# Modeling, analysis and design of an extensible planar parallel tendon actuated continuum robot

A Ouyoucef, Q Peyron, G Zheng, Frédéric Boyer

► To cite this version:

A Ouyoucef, Q Peyron, G Zheng, Frédéric Boyer. Modeling, analysis and design of an extensible planar parallel tendon actuated continuum robot. RoboSoft 2025: IEEE 8th International Conference on Soft Robotics, IEEE, Apr 2025, Lausanne, Switzerland. 10.1109/RoboSoft63089.2025.11020851. hal-05052145

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Submitted on 6 Jun 2025

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# Modeling, analysis and design of an extensible planar parallel tendon actuated continuum robot

A. Ouyoucef<fnref n="1" />, Q. Peyron<fnref n="1" />, G. Zheng<fnref n="1" /> and F. Boyer<fnref n="2" />

*Abstract*— In this paper, we introduce a novel design for a compressible/extensible planar continuum parallel manipulator, characterized by fixed joints and tendon-based actuation with parallel routing. The goal of this design is to generate a planar workspace without adding any rigid translational actuators by considering compressible legs. To optimize the robot's design, its static behavior is first modeled using a Jacobian-based geometric variable strain approach, providing a foundation for performance analysis. We then explore key performance metrics, including workspace, stability, and payload capacity, in relation to critical design parameters: the distance between the robot's legs and the bending-to-compression stiffness ratio. While the leg distance is a well-studied factor in parallel architectures, the influence of the stiffness ratio, despite its significant impact on performance, has received limited attention. Our work investigates the effects of these parameters, culminating in the development of a new modular prototype that leverages these insights to obtain optimal robot performances.

## I. INTRODUCTION

Parallel robots are closed-loop mechanical structures with better performance regarding accuracy, stiffness, and payload capacity [1] than serial ones. In the field of continuum robotics, this architecture is garnering growing interest in applications that require safe interactions with humans [2], including minimally invasive surgery, where precision and compliance are crucial; haptic feedback using wearable devices; and industrial tasks, such as pick-and-place tasks for delicate workpieces, which demand load-bearing capabilities, accuracy, and compliance.

The recent popularity of continuum parallel robots (CPRs) has induced the emergence of a multitude of new designs, differing in several aspects such as the degrees of freedom of their end-effectors, which can be either planar [3] or spatial [4]. In terms of actuation methods, we can find tendon actuation [5], direct translation [6], direct rotation [3], or pneumatic actuation [7]. Finally, concerning the joints between the legs, the base, and the platform, one can use fixed joints [7], revolute joints [8], prismatic joints [4], or spherical joints [5].

Designing a CPR optimally is a challenging task, as it requires a thorough performance analysis. This difficulty stems from two main factors. First, it involves a multi-criteria analysis. In the context of manipulation tasks, three key performance criteria must be considered: the workspace [9], the presence of singular and unstable configurations [10], and the payload capacity [11]. Second, these performances depend on a multitude of parameters, including not only the actuation strategy and the type of joints, but also the robot's geometry and mechanical properties. Interestingly, this dependence on mechanical properties is particularly important for achieving different kinematic behaviors. For instance, in [12], the authors investigate the effect of meta-materials on the kinematics of a CPR actuated by the direct rotation of the legs. In [13], a hybrid-driven continuum robot with a dual-plane backbone structure has been designed, enabling independent stiffness regulation and positioning. [14] focuses on addressing the instability problem of concentric-tube robots by leveraging the anisotropic characteristics of the tube's stiffness. In this work, the stability of the robot is shown to depend on the torsion-to-bending stiffness ratio.

In our work, we also exploit this dependency on mechanical parameters to design a CPR with tendon actuation and fixed joints. First, tendon actuation with parallel routing is one of the most commonly used methods for actuating continuum robots due to its many advantages [15]. This actuation strategy is well established within the continuum robotics community, with extensive research on its modeling [16] [17] and widely available technological solutions for integrating it into prototypes. One of its key properties is its ability to enable two modes of deformation: bending and compression. Several serial continuum robots feature designs that allow for compression and extension of the backbone [18] [19]. However, the design complexity of achieving high compression ratios for serial robots, in addition to the inherent complexity of parallel architectures, make designing a CPR with extensible/compressible legs more challenging. In the literature, many designs compensate for the absence of compression by adding additional actuators. For instance, in [5], a translating base is used to expand the robot's workspace. Concerning the fixed joints, they demonstrate interesting properties [7]. Firstly, they lead to simple and cost-effective structures, as they are easier and cheaper to manufacture and maintain due to the absence of moving parts that would require lubrication or regular adjustment. Secondly, they provide greater structural stiffness due to

