## Abstract

The demand for robots capable of performing collaborative tasks requiring interactions with the environment is on the rise. Safe interactions with the environment require attributes such as high dexterity and compliance around obstacles, while still maintaining the requisite stiffness levels for payload manipulation. Such attributes are inherent to biological musculoskeletal systems. Motivated by this realization, this paper proposes a cable-actuated spatial joint with variable stiffness, inspired by the tensegrity principles found in biological musculoskeletal systems. The paper provides a detailed analysis of the joint’s mobility and mechanism kinematics. Based on the limits of the actuation forces, the paper also presents the wrench-feasible workspace of the joint. The paper also outlines the conditions that the cable actuation forces must satisfy to maintain the static equilibrium of the joint. The stiffness modeling presented in this work demonstrates the modulation of stiffness bounds as a function of cable actuation forces. Furthermore, the stiffness modulation as a function of the geometrical parameters is also presented.

## 1 Introduction

Rigid robotic manipulators of serial and parallel architectures are well established in the industrial automation scene, owing to their inherent stiffness, heavy-duty high-speed payload manipulation capabilities, and high repeatability. Often, the purpose of such manipulators is to repeatedly perform tasks of industrial specifications and may not be suitable for other tasks, particularly those involving safe interactions with the environment. Despite the availability of anthropomorphic robotic arms, which draw inspiration from the human hand [1], they predominantly provide human-like dexterity, but lack attributes such as variable stiffness and obstacle avoidance, making them unsuitable for collaborative tasks. With the ever-increasing applications that demand human–robot–environment interactions, research has also expanded to develop compatible robotic systems.

Recent focus has shifted toward the development of robotic manipulators inspired by biological systems. These systems possess attributes such as high dexterity and flexibility, which are conducive to safe interactions with the environment, while maintaining the requisite stiffness for load-bearing capabilities.

A vast majority of the developments happening in bio-inspired robots fall under the broad category of soft robotics. The environment adaptability of the soft robots is made possible by the various kinds of soft actuators. For instance, the frog-inspired soft robot [2] utilizes articulated soft pneumatic actuators to connect the bones of its limbs. These actuators also serve as the driving component for emulating bionic swimming movements. In contrast, a collaborative robot proposed in the work of Stilli et al. [3] replaces conventional rigid links with variable stiffness links (VSLs) in a serial robot configuration. These VSLs interconnected using servo motors and actuated by a combination of pneumatics and tendons offer both extension–contraction capabilities and the ability to achieve desired stiffness levels at specific configurations. Similarly, an implementation employing both pneumatics and tendon-driven actuation for a continuum manipulator is put forth in Ref. [4]. Stiffness modulation in such manipulators is enabled by the antagonistic actuation of both tendons and pneumatics, making them particularly suitable for assisting in minimally invasive surgeries [5].

Beyond air pressure and tendon-driven systems, numerous other methods exist for energizing a continuum manipulator. Reference [6] does a comprehensive classification of soft actuators into two categories: (1) active (antagonistic arrangements) and (2) semi-active actuators. In active actuation/actuator, the active elements (of the same or different kinds, for example: tendons and shape memory materials) are applied to oppose each other, whereas in semi-active actuation/actuator, the variable stiffness is made possible by modulating the intrinsic mechanical properties of the material itself (example: granular jamming).

Another dimension of actuation categorization considers whether the actuation occurs within the manipulator itself, termed intrinsic actuation (e.g., using pneumatic and hydraulic chambers or shape memory effects), or if it takes place external to the manipulator, with forces transmitted through mechanical transmissions, referred to as extrinsic actuation (e.g., tendon-driven systems) [7]. Extrinsic actuation is often preferred for miniaturization purposes. A comprehensive review of actuation methods for soft robots is presented in Ref. [8]. Reference [9] made a comprehensive classification of applications, where continuum robots could be deployed.

Snake-like soft robots represent a distinct category, with further classification based on their backbone characteristics and chosen method of actuation [10]. These manipulators possess the unique capability to rotate around their own backbone axis, a vital feature for intricate medical suturing procedures. Depending on whether the backbones used are discrete or continuous, snake-like robots may be classified as continuum robots or discrete hyper-redundant robots (example: an elephant trunk-like manipulator [11]). However, with the variety of solutions that are being employed, the boundary that separates snake-like robots from continuum robots is indistinguishable [7].

As the loads to be supported by collaborative robots increase, the quest for stiffer solutions becomes imperative. Once again, bio-inspiration provides valuable insights. Larger terrestrial animals support their relatively heavy skulls and their own body weights with their spines. Drawing inspiration from this biological concept, spine-like manipulators [12] have been proposed. These manipulators are composed of hard vertebral elements and soft inter-vertebral disks, incorporating shape memory alloy coil springs for actuation.

