full paper single arm robotic garment folding path generation

October 3, 2017 Advanced Robotics adr2017

To appear in Advanced RoboticsVol. 31, No. 23, December 2017, 1–12

FULL PAPER

Single Arm Robotic Garment Folding Path Generation

Vladimır Petrık∗, Vladimır Smutny, Pavel Krsek, Vaclav Hlavac

Czech Institute of Informatics, Robotics, and Cybernetics

Czech Technical University in Prague(v1.0 released January 2017 - this is a preprint (prior to peer review) of an article whose final and definitive form has

been published in ADVANCED ROBOTICS 2017)

We address the accurate single arm robotic garment folding. The folding capability is influenced mostlyby the folding path which is performed by the robotic arm. This paper presents a new method for thefolding path generation based on the static equilibrium of forces. The existing approach based on asimilar principle confirmed to be accurate for one-dimensional strips only. We generalize the methodto two-dimensional shapes by modeling the garment as an elastic shell. The path generated by ourmethod prevents the garment from slipping while folding on a low friction surface. We demonstrate theaccuracy of this approach by comparing our paths (a) with the existing method when one-dimensionalstrips of different materials were modeled, and (b) experimentally with real robotic folding.

Keywords: robotic garment folding; path generation; Kirchhoff-Love shell

1. Introduction

The ability to fold a garment with a robot remains a challenging task due to the various garmentshapes and fabric materials. The folding consists of several independent folds performed in asequence. In each fold, the robot grasps the garment at specific positions and follows a foldingpath. The folding path is not unique. Several approaches how to design the folding path wereproposed. The simplest approaches [1, 2] depend on the garment shape only. However, ignoringthe material properties leads to an inaccurate fold. This inaccuracy increases with a number offolds and often results in a highly inaccurately folded garment. A physics-based model is usedin method [3]. The method provides better folding accuracy, but it is limited to one-dimensionalshapes only.

We extend the work [3] by using a physics-based model of two-dimensional shapes. We considermaterial properties and model the garment as the homogeneous isotropic elastic Kirchoff-Loveshell. The path is generated by calculating the static equilibrium of forces in every state of thepath. The effect of dynamics is neglected assuming that the robot movement is slow.

The key contribution of the paper is the method designing the folding path for the givengarment shape and material properties. Our path allows the folding on a low friction surface whilepreventing the garment slipping. For one-dimensional strips, our method generates path almostidentical to the state-of-the-art method, which was experimentally confirmed to yield accuratefolds. For two-dimensional garment shapes, we demonstrate the accuracy of our method in areal robotic garment folding. The experimental comparison with existing approaches is providedtoo.

∗Corresponding author. Email: [email protected]

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1.1 Task Formulation

The robotic garment folding starts with a garment, which lays freely on a horizontal desk. Thesingle arm folding is specified by a folding line and a grasping point. They are assumed to beknown in this paper. The folding line represents the axis of symmetry for the start and finalposition of the grasping point. The folding line is coincident with the folded state edge in a caseof infinitely flexible material. The goal of the robotic folding is to grasp the garment on a givenpoint and follow the folding path. The examples of the grasping points and folding lines areshown in Fig. 1. Our goal is to design the folding path which results in an expected folded stateaccurately and avoids the garment slipping at the same time.

(a) (b)

Figure 1.: Different folding scenarios for a square towel (a), and the real robotic folding (b).The start and the folded states are shown. Folding is specified by a folding line (black line)and by grasping point (blue circle). The final (red cross) and initial grasping point positions aresymmetric with respect to the folding line.

2. Related Work

The existing approaches for the folding path generation could be split into two categories: purelygeometrical algorithms and simulation-based algorithms. The geometrical methods have an ad-vantage of a low computation cost, but the result may be inaccurate due to the omitted materialparameters. The simulation-based methods are computationally more expensive and require thematerial properties to be known in advance, but the provided folding path results in an expectedfolded shape.

