CENTER FOR

MACHINE PERCEPTION

CZECH TECHNICAL

UNIVERSITY IN PRAGUE

RE

SE

AR

CH

RE

PO

RT

ISS

N12

13-2

365

Physics-Based Model of aRectangular Garment for

Robotic FoldingVladimır Petrık

Vladimır SmutnyPavel Krsek

Vaclav Hlavac

{vladimir.petrik, smutny, krsek, hlavac}@ciirc.cvut.cz

CTU–CMP–2016–06

August 12, 2016

Available atftp://cmp.felk.cvut.cz/pub/cmp/articles/petrik/Petrik-TR-2016-06.pdf

This work was supported by the Technology Agency of the Czech Republicunder Project TE01020197 Center Applied Cybernetics, the Grant Agencyof the Czech Technical University in Prague, grant No. SGS15/203/OHK3/3T/13.

Research Reports of CMP, Czech Technical University in Prague, No. 6, 2016

Published by

Center for Machine Perception, Department of CyberneticsFaculty of Electrical Engineering, Czech Technical University

Technicka 2, 166 27 Prague 6, Czech Republicfax +420 2 2435 7385, phone +420 2 2435 7637, www: http://cmp.felk.cvut.cz

Physics-Based Model of a RectangularGarment for Robotic Folding

Vladimır Petrık Vladimır Smutny Pavel KrsekVaclav Hlavac

August 12, 2016

Abstract

The ability to perform an accurate robotic fold is essential to obtain theproperly folded garment. Available solutions rely on a rough folding surfaceor on a comprehensive simulation, both preventing the garment to slip onthe table during folding. This paper proposes a new algorithm for a fold tra-jectory design respecting the garment material properties and preventing thegarment slipping. The folding trajectory is derived based on the equilibriumof forces under the simplifying assumptions of a rectangular and homoge-neous garment. This approach allows folding the rectangular garment on alow friction table surface as we demonstrated in the experiments performedby a dual arm robotic testbed.

1 IntroductionOne of the skills in the robotic garment folding is an ability to perform a singlefold. It involves the design of the robot gripper trajectory, which folds the garmentin the folding line as shown in Fig. 1. The robot gripper grasps the garment andfollows planned trajectory moving the garment into the required shape. If materialproperties are not considered in the design, the garment can slip on the foldingsurface, which later results in an inaccurate fold. The state of the art methods ofthe trajectory design assume the garment is infinitely flexible or rely on infinitefriction between the folding surface and the garment. These assumptions provedto be unrealistic in practice and result in inaccurate folds.

This paper extends the previous works by assuming more realistic materialproperties. This leads to a more complex model so we restrict our study to rectan-gular garments only, but the proposed method can be generalized to more complex

1

Figure 1: Folding of a garment. The initial position is shown on the left side. Thegreen dashed line represents the folding line and the red dotted line represents theexpected position of the grasped side of the garment. The robot grippers grasp thegarment at one side only. Expected fold is shown in the middle and inaccurate foldis shown on the right side. The inaccuracy is a result of incorrect manipulation,which causes the garment to slip or wrinkle. The folded garment shape near thefolding line depends on the material properties.

shapes. The assumptions neglect effects of dynamics and restrict the model to ahomogeneous rectangular garments with a constant bending stiffness. A physicalgarment model is used for computing the robot gripper trajectory. The methodcalculates the static equilibrium of forces in every state of the trajectory.

The key contribution of the paper is the algorithm for a folding trajectory de-sign respecting the material properties. Our solution prevents the garment slippingon the horizontal folding surface and is able to fold the rectangular garment on alow friction table surface. We demonstrated the accuracy of our approach in a realrobotic folding for different materials as well as for different folding surfaces.

2 Related WorkThe existing framework for the garment manipulation consists of bringing the gar-ment to the initial state followed by the folding. The former assures the garmentlies flattened on the table and it was studied in [1, 2, 3, 4, 5]. The folding is de-composed into the individual folds and the garment pose is estimated for each foldas shown in works [6, 7, 8].

