Design and Analysis of a Symmetric Joint Module for a Modular Wire-Actuated Robotic Arm with Symmetric Variable-Stiffness Units

Abstract

Collaborative robots are used in scenarios requiring interaction with humans. In order to improve the safety and adaptability of collaborative robots during human–robot interaction, this paper proposes a modular wire-actuated robotic arm with symmetric variable-stiffness units. The variable-stiffness unit is employed to extend the stiffness-adjustment range of the robotic arm. The variable-stiffness unit is designed based on flexure, featuring a compact and simple structure. The stiffness–force relationship of the variable-stiffness unit can be fitted by a quadratic function with an R-squared value of 0.99981, indicating weak nonlinearity. Based on the kinematics and stiffness analysis of the symmetric joint module of the robotic arm, the orientation of the joint module can be adjusted by regulating the length of the wires and the stiffness of the joint module can be adjusted by regulating the tension of the wires. Because of the actuation redundancy, the orientation and stiffness of the joint module can be adjusted synchronously. Furthermore, a direct method is proposed for the stiffness-oriented wire-tension-distribution problem of the 1-DOF joint module. A simulation is carried out to verify the proposed method. The simulation result shows that the deviation between the calculated stiffness and the desired stiffness was less than 0.005%.

1. Introduction

As a result of an aging population, needs for healthcare and assistance are growing. Due to the lack of workers, robots are being developed for healthcare and assistance [1]. There is a similar situation in catering, hotel and agriculture settings. In these applications, the robots usually work in unstructured environments and allow humans to work side by side with them or even collaborate with them. This kind of robots is called a collaborative robot, and they are specifically designed to work with humans in a shared workspace or assist humans in performing certain tasks safely [2]. Human–robot interaction (HRI) presents challenging issues regarding the need to guarantee humans’ safety during various tasks with robots [3]. Generally, robots can be divided into two types, hard robots or soft robots (including flexible robots), based on their compliance [4]. Hard robots always have a rigid structure, heavy weight and high stiffness, which limit their ability to interact with humans and their environment. Soft robots are made from soft materials or are actuated by pneumatic [5], fluid [6], or smart materials [7] (e.g., dielectric elastomers [8]). They are inherently compliant, adaptable, and safe. Due to these advantages, soft robots have great potential to interact with the humans and the environment more safely and more adaptively than traditional rigid robots [9,10].

Recent decades have seen the design, modelling, and control of soft robots. G. Runge et al. [11] presented a framework for the design and modelling of soft robots that effectively combined finite-element analysis, continuum robot modeling, and machine learning. H. Wang et al. [12] summarized the design, actuation, and function of magnetic soft robots. C. Zhang et al. [13] summarized modular units and connection mechanisms of modular soft robots and discussed their applications and challenges. T. Yoon et al. [14] proposed a kinematics-informed neural network (KINN) to model soft robots with pseudo-rigid bodies, which enabled the adoption of rigid-body kinematics in soft robots for sample-efficient model identification and control. B. Pawlowski et al. [15] proposed soft robots actuated by twisted-and-coiled actuators (TCAs) and established a general modelling framework for TCA-actuated soft robots. V. Subramaniam et al. [16] designed a soft gripper that consisted of three soft fingers and a stiff palm and investigated the control of grasp postures. Khairudin, M. et al. [17] investigated the dynamic modelling of a two-link flexible robot manipulator and discussed the effects of payload on the dynamic characteristics of the flexible manipulator. J. J. Jui et al. [18] proposed a new hybrid identification algorithm named average multiverse optimizer and sine cosine algorithm (AMVO-SCA), which achieved better performance in modelling of the flexible manipulator system than other methods.

In this paper, a soft robotic arm actuated by wires inspired by the human arm is proposed. The human arm is mainly composed of bones, joints, and muscles, and it is actuated by the soft muscles. The stiffness of the human arm can be adjusted by changing the muscle strength. The human arm can work safely and adaptively in daily life. The bio-inspired wire-actuated robotic arm has the potential to improve the safety and adaptability of human-robot interaction.

The wire-actuated robot is a type of soft parallel robots that employs soft wires instead of rigid links to actuate the robot. Compared with rigid robots, wire-actuated robots have the advantages of low inertia, large workspace, high safety, and adaptability. Because of these advantages, wire-actuated robots are applied widely in many fields, such as lifting and positioning [19,20], detection and maintenance [21,22,23], wearable rehabilitation and assistance [24,25,26,27,28,29], medical surgery [30,31,32], soft robotic arms and hands [33,34,35,36], and bionic robots [37]. Researchers have designed different wire-actuated robots and have conducted research on many subjects, such as kinematics and statics analysis [38,39,40,41,42], workspace analysis [43,44], wire tension analysis [45], stiffness analysis [46], dynamics analysis [47,48], path and trajectory planning [49,50,51], motion and compliance control [52,53,54,55,56], and so on. These publications are valuable for subsequent research work and applications of the wire-actuated robots.

