Tying knot precisely

Weifu Wang Devin Balkcom

Abstract— Simply pulling on the ends of string is suffi-cient to tighten simple knots such as the overhand knot,but many knots used for decoration or binding need tobe tightened at particular locations along the string. Forexample, a shoelace knot should have equal size bows, asshould a decorative cloverleaf knot. Knots tied in soundinglines (historically used to measure depth in the ocean) mustbe placed at precise distances apart. Precise control of tyingalso allows tangling to be avoided in complex knots. Thispaper explores an approach to tying knots precisely, so thatfriction locks occur at specified locations along the string.First, the knot is laid out on a fixture using an arm anda specialized gripper; then, the fixture contracts as stringis pulled to tighten the knot. This is the first work weknow of in robotic manipulation focused on precise knottightening (rather than loose arrangement), and presentstying of a fairly complex decorative knot, the Ruyi knot,as a proof of concept. The fixtures are each specific toparticular knots, but are designed automatically using thealgorithm presented.

I. INTRODUCTION

Knots are applied in many scenarios in daily life

ranging from decoration to binding. The exploration

of knot tying with robots started about three decades

ago [21].

Knot tying occurs in two phases: arrangement and

tightening. The most common approaches use general-

purpose robot arms and comprehensive sensor packages

to arrange string loosely into knot configurations [20],

[21], [34], [35], [36]; the knot is then typically tightened

by simply pulling on the ends, without particular concern

for precision of tightening. In contrast, in this work, use

of fixtures in the arrangement phase allows relatively

fast arrangement, without sensing; moving parts in the

fixtures then allow relatively precise tightening.

We present a general approach to tying knots without

simulation or sensing, while satisfying specified distance

constraints along the string. These distance constraints

are measured between string-string contact points, and

enforced at friction locks: locations where friction im-

mobilizes the string against external forces.

One of the central ideas is to manipulate string while

it is tautly held by a fixture, such that deformation of

the string is limited, allowing easy modeling and control.

Different fixtures for different knots are designed auto-

matically using the procedures proposed in this work.

(a) After arrangement, thecloverleaf knot is spanned onthe fixture.

(b) After pulling the ends fora small amount, some pinsmoves.

(c) The string are taut aroundthe rods, cannot be pulled fur-ther.

(d) The cloverleaf knot isfully tightened around thetightening fixture.

Fig. 1: Tightening a cloverleaf knot.

Even though new fixtures are required for new knots,

since the process can be automated, we consider our

approach general, which is not true for mechanisms

designed for specific knots [13], [14], [37], [1].

As motivating examples, we present fixtures for tying

sounding lines, cloverleaf knots, and Ruyi knots. Histori-

cally, sailors measured water depth using a sounding line

with knots tied at six foot (one fathom) intervals along

the line. Most decorative knots, including cloverleaf

knots (Figure 2a) and Ruyi knots, must be tied precisely.

(a) Machine-tied cloverleafknot.

(b) A hand-tied Ruyi knot.

Fig. 2: Decorative knots.

The first step of our approach is to arrange the knot on

the fixture that will be used to tighten the knot. Although

we have previously designed nearly passive arrangement

fixtures [39], [40], requiring only the application of

pressurized air, the larger size and complexity of knots

considered in this paper makes insertion of the string

difficult using that method. Therefore, we use a robot

arm with a special-purpose gripper to arrange knots

around fixtures.

The knot is then tightened on the fixture by pulling on

the ends of the string, as shown in Figure 1. The fixture

uses rods to immobilize the string during arrangement,

and to delay the friction locks from happening prema-

turely during tightening. Some rods are removed during

tightening, and others are allowed to move, leading

corresponding friction locks to appropriate locations.

This is an early study of the tightening problem, and

the primary contribution of this work is as a proof of

concept, showing that fixtures can allow arrangement

and fairly precise tightening of knots that are more

complex than those which have been previously studied

in the robotic manipulation community. As such, the

paper focuses on a few specific knots; although experi-

ments are still somewhat preliminary, these are the first

successful ties of the knots presented.

After a brief review of related work, we present a

brief overview of the knots tied in this work. We then

present a description of techniques for the arrangement

and tightening phases (Sections ?? and ??), and in

Section ??, describe the automated process for designing

fixtures.

A. Related work

We are not aware of work that explores tying complex

knots precisely in a general way. The use of fixtures to

tie knots has been explored in our previous work [10],

[39], both for arrangement and for tightening. Recently,

fixtures have been used in a system intended for cancer

treatment [30], with paths that guide ribbon cable to a

desired location without excessive bending or twisting.