<footnote>*This work was supported by French ANR COSSEROOTS under Grant ANR-20-CE33-0001, by Centrale Lille under Grant BQR22, and by Region Hauts-de-France. We also deeply thank Dr. Sebastien Briot for his guidance and advice in modeling the continuum parallel robot.</footnote>

<footnote n="1">¹A. Ouyoucef, Q. Peyron, and G. Zheng are with the CRISTAL Laboratory, Inria and Centrale Lille, France. azouaou.ouyoucef@inria.fr</footnote>

<footnote n="2">²F. Boyer is with LS2N Lab, Institut Mines Telecom Atlantique, 44307 Nantes, France frederic.boyer@imt-atlantique.fr</footnote>
increased constraints. Thirdly, they enable precise positioning by eliminating the mechanical play typically found in moving joints. On the other hand, fixed joints add constraints between the leg's deformations that are likely to increase the risk of singularity and buckling.

This paper includes several contributions, which can be summarized as follows:

* Introduction of a new design of compressible/extensible planar parallel tendon-actuated continuum robot, with legs clamped to the platform, along with its static model based on the Cosserat rod theory.

* Model-based analysis of its static performances in terms of workspace, stability, and payload capacity, and in relation to key geometrical and mechanical design parameters: the distance between the legs and the bending-to-compression stiffness ratio of the legs. For the first time, the critical importance of this stiffness ratio for parallel tendon-actuated continuum robots is highlighted.

* Experimental validation of the modeling and analysis results on a modular prototype.

## II. DESIGN OVERVIEW

We propose the structure shown in Fig. 1, which features a CPR, based on common tendon-actuated continuum robots (TACR). The design consists of two parallel continuum legs, clamped to a rigid platform and positioned at a distance *d* apart. These legs are designed with a simple rectangular cross-section, making them incompressible. Each leg is attached to compression springs at the other end and can slide along the x-axis, thanks to prismatic joints, enabling the compression and extension of the robot. For actuation, two tendons are placed on the outer sides of the legs, routed in parallel through spacer disks. The end-effector is located at the center of mass of the rigid platform, where the gravitational force from the manipulated object can be applied. Finally, to control the position of the end-effector, the tension in the tendons is used as the system's input.

## III. MODELING AND ANALYSIS

### A. Compression device

The behavior of the legs under tendon actuation, including their compression, can be modeled as a Cosserat rod [20]. This equivalence between a compressible rod and a sliding

<figure><img src="image_1.png" /><figcaption>Fig. 1. Design of the compressible/extensible planar parallel tendon actuated continuum robot.</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>Fig. 2. Equivalence between a sliding rigid leg attached to a spring in his end and a compressing clamped Cosserat rod</figcaption></figure>

incompressible leg attached to a spring at its end is valid as long as we consider a linear constitutive law in the Cosserat rod model and neglect the friction during the sliding operation. Based on Fig. 2, we can find the relation between the spring constant $K_s$ and the compression stiffness of the Cosserat rod $H_c$. We apply the same axial force $F_+$ at the distal section of the legs. Based on the action/reaction principle, this force $F_+$ in the case of the sliding leg, is equivalent to the force applied by the spring on the leg $F_s$:

$$F_+ = F_s = K_s dx a \quad (1)$$

where $dx$ is the displacement of the incompressible leg, $a$ is the unit vector along the X-axis of the inertial frame. Based on the boundary condition and the linear constitutive law, the tip force $F_+$ is equal to the elastic force at the tip. This is given by:

$$F_+ = \Lambda_{\text{elas}}(L) = H_c \frac{dx}{L} a \quad (2)$$

where $\Lambda_{\text{elas}}$ is the elastic internal force. The relationship between the spring constant and the compression stiffness of the beam used to model the leg is then given by:

$$H_c = K_s L. \quad (3)$$

### B. Geometrico-static model

To obtain the geometrico-static model of this CPR, one needs, first, to model the legs and the rigid platform separately. Note that we only model the leg's deformation along the parallel continuum manipulator and not the actuation section, as shown in Fig. 1.

The rigid platform is, first, modeled as a rigid body subjected to gravity. Here, we consider the Cartesian coordinates and orientation of the platform mass center $C_m$ in the global frame, $g_p = [\theta_p \ x_p \ y_p]^T$, as the generalized variable. The continuum legs are modeled as Cosserat rods using the Jacobian-based Geometric Variable Strain (GVS) approach, which proved valuable for modeling continuum robots [20]. Firstly, it takes the same Lagrangian form as the well-known rigid manipulator model. Secondly, the symbolic formulation of its matrices allows the analytical computation of the needed gradients for analyzing and solving the model. Finally, it allows a polynomial approximation of the strains, which makes a trade-off possible between the precision and the computation time by choosing the order of the
polynomial. The generalized variables in this model are the strain coordinates of the $i^{th}$ leg $q_{l,i}$. To model this robot, several assumptions have been considered. First, the tendons are completely constrained along the legs. Then, the friction between the tendons and the legs and between the legs and the base has been neglected. Finally, the legs can deform independently and cross each other without collisions.

