Suture Thread Modeling Using Control Barrier Functions for
Autonomous Surgery

Kimia Forghani1, Suraj Raval1, Lamar Mair2, Axel Krieger3, and Yancy Diaz-Mercado1

Abstract— Automating surgical systems enhances precision
and safety while reducing human involvement in high-risk
environments. A major challenge in automating surgical pro-
cedures like suturing is accurately modeling the suture thread,
a highly flexible and compliant component. Existing models
either lack the accuracy needed for safety-critical procedures
or are too computationally intensive for real-time execution.
In this work, we introduce a novel approach for modeling
suture thread dynamics using control barrier functions (CBFs),
achieving both realism and computational efficiency. Thread-
like behavior, collision avoidance, stiffness, and damping are
all modeled within a unified CBF and control Lyapunov
function (CLFs) framework. Our approach eliminates the need
to calculate complex forces or solve differential equations, sig-
nificantly reducing computational overhead while maintaining a
realistic model suitable for both automation and virtual reality
surgical training systems. The framework also allows visual
cues to be provided based on the thread’s interaction with
the environment, enhancing user experience when performing
suture or ligation tasks. The proposed model is tested on the
MagnetoSuture system, a minimally invasive robotic surgical
platform that uses magnetic fields to manipulate suture needles,
offering a less invasive solution for surgical procedures.

I. INTRODUCTION

Recent advancements in surgical robotics have under-
scored the potential of autonomous systems to revolutionize
healthcare by enhancing precision, reducing complications,
and ensuring consistent surgical outcomes [1]. The COVID-
19 pandemic further emphasized the need to minimize human
interaction in high-risk environments. Autonomous systems
address challenges like infection risks, communication delays
in telesurgery, and are ideal for tasks like suturing due to their
repetitive nature. The success of robotic platforms like the
da Vinci system reflects the medical community’s readiness
to embrace these technologies [2].

To achieve safe, fully autonomous surgeries, accurate
modeling of all components is instrumental. Understanding
suture thread behavior is a particularly important yet under-
studied component in surgical robotics [3]. This is because

1 Authors are with the Department of Mechanical Engineering, Uni-
versity of Maryland, College Park, MD 20742 (kimiaf, sraval,
yancy) @umd.edu

2 Author is with the Division of Magnetic Manipulation and Particle
Research, Weinberg Medical Physics, Inc., North Bethesda, MD 20852
lamar.mair@gmail.com

3 Author is with the Department of Mechanical Engineering, Johns
Hopkins University, Baltimore, MD 21218 axel@jhu.edu

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cable-like objects, such as threads, wires, and hair, are
challenging to model due to the inherent trade-off between
accuracy and real-time performance [4]. However, proper
suture thread modeling is key for simulating and performing
tasks like suturing, knot-tying, and ligation. Beyond surgery,
this model has applications in virtual reality training, textile
simulations, hair or cable animation, and soft robotics, where
realistic thread behavior is critical.

Recent advances in modeling suture threads are mainly
based on continuum mechanics approaches, finite element
methods [5], position-based dynamics (PBD) approaches [6],
[7], or a combination of them [8], [9]. Continuum mechanics
and finite element methods provide high accuracy, but are
computationally intensive due to the need to solve complex
equations governing deformation on top of applying con-
straints like inextensibility. Additionally, collision detection
and contact forces further increase computational demand,
especially for real-time simulations. PBD relies on solving
constraints iteratively in a Gauss-Seidel fashion to estimate
the thread’s position in each iteration. PBD does not solve
all constraints simultaneously but adjusts positions gradually
through successive approximations. This allows for quick
convergence, especially in systems with many constraints,
making PBD computationally efficient. However, the order
in which constraints are applied can significantly impact the
results, potentially leading to oscillations especially in over-
constrained scenarios. Although PBD is computationally effi-
cient and thus favored in applications like computer graphics
and video games, it sacrifices physical accuracy for speed.

Control barrier functions (CBFs) are a mathematical
framework used in control theory to enforce constraints while
ensuring system stability and safety [10]. They can be formu-
lated as a quadratic program (QP) for fast online computation
of safe control inputs. Unlike PBD, CBFs ensure that the
constraints are met by influencing the control actions of the
system, not by directly manipulating the positions of the
entities. Due to their safety assurances, barrier certificates
have been applied to various problems, including collision
avoidance for autonomous agents [11], [12], adaptive cruise
control [13], and lane keeping [14].

