# Stiffness Analysis and Tuning of a Four-Leg Tensegrity
Mechanism

Karol Muñoz, Mathieu Porez, Philippe Wenger

► To cite this version:

Karol Muñoz, Mathieu Porez, Philippe Wenger. Stiffness Analysis and Tuning of a Four-Leg Tensegrity Mechanism. Cablecon 2025 - Seventh International Conference on Cable-Driven Parallel Robots, The Chinese University of Hong Kong, Hong Kong SAR, China; University of Duisburg-Essen, Germany; University of Stuttgart, Germany, Jul 2025, Hong Kong, China. hal-05093495

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# Stiffness Analysis and Tuning of a Four-Leg Tensegrity Mechanism.

Karol Muñoz¹, Mathieu Porez², and Philippe Wenger³

Nantes Université¹,³, École Centrale Nantes¹,³, IMT Atlantique², CNRS³, LS2N, UMR 6004, F-44000 Nantes, France

karol.munoz-salas@ls2n.fr¹

mathieu.porez@imt-atlantique.fr²

philippe.wenger@ls2n.fr³

**Abstract.** This paper presents a stiffness analysis of a four-leg tensegrity mechanism. Tensegrity mechanisms are known for their simple and lightweight nature, making them suitable for a variety of applications, particularly in robotics. The study focuses on the ability of the mechanism to vary its stiffness through antagonistic actuation and maintain stability under specific actuation conditions. A reduced static formulation based on the Newton-Euler equations is used to model the stiffness matrix in generalized coordinates. The stiffness matrix is determined using numerical approximation methods according to specific conditions on the cable tensions. The results highlight the mechanism’s ability to adjust stiffness and ensure stability, providing valuable insights for the design and optimization of tensegrity structures for robotics.

**Keywords:** Tensegrity mechanism · Newton-Euler equations · Stiffness tuning and stability analysis.

## 1 Introduction

Tensegrity mechanisms are characterized by their lightweight, deployable, and reconfigurable nature, as well as their ability to exhibit variable stiffness properties [2]. The equilibrium in these mechanisms is achieved through the balanced interaction of compression bars and tensioned cables, ensuring that the forces and moments that act on the structure are balanced [4, 14]. In addition, their ability to maintain stability under external perturbations, due to their geometric configuration, inherent pre-stressing arrangements and adaptive stiffness, makes them an attractive option in various fields. This is particularly valuable in robotics, where soft compliance, adaptability, and robustness are critical for performance and durability [12, 16].

Taking into account the points mentioned above, the present work focuses on the study of the stiffness of the four-leg tensegrity mechanism, proposed in [10], with a particular emphasis on the manner in which the geometric configuration influences its behavior. To that end, a review of relevant studies on stiffness of
the tensegrity mechanisms will be presented. For example, Ramadoss [11] introduced the HEDRA robot, a pioneering design based on tensegrity that integrates variable stiffness and omnidirectional compliance. This mechanism combines tensioned cables, rigid elements, and bioinspired joints with up to three degrees of freedom (DoF), enabling modulation of directional stiffness that supports contraction, axial torsion, and flexion. The design strategically distributes stress and incorporates stiffness gradients, which have been shown to reduce motion coupling, simplify control strategies, and improve kinematic adaptability in dynamic and unstructured environments. Nan proposed a 2-DOF cable-driven mechanism with adjustable stiffness [7]. Building on similar biological principles, Muralidharan [8] proposed the X-module, which allows the modulation of stiffness through antagonistic forces. Similarly, Chevallereau and Wenger [17] introduced an innovative eccentric pulley design that overcomes the limitations of traditional circular pulleys. With this design, increased antagonistic forces produce increased joint stiffness, mimicking the behavior of biological joints. Compared to previous methods that rely on non-linear springs or elliptical pulleys, this approach simplifies the mechanisms while offering a wider range of modulation of stiffness, with promising applications in collaborative robotics and rehabilitation devices.