The first approach for the design of the robotic garment folding trajectory wasshown by Berg et al. in [9]. Authors designed a gravity based folding trajectoryassuming infinitely flexible material and infinite friction between the garment andthe folding surface. Such assumptions simplify the folding trajectory significantlybut are rarely met in a real environment. Nevertheless, the trajectory showed upto be a sufficient approximation if the requirements for folding precision were nothigh or if a table with high friction surface was used. The high friction surface isundesirable, because it is difficult to flatten the garment on such a surface and itallows undesirable tensions in the folded garment. The resulting gripper trajectory

2

consists of a sequence of linear trajectories and we will refer to this method as thelinear gravity based folding.

To increase the precision while folding the real garment, Petrık et al. [10]designed a circular folding trajectory. Instead of infinitely flexible garment, arigid material with frictionless joint located in the folding line was assumed. Theexperiments, comparing the linear and circular folding, were conducted, whilefolding the real garments on the low friction table surface. The better performancewas achieved in terms of precision, when real garments (jeans, towel) were folded.

Another trajectory for folding was designed by Li et al. in [11]. Authors useda simulated environment to evaluate the simulated folded shape. One parameter ofthe material, called shear resistance, was used to model the garment. For the grip-per trajectory, modeled by a Bezier curve, the resulting fold was evaluated. Thecurve parameters were perturbed until a sufficiently good fold was obtained. Au-thor observed that their method is less stable when folding denim material (jeans,pants) due to the relatively high shear resistance.

Our method describes the garment shape by a set of ordinary differential equa-tions (ODEs) and boundary conditions for each time during folding. The equa-tions were derived by considering the bending stiffness of the garment material.The gripper trajectory is then a sequence of solutions of these ODEs and the op-timisation across the trajectories is not required. The similar model was used forthe task of undersea pipes laying using the J-lay method [12] or for a pipelineinstallation and recovery in deepwater [13].

3 Garment ModelIn order to find the grippers trajectory, a physical garment model is derived. Themodel is restricted by several assumptions, which respect the folding requirementsand restrict the garment properties. One of the assumption is that the garmentmaterial is modeled according to Euler-Bernoulli beam theory [14]. The theorydescribes the relation between external forces, an internal stress and a resultingshape. The approximations considered in this theory are well satisfied for thinmaterials, such as garment. Furthermore, the garment is restricted to be rectan-gular and material is assumed to be homogeneous, so the density and bendingstiffness are constant. We also assumed that the folding line is parallel to thegrasped side of the garment. It results in the folding as shown in Fig. 1. Underthese assumptions, the garment can be represented in one dimension as shown inFig. 2.

The model incorporates a contact of the garment with the table and a contactof the garment with the garment itself. The garment is in the partial contact withthe table during the whole folding. The garment contact with itself occurs after

3

the upper layer touches the lower layer of the garment. Considering these twocontacts, we modelled the garment by four neighbouring sections. Two of thesesections are hanging in free space and are modelled as strings. They are describedby differential equations. The other sections are in contact and they affect theboundary conditions of the strings. The state where both contacts occurs is shownin Fig. 3.

3.1 String ModelThe string is supported on its ends only. The model of the string is derived basedon the equilibrium of forces acting on an element ds of the string. The element andthe acting forces are shown in Fig. 4. There is a tension force T (s) acting on theelement at the position s. The force acts in a direction of a tangent to the string atthe position s. The angle between the tangent and the horizontal axis x is denotedby θ(s). A force perpendicular to tension force is called shear force and denotedby V (s). Furthermore, the string bends under its own weight, which is expressedby a force ρA b g ds, where b is the garment width, ρA stands for the material areadensity, and g is a gravitational acceleration. For the string in equilibrium, thesum of forces must be zero which leads to the following equations:

d

ds(T (s) cos θ(s) + V (s) sin θ(s)) = 0 , (1)

d

ds(T (s) sin θ(s)− V (s) cos θ(s)) = ρA b g . (2)

The sum of moments is zero as well, which leads to the relation between the shearforce and the moment:

V (s) =dM(s)

ds. (3)

hb

l

Figure 2: One dimensional representation of the garment. A towel, with dimen-sions l, b, h, grasped by two grippers of a robot is shown on the left side. It’s onedimensional representation is shown on the right side.