In this paper, we propose using CBFs to model suture
threads both accurately and efficiently. We model the thread
as a line graph consisting of n single integrators, whose
velocities can be directly controlled and adjusted to meet all
safety constraints simultaneously. This is achieved through
solutions to a QP, which benefits from the inherent sparsity
of the constraints in a line graph structure. By leveraging
this property, the computational cost is significantly reduced.



This approach allows us to model connectivity, stiffness,
damping, and obstacle interactions, without the need to solve
Newtonian equations or sacrifice accuracy. The model is
validated experimentally using the MagnetoSuture system, a
robotic platform that magnetically controls needle position,
enabling tetherless minimally invasive surgeries.

The outline for the rest of the paper is as follows: Section
2 provides background on CBFs, and Section 3 explains the
suture thread model utilizing CBFs and CLFs. In Section 4,
the accuracy of the suture thread model is validated through
comparisons between model-based simulations and experi-
mental robot data of suture thread movement. Additionally
a surgical task simulation with visual feedback is provided.

II. BACKGROUND

In this section, we cover the basics of system safety and
control, beginning with a control model and introducing
CBFs for enforcing safety. We then explain how CBFs can
be integrated into a QP to ensure optimized safe operation.

A. System and Safety

Consider an affine control system:

ẋ = f(x) + g(x)u, (1)

with x ∈ Rd and u ∈ Rm, where f and g are locally
Lipschitz. The system is subject to input constraints u(t) ∈
U ⊆ Rm, the space of admissible control signals.

A set C ⊂ Rd, defined by:

C = {x ∈ Rd : h(x) ≥ 0},
∂C = {x ∈ Rd : h(x) = 0},
Int(C) = {x ∈ Rd : h(x) > 0}.

(2)

is forward invariant if, under a feedback controller π(x, t),
the solution x(t) of the closed-loop system:

ẋ = f(x) + g(x)π(x, t), (3)

remains in C for all time. The system is safe with respect
to C if C is forward invariant [10].

B. CBFs and CLFs

A control barrier function (CBF) ensures system safety
by keeping the state within a safe set defined by a function
h(x). The function h is a CBF if there exists a class K∞
[15] function α such that:

sup
u∈U

ḣ(x, u) ≥ −α(h(x)), (4)

for all x in the domain. This condition imposes a lower bound
on the evolution of h through the control policy so the system
remains in the safe set and maintains safety over time.

In addition to ensuring safety, we are also interested in
stabilizing the system to a desired state. This can be achieved
using a control Lyapunov function (CLF), which drives the
system to stability. A function V (x) is a CLF if it is positive
definite and there exists a class K∞ function γ such that

inf
u∈U

V̇ (x, u) ≤ −γ(V (x)), (5)

for all x in the domain. This condition imposes an upper
bound on the evolution of h through the control policy,
driving the system towards the desired equilibrium state over
time.

C. CBF-Based Control

Given a feedback controller u = k(x) for a control system,
we may encounter situations where k(x) does not ensure
safety according to the CBF criteria. Thus, we leverage the
fact that the safety condition, expressed as:

ḣ(x, u) = Lfh(x) + Lgh(x)u ≥ −α(h(x)), (6)

is affine in u, where Lfh(x) and Lgh(x) are the Lie
derivatives [15] with respect to f(x) and g(x), respectively.
This allows us to formulate a quadratic program (QP) that
finds the control input u(x) with the smallest deviation from
the desired control v(x) [10]:

u(x) = arg min
u∈Rm

1

2
∥u− v(x)∥2

s.t. ḣ(xk, uk) ≥ −αkh(xk)

V̇ (x, u) ≤ −γ(V (x)).

(7)

This QP-based approach ensures that the system remains
within the safe set by minimally perturbing the control input.
The same QP is also subject to the constraint in (5), which
drives the system to stability. This type of controller is
referred to as CLF-CBF-QP.

D. MagnetoSuture

The experiments included in this paper are performed by
the MagnetoSuture [16] system as shown in Fig. 1. This
surgical robotic system enables tetherless manipulation of
suture needles using magnetic fields, eliminating the need
for large robotic tools during minimally invasive surgery.
This approach reduces invasiveness, as the only components
introduced into the body are the needle and thread, lowering
the risk of tissue damage, scarring, and infections. The
system uses an electromagnetic coil array to guide NdFeB
(neodymium iron boron) suture needles for tasks like tissue
penetration, ligation, and complex suture patterns demon-
strated in successful experiments with ex vivo tissues.