Considering the progression of increasing stiffness, from continuum manipulators to hyper-redundant robots to spine-like soft robotic manipulators, the next in line would be the manipulators derived from the principles of tensegrity (a portmanteau of tensional integrity). Although the classical definition of tensegrity requires the stabilization of floating compression elements by a continuous network of tension elements (for the historical perspective, see Ref. [13]), the modern definition does not insist on the same. This led to the further classification of tensegrity based on the number of compression elements, connected at a node through torque-less joints [14].

According to Skelton and De Oliveira [14], a *class k* tensegrity system has *k* compression elements connected at a node. A notable example of this concept is the development of a class-changing variable stiffness tensegrity spine, as proposed in Ref. [15]. Tensegrity manipulators offer unique advantages over other bio-inspired solutions. They have the capability to exhibit compliance while still being capable of handling impact loads, as observed in the transfer of impact forces, akin to the woodpecker’s neck during pecking.

It is noteworthy that the actuation of biological musculoskeletal systems aligns with the principles of tensegrity. This alignment has been a rich source of inspiration for the development of bio-inspired manipulators [16,17]. Recent work in this field includes the development of a planar, bird-neck-like manipulator, employing planar anti-parallelogram (“X”) joints, as detailed in Refs. [18,19].

The planar “X” joints and their co-actuation characteristics [20,21] paved the way to the development of spatial two degrees-of-freedom (DOFs) cable-actuated wrist-like joints. A noteworthy example in this category is the quaternion joint [22,23]. In recent studies, the quaternion joints were stacked serially to synthesize lightweight, cable-actuated spatial manipulators with high flexibility and a good rigidity balance for precision operations [24]. A hyper-redundant robot composed of stacked quaternion joints was proposed recently for on-orbit servicing [25]. Evidently, the quaternion joints are widely adopted for developing spatial, cable-actuated lightweight manipulators.

According to the mobility analysis [22], the quaternion joint demonstrates two DOFs of instantaneous mobility in its home configuration, with the base and moving platform parallel to each other. At any other arbitrary configuration, it has been observed that each of the three limbs imposes two constraints on the moving platform. This results in a total of six independent constraints, which theoretically lead to a mobility of zero for the quaternion joint. However, despite this theoretically zero mobility, quaternion joints exhibit a practical mobility of two DOFs. In this case, the moving platform follows a motion akin to a sphere rolling over a fixed sphere. This practical mobility is achieved through specific sets of link lengths that impart the moving platform with the described rolling-like motion, with minimal error. This error is so negligible that the joint clearances and link flexures can accommodate it. In essence, for certain sets of link dimensions, some of the constraints that would theoretically immobilize the mechanism could become interdependent due to the influence of joint clearances and link flexures. As a result, the mechanism acquires practical mobility of 2 [26] at the price of internal stresses.

In this work, a new variant of the quaternion joint is introduced with its specific details described in Sec. 2. Section 3 will feature a mobility analysis based on screw theory, following a similar approach as in Ref. [22], to demonstrate finite mobilities at both the home and an arbitrary configuration. This analysis serves as a critical determination of whether the proposed joint qualifies as a true mechanism. In the subsequent sections, through the static analysis and stiffness modeling, this work presents the advantages of the joint in terms of its ability to be synthesized at desired proportions without constraints on the link lengths, a characteristic that sets it apart from the quaternion joint.

## 2 Description of the Mechanism

The mechanism, as depicted in Fig. 1, comprises two R⊥RS limbs (R for revolute, S for spherical, symbol ⊥ means that the axis of the next R joint is orthogonal to the previous one) and a single R⊥R‖R⊥R limb that connects the fixed base with a radius of *r* to a mobile platform of the same radius. Within the RRS limb, there are two links: a smaller link of length *h* is connected to the fixed base through an R-joint, while the distal end of this smaller link *h* is linked to the proximal end of a larger link *l* by another R-joint. The distal end of the larger link *l* is connected to the moving platform by an S-joint, positioned at a perpendicular offset of *h* from the platform frame {P}.

The RRRR limb consists of three limbs with link lengths of *h*, *l*, *h* connected in series by four R joints, extending from the base to the moving platform. Three local limb frames, denoted as {a}, {b}, and {c}, are attached to the proximal ends of these limbs. The *Z* axis of these frames aligns with the first revolute joint of their respective limbs. The first limb is an RRRR type, while the remaining two are RRS limbs. The distal ends of these limbs are linked with frames {d}, {e}, and {f}, respectively. The *X* axis of {d} aligns with the distal R joint of the RRRR limbs. The origins of {d}, {e}, {f}, and {P} lie in a plane parallel to the top platform. The mechanism is actuated using three cables, each equipped with parallel-mounted springs having spring constants denoted as “*k*_{spring}.” These cables are connected to the base and moving platform at radii of *r*′ at angular intervals of $90deg$, $210deg$, and $330deg$, respectively, measured from the +*X* axis.