In addition, Arsenault [1] analyzed a planar mechanism with two DoFs and developed a geometric stiffness model based on the spatial configuration of the mechanism, highlighting how stiffness indices vary across the workspace. Similarly, John [5] proposed a variable stiffness mechanism inspired by the principles of tensegrity. This mechanism modeled stiffness as a function of cable actuation forces and geometric parameters, demonstrating that the slenderness ratio is a key design variable that influences stiffness behavior. John's study showed that the stiffness and stability of the mechanism could be improved within certain limits by adjusting the actuation forces, emphasizing precise control through geometry-dependent stiffness adjustments.

Inspired by these investigations and given the importance of stiffness in the behavior of tensegrity mechanisms, the present study builds on the results of [10], which introduced an innovative four-leg tensegrity mechanism and analyzed the influence of key geometric parameters on the geometric workspace and the wrench-feasible workspace. Expanding on these findings, the present study examines the effect of the slenderness ratio on the stiffness of the four-leg tensegrity mechanism and its subsequent effect on performance and operational capabilities. A reduced static formulation based on the simplified Newton-Euler equations is used to model the stiffness. By integrating stiffness analysis, this research complements previous studies and aims to improve the efficiency of the mechanism, unleash its full potential, and broaden its applicability in various operational scenarios.
## 2 Mechanism description

The four-leg tensegrity mechanism consists of a fixed platform, a moving platform, three peripheral S-U legs<fnref n="1" />, and a central U-P-U leg (see Fig. 1). We assume that the bodies are rigid and that the joints are also rigid and frictionless. To describe the motions of the mechanism, we define the following two reference frames as shown in Fig. 1a: the fixed frame $\mathcal{F}_0 = (\mathbf{P}_0, \mathbf{s}_0, \mathbf{n}_0, \mathbf{a}_0)$ attached to the center $\mathbf{P}_0$ of the base platform and the mobile frame $\mathcal{F}_3 = (\mathbf{P}_3, \mathbf{s}_3, \mathbf{n}_3, \mathbf{a}_3)$ attached to the center $\mathbf{P}_3$ of the moving platform. In both frames, the vectors $\mathbf{s}$ and $\mathbf{n}$ are parallel to the plane of their respective platforms, while the vector $\mathbf{a}$ is orthogonal to this plane.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Four-leg tensegrity mechanism</figcaption></figure>

The configuration of the mechanism consists of the homogeneous transformation matrix $^0\mathbf{T}_3 \in SE(3)$ describing the position and orientation of the moving frame relative to the fixed one and the vector of joint coordinates $\mathbf{q} \in \mathbb{R}^{11}$ giving the position of the 11 internal joints (for which we consider the mechanism to be a tree-like multibody system with the S joints between the peripheral legs and the base platform cut off). The transformation matrix $^0\mathbf{T}_3$ is defined by:

$$ ^0\mathbf{T}_3 = \text{Trans}(^0\mathbf{P}_3) \text{Rot}(\mathbf{s}_0, \phi) \text{Rot}(\mathbf{n}_1, \theta) \text{Rot}(\mathbf{a}_2, \sigma), $$

<footnote n="1">¹ To simplify the design, the universal joints are replaced by spherical joints during manufacture, as shown in the various CAD representations of the mechanism throughout the document. This change does not affect the overall kinematic operation of the mechanism.</footnote>
and in its expanded form, is given by<fnref n="2" />:

$$ ^0\mathbf{T}_3 = \begin{bmatrix} C_\sigma C_\theta & -C_\theta S_\sigma & S_\theta & x \\ C_\phi S_\sigma + C_\sigma S_\phi S_\theta & C_\phi C_\sigma - S_\phi S_\sigma S_\theta & -C_\theta S_\phi & y \\ S_\phi S_\sigma - C_\phi C_\sigma S_\theta & C_\sigma S_\phi + C_\phi S_\sigma S_\theta & C_\phi C_\theta & z \\ 0 & 0 & 0 & 1 \end{bmatrix} . \quad (1) $$

where $(x, y, z)^T$ are the Cartesian coordinates of $\mathbf{P}_3$ in $\mathcal{F}_0$ and $(\phi, \theta, \sigma)$ are the ‘X-Y-Z’ Euler angles describing the orientation of the moving platform relative to $\mathcal{F}_0$. Note that at rest, $\phi = \theta = 0$ deg and $\sigma = \gamma$, where the angular parameter $\gamma$ is the anchoring angle around $\mathbf{a}_0$ of the central leg in the base platform.