Fig. 1. The MagnetoSuture system (left), and closed up look at the
workspace that includes the suture needle and thread (right). MagnetoSuture
is a magnetic manipulator robot which leverages vision-based feedback to
control magnetic tools (e.g., needle) via a coil-generated magnetic field. The
workspace used in this setup is a petri dish, 35mm in diameter.



III. MODELING SUTURE THREAD WITH CONTROL
BARRIER FUNCTIONS

In this section, we model the suture thread as a system
of n nodes and a lead node (needle), each represented as a
single integrator. The state dynamics of each node are:

ẋi = ui, xi ∈ D ⊂ Rd, i = 1, . . . , n,

ẋ0 = u0, x0 ∈ D ⊂ Rd
(8)

Here, ẋi represents the velocity of the i-th node, and ui is
the control input. The thread’s lead node corresponds to the
surgical needle, with ẋ0 and u0 representing its velocity and
control input, respectively. The nodes’ velocities (u0, . . . , un)
are determined using a QP, which minimizes deviations from
a desired velocity while ensuring the system remains in a
safe state. To guarantee forward invariance, the initial states
of all nodes, (x0, . . . , xn), are chosen within the safety set.
The desired control input for the lead node can either follow
a predefined path or be adjusted in real-time by the user
via a joystick. The unsafe states are described by the CBFs
discussed in this section. Fig. 2 shows a conceptual figure
of the thread model and its constraints.

A. CBF for Connectivity

This CBF ensures that the natural connectivity of the
thread is maintained by enforcing that the distance between
any two consecutive nodes does not exceed a maximum
allowable distance. We note that when the thread loops, it is
possible for two nodes to become coincident. For this reason,
we do not impose a minimum distance constraint.

For convenience, we define the ensemble vector of position
states as x = [xT

1 , . . . , x
T
n ]

T ∈ Rnd, and the ensemble vector
of control velocities as u = [uT

1 , . . . , u
T
n ]

T ∈ Rnd. For i =
2, . . . , n, we define a connectivity function as [17]:

hcon,i(x) =
1
2 (∆

2 − ∥xi − xi−1∥2), (9)

Fig. 2. Conceptual representation of the suture model, moving with needle
velocity of u0, and interacting with a circular obstacle. In the figure, z
corresponds to the distance between nodes, and d is the shortest distance
from a node to the obstacle boundary. The constraints encoded in hobs and
hcon enforce that the thread does not enter the obstacle and that it does not
extends past a maximum length, respectively.

where ∆ > 0 is the maximum node separation distance.
The connectivity CBF is applied only to the states of

the thread nodes and does not influence the control input
of the lead node. However, to ensures that the other nodes
follow the needle while maintaining thread connectivity, we
introduce an additional connectivity function related to the
needle’s position x0:

hcon,1(x, x0) =
1
2 (∆

2 − ∥x1 − x0(t)∥2), (10)

The safe state Ccon,i is defined as:

Ccon,i = {x ∈ D | hcon,i(x) ≥ 0}. (11)

The barrier function in (10) ensures that the lead node, or
needle, maintains connectivity with the rest of the thread.
However, since the model is implemented in discrete time,
depending on the user input, at each time step, the needle
can be positioned at a distance from the first node such that
it exits the safe state. In this case, the invariance of set C
isn’t guaranteed because we are not starting from a safe state.
However, even if the needle is positioned in an unsafe state
(i.e., hcon,1 < 0), after a few iterations, the system controls
the rest of the nodes to converge into a safe state. The smaller
the deviation of the needle’s position from the safe state is,
the faster the convergence will be. To account for this error,
slack variables are introduced as discussed in Section III-E.

Additionally, we enhance connectivity, by linking each
node i = 3, . . . , n, not only to node i − 1 but also to node
i−2. This approach increasing the algebraic connectivity (the
second smallest eigenvalue of the graph Laplacian matrix)
which has been shown to enhance system robustness [18].
For i = 3, . . . , n, we define this connectivity function as:

hcon enhanced,i−1(x) =
1
2 (∆

2 − ∥xi − xi−2∥2),
hcon enhanced,1(x) =

1
2 (∆

2 − ∥x2 − x0(t)∥2).
(12)

The safe state Ccon enhanced,i, corresponding to this connectiv-
ity function is defined as:

Ccon enhanced,i = {x ∈ D | hcon enhanced,i(x) ≥ 0}. (13)

B. CBF for Obstacle Avoidance

This section presents a CBF that maintains a minimum dis-
tance between each node and the nearest obstacle, preventing
penetration through the obstacle. When enforcing distance-
based constraints, obstacles are often approximated as ellip-
soids [19] or hyper-spheres [20] to avoid non-differentiability
caused by sharp corners. As the robot approaches a corner,
the closest point can shift abruptly between edges, caus-
ing discontinuities in the distance function. Approximating
these components as curved shapes ensure differentiability,
enabling use of control barrier functions. However, they tend
to overestimate obstacle and robot sizes.