## 3 Mobility Analysis of the 2-RRS-1-RRRR Mechanism

The mobility analysis of the proposed mechanism is conducted using a geometric approach based on screw theory. This approach was already employed to assess the mobility of the quaternion joint as detailed in Ref. [22]. Figure 2 illustrates the motion screws associated with the joints, the constraint screws imposed by the limbs on the moving platform, and the motion screws of the moving platform at two distinct configurations: (1) the home configuration or zero-orientation state and (2) an arbitrary configuration. The DOFs of the revolute and the spherical joints, which can be decomposed into three intersecting revolute joints, are depicted using zero-pitch motion screws represented as $$$_{ij} (*i*th joint of the *j*th limb). These motion screw axes align with the axes of the respective revolute joints. Various screw systems of the two configurations are analyzed in detail in the following sections.

### 3.1 Case 1: Home Configuration.

In Fig. 2(a), the motion screws are depicted using solid single-headed arrows (→). For the RRRR limb, the motion screws corresponding to the four R joints are labeled as $$$_{11}, $$$_{21}, $$$_{31}, and $$$_{41}. Reciprocal to the four-screw system, RRRR limb is a two-screw constraint system, consisting of a zero-pitch force constraint $$11c$ represented by a dashed single-headed arrow and an infinite-pitch couple constraint $$21c$ represented by a solid double-headed arrow ($\u27f7$). The force constraint $$11c$ lies along a straight line that connects the centers of the first and fourth R joints of the RRRR limb. The couple constraint $$21c$ is perpendicular to all four motion screws and lies perpendicular to the base and the top platforms. Regarding the RRS limbs, there exist zero-pitch constraint screws $$1jc$ (*j* = 2, 3), which are reciprocal to the five-screw motion systems of their respective limbs. The constraint screws $$1jc$ are depicted in Fig. 2(a) as dashed single-headed arrows. These force constraints intersect all the motion screws of the respective RRS limbs and are directed along a straight line that connects the centers of the first R joint and the S joint.

At a zero-orientation configuration, the zero-pitch constraints $$11c$ of the RRRR limb and $$12c$ and $$13c$ of the two RRS limbs intersect at a point. This alone would restrict all three translations of the mechanism, while the remaining couple constraint $$12c$ suppresses the instantaneous rotations about the vertical *Z* axis. Accordingly, the reciprocals of these constraint screws result in the zero-pitch motion screws of the top platform $$1p$ and $$2p$, which are depicted as thick single-headed arrows. At the home configuration, the motion screws are parallel to the *X* and *Y* axes of the base frame {B} but pass through a point where the three force constraints intersect. An exactly similar behavior was observed for the quaternion joint at its home configuration [22].

### 3.2 Case 2: Non-parallel Configuration.

For an arbitrary configuration as depicted in Fig. 2(b), the force constraints $$11c$ of the RRRR limb and $$12c$, $$13c$ of the two RRS limbs remain the same as previously described, with the only difference being that they no longer intersect at a single point. The couple constraint $$21c$ of the RRRR limb is no longer present; it has been replaced by an additional force constraint $$21c$ of the RRRR limb. This new constraint is illustrated in Fig. 2(b) as a dashed single-headed arrow. Its direction is parallel to the motion screws $$$_{21} and $$$_{31} (corresponding to the second and third joints of the RRRR limb) and passes through the intersection point of the motion screws $$$_{11} and $$$_{41} (associated with the first and fourth joints of the RRRR limb).

The motion screws of the moving platform, which are reciprocal to the four zero-pitch constraint forces $$11c$, $$21c$, $$12c$, and $$13c$, form a two-screw system consisting of two non-zero pitch screws, $$1p$ and $$2p$, represented with solid two-headed arrows ($\u21a0$). These non-zero pitch screws represent two rotations around the axes of $$1p$ and $$2p$, accompanied by parasitic translations along the same axes, characterized by their finite pitch values. However, the presence of these two motion screws confirms that the proposed mechanism possesses a mobility of two, contrary to the observations made with the quaternion joint. A brief matlab implementation of the above analysis, as presented in Appendix B, is used to identify three sets of screws, namely, the motion screws of all the limbs ($$$_{ij}, where *i* ranges from 1 to 4 for *j* = 1, and *i* ranges from 1 to 5 for *j* = 2, 3), the constraint screws imposed by each limb on the top platform ($$kjc$, where *k* ranges from 1 to 2 for *j* = 1 and *k* = 1 for *j* = 2, 3), and the instantaneous motion screws ($$mp$, where *m* = 1, 2) of the top platform.

## 4 Analytical Formulation of the Inverse Kinematics Problem

The proposed mechanism is actuated by three cables as shown in Fig. 1. This section determines the three cable lengths for a given set of two independent task-space variables. As shown in Fig. 1, two frames {B} and {P} are attached to the base and top platforms, respectively. The pose of the top platform with respect to the base platform is defined by six variables, three of which are the position coordinates $p\u2192=[pxpypz]T$ and the rest being the *Z*–*Y*–*X* Euler angles *ϕ* − *ψ* − *θ*. Since the mechanism has two DOFs, *θ* and *ψ* are selected as input variables for the inverse kinematics problem, while the rest of the task-space variables (*ϕ*, *p*_{x}, *p*_{y}, *p*_{z}) will be solved using the constraint equations presented in next subsection. Next, the cable lengths will be determined.

