<figure><img src="image_1.png" /></figure>

HAL
open science

Modeling and analysis of a four-leg tensegrity
mechanism

Karol Muñoz, Mathieu Porez, Philippe Wenger

► To cite this version:

Karol Muñoz, Mathieu Porez, Philippe Wenger. Modeling and analysis of a four-leg tensegrity mechanism. Mechanism and Machine Theory, 2025, 211, pp.106009. 10.1016/j.mechmachtheory.2025.106009. hal-05093485

HAL Id: hal-05093485

https://hal.science/hal-05093485v1

Submitted on 2 Jun 2025

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Modeling and Analysis of a Four-Leg Tensegrity Mechanism

Karol Muñoz¹, Mathieu Porez², and Philippe Wenger³

¹LS2N, Nantes Université, École Centrale Nantes - Petit Port, 1, rue de la Noë, B.P. 92101 44321 Nantes,
Cedex 3, France

³LS2N, CNRS, Nantes Université, École Centrale - Petit Port, 1, rue de la Noë, B.P. 92101 44321 Nantes,
Cedex 3, France

²LS2N, Institut Mines Télécom Atlantique - La Chantrerie, 4, rue Alfred Kastler, B.P. 20722 44307 Nantes,
Cedex 3, France

karol.munoz-salas@ls2n.fr, mathieu.porez@imt-atlantique.fr, philippe.wenger@ls2n.fr

## Abstract

In this paper, we present an innovative four-leg tensegrity mechanism designed for cooperative robotic applications. This lightweight mechanism has two degrees of freedom (2-DoFs) and is co-actuated by three cables. The novelty of the proposed mechanism is based on the use of a central leg to constrain torsion. This central leg eliminates torsional compliance and the need to use an additional motor to control the tension of a fourth central cable. Our study includes a detailed analysis of the kinematics, the Geometric Workspace (GW), the static behavior and the Wrench Feasible Workspace (WFW), together with the optimization of some geometric design parameters. We also present a modeling approach based on a reduced form of the static Newton-Euler equations derived from an analysis of the free-motion space. Through this work, we aim to address key challenges in tensegrity robotics, advancing the field, and opening up new avenues of exploration.

**Keywords:** Tensegrity mechanisms, cable actuation, kinetostatic modeling and analysis.

## Nomenclature

* $^i\mathbf{R}_j$ Rotation matrix that transforms vectors from basis $i$ to basis $j$.

* $^i\mathbf{T}_j$ Homogeneous transformation matrix from frame $i$ to frame $j$.

* $\delta_S$ Spherical joint angle limit.

* $\delta_U$ Universal joint angle limit.

* $\gamma$ Twist angle of the central leg.

* $\mathbf{\Gamma}$ Feasible wrench.

* $\mathbf{A}$ Kinematic constraint matrix.

* $\mathbf{B}_i$ Anchor point of the peripheral legs on the platforms.

* $\mathbf{C}_i$ Anchor point of the cables on the platforms.

* $\mathbf{F}_c$ Cable wrench.

* $\mathbf{F}_e$ External wrench.

* $\mathbf{H}$ Kernel of the admissible motion.

* $\mathbf{P}_0$ Point center of the lower (fixed) platform.

* $\mathbf{P}_3$ Point center of the upper (mobile) platform.

* $\mathbf{Z}$ Actuation matrix.

* $\mathcal{B}$ Body $i$.

* $\mathcal{F}_i$ Frame $i$ centered at $\mathbf{P}_i$ composed of the basis ($\mathbf{s}_i, \mathbf{n}_i, \mathbf{a}_i$).

* $\overline{\rho_w}$ Average of feasible wrench magnitudes.
$\phi, \theta, \sigma$ X-Y-Z Euler angles.

$\psi$ Orientation angle of anchored cables

$\rho_g$ Size of the geometric worspace.

$\rho_w$ Magnitude of the feasible wrench.

$b$ Side length of platforms.

$D$ Diameter of the central leg.

$d$ Diameter of the peripheral legs.

$f_{c_i}$ Tension in cable $i$.

$L$ Length of the peripheral legs.

$q_i$ Joint variables.

# 1 Introduction

Cooperative robotic applications have become an integral part of the modern industrial world. Cobots are capable of performing a wide range of tasks, such as assembling complex systems, operating in extreme environments, and handling heavy objects that need to be processed or stored [1, 2, 3, 4, 5, 6]. However, their versatility also introduces significant risks for humans. Differences in mass and stiffness between cobots and operators can cause severe injuries or damage in the event of a collision [7, 8, 9, 10]. To improve operator safety and optimize the risk-benefit ratio of robotics in industrial applications, it is essential to develop innovative adaptive kinematics that strike a balance between sufficient rigidity for precise operations in confined spaces and the required compliance for safe interactions in shared environments [11]. In addressing these challenges, our attention turns to tensegrity mechanisms, which are increasingly recognized in the field of robotics. The term "tensegrity" merges "tensile" and "integrity," describing a structural principle that employs isolated components under compression, supported by a continuous network of tension, resulting in stable and self-supporting structures [12]. Tensegrity structures are characterized by their lightweight design, where integrity is achieved through the balance between compressive bars and tensile cables. This configuration allows for precise geometric tuning, enhancing rigidity and integrity by modulating tension by co-actuation, a beneficial aspect of their functionality [13, 14, 15, 16, 17]. Such adaptability has made tensegrity an attractive choice in various applications, particularly in robotics, where flexibility and resilience are paramount.

Two key contributions to the field of tensegrity are the work of Snelson [18] and the research of Burkhardt [19]. Snelson introduced the X-shaped tensegrity structure (see Fig. 1a), which illustrates the balance between tension and compression. This configuration features a series of bars arranged in a cross pattern and connected by cables at their ends. The X-shape demonstrates the structural integrity that can be achieved with minimal use of material. On the other hand, Burkhardt reintroduced the concept of the tensegrity prism (see Fig. 1b), a simple yet fundamental spatial structure. His work focused on building this structure using dowels, fishing lines, and cylindrical coordinates. Both Snelson and Burkhardt made significant contributions to the development of tensegrity systems, their innovations being formally recognized through patents [20, 21]. Their works have been applied in civil engineering because of the numerous advantages they offer, including high stiffness-to-mass ratios, deployability, and reliability. These characteristics make tensegrity structures particularly suitable for a variety of applications, such as roofs, domes, and bridges [22, 23].

Inspired by the work of Snelson and Burkhardt, numerous studies have emerged in the field of robotics, but we can identify two main types of research relevant to our investigation: those that use linear actuators for actuation and those that rely on cables. In the type that uses linear actuators, for example, Marshall [24] introduced a parallel platform (a Gough-Stewart platform) device based on tensegrity principles, replacing elements with variable-length legs and integrating springs and connectors. His analysis confirms the configurability of the mobile platform using externally applied key coordinates. Similarly, Mirats and Camps [25] presented a tensegrity-based robot with a three-bar symmetric prismatic configuration anchored to the ground. Despite being underactuated, this robot demonstrates controlled motion within its workspace, highlighting the importance of singular equilibrium matrices for stable configurations and paving the way for future research on control laws and hyperactuated
mechanisms, thus illustrating the potential of tensegrity in the development of deformable and adaptive robots.

On the other hand, among the studies using cables, Arsenault and Gosselin [26] proposed a three-degrees-of-freedom (3-DoF) positional tensegrity mechanism with a modular design, reduced inertia, and a large workspace. This mechanism is treated as an assembly of independent elements based on the Snelson's X-shaped tensegrity system. The limitations of the actuator workspace highlight the challenges faced by mechanisms with three or more modules. Similarly, Muralidharan et al. [27] developed a planar 2-DoF manipulator with two X-joints and investigated two actuation schemes using three and four cables, respectively. Their comprehensive study compares performance metrics, including workspace, velocity, force, stiffness, and under-actuation, while addressing design considerations such as joint limits and spring feasibility, and concludes with an optimization and comparison of 2-X and 2-R manipulators. In a related paper, John et al. [28] performed a kinetostatic analysis of a spatial cable-actuated variable stiffness joint derived from the quaternion joint [29]. Their research focused on the ability of the joint to change its stiffness as a function of cable arrangement and applied tension. Through theoretical modeling and simulations, the study demonstrated how varying stiffness allows the mechanism to adapt its dynamic response to different loading conditions, improving both performance and stability. In the same context, the spatial tensegrity mechanism proposed in Muñoz et al. [30] (see Fig. 1c) introduces a design characterized by a simple architecture consisting of legs, cables, S-joints, and platforms, making it lightweight, simple, and well-suited for integration into cobotic applications. This research performed a static analysis of the mechanism in 3D space, while keeping its internal Degrees of Freedom (DoFs) fixed. The results improved our understanding of how external forces and torques affect the integrity of the mechanism, especially when integrated into a robotic manipulator. Specifically, the mechanism could be shown to rotate in 3D space without any cable slack; however, it was observed that the tension required to maintain this condition could become infinite at certain positions, requiring an improvement of the design. The research proposed constraining torsion to ensure the integrity of the mechanism by adding a fourth cable between the centers of the two platforms. As another example, [31] illustrates that the incorporation of a rigid RRPRR chain as a backbone in a continuous robot significantly increases the torsional stiffness and contributes to the overall integrity of the mechanism.

<figure><img src="image_3.png" /><figcaption>Figure 1: Inspirational works for current research.</figcaption></figure>

Inspired by these investigations, the present research extends the work of [30] by introducing a rigid U-P-U leg between the centers of the two platforms instead of a cable, resulting in a zero-torsion constraint. This innovative design aims to address several issues discussed in [30], which we summarize below. As shown in [19], the height of the upper platform changes with its torsion angle. In other words, without a central cable or a central leg, the upper platform would move up and down under the effect of a torsional motion. However, the upper platform will also experience (small) vertical displacements as it tilts (see Fig. 6a in the end of Sec. 3). Thus, the center cable tension will vary during these tilting motions if its length remains fixed. Therefore, the length of the central cable must be carefully controlled during motion, which requires an additional motor. Replacing the central cable with a U-P-U central leg is a simpler solution that ensures mechanical blocking of the torsion. In short,
the central leg eliminates torsional compliance as well as the need to use and control an additional motor to manage the tension of the central cable.

The main objective of this work is to analyze how geometric design parameters affect the performance of the mechanism and its operational capabilities by studying both the Geometric Workspace (GW) and the Wrench Feasible Workspace (WFW). In addition, this study presents a first-order derivative modeling approach derived from the reduced form of the static Newton-Euler equations. This new approach paves the way for future research by enabling various analyses, such as the calculation of the mechanism dynamics, which provide valuable insights into the control and trajectory planning of the mechanism.

The article is organized as follows: Section 2 describes the proposed four-leg tensegrity mechanism, detailing its configuration, DoFs, geometric design parameters and joint variables. Sec. 3 presents the inverse kinematic model, including the definition of the zero-extension and zero-torsion constraints to determine the spatial positions and joint variables of the mechanism. The analysis of the GW is developed in Section 4. A methodology for determining the feasible wrench (FW) is proposed in Sec. 5, which serves as the basis for the WFW analysis detailed in Sec. 6. Finally, the article presents the results and conclusions, summarizing the findings and suggesting possible directions for future research in Secs. 7 and 8.