### 1) Leg's model:

In this paper, we directly introduce the obtained Jacobian based GVS model in the space of strain coordinates $q_{l,i}$. For more details about the passage from the strong form to the weak form, please refer to [20]. The static model is given by

$$Q_{g,i}(q_{l,i}) + K_i q_{l,i} - Q_{act,i}(q_{l,i}) = 0 \quad (4)$$

where $Q_{g,i}(q_{l,i})$ is the vector of generalized gravity force, $Q_{act,i}(q_{l,i})$ is the vector of generalized actuation force and $K_i q_{l,i}$ is the vector of generalized elastic force. These vectors are given by

$$Q_{g,i}(q_{l,i}) = - \int_{0}^{L} J_{i}^{T} \bar{\mathcal{F}}_{e,i} dX \quad (5)$$

$$Q_{act,i}(q_{l,i}) = - \int_{0}^{L} \Phi^{T} \Lambda_{act,i} dX = H_{i} \tau_{i} \quad (6)$$

$$K_i = \int_0^L \Phi^T B^T \mathcal{H}_i B \Phi dX \quad (7)$$

$\bar{\mathcal{F}}_{e,i}$ is the external wrench, $J_i$ is the Jacobian matrix given by the analytical solving of the kinematic model, $\Phi$ is the matrix of basis functions or "strain modes". In this work, the orthogonal polynomial basis of Legendre has been chosen as a strain base. Its orthogonality results in a diagonal stiffness matrix after the projection. The strain of each leg is then approximated by a second order polynomial. $\Lambda_{act,i}$ is the contribution of the tendon actuation in the internal force, $H_i$ is the actuation matrix, $\tau_i$ is the cable tension, and $B$ is a matrix used for the selection of the allowed modes of deformation of the rods. In this work, Z-axis bending and X-axis compression are considered, such as $B^T \mathcal{H}_i B = \text{diag}(\mathcal{H}_b, \mathcal{H}_c)$ is the reduced cross-sectional Hooke tensor expressed in the cross-section frames along the rod. This Hooke tensor can be expressed with respect to a bending-to-stiffness ratio, denoted $u$, as $B^T \mathcal{H}_i B = \text{diag}(\mathcal{H}_{bi}, \frac{\mathcal{H}_{bi}}{u})$.

### 2) Constraints:

The first set of constraints that must be considered is the one due to the fixed joints between the legs and the rigid platform. Its algebraic equation is given by:

$$\Psi = A \begin{bmatrix} (R_p R_{c,1} R_1^T(L) - R_1(L) R_{c,1}^T R_p^T)^\mathrm{T} \\ r_p - r_1(L) - R_p \begin{pmatrix} \frac{l}{2} \\ \frac{d}{2} \\ 0 \end{pmatrix} \\ (R_p R_{c,2} R_2^T(L) - R_2(L) R_{c,2}^T R_p^T)^\mathrm{T} \\ r_p - r_2(L) - R_p \begin{pmatrix} \frac{l}{2} \\ -\frac{d}{2} \\ 0 \end{pmatrix} \end{bmatrix} = 0 \quad (8)$$

$R_p$ is the rotation matrix of the rigid platform with respect to the inertial frame, $r_p = [x_p \ y_p \ z_p]^\mathrm{T}$ is the Cartesian coordinates of $C_m$ in the inertial frame, $R_{c,i}$ is the rotation matrix of the rigid platform with respect to the last body frame of the $i^{th}$ leg, $R_i(L)$ and $r_i(L)$ are, respectively, the rotation matrix and the Cartesian coordinates of the last body frame of the $i^{th}$ leg with respect to the inertial frame, $l$ is the width of the rigid platform, $d$ is the distance between the legs at their base and the platform length and $A$ is a selection matrix that eliminates the constraints out of the plan X-Y.