*c*

_{.}and

*s*

_{.}represent $cos(.)$ and $sin(.)$, respectively. The sub-matrix composed of the first three rows and columns of

^{B}

**T**

_{P}is the rotation matrix

**R**, corresponding to the

*Z*–

*Y*–

*X*Euler-angles.

*a*)

*b*)

*c*)

*a*)

*b*)

*c*)

The following formulation of constraint equations considers the offset between all of the R⊥R joints, “*h*,” as zero. The offset “*h*” was an artifact carried over from the quaternion joint, where it was essential for the mechanism to closely approximate a pure spherical rolling motion, thus exhibiting a practically observable mobility of 2. Considering the offset “*h*” to be zero does not invalidate the generality of the constraint equations, nor does it affect the mobility of the proposed mechanism. It however simplifies the formulation of the constraint equations, which are discussed below.

### 4.1 Constraint Equations.

With “*h*” considered to be zero, the RRRR limb becomes a U-U (U-universal joint) limb, which has the following two constraints [27]:

- (C.1)
The length between the centers of the two U-joints is constant and is equal to the link length “

*l*.” - (C.2)
The origins of the frames (see Fig. 1) {B}, {a}, {d}, and {P} lie on a planar quadrangle.

The five-DOF RRS limb, when “

*h*” is considered 0, becomes a U-S joint, which has a single constraint. - (C.3)
The length between the centers of the U and the S joints is constant and is equal to the link length “

*l*.”

*a*)

*b*)

*c*)

*a*)

*b*)

*c*)

*a*

_{i},

*b*

_{i},

*c*

_{i},

*i*= 1, 2, 3, are detailed in Appendix C. These spheres are projected onto the plane defined by the frames {d}, {e}, and {f}, resulting in the following circles:

*a*)

*b*)

*c*)

*t*/(1 +

*t*

^{2}) and (1 −

*t*

^{2})/(1 +

*t*

^{2}) as per the tangent half-angle substitutions, where $t=tan(\varphi /2)$. Subtracting Eq. (6

*b*) from Eq. (6

*a*) and Eq. (6

*c*) from Eq. (6a) yields the following system of linear equations:

*p*

_{x}(

*θ*,

*ψ*,

*p*

_{z},

*t*) and

*p*

_{y}(

*θ*,

*ψ*,

*p*

_{z},

*t*) as functions of unknowns

*p*

_{z}and

*t*and known parameters

*θ*and

*ψ*. Back-substituting

*p*

_{x}(

*θ*,

*ψ*,

*p*

_{z},

*t*) and

*p*

_{y}(

*θ*,

*ψ*,

*p*

_{z},

*t*) into the first circle equation (Eq. (6a)) yields a polynomial of degree two in

*p*

_{z}and

*t*as

*p*

_{z}. From Eq. (10),

*p*

_{z}can be expressed as a function of

*θ*,

*ψ*,

*t*, which can then be back-substituted into Eq. (9). The resulting expression is a polynomial expression that can be written as a product of four factors of the form

*p*(

*t*) is a polynomial of degree 10 in

*t*whose expression is very lengthy and cannot been reproduced here. Clearly, the first two factors are strictly positive and the third one cannot vanish when

*ϕ*,

*ψ*,

*θ*∈ ] − (

*π*/2)(

*π*/2)[. Once the solution is solved from

*p*(

*t*) = 0, expressions for

*p*

_{z},

*p*

_{y}, and

*p*

_{x}can be obtained by back-substitutions into Eqs. (9) and (8).

### 4.2 Actuation Cable Lengths.

### 4.3 Numerical Example.

The geometric parameters of the mechanism for this example are selected as [*l*, *r*, *r*′] = [400 mm, 100 mm, 150 mm]. For the particular case of $[\theta ,\psi ]=[0deg,40deg]$, the real, feasible solutions of the remaining four task-space variables *ϕ*, *p*_{x}, *p*_{y}, *p*_{z} and the actuation cable lengths (*l*_{c})_{i} are given in Table 1. The first column gives the real roots of *p*(*t*) = 0. The second column gives the feasible solutions of *p*_{z}, which are obtained by solving Eq. (9) after back-substituting the value of *t*. The solutions satisfying the constraint equations (Eqs. (5a), (5b), (5c), and (10)) are considered feasible. The third and fourth columns give *p*_{x} and *p*_{y}, obtained by back-substitutions of numerical values *p*_{z} and *t* in Eq. (8). Columns 5, 6, and 7 give the cable actuation lengths obtained from Eq. (12). Considering the solutions with *ϕ* ∈ ] − (*π*/2)(*π*/2)[ (where $\varphi =2\u22c5arctan(t)$) and *p*_{z} > 0, only the solution corresponding to the third row of Table 1 is practical. The four theoretical solutions are shown in Fig. 3. The sole practical solution is given in Fig. 3(c).