## 2 Mechanism description

The kinematic scheme of the proposed mechanism is shown in Fig. 2a. The mechanism consists of two platforms: a fixed lower platform and a mobile upper platform, named $B_0$ and, $B_3$ respectively. As shown in Fig. 2b, we attach at the center $P_0$ of the fixed platform (resp. at the center $P_3$ of the mobile platform) the frame $\mathcal{F}_0 = (\mathbf{P}_0, \mathbf{s}_0, \mathbf{n}_0, \mathbf{a}_0)$ (resp. the frame $\mathcal{F}_3 = (\mathbf{P}_3, \mathbf{s}_3, \mathbf{n}_3, \mathbf{a}_3)$), where the vectors $\mathbf{s}_0$ and $\mathbf{n}_0$ (resp. $\mathbf{s}_3$ and $\mathbf{n}_3$) are parallel to the platform and the vector $\mathbf{a}_0$ (resp. the vector $\mathbf{a}_3$) is orthogonal. According to Fig. 2c, the pose of $B_3$, according to $\mathcal{F}_0$, is described by the following homogeneous transformation matrix:

$$ ^0\mathbf{T}_3 = \mathrm{Trans}(^0\mathbf{P}_3) \mathrm{Rot}(\mathbf{s}_0, \phi) \mathrm{Rot}(\mathbf{n}_1, \theta) \mathrm{Rot}(\mathbf{a}_2, \sigma), \quad (1) $$

which can be expressed as (for now on, $C_\alpha$ denotes the cosine of $\alpha$, and $S_\alpha$ denotes the sine of $\alpha$):

$$ ^0\mathbf{T}_3 = \begin{bmatrix} C_\sigma C_\theta & -C_\theta S_\sigma & S_\theta & x \\ C_\phi S_\sigma + C_\sigma S_\phi S_\theta & C_\phi C_\sigma - S_\phi S_\sigma S_\theta & -C_\theta S_\phi & y \\ S_\phi S_\sigma - C_\phi C_\sigma S_\theta & C_\sigma S_\phi + C_\phi S_\sigma S_\theta & C_\phi C_\theta & z \\ 0 & 0 & 0 & 1 \end{bmatrix}, \quad (2) $$

where the coordinates $(x, y, z)^T$ are those of $\mathbf{P}_3$ expressed in $\mathcal{F}_0$ and $\phi, \theta$ and $\sigma$ are the X-Y-Z Euler angles used here to describe the orientation of the mobile platform<fnref n="1" />.

Moreover, as shown in Fig. 2b, we define a set of anchor points on the platforms: $B_1, B_2$ and $B_3$ on $B_0$ and $B_4, B_5$ and $B_6$ on $B_3$. The triangles $B_1B_2B_3$ and $B_4B_5B_6$ (according to the trigonometric orientation of the space) are equilateral with a side length of $b$, centered on $P_0$ and $P_3$ respectively. Furthermore, $B_1B_2$ and $B_4B_5$ are parallel to $\mathbf{s}_0$ and, $\mathbf{s}_3$ respectively.

As shown in Fig. 2a, the two platforms $B_0$ and $B_3$ are connected by four legs: three S-U peripheral legs denoted by $B_5, B_7$ and $B_9$ respectively, each of equal length $L$, connecting $B_1$ to $B_6, B_2$ to $B_4$, and $B_3$ to $B_5$, while a central U-P-U leg, composed of the bodies $B_{11}, B_{12}$ and $B_{14}$, anchors $P_0$ to $P_3$. Note that the U-P-U leg is rigidly attached to the lower platform at an angle $\gamma$ (see Fig. 2d). The bodies $B_4, B_6, B_8$ and $B_{13}$ (not shown in Fig. 2a) are the internal cross shafts of the U-joints of the mechanism. Moreover, $B_1$ and $B_2$ (also not shown) correspond to virtual bodies to describe the configuration in space, especially the orientation, of the body $B_3$ according to the frame $\mathcal{F}_0$. Finally, three cables complete the mechanism, more details on their role and configuration can be found in Sec. 5.4. To describe the internal kinematic relationships between the bodies, we use the modified Denavit-Hartenberg (DH) convention [33]. According to the joint variables $\mathbf{q} = (q_4, \dots, q_{14})^T$ described

<footnote n="1">¹We could have used the tilt and torsion angles [32] for such a zero-torsion mechanism, as we did in our previous work [30]. The X-Y-Z Euler angles were found to be more suitable for future stiffness analyses.</footnote>
<figure><img src="image_4.png" /><figcaption>(a) Kinematic scheme.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(c) X-Y-Z Euler parametrization.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(b) Geometrical design parameters (home configuration).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(d) Top view of the mechanism platform (home configuration).</figcaption></figure>

<caption>Figure 2: Schematic representation of the four-leg tensegrity mechanism and its orientation parametrization.</caption>

in Fig. 3 and the parameters from Tab. 1, for {$i \in \mathbb{N}: 4 \le i \le 14$}, the homogeneous transformation $^{a_i}\mathbf{T}_i$ of the body $\mathcal{B}_i$ with respect to $\mathcal{B}_{a_i}$ is expressed as follows:

$$^{a_i}\mathbf{T}_i = \mathrm{Rot}(Z, \beta_i) \mathrm{Rot}(X, \alpha_i) \mathrm{Trans}(X, d_i) \mathrm{Rot}(Z, \eta_i) \mathrm{Trans}(X, r_i)^2, \quad (3)$$

where $a_i$ is the index of the previous link of $\mathcal{B}_i$ in the kinematic chain while $\beta_i$, $\alpha_i$, $d_i$, $\eta_i$ and $r_i$ are the modified DH parameters [33].

As far as the mobility of the mechanism is concerned, a closed-loop structure is described geometrically by an equivalent tree obtained by cutting each closed-loop joint. Specifically in this case, cutting the three S joints of the mechanism at the anchor points of the peripheral legs (highlighted in red in

<footnote n="2">²X, Y, and Z are the principal axes of the transformation system [33].</footnote>
<figure><img src="image_8.png" /><figcaption>(a) Leg connecting anchor points $B_1$ and $B_6$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(b) Leg connecting anchor points $B_2$ and $B_4$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(c) Leg connecting anchor points $B_3$ and $B_5$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(d) Joint variables $q_{10}$ to $q_{14}$.</figcaption></figure>

<caption>Figure 3: Joints variables of the four legs from $q_4$ to $q_{14}$.</caption>

<table><caption>Table 1: DH parameters for the four-leg tensegrity mechanism.</caption><thead><tr><th>$i$</th><th>$a_i$</th><th>$\beta_i$</th><th>$\alpha_i$</th><th>$d_i$</th><th>$\eta_i$</th><th>$r_i$</th></tr></thead><tbody><tr><td>4</td><td>3</td><td>$-\pi/2$</td><td>$\pi/2$</td><td>$-b\sqrt{3}/3$</td><td>$-\pi/2 + q_4$</td><td>0</td></tr><tr><td>5</td><td>4</td><td>0</td><td>$-\pi/2$</td><td>0</td><td>$q_5$</td><td>0</td></tr><tr><td>6</td><td>3</td><td>$5\pi/6$</td><td>$\pi/2$</td><td>$-b\sqrt{3}/3$</td><td>$-\pi/2 + q_6$</td><td>0</td></tr><tr><td>7</td><td>6</td><td>0</td><td>$-\pi/2$</td><td>0</td><td>$q_7$</td><td>0</td></tr><tr><td>8</td><td>3</td><td>$\pi/6$</td><td>$\pi/2$</td><td>$-b\sqrt{3}/3$</td><td>$-\pi/2 + q_8$</td><td>0</td></tr><tr><td>9</td><td>8</td><td>0</td><td>$-\pi/2$</td><td>0</td><td>$q_9$</td><td>0</td></tr><tr><td>10</td><td>3</td><td>$-\pi/2$</td><td>$\pi/2$</td><td>0</td><td>$\pi/2 + q_{10}$</td><td>0</td></tr><tr><td>11</td><td>10</td><td>0</td><td>$-\pi/2$</td><td>0</td><td>$q_{11}$</td><td>0</td></tr><tr><td>12</td><td>11</td><td>0</td><td>0</td><td>0</td><td>$\pi$</td><td>$q_{12}$</td></tr><tr><td>13</td><td>12</td><td>0</td><td>0</td><td>0</td><td>$\pi + q_{13}$</td><td>0</td></tr><tr><td>14</td><td>13</td><td>0</td><td>$\pi/2$</td><td>0</td><td>$q_{14}$</td><td>0</td></tr></tbody></table>

Fig. 2a) results in 17 DoFs: the mobile platform contributes six DoFs, while the upper joints of the three S-U legs add another six DoFs and the U-P-U leg contributes another five DoFs through its joints. From a constraint perspective, six kinematic constraints are imposed by the fixed platform anchored to the ground, while a further nine constraints are imposed by the three S-joints attached to the fixed platform, giving a total of 15 kinematic constraints which reduce the number of DoFs in the
mechanism to two. In the following, these two DoFs are represented by $\phi$ and $\theta$. In addition to this analysis, we considered applying the Tchebyshev Grübler-Kutzbach mobility formula, which provides a more formal approach to computing the DoFs of the mechanism. Using this formula, the mobility $D$ is given by:

$$D = 6(n - 1 - g) + \sum_{i=1}^{g} b_i = 6(7 - 1 - 9) + 20 = 2, \quad (4)$$

where $n$ represents the total number of links in the mechanism, which, in this case, is 7. This includes the mobile platform and all connected legs. $g$ denotes the number of joints or constraints in the system. For this system, $g = 9$, which reflects the number of joints or connections that limit movement between the links. Lastly, $\sum_{i=1}^{g} b_i$ represents the sum of the DoFs contributed by each individual joint $i$. In this case, the sum is 20, accounting for the mobility provided by the various joints in the mechanism. This calculation confirms that the mechanism has 2 DoFs, consistent with our previous analysis.

Finally, in order to assess how the design affects the behavior of the mechanism, we study 16 slenderness ratios, defined by $L/b$ and keeping $L$ constant while varying $b$. As examples, three specific ratios 1.5, 2.0 and 3.0 are shown in Fig. 4 according to the parameters in Tab. 2. The goal of this comparison is to identify the slenderness ratio that improves the performance and operational capabilities of the four-leg tensegrity mechanism.