In terms of assembly, the anchor points of the legs are located at $\mathbf{B}_1$, $\mathbf{B}_2$, and $\mathbf{B}_3$ on the base platform, and at $\mathbf{B}_4$, $\mathbf{B}_5$, and $\mathbf{B}_6$ on the moving platform (see Fig. 1b). The triangles formed by these points, namely $\mathbf{B}_1\mathbf{B}_2\mathbf{B}_3$ and $\mathbf{B}_4\mathbf{B}_5\mathbf{B}_6$, are equilateral with a side length of $b$ and are centered at $\mathbf{P}_0$ and $\mathbf{P}_3$, respectively. The three peripheral legs, each of length $L$, are connected $\mathbf{B}_1$ to $\mathbf{B}_6$, $\mathbf{B}_2$ to $\mathbf{B}_4$, and $\mathbf{B}_3$ to $\mathbf{B}_5$ respectively. The central leg anchors $\mathbf{P}_0$ to $\mathbf{P}_3$ and imposes, due to its internal kinematics, a ‘zero-torsion’ condition on the motion of the mobile platform according to the base, in other words:

$$ \mathbf{s}_3^T (\mathbf{P}_3 \times \mathbf{s}_\gamma) = 0, \text{ with } \mathbf{s}_\gamma = \begin{bmatrix} C\gamma & -S\gamma & 0 \\ S\gamma & C\gamma & 0 \\ 0 & 0 & 1 \end{bmatrix} \mathbf{s}_0. \quad (2) $$

In addition, the base platform and the moving platform are connected by cables (not shown in Fig. 1c), which are assumed to be inelastic and anchored at designated points: $\mathbf{C}_1$, $\mathbf{C}_2$, and $\mathbf{C}_3$ on the base platform and $\mathbf{C}_4$, $\mathbf{C}_5$, and $\mathbf{C}_6$ on the moving platform. Each cable connects the following anchor points: $\mathbf{C}_1$ to $\mathbf{C}_4$, $\mathbf{C}_2$ to $\mathbf{C}_5$, and $\mathbf{C}_3$ to $\mathbf{C}_6$ (see Fig. 1c). This figure also illustrates the geometric design parameter $\psi$, which represents an angle that defines the configuration of the cables connecting the two platforms. The reader will find more information in [10].

Finally, according to the configuration and assembly described above, the connections of the peripheral legs to the base platform and those of the base platform to the earth impose 15 kinematic constraints on the mechanism (i.e. 3 per peripheral legs at the spherical joint and 6 for the base platform) which can be expressed simply as follows:<fnref n="3" />

$$ \mathbf{A} \begin{pmatrix} \mathbf{V} \\ \mathbf{\Omega} \\ \dot{\mathbf{q}} \end{pmatrix} = \mathbf{0}_{15}, \quad (3) $$

<footnote n="2">² C and S denote cos and sin, respectively, with subscripts corresponding to the relevant angles</footnote>