t_{roots} | (p_{z})_{roots} | (p_{x})_{roots} | (p_{y})_{roots} | (l_{1})_{c} | (l_{2})_{c} | (l_{3})_{c} |
---|---|---|---|---|---|---|

−17.7004 | 0.0000 | −107.6541 | −381.2781 | 686.3509 | 261.7253 | 425.2130 |

0 | −323.8528 | −134.1442 | 7.3223 | 350.6122 | 251.2184 | 450.0194 |

0 | 323.8528 | 134.1442 | 7.3223 | 350.6122 | 450.0194 | 251.2184 |

−17.7004 | −0.0000 | 107.6541 | −381.2781 | 686.3509 | 425.2130 | 261.7253 |

t_{roots} | (p_{z})_{roots} | (p_{x})_{roots} | (p_{y})_{roots} | (l_{1})_{c} | (l_{2})_{c} | (l_{3})_{c} |
---|---|---|---|---|---|---|

−17.7004 | 0.0000 | −107.6541 | −381.2781 | 686.3509 | 261.7253 | 425.2130 |

0 | −323.8528 | −134.1442 | 7.3223 | 350.6122 | 251.2184 | 450.0194 |

0 | 323.8528 | 134.1442 | 7.3223 | 350.6122 | 450.0194 | 251.2184 |

−17.7004 | −0.0000 | 107.6541 | −381.2781 | 686.3509 | 425.2130 | 261.7253 |

## 5 Static Analysis and Wrench-Feasible Workspace

This section presents the expressions of static equilibrium of the 2-RRS-1-RRRR mechanism. The mechanism achieves static equilibrium when the wrenches contributed by the mechanism’s springs and gravity potentials are balanced by the wrenches contributed by the cable-actuation forces. Here, the wrench of potential is defined as the partial derivatives of the potential, with respect to the independent task-space coordinates [*θ*, *ψ*]. The section also introduces the concept of the wrench-feasible workspace (WFW) for the mechanism. The WFW is defined as the set of poses that satisfy the above-mentioned constraints and in which the manipulator can balance a bounded set of external wrenches [28]. In our case, the external wrenches are contributed by the springs and the gravity effects only. The expressions of the potential energy contributed by gravity, springs, and cables are outlined below.

### 5.1 Gravitational Potential Energy.

*l*” with respect to the base frame {B} are denoted as “

*h*

_{1},” “

*h*

_{2},” and “

*h*

_{3}” for the RRRR limb and the two RRS limbs, respectively. The center of gravity of the top platform is situated at a height of “

*h*

_{4}.” The expressions for

*h*

_{1}· · ·

*h*

_{4}are provided below:

*a*)

*b*)

*c*)

*d*)

*h*of these connecting limbs is zero and non-zero will be considered in the static analysis. When

*h*= 0, their centers of gravity are at the elevations of

*h*

_{5}= 2

*h*

_{1},

*h*

_{6}= 2

*h*

_{2}, and

*h*

_{7}= 2

*h*

_{3}, respectively. Moreover, the intermediate links connecting the three limbs to the base platform have their center of gravity at zero elevation when

*h*= 0 (

*h*

_{8}=

*h*

_{9}=

*h*

_{10}= 0). The total gravitational potential energy of the entire mechanism is given as

*g*= 9.81 m/s

^{2}. The individual masses

*m*

_{i}are given in Appendix A. The plot of the gravitational potential energy

*U*

_{g}against the independent input parameters [

*θ*,

*ψ*] governed by Eq. (14) is shown in Fig. 4.

### 5.2 Spring and Cable Potential Energy.

*l*

_{c})

_{i}are the cable lengths as per Eq. (12). The spring constants

*k*

_{s}are given in Appendix A. The plot of

*U*

_{s}against the independent parameters [

*θ*,

*ψ*] as per Eq. (15) is shown in Fig. 5. The potential energy contributed by the cables is

*F*

_{i}are the cable actuation forces.