<figure><img src="image_12.png" /><figcaption>Figure 4: Slenderness ($L/b$) ratios for the four-leg tensegrity mechanism.</figcaption></figure>

<table><caption>Table 2: 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 <b>B</b><sub>i</sub> (based on slenderness ratio)</td><td><i>b</i></td><td>120, 90, 60 mm</td></tr><tr><td>S-U peripheral leg length</td><td><i>L</i></td><td>180 mm</td></tr><tr><td>S-U peripheral leg diameter</td><td><i>d</i></td><td>4 mm</td></tr><tr><td>U-P-U central leg diameter</td><td><i>D</i></td><td>8 mm</td></tr><tr><td>Spherical joint angle limit</td><td>δ<sub>S</sub></td><td>75°</td></tr><tr><td>Universal joint angle limit</td><td>δ<sub>U</sub></td><td>45°</td></tr><tr><td>Minimum cable tension</td><td><i>f</i><sub>min</sub></td><td>1 N</td></tr><tr><td>Maximum cable tension</td><td><i>f</i><sub>max</sub></td><td>20 N</td></tr></tbody></table>

## 3 Inverse kinematic model

To analyze the GW and WFW of the proposed tensegrity mechanism, it is important to determine the configurations of the four legs as functions of $\phi$ and $\theta$. This requires obtaining their joint variables $\mathbf{q} = (q_4, \dots, q_{14})^T$, which are functions of the mechanism's DoFs. An inverse kinematic model will allow us to achieve this. Therefore, in this section we extend the framework previously developed in [30], where the zero-extension constraint defined by the prism (composed of the two platforms and
the peripheral legs) was identified, and a trilateration approach was used to solve it. However, with the introduction of the central leg connecting $P_0$ and $P_3$, it becomes essential to modify the initial strategy to effectively address the zero-torsion constraint.

The general structure of the inverse kinematic model is shown in Fig. 5. Given the input angles $\phi$ and $\theta$, the zero-torsion constraint (described in detail below) and the trilateration method (as described in [30]) are applied within MATLAB's `fsolve` function (or any other nonlinear solver) to iteratively determine the torsion value $\sigma$ that satisfies the zero-torsion constraint. Since this solution is not unique, only the one feasible within the mechanism's operational workspace is retained. The variables $\phi$, $\theta$ and $\sigma$ are then used to perform a final trilateration, yielding the coordinates of $P_3$ according to $F_0$. With $\phi$, $\theta$, $\sigma$ and $P_3$ known, the coordinates of the anchor points $B_i$ ($i \in N: 1 \le N \le 6$) for the three legs are calculated. These values are then used to determine the joint variables $q$ using Paul's method [34]. Details of the calculation process, together with the results for the joint variables, are given in the Appendix A.

<figure><img src="image_13.png" /><figcaption>Figure 5: Inverse kinematic model for the four-leg tensegrity mechanism incorporating zero-torsion constraint via $g(\sigma)$.</figcaption></figure>

The zero-torsion constraint is inspired by the methodology presented in [28] and serves to limit the torsional motion of the mobile platform. As shown in Fig. 7, the constraint mandates that the vector $s_3$ remains parallel to the vector $s_\gamma$. Here, $s_\gamma$ denotes the vector $s_0$ rotated about $a_0$ by the angle $\gamma$ (see Sec. 2) described by the following equation:

$$ \mathbf{s}_{\gamma} = \begin{bmatrix} C\gamma & -S\gamma & 0 \\ S\gamma & C\gamma & 0 \\ 0 & 0 & 1 \end{bmatrix} \mathbf{s}_{0}. \qquad (5) $$

This results in the zero-torsion constraint, called $g(\sigma)$, which is defined as:

$$ g(\sigma) = \mathbf{s}_3^T \cdot (\mathbf{P}_3 \times \mathbf{s}_\gamma) = 0. \qquad (6) $$

To solve for $\sigma$, we use MATLAB's `fsolve` function, with $\sigma_0$ as the initial guess for $\sigma$, which is the value of $\gamma$. If no feasible value of $\sigma$ is found, `fsolve` repeats the process iteratively until a satisfactory value is found.

As an example, Fig. 6 shows the evolution of the outputs of the inverse kinematic model $\{\sigma, P_3\}$ according to the DoFs $\phi$ and $\theta$. In Fig. 6a, for a ratio of $L/b = 2$ and $\gamma = 0$ deg, we observe that the distance between the platforms increases as the mobile platform moves away from the rest position (i.e. for $\phi = 0^\circ$ and $\theta = 0^\circ$). As far as the angle $\sigma$ is concerned (see Fig. 6b), it has an anisotropic behavior. The main axis of variation is the diagonal of the geometric workspace where the sign of $\phi$ is opposite to that of $\sigma$. Along the other diagonal of the workspace, $\sigma$ varies little from the value taken in the rest configuration.
<figure><img src="image_14.png" /><figcaption>Figure 6: Distance between the platforms (a) and the angle $\sigma$ of the mobile platform (b) as a function of $\phi$ and $\theta$ when $L/b = 2$ and $\gamma = 0$ deg.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 7: Zero-torsion constraint for the four-leg tensegrity mechanism.</figcaption></figure>

# 4 Geometric workspace analysis

This section examines how the design of the mechanism affects the size of the GW, highlighting the slenderness ratio and the optimal angle $\gamma$ (see Sec. 2) as key factors in improving performance. Building on the inverse kinematic framework in Sec. 3, the GW is constrained by collisions between each possible pair of legs, computed by evaluating the distances between them, as detailed in [30]. These distances depend on the peripheral leg diameter $d$ and the central leg diameter $D$. Expressing the distances as functions of $\phi$ and $\theta$ allows us to identify configurations where the distances exceed these diameters, thereby defining the angular range within which the mechanism operates without collision. The GW analysis also includes two joint angle constraints. The spherical joint angle constraint, $\delta_{S_i}$, is the angle between the platform normal vectors and the peripheral leg orientation vectors, giving six constraints for the joints of the three legs. The universal joint angle constraint, $\delta_{U_i}$, is the angle between the platform normal vectors and the central leg orientation vector, giving two constraints of this type. These joint angle limits are essential to ensure smooth operation of the U-joint, allowing the internal cross shaft to rotate without interference.
The geometric parameters used for the computations are given in Tab. 2.

The results of the GW analysis using the various parameters outlined in Tab. 2 are shown in Figs. 8, 9 and 10. Figs. 8 and 10 illustrate the workspace boundaries for the selected slenderness ratio from 16 studied ones, which are color-coded: blue indicates peripheral leg collisions, orange indicates collisions between peripheral and central legs, green indicates the boundaries of the S-joint, and red indicates the boundaries of the U-joint. In both figures, the largest centered square (shown in dashed lines) is determined. Let $ρ_g$ be defined as the side length of this square. This value is used as a measure of the workspace size.

Fig. 8 shows the GW for the non-optimized cases A, B, and C at $γ = 0$ for the three slenderness ratios. This figure illustrates a trend to be confirmed: as the slenderness ratio ($L/b$) increases, the value of $ρ_g$ within the tensegrity mechanism increases. Specifically, in case A (slenderness ratio 1.5), $ρ_g$ is approximately 94.4°, while in case B (slenderness ratio 2.0) it increases to 97.2°, and in case C (slenderness ratio 3.0) it further reaches 98.4°.

<figure><img src="image_16.png" /><figcaption>Figure 8: Geometric workspace for the three non-optimized cases A, B, and C with $γ = 0$.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 9: Workspace size $ρ_g$ as a function of design parameters: (a) the contour plot in blue and the optimal cases in purple, (b) the optimal cases vs. the slenderness ratio.</figcaption></figure>

Fig. 9 shows the variation of the workspace size as a function of $γ \in [-25, 5]°$ and the 16 slenderness ratios $L/b \in [1.5, 3]$. First, the plot (see Fig. 9b) confirms the previous observation that a higher slenderness ratio improves mobility and operating range but only up to a ratio of 2.7, at which point the size of the GW decreases. Second, we can identify an optimal value of $γ$ for each aspect ratio included in the range studied. In Fig. 9a, these values are represented by a purple dotted line, while in Fig. 9b, the corresponding values $γ$ and $ρ$ are shown. We have identified by D, E and F the optimized
designs for the three selected ratios 1.5, 2 and 3, whose GWs for these optimal cases are shown in Fig. 10. In addition, the design identified by the letter G ($\gamma = -24°$ and $L/b = 2.7$) in red in Fig. 9 gives the best geometric workspace, i.e. $\rho_g = 123.8°$.

The GW analysis for the optimized cases D, E and F, evaluated at their respective optimal angles $\gamma$, provides significant insight into the influence of different slenderness ratios. In case D, with a slenderness ratio of 1.5 at $\gamma = -4.0°$, the maximum workspace size $\rho_g$ is approximately 98.0°. Case E, with a slenderness ratio of 2.0 at $\gamma = -19.5°$, has an increase in $\rho_g$ of 117.0°, suggesting an improvement in workspace capability due to a more optimal configuration. Finally, case F, with a slenderness ratio of 3.0 at $\gamma = -20.5°$, achieves the largest $\rho_g$ of 123.0° for the selected designs. The GW size obtained is very close to that of the optimal design G. This trend highlights the fact that an optimized setting of the slenderness ratio in conjunction with the angle $\gamma$ effectively affects the GW, thus improving the operational performance of the mechanism.

<figure><img src="image_18.png" /></figure>

(a) Case D: ratio of 1.5 at $\gamma = -4.0°$, for $\rho_g \approx 98.0°$.

(b) Case E: ratio of 2.0 at $\gamma = -19.5°$, for $\rho_g \approx 117.0°$.

(c) Case F: ratio of 3.0 at $\gamma = -20.5°$, for $\rho_g \approx 123.0°$.

<caption>Figure 10: Geometric workspace for the three optimized cases D, E, and F at optimal $\gamma$.</caption>

Moreover, Figs. 8 and 10 illustrate the different types of collisions that define the limits and maximum size of the GW for each slenderness ratio. In both non-optimal and optimal scenarios, it is evident that most collisions occur at the boundaries of the U-joints (increasing the stroke of the cross shaft would increase the GW, although this design issue is beyond the scope of this paper). This observation suggests that the primary factor influencing the GW of the mechanism is $\delta_U$. Notably, for a slenderness ratio of 1.5, significant collisions are also observed at the S-joint boundaries in both the optimal and non-optimal cases. In the optimal scenario for a slenderness ratio of 2.0, the peripheral leg collisions follow the U-joint collisions. In contrast, for a slenderness ratio of 3.0, collisions between the peripheral legs and the central leg are observed after the U-joint collisions.

The GW analysis highlights the critical influence of the slenderness ratio and the optimal $\gamma$ value on the performance of the proposed tensegrity mechanism. By evaluating different configurations, we have determined how optimizing $\gamma$ can maximize the GW for different slenderness ratios. This shows that careful tuning of these parameters can significantly improve the operational capabilities and range of motion of the mechanism. But this raises the question: what about the influence of the tension in the cables on the performance of the mechanism?