In this paper, we adopt a modular design method to design a wire-actuated human-like robotic arm that can simplify the structure of the robotic arm and make it easy to analyze, control, reconstruct, and maintain. The modular wire-actuated robotic arm consists of three joint modules in series, i.e., a 3-DOF shoulder joint module, a 1-DOF elbow joint module, and a 3-DOF wrist joint. The wire actuation units are located at the base of the robotic arm. Since the wires have the property of unidirectional force transmission, the wire-actuated robotic arm is redundantly actuated and the stiffness of the robotic arm can be adjusted by regulating the wire tensions. These features give the proposed modular wire-actuated robotic arm low inertia, high load-to-weight ratio, large workspace, variable stiffness, and high safety. Due to the limitation of the wire stiffness, the stiffness variation of the wire-actuated robotic arm is small. To increase the stiffness-adjustment range, a variable-stiffness unit (VSU) is designed to be placed in the wire actuation system [57]. However, the existing VSUs generally have the following shortcomings [58,59,60]: (1) the VSU is made up with several parts, making it complex, large, and heavy; (2) the assembly of the VSUs can not be standardized; (3) the stiffness–force relationship of the VSU is highly nonlinear. In this paper, a novel VSU is proposed based on the flexure with a simple, symmetric, and compact structure and a slightly nonlinear stiffness–force relationship. Furthermore, because of the redundant actuation property, the wires of the joint modules and the robotic arm can be divided into two groups: wires for orientation adjustment and wires for stiffness adjustment. The orientation can be adjusted by regulating the length of the wires, and the stiffness can be adjusted by regulating the tension of the wires. Thus, the orientation and stiffness of the joint modules and robotic arm can be adjusted synchronously. In order to achieve the desired stiffness of the joint module or robotic arm, it is necessary to solve the stiffness-oriented wire tension distribution (SWTD) problem. For multiple-DOF joint module, since the stiffness model is complicated [61], it is difficult to solve the SWTD problem directly. The SWTD problem is thus reformulated as an optimization problem [62]. In this paper, we focus on the kinematics, statics, stiffness, and wire-tension analysis of the 1-DOF joint module and the stiffness analysis of the VSU. The SWTD problem of the 1-DOF joint module is solved directly based on the statics and stiffness analysis. A simulation is carried out to verify the proposed analysis.

The main contributions of this paper are as follows: (1) the design of a modular wire-actuated robotic arm with variable-stiffness units to improve the safety and adaptability of collaborative robots; (2) the establishment of a stiffness model of the joint module and a proposed direct algorithm for the stiffness-oriented wire-tension-distribution problem of the joint module.

2. Design of a Modular Wire-Actuated Robotic Arm with Symmetric Variable-Stiffness Devices

This paper describes the design of a wire-actuated robotic arm inspired by the human arm in which rigid links are employed instead of the skeleton of the human arm and flexible steel wires are employed instead of the muscles of the human arm. For the convenience of fabrication and maintenance, the wire-actuated robotic arm is made up of three joint modules: the shoulder-joint module, the elbow-joint module, and the wrist-joint module. The three joint modules are connected in series to build a modular wire-actuated robot arm, as shown in Figure 1. The end of each wire for the robotic arm is connected with the wire actuation unit. All the wire actuation units are installed on the base of the robotic arm. This arrangement can reduce the moment of inertia and improve the security of the robot.

The three joint modules of the modular wire-actuated robotic arm can be divided into two types: one-degree-of-freedom (1-DOF) joint modules and three-degree-of-freedom (3-DOF) joint modules. The conceptual designs of the 1-DOF and 3-DOF joint modules are shown in Figure 2. The 1-DOF joint module has a symmetric structure and consists of a movable platform, a static platform, a revolute joint, and wires. The 3-DOF joint module consists of a movable platform, a static platform, a spherical joint, and wires. Since the variation range of the steel wire is limited, a variable-stiffness unit (VSU) is developed and placed along the wire to extend the variable-stiffness range of the robotic arm and improve its flexibility and safety.

However, the existing VSUs generally have complex structures, large volumes, heavy masses, and highly nonlinear stiffness–force relationships. In order to overcome these disadvantages, in this paper, a novel VSU is proposed based on flexure. As shown in Figure 3, the proposed VSU has a simple, symmetric, and compact structure. The VSU has a rhombus frame, denoted by ABCD. When the force $t_{\mathrm{v}}\in ℝ$ acted at points A and C, the VSU will deform. Due to the symmetric structure, we need only to analyze the deformation behavior of one side of the VSU. In this paper, the finite-element method is employed to analyze the deformation of the VSU. The deformations of one side of the VSU are investigated under different forces. The results of the finite-element analysis are shown in Figure 4.

Based on the FEA results, the stiffness–force relationship of the VSU yielded the following fitted function:

$$k_{\mathrm{v}}=0.47t_{\mathrm{v}}^2+118.05t_{\mathrm{v}}+\mathrm{21,852.34}$$

(1)

where $k_{\mathrm{v}}\in ℝ$ is the VSU stiffness and $t_{\mathrm{v}}\in ℝ$ is the force acting on the VSU. The weakly nonlinear curve of the fitted function is shown in Figure 5.

