# Optimization-Based TDA for CDPRs with Elastic Cables: Twice Continuously Differentiable Cable Tensions

Ahmed Nassim Bellila, Hélène Chanal, Olivier Company, Marc Gouttefarde, Chedli Bouzgarrou

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Ahmed Nassim Bellila, Hélène Chanal, Olivier Company, Marc Gouttefarde, Chedli Bouzgarrou. Optimization-Based TDA for CDPRs with Elastic Cables: Twice Continuously Differentiable Cable Tensions. CableCon 2025 - 7th International Conference on Cable-Driven Parallel Robots, The Chinese University of Hong Kong, Hong Kong SAR, China, Jul 2025, Hong Kong, China. hal-05369045

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Distributed under a Creative Commons Attribution 4.0 International License
Optimization-Based TDA for CDPRs with Elastic
Cables: Twice Continuously Differentiable Cable
Tensions

Ahmed Nassim Bellila¹,², Hélène Chanal¹, Olivier Company², Marc Gouttefarde², and Chedli Bouzgarrou¹

¹ Université Clermont Auvergne, Clermont Auvergne INP, CNRS, Institut Pascal, F-63000, France

² LIRMM, Univ Montpellier, CNRS, Montpellier, France
ahmed.bellila@lirmm.fr

**Abstract.** The need for $C^2$-continuous cable tensions profiles arises in torque-based control of Cable-Driven Parallel Robots (CDPRs), when considering cable elasticity, as discontinuous profiles can hinder smooth motor torques and overall system performance. To tackle this, we propose a novel Tension Distribution Algorithm (TDA) formulated as an optimization problem. The method incorporates a 9th-degree polynomial trajectory generation to ensure $C^2$-class continuity in desired motions while generating tension profiles that respect system constraints and avoid abrupt changes. Using an inverse quadratic penalty-based cost function, the approach guarantees regularity and smoothness, as demonstrated through rigorous mathematical proofs. Simulation results validate the method on a 6-DoF CDPR with eight actuating cables. This contribution lays the groundwork for achieving enhanced trajectory tracking and precise control in future applications.

**Keywords:** Cable-driven parallel robot · Tension distribution · Torque-based control.

# 1 Introduction

CDPRs stand out for their mechanical simplicity and flexibility, with the ability to cover large workspaces by spooling cables on winches and routing them through pulleys. The CDPR end-effector is commonly referred to as the Mobile Platform (MP).

Thanks to their adaptability and efficiency in handling loads, CDPRs have been applied across a range of industries and tasks, from 3D printing [1] and rapid pick-and-place operations [2] to rescue missions after natural disasters [3]. They are also seen in high-speed applications, human assistance systems [4], and popular setups like skycams [5]. Additionally, their ability to handle heavy payloads with minimal structure makes them ideal for scenarios requiring robust load-bearing capabilities [6].
A distinctive feature of CDPRs is their reliance on tensile forces, as cables can only pull. To manage this, CDPRs often incorporate more cables than the degrees of freedom of the MP, leading to redundant actuation. This redundancy allows for multiple feasible tension distributions to apply a specific wrench on the MP. Consequently, optimizing tension distribution has been a significant area of research, aiming to enhance both performance and stability [7] of CDPRs.

To accurately model CDPRs and achieve high positioning precision, particularly in applications like Additive Manufacturing that demand precise positioning and motion control, it is essential to account for the behavior of each of their components. Indeed, the behavior of cables change under the influence of applied tensions. Thus, various cable properties, such as material composition, density, and elasticity, significantly influence the static and dynamic behavior of CDPRs [8]. Among these, cable mass and elasticity are frequently analyzed in the literature. While the mass of a cable can be neglected when its tension substantially exceeds its weight [8,9], elasticity must be accounted for to predict deformation under load, which can critically affect positioning accuracy. Controlling system stiffness is crucial for the operational efficiency of CDPRs. Enhanced stiffness generally ensures greater precision. Algorithms that optimize tension distribution play a key role in maintaining sufficient stiffness along the CDPR trajectory. For instance, in 3D printing, such optimization is essential for maintaining high precision and quality in the printed structures [10,11].