# 5 Static analysis

To answer this last question, we present a methodology to determine the FW of the proposed tensegrity mechanism through a static analysis. The following subsections detail the individual steps of the methodology, and in the final part of this section, we demonstrate how these steps are integrated into a cohesive framework.
## 5.1 Static model

The static equation for the mechanism, derived from the Newton-Euler formulation, is expressed at the center of the mobile platform $\mathbf{P}_3$. According to [35, 36], this equation is given by

$$ \begin{pmatrix} \mathbf{F}_e \\ \mathbf{Q}_e \end{pmatrix} + \begin{pmatrix} \mathbf{F}_c \\ \tau \end{pmatrix} + \mathbf{A}^T \mathbf{\lambda} = \mathbf{0}, \qquad (7) $$

where $\mathbf{F}_c$ is the cable wrench, $\mathbf{F}_e$ is the external wrench acting on the mechanism, $\mathbf{Q}_e$ is the external torques and forces acting on the joints and $\mathbf{A}$ is the kinematic constraint matrix. The term $\tau$ corresponds to the internal torques acting on the joints. In this case, both $\mathbf{Q}_e$ and $\tau$ are set to zero because the joints do not interact with the exterior and are passive (not actuated). The term $\mathbf{\lambda}$ consists of the Lagrange multipliers that quantify the internal torques and forces required to satisfy the kinematic constraints enforced by the constraint matrix $\mathbf{A}$. These constraints are essential to ensure that the kinematic loops of the mechanism are closed and that the legs are properly anchored to the fixed platform.

To simplify the static equilibrium analysis by projecting onto the free space, we eliminate the internal torques and forces $\mathbf{\lambda}$. This simplification results in a reduced static model that focuses exclusively on the external and cable wrenches $\mathbf{F}_c$. To achieve this, we need to identify the kernel of allowed motion $\mathbf{H}$ [37]. This kernel encapsulates the part of the motion that is free and unaffected by the constraint matrix $\mathbf{A}$ (i.e. the closed loops) and is computed as follows:

$$ \mathbf{H} = \mathrm{null}(\mathbf{A}). \qquad (8) $$

By multiplying Eq. (7) by the transposed kernel of the admissible motion $\mathbf{H}^T$, we effectively eliminate the internal torques and forces $\mathbf{\lambda}$ and simplify the equilibrium equation to obtain the reduced static model:

$$ \mathbf{H}^T \begin{pmatrix} \mathbf{F}_e \\ \mathbf{0} \end{pmatrix} + \mathbf{H}^T \begin{pmatrix} \mathbf{F}_c \\ \mathbf{0} \end{pmatrix} + \mathbf{H}^T \mathbf{A}^T \mathbf{\lambda} = \mathbf{0}. \qquad (9) $$

Since $\mathbf{H}^T \mathbf{A}^T = \mathbf{0}$, the equation simplifies further to:

$$ \mathbf{H}^T \begin{pmatrix} \mathbf{F}_e \\ \mathbf{0} \end{pmatrix} + \mathbf{H}^T \begin{pmatrix} \mathbf{F}_c \\ \mathbf{0} \end{pmatrix} = \mathbf{0}. \qquad (10) $$

Finally, according to Eq. (10), we introduce the reduced feasible wrench $\mathbf{\Gamma}$ as:

$$ \mathbf{\Gamma} = \mathbf{H}^T \begin{pmatrix} \mathbf{F}_c \\ \mathbf{0} \end{pmatrix}, \qquad (11) $$

which represents the two-dimensional torque vector that the mechanism produces to balance $\mathbf{F}_e$. Its components $\Gamma_x$ and $\Gamma_y$ are torques applied around $\mathbf{s}_3$ and $\mathbf{n}_3$ respectively.

## 5.2 Kinematic constraints

To calculate the kinematic constraint matrix $\mathbf{A}$, we first consider the twist system of the four legs, reduced at their respective anchor points. Since the static model is evaluated at $\mathbf{P}_3$, we introduce the linear and angular velocities of the mobile platform, denoted by $\mathbf{V}_3$ and $\mathbf{\Omega}_3$ respectively. These velocities serve as reference components to understand the kinematic behavior of the legs. The kinematic constraint equations of the mechanism can then be defined as follows:

$$ \mathbf{A} \begin{pmatrix} \mathbf{V}_3 \\ \mathbf{\Omega}_3 \\ \dot{\mathbf{q}} \end{pmatrix} = \mathbf{0}, \qquad (12) $$

According to the velocities of the mobile platform and joints, we introduce the 15 kinematic constraints (see Sec. 2), which can be categorized into two parts (see Sec. 3): those imposed by the central leg, called the zero-torsion constraint, and those imposed by the prism (the two platforms and the three
peripheral legs), called the zero-extension constraint. These constraints are defined by the following expression:

$$ \begin{bmatrix} P \mathbf{A} \\ L \mathbf{A} \end{bmatrix} \begin{pmatrix} \mathbf{V}_3 \\ \mathbf{\Omega}_3 \\ \dot{\mathbf{q}} \end{pmatrix} = \mathbf{0}, \qquad (13) $$

where $P \mathbf{A}$ and $L \mathbf{A}$ are the constraint matrices of the prism and the central leg, respectively.

For the prism, the kinematic constraints arise from the closed loop structure that enforces zero velocity at the bottom anchor points of the three peripheral legs, specifically $\mathbf{V}_{B_1} = \mathbf{V}_{B_2} = \mathbf{V}_{B_3} = \mathbf{0}$. These conditions result in nine constraints and can be expressed, using the principles outlined in [33], as:

$$ \begin{align} \mathbf{V}_{B_1} &= \mathbf{0} &\Rightarrow \mathbf{V}_3 + (\mathbf{a}_4\dot{\mathbf{q}}_4 + \mathbf{a}_5\dot{\mathbf{q}}_5) \times (\mathbf{B}_1 - \mathbf{B}_6) &= \mathbf{0}, \nonumber \\ \mathbf{V}_{B_2} &= \mathbf{0} &\Rightarrow \mathbf{V}_3 + (\mathbf{a}_6\dot{\mathbf{q}}_6 + \mathbf{a}_7\dot{\mathbf{q}}_7) \times (\mathbf{B}_2 - \mathbf{B}_4) &= \mathbf{0} \text{ and} \nonumber \\ \mathbf{V}_{B_3} &= \mathbf{0} &\Rightarrow \mathbf{V}_3 + (\mathbf{a}_8\dot{\mathbf{q}}_8 + \mathbf{a}_9\dot{\mathbf{q}}_9) \times (\mathbf{B}_3 - \mathbf{B}_5) &= \mathbf{0}. \tag{14} \end{align} $$

For the central leg, both the linear velocity $\mathbf{V}_{P_0}$ and the angular velocity $\mathbf{\Omega}_{P_0}$ are zero at the bottom anchor point of this leg, i.e., $\mathbf{V}_{P_0} = \mathbf{0}$ and $\mathbf{\Omega}_{P_0} = \mathbf{0}$. This introduces six additional constraints, given by:

$$ \begin{align} \mathbf{V}_{P_0} &= \mathbf{0} &\Rightarrow \mathbf{V}_3 + (\mathbf{a}_{10}\dot{\mathbf{q}}_{10} + \mathbf{a}_{11}\dot{\mathbf{q}}_{11}) \times (\mathbf{P}_0 - \mathbf{P}_3) + \mathbf{s}_{12}\dot{\mathbf{q}}_{12} &= \mathbf{0} \text{ and} \nonumber \\ \mathbf{\Omega}_{P_0} &= \mathbf{0} &\Rightarrow (\mathbf{\Omega}_3 + \mathbf{a}_{10}\dot{\mathbf{q}}_{10} + \mathbf{a}_{11}\dot{\mathbf{q}}_{11} + \mathbf{a}_{13}\dot{\mathbf{q}}_{13} + \mathbf{a}_{14}\dot{\mathbf{q}}_{14}) &= \mathbf{0}. \tag{15} \end{align} $$

Finally, according to (13), by factoring Eqs. (14) and (15) with respect to $\mathbf{V}_3$, $\mathbf{\Omega}_3$ and $\dot{\mathbf{q}}$, the components of the constraint matrix $\mathbf{A}$ can be determined.

## 5.3 Kernel composition

Building on the definitions established in Secs. 5.1 and 5.2, we can compute, $\mathbf{H}$ derived from the constraint matrix $\mathbf{A}$ (see Eq. (8)). This kernel establishes a direct relationship between the generalized velocities of the mechanism and the angular velocities $\mathbf{\Omega}_{3x}$ and $\mathbf{\Omega}_{3y}$, which are dual to $\Gamma_x$ and $\Gamma_y$. The motion equation of the mechanism is expressed as:

$$ \begin{pmatrix} \mathbf{V}_3 \\ \mathbf{\Omega}_3 \\ \dot{\mathbf{q}} \end{pmatrix} = \mathbf{H} \begin{pmatrix} \mathbf{\Omega}_{3x} \\ \mathbf{\Omega}_{3y} \end{pmatrix}. \qquad (16) $$

In this context, the matrix $\mathbf{H}$ serves as a transformation that converts variations in the angular velocities $\mathbf{\Omega}_{3x}$ and $\mathbf{\Omega}_{3y}$ into the corresponding linear and angular velocities of the platform. This illustrates how changes in the angular components directly affect $\mathbf{V}_3$ and $\mathbf{\Omega}_3$. By deriving $\mathbf{H}$ from the null space of $\mathbf{A}$, we encapsulate the allowable configurations of the mechanism and allow for the identification of motion combinations that satisfy the kinematic constraints. This equation can be expanded as follows:

$$ \begin{pmatrix} V_{3x} \\ V_{3y} \\ V_{3z} \\ \mathbf{\Omega}_{3x} \\ \mathbf{\Omega}_{3y} \\ \mathbf{\Omega}_{3z} \\ \dot{\mathbf{q}}_4 \\ \vdots \\ \dot{\mathbf{q}}_{14} \end{pmatrix} = \begin{bmatrix} H_{11} & H_{12} \\ H_{21} & H_{22} \\ H_{31} & H_{32} \\ 1 & 0 \\ 0 & 1 \\ H_{61} & H_{62} \\ H_{71} & H_{72} \\ \vdots & \vdots \\ H_{171} & H_{172} \end{bmatrix} \begin{pmatrix} \mathbf{\Omega}_{3x} \\ \mathbf{\Omega}_{3y} \end{pmatrix}. \qquad (17) $$