There are two geometrical methods for path generation: a linear gravity based folding proposedby Berg et al. [1] and a circular folding proposed by Petrık et al. [2]. The linear path was designedfor an infinitely flexible material under the assumption of an infinite friction between the garmentand the desk. The circular path assumes the rigid garment with an infinite flexibility in thefolding line only. Both paths were experimentally compared in work [2] while folding on differentsurfaces.

The first simulation-based folding path generation was proposed by Li et al. [4]. The authorsused a simulation software Autodesk Maya to evaluate the folded state for a given path virtually.The garment model in the Maya software utilizes a mass-spring network which is parametrized bymany coefficients. The authors observed that the shear resistance coefficient influences the folded

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state in the most significant way. Other parameters were fixed to their default values providedby the Maya software. The shear resistance coefficient and the friction coefficient between thegarment and the desk were measured manually in advance. After the needed coefficients hadbeen estimated, an arbitrary path was followed. The obtained folded state was compared tothe required state. The path was then perturbed until the folded state satisfied the foldingrequirements. The version for both the single arm and the double arm robotic folding wasstudied in the paper.

The second simulation based method is based on Euler-Bernoulli beam theory and was pro-posed by Petrık et al. [3]. The authors considered homogeneous garments described by a one-dimensional string. It allowed to fold narrow one-dimensional strips or rectangular garmentswith folding line aligned with an edge of the rectangle. Under these assumptions, a physics-based garment model was formulated, and it was shown that the material is described by asingle parameter. The parameter was estimated from a manual measurement in advance. Thefolding path generation was divided into the sequence of states which were in static equilib-rium. The equilibrium was found by solving a boundary value problem described by Kierzenkaand Shampine [5]. The forces on the boundary were specified for each state according to thefolding requirements while the horizontal force was kept minimal. The minimal horizontal forceincreases the range of friction coefficients which still prevents the garment slipping on the desk.The method was experimentally verified and outperformed both the linear as well as the circularpath. An additional experimental analysis of the method was provided by Petrık et al. [6].

The method [3] allows folding of one-dimensional strips only. This paper generalizes thismethod to two-dimensional shapes.

3. Model Description

The direct extension of Euler-Bernoulli beam theory is Kirchoff-Love shell theory described ina series of works written by Simo et al. [7–9]. The shell theory neglects transverse shear andtherefore is suitable for thin materials only. The typical fabric is thin enough to satisfy thecondition. A shell surface geometry is described by NonUniform Rational B-Splines (NURBS),which allows formulating a finite element method for the Kirchoff-Love shell as described byKiendl et al. [10]. The same method was used for dynamic garment simulation by Lu andZheng [11]. To generate a folding path, we computed a sequence of states, such that each statesatisfies the condition of a static equilibrium of forces.

3.1 Equilibrium Equation

We restrict our analysis to isotropic elastic material. Such material is described by Young’smodulus E, Poisson’s ratio ν, area mass density ρ, and thickness h. The weak form of the staticequilibrium equation is given by Belytschko et al. [12]:∫

Ωδu> ρ g dA−

∫Ω

(hE

1− ν2δε>D ε+

h3E

12 (1− ν2)δκ>Dκ

)dA = 0, (1)

where Ω is a NURBS knot span [13], dA is the differential area of the surface, g represents thegravitational acceleration, δu is the variation of the displacement of dA, ε is membrane strainand κ is curvature. The matrix D depends on the Poisson’s ratio and represents the isotropicmaterial properties [10]. According to Lu and Zheng [11], the parameters h,E, and ν used inthe equation (1) are not suitable for the fabric simulation due to the internal fabric structure.They proposed to model the membrane stiffness and the bending stiffness by two independentfunctions. In our analysis, we approximate them by linear functions specified by the constantmembrane and bending stiffnesses. By introducing weight-to-membrane-stiffness ratio ηm and

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weight-to-bending-stiffness ratio ηb, equation (1) reads:∫Ωδu> g dA−

∫Ω

(1

ηmδε>D ε+

1

ηbδκ>Dκ

)dA = 0. (2)

The fabric model characterised by ηm and ηb is rather phenomenological description of the fabricbehaviour avoiding the details of threads and yarns interactions. The model is thus parametrizedby the garment shape described by NURBS and by the material properties given by the Poisson’sratio and two weight-to-stiffness ratios.