Another constraint that must be considered is the actuation limit. In the case of tendon actuation, the cable's tension can only be positive or null and its maximum value $\tau_{max}$ depends on the performances of the actuating motor.

$$0 \le \tau \le \tau_{max} \quad (9)$$

### 3) Parallel robot model:

To constrain the legs and the platform models, one can use the Lagrange multipliers. As a result, the geometric-static model is composed of the following two equations for the legs and the platform respectively:

$$Q_{g,i}(q_{l,i}) + K_i q_{l,i} - Q_{act,i}(q_{l,i}) + \nabla_{q_{l,i}}^T \Psi \lambda = 0 \quad (10)$$

$$F_p(g_p) + \nabla_{g_p}^T \Psi \lambda = 0. \quad (11)$$

where $i = 1, 2$, $\nabla$ is the gradient operator, and $F_p(g_p) = [0 \ mg \ 0]^\mathrm{T}$ is the external planar wrench applied to the rigid platform. In this work, we consider only gravity as the external wrench, where $m$ is the combined mass of the platform and the manipulated load, and $g$ is the gravitational acceleration. $\lambda$ is the vector of Lagrange multipliers. The equations (10-11) can be written in a compact form

$$\varepsilon = Q_g(q) + K q_l - Q_{act}(q_l) + \nabla_q^T \Psi \lambda = 0 \quad (12)$$

$$\Psi = 0 \quad (13)$$

where $q_l = [q_{l,1}^T \ q_{l,2}^T]^\mathrm{T}$, $q = [q_l^T \ g_p^T]^\mathrm{T}$, $Q_g(q) = [Q_{g,1}^T(q_{l,1}) \ Q_{g,2}^T(q_{l,2}) \ F_p^T(g_p)]^\mathrm{T}$, $Q_{act}(q) = [(H_1 \tau_1)^T \ (H_2 \tau_2)^T \ 0]^\mathrm{T}$ and $K q_l = [(K_1 q_{l,1})^T \ (K_2 q_{l,2})^T \ 0]^\mathrm{T}$.

### C. Workspace generation

In order to generate the workspace of the robot, several methods exist in the literature. These can be classified into two main categories: forward approaches, where the actuation space is discretized [21], and backward approaches, where the end-effector space is discretized [22]. In this work, a classical backward approach has been used. The workspace plan is first discretized into a regular grid, with each node of this grid corresponding to a possible mass center position of the robot's platform $P_{ref}$. Subsequently, we define and solve, for each node of the regular grid, a constrained optimization problem to find the configuration vector $\mathbf{x} =$
$[q_l^T \ \theta_p \ x_p \ y_p \ \tau^T \ \lambda^T]^T$. The workspace generation problem is defined as:

$$x^* = \arg\min_{x} \| [x_p \ y_p]^T - P_{ref} \| \quad (14)$$

Subject to:

$$Q_g(q) + Kq_l - Q_{act}(q_l, \tau) + \nabla_q^T \Psi \lambda = 0 \quad (15)$$

$$\Psi(q) = 0 \quad (16)$$

### D. Kinemato-static model

The kinemato-static model of the robot can be derived by computing the variation of the equations (12-13) with respect to all the variables $q_l$, $g_p$, $\lambda$ and $\tau$. Thus, the kinemato-static model is given by

$$\Delta\varepsilon = L_{\varepsilon}\Delta q_l + P_{\varepsilon}\Delta g_p + T_{\varepsilon}\Delta\tau + \Lambda_{\varepsilon}\Delta\lambda = 0 \quad (17)$$

$$\Delta\Psi = L_{\Psi}\Delta q_l + P_{\Psi}\Delta g_p = 0 \quad (18)$$

where $L_{\varepsilon} = \frac{\partial\varepsilon}{\partial q_l}$, $P_{\varepsilon} = \frac{\partial\varepsilon}{\partial g_p}$, $T_{\varepsilon} = \frac{\partial\varepsilon}{\partial\tau}$, $\Lambda_{\varepsilon} = \frac{\partial\varepsilon}{\partial\lambda}$, $L_{\Psi} = \frac{\partial\Psi}{\partial q_l}$ and $P_{\Psi} = \frac{\partial\Psi}{\partial g_p}$.

### E. Singularity conditions

Based on the analysis made in [10] [23], in order to deduce the singularity conditions from the kinemato-static model, one should remove the variations of the Lagrange multipliers $\Delta\lambda$ from (17). It can be done by multiplying (17) by the matrix $Z^T$ with $Z$ spanning the nullspace of the matrix $\Lambda_{\varepsilon}^T$, i.e. $\Lambda_{\varepsilon}^T Z = 0$.

$$Z^T(L_{\varepsilon}\Delta q_l + P_{\varepsilon}\Delta q_p + T_{\varepsilon}\Delta\tau + \Lambda_{\varepsilon}\Delta\lambda) = 0 \quad (19)$$