However, due to the complexity of the mechanism, direct computation of the kernel $\mathbf{H}$ (whether by hand or using symbolic math software) proved infeasible. To overcome this challenge, an alternative approach has been developed. The proposed solution is to perform kernel composition, which involves treating the prism and the central leg as separate entities. Consequently, the kernels for each component are computed independently:

$$ P \mathbf{H} = \text{null}(P \mathbf{A}) \text{ and } L \mathbf{H} = \text{null}(L \mathbf{A}), \qquad (18) $$
where $^P H$ is the kernel of the prism and $^L H$ is the kernel of the central leg. Subsequently, the kernel composition enables us to combine both kernels to determine the components of the admissible motion space for the four-leg tensegrity mechanism. The first step is to compute the motion equation for the central leg mechanism, which relates the linear velocity $V_3$ and angular velocity $\Omega_3$ of the mobile platform to its five DoFs:

$$ \begin{pmatrix} V_{3x} \\ V_{3y} \\ V_{3z} \\ \Omega_{3x} \\ \Omega_{3y} \\ \Omega_{3z} \\ \dot{q}_{10} \\ \vdots \\ \dot{q}_{14} \end{pmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ {}^{L}H_{61} & {}^{L}H_{62} & {}^{L}H_{63} & {}^{L}H_{64} & {}^{L}H_{65} \\ {}^{L}H_{71} & \dots & \dots & \dots & {}^{L}H_{75} \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ {}^{L}H_{111} & \dots & \dots & \dots & {}^{L}H_{115} \end{bmatrix} \begin{pmatrix} V_{3x} \\ V_{3y} \\ V_{3z} \\ \Omega_{3x} \\ \Omega_{3y} \\ \Omega_{3z} \\ \dot{q}_{4} \\ \vdots \\ \dot{q}_{9} \end{pmatrix}. \quad (19) $$

Also, we have the motion equation for the prism mechanism, which relates the linear and angular velocities of the mobile platform, $V_3$ and $\Omega_3$, respectively, with its three DoFs:

$$ \begin{pmatrix} V_{3x} \\ V_{3y} \\ V_{3z} \\ \Omega_{3x} \\ \Omega_{3y} \\ \Omega_{3z} \\ \dot{q}_{4} \\ \vdots \\ \dot{q}_{9} \end{pmatrix} = \begin{bmatrix} {}^{P}H_{11} & {}^{P}H_{12} & {}^{P}H_{13} \\ {}^{P}H_{21} & {}^{P}H_{22} & {}^{P}H_{23} \\ {}^{P}H_{31} & {}^{P}H_{32} & {}^{P}H_{33} \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ {}^{P}H_{71} & \dots & {}^{P}H_{73} \\ \vdots & \ddots & \vdots \\ {}^{P}H_{121} & \dots & {}^{P}H_{123} \end{bmatrix} \begin{pmatrix} \Omega_{3x} \\ \Omega_{3y} \\ \Omega_{3z} \end{pmatrix}. \quad (20) $$

From Eq. (19), we derive the expression for $\Omega_{3z}$:

$$ \Omega_{3z} = {}^{L}H_{61}V_{3x} + {}^{L}H_{62}V_{3y} + {}^{L}H_{63}V_{3z} + {}^{L}H_{64}\Omega_{3x} + {}^{L}H_{65}\Omega_{3y}. \quad (21) $$

The replacement of Eq. (21) in Eq. (20) gives the following equation:

$$ \begin{pmatrix} V_{3x} \\ V_{3y} \\ V_{3z} \\ \Omega_{3x} \\ \Omega_{3y} \\ \Omega_{3z} \\ \dot{q}_{4} \\ \vdots \\ \dot{q}_{9} \end{pmatrix} = \begin{bmatrix} {}^{P}H_{11} & {}^{P}H_{12} \\ {}^{P}H_{21} & {}^{P}H_{22} \\ {}^{P}H_{31} & {}^{P}H_{32} \\ 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ {}^{P}H_{71} & {}^{P}H_{72} \\ \vdots & \vdots \\ {}^{P}H_{121} & {}^{P}H_{122} \end{bmatrix} \begin{pmatrix} \Omega_{3x} \\ \Omega_{3y} \end{pmatrix} + \begin{bmatrix} {}^{P}H_{13} \\ {}^{P}H_{23} \\ {}^{P}H_{33} \\ 0 \\ 0 \\ 1 \\ {}^{P}H_{73} \\ \vdots \\ {}^{P}H_{123} \end{bmatrix} \begin{pmatrix} {}^{L}H_{61} & \dots & {}^{L}H_{65} \\ \end{pmatrix} \begin{pmatrix} V_{3x} \\ V_{3y} \\ V_{3z} \\ \Omega_{3x} \\ \Omega_{3y} \end{pmatrix}. \quad (22) $$

Starting from Eq. (22), we calculate the components of the linear velocity vector $V_3$:

$$ \begin{pmatrix} V_{3x} \\ V_{3y} \\ V_{3z} \end{pmatrix} = \begin{bmatrix} 1_3 - \begin{pmatrix} {}^{P}H_{13} \\ {}^{P}H_{23} \\ {}^{P}H_{33} \end{pmatrix} ( {}^{L}H_{61} \quad \dots \quad {}^{L}H_{63} ) \\ \begin{bmatrix} {}^{P}H_{11} & {}^{P}H_{12} \\ {}^{P}H_{21} & {}^{P}H_{22} \\ {}^{P}H_{31} & {}^{P}H_{32} \end{bmatrix} + \begin{pmatrix} {}^{P}H_{13} \\ {}^{P}H_{23} \\ {}^{P}H_{33} \end{pmatrix} ( {}^{L}H_{64} \quad {}^{L}H_{65} ) \end{bmatrix} \begin{pmatrix} \Omega_{3x} \\ \Omega_{3y} \end{pmatrix}, \quad (23) $$

where the equivalents of the components for the first three rows of the kernel of the four-leg tensegrity
mechanism, $H_{11}$, $H_{12}$, $H_{21}$, $H_{22}$, $H_{31}$, and $H_{32}$ (see Eq. (17)), can be identified as follows:

$$ \begin{bmatrix} H_{11} & H_{12} \\ H_{21} & H_{22} \\ H_{31} & H_{32} \end{bmatrix} = \begin{bmatrix} \mathbf{1}_3 - \begin{pmatrix} {}^P H_{13} \\ {}^P H_{23} \\ {}^P H_{33} \end{pmatrix} ( {}^L H_{61} \quad \dots \quad {}^L H_{63} ) \\ \begin{bmatrix} {}^P H_{11} & {}^P H_{12} \\ {}^P H_{21} & {}^P H_{22} \\ {}^P H_{31} & {}^P H_{32} \end{bmatrix} + \begin{pmatrix} {}^P H_{13} \\ {}^P H_{23} \\ {}^P H_{33} \end{pmatrix} ( {}^L H_{64} \quad {}^L H_{65} ) \end{bmatrix}^{-1} \quad (24) $$

Next, also from Eq. (22), we calculate the components of the angular velocity vector $\Omega_3$:

$$ \begin{pmatrix} \Omega_{3x} \\ \Omega_{3y} \\ \Omega_{3z} \end{pmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ {}^L H_{64} & {}^L H_{65} \end{bmatrix} \begin{pmatrix} \Omega_{3x} \\ \Omega_{3y} \end{pmatrix} + \begin{bmatrix} {}^L H_{61} & {}^L H_{62} & {}^L H_{63} \\ {}^L H_{61} & {}^L H_{62} & {}^L H_{63} \\ {}^L H_{61} & {}^L H_{62} & {}^L H_{63} \end{bmatrix} \begin{pmatrix} V_{3x} \\ V_{3y} \\ V_{3z} \end{pmatrix} \quad (25) $$

We substitute the vector $V_3$ from Eq. (17) into Eq. (25), which results in the following expression:

$$ \begin{pmatrix} \Omega_{3x} \\ \Omega_{3y} \\ \Omega_{3z} \end{pmatrix} = \begin{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ {}^L H_{64} & {}^L H_{65} \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ {}^L H_{61} & {}^L H_{62} & {}^L H_{63} \end{bmatrix} \begin{bmatrix} H_{11} & H_{12} \\ H_{21} & H_{22} \\ H_{31} & H_{32} \end{bmatrix} \end{bmatrix} \begin{pmatrix} \Omega_{3x} \\ \Omega_{3y} \end{pmatrix}, \quad (26) $$

where the equivalents of the components for the sixth row of the kernel $\mathbf{H}$, $H_{61}$ and $H_{62}$ (see Eq. (17)), can be identified as follows:

$$ (H_{61} \quad H_{62}) = \left( {}^L H_{64} + \sum_{i=1}^{3} {}^L H_{6i} H_{i1} \quad {}^L H_{65} + \sum_{i=1}^{3} {}^L H_{6i} H_{i2} \right). \quad (27) $$

Using the Eqs. (24) and (27), we have derived the complete set of coefficients defining the kernel $\mathbf{H}$ (from $H_{11}$ to $H_{62}$). These coefficients correspond to both the linear velocity vector $\mathbf{V}_3$ and the angular velocity vector, $\Omega_3$ as described in Eq. (17).

## 5.4 Cable routing

<figure><img src="image_19.png" /><figcaption>Figure 11: Cable routing for the four-leg tensegrity mechanism.</figcaption></figure>

Another critical factor for obtaining the FW, as explained in Eq. (11), is the cable wrench $\mathbf{F}_c$. To derive the cable wrench, we detail the cable routing (see Fig. 11), which elucidates the arrangement of the cables within the four-leg tensegrity mechanism along with all associated parameters. The lower (fixed) platform has three anchor points where the cables are anchored: $\mathbf{C}_1$, $\mathbf{C}_2$, and $\mathbf{C}_3$. Correspondingly, the upper (mobile) platform has three anchor points: $\mathbf{C}_4$, $\mathbf{C}_5$, and $\mathbf{C}_6$. The geometric design parameter $\psi$ is an angle that delineates the configuration of the cables connecting the two platforms, as illustrated
in Fig. 11b and Fig. 11c. The value of $\psi$ is one of the parameters that will be optimized as described in Sec. 6. The three cables connect the anchor points as follows: $C_1$ to $C_4$, $C_2$ to $C_5$, and $C_3$ to $C_6$, with their lengths denoted by $l_1$, $l_2$, and $l_3$, respectively.

Based on the cable routing described above, and using the equilibrium equation, we can express the cable wrench $\mathbf{F}_c$ as a function of the cable tensions $f_{c_1}$, $f_{c_2}$, and $f_{c_3}$ as follows:

$$ \mathbf{F}_c = \mathbf{Z} \begin{pmatrix} f_{c_1} \\ f_{c_2} \\ f_{c_3} \end{pmatrix}, \qquad (28) $$

where $\mathbf{Z}$ represents the actuation matrix that contains the coefficients related to the cable tensions and is defined as follows<fnref n="3" />:

$$ \mathbf{Z} = \begin{pmatrix} \mathbf{1}_3 \\ \hat{\mathbf{P}}_3 \end{pmatrix} \begin{bmatrix} \frac{(\mathbf{C}_4 - \mathbf{C}_1)^T}{l_1} & \frac{(\mathbf{C}_5 - \mathbf{C}_2)^T}{l_2} & \frac{(\mathbf{C}_6 - \mathbf{C}_3)^T}{l_3} \end{bmatrix}. \qquad (29) $$

## 5.5 Actuation boundaries

In order to compute the FW, and in accordance with (28), our goal is to determine the actuation boundaries for any given pose of the mechanism. As discussed in previous sections, Eq. (11) establishes the reduced static model that represents the relationship between the cable wrench and the resulting wrench acting on the mechanism. Replacing Eq. (28) in Eq. (11), we obtain:

$$ \begin{pmatrix} \Gamma_x \\ \Gamma_y \end{pmatrix} = \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \end{bmatrix} \begin{pmatrix} f_{c_1} \\ f_{c_2} \\ f_{c_3} \end{pmatrix}, $$

where the coefficients $m_{11}$, $m_{12}$, $m_{13}$, $m_{21}$, $m_{22}$, and $m_{23}$ relate the cable tensions $f_{c_1}$, $f_{c_2}$, and $f_{c_3}$ to the FW.

