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Abstract

The design of variable stiffness mechanism is of great importance to wheelchair-mounted robotic manipulators. In this article, the authors propose a cable-driven parallel–series hybrid joint toward wheelchair-mounted robotic manipulators with adjusting the system stiffness actively. The joint is a cable-driven, compression-spring-supported hybrid mechanism. Then, the kinematic analysis and cable tension analysis through static modeling are established to derive the joint stiffness model. Finally, cable-driven parallel–series hybrid joint stiffness adjustment ability analysis is carried out in four different load cases with MATLAB. Featured with light structure, stiffness adjustability, and large workspace, the proposed cable-driven parallel–series hybrid joint proves to be of great potential use for wheelchair-mounted robotic manipulators.

Keywords

Wheelchair-mounted robotic manipulator cable-driven hybrid mechanism static stiffness analysis stiffness adjustment

Introduction

Wheelchair-mounted robotic manipulator (WMRM) is a typical type of the assistive robot to help disabled elderly people with their activities of daily living (ADLs).^1,2 The research and development (R&D) of WMRMs can be traced back to the1960s.² Over the past 50 years, nearly a dozen WMRMs have been developed. However, the products are not widely available in the market, owing to poor usability, low payload, and high cost.^1,2 These problems can be ameliorated by the cable-driven mechanism (CDM), thanks to its low inertia, large workspace, high payload-to-weight ratio, good transportability, fully remote actuation, and ideal reconfigurability.³ Meanwhile, physical human-robot interaction is the core of a WMRM. The key of the interaction lies in the compliant behavior of the robot, namely, in which the variable stiffness mechanisms are used. Multiple mechanical design idea to achieve variable stiffness has been developed. From the arrangement of the actuators, Nam et al.,⁴ Petit et al.,⁵ and Tagliamonte et al.⁶ pointed out that there are two categories. The first is in serial configurations: a bigger motor to adjust the joint torque/position and another smaller motor to adjust the joint stiffness. The drawback of this configuration is that the maximum torque is limited by the smallest motor.^7–9 The second is in parallel configurations or antagonistic variable stiffness mechanism inspired by the musculoskeletal system.^4,5 This actuation architecture causes two significant drawbacks: the necessity of using complex control algorithms to achieve the desired behaviors and a reduced energetic efficiency.⁶ Azadi et al.¹⁰ and Boehler et al.¹¹ proposed CDM as antagonistic variable stiffness mechanism which could provide a faster response, coming from the level of internal forces (prestress) in the links of the mechanism. The use of the prestress modulation to generate variable stiffness can be very effective to obtain large stiffness variations without modifying the equilibrium configuration.¹⁰ From physically controllable impedance,⁶ a number of variable stiffness mechanism have been developed by changing the shape or geometry of the mechanism using mechanical methods.¹⁰ Most of the mechanical methods make use of complex nonlinear spring mechanisms,^12,13 such as controlling the effective number of active coils in a coil spring, using adjustable pitching stiffness, and adjusting the gap between two leaf springs. Hence, CDM coupled with stiffness adjustability may address some safety issues in the physical human-machine interaction¹⁴ with relatively low energy consumption and low cost.

CDM stiffness analysis is divided into static stiffness analysis (SSA) and dynamic stiffness analysis (DSA).^15–17 The SSA of CDM is aiming to analyze and improve the static positioning accuracy,^16–18 especially for the pick-and-place application,¹⁹ such as the CoGiRo.¹⁸ The purpose of DSA of CDM is to analyze the vibration of those applications^15–17 requiring high performances, especially dynamic performances,¹⁶ such as high-speed FALCON²⁰ and large radio telescope FAST (Five-hundred-meter Aperture Spherical radio Telescope).²¹ Whether SSA or DSA, cable modeling should be studied first. Generally, there are three different models of cable, namely, ideal lines which have no mass and no flexibility, a massless spring, and the sagging cable model that considers cable mass and elasticity.^15,16 The first model is the simplest cable model used in the theoretical analysis of CDM. The second model is valid and efficient for light-weight,¹⁸ low-speed,¹⁵ and small-size robots.^22,23 The third model is used for high-speed, large-dimension, or heavy-load CDM.^16,23–28 Previous researches have shown that the sagging cable model is divided into two-dimensional inclined cable^16–19,29 and three-dimensional inclined cable.^15,23,30 Riehl et al.²⁹ analyzed the influence of the sagging cable model on the end-effector positioning, the control lengths of the cables, and the cable tensions. They pointed out that for large working volume manipulators, this aspect needs to be taken into account at the design level in order to achieve good positioning and accuracy. Comparing these three models, the third is the most accurate, but numerous CDMs designed under the second model are working well in practice, and it is believed that this model is practically valid.³¹