3. Kinematics Modelling and Analysis of the Symmetric Joint Module

In the 1-DOF joint module, the movable platform rotates around the center O of the revolute joint with respect to the static platform. In order to describe the orientation of the movable platform relative to the static platform, two coordinate systems are established at the center O of the revolute joint: coordinate system {M}, which is attached to the movable platform, and coordinate system {S}, which is attached to the static platform, as shown in Figure 6. Based on the two coordinate systems, the orientation of the movable platform relative to the static platform can be described by the orientation of coordinate system {M} relative to coordinate system {S}, which can be expressed mathematically by the rotation matrix $R\in SO(3)$. Here $SO(3)$ represents the special orthogonal group. When the two coordinate systems coincide with each other, the joint module is said to be in the home pose, also known as the initial pose.

Since the 1-DOF joint module has only one rotational degree of freedom and the rotation occurs only in the $\mathrm{OYZ}$ plane of coordinate system {S}, the orientation of the movable platform relative to the static platform can be described by the rotational angle ${\theta }_{\mathrm{x}}\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ about the X axis of {S}. That means the rotation matrix $R\in SO(3)$ can be determined by only one parameter, ${\theta }_{\mathrm{x}}$, which satisfies the following condition:

$$R=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos {\theta }_{\mathrm{x}}& -\sin {\theta }_{\mathrm{x}}\\ 0& \sin {\theta }_{\mathrm{x}}& \cos {\theta }_{\mathrm{x}}\end{array}\right)$$

(2)

When the 1-DOF joint module is in the home pose, ${\theta }_{\mathrm{x}}=0$. When the movable platform rotates in the positive direction of the X axis, ${\theta }_{\mathrm{x}}>0$. When the movable platform rotates in the negative direction of the X axis, ${\theta }_{\mathrm{x}}<0$.

Writing $m_i=\overline{{\mathrm{OM}}_i}\in ℝ$$(i=1,2)$ as the distance between two points $\mathrm{O}$ and ${\mathrm{M}}_i$ and $s_i=\overline{{\mathrm{OS}}_i}\in ℝ$ $(i=1,2)$ as the distance between two points $\mathrm{O}$ and ${\mathrm{S}}_i$, then for a given 1-DOF joint module, the parameters $m_i$ and $s_i$ are constant. Writing ${\phi }_i=\angle {\mathrm{M}}_i{\mathrm{OS}}_i\in ℝ$ $(i=1,2)$ as the angle between the two lines ${\mathrm{OM}}_i$ and ${\mathrm{OS}}_i$, when the 1-DOF joint module is in the home pose, the initial value of ${\phi }_i$ is written as ${\phi }_{i,0}$, which satisfies

$${\phi }_{i,0}=\mathsf{\pi }-\angle {\mathrm{M}}_i{\mathrm{OO}}_{\mathrm{M}}-\angle {\mathrm{S}}_i{\mathrm{OO}}_{\mathrm{S}}$$

(3)

For a given 1-DOF joint module, ${\phi }_{i,0}$ is constant. When the movable platform rotates by the angle ${\theta }_x$ starting from the home pose, ${\phi }_i(i=1,2)$ yields the following:

$${\phi }_1={\phi }_{1,0}+{\theta }_{\mathrm{x}}$$

(4)

$${\phi }_2={\phi }_{2,0}-{\theta }_{\mathrm{x}}$$

(5)

Writing $c_i=\overline{{\mathrm{M}}_i{\mathrm{S}}_i}\in ℝ$$(i=1,2)$ as the wire length between two points ${\mathrm{M}}_i$ and ${\mathrm{S}}_i$, then the parameters $c_i$ and ${\phi }_i$ yields the following:

$$c_i^2=m_i^2+s_i^2-2m_i{s}_i\cos {\phi }_i$$

(6)

Substituting (4) and (5) into (6), then the wire length $c_i (i=1,2)$ satisfies the following equations, i.e.,

$$c_1^2=m_1^2+s_1^2-2m_1{s}_1\cos {\phi }_1=m_1^2+s_1^2-2m_1{s}_1\cos ({\phi }_{1,0}+{\theta }_{\mathrm{x}})$$

(7)

$$c_2^2=m_2^2+s_2^2-2m_2{s}_2\cos {\phi }_2=m_2^2+s_2^2-2m_2{s}_2\cos ({\phi }_{2,0}-{\theta }_{\mathrm{x}})$$

(8)

From (7) and (8), it can be concluded that the wire length $c_i (i=1,2)$ is determined by the angle ${\theta }_{\mathrm{x}}$. On the other hand, the desired orientation ${\theta }_{\mathrm{x}}$ of the 1-DOF joint module can be reached by regulating the wire length $c_i (i=1,2)$.