### 5.3 Wrench-Feasible Workspace.

*θ*,

*ψ*]. This relationship can be expressed as

The WFW can be computed as the set of platform poses (*θ*, *ψ*) such that the wrenches contributed by the springs and gravity (*G*_{k}, where *k* = *θ*, *ψ*) are fully bounded by the maximum and minimum limits of the cable actuation wrenches ($[\Gamma kmin,\Gamma kmax]$, where *k* = *θ*, *ψ*). This relationship can be expressed as $\Gamma kmin<Gk<\Gamma kmax$. The bounds Γ_{min} and Γ_{max} can be determined by appropriately selecting values for *F*_{i} based on the signs of the coefficients of *F*_{i} in Eq. (18). To estimate Γ_{max}, if the coefficient (− (∂(*l*_{c})*i*/∂*k*)) is greater (respectively less) than 0, *F*_{i} is set to *F*_{max} (respectively *F*_{min}). A similar approach can be used to estimate the lower bound Γ_{min}. The boundaries of WFW corresponding to the variables *θ* and *ψ* are shown in Fig. 6. The boundaries where *G*_{θ} (respectively *G*_{ψ}) breaches the bounds $[\Gamma \theta min\Gamma \theta max]$ (respectively $[\Gamma \psi min\Gamma \psi max]$) projected on the *θψ* plane are presented in Fig. 6(c). A circle centered at the origin of the *θ* − *ψ* plane with a maximal radius, within the WFW bounds, is also presented in Fig. 6(c). The maximally inscribed origin-centered circle (MIOC) can be used as a performance metric for optimizing the link dimensions, spring constants, and actuation force limits in future work.

### 5.4 Static Equilibrium Conditions.

*θ*,

*ψ*]. The conditions for static equilibrium are expressed as

*G*

_{k}and Γ

_{k}are obtained from Eqs. (17) and (18). Upon fixing one of the actuation forces, the remaining two forces can be determined by simultaneously solving the set of equations expressed as Eq. (19), such that the mechanism remains in static equilibrium. Six such conditions are presented below. In the first three conditions,

*F*

_{1},

*F*

_{2}, and

*F*

_{3}are, respectively, kept at $Fmax$, and the rest are determined using Eq. (19). Cases 4–6 are similar except that the forces

*F*

_{1},

*F*

_{2}, and

*F*

_{3}are kept at $Fmin$. The expressions for the actuation forces for the six cases are as follows:

Case 1: $F1=Fmax$

*a*)

*b*)

Case 2: $F2=Fmax$

*a*)

*b*)

- Case 3: $F3=Fmax$(22$F1=\u2212\theta 2\psi 3Fmax\u2212\theta 3\psi 2Fmax+\theta 2G\psi \u2212\psi 2G\theta \theta 2\psi 1\u2212\theta 1\psi 2$
*a*)(22$F2=\u2212\u2212\theta 1\psi 3Fmax+\theta 3\psi 1Fmax+\theta 1(\u2212G\psi )+\psi 1G\theta \theta 2\psi 1\u2212\theta 1\psi 2$*b*) Case 4: $F1=Fmin$

Case 5: $F2=Fmin$

Case 6: $F3=Fmin$

The terms *θ*_{i} and *ψ*_{i}, respectively, denote the partial derivatives ∂(*l*_{c})_{i}/∂*θ* and ∂(*l*_{c})_{i}/∂*ψ*, for *i* = 1, 2, 3. One-dimensional representations of the six cases of the static-equilibrium conditions for a specific instance where *θ* = 0 and *ψ* ranges from −*ψ*_{lim} to *ψ*_{lim} (where $\psi lim$ is considered as the radius of the MIOC) are presented in Fig. 7. It is evident from all the six cases that not all regions within $[\u2212\psi lim\psi lim]$ are feasible when the actuation force limits $Fi\u2208[FminFmax]$ are enforced. Taking case 1 (Fig. 7(a)) as an example, all the three forces *F*_{i}, *i* = 1, 2, 3, are equal to $Fmax$ at *ψ* = 0 or when the top and base platforms are parallel to each other. Any tilt in the direction of +*ψ* (respectively −*ψ*) would result in $F3>Fmax$ (respectively $F2>Fmax$), rendering such configurations infeasible as one of the actuation forces going beyond the actuation bounds exceeds the bounds *F*_{min}*F*_{max}. Figure 7 also outlines the feasible and infeasible regions of the WFW for all six cases, based on the actuation bounds.

## 6 Stiffness Modelling and Stability of the Mechanism Wrench-Feasible Workspace

*θ*,

*ψ*). The general expressions for the four stiffness parameters are

*θ*= 0 and $\psi \u2208[\u2212\psi lim\psi lim]$, where $\psi lim$ represents the radius of the MIOC. Each of the four parameters is evaluated at two limits, labeled as $Case_max$ and $Case_min$. In $Case_max$ (respectively $Case_min$), at least one of the three actuation forces is kept at $Fmax$ (respectively $Fmin$), while the remaining are determined by simultaneously solving the two static-equilibrium conditions. Referring to the static-equilibrium plots in Fig. 7, $F1=Fmax$ (respectively $F1=Fmin$) can never be maintained in the 1D WFW except at

*ψ*= 0. For the range of WFW $\psi \u2208[\u2212\psi lim,0]$, the limits of the stiffness parameters $Case_max$ (respectively $Case_min$) are evaluated by fixing $F2=Fmax$ (respectively $F2=Fmin$) while the rest are determined from the static-equilibrium conditions. For the range of WFW $\psi \u2208[0,\psi lim]$, the limits of the stiffness parameters $Case_max$ (respectively $Case_min$) are evaluated by maintaining $F3=Fmax$ (respectively $F3=Fmin$) while determining the remaining two using the static-equilibrium conditions.