Special machines that can tie specific knots have been

invented [13], [14], [37], [1]. Fixture-based manipulation

of cartons by Lu and Akella [26] also inspired our

current approach, as has work on robotic folding of

origami [41], [6]. This section provides a short survey

on robotic knot tying; a lengthier survey can be found

in [9].

Mathematical knot diagrams [2], [3], [24], [33] are

typically used to describe the structure of a knot. A

knot diagram is a regular projection of a loose knot (or

unknot) onto a plane [25], where broken lines indicate

one segment of the string under-crosses some other

segment (crossing).

Various approaches have been used to achieve par-

ticular string manipulation tasks. Wires and string are

modeled as elastic rods for manipulation in [38], and

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(d) Tight cloverleaf knot.

Fig. 3: Contact diagrams on loose knot diagrams and

machine-tightened sounding line and cloverleaf knot.

The range between the brackets indicates the knot units,

and the black circles indicates the crossings and their

corresponding locations along the string.

2

devices have been built to suture and tie string in medical

applications [22]. String manipulation has also been used

as a challenge problem to study single- and multi-arm

coordination [21], [23].

When is a knot tight? This question has been studied

in applied mathematics and physics [31], [8], [27]. It is

useful to measure distances along the string in units of

string thickness; such a measure is referred to as rope

length in [4].

When pulled taut around rods on tightening fixture,

string can be modeled and controlled relatively easily.

The taut configuration of string among rods has been

studied in computational geometry community as the

shortest curves in a particular homotopy class among

point obstacles [12], [15], [17], [18], [19].

Friction exists between the contacts of string and

the fixture, and even between the contacts of string

and string. Rather than relying on analysis of these

frictional forces (as in, for example, [7], [28], [11]) we

mitigate friction by mounting all rods on low friction

ball bearings, following the direction explored by Furst

and Goldberg [16]. Motion planning studies of flexible

bodies’ minimum energy configurations [29], [32] are

another source of inspiration leading us to manipulate

the string when it is taut.

II. KNOTS TIED

We will discuss the tying of three different knots as a

proof of concept: the sounding line, cloverleaf knot and

Ruyi knot.

The sounding line is a sequence of overhand knots

(Figure 3b) with equal distance between adjacent units.

We consider the sounding line to be a compound knot in

the sense that it is a collection of knot units: segments

of string within which all labels in the Gauss code have

appeared twice. Note that there will be no more than

O(n3) knot units in a knot with n crossings; we will

primarily be concerned with atomic knot units that do

not contain any other non-empty knot units.

The cloverleaf knot (Figure 2a) has three bows ar-

ranged in a single knot unit; the Ruyi knot (Figure 2b)

is a compound knot consisting of three cloverleaf knot

units, each on the bow of a larger cloverleaf knot.

To describe the distance constraints on a knot, we

introduce the contact diagram, which is shown on the

bottom of every subfigure in Figure 3. A contact diagram

marks the distances of all crossings along the string;

matching square brackets indicate the atomic knot units.

Precise tightening is the transformation of a knot from

a loose state such as that in Figure 3a or 3c to a goal

state: Figure 3b or 3d.

Fig. 5: The first knot unit along the sounding line is

tightened around the stationary pin.

b

c d e

f

ga h

Fig. 6: The knot diagram of a cloverleaf knot.

We attempted to place adjacent knot units of a sound-

ing line at 78mm (rather than one fathom) apart. The

machine-tied sounding line (with three knot units) had

average distances 75mm and 77mm between adjacent

knot units. Figure 4 shows the fully automated arranging

process of tying a sounding line, and a snapshot of the

automated tightening process is shown in Figure 5. For

both the sounding line and cloverleaf knots, distance

errors (measured along the string) were less than 5%,

averaged over 100 trials.

A tied cloverleaf knot is shown in Figure 2a, and the

tightening process is shown in Figure 1. This knot has

16 crossings, and its knot diagram is shown in Figure 6.

Three bows of the cloverleaf knot between b and c,

between d and e, and between f and g should have

equal lengths. On the fixture, we enforce the distance

to be 200mm, and the machine-tied knots had average

lengths of 189mm, 190mm and 191mm for the three

bows in our experiments. We believe that much of the

error is due to the stretching of the yarn, which we do

not model.

A. Ruyi knots

Initial attempts to tie a Ruyi knot were unsuccessful

using a fixture with purely passive rods like those used

to tie the cloverleaf knot. We observed that regardless

of the amount of force exerted at the ends of the string,

3

(a) Before re-grasp task in thefirst unit.