<footnote n="3">³ The ‘hat’ notation is $(3 \times 3)$ asymmetric tensor associated to $(3 \times 1)$ mentioned vector. For any vectors $U$ and $V$ of $\mathbb{R}^3$, $\hat{U}$ is defined such that $\hat{U}V = U \times V$.</footnote>
with:

$$ \begin{bmatrix} \dot{\Omega} & \mathbf{V} \\ 0 & 1 \end{bmatrix} = {}^0\mathbf{T}_3^{-1} {}^0\dot{\mathbf{T}}_3, \qquad (4) $$

where the matrix $\mathbf{A} \in \mathbb{R}^{15 \times 17}$ is the kinematic constraint matrix while $\mathbf{V} \in \mathbb{R}^3$ and $\Omega \in \mathbb{R}^3$ are the linear and angular components of the twist of the origin of $\mathcal{F}_3$ (i.e. the moving platform) obtained by time differentiation of Eq. (1). Once we have subtracted the 15 constraints imposed by the legs anchored to the base platform from the 17 variables of the mechanism, i.e., the 6 spatial DoFs of the moving platform and 11 joint variables, only 2 degrees of freedom remain for the mechanism. In the following, we will consider the set $\Theta = (\phi, \theta)^T \in \mathbb{R}^2$ as the generalized coordinates or DoFs of our mechanism. It should be noted that knowing $\Theta$, by applying the trilateration method as detailed in [9], the classic tools of robotics (see [6]) and taking into account Eq. (2), the geometric configuration of the mechanism can be calculated (see [10]).

## 3 Stiffness analysis of the mechanism

As discussed previously, the main objective of this work is to establish and analyze the stiffness matrix $\mathbf{K} \in \mathbb{R}^{2 \times 2}$ according to the generalized coordinates $\Theta$. The stiffness can be defined as follows:

$$ \mathbf{K} = \frac{\delta \Gamma}{\delta \Theta}, \qquad (5) $$

where $\delta \Gamma \in \mathbb{R}^2$ is the variation of an (external) torque $\Gamma$ applied to the moving platform of the mechanism and $\delta \Theta$ is the (angular) deflection produced on $\Theta$. Note that to evaluate $\mathbf{K}$, the applied torque must be expressed in the same space as the deflection, namely, in the space of generalized coordinates, or, in other words, along the axes $\mathbf{s}_0$ and $\mathbf{n}_1$ of angles $\phi$ and $\theta$.

To compute $\mathbf{K}$, we start with the static model of the mechanism in the task space, i.e., in $\mathcal{F}_3$. Then, a reduced form is derived by projecting the static model onto the kernel of allowable motions as developed in [10]. Finally, the reduced static model is reformulated in the generalized coordinate space.

The static equilibrium model of the four-leg tensegrity mechanism in the task space, derived using the Newton-Euler formulation, is expressed in the center of the moving platform $\mathbf{P}_3$ in the frame $\mathcal{F}_3$ as follows [3, 13]:

$$ \left( \begin{matrix} \mathbf{F}_e + \mathbf{F}_g \\ \mathbf{Q}_e \end{matrix} \right) + \left( \begin{matrix} \mathbf{F}_c \\ \tau \end{matrix} \right) + \mathbf{A}^T \mathbf{\lambda} = \mathbf{0}_{17}, \qquad (6) $$

where $\mathbf{F}_e \in \mathbb{R}^6$ (resp. $\mathbf{F}_e$ and $\mathbf{F}_g$) represents cable (resp. external and gravity) wrenches acting on the moving platform, $\tau \in \mathbb{R}^{11}$ is the vector of the internal (actuation) torques, $\mathbf{Q}_e \in \mathbb{R}^{11}$ is the vector of external forces applied on the joints, and $\mathbf{\lambda} \in \mathbb{R}^{15}$ is the vector of the Lagrange multipliers quantifying the internal torques and forces required to satisfy the 15 kinematic constraints of
Eq. (3). Note that due to the passive nature of the joints, $\tau$ and $Q_e$ are zero vectors.