CDPRs geometric stiffness was notably introduced in [12]. This property is fundamentally linked to their precision. Several notable studies have advanced the understanding of stiffness in CDPRs [8]. Verhoeven [13] formulated a linear elastic stiffness model, where cables are modeled as linear springs. The stiffness of a CDPR is strongly linked to Wrench-Feasible-Workspace (WFW) concept [12], as it depends on both the cable tension distribution and the robot geometric configuration. It is composed of two components: passive stiffness, which comes from the elasticity of the cables and CDPR configuration, and active stiffness, which is influenced by the cable tensions.

Besides, tension distribution in CDPRs is a critical process that significantly influences the system performance and functionality. It affects system equilibrium (static or dynamic), determining cable tensions along trajectories, designing controllers, and maintaining system stiffness. The main TDAs used to calculate the tension distribution among the cables in CDPRs are: quadratic programming [14,15], linear programming techniques [16], the Dykstra method for norm minimization [17], and gradient descent based distribution [14]. In [18], a computationally efficient algorithm is introduced, ensuring smooth tension transitions within the WFW. Boumann and Bruckmann [19] proposed the Nearest Corner Method, a geometric approach providing continuity near hypercube corners. Côté et al. [20] used quadratic optimization with slack variables to keep tensions smooth. Gouttefarde et al. [21] define a feasible polygon approach for *n*-DOF robots with *n* + 2 cables, representing sets that satisfy cable tensions limits.

In [13], the continuity of cable tensions was analyzed, proving that solutions ensuring only $C^0$ continuity exist. Later, in [22], the problem of tension distribu-
tion in overconstrained CDPRs is formulated as an optimization problem. The algorithm focuses on maintaining continuously differentiable $C^1$ cables tensions, enabling smooth control and operation. To this end, small, penalized errors in the resultant MP wrench are allowed, with bounds placed on these errors under specific conditions. The method introduces slack variables into the optimization process. Additionally, a new cost function that ensures $C^1$ continuity of cable tensions and addresses small errors in force equilibrium. This cost function uses logarithmic barrier functions to keep cable tensions within predefined limits, thus avoiding the need for strict inequality constraints. The approach has been validated with simulation based evaluations, highlighting its computational efficiency and real time suitability.

To the best of our knowledge, the existing studies on TDAs primarily emphasize minimizing computation time and ensuring continuous tension adjustments along a desired MP trajectory. In this context, the main contribution of this paper is to demonstrate the necessity of having $C^2$ continuous cable tensions for motor torque-based control of CDPRs, when considering cable elasticity. In response to this need, we propose a TDA that ensures smooth $C^2$ tensions profiles, aiming to enhance precision in trajectory tracking tasks.

This paper is organised as follows. Section 2 introduces CDPR kinematic and dynamic modeling. Section 3 explains the main tension and motor torque continuity issue dealt with in this work. Section 4 introduces the proposed TDA while Section 5 concludes the paper.

# 2 Kinematic and dynamic analysis

## 2.1 Inverse Kinematics

The pose of the MP of a CDPR is given by the vector $\mathbf{x} = [\mathbf{p}^T \psi^T]^T$ where the MP position vector is $\mathbf{p} = [p_x p_y p_z]^T$ and its orientation is represented by a vector of three angles $\psi = [\alpha \ \beta \ \gamma]^T$. As shown in Figure 1, $\mathbf{a}_i$ is the position vector of the drawing point of the $i$th cable. This vector is written in the CDPR base frame $R_b$. Similarly, $\mathbf{b}_i$ is the position vector of the attachment point of the $i$th cable on the MP, expressed in the MP frame $R_p$.

The inverse kinematics consists in finding the cable lengths given a pose of the platform, i.e. in computing $\mathbf{l}$ for a given pose, as follows:

$$l_i^2 = \|Q\mathbf{b}_i + \mathbf{p} - \mathbf{a}_i\|_2, \quad i \in \{1, 2, \dots, m\} \qquad (1)$$

where $m$ is the number of cables and $Q$ the rotation matrix defined as $Q = R_z(\gamma)R_y(\beta)R_x(\alpha)$.

Assuming a linear relationship between the tension-free cable length $l_{0i}$ and the winch position $q_i$ (rad), the relationship for the $i$th winch can be expressed as:

$$l_{0i} = \pm r_i q_i, \quad i \in \{1, 2, \dots, m\} \qquad (2)$$
<figure><img src="image_3.png" /><figcaption>Fig. 1: CDPR kinematic notations.</figcaption></figure>

where $r_i$ is a constant. Consequently, the vectorized form of the relationship between the tension-free cable lengths and the winch positions is given by $l_0 = R_q$, where $R$ is a diagonal matrix containing the winch radius $r_i$.