(b) After the re-grasp task inthe first unit.

(c) About to finish arrangingfor the first unit.

(d) Finished arranging threeunits of sounding line.

Fig. 4: Sounding line arrangement process.

inner segments of string remained loose. Tension along

the string caused rods to tilt away from the axis of

rotation permitted by ball bearings. To distribute tension

evenly along the string, we added motors that spin some

of the rods.

How should powered rods be placed? The amount

of tension that a powered spinning rod can distribute

depends on the frictional force between the rod and

the string, which can be modeled using capstan equa-

tions [5] that relate the wrapping angle of the string

around a rod to the maximum difference in tensions

on each side of the string before slipping, based on

the coefficient of friction. A few simple experiments

validated the applicability of this model – for example,

as the wrapping angle changed from 30deg to 150deg,

maximum tension differences increased from less than

1N to about 16N, which is consistent with predictions

of the capstan equations. Qualitatively, larger wrapping

angles on powered rods are more effective at distributing

tension.

We therefore placed powered rods in such a way as to

ensure large wrapping angles so that sufficient tension

could be distributed along the string to allow consistent

tightening throughout the string as the string is pulled

from the ends. The layout can be seen in Figure 7, where

the powered rod locations are labeled with yellow dots,

and all other rod locations are labeled with red dots.

After successful tightening of the Ruyi knot, distances

measured along the string were all within 6% of target

distances.

III. KNOT ARRANGEMENT

In previous work [10], [39], we showed that fixtures

can in principle be used to arrange any knot, and the

approach was successful in practice for many simple

knots. However, the physical knots we consider in this

work are too complex to allow easy threading of string

of string through such fixtures. We use a robot arm to

arrange knots on tightening fixtures.

Fig. 7: The initial layout of the fixtures and the rotating

rods for tying Ruyi knot.

We used a spool to lay out the knot. The spool is

mounted on a ball bearing, so the string can be laid out

during arrangement with minimum friction as the robot

arm moves the spool around the tightening fixture.

We designed a gripper that can re-grasp the spool

without completely dropping the spool. This design

allows the spool to pass under segments of the string

as needed. Figure 8 shows the gripper configurations for

re-grasping. The gripper uses two linear motors, with an

electromagnet attached to the tip of each linear motor.

The spool is attached to a small handle. There are two

metal plates on top of the handle, allowing the gripper

to release or grasp the handle using the electromagnets.

The robot arm follows the knot diagram to lay out the

knot on the fixture, performing re-grasps as needed. An

example arrangement of the sounding line can be seen

in Figure 4.

During arrangement, if two segments of string are too

close to allow the spool to pass through, arrangement

cannot be completed. Section V will discuss how to

adjust the design of the fixture to solve this issue.

IV. KNOT TIGHTENING

The tightening fixture contains a collection of straight

rods, some of which can move along straight slots which

guide the rods to predefined locations in which desired

4

Fig. 8: Two different configurations of the gripper grasp-

ing the spool used to arrange the knots.

Fig. 9: The slider used for moving rods. The base has

four wheels that are made of small ball bearings, while

the rod is also on a ball bearing.

cells reach minimum size.

Moving rods are mounted on sliders, as shown in

Figure 9. Each slider uses four small ball bearings that

allow the slider to ride along a track inset in the fixture;

an algorithm for automatic placement of these tracks

during the design process is described in Section V. Each

rod is also mounted on a ball bearing, so that the rod

can rotate during tightening. Two motors pull the open

ends of the knot to tighten, while the tension along the

string increases and forces the sliders along their tracks.

After the rods stops moving, the knot is almost tight,

and must be lifted away from the moving rods (using,

for example, a plate under the string) to be tightened

further, forming tight friction locks. This last tightening

is not controlled, and was a source of error.

V. AUTOMATIC FIXTURE DESIGN

As the tightening fixture transforms the knot from a

loose configuration to a tight configuration, the fixture

needs to be able to support both configurations of the

knot. This section describes a procedure that takes a

loose knot diagram as input, as well as the required

distance between contacts.

The outline of the approach is as follows. First, the

procedure needs to determine where to place rods to

support the loose knot. Based on the required distance

between contacts, the procedure then needs to identify

the rods that will remain in position as the knot is tight-

ened (e.g., to hold the bows of a shoelace in place). If

there are rods that contact more than one strand of string,

Fig. 10: When multiple strands of string contact the same

rod, double the rod.

the procedure duplicates these rods. The procedure then

compares the knot layout and the size of the spool, ex-

panding the layout if necessary to allow the arrangement.