To mitigate this, we propose a method which is applicable
to both convex and non-convex obstacles. In this approach
we smooth the corners by estimating curves that approxi-
mate only the sharp regions, reducing the abrupt direction
changes while keeping the integrity of the obstacle shape.
Although the distance function remains non-differentiable,



Fig. 3. Smoothing and triangulation of a non-convex shape (left) using
the Delaunay method. The polygon vertices and edges are depicted in blue,
while the triangulation is highlight by arbitrary colors (right).

the transition between edges becomes much smoother, and
the changes in direction are less drastic. Then the obstacles
are divided into triangles using the Delaunay triangulation
method [21]. Fig. 3 illustrates an examples of a non-convex
shape that has been smoothed and triangulated.

Furthermore, for each node, the closest point on each
triangle is determined by projecting the node’s position onto
the triangle’s edges and clamping the projection to ensure the
closest point remains on the edge. The overall closest point
on the obstacle is then selected by finding the minimum
distance across all triangles, and this distance is used to
compute the CBF below. This method efficiently handles
large sets of points and triangles by minimizing loop usage
and relying on vectorized operations.

For nodes i = 1, . . . , n and obstacles O = 1, . . . ,M , we
define an obstacle avoidance function as:

hobs,i,O(x, p) =
1
2 (∥xi − pO∥2 − ρ2), (14)

where pO is the position of the closest point on obstacle O
to node i, and ρ is the minimum allowable distance between
the two points. Similarly, we define an obstacle avoidance
function for the lead node as:

hobs,needle,O(x, p) =
1
2 (∥x0 − pO∥2 − ρ2), (15)

As opposed to the connectivity CBF, the obstacle avoidance
CBF affects the needle control input to stop it from pene-
trating obstacles, despite the user input or a defined path.

The safe state Cobs, corresponding to this collision avoid-
ance function is defined as:

Cobs,i = {x ∈ D | hobs,i(x) ≥ 0}. (16)

C. CLF for Stiffness

For suture threads, which typically exhibit memory effects,
the thread naturally tends to return to its original shape
when no additional forces are provided. This behavior is
modeled using a CLF rather than a CBF, as the thread has
an equilibrium state it seeks to maintain, rather than a safe
set to stay within. The memory is modeled as the distance
between node i and node i−2 for i = 3, . . . , n in the natural
state. For example, if the memory of the suture thread is for
it to be a straight line, then the natural state distance between
nodes would be 2×∆, where ∆ is the maximum separation
distance between two adjacent nodes.

For the general case, the CLF is defined as:

Vi−1(x) =
1
2

(
∥xi − xi−2∥2 − (δi)

2
)2

,

V1(x) =
1
2

(
∥x2 − r(t)∥2 − (δ2)

2
)2

,
(17)

Fig. 4. The CLF in (17) acts analogous to a stiff mechanical spring between
nodes i and i+ 2. Nodes are represented by gray circles.

where δi is distance of the respective 2 nodes in the natural
state. Using this CLF in the controller ensures that the thread
naturally returns to its original shape over time. This concept
is equivalent to having a stiff spring attached between node
i and i + 2, whose equilibrium state is achieved when
the two nodes are at distance δ. Fig. 4 illustrates when
the thread’s curvature exceeds its natural state, the ‘spring’
compresses, pushing the nodes apart to restore its natural
state. Conversely, when the nodes are too far apart, the spring
stretches, pulling them back toward the natural equilibrium,
maintaining the thread’s overall shape.

D. Damping

Up until this point, it is assumed that the thread moves
on a high-friction surface, which implies that, in the absence
of control input, the thread remains stationary. However, in
our experimental setup, the thread floats on a viscous fluid
as part of the MagnetoSuture system. In this case, a new
damping assumption is required.

Instead of assuming that the desired velocity of each node,
which we optimize to minimize deviation from, is zero, we
assume it is a fraction κ of the node’s current velocity. The
damping percentage was determined experimentally to be
κ = 0.95 by tuning the simulation to match the experimental
results of the thread moving in glycerin.