We can group (18) and (19) in a more compact form

$$L\Delta q_l + P\Delta g_p + T\Delta\tau = 0 \quad (20)$$

where $L = \begin{bmatrix} Z^T L_{\varepsilon} \\ L_{\Psi} \end{bmatrix}$, $P = \begin{bmatrix} Z^T P_{\varepsilon} \\ P_{\Psi} \end{bmatrix}$ and $T = \begin{bmatrix} Z^T T_{\varepsilon} \\ 0 \end{bmatrix}$

1) Type 1 singularity: One can write the equation of the variations of $q_l$ and $\tau$ with respect to the variation of $g_p$

$$\begin{bmatrix} \Delta q_l \\ \Delta \tau \end{bmatrix} = - \begin{bmatrix} L & T \end{bmatrix}^{-1} P \Delta g_p \quad (21)$$

If $[L \ T]$ is singular with $L$ and $T$ full rank, it exists variations $\Delta q_l$ and $\Delta \tau$ non null for which the variation $\Delta g_p$ is null. This means that the end-effector loses the ability to move along some desired controlled directions. The analysis of this singularity allows to find a part of the workspace borders. The other part is defined by the actuation limits, as the tendon's tension allows only the compression of the leg and doesn't allow its extension, and its value is limited to a certain maximum.

2) Type 2 singularity: One can write the equation of the variations of $q_l$ and $g_p$ with respect to the variation of $\tau$

$$\begin{bmatrix} \Delta q_l \\ \Delta g_p \end{bmatrix} = - \begin{bmatrix} L & P \end{bmatrix}^{-1} T \Delta \tau \quad (22)$$

If $[L \ P]$ is singular with $L$ and $P$ full rank, it exists variations $\Delta q_l$ and $\Delta g_p$ non null for which the variation $\Delta \tau$ is null. For CPR, this means the robot reaches a stability limit beyond which snapping and buckling occur.

### F. Stability of the model solutions

Based on the analysis made on [10], the solutions of the geometric-static model (12-13) are stable if they locally minimize the potential energy of the system. One can analyse the reduced Hessian matrix $H^r$ of the potential energy given by

$$H^r = Z^T H Z \quad (23)$$

where $H = [L_{\varepsilon} \ P_{\varepsilon}]$ is the hessian matrix of the potential energy. If all the eigenvalues of $H^r$ are positive, the solution of the model is a minimum of the potential energy (stable). If all the eigenvalues of $H^r$ are negative, the solution of the model is a maximum of the potential energy (unstable). Then, if at least one eigenvalue of $H^r$ is negative, the model solution is a saddle point of the potential energy and the configuration is unstable. Finally, if at least one eigenvalue of $H^r$ is zero, it is a type 2 singularity case.

## IV. PERFORMANCE ANALYSIS

For this study, we consider the legs' length to be $L = 200$ mm. Their cross-section is rectangular with 30 mm width and 1 mm thickness and their Young's modulus is equal to $E = 3 \times 10^9$ Pa. The actuating tendons are routed in a parallel path, at a distance 15 mm from the leg. The weight of each leg is approximately 20 g.

The distance $d$ between the legs and their bending to compression stiffness ratio $u = \frac{H_b}{H_c}$ have been identified as the most crucial parameters to analyze due to their effects on the robot performance robot. Analyzing the parameter $d$ is important as it defines how close ($d \ll 1$) or far away ($d \gg 1$) the CPR is from the serial architecture. However, the importance of the stiffness ratio $u$ compared to the legs' bending and compression stiffness is less intuitive. Fig. 3 shows three workspace maps with a color bar representing the smallest eigenvalue of $H^r$ to indicate the stability of the robot. We consider a reference case (A) where $u = u_{ref} = 1.5 \times 10^{-4}$ without unstable and singular configurations. In case (B), the compression and bending stiffness are multiplied by 100 while $u = u_{ref}$. This gives a map similar to the reference case (A), without any unstable and singular configurations. The only difference is the amount of force required to reach a point in the workspace. However, in case (C), where the value of $u$ has been multiplied by 0.1 ($u = 0.1 \times u_{ref}$), the obtained map is different in terms of workspace and the existence of unstable and singular configurations. This means that the workspace and the stability of the robot are impacted by the variation of the bending stiffness and compression stiffness only when the ratio $u$ between them varies.

The results of the influence of these design parameters on the performance of the robot are reported in Fig. 4-5.