Then, the procedure moves all non-stationary rods closer

towards their geometric center. The rope length between

contacts are then calculated, and adjustments are made

to satisfy the required distances. The trajectories of the

individual rods become the tracks.

We now discuss some further details of the steps of

the design process. In previous work [39], we placed one

rod per cell to tie some simple knots (without particular

regard for precision of tightening). In the current work,

we avoid using a rod for each of the 16 cells in the

cloverleaf knot, or for each of the 64 cells in the Ruyi

knot. Procedure RodPlacement first identifies the cells,

placing a rod in each cell. We then find an approximately

shortest curve among these rods, using an algorithm

for finding the shortest curve of a particular homotopy

class among points sampled along the boundaries of

discs representing rods [12]. We remove rods not in

contact with this curve. For the cloverleaf knot, 7 rods

are placed.

We also would like to avoid letting multiple strands

of string contact the same rod, since two strands might

move at different velocities relative to the rod, prevent-

ing free rotation. We separate contacting strands of string

by doubling and separating such rods; an example is

shown in Figure 10.

How should duplicate rods be placed? Denote the

rod as r and the two strands of string as arb and crd.

For arb, let the new rod be r′, such that the vector rr′

is perpendicular to vector ab. We attempt to minimize

friction around unpowered rods by minimizing wrapping

angles around the rods, to allow tension to be distributed

evenly throughout the string.

Theorem 1: Moving a duplicated rod along the direc-

tion pointing from the original rod location to the middle

of the original contact arc with a distance d minimizes

the angle that string wraps around the new rod.

Proof: Consider the extreme case where the radius

of the rod is 0, with only one strand of string (the other

strand is symmetric). Choose coordinates so that the

location of the original rod is located at O = (0,0). The

vertex Q is on the circle with radius d with P and R

5

Fig. 11: The notation for the triangle where the string

wraps around the moved rod.

fixed as shown in Figure 11.

Let the length of PR be |PR|= c, the distance from O

to line PR is h, |PO|= |OR|= e, and |OQ|= |OB|= d.

Let OB be perpendicular to PR, where B is achieved by

moving O along the direction pointing from O towards

the midpoint of the original contact arc with a distance

d. The angle ∠PQR = α , ∠QOB = β and ∠BOR =∠BOP = γ .

Using the notation in Figure 11, denote the area of

the triangle PQR as A. We have the following geometric

relations:

2A = c · (h+d · cosβ ) (1)

= xy · sinα (2)

cosα = (x2 + y2− c2)/2xy (3)

xy = (x2 + y2− c2)/2cosα (4)

cos(γ−β ) = (e2 +d2− x2)/2ed (5)

cos(γ +β ) = (e2 +d2− y2)/2ed (6)

cosγ = h/e. (7)

Adding Equation 6 to Equation 7, we have

x2 + y2 = 2e2 +2d2−4dhcosβ . (8)

Combining Equation 8 and Equation 3,

2A = (e2 +d2− (c2)/2−2hd · cosβ ) · tanα. (9)

Combining Equation 2 and Equation 9,

(e2 +d2−c2

2−2hd · cosβ ) · tanα = c · (h+d cosβ )

⇒c

tanα=

e2 +d2− (c2)/2−2hd · cosβ

h+d cosβ

⇒c

tanα=

d2 + e2 +2h2− c2

2

h+d cosβ−2h (10)

(a) The polygonal approxima-tion of the cloverleaf knot di-agram. (b) Place a rod in each cell.

(c) Delete all rods that do notcause cell degeneration.

(d) Split rods to let each rodcontact only one strand ofstring.

(e) Reduce the ropelength. Result after Pro-cedure ReduceRopeLength.

(f) Design the tracks by con-necting corresponding rodsusing a line segment.

Fig. 12: Automated design of the fixture for the clover-

leaf knot using presented procedures

Since d2 + e2 + 2h2 − (c2)/2 and c are constants,

when α is increasing, the left side of Equation 10 is

decreasing, which leads to the increase of cosβ and

decrease of angle β . Therefore, when α increases, βdecreases. When the angle β decreases, area A increases.

When β reaches 0, the area A reaches a maximum, as

does α , minimizing the wrapping angle of the string

around the rod.

The fixture designed as described can be used to

support the knot during arrangement, but some distances

between string segments can be too small for the gripper

to re-grasp the spool. We can add certain size con-

straints to each cell where a re-grasp might happen.