Differentiating on both sides of (7) and (8), we have the following results:

$$\stackrel{•}{c_1}=\frac{m_1{s}_1}{c_1}\sin ({\phi }_{1,0}+{\theta }_{\mathrm{x}})\stackrel{•}{{\theta }_{\mathrm{x}}}$$

(9)

$$\stackrel{•}{c_2}=-\frac{m_2{s}_2}{c_2}\sin ({\phi }_{2,0}-{\theta }_{\mathrm{x}})\stackrel{•}{{\theta }_{\mathrm{x}}}$$

(10)

Writing $l_{\mathrm{c}}={\left(\begin{array}{cc}{c}_1& c_2\end{array}\right)}^{\mathrm{T}}\in {ℝ}^2$, where $c_i (i=1,2)$ is the wire lengths of the 1-DOF joint module, then (9) and (10) can be written in the following matrix form:

$${\stackrel{•}{l}}_{\mathrm{c}}=J_{\mathrm{x}}\stackrel{•}{{\theta }_{\mathrm{x}}}$$

(11)

Here $J_{\mathrm{x}}\in {ℝ}^{2\times 1}$ is the Jacobian matrix, which yields the following:

$$J_{\mathrm{x}}=\left(\begin{array}{c}\frac{m_1{s}_1}{c_1}\sin ({\phi }_{1,0}+{\theta }_{\mathrm{x}})\\ -\frac{m_2{s}_2}{c_2}\sin ({\phi }_{2,0}-{\theta }_{\mathrm{x}})\end{array}\right)$$

(12)

Writing ${\omega }_{\mathrm{x}}=\stackrel{•}{{\theta }_{\mathrm{x}}}\in ℝ$ as the angular velocity of coordinate system {M} relative to coordinate system {S} about the X axis of {S}, then (11) can be written as:

$${\stackrel{•}{l}}_{\mathrm{c}}=J_{\mathrm{x}}{\omega }_{\mathrm{x}}$$

(13)

From (13), for a given orientation of the 1-DOF joint module, the change rate of the wire lengths, ${\stackrel{•}{l}}_{\mathrm{c}}$, is determined by the angular velocity, ${\omega }_{\mathrm{x}}$. On the other hand, the rotational velocity ${\omega }_{\mathrm{x}}$ can be adjusted by regulating the change rate of the wire lengths.

4. Stiffness Modelling and Analysis of the Symmetric Joint Module

As shown in Figure 6, the external force and torque act on the movable platform of the 1-DOF joint module are denoted as $f_{\mathrm{ex}}\in {ℝ}^3$ and ${\mathsf{\tau }}_{\mathrm{ex}}\in {ℝ}^3$, the tension of the $i\mathrm{th} (i=1,2)$ wire is denoted as $t_i\in {ℝ}^3$, the force and torque acted on the revolute joint of the movable platform are denoted as $f_{\mathrm{in}}\in {ℝ}^3$ and ${\mathsf{\tau }}_{\mathrm{in}}\in {ℝ}^3$, respectively. In coordinate system {S}, the movable platform of the 1-DOF joint module satisfies the following equilibrium equations:

$$f_{\mathrm{ex}}+f_{\mathrm{in}}+{\displaystyle \sum _{i=1}^2{t}_i}=0$$

(14)

$${\mathsf{\tau }}_{\mathrm{ex}}+{\mathsf{\tau }}_{\mathrm{in}}+{\displaystyle \sum _{i=1}^2{m}_i\times t_i}=0$$

(15)

where $m_i=\overrightarrow{{\mathrm{OM}}_i}\in {ℝ}^3$$(i=1,2)$.

Writing $c_i=\overrightarrow{{\mathrm{M}}_i{\mathrm{S}}_i}\in {ℝ}^3$$(i=1,2)$ and defining $u_i=\frac{c_i}{c_i}\in {ℝ}^3$ as the unit vector along the $i\mathrm{th} (i=1,2)$ wire of the 1-DOF joint module, then the tension $t_i$ of the $i\mathrm{th} (i=1,2)$ wire can be written as

$$t_i=t_i{u}_i$$

(16)

where $t_i=\left|t_i\right|\in ℝ$ represents the value of the wire tension $t_i$. Thus, the equilibrium Equations (14) and (15) can be written as follows:

$$f_{\mathrm{ex}}+f_{\mathrm{in}}=-{\displaystyle \sum _{i=1}^2{u}_i{t}_i}=-U T$$

(17)

$${\mathsf{\tau }}_{\mathrm{ex}}+{\mathsf{\tau }}_{\mathrm{in}}=-{\displaystyle \sum _{i=1}^2{m}_i\times t_i}=-{\displaystyle \sum _{i=1}^2\left(m_i\times u_i\right)t_i}=J^{\mathrm{T}} T$$