## 2.2 Cable elasticity

In this paper, cable mass is neglected, focusing solely on cable elasticity. The common approach to model cable behavior is the linear spring model [23]

$$t_i = k_i \cdot \Delta l_i \qquad (3)$$

where $t_i$ is the tension in the i-th cable, $\Delta l_i = l_i - l_i^0$ represents the length change from the tension-free state $l_i^0$, and $k_i = \frac{AE}{l_i^0}$ is the stiffness constant, determined by Young's modulus $E$ and the cable cross-sectional area $A$.

## 2.3 Platform dynamics

The general motion equation for the CDPR MP is given by:

$$M\ddot{x} + C(x, \dot{x})\dot{x} = G + Wt + f_{ext} \qquad (4)$$

where $M$ is the mass inertia matrix of the MP, $C(x, \dot{x})\dot{x}$ represents Coriolis and centrifugal effects, $G$ is the gravitational vector, $f_{ext}$ includes external wrenches and other disturbances that may be applied to the MP, and $Wt$ represents the wrenches exerted by the cables on the platform [21,24], with $t$ being the vector of cable tensions. $W$ is the wrench matrix which relates the applied wrench to cable tensions.
## 2.4 Winch dynamics

The dynamics of the winches can be expressed as

$$ \tau_m = \mathbf{I}_m \ddot{\mathbf{q}} + \mathbf{f} + \mathbf{R} \mathbf{t} \qquad (5) $$

where $\mathbf{f} \in \mathbb{R}^m$ is the vector of torques due to friction, $\mathbf{R} \mathbf{t}$ is a vector representing the torques applied by the cables on the winch drum, $\tau_m$ is the vector of motor torques, $\mathbf{I}_m$ is a diagonal matrix containing the winch inertia moments and $\ddot{\mathbf{q}}$ is the motor angular acceleration.

## 3 Cable elasticity and tension continuity

Let us consider the open-loop control scheme depicted in Figure 3. First, the trajectory block (Traj) provides the desired pose, velocity and acceleration of the MP. This desired pose $\mathbf{x}_d$ is transformed into cable lengths $l_d$ using the inverse kinematics (IK) block. To ensure accurate trajectory execution, the platform dynamics compensation block computes the desired wrench $\mathbf{w}_{ref}$ to be applied to the MP. This wrench is then passed to the TDA, which determines the cable tensions $\mathbf{t}_d$ required to achieve the desired wrench while satisfying minimum and maximum tension limits. The computed tensions, along with the cable lengths $l_d$, are used in the inverse elastic model to calculate the tension-free cable lengths $l_{0d}$, which account for cable elasticity. These lengths are linearly related through matrix $\mathbf{R}$ to the winch positions $\mathbf{q}_d$. Subsequently, the winch dynamics compensation block determines the motor torques $\tau_{md}$ needed to achieve the desired trajectory, considering the dynamic behavior of the winches.

<figure><img src="image_4.png" /><figcaption>Fig. 2: Open-loop control scheme as implemented in simulation.</figcaption></figure>

To achieve torque control of the CDPR motors, as illustrated in the open-loop control scheme shown in Figure 2, it is essential to generate continuous torques, which are derived from the tension-free cable lengths. As shown in Eq. (5), the motor torques $\tau_{md}$ depend on the motors accelerations $\ddot{\mathbf{q}}$. The relationship between the winch position $q_i$ and the tension-free cable length $l_{0i}$ is given by Eq. (2). Thus, according to Eq. (2) and Eq. (3), the desired winch position can be written as

$$ q_{i_d} = r_i^{-1} \frac{AE}{t_i + AE} l_{0i} \qquad (6) $$
Here, $q_{id}$ is influenced by the cable tension $t_i$ and cable length $l_{id}$, taking into account cable elasticity. Since $\tau_{md}$ is a function of the motors accelerations $\ddot{q}_d$ (refer to Eq. (5)), ensuring that $\tau_{md}$ remains continuous requires $\ddot{q}_d$ to be continuous. Therefore, for $\ddot{q}_{id}$ to be continuous, according to Eq. (6), the tension $t_i$ must be of class $C^2$. Thus, achieving precise and smooth motor torque control of CDPRs accounting for cable elasticity fundamentally depends on maintaining $C^2$ continuous cable tensions.