To eliminate $\lambda$ in Eq. (6), Eq. (6) is projected onto the kernel denoted $\mathbf{H}$ of allowed motions, as announced before. The kernel is defined as the null space of $\mathbf{A}$, i.e. $\mathbf{H} = \text{null}(\mathbf{A}) \in \mathbb{R}^{17 \times 2}$. The matrix $\mathbf{H}$ relates the first derivatives of the configuration variables $(\mathbf{V}, \mathbf{\Omega}, \dot{\mathbf{q}})^T$ to two independent velocities chosen for the convenience of the study, that is,

$$ \begin{pmatrix} \mathbf{V} \\ \mathbf{\Omega} \\ \dot{\mathbf{q}} \end{pmatrix} = \mathbf{H} \begin{pmatrix} \Omega_x \\ \Omega_y \\ \end{pmatrix}, \text{ with } \mathbf{H}^T = \begin{bmatrix} H_{11} & H_{21} & H_{31} & 1 & 0 & H_{61} & \dots & H_{171} \\ H_{12} & H_{22} & H_{32} & 0 & 1 & H_{62} & \dots & H_{172} \end{bmatrix}, $$

where $\Omega_x$ and $\Omega_y$ are the first two components of the angular velocity $\mathbf{\Omega}$ around $\mathbf{s}_3$ and $\mathbf{n}_3$. The projection leads to the reduced static model, expressed as:

$$ \mathbf{H}^T \begin{pmatrix} \mathbf{F}_e + \mathbf{F}_g \\ \mathbf{0}_{11} \end{pmatrix} + \mathbf{H}^T \begin{pmatrix} \mathbf{F}_c \\ \mathbf{0}_{11} \end{pmatrix} = \mathbf{0}_2. \qquad (7) $$

Using the cable routing described in Sec. 2, the cable wrench $\mathbf{F}_c$ is expressed simply as a function of the cable tension vector $\mathbf{f} = (f_1, f_2, f_3)^T$:

$$ \mathbf{F}_c = \mathbf{Zf}, \qquad (8) $$

with $\mathbf{Z}$ the actuation matrix:

$$ \mathbf{Z} = \begin{pmatrix} \mathbf{1}_3 \\ \dot{\mathbf{P}}_3 \end{pmatrix} \begin{bmatrix} \frac{(\mathbf{C}_4 - \mathbf{C}_1)^T}{l_1} & \frac{(\mathbf{C}_5 - \mathbf{C}_2)^T}{l_2} & \frac{(\mathbf{C}_6 - \mathbf{C}_3)^T}{l_3} \\ \end{bmatrix}, \qquad (9) $$

where $l_1, l_2$ and $l_3$ are the cable lengths. Substituting Eq. (8) into Eq. (7) yields the following reduced static model in the task space:

$$ \begin{pmatrix} \Gamma_x \\ \Gamma_y \end{pmatrix} = -\mathbf{H}^T \begin{pmatrix} \mathbf{F}_g \\ \mathbf{0}_{11} \end{pmatrix} - \mathbf{H}^T \begin{bmatrix} \mathbf{Z} \\ \mathbf{0}_{11 \times 3} \end{bmatrix} \begin{pmatrix} f_1 \\ f_2 \\ f_3 \end{pmatrix}. \qquad (10) $$

where $\Gamma_x$ and $\Gamma_y$ are the first two torque components of $\mathbf{F}_e$ in $\mathcal{F}_3$, that is, about $\mathbf{s}_3$ and $\mathbf{n}_3$. Since the objective is to determine the torques in the generalized space to derive the stiffness matrix $\mathbf{K}$ of the mechanism, as seen in Eq. (5). From Eq. (4) the following relationship is defined:

$$ \begin{pmatrix} \Omega_x \\ \Omega_y \end{pmatrix} = \mathbf{G} \begin{pmatrix} \dot{\phi} \\ \dot{\theta} \end{pmatrix} \implies \mathbf{G}^T \begin{pmatrix} \Gamma_x \\ \Gamma_y \end{pmatrix} = \begin{pmatrix} \Gamma_\phi \\ \Gamma_\theta \end{pmatrix}, \text{ with } \mathbf{G} = \begin{bmatrix} C_\sigma C_\theta & S_\theta \\ -C_\theta S_\sigma & C_\sigma \end{bmatrix}, \qquad (11) $$