4

q−1

q+1 q−2

q+2

l1l2

Figure 3: The state of the model with both types of contacts: the contact betweenthe garment and the folding surface, and between the garment itself. Two stringswith length l1 and l2 are used to model this state. The effects of the contacts areincorporated through the string boundary conditions qi.

The combination of the moment/curvature relation [14] and the angle/curvaturerelation [15] leads to:

κ(s) =dθ(s)

ds=M(s)

K, (4)

where κ(s) is the string curvature and the constantK is introduced to represent thebending stiffness of the material. For isotropic materials without internal structureK = E I , where E represents Young’s modulus and I is the second moment ofarea of the string cross-section. Neither E nor I has good meaning for the fabricmaterials. Combination of (3) and (4) results in:

V (s) = Kd2θ(s)

ds2. (5)

T + dT

V + dV

M + dM

T

V

M

θ + dθ

ds

ρAbgdsθ

Figure 4: The element of the string and the forces acting on it.

5

The formulation is extended by the additional two equations representing theCartesian position of the string elements:

dx(s)

ds= cos θ(s) ,

dy(s)

ds= sin θ(s) . (6)

The geometry of the garment depends on a single parameter weight to stiffnessratio only. Let us introduce new variables:

M(s) =M(s)

ρA b g, V (s) =

V (s)

ρA b g, T (s) =

T (s)

ρA b g. (7)

By substituting (5) and (7) into (1) and (2), we obtain:

d

ds

(T (s) cos θ(s) +

1

η

d2θ(s)

ds2sin θ(s)

)= 0 , (8)

d

ds

(T (s) sin θ(s)− 1

η

d2θ(s)

ds2cos θ(s)

)= 1 , (9)

whereη =

ρA b g

K

is the weight to stiffness ratio which describes the material properties. The sub-stitution (7) can be used because the robot controls the grippers position indepen-dently of the force required. If we need to know values of M(s), V (s) and T (s),the constant K or ρA has to be measured in addition to η.

A system of the first order ordinary differential equations (ODE) is createdfrom the higher order ODEs (8-9). A new variable representing the element stateis introduced:

q(s) =[θ(s), M(s), V (s), T (s), x(s), y(s)

]>. (10)

The system of the first order ODEs is:

dq(s)

ds=

η q2q3

− cos q1 + η q2 q4sin q1 − η q2 q3

cos q1sin q1

, (11)

where qi is the i-th element of the vector q(s).

6

3.2 Effect of ContactsThe contacts which occur during folding affect the boundary conditions of thestrings. Two strings are used, which implies four boundaries in total. The bound-aries for the first string are denoted:

q1(0) = q−1 , q1(l1) = q+1 , (12)

where superscript ‘−’ denotes the left boundary condition and superscript ‘+’ de-notes the right boundary condition, similarly for the second string and other vari-ables (T−1 , θ

+1 , etc.). The contact of the garment with table affects only the left

boundary condition of the first string (i.e. q−1 ). The contact between the garmentlayers affects the boundary conditions q+

1 and q−2 and causes the discontinuitybetween the strings.

The garment state is determined by state of all its elements q(s), for s ∈ (0, l).Two garment state situations need to be distinguished during folding: the state, inwhich the contact between the lower and the upper layer does not exists (Fig. 5aand Fig. 5b) and the state where it does (Fig. 5c and Fig. 5d). In the formersituation, only one string is modelled.