### A. Workspace

The first performance criterion to analyze is the workspace area, for both payload-free and payload-bearing cases of a robot suspended parallel to gravity. In the payload-free scenario, the blue area in Fig. 4 demonstrates that reducing the distance between the legs increases the workspace area.
<figure><img src="image_3.png" /><figcaption>Fig. 3. Evolution of the workspace and stability map under the variation of the compression and bending stiffness with a constant stiffness ratio and variable one. (A) A chosen reference case where $u_{ref} = 1.5e-5$, (B) The bending and compression stiffness are multiplied by 100 ($u = u_{ref}$), (C) The compression stiffness is multiplied by 10 ($u = 0.1 \times u_{ref}$)</figcaption></figure>

This supports the observation that a serial robot typically has a larger workspace than a parallel one when no payload is present, as making *d* tend to 0 effectively transforms the parallel architecture into a serial one. However, the influence of the stiffness ratio, *u*, is not as straightforward. For any given distance *d*, there exists a unique value, $u_{opt} = 1.5 \times 10^{-4}$, that maximizes the workspace. When a payload is introduced, Fig. 4 shows that the robot's workspace area decreases as the platform mass increases. In this scenario, the parallel architecture becomes increasingly important, as the optimal distance $d_{opt}$, which maximizes the workspace area, increases with the payload. For instance, for a payload of 100 g, $d_{opt} = 2.5$ cm, and for a payload of 400 g, $d_{opt} = 3$ cm. Nonetheless, the optimal value of the stiffness

ratio, *u*, remains constant.

## B. Instability

The second performance criterion to analyze is the instability zone area of the system, which corresponds to the set of end-effector positions where the configurations become unstable. In the payload-free scenario, Fig. 5 shows that the instability zone is influenced by the values of *u* and *d*. For a fixed leg distance, *d*, instability emerges when *u* exceeds a certain threshold, which varies depending on *d*. When *d* is sufficiently small, instabilities disappear entirely, regardless of the value of *u*. This is because reducing the leg distance drives the parallel architecture closer to a serial one, eliminating the singularities and instabilities typically associated with parallel robots. Conversely, when *d* approaches 6 cm, the threshold value of *u* that triggers instability reaches its maximum. Moreover, Fig. 5 shows that the parameters that optimize the workspace also result in zero instability, meaning these parameters are optimal for both workspace maximization and stability.

In the case where a payload is applied, Fig. 5 shows that increasing the end-effector load reduces the size of the unstable zones compared to the payload-free case. Additionally, we note that the parameters that optimize the workspace also result in zero instability in the payload-bearing cases.

## C. Results interpretation

Based on the analyses made in sections IV-A and IV-B, we generated the workspace and instability zone map for three cases with $d = 4$ cm and different values of *u*, as shown in Fig. 6. In this figure, map (B) results from using the optimal value of *u* obtained from the previous performance analysis, $u = 1.5 \times 10^{-4}$. This case allows a large workspace free of unstable configurations. If we increase the value of *u* by reducing the compression stiffness, as in map (C) where $u = 5 \times 10^{-3}$, the resulting workspace area decreases, as the compression mode becomes more dominant, which reduces the width of the workspace. However, it remains singularity-free. On the other hand, if we decrease *u* by increasing the

<figure><img src="image_4.png" /><figcaption>Fig. 4. Influence of the parameters *u* and *d* on the workspace area of the CPR for several payloads. (0g in blue, 100g in magenta, 200g in red, 300g in black and 400g in green). (A) 3D representation, (B) Projection on (u,Ws) plan, (C) Projection on (d,Ws) plan. Circles represent the curves' maximum.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 5. Influence of the parameters *u* and *d* on the instability zone area of the parallel robot under several Payloads (0g in blue, 100g in magenta, 200g in red, 300g in black and 400g in green)</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 6. (A) Robot's initial configuration, (B) Workspace and stability zones for the design parameters $d = 4\text{cm}$, $u = 1.5 \times 10^{-4}$, (C) Workspace and stability zones for the design parameters $d = 4\text{cm}$, $u = 5 \times 10^{-3}$, (D) Workspace and stability zones for the design parameters $d = 4\text{cm}$, $u = 2.5 \times 10^{-5}$. The color bar indicates the smallest eigen-value of the Hessian matrix of the potential energy. The blue regions are unstable and the red ones are stable. The null values indicate a type 2 singularity.</figcaption></figure>

compression stiffness, as in map (D) where $u = 2.5 \times 10^{-5}$, the bending mode becomes more significant. However, the force required for the robot's compression increases, and the workspace becomes limited by the actuation constraints. The high compression stiffness of the legs, combined with the constraints induced by the fixed joints, leads to buckling.