<table><caption>Table 3: Actuation boundaries of cable tensions.</caption><thead><tr><th>Boundary</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th></tr></thead><tbody><tr><td><i>i</i></td><td>1</td><td>1</td><td>2</td><td>2</td><td>3</td><td>3</td></tr><tr><td><i>j</i></td><td>2</td><td>3</td><td>1</td><td>3</td><td>1</td><td>2</td></tr><tr><td><i>k</i></td><td>3</td><td>2</td><td>3</td><td>1</td><td>2</td><td>1</td></tr></tbody></table>

To determine the actuation boundaries, we analyze different combinations of the cable tensions where two of the tensions are fixed at their extreme values: the maximum ($f_{max}$) and minimum ($f_{min}$) values, while the third tension varies within these limits. The cable tensions are referred to as $f_{c_i}$, $f_{c_j}$ and $f_{c_k}$. The variable tension $f_{c_i}$, which depends on the two fixed tensions $f_{c_j}$ and $f_{c_k}$, can be described as follows:

$$ f_{c_i} = \frac{1}{m_{1i}^2 + m_{2i}^2} \left( m_{1i} (\Gamma_x - m_{1j} f_{c_j} - m_{1k} f_{c_k}) + m_{2i} (\Gamma_y - m_{2j} f_{c_j} - m_{2k} f_{c_k}) \right), \qquad (30) $$

subject to the following conditions:

$$ f_{c_j} = f_{\max}, \quad f_{c_k} = f_{\min} \quad \text{and} \quad f_{\min} \le f_{c_i} \le f_{\max}. \qquad (31) $$

Each combination of the indices $i$, $j$, and $k$ defines a unique actuation boundary for cable tensions, as summarized in Tab. 3. By solving Eq. (30) for $f_{c_i}$ in each actuation boundary and considering Eq. (31), only one of them will satisfy the condition, ensuring that the mechanism operates within the physical constraints of the tensegrity system. This approach provides a comprehensive framework for choosing the correct actuation boundary that achieves the required FW while maintaining the equilibrium of the mechanism. Given the definition of the actuation boundaries for the cable tensions (see Tab. 3)

<footnote n="3">³The 'hat' notation is the $(3 \times 3)$ asymmetric tensor associated with the $(3 \times 1)$ vector mentioned above. For arbitrary vectors $U$ and $V$ of $\mathbb{R}^3$, $\hat{U}$ is defined such that $\hat{U}V = U \times V$.</footnote>
<figure><img src="image_20.png" /><figcaption>Figure 12: Top views of the four tested configurations in the geometric workspace for an optimized case of the four-leg tensegrity mechanism with a L/b ratio of 3.0 and $\gamma = -20.5^\circ$.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Figure 13: The polytopes related to the geometric configurations for an optimized case of the four-leg tensegrity mechanism with a L/b ratio of 3 and $\gamma = -20.5^\circ$ illustrated in Fig. 12. For each configuration, the colors are related to the 6 actuation boundaries defined in Tab. 3 and the dashed square represents the restricted FW workspace, $\rho_w$, according to our requirement.</figcaption></figure>

and the condition (see Eq. (31)) that must be satisfied, we can proceed to an example of obtaining the FW for an optimized case of the four-leg tensegrity mechanism with a L/b ratio of 3.0 (where
$\gamma = -20.5^{\circ}$). To illustrate this, we develop the FW for four specific geometric configurations of the four-leg tensegrity mechanism, as shown in Fig. 12. The FW for these four geometric configurations is shown in Fig. 13 for minimum and maximum cable tensions $f_{min}$ and $f_{max}$ given in Tab. 2. As shown in each of the four figures, the six actuation boundaries are plotted according to the values of $\Gamma_x$ and $\Gamma_y$, forming a polytope (a polygon in the case considered) that allows us to determine the FW. The largest centered square is obtained by expanding the FW to the first boundary. The dashed square represents the maximum size of the bounded FW, with $\rho_w$ defined as the magnitude of this square.

## 5.6 Synthesis of the methodology

<figure><img src="image_22.png" /><figcaption>Figure 14: Framework for feasible wrench analysis in a four-leg tensegrity mechanism.</figcaption></figure>

In summary, the flowchart in Fig. 14 illustrates the synthesis of the methodology used to derive the feasible wrench (FW) applicable to the four-leg tensegrity mechanism. The analysis starts with the input parameters, {$\phi, \theta$}. Subsequently, the **Inverse kinematic model** (see Sec. 3) allows determining $\sigma$, $\mathbf{P}_3$ and the joint variables $\mathbf{q}$. This information is then used in the **Kinematic constraints** stage (see Sec. 5.2) to compute the constraint matrix **A**. Subsequently, the **Kernel composition** (see Sec. 5.3) allows the evaluation of the kernel of the admissible motion, referred to as **H**, which identifies the two DoFs of the mechanism. In parallel, **Cable routing** (see Sec. 5.4) establishes the relationships between cable tensions and their respective actuation matrix, referred to as **Z**. The next phase involves projecting the **Static model** (see Sec. 5.1) onto **H**, which simplifies the analysis by focusing on the primary forces and torques while neglecting the internal ones, resulting in the reduced static model. Finally, the framework culminates in the formulation of **Actuation boundaries** (see Sec. 5.5), which leads to the comprehensive determination of the feasible set of torques, denoted by $\Gamma_x$ and $\Gamma_y$. By synthesizing these elements, we achieve a robust methodology for computing the FW produced by the four-leg tensegrity mechanism according to a set of tensions. This approach ensures that all relevant factors are considered, resulting in a more accurate and effective determination of the mechanism's capabilities and performance.
# 6 Wrench feasible workspace analysis

Having established the methodology for obtaining the FW (see Fig. 14), our aim is now to determine the WFW. To achieve this, we will analyze the impact of three geometric design parameters: the slenderness ratio $L/b$, the twist angle of the central leg $\gamma$ and the mounting angle of cables $\psi$.

<figure><img src="image_23.png" /><figcaption>Figure 15: Feasible wrench for the three non-optimized cases H, I, and J with $\psi = 0$.</figcaption></figure>

The geometric parameter values used in this analysis are listed in Tab. 2. As a reference, we first compute the WFW for the three selected slenderness ratios by setting $\psi = 0$ and $\gamma$ to the values found in Sec. 4 (see cases D, E and F in Fig. 9b). In Fig. 15, the values of $\rho_w$, which represent the magnitude of the FW for each possible geometric configuration in the GW (see Sec. 5.5), are shown for slenderness ratio 1.5 (Scenario H), 2 (Scenario I) and 3 (Scenario J). The calculated magnitudes show that the stockiest mechanism ($L/b = 1.5$) produces more torque than the slenderest mechanism ($L/b = 3$). There may be an inverse proportional relationship between $\rho_w$ and the slenderness ratio.

<figure><img src="image_24.png" /><figcaption>Figure 16: $\overline{\rho_w}$ as a function of the angles $\psi$ and $\gamma$ for the three slenderness ratios selected: (a) 1.5, (b) 2 and (c) 3. Over each contour plot, the dotted green lines represent $\rho_g$ and the optimal cases are in dotted purple.</figcaption></figure>

To optimize the WFW, one of our objectives, we compute the average of $\rho_w$, denoted $\overline{\rho_w}$, over the GW according to the results of Sec. 4, as a function of $\psi$ and $\gamma$ for the three selected slenderness ratios. The averaging process consists of discretizing the GW (a square of side length $\rho_g(\gamma, L/b)$) into a regular grid of 10 × 10 geometric configurations, then, calculating $\rho_w$ for each configuration, and finally, by a weighted sum, evaluating the average value $\overline{\rho_w}$. This result is shown in Fig. 16, where the non-optimized cases H, I, and J ($\psi = 0$) and the optimized cases K, L, and M are identified. Cases K, L, and M were obtained by taking into account the values of $\gamma$ angles found in Sec. 4 (see cases D,
E and F in Fig. 9b), their WFW are shown in Fig. 17. In fact, in order to observe trends, we have chosen to give priority to the GW over the WFW.

<figure><img src="image_25.png" /><figcaption>Figure 17: Feasible wrench for the three optimized cases K, L, and M at optimal $\psi$.</figcaption></figure>

From this analysis, we observe that for a given $\gamma$, we can find a maximum value of $\overline{\rho}_w$ by adjusting the value of $\psi$. Note that for the three models studied, $\gamma$ also affects the WFW. According to the purple plots in Fig. 16, if we deviate from the value taken by $\gamma$ to optimize the GW, the WFW increases, but at the expense of the angular range of the mechanism. The mechanism will therefore be adjusted based on a compromise between GW and WFW, depending on the task at hand. For example, if we want to design a mechanism with a stroke of more than 55° (i.e. for $\rho_g$ > 110°), producing a torque of more than 0.5 Nm per axis (i.e. for $\rho_w$ > 1 Nm), having an aspect ratio among the 16 ratios selected between 1.5 and 3, and maximum tension $F_{max}$ = 20 N, the designs that meet these criteria would be those shown in Fig. 18. Only slenderness ratios of 1.7 and 1.8 solve the problem. Assuming that we want the most compact mechanism and that we prioritize GW in the selection, the mechanism (letter O in Fig. 18) that would meet our requirements would have the following parameters: $L/b = 1.7$, $\gamma = -13.5°$ and $\psi = -28.0°$.