(18)

where $T={\left(\begin{array}{cc}{t}_1& t_2\end{array}\right)}^{\mathrm{T}}\in {ℝ}^2$ is a vector composed of the values of the wire tension $t_i (i=1,2)$, and $U=\left(\begin{array}{cc}{u}_1& u_2\end{array}\right)\in {ℝ}^{3\times 2}$ is a matrix composed of the unit vector $u_i (i=1,2)$. $J\in {ℝ}^{2\times 3}$ represents the Jacobian matrix, and its transposed matrix $J^{\mathrm{T}}\in {ℝ}^{3\times 2}$ yields the following:

$$J^{\mathrm{T}}=-\left(\begin{array}{cc}{m}_1\times u_1& m_2\times u_2\end{array}\right)$$

(19)

In this paper, the friction of the revolute joint is ignored. Then, the component of the torque ${\mathsf{\tau }}_{\mathrm{in}}$ in the X-axis direction, denoted as ${\tau }_{\mathrm{in-x}}\in ℝ$, yields the following:

$${\tau }_{\mathrm{in-x}}=0$$

(20)

Substituting (20) into (18), we have the following equation:

$${\tau }_{\mathrm{ex-x}}-\frac{m_1{s}_1\sin ({\phi }_{1,0}+{\theta }_{\mathrm{x}})}{c_1}{t}_1+\frac{m_2{s}_2\sin ({\phi }_{2,0}-{\theta }_{\mathrm{x}})}{c_2}{t}_2=0$$

(21)

where ${\tau }_{\mathrm{ex-x}}\in ℝ$ represents the component of the torque ${\mathsf{\tau }}_{\mathrm{ex}}$ in the X-axis direction.

That is

$$\begin{array}{ll}{\tau }_{\mathrm{ex-x}}& =\frac{m_1{s}_1\sin ({\phi }_{1,0}-{\theta }_{\mathrm{x}})}{c_1}{t}_1-\frac{m_2{s}_2\sin ({\phi }_{2,0}-{\theta }_{\mathrm{x}})}{c_2}{t}_2\\ & =\left(\begin{array}{cc}\frac{m_1{s}_1\sin ({\phi }_{1,0}+{\theta }_{\mathrm{x}})}{c_1}& -\frac{m_2{s}_2\sin ({\phi }_{2,0}-{\theta }_{\mathrm{x}})}{c_2}\end{array}\right)\left(\begin{array}{c}{t}_1\\ t_2\end{array}\right)\\ & =J_{\mathrm{x}}^{\mathrm{T}} T\end{array}$$

(22)

where $J_{\mathrm{x}}$ is given in (12).

When the component of torque acted on the movable platform of 1-DOF joint module increase from ${\tau }_{\mathrm{ex-x}}$ to ${\tau }_{\mathrm{ex-x}}+\mathrm{d}{\tau }_{\mathrm{ex-x}}$, the movable platform will rotate slightly around the X axis of {S}, and the increment of the angular displacement will be denoted as $\mathrm{d}{\theta }_{\mathrm{x}}$. The torque increment $\mathrm{d}{\tau }_{\mathrm{ex-x}}$ and the angular increment $\mathrm{d}{\theta }_{\mathrm{x}}$ yield [61]:

$$\mathrm{d}{\tau }_{\mathrm{ex-x}}=K_{{\theta }_{\mathrm{x}}}\mathrm{d}{\theta }_{\mathrm{x}}$$

(23)

where $K_{{\theta }_{\mathrm{x}}}\in ℝ$ represents the stiffness of the 1-DOF joint module around the X axis of {S}.

Differentiating on both sides of (22) yields the following:

$$\mathrm{d}{\tau }_{\mathrm{ex-x}}=\mathrm{d}{J}_{\mathrm{x}}^{\mathrm{T}} T+J_{\mathrm{x}}^{\mathrm{T}} \mathrm{d}T$$

(24)

Writing $\mathrm{d}{J}_{\mathrm{x}}^{\mathrm{T}} T$ as

$$\mathrm{d}{J}_{\mathrm{x}}^{\mathrm{T}} T=D_{\mathrm{x}} \mathrm{d}{\theta }_{\mathrm{x}}$$

(25)

where $D_{\mathrm{x}}\in ℝ$ is defined as

$$D_{\mathrm{x}}=\frac{\mathrm{d}{J}_{\mathrm{x}}^{\mathrm{T}}}{\mathrm{d}{\theta }_{\mathrm{x}}}T$$

(26)

substituting (12) into (26) yields the following:

$$\begin{array}{ll}{D}_{\mathrm{x}}=& \frac{m_1{s}_1{c}_1^2\cos ({\phi }_{1,0}+{\theta }_{\mathrm{x}})-m_1^2{s}_1^2{\sin }^2({\phi }_{1,0}+{\theta }_{\mathrm{x}})}{c_1^3}{t}_1\\ & -\frac{m_2{s}_2{c}_2^2\cos ({\phi }_{2,0}-{\theta }_{\mathrm{x}})+m_2^2{s}_2^2{\sin }^2({\phi }_{2,0}-{\theta }_{\mathrm{x}})}{c_2^3}{t}_2\end{array}$$

(27)