3.3 State of the ModelThe garment model consists of two strings and individual states are distinguishedby the boundary conditions. Fourteen conditions have to be specified in order tofind a garment state. They restricts two times six first order ODEs (11) and twounknown lengths of the strings. Finding the solution leads to a boundary valueproblem [16]. More specifically, a multipoint boundary value problem is formedand we used tool [17] to solve it. The solver requires known limits of integrationand since lengths are not known a priori, we reformulate the string model into thedimensionless formulation. It was achieved by substitution s = ε l, where ε is adimensionless position of the element.

The example of solving the garment state is shown for a completely foldedstate. The seven boundary conditions, which describe the folded state, are listedin the first column of Table 1. The individual rows of the table correspond toequations representing the boundary conditions. For example, the first row for thefirst column of the table represents the boundary condition:

θ−1 = 180◦ . (13)

It means that the angle at the ‘touchdown’ point is restricted to be 180◦ in thefolded state. Note, that boundary conditions are specified in such a way that thefolded garment layers overlay. The length l can be easily adjusted, if differentfolding is desired (e.g. as in Fig. 1).

7

4 Folding TrajectoryThe individual states of the model are determined by the boundary conditions.The boundary conditions specify the folding task respecting the restrictions ofgeometry and applied forces. For example, one of the conditions restricts themoment to be zero at the beginning of the first section (M−

1 = 0). Evolution ofthe garment states in time is called the garment trajectory and denoted by:

q(s, t) , t ∈ (0, te) , (14)

where te is specified manually and has to be large enough so that the robot movesslowly and the effect of dynamics can be neglected. In fact, the gripper should passthrough the sequence of trajectory points and actual time between consecutivepoints is not important.

While folding, four groups of states are distinguishable and we call them fold-ing phases. The individual phases are shown in Fig. 6. The garment states in aphase are determined by the constant boundary conditions except one, which wewill call a control variable δ(t). The control variable is monotonic function oftime during folding.

In the Phase 1, the garment is lifted up until its ‘touchdown’ point reachesthe expected position. The Phase 2 bends the garment until the contact betweenthe garment layers occurs. In the first two phases, there is no contact between thegarment layers and thus only the second string is modelled. The contact introducesanother seven conditions for the first string and it is examined in the Phase 3 andPhase 4. In the former case, there is one point contact only and the point of thecontact moves in the direction of folding. At the end of the Phase 3, the first stringreaches its final shape. In the Phase 4, the second string is then put on the lowerlayer until its length is zero.

Some of the boundary conditions depend on the values computed in a specificphase. For example, the expected position of the ‘touchdown’ point is determinedby the folded garment state. The phases are thus solved in the mixed order. First,the folded state is solved followed by the phases solving in order: 1, 4, 3, 2. Itensures that all required values are known before solving the phase. The Phase 3and Phase 4 are solved backward in time.

Folded State Only the first string is modelled in the folded state. The boundaryconditions ensure that the string is supported by the table at the beginning ofthe string and by the lower layer at the end of the string. The constants 3h

2and h

2

appear because the string model represents a neutral axis. The neutral axis is in themiddle of the string due to the assumption of constant density and the rectangularshape. The constant h is negligible unless multiple folds are stacked.

8

(a) Phase 1 (b) Phase 2

(c) Phase 3 (d) Phase 4

Figure 5: Individual folding phases. The linear / circular trajectory is shown inblue color.

Phase 1 The garment trajectory starts from a state, in which the garment lies onthe table and with the second string length equals to zero. The length of the stringthen increases linearly with time, which results in the garment lifting. During thelifting, the ‘touchdown’ point is moving in the direction of folding. The liftingcontinues until the ‘touchdown’ point reaches it’s final position, which was com-puted in advance based on the folded state of the garment. The variable δ is in therange:

δ ∈(0, x−1 (te)

), (15)

where te represents the time at the end of the folding and x−1 (te) is the boundaryvalue of the folded state.

Phase 2 In Phase 2, the garment should rather bend than be lifted. The length ofthe lifted section is fixed. The shear force V −2 controls Phase 2, as we observed it

9

is increasing monotonically during this phase. The shear force is increasing untilthe upper layer touches the lower layer of the garment. The control variable δ isfrom a range:

δ ∈(V −2 (t1e), V

−1 (t3s)

), (16)

where t1e is the time at the end of the Phase 1 and t3s is the time at the start of thePhase 3. The value of V −1 (t3s) is known from the Phase 3.