E. Controller

The overall thread safety set is the intersection of all L =
3n−2+M(n+1) safety sets which include: n connectivity
constraints, n−1 enhanced connectivity constraints, M(n+
1) obstacle constraints, and n− 1 stiffness constraints:

Cthread =
⋂3n−1+M(n+1)

ℓ=1
Cℓ (18)

The control input is determined by solving the following QP:

min
u∈Rnd,u0∈Rd

s∈R3n−2

1
2∥u− v(x)∥2 + 1

2∥u0 − v0(t)∥2 + ϕ(s) (19)

s.t. Aobs [u
T
0 , u

T ]T ≥ bobs

Acon u ≥ bcon − [s1, . . . , sn]
T

Acon,enh u ≥ bcon,enh − [sn+1, . . . , s2n−1]
T

Astiff u ≤ bstiff + [s2n, . . . , s3n−2]
T

s ≥ 0

Here, the desired value for node and needle velocities,
v(x) and v0(x) are adjusted based on the environment
damping properties and the user input. The slack variables
sT = [s1, . . . , s3n−2] are introduced to relax connectivity
and stiffness constraints, ensuring the existence of a feasible



Fig. 5. Visual comparison of (a) the experimental behavior of a 19 mm long
Polyamide suture thread in the MagnetoSuture system and (b) the behavior
of the proposed model for four instances in time (i-iv). For simplicity, the
needle is not depicted; instead, its endpoint is represented by a red circle
around the lead node.

solution even when constraints must be violated due to the
needle’s input. Note that even if this is the case, the properties
of the CLF-CBF-QP eventually drive the system back to the
safe region. Additionally, the slack penalty function ϕ(s) =
1
2s

TWs, where W is a diagonal matrix of positive weights,
enables constraint prioritization. Connectivity constraint is
heavily enforced with Wcon ≫ Wstiff. Stiffness constraint is
assigned lower weights, empirically tuned based on suture
material. Obstacle avoidance is treated as the highest priority,
with no slack allowed. The cost function is normalized such
that W = 1 for terms that minimize velocity deviations.

Due to our simplified dynamics, the entries for the con-
straint matrices are given by partial derivatives and evalu-
ation of the constraints functions. For example, for the ith

connectivity constraint function, we get

[Acon]i,j =
∂hi,con

∂xj
, [bcon]i = −αkhcon,i (20)

As these functions only rely on local state information (e.g.,
xi for i = 1, ..., n, and xi−1 for i = 2, ..., n), the majority
of the entries in Acon will be zero. This is true for the
other constraints as well. Thus, for UT = [uT , uT

0 , s
T ], it

is possible to represent all the constraints in (19) as a single
highly sparse inequality

AU + b ≥ 0, (21)

which significantly reduces the computational complexity of
this optimization task.

IV. RESULTS

In this section, experimental maneuvering tasks were con-
ducted on the MagnetoSuture system to compare with the
proposed simulation model, and both robotic and simulated
tasks were video-recorded to assess suture thread behavior.

A. Thread Behavior Test

The proposed model is validated against experiments using
two suture threads: a USP 7-0 polyamide monofilament and
a USP 6-0 silk braided suture, both maneuvered in a glycerin
medium. The silk thread was tested in a larger workspace to
assess adaptability. These threads were selected to evaluate
the model’s ability to capture varying stiffness, as polyamide

Fig. 6. Visual comparison of (a) the experimental behavior of a 150 mm
long silk suture thread in the MagnetoSuture system and (b) the behavior of
the proposed model. The needle shown in (b) is a visual copy of the needle
from (a), to improve clarity, and is not part of the simulation output.

is inherently stiffer than silk. Fig. 5 and Fig. 6 compare
simulated and experimental thread behavior along arbitrary
paths. The simulation runs at 66 Hz.

Fig. 7 illustrates the mean error relative to the thread length
for various lengths and materials. The silk suture thread
exhibits minimal error, primarily due to two factors. First,
silk has lower inherent stiffness compared to polyamide,
making it easier to model accurately. Second, the method of
attachment to the needle affects performance. The polyamide
thread is glued to the needle, creating increased rigidity near
the needle, causing the thread to assume a specific angle that
the model does not account for (this is not the conventional
method of attachment). In contrast, the silk thread is passed
through a hole in the needle, allowing it to move freely.
Future experiments will involve silk sutures performing tasks
such as suturing for further model validation.