In addition, according to (11), we have

$$\mathrm{d}{l}_{\mathrm{c}}=J_{\mathrm{x}} \mathrm{d}{\theta }_{\mathrm{x}}$$

(28)

Then, $\mathrm{d}T$ can be written as:

$$\mathrm{d}T=K_{\mathrm{diag}}\mathrm{d}{l}_{\mathrm{c}}=K_{\mathrm{diag}} J_{\mathrm{x}} \mathrm{d}{\theta }_{\mathrm{x}}$$

(29)

where $K_{\mathrm{diag}}=\left(\begin{array}{cc}{k}_1& 0\\ 0& k_2\end{array}\right)\in {ℝ}^{2\times 2}$. Here, $k_i$ represents the stiffness of the $i\mathrm{th} (i=1,2)$ wire with the VSU in series. Since the stiffness of the wire is far greater than the stiffness of the VSU, then $k_i\approx k_{\mathrm{v}i} (i=1,2)$, where $k_{\mathrm{v}i}\in ℝ$ is the stiffness of the VSU placed in the $i\mathrm{th} (i=1,2)$ wire.

Substituting (25) and (29) into (24) yields the following:

$$\mathrm{d}{\tau }_{\mathrm{ex-x}}=D_{\mathrm{x}} \mathrm{d}{\theta }_{\mathrm{x}}+J_{\mathrm{x}}^{\mathrm{T}} K_{\mathrm{diag}} J_{\mathrm{x}} \mathrm{d}{\theta }_{\mathrm{x}}=\left(D_{\mathrm{x}}+J_{\mathrm{x}}^{\mathrm{T}} K_{\mathrm{diag}} J_{\mathrm{x}}\right)\mathrm{d}{\theta }_{\mathrm{x}}$$

(30)

Then, the stiffness of the 1-DOF joint module around the X axis of {S}, $K_{{\theta }_{\mathrm{x}}}\in ℝ$, can be written as follows:

$$K_{{\theta }_{\mathrm{x}}}=D_{\mathrm{x}}+J_{\mathrm{x}}^{\mathrm{T}} K_{\mathrm{diag}} J_{\mathrm{x}}$$

(31)

For a certain orientation of the 1-DOF joint module, the matrix $J_{\mathrm{x}}$ is constant. According to (22), when the component of the external torque, ${\tau }_{\mathrm{ex-x}}$, is given and that of the wire tension is determined, another wire tension can be calculated. According to (31), the stiffness of the 1-DOF joint module can be adjusted by regulating the wire tensions.

5. Wire-Tension Analysis of the Symmetric Joint Module

Since the wire can only pull, but not push, the wire-actuated joint module is actuated redundantly. According to the kinematics and stiffness analyses of the 1-DOF joint module, the wires of the 1-DOF joint module can be divided into two groups. Each group included one wire. The orientation of the 1-DOF joint module can be adjusted by regulating the length of one wire, and the stiffness of the 1-DOF joint module can be adjusted by regulating the tension of the other wire. Thus, the orientation and stiffness of the 1-DOF joint module can be adjusted synchronously. For a desired orientation of the 1-DOF joint module, ${\theta }_{\mathrm{x-des}}\in ℝ$, the corresponding wire lengths can be solved from (7) and (8) easily. For a desired stiffness of the 1-DOF joint module, $K_{{\theta }_{\mathrm{x-des}}}\in ℝ$, the issue of finding the corresponding wire tensions is called the stiffness-oriented wire-tension-distribution (SWTD) problem.

According to the stiffness model (31) of the 1-DOF joint module, the desired stiffness $K_{{\theta }_{\mathrm{x-des}}}$ is represented as follows:

$$K_{{\theta }_{\mathrm{x-des}}}=D_{\mathrm{x}}+J_{\mathrm{x}}^{\mathrm{T}}{K}_{\mathrm{diag}}{J}_{\mathrm{x}}$$

(32)

According to equilibrium Equation (22) of the 1-DOF joint module, when the orientation and external force are given, $J_{\mathrm{x}}$ and ${\tau }_{\mathrm{ex-x}}$ are both constant. The wire tension $t_1$ can be expressed by a function of the wire tension $t_2$, i.e.,

$$t_1=\frac{m_2{s}_2{c}_1\sin ({\phi }_{2,0}-{\theta }_{\mathrm{x}})}{m_1{s}_1{c}_2\sin ({\phi }_{1,0}+{\theta }_{\mathrm{x}})}{t}_2+\frac{c_1{\tau }_{\mathrm{ex-x}}}{m_1{s}_1\sin ({\phi }_{1,0}+{\theta }_{\mathrm{x}})}$$

(33)

Substituting (33) into (32), we have a quadratic equation with one variable, $t_2$. Solving for $t_2$ from the quadratic equation and substituting $t_2$ into (33), the wire tension $t_1$ can be obtained. Then, the two wire tensions $t_1$ and $t_2$ are both found for the desired stiffness $K_{{\theta }_{\mathrm{x-des}}}$. Thus, the SWTD problem of the 1-DOF joint module is solved.