Phase 3 In Phase 3, there is a point contact between the lower and the upperlayer of the garment. The contact results in a discontinuity between the shearforces V +

1 and V −2 . The control variable δ represents this discontinuity. The rangeof the control variable is:

δ ∈(V +1 (t4s)− V −2 (t4s), 0

). (17)

In Phase 3, only a single point of the upper layer is in the contact. The position ofthis point changes with the control variable.

Phase 4 The last phase represents the lowering of the garment. The first stringreached the equilibrium and does not move in this phase. The control variablerepresents the length of the second string. It is from the range:

δ ∈(0, l − x+1 (te)

). (18)

The last state of this phase is equal to the folded garment state.

10

Table 1: The boundary conditions for the individual folding phases. The symbol× represents that variable is not constrained. The symbol | indicates that the stringis not modelled. The symbol δ parametrizes the phase.

Folded State Phase 1 Phase 2 Phase 3 Phase 4θ−1 180◦ | | 180◦ |M−

1 0 | | 0 |V −1 × | | × |T−1 0 | | × |x−1 l + l1 − x+1 | | x−1 (te) |y−1

h2

| | h2

|θ+1 0 | | 0 |M+

1 × | | × |V +1 × | | × |T+1 × | | × |x+1 × | | × |y+1

3h2

| | 3h2

|θ−2 | 180◦ 180◦ 0 0

M−2 | 0 0 M+

1 0

V −2 | × δ V +1 − δ ×

T−2 | 0 × T+1 0

x−2 | l2 x−1 (te) x+1 l − l2y−2 | h

2h2

3h2

3h2

θ+2 | × × × ×M+

2 | 0 0 0 0

V +2 | × × × ×T+2 | × × × ×x+2 | × × × ×y+2 | × × × ×l1 × | | × |l2 | δ x−1 (te) x−1 (te)−l1 δ

11

5 Reducing Number of PhasesThe four phases introduced in the previous section were caused by the four neigh-bouring sections of the model used for the contact modelling. To reduce the num-ber of phases, we can utilize another approach of contact incorporation. Thisapproach adds a nonlinear force into the model and this force acts in a directionopposite to gravitational acceleration. The model ODEs (8-9) are modified:

d

ds

(T (s) cos θ(s) +

1

η

d2θ(s)

ds2sin θ(s)

)= 0 , (19)

d

ds

(T (s) sin θ(s)− 1

η

d2θ(s)

ds2cos θ(s)

)= 1− f(y(s)) , (20)

(21)

where f(y(s)) is a force used to incorporate the contact of the element s with thetable. The contact force f(y(s)), which supports the part of the string laying onthe table is modeled as:

f(y(s)) =1

y2c [m], (22)

which implies that the folding surface is horizontal with the zero height. Theforce is already normalized by the term ρA b g. The force was chosen such that thefollowing condition is satisfied:

∞∫0

f(y(s))dy =∞ , (23)

which means that energy required to pass through the zero height has to be infinite.In practice, the string ’levitates‘ over the table and the height of the levitationis controlled by the factor c in the contact force term. In our model, we usedc = 10−6. Note, that the thickness of the string is ignored in the contact force.

Only two phases are required with such a model, the forward and the backwardfolding phase. The forward folding phase starts from the string which lies on thetable initially. The height y+ is then increased linearly which results in the stringlifting as shown in Fig. 6a. The lifting stops after the touchdown point (TDP)reaches its final position, which was computed from the folded state. The max-imum tension force is then stored and used in the backward folding phase. Theboundary conditions are:

θ− = 180◦ ,dθ−

ds= 0 , T− = 0 , x− = l ,

dθ+

ds= 0 , y+ = δ , (24)

12

(a) Forward folding phase.

↑touch down point

(b) Backward folding phase.