where $\Gamma_\phi$ and $\Gamma_\theta$ are the components of the external torque expressed in the generalized space (that is, about $\mathbf{s}_0$ and $\mathbf{n}_1$) and $\mathbf{G}$ is a transformation matrix from the basis $(\mathbf{s}_0, \mathbf{n}_1)$ of the axis of angles $\phi$ and $\theta$ to the basis $(\mathbf{s}_3, \mathbf{n}_3)$ of the task space. Upon multiplying Eq. (10) by $\mathbf{G}^T$, the following equation is obtained:

$$ \begin{pmatrix} \Gamma_\phi \\ \Gamma_\theta \end{pmatrix} = -\mathbf{\Gamma}_g + \mathbf{Wf}, \qquad (12) $$
with:

$$ \Gamma_g = G^T H^T \begin{pmatrix} F_g \\ 0_{11} \end{pmatrix}, \quad \text{and} \quad W = -G^T H^T \begin{bmatrix} Z \\ 0_{11 \times 3} \end{bmatrix}, $$

where $\Gamma_g$ is the torque due to the mass of the moving platform and $Wf$ corresponds to the torque generated by the cable tensions, both expressed in the generalized coordinate space.

Finally, based on Eqs. (5) and (12) and for constant input cable tensions, the stiffness matrix $K$ in generalized coordinates can be expressed as follows:

$$ K = -\frac{\partial \Gamma_g}{\partial \Theta} + \frac{\partial W}{\partial \Theta} f. \qquad (13) $$

# 4 Results and discussions

We now focus on some relevant numerical results showing the ability of the mechanism: 1) to vary its stiffness by co-actuation, and 2) to be stable under certain actuation conditions. The parameter values of our four-leg tensegrity mechanism used for these simulations are summarized in Tab. 1. Note that these values were optimized in [10] with respect to the workspace size. Before introducing the results, let us focus on the algorithm used to compute and analyze the stiffness matrix $K$.

<table><caption>Table 1: Parameters, symbols, and values for the proposed mechanism.</caption><thead><tr><th>Parameter</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>Distance between anchor points</td><td><i>b</i></td><td>90 mm</td></tr><tr><td>U-S peripheral leg length</td><td><i>L</i></td><td>180 mm</td></tr><tr><td>Anchorage angle of the center leg</td><td><i>γ</i></td><td>-19,5 deg</td></tr><tr><td>Anchorage angle of cables</td><td><i>ψ</i></td><td>-31,7 deg</td></tr><tr><td>Moving platform mass</td><td><i>m</i></td><td>250 g</td></tr><tr><td>Minimal cable tension</td><td><i>f</i><sub>min</sub></td><td>1 N</td></tr><tr><td>Maximal cable tension</td><td><i>f</i><sub>max</sub></td><td>25 N</td></tr><tr><td>Minimal angular travel</td><td><i>φ</i><sub>min</sub>, <i>θ</i><sub>min</sub></td><td>-58,5 deg</td></tr><tr><td>Maximal angular travel</td><td><i>φ</i><sub>max</sub>, <i>θ</i><sub>max</sub></td><td>58,5 deg</td></tr><tr><td>Number of discretizing elements</td><td><i>N</i>, <i>M</i></td><td>200</td></tr></tbody></table>