<figure><img src="image_26.png" /><figcaption>Figure 18: Example of design optimization. The areas colored blue and red are the areas containing the designs which are solutions to the optimization problem.</figcaption></figure>

# 7 Results and discussions

The results in Tab. 4 provide a comparison between optimal and non-optimal configurations for the selected slenderness ratios 1.5, 2 and 3 of the 16 ratio studied. Specifically, the table shows how the geometric design parameters $\gamma$, $\psi$, $\rho_g$, and $\overline{\rho}_w$ vary according to the aspect configurations selected.
<table><caption>Table 4: Comparative results of $\rho_g$ and $\overline{\rho_w}$ for optimal and non-optimal configurations across different ratios of $L/b$ based on $\gamma$ and $\psi$.</caption><thead><tr><th>Ratio L/b</th><th colspan="2">1.5</th><th colspan="2">2</th><th colspan="2">3</th></tr><tr><th>Case</th><th>Non-opt.</th><th>Opt.</th><th>Non-opt.</th><th>Opt.</th><th>Non-opt.</th><th>Opt.</th></tr></thead><tbody><tr><td>γ [deg]</td><td>0.0</td><td>-4.0</td><td>0.0</td><td>-19.5</td><td>0.0</td><td>-20.5</td></tr><tr><td>ψ [deg]</td><td>0.0</td><td>-21.0</td><td>0.0</td><td>-31.3</td><td>0.0</td><td>-33.8</td></tr><tr><td>ρ<sub>g</sub> [deg]</td><td>94.4</td><td>98.0</td><td>97.2</td><td>117.0</td><td>98.4</td><td>123.0</td></tr><tr><td>ρ<sub>w</sub> [Nm]</td><td>0.97</td><td>1.07</td><td>0.76</td><td>0.93</td><td>0.58</td><td>0.71</td></tr></tbody></table>

For the slenderness ratio $L/b = 1.5$, the optimized configuration with $\gamma = -4.0^{\circ}$ and $\psi = -21.0^{\circ}$ gives a $\rho_g$ of $98.0^{\circ}$ compared to $94.4^{\circ}$ in the non-optimized case, an improvement of **3.8%**. Similarly, $\overline{\rho_w}$ improves from 0.92 Nm in the non-optimized configuration to 1.07 Nm in the optimized configuration, an increase of **16.3%**. For a slenderness ratio of $L/b = 2.0$, the optimal configuration achieves a $\rho_g$ of $117.0^{\circ}$, an increase of **20.4%** from the non-optimized value of $97.2^{\circ}$. Accordingly, the $\overline{\rho_w}$ increases by **22.4%**, from 0.76 Nm to 0.93 Nm. Finally, for $L/b = 3.0$, the optimized case with $\gamma = -20.5^{\circ}$ and $\psi = -33.8^{\circ}$ shows the most significant improvements. The $\rho_g$ increases by **25.0%**, from $98.4^{\circ}$ in the non-optimal case to $123.0^{\circ}$ in the optimized case. Meanwhile, $\overline{\rho_w}$ increases by **22.4%**, from 0.58 to 0.71 Nm.

The slenderness ratio $L/b$ plays a crucial role in influencing the performance metrics of the mechanism. In particular, there is a direct proportional relationship between the slenderness ratio and the geometric performance metric $\rho_g$, which means that an increase in $L/b$ leads to an increase in $\rho_g$. In contrast, the slenderness ratio has an inverse relationship with the wrench performance metric $\overline{\rho_w}$, with a higher $L/b$ resulting in a decrease in $\overline{\rho_w}$. In this context, for the three selected slenderness ratio (where the optimal configurations are shown in Fig. 19), while the configuration with $L/b = 3.0$ achieves a higher $\rho_g$, it shows a relatively smaller improvement in $\overline{\rho_w}$; the opposite is true for the configuration with $L/b = 1.5$, leading to an unbalanced performance in these two cases. In contrast, the configuration with $L/b = 2.0$ maintains a more optimal balance between $\rho_g$ and $\overline{\rho_w}$. Thus, the slenderness ratio $L/b = 2.0$ emerges as the most favorable option for maximizing the overall performance of the mechanism. However, even if this design is interesting, as we have shown and explained in the last example in the previous section, the choice of design largely depends on the expected requirements in terms of angular stroke and achievable torque and will therefore have to be adapted to these needs.

<figure><img src="image_27.png" /><figcaption>Figure 19: The different optimized mechanisms according to the optimized parameters of Tab. 4</figcaption></figure>
# 8 Conclusions and future works

In this study, we proposed an innovative four-leg tensegrity mechanism, with an emphasis on analyzing the influence of some geometric parameters of the mechanism on its performance, which provides valuable insight to improve its operational capabilities. The novelty of the proposed mechanism consists in using a central leg instead of a cable. This choice eliminates any torsional compliance and avoids the use and control of an additional motor to manage the central cable tension.

We have developed a comprehensive methodology for analyzing the GW and the WFW. The slenderness ratio $L/b$ plays a critical role in shaping both the GW and the WFW, demonstrating a direct proportional relationship between the geometric performance metric $\rho_g$ and the slenderness ratio, as well as an inverse relationship in terms of its effect on the wrench performance metric $\rho_w$. This relationship is consistent for both optimized and non-optimized configurations. We also found that the U-joint angle limit $\delta_U$ acts as a significant constraint on the geometric capacity of the mechanism. In particular, increasing the stroke of the cross shaft of the U-joint would increase the GW, suggesting a potential direction for future work. These results underscore the importance of carefully selecting geometric design parameters to achieve an optimal workspace for practical applications, allowing for a more accurate and effective assessment of the mechanism's capabilities and overall performance. These analyses are preliminary, and future work will complement this study with additional analyses such as stability and stiffness evaluation, dynamic performance, control strategies, and other performance-related factors.

Furthermore, our modelling approach uses a first-order derivative technique derived from the reduced form of the static Newton-Euler equations. This approach lays the groundwork for future research by simplifying the calculation of the velocities and accelerations of the cable lengths in the mechanism, which is essential for motion control and trajectory planning.

As potential future work, we will develop a stiffness model of our mechanism as well as of the mechanism with the central cable analyzed in [30]. Comparison analyzes of the stiffness of the two mechanisms will be conducted. We will verify that the torsional stiffness of the mechanism with the central cable will be lower than the proposed mechanism with the central leg. Later, we will analyze the GW and WFW for a stack of these mechanisms to evaluate their performance and operational capabilities with the goal of developing a robotic manipulator. By addressing these areas, future studies can further enhance the capabilities and applications of the four-leg tensegrity mechanism, making it a more versatile and effective solution for a wide range of engineering challenges in cobotics.

**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).

# References

[1] J. Colgate, W. Wannasuphoprasit, and M. Peshkin. Cobots: Robots for collaboration with human operators. *Dynamic Systems and Control*, 1996.

[2] A. Cherubini, R. Passama, A. Crosnier, A. Lasnier, and P. Fraisse. Collaborative manufacturing with physical human-robot interaction. *Robotics and Computer-integrated Manufacturing*, 40:1-13, 2016.

[3] I. El Makrini, S. A. Elprama, J. Van den Bergh, B. Vanderborght, A.-J. Knevels, C. I. Jewell, F. Stals, G. De Coppel, I. Ravyse, J. Potargent, et al. Working with walt: How a cobot was developed and inserted on an auto assembly line. *IEEE Robotics & Automation Magazine*, 25(2):51-58, 2018.

[4] O. Salunkhe, O. Stensöta, M. Åkerman, Å. Fasth Berglund, and P.-A. Alveflo. Assembly 4.0: Wheel hub nut assembly using a cobot. *IFAC-PapersOnLine*, 2019.

[5] A. Weiss, A. K. Wortmeier, and B. Kubicek. Cobots in industry 4.0: A roadmap for future practice studies on human-robot collaboration. *IEEE Transactions on Human-Machine Systems*, 51(4):335-345, 2021.
[6] Y. Cohen, S. Shoval, M. Faccio, and R. Minto. Deploying cobots in collaborative systems: major considerations and productivity analysis. *International Journal of Production Research*, 60:1815 – 1831, 2021.

[7] E. Colgate, A. Bicchi, M. A. Peshkin, and J. E. Colgate. Safety for physical human-robot interaction. In *Springer Handbook of Robotics*, pages 1335–1348. 2008.

[8] F. Hernoux, E. Nyiri, and O. Gibaru. Virtual reality for improving safety and collaborative control of industrial robots. *Proceedings of the 2015 Virtual Reality International Conference*, 2015.

[9] Y. Ayoubi, M. Laribi, S. Zeghloul, and M. Arsicault. V2som: A novel safety mechanism dedicated to a cobot's rotary joints. *Robotics*, 8:18, 2019.

[10] R. Stone, S. Pujari, A. Mumani, C. W. Fales, and M. Ameen. Cobot and robot risk assessment (carra) method: an automation level-based safety assessment tool to improve fluency in safe human cobot/robot interaction. *Proceedings of the Human Factors and Ergonomics Society Annual Meeting*, 65:737 – 741, 2021.

[11] G. Michalos, S. Makris, P. Tsarouchi, T. Guasch, D. Kontovrakis, and G. Chryssolouris. Design considerations for safe human-robot collaborative workplaces. *Procedia CIRP*, 37:248–253, 2015.

[12] R. B. Fuller. Tensile-integrity structures, 1962.

[13] J. Pinaud. An introduction to the mechanics of tensegrity structures. *Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228)*, 5:4254–4259 vol.5, 2001.

[14] C. Gosselin and J. Angeles. A manipulator based on a tensegrity structure. *IEEE Transactions on Robotics and Automation*, 22(3):519–524, 2006.

[15] A. Thoma and C. Gosselin. Design of a tensegrity manipulator. In *Proceedings of the IEEE International Conference on Robotics and Automation*, pages 1226–1231. IEEE, 2012.

[16] R. Goyal, M. Chen, M. Majji, and R. Skelton. Motes: Modeling of tensegrity structures. *J. Open Source Softw.*, 4:1613, 2019.

[17] D. S. Shah, J. W. Booth, R. L. Baines, K. Wang, M. Vespignani, K. E. Bekris, and R. Kramer-Bottiglio. Tensegrity robotics. *Soft robotics*, 2021.

[18] K. D. Snelson. Tensegrity structures, 1965.

[19] R. W. Burkhardt. *A Practical Guide to Tensegrity Design*. Cambridge, USA, 2008.

[20] K. D. Snelson. Continuous tension, discontinuous compression structures, February 1965.

[21] A. Burkhardt. Tensegrity structure and method for producing the same, April 2008.

[22] L. Rhode-Barbarigos, N. B. H. Ali, R. Motro, and I. Smith. Designing tensegrity modules for pedestrian bridges. *Engineering Structures*, 32:1158–1167, 2010.

[23] W. Gilewski, J. Kłosowska, and P. Obara. Applications of tensegrity structures in civil engineering. *Procedia Engineering*, 111:242–248, 2015.

[24] M. Q. Marshall. *Analysis of tensegrity-based parallel platform devices*. PhD thesis, University of Florida, 2003.

[25] J. M. Mirats-Tur and J. Camps. A three-dof actuated robot. *IEEE Robotics and Automation Magazine*, 18(3):96–103, 2011.

[26] M. Arsenault and C. M. Gosselin. Kinematic and static analysis of a three-degree-of-freedom spatial modular tensegrity mechanism. *The International Journal of Robotics Research*, 27(8):951–966, 2008.

[27] V. Muralidharan, P. Wenger, and C. Chevallereau. Design considerations and workspace computation of 2-x and 2-r planar cable-driven tensegrity-inspired manipulators. *Mechanism and Machine Theory*, 195:104189, 2024.
[28] I. John, S. Mohan, and P. Wenger. Kinetostatic analysis of a spatial cable-actuated variable stiffness joint. ASME. J. Mechanisms Robotics, 16(9):091003, September 2024.

[29] Yong-Jae Kim, Jong-In Kim, and Wooseok Jang. Quaternion joint: Dexterous 3-dof joint representing quaternion motion for high-speed safe interaction. In 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 935–942. IEEE, 2018.