Through simulations, we attempted to address this issue using algorithms from the literature, such as the TDA approach in [21], applying multiple possible tension distributions. However, these methods did not produce satisfactory results, as illustrated in Figure 3.

<figure><img src="image_5.png" /><figcaption>Fig. 3: Example of simulated cable tension profiles using turnaround algorithm from [21] with the centroid distribution.</figcaption></figure>

Linear elastic cable model is considered and the trajectory generation block uses 5th degree polynomial pose to pose interpolation. As one can see on the Figure 3, the second derivative of each cable tension is not smooth along the trajectory, generating discontinuous motor torques using the open-loop control scheme of Figure 2. To address the discontinuities identified in cable tensions, we propose in the following section an optimization-based TDA.

# 4 Tension Distribution Algorithm

Based on the findings in [22] and further analysis through simulations, it was observed that the problem of discontinuity in cable tensions tends to arise whenever tension limit inequality constraints change their state. Specifically, this occurs
when the tension in a cable shifts from being within the permissible range (inac- tive constraint) to exactly reaching its upper or lower limit (active constraint). These state changes can cause sudden shifts in cable tension derivatives, disrupt- ing the smooth and stable control of the system. As a result, the TDA should be designed to minimize or prevent these transitions, ensuring consistent and sufficiently continuous tension profiles.

## 4.1 Problem Formulation

We formalize the TDA as an optimization problem by combining a cost function and constraints. The goal is to find the cable tensions vector **t** that minimizes the cost function while satisfying both equality and inequality constraints:

(i) **Equality constraint:** $W\mathbf{t} = \mathbf{w}_{\text{ref}}$ where $\mathbf{w}_{\text{ref}}$ is the wrench to be applied on MP.

(ii) **Inequality constraints:** Strict inequality constraints on the cable tensions $\mathbf{t}_{\min} < \mathbf{t} < \mathbf{t}_{\max}$

We aim at achieving $C^2$ class cable tensions. To this end, we should en- sure that the optimization problem (defined below in Eq. (8)) generates smooth and continuous transitions in the cable tensions profiles. This approach prevents abrupt or rapid changes in tension values by also minimizing the likelihood of activating inequality constraints, which can disrupt the continuity of the solu- tion. To ensure that the minimizer of the optimization problem (8) belongs to the $C^2$ class, we use the method of inverse quadratic penalties [25] to define the cost function. This approach ensures coercivity<fnref n="3" /> and smoothness. We define the cost function $J(\mathbf{t})$ as:

$$J(\mathbf{t}) = \sum_{i=1}^{m} \left( c_1 \frac{1}{(t_i - t_{i, \min})^2} + c_2 \frac{1}{(t_{i, \max} - t_i)^2} \right) \quad (7)$$

where $c_1, c_2 > 0$ are constants.

Thus, the optimization-based TDA can be written as follows:

$$\left\{ \begin{array}{ll} \underset{\mathbf{t}}{\min} & J(\mathbf{t}) \\ \text{s.t.} & W\mathbf{t} = \mathbf{w}_{\text{ref}} \\ & \mathbf{t}_{\min} < \mathbf{t} < \mathbf{t}_{\max} \end{array} \right. \quad (8)$$

## 4.2 Methodology and Mathematical Proof

Based on the aforementioned properties, as well as on Karush-Kuhn-Tucker (KKT) conditions, this subsection demonstrates the optimality and regularity of the solution.

<footnote n="3">³ A function $f: \mathbb{R}^n \to \mathbb{R}$ is said to be *coercive* if $f(\mathbf{x}) \to \infty$ whenever $\|\mathbf{x}\| \to \infty$.</footnote>
**Motivation for Using the Inverse Quadratic Penalty Method:** The inverse quadratic penalty method is designed to ensure both coercivity and stability<fnref n="4" /> by enforcing strict adherence to the inequality constraints $t_{min} < t < t_{max}$. The penalty terms cause the cost function $J(t)$ to grow sharply as $t_i$ approaches its limits $t_{i,min}$ or $t_{i,max}$, driving $J(t)$ to infinity when $t$ nears $t_{min}$ or $t_{max}$. This behavior effectively prevents the cable tensions from reaching its exact limits, keeping the solution well within the feasible region [25,26].