# V. EXPERIMENTAL VALIDATION

## A. Experimental setup

The experimental setup, as represented in Fig. 7, is composed of legs with rectangular sections to constrain their deformation in a plane, and spaced rigid disks. The legs are 3D-printed using the fused deposit filament process on a large-workspace printer (RAISE3D Pro3, "RAISE3D"), their flat part resting on the building plate. They are fabricated out of PLA, chosen for its wide availability, good bending deformability for small thickness values (1mm here), and high compression stiffness. The standard printing parameters for PLA were used (printing head: 210°C, building plate: 70°C). The legs are also designed to avoid contact between them. The legs are designed to avoid contact between them. For that purpose, one of the legs is divided into two strips, positioned on either side of the opposite leg. Each leg is fixed to a modular actuation unit, comprising a motor, a linear bearing, and springs. This allows the legs to be positioned at a specific distance from each other to adjust this distance parameter $d$. Due to design constraints imposed by the size of the aluminum sections used for the structure (15 mm × 15 mm), and the bearings used for minimizing friction, we have $d_{min} = 40$ mm.

The actuation is carried out by tendons routed in a parallel path through the rigid disks and attached to the ends of the legs. It is driven by the rotation of Maxon EC 60 flat brushless motors, rated at 150 W. Each motor actuates a pulley that winds up the tendon, and, thanks to the direct transmission (no gears), we can control the current and,

consequently, the tension in the tendons. The current control and power electronics are managed by the EPOS4 70/15 driver provided with the motors, with the control gains calibrated using EPOS Studio software. To ensure accurate torque values, the torque constant $K_t$, which relates the current $I$ to torque $T = K_t.I$, was configured based on the motor datasheet $K_t = 52.5\text{mNm/A}$.

The choice of the compression springs is based on the analysis made in Section IV and equation (3). The objective is to choose the spring constant $K_s$ that gives the optimal stiffness ratio. The difficulty of this part is finding commercially available springs with the exact dimensions and properties we need. To obtain satisfying spring properties, we combined four similar steel springs in series. The spring constant of each one is approximately $K_{s1} = 1032.6 \text{ N/m}$. The serial combination of the four springs has a constant of $K_s = \frac{K_{s1}}{4} = 258.15 \text{ N/m}$. Based on (3), the equivalent compression stiffness of the Cosserat rod is $H_c = 52.41 \text{ N}$. If we consider a Young's modulus of $E = 3 \times 10^9 \text{ Pa}$, the obtained stiffness ratio is $u = 1.43 \times 10^{-4}$.

## B. Calibration

Although theoretical values of the leg's mechanical properties can be found, their value can vary in practice. The Young's modulus of 3D-printed PLA parts typically ranges between 2.0 and 3.5 GPa, this variability being due to 3D printing parameters and environmental conditions. Therefore, we calibrated them from experiments using the prototype. The calibration process is based on a series of position measurements of a single marker, located at the mass center of the rigid platform, under varying tendon tensions. These positions are recorded using an OptiTrack system. Calibration is then performed by solving a constrained, nonlinear optimization problem aimed at determining the set of parameters (E,u) that minimizes the total positional errors at the platform's mass center across 10 different experimental trials. The optimization variable is defined as $x = [E \ u \ q_1^T \ \lambda_1^T \ ... \ q_{10}^T \ \lambda_{10}^T]^T$. The optimization problem is formulated as:

$$x^* = \arg \min_x \sum_{k=1}^{10} \|e_k(x)\|^2 \quad (24)$$

Subject to:

$$Q_g(q_k) + K(E, u)q_{lk} - Q_{act}(q_{lk}) + \nabla_{q_k}^T \Psi \lambda_k = 0 \quad (25)$$

$$\Psi(q_k) = 0 \quad (26)$$

where $||e_k(x)|| = ||[x_{p,k} \ y_{p,k}]^T - P_{exp,k}||$ is the Euclidean distance between the model prediction of the platform mass center position and the measurement. $q_k = [q_{lk}^T \ \theta_{p,k} \ x_{p,k} \ y_{p,k}]^T$, $q_{lk} = [q_{l,1k}^T \ q_{l,2k}^T]^T$, and $\lambda_k$ are, respectively, the vector of the legs and platform generalized variables, the vector of the legs generalized variables and the vector of the lagrange multipliers, in the $k^{th}$ configuration. To perform this minimization, we used the active-set algorithm as implemented by MATLAB's fmincon function.
<figure><img src="image_7.png" /><figcaption>Fig. 7. Experimental platform</figcaption></figure>

Solving the optimization problem gives a value of Young's modulus equal to $E = 3.000205601 \times 10^9$ Pa, and the value of the stiffness ratio given by the chosen compression springs is $u = 1.697 \times 10^{-4}$. Based on Fig. 4,

these parameters allow a workspace area close to the one obtained with the optimal value found in simulation.