As evident from the two limits, increasing the actuation forces has a positive effect on the stiffness limits. Co-contraction (respectively co-relaxation) of the cable tension would increase (respectively decrease) the stiffness of the joints.

### 6.1 Range of Stability Within the Wrench-Feasible Workspace.

A configuration is considered stable when the stiffness matrix **K** is positive definite. These conditions for positive definiteness are expressed as

*a*)

*b*)

## 7 Effects of Geometrical Parameters on Stiffness Bounds

The stiffness analysis suggests that actuation forces have a positive impact on the stiffness and stability bounds of the mechanism. This section explores the influence of geometric dimensions on the stiffness bounds of the mechanism. For the purposes of the following analysis, the spring constant *k*_{spring}, the actuation force bounds $[FmaxFmin]$, and the mass of each link are given (see Appendix A). A new parameter henceforth referred to as “slenderness ratio” *s*_{r} is introduced, which is defined to be *s*_{r} = *l*/*r*. The remaining link dimension “*r*′” is kept at *r* + 50. The variation of the diagonal stiffness parameters *K*_{θθ}, *K*_{ψψ} at the home configuration $(\theta ,\psi )=(0deg,0deg)$ is plotted in Fig. 9(a). For the two diagonal stiffness parameters, two cases $Case_max$ and $Case_min$ are considered, as in the previous section. $Case_max$ (respectively $Case_min$) is when one of the actuation forces is kept at $Fmax$ (respectively $Fmin$) and the remaining two being determined from the static-equilibrium conditions. It should be noted from Fig. 9(a) that as the slenderness of the joint increases, the gap between the two cases $Case_min$ and $Case_max$ decreases until the parameters become less than zero, rendering the mechanism unstable.

The same analysis is repeated when the offset distance between the R⊥R joints is *h* = 0.5*r*. Figure 9(b) corresponds to the variation of the diagonal stiffness parameters with respect to *s*_{r}. It is observed that the stiffness parameters *K*_{θθ} increase with increasing actuation forces for mechanisms up to a specific slenderness ratio, labeled in Fig. 9(b) as limit 1, beyond which cross-over occurs between $K\theta \theta Case_max$ and $K\theta \theta Case_min$. Beyond limit 1, the stiffness parameter *K*_{θθ} decreases with an increase in actuation forces. A similar cross-over trend at limit 2 is observed for the second diagonal stiffness parameter *K*_{ψψ}. For the range of slenderness ratio between limit 1 and limit 2, with the increase in the actuation forces, the stiffness parameter *K*_{θθ} decreases whereas the *K*_{ψψ} increases. This means that the joint becomes compliant in one of its two degrees-of-freedom and stiff in the other.

## 8 Conclusions

This paper proposed a spatial, cable-actuated two-DOF joint having variable stiffness capabilities. By stacking several such mechanisms in series, remotely actuated by cables, a lightweight manipulator suitable for safe human interaction can be built. Compared to continuum or origami manipulators, the proposed solution can transmit higher forces. The work presented in this paper covers a screw-theory-based mobility analysis and the analytical formulation of inverse kinematics. The paper also covers the static analysis that estimates the WFW for the mechanism and the conditions for actuation forces to maintain the mechanism in static equilibrium. In furthering the static modeling, the stiffness model of the mechanism was also detailed, where the variable stiffness capabilities of the mechanism were also demonstrated. Although presented in 1D, the stability limits determined based on the positive definiteness of the stiffness matrix can be extended to the entirety of the WFW.

Figures 9(a) and 9(b) demonstrate how the geometry can influence the variable stiffness capabilities of the mechanism. A special case of the mechanism, with the offset of *h* ≠ 0, can be made stiff along one independent coordinate and compliant along the other, using the same set of actuation forces. However, challenges remain, such as identifying the optimal link dimensions based on the desired range of the WFW and achieving maximum and minimum stiffness bounds. Future work will involve addressing these challenges, prototyping the mechanism, and potentially extending it into multiple stages to develop a variable stiffness cable-actuated manipulator. A 3D printed prototype of the mechanism of the following dimension is presented in Fig. 10. The dimensions *l* = 108 mm, *r* = 48 mm, and *h* = 12 mm correspond to a slenderness ratio *s*_{r} = 2.25, which has its diagonal stiffness parameters at zero-orientation, to the left of limit 1 (see Fig. 9(b)). The mechanism is stabilized with three springs. However, the experimental verification of the behavior of the stiffness parameters, as per the region to the left of limit 1 in Fig. 9(b), will be presented in future work.

The potential of the proposed mechanism for human–robot interactions can be utilized in the following ways: (1) as a standalone joint and (2) through the serial stacking of several such joints. The applications cover using the variable stiffness cable-actuated mechanism as a standalone joint, or as part of a larger lightweight robotic manipulator, or serially stacking several such joints to form a hyper-redundant snake-like manipulator for industrial inspection, search, and rescue, and minimally invasive surgeries. We also propose a few relatively unexplored applications, where the variable stiffness joint can be integral in making the manipulator lightweight and suitable for human and environment interactions.