**Regularity conditions:** To ensure necessary conditions for optimality and continuous differentiability, we make the following regularity conditions:

(i) The reference wrench $\mathbf{w}_{\text{ref}}$ is a continuous and twice differentiable $C^2$ function. It is given by $\mathbf{w}_{\text{ref}} = \mathbf{M}\ddot{\mathbf{x}} + \mathbf{C}(\mathbf{x}, \dot{\mathbf{x}})\dot{\mathbf{x}} + \mathbf{G} + \mathbf{f}_{\text{ext}}$.

(ii) The wrench matrix $\mathbf{W}$ has full rank, ensuring that the equality constraint $\mathbf{W}\mathbf{t} = \mathbf{w}_{\text{ref}}$ is regular.

**Proof of the regularity:** The vector $\mathbf{w}_{\text{ref}}$ represents the wrench that must be applied to the MP to make it follow the desired trajectory. To ensure that $\mathbf{w}_{\text{ref}}$ is sufficiently smooth and belongs to the $C^2$ class, it is essential that the acceleration $\ddot{\mathbf{x}}$ is also of class $C^2$.

To achieve $C^2$ class continuity in acceleration, we employ trajectory generation using a 9th-order polynomial, which allows us to define a smooth curve linking two poses while satisfying boundary conditions on velocity, acceleration, jerk, and jounce. This approach, illustrated in Figure 4, ensures that the trajectory is sufficiently smooth across all derivatives up to jounce. The figure illustrates an example of a smooth trajectory, highlighting the continuity of motion achieved with this polynomial formulation.

Ensuring the trajectory remains within the WFW [8,24] of the CDPR, guarantees that the wrench matrix $\mathbf{W}$ is always full rank. Hence, regularity conditions hold.

**Existence of a Minimizer:** The existence of a minimizer is guaranteed by the coercivity and lower semi-continuity of $J(t)$. Coercivity is ensured by the inverse quadratic penalty terms, which grow unbounded as $t_i \to t_{i,min}$ or $t_i \to t_{i,max}$. The lower semi-continuity of $J(t)$ follows from the continuity of each of its components.

**Differentiability of the Cost Function:** The cost function $J(t)$ is of class $C^2$ because the quadratic penalty terms $\frac{1}{(t_i - t_{i,min})^2}$ and $\frac{1}{(t_{i,max} - t_i)^2}$ are differentiable within the feasible region and possess well-defined first and second derivatives.

<footnote n="4">⁴ In this context, *stability* refers to numerical stability, ensuring that the optimization algorithm avoids ill-conditioning or oscillations as cable tensions approach their limits.</footnote>
<figure><img src="image_6.png" /><figcaption>Fig. 4: Example of a smooth continuous trajectory generated using a 9th-order polynomial.</figcaption></figure>

**KKT Conditions and Optimality:** KKT conditions hold because the cost function is smooth $C^2$ class and coercive, the equality constraint $Wt = w_{ref}$ is linear, and the inequality constraints $t_{min} < t < t_{max}$ are simple box constraints [25,26].

**Second Order Sufficient Conditions:** The second order sufficient conditions for optimality require the Hessian $\nabla_t^2 \mathcal{L}(t, \lambda, \mu, \nu)$ to be positive definite on the subspace defined by the active constraints at the minimizer. Since $J(t)$ is $C^2$, coercive, and $W$ has full rank, the Hessian is positive definite, ensuring a unique minimizer that is $C^2$. The Hessian $\nabla_t^2 \mathcal{L}(t, \lambda, \mu, \nu) = \nabla_t^2 J(t)$ is a diagonal matrix written as:

$$ \nabla_t^2 J(t) = \text{diag} \left( \frac{6c_1}{(t_1 - t_{1,min})^4} + \frac{6c_2}{(t_{1,max} - t_1)^4}, \dots, \frac{6c_1}{(t_m - t_{m,min})^4} + \frac{6c_2}{(t_{m,max} - t_m)^4} \right) \quad (9) $$

each diagonal entry $\frac{\partial^2 J}{\partial t_i^2} = \frac{6c_1}{(t_i - t_{i,min})^4} + \frac{6c_2}{(t_{i,max} - t_i)^4}$ is positive within the feasible region $t_{i,min} < t_i < t_{i,max}$ because $c_1, c_2 > 0$. Therefore, $\nabla_t^2 J(t)$ is positive definite within this region, confirming that $J(t)$ is strictly convex and the cost function has a unique minimizer.