Verhoeven et al.,³² Behzadipour and Khajepour,³³ and Anson et al.³⁴ took cables as massless springs and pointed out that the static stiffness depends on the stiffness of the cables and the geometrical arrangement of the cables and can be always improved by increasing the antagonistic forces in the cables. All of them use the Jacobian-based stiffness analysis method^16,35 to explain the Cartesian stiffness matrix as the function of a manipulator’s configuration and joint stiffness values. Actually, cables as massless springs just consider axial stiffness of cables; however, if the cable profile is a sagging curve and it needs to consider transversal stiffness, Jacobian matrix cannot be used to calculate the Cartesian stiffness matrix.¹⁶ Amare et al.¹⁵ made SSA and DSA of the CDM in three-dimensional inclined plane with external forces exerted by hydraulic cylinder on the system. Yuan and colleagues^16,17 studied the variation of pose error with external load for evaluating the static stiffness performance of CDM and used dynamic stiffness matrix method to identify the system natural frequencies to solve the vibration problems of structures. Nguyen et al.¹⁸ proposed a simplified sagging cable model to establish a new expression of the cable unstrained length to make SSA which was verified to help to improve the positioning accuracy of a large-dimension CDM in simulation and experiments.

This article focuses on the SSA and stiffness adjustment of a cable-driven parallel–series hybrid joint (CDPSHJ) toward WMRM inspired by the three types of humanoid joints.^36–39 CDPSHJ is a cable-driven, compression-spring-supported hybrid mechanism. According to the application characteristics of WMRM, it is appropriate to model the cable as a massless spring. Linear helical compression spring provides nonlinear stiffness characteristics under combined bending and compression effects;^39–41 meanwhile, CDPSHJ stiffness adjustment is achieved by the additional translation motion; hence, CDPSHJ is the simplification of nonlinear stiffness mechanism. That is to say, the CDPSHJ provides a simple, compact solution to the stiffness adjustment problem. To the best knowledge of the authors, there are only a few studies analyzing the effect of lateral bending and compression of spring. In this article, concerning the elastic stability under lateral bending and compression, the coil spring can be treated as an elastic beam,^38,41 so the spring lateral bending and compression model based on the elliptic integral solution is introduced. Compared with other methods, this method solves the large-deflection problems in compliant mechanisms more accurately.⁴² Then, based on the statics, the Cartesian stiffness matrix is deduced by the Jacobian-based stiffness analysis method to analyze the joint variable stiffness, which is found to be affected by the compression spring in addition to the cable stiffness and antagonistic forces.

This article is organized as follows: The concept of CDPSHJ is presented in the “CDPSHJ description” section; then, the kinematic modeling is presented in the “Velocity Jacobian matrix” section; next, the static modeling and lateral bending and compression modeling of the spring are given in the “Tension analysis” section; after that, the stiffness adjustment of the mechanism is investigated in the “CDPSHJ variable stiffness analysis” section; and finally, conclusions obtained from the results are presented in the “Conclusion” section.

CDPSHJ description

The human elbow joint is controlled by the biceps and triceps muscles. Flexion (Figure 1(a)) occurs when the biceps contracts and the triceps relaxes, and extension (Figure 1(b)) is achieved in the opposite manner. Based on the musculoskeletal mechanism, an elbow joint has been designed for WMAMs. The keywords of design idea of CDPSHJ are cable-driven and stiffness adjustment. As shown in Figure 2(a), cable 1 and cable 2 imitate biceps brachii and triceps brachii, respectively, and drive the upper platform (moving platform); a compression spring and cables 1 and 2 support the two platforms, which is regarded as the parallel part; two rigid shafts with a revolute pair are in the middle of the spring and rigid shaft 1 passes through the upper platform, which is regarded as the series part. Linear helical compression spring provides nonlinear stiffness characteristics under combined bending and compression effects, so the CDPSHJ is a variable stiffness mechanism and it has a total of 2 degrees of freedom. One is rotation and the other one is translational motion, which is designed to adjust the stiffness of the spring to determine the joint system stiffness. Therefore, CDPSHJ is the simplification of nonlinear stiffness mechanism. That is to say, the CDPSHJ with stiffness adjustability toward WMRM may address some safety issues in the physical human-machine interaction¹⁴ with relatively low energy consumption and low cost.