Figure 6: Folding phases. The linear and circular trajectories are shown in blue.

where δ denotes the control variable, which parametrizes the individual statesduring the forward phase.

The backward folding phase starts from the completely folded state. Thetouchdown point is fixed during the backward phase. The fixation is achievedby specifying the interface point for the element at the TDP. The following BCsare used for the phase:

θ− = 180◦ ,dθ−

ds= 0 , x− = l ,

dθ+

ds= 0 , T+ = δ , xTDP = c , (25)

where c is the x position of the TDP in the completely folded state and δ is in-terpolated linearly between zero and the maximum tension force achieved in theforward phase. The backward phase is then reversed and both phases are concate-nated to achieve the folding trajectory.

6 ExperimentsWe performed the experiment measuring the quality of the single fold for variousgarment materials and folding surfaces. The quality of the fold is measured basedon the slipping of the lower layer from its initial position and based on the thedisplacement of the planned and real position of the grasped side of the foldedgarment. The signed displacement is used in a way that positive values are used ifthe grasped side extends beyond the planned position (Fig. 7).

The value of the weight to stiffness ratio was not known exactly and was es-timated by the operator. The estimation was based on the maximal height of themanually folded garment (Fig. 7). The relation between the weight to stiffness ra-tio η and the maximal height hm (Fig. 8) was obtained by simulation of the folded

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Table 2: The properties of used materials. The lengths l, b, h are garment dimen-sions. The symbol hm represents the maximal height of the folded garment andsymbols ρA, K and η stands for material properties.

l / b / h hm ρA K η

[mm] [mm] [kgm−2] [Nm2] [m−3]synthetic 525 / 57 / 1 10 0.75 5.3·10−7 800000

denim 525 / 110 / 1 30 0.70 2.6·10−5 30000rubber1 995 / 36 / 3 70 3.95 5.9·10−4 2400rubber2 995 / 59 / 4 89 5.25 2.7·10−3 1150rubber3 995 / 50 / 5 99 6.55 3.9·10−3 850towel1 590 / 360 / 1 24 0.30 1.9·10−5 58000towel2 560 / 490 / 2 28 0.60 7.6·10−5 38500

state for different values of η. In the experiment, the garment was folded manu-ally by the operator and the maximal height was measured. The weight to stiffnessratio η was then estimated from the experimentally obtained relation (Fig. 8).

Two folding surfaces were used: a melamine faced chipboard table surfaceand a table covered by a rough tablecloth. Furthermore, two types of garmentswere used: narrow strips and real rectangular garments - towels. The propertiesof the used materials are shown in Table. 2.

6.1 Narrow StripsThe narrow strips were folded by one gripper only. Two fabrics were used forthe strips: synthetic and denim. Furthermore, the strips with lower weight tostiffness ratio η were used: a Ethylene Propylene Diene Monomer rubbers withvarious thicknesses. Different thickness of the strips represents the different η inthe experiment. Since the friction between the rubber and the table is large, wealso conducted experiments with the paper sheet inserted between the rubber and

d

hm

Initial:

Folded:

e

Figure 7: The displacement measurement. The slippage e and the displacement dmeasure the quality of the fold and the maximal height hm is used to estimate thematerial property.

14

210 3 10 4 10 5 10 6 10 7

hm

[mm

]

0

20

40

60

80

100

Figure 8: The relation between the maximum height hm and the weight to stiffnessratio η.

Table 3: The measured lower layer slippage e and the measured layers displace-ment d. The upper part of the table shows the displacements for the narrow stripsand the rest shows displacements for the real garments. The garments are orderedbased on the weight to stiffness ratio η. The measured static friction is denotedby µ.