This step (Procedure IncreaseCellSizes) happens after

Procedure RodPlacement.

After all the rods are placed, we find a set of line

segments (tracks) that guide the moving rods to a nearly

tight configuration using Procedure ReduceRopeLength.

Finally, it is important to control the distance between

contact points for precise tightening. The distance is

6

Procedure IncreaseCellSizes

Input: Crossing sequence: S= {s1,s2, . . . ,sn}, mapping f (si) : N→ R2

for under-crossing (over-crossing) si and next over-crossing (under-crossing) s j do

if distance between f (si) and f (s j) too small then

Find previous rod ri,p and next rod ri,n of si;

Find previous rod r j,p and next rod r j,n of s j;

if si and s j shares a common adjacent rod then

Find the common rod f (ri j), and the middle point f (si, j) between f (si) and f (s j);Create rod f (r′i) and f (r′j), such that f (r′i) f (r′j) is perpendicular to f (si, j) f (ri j) and pass through

f (ri j), let f (r′i) f (ri j) = f (r′j) f (ri j) = d;

Denote the other rods adjacent to si and s j as ri,o and r j,o;

Move f (ri,o) to f (r′i,o) and f (r j,o) to f (r′j,o), such that f (ri,o) f (r′i,o) is parallel to f (si) f (s j), and

f (r j,o) f (r′j,o) parallel to f (si) f (s j), let f (ri,o) f (r′i,o) = d and f (r j,o) f (r′j,o) = d;

Let 2d + f (si) f (s j)> D where D is larger than the spool size;

else

Move f (ri,p) to f (r′i,p) so that f (ri,p) f (r′i,p) = d and f (ri,p) f (r′i,p) parallel to f (si) f (s j);

Move f (ri,n) to f (r′i,n) so that f (ri,n) f (r′i,n) = d and f (ri,n) f (r′i,n) parallel to f (si) f (s j);

Move f (r j,p) to f (r′j,p) so that f (r j,p) f (r′j,p) = d and f (r j,p) f (r′j,p) parallel to f (si) f (s j);

Move f (r j,n) to f (r′j,n) so that f (r j,n) f (r′j,n) = d and f (r j,n) f (r′j,n) parallel to f (si) f (s j);

Let 2d + f (si) f (s j)> D where D is larger than the spool size;

Procedure RodPlacement

Input: polygonal drawing D ; cells of D : Ci;

for each cell Ci doFind the largest containing disc with center di

and radius ri;

r←min∀i ri;

Place a disc of radius r at each di;

Find the shortest homotopy curve among these

discs;

for disc centered at di do

remove the disc, detect if any cell degenerates;

if cell degenerates then

place disc with radius r centered at di;

while ∃ disc at di contacting two stands of string

dodouble the disc so each disc only contacts one

strand of string;

calculated between two nearest crossings using the com-

puted shortest curve length. If the distance is not yet

satisfied around the rods, we move the corresponding

static rods parallel to the direction of the contact normal

at mid-point of the string contact to meet the distance

constraint.

All fixtures presented were designed using the de-

scribed process. Figure 12 shows the important steps of

Procedure ReduceRopeLength

Input: rope thickness Thi(γ); small clearance a,

geometric center of all rods c; a set of

stationary rods D

Output: Tight ropelength Lp; disc locations D

for disc di 6∈ D do

connect di and c;

k← number of intersections between ~did j with

the knot drawing;

move di along ~dic to d′i such that

d(d′i ,c) = k ·Thi(γ)+a;

di← d′i ;

Calculate distance between contact points;

Move rods if distance constraint is not met;

Calculate ropelength Rop(γ);Lp← Rop(γ);

the fixture design process for the cloverleaf knot. We

printed the fixture after some work with a 3D modeling

package, Solidworks, to model tracks and create sockets

for static rods.

VI. CONCLUSIONS AND ACKNOWLEDGMENTS

In this work, we demonstrated some fixtures (designed

automatically) for tying knots precisely. The main idea is

to delay friction locks, while moving all desired contact

7

points into position. Before tightening, a robot arm is

used to arrange the knot using a specially designed

gripper. An automated design procedure for the fixtures

was also presented.

Using the fixtures, we tied sounding lines, cloverleaf

knots and Ruyi knots fairly precisely. Obvious sources

of error include the stretch of the string and final short

uncontrolled tightening. Future work includes the design

of better mechanical systems to allow faster and more

precise tightening. This work was supported by NSF

grant IIS-1217447.

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