In order to avoid slackness in the actuation wire, a lower limit of the wire tension $t_{\min }\in ℝ$ is set. In order to avoid the tension of the actuation wire exceeding the capacity of the wire driving unit and the VSU, an upper limit of the wire tension $t_{\max }\in ℝ$ is set. In this paper, we set $t_{\min }=100 \mathrm{N}$ and $t_{\max }=900 \mathrm{N}$. The solved wire tensions in the safe interval $\left(t_{\min },t_{\max }\right)$ are called the feasible wire tensions.

6. Simulation Verification

To verify the proposed analysis and method, a simulation example is carried out. The 1-DOF joint module for the simulation is shown in Figure 6, and its dimension parameters are given below:

(a) the distance between the wire hole ${\mathrm{S}}_i (i=1,2)$ and the center ${\mathrm{O}}_{\mathrm{S}}$ of the static platform:

$$\overline{{\mathrm{O}}_{\mathrm{S}}{\mathrm{S}}_1}=\overline{{\mathrm{O}}_{\mathrm{S}}{\mathrm{S}}_2}=0.070 \mathrm{m}$$

(b) the distance between the wire hole ${\mathrm{M}}_i (i=1,2)$ and the center ${\mathrm{O}}_{\mathrm{M}}$ of the movable platform:

$$\overline{{\mathrm{O}}_{\mathrm{M}}{\mathrm{M}}_1}=\overline{{\mathrm{O}}_{\mathrm{M}}{\mathrm{M}}_2}=0.050 \mathrm{m}$$

(c) the distance between the center ${\mathrm{O}}_{\mathrm{S}}$ of the static platform and the center O of the revolute joint:

$$h_{\mathrm{S}}=\overline{{\mathrm{OO}}_{\mathrm{S}}}=0.080 \mathrm{m}$$

(d) the distance between the center ${\mathrm{O}}_{\mathrm{M}}$ of the movable platform and the center O of the revolute joint:

$$h_{\mathrm{M}}=\overline{{\mathrm{OO}}_{\mathrm{M}}}=0.060 \mathrm{m}$$

Based on the dimension parameters of the 1-DOF joint module given above, the parameters $m_i=\overline{{\mathrm{OM}}_i}$, $s_i=\overline{{\mathrm{OS}}_i}$, ${\phi }_{i,0}$$(i=1,2)$ can be calculated as follows:

$$m_i=\overline{{\mathrm{OM}}_i}=\sqrt{{\overline{{\mathrm{O}}_{\mathrm{M}}{\mathrm{M}}_i}}^2+{\overline{{\mathrm{OO}}_{\mathrm{M}}}}^2}\approx 0.0781 \mathrm{m}$$

(34)

$$s_i=\overline{{\mathrm{OS}}_i}=\sqrt{{\overline{{\mathrm{O}}_{\mathrm{S}}{\mathrm{S}}_i}}^2+{\overline{{\mathrm{OO}}_{\mathrm{S}}}}^2}\approx 0.1063 \mathrm{m}$$

(35)

$${\phi }_{i,0}=\mathsf{\pi }-\angle {\mathrm{M}}_i{\mathrm{OO}}_{\mathrm{M}}-\angle {\mathrm{S}}_i{\mathrm{OO}}_{\mathrm{S}}\approx 1.7280 \mathrm{rad}$$

(36)

When the orientation of the 1-DOF joint module, ${\theta }_{\mathrm{x}}$, is given, the parameters ${\phi }_i=\angle {\mathrm{M}}_i{\mathrm{OS}}_i$, $c_i=\overline{{\mathrm{M}}_i{\mathrm{S}}_i}$$(i=1,2)$ can also be calculated according to (4), (5), (7), and (8), respectively.

In order to verify the solution to the SWTD problem of the 1-DOF joint module, two simulation cases are set for different orientations and external loads of the 1-DOF joint module and named Case 1 and Case 2, respectively. For Case $i (i=1,2)$, two sub-cases are set for different desired degrees of stiffness and named Case $i$-a and Case $i$-b $(i=1,2)$, respectively. The cases of the simulation are listed in

Table 1. Simulation cases for the SWTD problem of the 1-DOF joint module.

Case	External Loads ${\mathit{\tau }}_{\mathbf{ex-x}}$ (Nm)	Orientation ${\mathit{\theta }}_{\mathbf{x}}$ (rad)	Desired Stiffness ${\mathit{K}}_{{\mathit{\theta }}_{\mathbf{x-des}}}$ (Nm/rad)	Wire Tension $\mathit{T}={\left(\begin{array}{cc}{\mathit{t}}_1& {\mathit{t}}_2\end{array}\right)}^{\mathbf{T}}$ (N)	Calculated Stiffness ${\mathit{K}}_{{\mathit{\theta }}_{\mathbf{x-cal}}}$ (Nm/rad)	Deviation $\mathit{\eta }$
Case 1-a	2	0.0524	366.33	${\left(\begin{array}{cc}194.77& 149.99\end{array}\right)}^{\mathrm{T}}$	366.32	0.0027%
Case 1-b	2	0.0524	313.19	${\left(\begin{array}{cc}162.93& 119.99\end{array}\right)}^{\mathrm{T}}$	313.18	0.0032%
Case 2-a	−2	0.0349	322.80	${\left(\begin{array}{cc}131.29& 160.00\end{array}\right)}^{\mathrm{T}}$	322.81	0.0031%
Case 2-b	−2	0.0349	290.32	${\left(\begin{array}{cc}110.47& 140.00\end{array}\right)}^{\mathrm{T}}$	290.33	0.0034%

.