Surface Garment µ [-]Slippage [mm] / Displacement [mm]

Folding trajectoryLinear Circular Our Model

table

synthetic 0.20 6 / -24 -10 / 46 0 / -5denim 0.26 15 / -28 -14 / 40 0 / -1

rubber1 0.57 31 / -46 -21 / 68 0 / -2rubber2 0.57 28 / -45 -18 / 47 0 / 4rubber3 0.57 30 / -50 -11 / 32 0 / 5

table/paperrubber1 0.31 39 / -55 -32 / 62 0 / -3rubber2 0.31 64 / -82 -35 / 53 0 / 3rubber3 0.31 70 / -91 -24 / 37 0 / 5

tablecloth

synthetic 0.53 0 / -19 0 / 31 0 / -7denim 0.81 0 / -10 0 / 28 0 / -4

rubber1 0.80 23 / -42 -8 / 43 0 / -2rubber2 0.80 25 / -46 -8 / 39 0 / 2rubber3 0.80 30 / -50 -5 / 22 0 / 3

tabletowel1 0.28 17 / -35 -10 / 56 0 / 5towel2 0.24 28 / -40 -10 / 43 0 / 3

tableclothtowel1 1.23 0 / -23 0 / 41 0 / 4towel2 1.20 0 / -25 0 / 30 0 / 3

15

the folding surface and denoted them as table/paper surface. Strips slide on thepaper-table interface if horizontal force is larger than friction force between thepaper and the table surface. The measured results are shown in the upper part ofTable 3.

Table 3 indicates that there is a relation between the slippage and displacementof the linear trajectory and the folding surface friction. The lower slippages / dis-placements were measured for the higher friction. Furthermore, the lower weightto stiffness ratio η results in the higher slippage / displacement as well. This ful-fils the expectation of the linear trajectory as it was designed for infinitely flexiblegarments (infinite η) and for infinite surface friction. Based on the oriented dis-placements, it can be concluded that linear trajectory has tendency to push thegarment in the direction of folding.

On the contrary, the circular trajectory tends to pull the garment. The circulartrajectory outperformed the linear trajectory in the situations with low η only.

Our model based trajectory outperformed both state of the art methods: thelinear as well as the circular folding trajectory. For our trajectory, there is norelation between the friction and the measured displacement in the experiment. Itis a result of minimization of the horizontal force, which prevents the slipping ofthe garment in the most cases. We did not observed any slipping in the experiment.

6.2 Real Rectangular GarmentsTwo real towels were folded in order to show that the model is sufficient for rect-angular garments folded by two grippers. The towel was grasped by its cornersas shown in Fig. 1. A single trajectory was designed using our one dimensionalrepresentation and this trajectory was performed by both grippers simultaneously.We observed that the model error is negligible small, when folding with two grip-pers. The measured results are shown in the lower part of Table 3. The errorsare similar to strips folding: linear trajectory outperformed circular trajectory andthe displacement of the linear trajectory can be lowered by the higher friction.Our trajectory outperformed both of them. The results indicate that our trajec-tory can be used, when folding rectangular garment with a dual arm robot withoutexplicitly modelling of two dimensions.

7 ConclusionThis paper examined the folding of the garment performed by the robotic arms.The folding trajectory was designed respecting the material properties but restrict-ing the garment to rectangular shape only. The designed trajectory prevents thegarment from slipping when folded on the low friction surface, which results in a

16

better folding accuracy. The main advantage of our method is an ability to fold thegarment on the table without using high friction folding surface. We performedexperiments for several garments and folding surfaces, comparing our trajectorywith two state-of-the-art methods for folding trajectory design. Our trajectory out-performed the state of the art methods in term of the precision and it was experi-mentally shown that it could be used for folding on the low friction table surface.It has been demonstrated that the folding trajectory cannot ignore the garment ma-terial properties. The experiments show that violations of our model assumptionsby real garments do not result in inaccurate folds.

The reported work considered the folding of the homogeneous, rectangularshape garments only. In the future work, the rectangular shape assumption shouldbe relaxed which will extend the set of the modelled garments. An extension tothe non-rectangular shape could be achieved by simulating shell models insteadof strings. We also assumed the material properties are known or are measuredin advance by the operator. In future, these properties can be estimated in thecourse of the folding by using the camera recognising the garment shape duringthe folding.

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