## 4.1 Computing algorithm

The evaluation of the stiffness matrix $K$, according to Eq. (13), involves the computation of partial derivatives of $\Gamma_g$ and $W$ with respect to the generalized coordinates. Since we do not have explicit analytical expressions for these quantities with respect to $\phi$ and $\theta$, we approximate them numerically using the finite difference approximation in 2 dimensions. To do this, we define a rectangular domain $D = \{(\phi, \theta) \in \mathbb{R}^2 : \phi_{\min} \le \phi \le \phi_{\max} \text{ and } \theta_{\min} \le \theta \le \theta_{\max}\}$
where we discretize $\phi$ and $\theta$ into $N \ge 3$ and $M \ge 3$ elements, respectively, given $NM$ nodes. Let $\{(\alpha, \beta) \in \mathbb{N}^2 : \alpha \le N$ and $\beta \le M\}$ the set of indices of the nodes, the coordinates of any node $(\phi_\alpha, \theta_\beta)$ are defined by $\phi_\alpha = \phi_{\min} + \alpha\Delta\phi$ and $\theta_\beta = \theta_{\min} + \beta\Delta\theta$ with $\Delta\phi = (\phi_{\max} - \phi_{\min})/N$ and $\Delta\theta = (\theta_{\max} - \theta_{\min})/M$ the angular thresholds.

Once the computing domain discretized, at each node, according to the inverse kinematic model introduced in [9, 10], the configuration of the mechanism $\{^0\mathbf{T}_3, \mathbf{q}\}$ is determined. Then, the components of the kernel $\mathbf{H}$ from Eq. (3), the components of the actuation matrix $\mathbf{Z}$ from Eq. (9) and those of the transformation matrix $\mathbf{G}$ from Eq. (11) are calculated. The reduced static model, Eq. (10), is solved to determine the actuation forces $\mathbf{f}$. Their $i^{th}$, $j^{th}$ and $k^{th}$ components, with $\{(i, j, k) \in \{1, 2, 3\}^3 : i \neq j, j \neq k$ and $k \neq i\}$, must satisfy:

1. the minimum actuation condition:

$$f_i = f_{\min}, \text{ and } f_i \le f_j \le f_k, \text{ or} \quad (14)$$

2. the maximum actuation condition:

$$f_k = f_{\max}, \text{ and } f_i \le f_j \le f_k. \quad (15)$$

If $f_i \le f_{\min}$ or $f_k \ge f_{\max}$: the mechanism cannot be operated.

Finally, the components of the stiffness matrix $\mathbf{K}$ are determined according to Eq. (13) and the following approximations of the first partial derivatives of a function $f: \mathcal{D} \to \mathbb{R}$ discretized on $\mathcal{D}$:

$$\frac{\partial f}{\partial \phi}(\phi_{\alpha}, \theta_{\beta}) \approx \frac{f(\phi_{\alpha+1}, \theta_{\beta}) - f(\phi_{\alpha-1}, \theta_{\beta})}{2\Delta\phi}, \text{ and}$$

$$\frac{\partial f}{\partial \theta}(\phi_{\alpha}, \theta_{\beta}) \approx \frac{f(\phi_{\alpha}, \theta_{\beta+1}) - f(\phi_{\alpha}, \theta_{\beta-1})}{2\Delta\theta}.$$

To facilitate the analysis, the vector of the eigenvalues of $\mathbf{K}$, denoted by $\mathbf{\Lambda}_K$ is calculated. Depending on the sign of these eigenvalues, the mechanism will be considered stable if they are positive. A such region of $\mathcal{D}$ where this is the case is called the stability region.

## 4.2 Stiffness, actuation tuning and stability analysis

In accordance with Tab. 1 and the methodology presented in Sec. 4.1, we have developed a simulator in MATLAB [15] to analyze the stiffness, the antagonistic tensions tuning, and the stability of our four-leg tensegrity mechanism.

Fig. 2 shows the evolution of $\mathbf{\Lambda}_K$ according to the generalized coordinates $\phi$ and $\theta$ under the actuation conditions Eq. (14) and Eq. (15). In Fig. 2a and Fig. 2b, we observe that, when we impose the minimum actuation condition on the mechanism, only one eigenvalue is positive over a large part of $\mathcal{D}$ while the
<figure><img src="image_3.png" /><figcaption>Fig. 2: The eigenvalues of **K** for the minimum (a)-(b) and maximum (c)-(d) co-actuation conditions.</figcaption></figure>

other remains negative. Under this actuation condition, the mechanism should be actively controlled to maintain its position and its integrity for safe use. Once we have increased the cable tension until the maximum actuation condition is reached, see Fig. 2c and Fig. 2d, both eigenvalues become positive over a significant part of $D$, i.e. for a range of up 33.4 deg around both axes $\phi$ and $\theta$ (represented in Figs. 2c and 2d by a dotted blue square). This result is interesting because it shows that the mechanism will be stable if $f_i$, the tension of the least taut cable, is greater than a certain value.