3.2 Model State

The different states of the model are described by boundary conditions. The boundary conditionrestricts selected NURBS control point to be at the specific position. The example is shown inFig. 2(a), where the garment is grasped at a single corner. The holding simulates the roboticgrasping: the corner position as well as orientation are fixed by the gripper.

To generate a folding path, a contact with a folding desk needs to be modeled. The desk isassumed to be horizontal. The contact of the garment with the desk is frictionless and modeledas described by Kopacka et al. [14]. The example of the garment held by a corner with partialfrictionless contact with the desk is shown in Fig. 2(c). However, the friction is required for thefolding process to be successful [3]. The horizontal forces, which simulate the friction and arerequired for the path generation, are treated individually via boundary conditions.

(a) In the air (b) In the air (c) In a contact with a desk

Figure 2.: The model state examples of a garment held at a single corner. Three cases aredistinguished: (a) the flexible (ηb = 105 m−4 s2) garment hanging in the air, (b) the stiff (ηb =104 m−4 s2) garment hanging in the air, and (c) the state where the stiff garment is in a partialcontact with a desk.

4. Folding Path

The goal of the folding path generation is to design a path, which brings the garment fromthe initial state to its folded state while avoiding the garment slipping. To achieve that, weapproximate the path by a sequence of points in which the garment state is computed. Thematerial properties are assumed to be known.

4.1 Folding Specification

It is assumed that the garment is laying flat on the desk in the initial state. The folding line andthe grasping point determine the folding task as shown in Fig. 3(a). The grasping point position

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is denoted:

xg(τ) =(xg(τ), yg(τ), zg(τ)

)>, τ ∈ 〈τs, τe〉 , (3)

where τ is a dimensionless monotonic function of time, and τs and τe represent the start andthe end of the folding path, respectively. The positions xg(τe) and xg(τs) are symmetrical withrespect to the folding line. The force acting by the gripper in the grasping point is denoted

fg(τ) =(fx(τ), fy(τ), fz(τ)

)>.

All positions are specified with respect to a Cartesian coordinate system shown in Fig. 3.The x-axis points towards the xg(τe), the y-axis coincide with the folding line, and the z-axis isperpendicular to the desk and points upward.

Further, we specify a set of holding points - points which simulate the frictional forces. Theholding points are selected garment parts, which are not expected to move while folding. Becausewe use NURBS parametrization, holding points are selected from a set of NURBS control points.The i-th holding point is denoted:

xih(τ) =(xih(τ), yih(τ), 0

)>, (4)

where zero z-coordinate indicates that point lies on the desk. For the path generation, all holdingpoints are fixed to their initial positions. The forces f ih(τ), required to hold the points in theirpositions, are analyzed later to check the garment does not move.

xy

(a) Initial state

y

(b) Folded state

y

z

(c) Folded state, 3D view

Figure 3.: The initial and folded state visualization. The holding points, simulating the frictionalforces, are shown as the green circles. The blue circle represents the grasping point, and the redcross shows the final grasping point position. The folding line is shown in black.

4.2 Folded State

Our folding path generation procedure starts with the folded state computation. The folded statedepends on the material properties, and we require it to evaluate the folded state touch downpoint as shown in Fig. 4. That point is used to guide the path generation procedure to finish inthe desired folded state. The touch down point lies in the plane %, which is perpendicular to thedesk and passes through the points xg(τs) and xg(τe). Our path generation procedure requiresthe touch down position to be equal to the folded state touch down position. To achieve that, apoint on the shell xd(τ) is computed such that the position of the point in the folded state xd(τe)is not in contact with the desk and is the closest to the folded state touch down position. Fixingthe coordinate zd(τ) = zd(τe) causes the touch down point position to be determined.

