each joint sequence, find all feasible δ discretized values of for each joint O n ! s n ( ) 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

Folding Sequence folded carton unfolded carton 3 7 4 6 2 5 1

Folding Sequence folded carton unfolded carton 3 7 4 6 2 5 1

Multiple Angular Velocity Formulation Permit different carton joints to rotate at different angular velocities For the k th joint: = + θ ik t ( ) γ ik ω t start end ≤ ≤ low ik t k t t k – 1 k δ j 1 ∑ γ ik ≥ start = - - - - - - - - - - t k γ i j ω = 1 j – high ik low ik end = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + start t k t k γ ik ω

Motion Constraints Initial constraints: = θ ik t ( ) start = 1 … n < , , , low ik t t k k Goal constraints: end k = = 1 … n θ ik t ( ) > , , , high ik t t k θ 3 q goal l θ 5 q 2 δ init 2 l l 1 4 l l δ 3 1 2 θ 1

Implemented Folding Sequence folded carton unfolded carton 3 4 7 6 2 5 1

Implementation AdeptOne robot moves carton blank through the fixture Robot Holding Plate Flat Carton Blank Fixture �� �� � ���� ���� � � � � Actuator Actuator � �

Carton Styles Radio carton HP carton Slope carton

Exploiting Carton Symmetry HP carton robot with 7 joints θ 7 θ θ θ 6 4 3 θ 5 θ 2 θ 1 HP carton robot with 5 joints θ θ 4 3 θ 5 θ 2 θ 1

Folding Sequence: HP Carton folded carton unfolded carton 6 7 3 4 5 2 1

Folding Sequence and Fixture for HP Carton θ θ 7 6 θ θ θ 5 4 3 θ 2 θ 1 ( 7 ) θ θ θ 1 5 Fixture for the HP Carton ( ) θ θ 2 6 ( ) θ 3

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