## Acknowledgment

The first author would like to express their sincere gratitude to Professor Philippe Wenger from Nantes Université, Ecole Centrale de Nantes, CNRS, LS2N for hosting and facilitating the research presented in this paper. The first author also extends their appreciation to Professor Mathieu Porez for his valuable support in developing the analytical inverse kinematics model. Additionally, the author would like to acknowledge the contributions of research scholars Vimalesh Muralidharan and Nicolas Testard for their dedicated time and insightful discussions, which greatly enriched the research process.

## Funding Data

The first author is financially supported through the Prime Minister’s Research Fellowship (PMRF) scheme funded by the Ministry of Education (MoE), India.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Appendix A: Mechanism Data

The mechanism data used in this study are detailed in Table 2.

Variable | Description | Numerical value |
---|---|---|

l | Link length connecting the base and top platform | 400 mm |

r | Radius of base and top platform | 100 mm |

r′ | Radius of cable attachment points on base and top platforms | r + 50 mm |

k_{spring} | Spring constant | 500 N/m |

$Fmin$ | Minimum cable actuation forces | 0 N |

$Fmax$ | Maximum cable actuation forces | 150 N |

Masses of all links of the joint | ||

m_{1} | Mass of the larger l link of the RRRR limb | 718 g |

m_{2}, m_{3} | Mass of the larger l link of the RRS limb | 685 g |

m_{4} | Mass of the moving platform | 2911 g |

m_{5} | Mass of the intermediate link h connecting RRRR limb to the top platform | 132 g |

m_{6}, m_{7} | Mass of the intermediate link h connecting RRS limb to the top platform | 104 g |

m_{8}, m_{9}, m_{10} | Mass of the intermediate link h connecting the three limbs to the base platform | 132 g |

Variable | Description | Numerical value |
---|---|---|

l | Link length connecting the base and top platform | 400 mm |

r | Radius of base and top platform | 100 mm |

r′ | Radius of cable attachment points on base and top platforms | r + 50 mm |

k_{spring} | Spring constant | 500 N/m |

$Fmin$ | Minimum cable actuation forces | 0 N |

$Fmax$ | Maximum cable actuation forces | 150 N |

Masses of all links of the joint | ||

m_{1} | Mass of the larger l link of the RRRR limb | 718 g |

m_{2}, m_{3} | Mass of the larger l link of the RRS limb | 685 g |

m_{4} | Mass of the moving platform | 2911 g |

m_{5} | Mass of the intermediate link h connecting RRRR limb to the top platform | 132 g |

m_{6}, m_{7} | Mass of the intermediate link h connecting RRS limb to the top platform | 104 g |

m_{8}, m_{9}, m_{10} | Mass of the intermediate link h connecting the three limbs to the base platform | 132 g |

Note: The mass properties are computed from a computer-aided design model developed in ONSHAPE®. Every link is assigned a material ABS with density 1000 kg/m^{3}.

## Appendix B: A Numerical Simulation for Identification of Screw Systems

The inverse kinematics derived from numerical methods of the 2-RRS-1-RRRR joint was presented in our preceding work [29]. For a given set of independent task-space variables [*θ*, *ψ*], the numerical inverse kinematics scheme fully determines the configuration of the joint, and thus the location and direction of all the motion and constraint screws of the joint.

Figure 11(a) depicts the motion screw system for each of the limbs. The nomenclature used $$$_{ij} points to the *i*th screw of the *j*th limb. Here, *i* = 1…4 for the RRRR limb, *i* = 1…5 for the RRS limbs, *j* = 1 for the RRRR limb, and *j* = 1, 2 for the RRS limbs. Figure 11(b) represents the constraint screw systems, which are reciprocal to each of the limb motion-screw-systems, depicted with dashed lines. For the four-screw motion system of the RRRR limb, their reciprocal two-screw constraint system is depicted as $_{11}^{c} and $_{21}^{c} in Fig. 11(b). The constraint $$11c$ is along the longitudinal axis of the RRRR limb and $$21c$ is parallel to both $$$_{21} and $$$_{41}, while passing through the point of intersection of the screws $$$_{11} and $$$_{31}. The five-screw motion system of the RRS limbs exerts a reciprocal one-screw constraint system, depicted as $_{12}^{c} for limb 2 and $_{13}^{c} for limb 3, in Fig. 11(b). Both of these screws, $$12c$ and $$13c$, are along the longitudinal axes of the respective RRS limbs. The two motion screws (finite pitch) of the moving platform, reciprocal to all four constraints are presented with solid lines, labelled as $_{1}^{p} and $_{2}^{p} in Fig. 11(c).

## Appendix C: Expressions for Inverse Kinematics

*a*)

*b*)

*c*)

*d*)

*e*)