Finding a state in a finite element analysis corresponds to solving a set of nonlinear equations.The solution is typically not unique. To select the appropriate one, one has to specify the start

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%

(a) The plane perpendicular to the

folding line

%

Touch down point

xd x

z

(b) The folded state touch down point

Figure 4.: The folded state touch down point position. The point xd is used to fix the touchdown point position.

state, which is close to the desired solution. In the case of folded state computation, we startfrom the spread garment state and follow a circular path [2], which is generated for the graspingpoint. The circular folding path is not optimal. However, it provides a correct estimation of thefolded state if three conditions are satisfied: (a) the holding forces could be arbitrarily large, (b)there is no friction between the garment layers, and (c) the grasping point is fixed to the circularpath during the whole movement and in the folded state too. The first condition prevents thegarment from slipping on the desk. The second allows sliding of the upper garment layer on thetop of the lower garment layer. Sliding is required for the correction of the upper layer positionafter both layers touch each other (see [6] for details). All conditions are satisfied in our model,and the folded state is visualized in Fig. 3(b) and Fig. 3(c).

With the computed folded state, we are able to determine which parts of the surface remainfixed. This information is used to verify that the selected holding points do not move whilethe path is executed. If the verification fails, new holding points are selected, and the foldedstate is recomputed. The new holding points are further away from the folding line in the x-axisdirection.

4.3 Path Generation

Each garment state on the path is used to compute gripper orientation and position. The gripperposition is equal to the grasping point position. The gripper orientation is computed from thegarment normal at the grasping point. The rotation of the gripper around the normal can bearbitrary but has to be fixed on the whole path. It corresponds to the azimuth relaxation asdescribed in [2].

The normal in the grasping point is not constrained during garment state computation. Thegripper with the orientation following the garment normal does not apply moment on the gar-ment. The grasping point position is restricted to be in the plane %, which is perpendicular to thedesk and passes through the points xg(τs) and xg(τe) (Fig. 4(a)). With the coordinate systemspecified above, the boundary condition restricting the position to be in the plane is: yg(τ) = 0.

Different boundary conditions are used for different path phases to compute the coordi-nates xg(τ) and zg(τ). We divide the path into three phases such that the boundary conditionsin a particular phase differ in a single variable only. This boundary condition is controlled by acontrol variable δi(τ), which is a monotonic function of τ for the Phase i. The phases are shownin Fig. 5 and their boundary conditions are shown in Table 1.

4.3.1 Phase 1

The Phase 1 (Fig. 5(a)) starts from the garment initial state. The grasping point is lifted upwith the force fx(τ) = 0. The control variable δ1(τ) = zg(τ) controls the lifting and starts fromthe zero value. It increases linearly with τ until the touch down point (Fig. 4(b)) reaches its finalposition, which is evaluated on the folded state.

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(a) Phase 1 (b) Phase 2 (c) Phase 3 (d) The whole path

Figure 5.: The paths (bold red line) for the individual phases (a)-(c) with the visualisation ofthe state at the end of the phase, and the whole trajectory (d) with the state where the upperlayer touches the lower layer for the first time.

4.3.2 Phase 2

In Phase 2 (Fig. 5(b)), the garment should rather bend than be lifted. To achieve that, thecontrol variable is δ2(τ) = xg(τ) and starts from the last value computed in Phase 1. Theforce fx(τ) cannot be zero anymore [3]. Instead, the touch down point is fixed, which is achievedby constraining the position zd. Note, that there is no additional force acting on the touch downpoint. It is a purely geometrical constraint used to select the solution from the space of alladmissible solutions. The control variable increases linearly with τ while ∆xg > −∆zg, where ∆represents differences between subsequent states.

4.3.3 Phase 3

In Phase 3 (Fig. 5(c)) the touch down point is fixed as in the Phase 2. The control variableis δ3(τ) = zg(τ), and the position xg is not constrained. The phase ends when δ3(τ) = 0, whichcorresponds to the folded state.

Table 1.: The boundary conditions for the phases. The symbol × indicates that variable is notconstrained. The variable δi parametrizes the Phase i.

fx fy fz xg yg zg xd yd zd fh xhPhase 1 0 × × × 0 δ1(τ) × × × × xh(τs)Phase 2 × × × δ2(τ) 0 × × × zd(τe) × xh(τs)Phase 3 × × × × 0 δ3(τ) × × zd(τe) × xh(τs)

4.4 Folding Limitations

Our path generation procedure assumes the folded state is stable, i.e. the garment remains fixedafter the gripper releases the garment at the end of the folding. The stability is affected by thefolding line distance from the initial grasping point position, and by the material properties. Thestiffer material requires the bigger folding line distance. The example of the nonstable foldedstate is shown in Fig. 6(a).