To illustrate the possibility of tuning the stiffness of the mechanism using the cable tension $f_i$, we chose four arbitrary configurations, $\Theta_H = (0,0)^\circ$, $\Theta_A = (30,0)^\circ$, $\Theta_B = (0,30)^\circ$, and $\Theta_C = (30,30)^\circ$, $\Theta_B = (0,30)^\circ$, and $\Theta_C = (30,30)^\circ$ over $D$ for which we computed the evolution of **Λ**_K by varying $f_i$ from $f_{min}$ until the tension $f_k$ of the tightest cable reaches $f_{max}$. Fig. 3 shows the results
<figure><img src="image_4.png" /><figcaption>(a) CAD view of the configuration H.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b) $\Lambda_K$ components according to $f_i$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(c) Cable tensions according to $f_i$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(d) CAD view of the configuration A.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(e) $\Lambda_K$ components according to $f_i$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(f) Cable tensions according to $f_i$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(g) CAD view of the configuration B.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(h) $\Lambda_K$ components according to $f_i$.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(i) Cable tensions according to $f_i$.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(j) CAD view of the configuration C.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(k) $\Lambda_K$ components according to $f_i$.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(l) Cable tensions according to $f_i$.</figcaption></figure>

<caption>Fig. 3: Stiffness analysis as a function of $f_i$ in H (a-c), A (d-f), B (g-i) and C (j-l)</caption>
of this study. For the four configurations studied, the eigenvalues are increasing
monotonic functions of $f_i$. The stability region is reached at $f_i = 8.28$ N (see
Fig. 3b), $f_i = 7.06$ N (see Fig. 3e), $f_i = 12.1$ N (see Fig. 3h) and $f_i = 9.24$ N (see
Fig. 3k) for H, A, B and C respectively. The corresponding $i^{th}$ cable is colored
in the CAD views, see Figs 3a-3d-3g-3j, according to the legends of Figs. 3c-3f-
3i-3l, the cable tensions $f_1, f_2$ and $f_3$ are also increasing monotonic functions of
$f_i$.

<figure><img src="image_16.png" /><figcaption>Fig. 4: Minimal antagonistic tension conditions to reach the stability region.</figcaption></figure>

Finally, Fig. 4 shows the minimum antagonistic tension conditions to reach the
stability region. The minimum value is $f_i = 4.5$ N at $\Theta = (33.4, -33.4)$ deg and
the maximum is $f_i = 20.2$ N at $\Theta = (-33.4, 33.4)$ deg. This opposite pattern
over $\mathcal{D}$ is probably due to the specific routing of the cables used in this study.

# 5 Conclusions and future works

This paper investigates the stiffness of a 2-DOF tensegrity mechanism actuated
by 3 cables. The mechanism consists of 3 peripheral legs and a central leg. A
reduced static formulation based on the Newton-Euler equations was used to
model the stiffness matrix. We have shown that the spatial arrangement of the
peripheral legs imparts an interesting property to the mechanism, namely the
ability to tune the stiffness. More interestingly, we have shown that the cable
actuation forces have a positive effect on the stiffness. This property, known as
coactivation in biological joints [8], is of great interest for cobotic applications.
Future work will focus on experimental validation on a prototype. The ultimate
goal of this mechanism is to be used as a joint in a spatial tensegrity manipulator.
**Acknowledgment:** This work has been partially supported by the Agence Nationale de la Recherche under the France 2030 program, reference PEPR O2R - PI1 Reivent (ANR-22-EXOD-0002) and AS1 (ANR-22-EXOD-0005).

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