## 4.3 Results and Discussions

Simulations were carried out in MATLAB/Simulink, where the HRPCable, a 6-DoF fully-constrained CDPR with eight cables described in [27], was considered. The robot base dimensions are 3 m × 4 m × 10 m. The MP is modeled as a cube of side length 0.5m with a mass of $m_p = 25$ kg. Cable tensions were simulated
considering the linear elastic model of Eq. (3) and system dynamics to assess
the performance of the proposed TDA.

The tests were performed for trajectories from an initial pose **x**<sub>i</sub> to a final
pose **x**<sub>f</sub>, staying entirely within the wrench-feasible workspace of the CDPR.
The test task is a trajectory between the static equilibrium pose **x**<sub>i</sub> to the static
equilibrium pose **x**<sub>f</sub>, performed over 10 seconds between t = 2 s and t = 12 s.
From t = 0 s to t = 2 s and from t = 12 s to t = 14 s, the CDPR remains in
static equilibrium. To define the trajectory, a 9th-order polynomial was used,
as shown in Figure 4, ensuring C<sup>2</sup> continuity in the desired wrench **w**<sub>ref</sub>. The
cable tensions were constrained to remain within predefined limits of t<sub>min</sub> =
10 N and t<sub>max</sub> = 4000 N. These limits ensure the cables are always in tension.
The parameters c<sub>1</sub> and c<sub>2</sub> directly influence the cost function by controlling
penalties for approaching tension limits. Specifically, c<sub>1</sub> increases the penalty as
tensions near the lower limit t<sub>min</sub>, ensuring they stay well within the feasible
range. Similarly, c<sub>2</sub> amplifies the penalty as tensions approach the upper limit
t<sub>max</sub>. Together, these parameters shape the cost function profile to maintain C<sup>2</sup>
continuity in cable tensions, even under strict constraints. In our tests, the values
c<sub>1</sub> = c<sub>2</sub> = 0.1 were chosen based on empirical adjustments through tests. On
the one hand, high values of c<sub>1</sub>, c<sub>2</sub> overly penalize near-limit tensions since the
shape of the cost function becomes steeper. On the other hand, low values of c<sub>1</sub>, c<sub>2</sub>
do not sufficiently enforce smoothness. The chosen values c<sub>1</sub> = c<sub>2</sub> = 0.1 have
thus been selected as a balance between computational efficiency and tension
regularity.

<figure><img src="image_7.png" /><figcaption>Fig. 5: Cable tensions and their second derivatives with tension limits (t<sub>min</sub> = 10 N, t<sub>max</sub> = 4000 N).</figcaption></figure>
The results of the TDA are depicted in Figures 5 where the tension profile of each cable, along with their second derivatives, demonstrate $C^2$ continuity while adhering to the tension limits. To further evaluate of the proposed TDA, the tension limits were reduced to $t_{min} = 10$ N and $t_{max} = 2500$ N, as shown in Figure 6. Even under these stricter constraints, the algorithm successfully maintained $C^2$ continuity of the cable tensions, highlighting the effectiveness of the penalty-based cost function. The optimization problem was solved at each simulation step using an interior-point solver in MATLAB/Simulink on a standard workstation (Intel Core i7, 1.9 GHz, 32 GB RAM). The measured computation time per iteration showed an average execution time of 11.6 ms and a maximum execution time of 46.2 ms over the entire simulation.

<figure><img src="image_8.png" /><figcaption>Fig. 6: Cable tensions and their second derivatives with reduced tension limits ($t_{min} = 10$ N, $t_{max} = 2500$ N).</figcaption></figure>

# 5 Conclusion

This paper addressed the challenge of achieving $C^2$-continuous tension profiles for torque-based control of CDPRs with consideration of cable elasticity. We proposed a TDA formulated as an optimization problem, using a 9th-degree polynomial for trajectory generation to ensure smooth tension profiles while respecting cable tensions limits. Through simulations, the method demonstrated its ability to generate $C^2$-continuous tensions even under strict tension constraints. The findings underline the importance of $C^2$-continuity in tension profiles for motor torque control of CDPRs used in high-accuracy applications, such as additive manufacturing. Future work will focus on experimental validation to confirm the practical effectiveness of the proposed approach. Additionally, integrating
the algorithm into real-time control frameworks and exploring its application to
more complex trajectories will further expand its potential use cases.

**Acknowledgments.** This research was supported by ANR CABTIVE, project num-
ber ANR-22-CE10-0008.

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