B. Hernia Repair Simulation with Visual Feedback

We use the proposed model to simulate an inguinal hernia
repair procedure. The hernia is modeled as a ring with an
outer diameter of do = 1.2× 10−2 m and an inner diameter
of di = 9 × 10−3 m. The needle wraps around the hernia
ring, pulling and tightening the suture thread. Fig. 8 shows
the simulation at various stages. As the needle penetrates
tissue and wraps around the hernia, the thread nodes follow,
tightening around the ring.

In this simulation, the needle’s position (represented by the
lead node) is controlled in real-time by the user via keyboard
arrows. The simulation is computed at the speed of 33Hz on
a Dell Inspiron laptop, with specifications including 8GB
RAM, and Intel i7 processor. Despite the modest capabili-
ties of this system, the simulation performs efficiently, and

Fig. 7. Mean error percentage relative to thread length for two experiments.
Number of simulated nodes is n = 25 (Polyamide) and n = 81 (Silk).



Fig. 8. Six time instances from the real-time simulation of an inguinal
hernia repair using the proposed suture thread model. The pink geometry
with a ring is the inguinal hernia in tissue, and the colored line is the
suture thread with 40 nodes. i) The thread moves freely towards the obstacle
(T=0s). ii) The thread turns orange, indicating friction in the penetration
path, slowing the needle (T=12s). iii) The lead node moves left, not pulling
the part of the thread under friction, turning it blue (T=17s). iv-vi) The
thread wraps around the obstacle, changing color and slowing further as
friction increases (T=27s, 35s, 55s).

Fig. 9. The obstacle constraint function hobs corresponding to the first five
nodes of the thread for the three obstacles used in the hernia simulation.

the results remain realistic. With more advanced hardware
typically used for real-time applications, the computational
speed would be even higher, but this model demonstrates its
realism and functionality even on basic consumer hardware.

Visual feedback is implemented as described in Algo-
rithm 1. As the needle moves, the obstacle avoidance barrier
function for nodes approaching the tissue decreases towards
zero. If the barrier function of node i approaches zero
for two obstacles, the system detects a penetration path,
triggering visual feedback to slow the needle, simulating
increased resistance. Additionally, the needle turns orange to
indicate friction, with the shade of orange and the degree of

Algorithm 1 Real-Time Thread Sim with Visual Feedback
1: Initialize obstacles by smoothing edges and triangulating
2: Initialize states and reference velocity xi = x0

i , v = v0i
3: Get initial hobs

4: loop
5: Set v0 ← User Input
6: index ← i s.t. two elements of hobs,i < threshold
7: if index is empty then
8: Set Thread.Color ← green
9: else if vi ̸= 0 then

10: Decrease needle speed: v0 ← v0×βlength(index)

11: Set Thread.Color ← Scale(red, length(index))
12: else
13: Set Thread.Color ← blue
14: end if
15: Damp reference velocity vi ← κvi
16: Determine constraints (21)
17: Determine safe velocities [ui, u0] according to (19)
18: Update dynamics according to (8)
19: end loop

slowdown proportional to the number of nodes experiencing
friction. If part of the thread is under friction but not being
pulled, it turns blue. When the thread moves freely, it turns
green. This visual feedback is entirely based on the thread’s
interaction with the environment, as captured by the barrier
function values, without explicitly calculating forces. The
barrier functions for this simulation are plotted over time
in Fig. 9.

Each node i = 1, . . . , n has velocity ui and position
xi. Needle velocity and position are represented by u0

and x0 respectively. Elements of hobs,i corresponds to the
barrier function value of node i in respect to each obstacle.
0 < β < 1 is a scaling factor for reducing the velocity
based on friction. In the simulation here β = 0.9 is used.
When applying Equation (19), the slack function weights
are empirically set as Wcon = 105 and Wstiff = 1.

V. CONCLUSIONS

In this work, we presented a novel approach to modeling
cable-like objects, specifically suture threads, that is both
computationally efficient and accurate. By utilizing control
barrier functions and optimization techniques, our method
avoids solving complex equations of motion, enabling real-
time applications in surgical robotics and training simula-
tions. Validation against experimental data, particularly with
silk sutures, highlights its potential to advance autonomous
surgical tasks. Future work will focus on extending the model
to handle more complex procedures, including thread self-
collision, such as suturing and ligation tasks.

ACKNOWLEDGMENT

The authors would like to thank Daniel (Qinhan) Wang
for providing videos of the MagnetoSuture system.



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