Another example of the folding limitation is the motion of the garment towards the graspingpoint as shown in Fig. 6(b). This garment movement results in the collapsed folded state.

Both limitations have the origin in the physical behavior of the garment. It is not a limitationof our folding path generation method. The single arm folding path, which folds such garments,does not exist. The collapse could be avoided if an additional robotic arm is used. However, thestability of the folding has not been analyzed yet, and it is the subject of our future work.

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(a) Fold too small (b) Fold too big

Figure 6.: Scenarios in which the folding is not possible. In the scenario (a), the folded state isnot stable. After the gripper releases the garment, the garment returns into the spread state.The scenario (b) shows the collapsed state of the garment.

5. Experiments

5.1 Comparison with the 1D Case

The first experiment compares our path with the state-of-the-art method, which is based on thesimilar principle [3]. The method generates a folding path for the one-dimensional inextensiblestring with material described by a weight-to-bending-stiffness ηb ratio. This path can be usede.g. for a single arm folding a narrow strip. The method was experimentally verified by foldingfabric strips of different materials [6]. For the comparison, we generated paths for the 1 m longstrips with the ηb set to value from 103 to 106 m−4 s2. The folding line was placed into the middleof the strip.

Our path generation method should provide result equal to the state-of-the-art path for thesame material. In order to describe an inextensible material with our model, we set the weight-to-membrane-stiffness ratio to the small value. The other material and geometrical properties wereset to be equal to the state-of-the-art path generation method. The generated paths are shownin Fig. 7. It can be seen that the both methods provide similar results. The small differencesmight be attributed to the different garment shape representations.

(a) ηb = 103 m−4 s2 (b) ηb = 104 m−4 s2 (c) ηb = 105 m−4 s2 (d) ηb = 106 m−4 s2

Figure 7.: The comparison of trajectories for a narrow strip folding. The black line shows the pathgenerated by our method. The red crosses visualizes the path generated by the state-of-the-artmethod [3].

5.2 Robotic Experiments

We used three garment samples of rectangular shape with dimensions: 500 × 500 mm. Threerepresentatives of the commonly used fabrics were selected for the samples: lining, chanel, anddenim. To measure the accuracy of the folding, we generated the folding path for various foldinglines. We parametrized the folding line by its distance to the initial corner position and by theangle between the x-axis and the garment axis of symmetry, which passes through the graspingpoint initial position as shown in Fig. 8(a). The folding path was performed by the robot, and all

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folded state corners positions were measured. We used a melamine faced chipboard desk surfacefor all experiments. The same accuracy measurement was performed for the linear and circularpaths.

α

d

(a) (b) α = 0, d = 250 mm (c) α = 10, d = 220 mm

Figure 8.: The folding line parametrization (a), and the folding lines used in the experiments (b-c). The folding line is parametrized by its distance to the initial grasping point position, andby the angle α measured between the x-axis and the garment axis of symmetry (dashed line),which passes through the grasping point initial position.

5.2.1 Material Properties Estimation

Our path generation method requires material properties to be known. We estimated theweight-to-stiffness-ratios ηb and ηm before folding by comparing manually and virtually foldedgarment. One corner of the rectangle was folded around the folding line with parameters: α = 0

and d = 250 mm (Fig. 8(b)). Two quantities were measured on the real folded state: the maximalheight hm and the overhang ho as shown in Fig. 9. The relations between these quantities andthe weight-to-stiffness-ratios were obtained by the simulation and are shown in Fig. 10. Themeasured and estimated quantities for our samples are shown in Table 2.

hm

ho

Figure 9.: Material properties estimation from the folded state. The maximal height hm and theoverhang ho are measured on the folded state and are used for the estimation.