Figure 1.

Elbow joint motion mechanism:⁴³ (a) elbow flexion and (b) elbow extension.

Figure 2.

Diagram of the CDPSHJ: (a) 3D joint mechanism and (b) 2D joint diagram.

Velocity Jacobian matrix

Figure 2(b) is the diagram of the CDPSHJ. The upper and lower platforms are designed as homogeneous disks with a radius of b and a, respectively. The two Cartesian coordinate systems {O₁x₁y₁z₁} and {O₂x₂y₂z₂} are fixed to the centroid of the two platforms, respectively, and {O₁x₁y₁z₁} is the global coordinate system. The connecting points of cables 1 and 2 are denoted as A₁, B₁, A₂, and B₂, respectively, and the distance from O₁ to the revolute pair center is d. The spring is simply drawn as an arc.

As mentioned in the previous section, the CDPSHJ has a total of 2 DoFs. One is the rotation around the Z-axis and the other is the translational motion on the X-Y plane. Due to the rigid restraint of the rotation pair in the middle, the relationship between the motion relative to O₂ along the X-axis and the Y-axis is as follows

x = (y - d) \tan θ

(1)

$l_{m}$ is the vector defining the mth cable, which is represented in the global frame; $L_{O_{1} O_{2}}$ is $\vec{O_{1} O_{2}}$ , which is represented in the global frame; $L_{O_{1} A_{m}}$ is $\vec{O_{1} A_{\begin{matrix} m \end{matrix}}}$ , which is represented in the global frame; and $B_{m}^{o_{2}}$ is $\vec{O_{2} B_{m}}$ , which is represented in the local frame. The kinematic relationship between the input (l₁, l₂) and output (y, θ) can be obtained by the closed-loop vector method

l_{m} = L_{O_{1} O_{2}} + {}_{O_{2}}^{O_{1}}R B_{m}^{o_{2}} - L_{O_{1} A_{m}}

(2)

where ${}_{O_{2}}^{O_{1}}R = \begin{matrix} [\begin{matrix} \cos θ & \sin θ & 0 \\ - \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{matrix}] \end{matrix}$ . Since m = 1, 2, the above formula can be expanded as $l_{1} = [\begin{matrix} x - b \cos θ + a \\ y + b \sin θ \\ 0 \end{matrix}]$ and $l_{2} = [\begin{matrix} x + b \cos θ - a \\ y - b \sin θ \\ 0 \end{matrix}]$ . Hence, the cable length is $l = [\begin{matrix} l_{1} \\ l_{2} \end{matrix}]$ ; $l_{1} = ‖ l_{1} ‖$ and $l_{2} = ‖ l_{2} ‖$ .

The first-order differential kinematics is necessary for singularity analysis and kinematic control, and the velocity Jacobian matrix is necessary for the analysis of mechanism velocity and stiffness. It is the nonlinear mapping between the position change of the moving platform and the length change of the cable. It can be known from equation (2)

\begin{matrix} l_{m}^{2} = {(L_{O_{1} O_{2}} + {}_{O_{2}}^{O_{1}}R B_{m}^{o_{2}} - L_{O_{1} A_{m}})}^{T} \end{matrix} (L_{O_{1} O_{2}} + {}_{O_{2}}^{O_{1}}R B_{m}^{o_{2}} - L_{O_{1} A_{m}})

(3)

Differentiating equation (3) with respect to time, we obtain

{\overset{\cdot}{l}}_{m} = {\hat{l}}_{m}^{T} ({\overset{\cdot}{L}}_{O_{1} O_{2}} + {}_{O_{2}}^{O_{1}}{\overset{\cdot}{R}} B_{m}^{o_{2}})