[30] K. Muñoz, M. Porez, and P. Wenger. Kinematic and static analyses of a 3-dof spatial tensegrity mechanism. In J. Lenarčič and M. Husty, editors, Advances in Robot Kinematics 2024, volume 31 of Springer Proceedings in Advanced Robotics, page 36. Springer, 2024.

[31] K. Xu, M. Fu, and J. Zhao. An experimental kinematic comparison between continuum manipulators with structural variations. In 2014 IEEE International Conference on Robotics and Automation (ICRA), pages 3258–3264, Hong Kong, China, 2014.

[32] I. A. Bonev, D. Zlatanov, and C. M. Gosselin. Advantages of the modified euler angles in the design and control of pkms. In 2002 Parallel Kinematic Machines International Conference, pages 171–188, Chemnitz, Germany, 2002.

[33] W. Khalil and E. Dombre. Modeling Identification and Control of Robots. Taylor & Francis, 2002.

[34] R. P. Paul. Robot Manipulators: Mathematics, Programming, and Control: The Computer Control of Robot Manipulators. Richard Paul, 1981.

[35] F. Boyer and A. Belkhiri. Reduced locomotion dynamics with passive internal dofs: Application to nonholonomic and soft robotics. IEEE Transactions on Robotics, 30(3):578–592, 2014.

[36] B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo. Robotics: Modeling, Planning and Control. Springer, Berlin, Heidelberg, 1 edition, 2009.

[37] F. Boyer, M. Porez, and J. Mauny. Reduced dynamics of the non-holonomic whipple bicycle. Journal of Nonlinear Science, 28(3):943–983, 2018.

A Computation of the joint variables.

To perform the computation of the joint variables of the mechanism according to Paul's method [34], we follow an approach that includes two main steps. Firstly, we analyze the three peripheral legs, followed by an examination of the central leg. We determine the configuration of the peripheral legs by calculating six joint variables, referred to as {$q_4, \dots, q_9$}. These angles define the angular position of the U-joints that form the peripheral legs attached to the mobile platform (see Fig. 2a).

We start this step by calculating the unit vectors $^3\mathbf{u}$, $^3\mathbf{v}$ and $^3\mathbf{w}$ expressed in $\mathcal{F}_3$, which represent the orientations of the peripheral legs and are defined as follows:

$$ ^3\mathbf{u} = {}^0\mathbf{T}_3 \frac{^0\mathbf{B}_1 - ^0\mathbf{B}_6}{\| ^0\mathbf{B}_1 - ^0\mathbf{B}_6 \|} , \quad ^3\mathbf{v} = {}^0\mathbf{T}_3 \frac{^0\mathbf{B}_2 - ^0\mathbf{B}_4}{\| ^0\mathbf{B}_2 - ^0\mathbf{B}_4 \|} , \quad \text{and} \quad ^3\mathbf{w} = {}^0\mathbf{T}_3 \frac{^0\mathbf{B}_3 - ^0\mathbf{B}_5}{\| ^0\mathbf{B}_3 - ^0\mathbf{B}_5 \|} . \quad (32) $$

In parallel, the unit vectors $^3\mathbf{u}$, $^3\mathbf{v}$ and $^3\mathbf{w}$ are related to the transformation matrices $^3\mathbf{T}_5$, $^3\mathbf{T}_7$, and $^3\mathbf{T}_9$ (defined using the DH parameters from Tab. 1), with $\mathbf{e}_1 = (1, 0, 0, 0)^T$, as follows:

$$ \begin{align} ^3\mathbf{u} = ^3\mathbf{T}_5 \mathbf{e}_1 &\implies \begin{pmatrix} ^3u_x \\ ^3u_y \\ ^3u_z \end{pmatrix} = \begin{pmatrix} -S_{q_5} \\ -C_{q_5} S_{q_4} \\ -C_{q_4} C_{q_5} \end{pmatrix}, \notag \\ ^3\mathbf{v} = ^3\mathbf{T}_7 \mathbf{e}_1 &\implies \begin{pmatrix} ^3v_x \\ ^3v_y \\ ^3v_z \end{pmatrix} = \begin{pmatrix} -S_{q_7} \\ -C_{q_7} S_{q_6} \\ -C_{q_6} C_{q_7} \end{pmatrix}, \text{ and} \tag{33} \\ ^3\mathbf{w} = ^3\mathbf{T}_9 \mathbf{e}_1 &\implies \begin{pmatrix} ^3w_x \\ ^3w_y \\ ^3w_z \end{pmatrix} = \begin{pmatrix} -S_{q_9} \\ -C_{q_9} S_{q_8} \\ -C_{q_8} C_{q_9} \end{pmatrix}. \notag \end{align} $$
The joint angles $q_4, q_5, q_6, q_7, q_8$, and $q_9$, are then obtained as, according to the usual solutions of the inverse geometric model equations [30]:

$$ 
\begin{aligned}
q_5 &= \pm \operatorname{atan2} \left( -{}^3u_x, \sqrt{1 - {}^3u_x^2} \right), & q_4 &= \operatorname{atan2} \left( -\frac{{}^3u_y}{C_{q_5}}, -\frac{{}^3u_z}{C_{q_5}} \right), \\
q_7 &= \pm \operatorname{atan2} \left( -{}^3v_x, \sqrt{1 - ({}^3v_x)^2} \right), & q_6 &= \operatorname{atan2} \left( -\frac{{}^3v_y}{C_{q_7}}, -\frac{{}^3v_z}{C_{q_7}} \right), \\
q_9 &= \pm \operatorname{atan2} \left( -{}^3w_x, \sqrt{1 - ({}^3w_x)^2} \right), \text{ and} & q_8 &= \operatorname{atan2} \left( -\frac{{}^3w_y}{C_{q_9}}, -\frac{{}^3w_z}{C_{q_9}} \right).
\end{aligned}
$$

Finally, we can substitute Eq. (32) into Eq. (33) to derive the joint variables for the peripheral legs, referred to as $q_4$ through $q_9$.

Having completed the analysis of the peripheral legs, we now proceed to the second step: determining the configuration of the central leg. In this phase, we calculate the joint variables referred to as $q_{10}, \dots, q_{14}$, which define the angular position of the two U-joints and the prismatic joint constituting the central leg. To achieve this, we decouple the position and orientation components of the mechanism (usual methodology in robotics to derive the inverse kinematic model of a 6-DoF serial robot with a spherical wrist) and use the following transformation matrix ${}^3\mathbf{T}_0$ (defined using the DH parameters from Tab. 1 and Fig. 3d):

$$ {}^3\mathbf{T}_0 = \operatorname{Rot}(\mathbf{a}_{10}, q_{10}) \operatorname{Rot}(\mathbf{a}_{11}, q_{11}) \operatorname{Trans}(\mathbf{s}_{12}, q_{12}) \operatorname{Rot}(\mathbf{a}_{13}, q_{13}) \operatorname{Rot}(\mathbf{a}_{14}, q_{14}) \operatorname{Rot}(\mathbf{a}_0, \gamma). \quad (34) $$

Firstly, for the position component, which will help us find $q_{10}, q_{11}$, and $q_{12}$, we consider the transformation matrix ${}^3\mathbf{T}_{12}$ derived from Eq. (34), which is expressed as:

$$ {}^3\mathbf{T}_{12} = \operatorname{Rot}(\mathbf{a}_{10}, q_{10}) \operatorname{Rot}(\mathbf{a}_{11}, q_{11}) \operatorname{Trans}(\mathbf{s}_{12}, q_{12}). \quad (35) $$

The position vector ${}^3\mathbf{P}_0$ relative to the homogeneous transformation matrix ${}^3\mathbf{T}_{12}$ is given by:

$$ ( {}^3\mathbf{P}_0, 1)^T = {}^3\mathbf{T}_{12} (0, 0, 0, 1)^T. \quad (36) $$

Thus, the vector ${}^3\mathbf{P}_0$ is expressed as:

$$ {}^3\mathbf{P}_0 = \begin{pmatrix} {}^3P_{0x} \\ {}^3P_{0y} \\ {}^3P_{0z} \end{pmatrix} = \begin{pmatrix} q_{12} S_{q_{11}} \\ -q_{12} C_{q_{11}} S_{q_{10}} \\ -q_{12} C_{q_{11}} C_{q_{10}} \end{pmatrix}. \quad (37) $$

To compute the joint variables $q_{12}, q_{11}$, and $q_{10}$, we apply the following inverse kinematic formulas:

$$ 
\begin{aligned}
q_{12} &= \sqrt{({}^3P_{0x})^2 + ({}^3P_{0y})^2 + ({}^3P_{0z})^2}, \\
q_{11} &= \pm \operatorname{atan2} \left( \frac{{}^3P_{0x}}{q_{12}}, \sqrt{1 - \left(\frac{{}^3P_{0x}}{q_{12}}\right)^2} \right) \text{ and} \\
q_{10} &= \operatorname{atan2} \left( -\frac{{}^3P_{0y}}{q_{12} C_{q_{11}}}, -\frac{{}^3P_{0z}}{q_{12} C_{q_{11}}} \right).
\end{aligned}
$$

Then, the orientation component is used to calculate $q_{13}$ and $q_{14}$ based on Eq. (34) and the known variables $q_{10}, q_{11}$, and $q_{12}$. This results in:

$$ \operatorname{Trans}(\mathbf{s}_{12}, -q_{12}) \operatorname{Rot}(\mathbf{a}_{11}, -q_{11}) \operatorname{Rot}(\mathbf{a}_{10}, -q_{10}) {}^3\mathbf{T}_0 = \operatorname{Rot}(\mathbf{a}_{13}, q_{13}) \operatorname{Rot}(\mathbf{a}_{14}, q_{14}) \operatorname{Rot}(\mathbf{a}_0, \gamma). \quad (38) $$

Based on (38), the orientation vector $\mathbf{t}$ relative to the equation presented before is given by:

$$ (\mathbf{t}^T, 0)^T = \operatorname{Trans}(\mathbf{s}_{12}, -q_{12}) \operatorname{Rot}(\mathbf{a}_{11}, -q_{11}) \operatorname{Rot}(\mathbf{a}_{10}, -q_{10}) {}^3\mathbf{T}_0(0, 0, 1, 0)^T, \quad (39) $$

This represents the transformation to the position at the center of the central leg (see Fig. 3d). At this point, the components of the vector $\mathbf{t}$ are given by:

$$ \mathbf{t} = \begin{pmatrix} t_x \\ t_y \\ t_z \end{pmatrix} = \begin{pmatrix} -C_{q_{13}} & C_{q_{14}} \\ -C_{q_{14}} & S_{q_{13}} \\ -S_{q_{14}} & 0 \end{pmatrix}. \quad (40) $$
Finally, the angles $q_{14}$ and $q_{13}$ are determined:

$$q_{14} = \pm \operatorname{atan2}\left(-t_z, \sqrt{1-(t_z)^2}\right) \quad \text{and} \quad q_{13} = \operatorname{atan2}\left(-\frac{t_y}{C_{q_{14}}}, -\frac{t_x}{C_{q_{14}}}\right).$$