Our material is further described by a Poisson’s ratio ν. However, for the relatively smallweight-to-membrane-stiffness-ratios of our samples, the Poisson’s ratio has negligible influenceon the generated path. We used constant value ν = 0 for all materials in our experiment.

Table 2.: The material properties of used materials.

hm [mm] ho [mm] ηm [m−2 s2] ηb [m−4 s2] ν[−]lining 16.8 3.5 2.51 · 10−3 2.60 · 105 0

chanel 27.7 6.6 2.99 · 10−3 6.39 · 104 0denim 33.8 1.3 1.72 · 10−4 2.43 · 104 0

9


0

10 -2

20h

m [m

m]

40

etam

[m -2s2 ]

10 -3

60

10 6

etab [m -4s2 ]

10 510 -4

10 4

(a)

-5

10 -2

0

5

ho [m

m]

etam

[m -2s2 ]

10 -3

10

15

10 6

etab [m -4s2 ]

10 510 -4

10 4

(b)

Figure 10.: The relations between the weight-to-stiffness-ratios ηm and ηb and the maximalheight hm and the overhang ho measured on the folded state.

5.2.2 Results

Two folding line positions were tested: (a) with distance 250 mm and angle 0, and (b) withdistance 220 mm and angle 10. The folding scenarios are shown in Fig. 8. We measured dis-placements of all garment corners from their expected positions in the folded state. One cornercorresponds to the grasping point, and other were used as the holding points. Ideally, all dis-placements would be zero.

The measured displacements of the grasping point from its expected positions are shown inTable 3. Our path outperformed the linear and circular paths in all experiments because thedisplacement caused by our path was smaller than the displacement caused by other paths. Thedisplacement of the linear path is higher for stiffer material which corresponds to the violationof the infinite flexibility assumption. The circular path shows the opposite relation: the stiffermaterial displacement is lower.

The holding points displacement was zero for the linear path as well as for our path. Thefriction between the garment and the desk was thus sufficient. The displacement for the circularpath was non-zero which was caused by the garment slipping. The displacements for the circularpath was less then 10 mm in all experiments.

Table 3.: The measured grasping point displacements from the expected position.

Grasping point displacements

Folding Line MaterialLinear path Circular path Our path

x [mm] y [mm] x [mm] y [mm] x [mm] y [mm]

α = 0

d = 250 mm

lining 20 0 -62 10 3 1chanel 16 2 -25 1 3 0denim 26 7 -7 4 -2 0

α = 10

d = 220 mm

lining 20 4 -40 22 10 0chanel 21 1 -19 4 1 3denim 28 5 -5 3 -3 -2

6. Conclusion

We presented the new method for the robotic garment folding path design. Our method ex-tends the state-of-the-art one-dimensional path generation technique [3] to two dimensions. The

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method respects the garment shape and material properties by modeling the garment as an elas-tic shell. The designed path prevents the garment from slipping when folding on a low frictionsurface.

We demonstrated the accuracy of our method by: (a) comparing with the state-of-the-artmethod in case of the one-dimensional strips, and (b) by real robotic garment folding. In theformer case, our generated path is almost identical to the state-of-the-art method [3], whichwas experimentally confirmed to be accurate. The latter case demonstrated the accuracy forfolding scenarios, which are not possible with the state-of-the-art method [3]. Our experimentsconfirmed the accuracy of the generated path and showed that our method outperforms thelinear [1] and circular [2] paths.

The reported work designed a folding path for a garment of given shape and material proper-ties. The shape can be easily measured automatically by e.g. a camera in advance of the folding.On the other hand, the material parameters are measured manually by the operator. In future,these properties could be estimated automatically too, either before folding or in the course offolding.

Acknowledgment

This work was supported by the Technology Agency of the Czech Republic under Project TE01020197Center Applied Cybernetics; the Grant Agency of the Czech Technical University in Prague, grant No.SGS15/203/OHK3/3T/13; RadioRoSo, part of project Echord++ [number FP7-ICT-601116]; and theEuropean Regional Development Fund under the project IMPACT (reg. no. CZ.02.1.01/0.0/0.0/15 003/0000468).

References

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