(4)

where ${\hat{l}}_{m} = l_{m} / l_{m}$ , m = 1, 2; ${}_{O_{2}}^{O_{1}}{\overset{\cdot}{R}} = S (ω) {}_{O_{2}}^{O_{1}}R$ , where $S (ω)$ is the skew symmetric matrix of the angular velocity vector $ω$ , namely, $S (ω) = ω \times = [\begin{matrix} 0 & - ω_{z} & ω_{y} \\ ω_{z} & 0 & - ω_{x} \\ - ω_{y} & ω_{x} & 0 \end{matrix}]$ . Substituting it into equation (4) and simplifying, we obtain

{\overset{\cdot}{l}}_{m} = {\hat{l}}_{m}^{T} {\overset{\cdot}{L}}_{O_{1} O_{2}} + {\hat{l}}_{m}^{T} ω \times {}_{O_{2}}^{O_{1}}R B_{m}^{o_{2}}

(5)

Using $a \cdot (b \times c) = b \cdot (c \times a) = (c \times a) \cdot b$ , equation (5) can become

l_{m} {\overset{\cdot}{l}}_{m} = l_{m}^{T} {\overset{\cdot}{L}}_{O_{1} O_{2}} + ({}_{O_{2}}^{O_{1}}R B_{m}^{o_{2}} \times l_{m})^{T} ω

(6)

The above formula can be expanded as

[\begin{matrix} l_{1} & 0 \\ 0 & l_{2} \end{matrix}] [\begin{matrix} {\overset{\cdot}{l}}_{1} \\ {\overset{\cdot}{l}}_{2} \end{matrix}] = [\begin{matrix} l_{1 x} & l_{1 y} & Φ_{1} \\ l_{2 x} & l_{2 y} & Φ_{2} \end{matrix}] [\begin{matrix} \overset{\cdot}{x} \\ \overset{\cdot}{y} \\ \overset{\cdot}{θ} \end{matrix}]

(7)

where $Φ_{1} = b \cos θ l_{1 y} + b \sin θ l_{1 x}$ and $Φ_{2} = - b \cos$ $θ l_{2 y} - b \sin θ l_{2 x}$ . Differentiating equation (1) with respect to time, we obtain

\overset{\cdot}{x} = \overset{\cdot}{y} \tan (θ) + (y - d) \sec^{2} θ \overset{\cdot}{θ}

(8)

Substituting equation (8) into equation (7) and simplifying it, we obtain

B \overset{\cdot}{q} = A \overset{\cdot}{X}

(9)

where $B_{2 \times 2} = [\begin{matrix} l_{1} & 0 \\ 0 & l_{2} \end{matrix}]$ , ${\overset{\cdot}{q}}_{2 \times 1} = [\begin{matrix} {\overset{\cdot}{l}}_{1} \\ {\overset{\cdot}{l}}_{2} \end{matrix}]$ , $A_{2 \times 2} = [\begin{matrix} l_{1 x} \tan θ + l_{1 y} l_{1 x} (y - d) \sec^{2} θ + b \cos θ l_{1 y} + b \sin θ l_{1 x} \\ l_{2 x} \tan θ + l_{2 y} l_{2 x} (y - d) \sec^{2} θ - b \cos θ l_{2 y} - b \sin θ l_{2 x} \end{matrix}]$ , and ${\overset{\cdot}{X}}_{2 \times 1} = [\begin{matrix} \overset{\cdot}{y} \\ \overset{\cdot}{θ} \end{matrix}]$ .

Hence

\dot{X} = J \dot{q}

(10)

where $J = A^{- 1} B$ .

So, now, the velocity Jacobian matrix $J$ is obtained, which is the bridge connecting the joint space and the operating space.