### C. Experimental and Model Comparison

The calibrated model has been used for the simulation of 29 experiments, including compression and bending tests. In the compression tests, tensions of 2.10 N, 3.15 N, 4.20 N, 5.25 N, 6.30 N, 7.35 N, 8.40 N, 9.45 N, 10.50 N, 11.55 N, and 12.60 N were applied simultaneously to both tendons of the robot. Then, for the bending tests, tensions of 2.10 N, 3.15 N, 4.20 N, 5.25 N, 6.30 N, 7.35 N, 8.40 N, 9.45 N, and 10.50 N were applied individually to each tendon. The average tip error that has been obtained is 8.87 mm, which is 4% of the total length of the non-actuated parallel continuum manipulator. This error may be due to the presence of friction between the legs and the base, as the robot's legs slide rather than compress, and also between the tendons and the rigid discs. Fig. 8 shows a comparison between the positions of the markers with respect to the simulation in six chosen configurations. Specifically, $m_0$ and $m_p$ are, respectively, the markers attached to the base and the center of the mass of the platform. Meanwhile, $m_{1i}$ and $m_{2i}$ denote the markers positioned along the first and second legs, respectively. It is important to note that these latter markers are placed on the rigid discs along the tendon paths and are used solely for visualization purposes. We can see that the prototype's behavior closely aligns with the simulation results. The calculated plateform position is close to the measured one, and the tendon path passes as expected by the intermediate markers along the leg as expected. This demonstrates the accuracy of the geometrico-static model, as well as the robot performances. A large workspace can be achieved without encountering singularities or instabilities. However, when the stiffness ratio is reduced, singular configurations arise, leading to buckling. Fig. 9 presents a comparison between

<figure><img src="image_8.png" /><figcaption>Fig. 8. Comparison between simulation and real data in 6 cases with different values of tendon's tension.</figcaption></figure>

two configurations: in the first, the optimal stiffness ratio is used, while in the second, the ratio is reduced by completely blocking the sliding of the legs. This last result shows the importance of the compression mode in the CPR stability.

## VI. CONCLUSION AND PERSPECTIVES

In this paper, we presented several new contributions toward the design of a novel extensible planar CPR with fixed joints and tendon actuation. We derived the geometrico-static and kinemato-static models based on the GVS Cosserat rod model, which were instrumental in analyzing the robot's performance in relation to key design parameters.

Our analysis revealed the existence of optimal values for the leg distance, *d*, and the bending-to-compression stiffness ratio, *u*, which maximize the workspace while avoiding
<figure><img src="image_9.png" /><figcaption>Fig. 9. Comparison between two configurations in an optimal and non optimal cases. (A) Configuration with a buckled leg in a case of small stiffness ratio in the real robot. (B) Configuration with a buckled leg in a case of small stiffness ratio in the simulation. (C) Configuration without buckling in the case of optimal stiffness ratio in the real robot. (D) Configuration without buckling in the case of optimal stiffness ratio given by simulation</figcaption></figure>

unstable or singular configurations, both with and without payload. Note that this analysis is based on a 2D model but has also been pre-validated using a 3D model to ensure the absence of any potential singularity outside the plane, as it has been observed in [3]. Additionally, we developed and constructed a new prototype capable of reconfiguring these design parameters. Calibration of the model against experimental data, using selected compression springs, yielded a stiffness ratio $u = 1.697 \times 10^{-4}$, closely matching the optimal value from our theoretical analysis. Despite this, the platform's center of mass exhibited a relative positioning error of 4% of the total length.

From the obtained results, we observe that, with an optimal stiffness ratio, the curvatures of the robot's legs are similar. In this case, the parallel robot resembles a serial one, as shown in Fig. 8. Nevertheless, it remains important to maintain a parallel architecture to achieve a better payload capacity while ensuring that the legs are sufficiently slim to deform without breaking.

As future perspectives, further iterations of the prototype will focus on improving the compression system. It is challenging to find springs with the precise geometric and mechanical properties required. An active system, such as a pneumatic one, could be used instead of passive, commercially available springs. This would enable real-time regulation of the stiffness ratio.

Thanks to the modularity of the prototype design and the genericity of the model and performance criteria, this work can also be extended to a 3D case. The transition from 2D to 3D raises several research questions regarding performance analysis. First, due to the existence of more deformation modes, there may be a greater occurrence of singular configurations. Then, the robot's performance may become sensitive to a broader range of design parameters, making the selection of appropriate parameters for analysis

more challenging. Finally, the workspace evaluation will require more computational time. This is due to the high number of possible configurations (spatial workspace), and the dimension of the system that increases with the number of legs and deformation modes.

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