Here, we take Case 1-a as an example to illustrate the solution to the SWTD problem of the 1-DOF joint module. For the given orientation ${\theta }_{\mathrm{x}}=0.0524 \mathrm{rad}$, the parameters $D_{\mathrm{x}}$ and $J_{\mathrm{x}}$ can be calculated according to (27) and (12), respectively:

$$D_{\mathrm{x}}=-0.0339t_1-0.0257t_2+0.0063$$

(37)

$$J_{\mathrm{x}}={\left(\begin{array}{cc}0.0562& -0.0597\end{array}\right)}^{\mathrm{T}}$$

(38)

Substituting ${\tau }_{\mathrm{ex-x}}=2 \mathrm{Nm}$ and ${\theta }_{\mathrm{x}}=0.0524 \mathrm{rad}$ into (33), the wire tension $t_1$ can be obtained from the expression wire tension $t_2$:

$$t_1=1.0614t_2+35.5662$$

(39)

Substituting the desired stiffness $K_{{\theta }_{\mathrm{x-des}}}=366.33 \mathrm{Nm}/\mathrm{rad}$ and (39) into (32), a quadratic equation of $t_2$ is obtained:

$$0.0033t_2^2+0.8673t_2-205.4286=0$$

(40)

Solving (40), we have the wire tension $t_2=149.99 \mathrm{N}$. Substituting $t_2$ into (39), we obtain the wire tension $t_1=194.77 \mathrm{N}$. The wire tensions $t_1$ and $t_2$ are both in the safe interval $\left(t_{\min },t_{\max }\right)$. In summary, for Case 1-a, the feasible wire tensions for the desired stiffness $K_{{\theta }_{\mathrm{x-des}}}=366.33 \mathrm{Nm}/\mathrm{rad}$ are $t_1=194.77 \mathrm{N}$ and $t_2=149.99 \mathrm{N}$. Based on the solved wire tensions, the calculated stiffness $K_{{\theta }_{\mathrm{x-cal}}}$ of the 1-DOF joint module can be obtained via (32), which yields $K_{{\theta }_{\mathrm{x-cal}}}=366.32 \mathrm{Nm}/\mathrm{rad}$. A parameter $\eta \in ℝ$ is defined to evaluate the deviation between the calculated stiffness $K_{{\theta }_{\mathrm{x-cal}}}$ and the desired stiffness $K_{{\theta }_{\mathrm{x-des}}}$, i.e.,

$$\eta =\left|\frac{K_{{\theta }_{\mathrm{x-cal}}}-K_{{\theta }_{\mathrm{x-des}}}}{K_{{\theta }_{\mathrm{x-des}}}}\right|\times 100\%$$

(41)

For Case 1-a, $\eta =0.0027\%$. Thus, the calculated stiffness is close to the desired stiffness.

As in Case 1-a, the wire-tension solutions to the SWTD problem of the 1-DOF joint module for the other simulation cases can be obtained. The results of the simulation are listed in Table 1. The results show that the proposed method is effective.

7. Conclusions

In this paper, a modular wire-actuated robotic arm is proposed to improve the safety and adaptability of collaborative robots during human-robot interaction. This robotic arm consists of three symmetric joint modules in series. To extend the stiffness-adjustment range of the robotic arm, a variable-stiffness unit (VSU) is designed based on flexure and has a simple, symmetric and compact structure. The FEA result shows that the stiffness–force relationship of the VSU can be fitted by a quadratic function with an R-squared value of 0.99981, yielding a weakly nonlinear stiffness–force relationship. In this paper, we focus on the research on the 1-DOF joint module, including kinematics, statics, stiffness, and wire-tension analysis. Thus, the orientation of the joint module can be adjusted by regulating the length of the wires and the stiffness of the joint module can be adjusted by regulating the tension of the wires. Due to the actuation redundancy, the wires can be divided into two groups and the orientation and stiffness of the joint module can be adjusted synchronously. For a given desired stiffness, in order to find the corresponding wire tensions, it is necessary to solve the stiffness-oriented wire-tension-distribution (SWTD) problem. In this paper, a direct method is proposed for the SWTD problem of the 1-DOF joint module. A simulation is carried out to verify the effectiveness of the proposed analysis and method. The result shows that the deviation between the calculated stiffness and the desired stiffness is less than 0.005%.