Tension analysis

In order to study whether the system has enough stiffness to withstand certain loads, cable tension analysis is carried out in load case. The static analysis model of point O₂ is shown in Figure 3. Due to the small mass of the upper platform, the gravity of the upper platform G, positive pressure N, and friction F_µ are negligible. Thus, the static equilibrium equations can be expressed as

\sum_{m = 1}^{2} T_{m} (- {\hat{l}}_{m}) = - F_{ext}

(11)

\sum_{m = 1}^{2} {}_{O_{2}}^{O_{1}}R B_{m}^{o_{2}} \times T_{m} (- {\hat{l}}_{m}) = - M_{ext}

(12)

where $T_{m}$ is the size of cable tension, $- {\hat{l}}_{m}$ is the direction of cable tension, $F_{ext} = [\begin{matrix} - F_{1} + F_{px} \\ F_{2} + F_{py} \\ 0 \end{matrix}]$ , and $M_{ext} = [\begin{matrix} 0 \\ 0 \\ M - M_{p} \end{matrix}]$ . Denote ${}_{O_{2}}^{O_{1}}R B_{1}^{o_{2}} \times {\hat{l}}_{1}$ as r₁ and ${}_{O_{2}}^{O_{1}}R B_{2}^{o_{2}} \times {\hat{l}}_{2}$ as r₂. The two equations can be expanded and merged into

ST = - W

(13)

where ${\hat{l}}_{m} = [\begin{matrix} {\hat{l}}_{mx} \\ {\hat{l}}_{my} \end{matrix}]$ , $S_{3 \times 2} = - [\begin{matrix} {\hat{l}}_{1 x} & {\hat{l}}_{2 x} \\ {\hat{l}}_{1 y} & {\hat{l}}_{2 y} \\ r_{1 z} & r_{2 z} \end{matrix}]$ , $T_{2 \times 1} = [\begin{matrix} T_{1} \\ T_{2} \end{matrix}]$ , and $W_{3 \times 1} = [\begin{matrix} - F_{1} + F_{px} \\ F_{2} + F_{py} \\ M - M_{p} \end{matrix}]$ .

Figure 3.

Static analysis model of point O₂. G is the gravity of the upper platform; N is the positive pressure of rigid shaft 1 at point O₂; F_µ is the friction of rigid shaft 1 at point O₂; F₁, F₂, and M are the forces/moment with the compression spring acting on point O₂; F_px, F_py, and M_p are the external forces/moment acting on point O₂; and T₁ and T₂ are tension forces on cables 1 and 2, respectively.

Static analysis of lateral bending and compression spring

Timoshenko and Gere⁴¹ pointed out that the coil spring under lateral bending and compression could be treated as an elastic beam, but it should consider the change in the length of the spring due to bending and compression. To solve the large deflection problems in compliant mechanisms, the elliptic integral solution is often considered to be the most accurate method for modeling large deflections of beams.⁴² According to this idea, force analysis is performed on the bending and compression spring of the CDPSHJ, as shown in Figure 4. The end forces of the spring satisfy $F'_{1} = - F_{1}, F'_{2} = - F_{2}, and M' = - M$ . According to the Bernoulli–Euler beam theory, the end point of the spring (coinciding with O₂) meets the following equation

β \frac{d ρ}{dp} = - F'_{1} y - F'_{2} x - M' = F_{2} x + F_{1} y + M

(14)

where β is the bending stiffness of the spring and $d ρ / dp$ is the curvature of the elastic beam with large deformation. Based on equation (14), the following equations regarding F₁, F₂, and M can be derived⁴⁴

M = - F_{2} x - F_{1} y

(15)

xk = x_{_1} + x_{_2}

(16)

where $x_{_1} = \cos φ \int_{0}^{\frac{π}{2}} 2 \sqrt{2} q \sin ε d ε = 2 \sqrt{2} q \cos φ$ ; $φ \int_{0}^{\frac{π}{2}} (\sqrt{2} - 2 \sqrt{2} q^{2} si n^{2} ε) / (\sqrt{1 - q^{2} si n^{2}} ε) d ε$ ; $k = \sqrt{(2 / β) \sqrt{F_{1}^{2} + F_{2}^{2}}}$ ; $\cos φ = F_{2} / \sqrt{F_{1}^{2} + F_{2}^{2}}$ ; $\sin φ = F_{1} /$ $\sqrt{F_{1}^{2} + F_{2}^{2}}$ ; $q = \sin ((θ + φ) / 2)$ ; and $ε \in [0, π / 2]$ .

Figure 4.

Force analysis diagram of the compression spring.

Because $x_{_2}$ is an elliptic integral, there is no analytical solution but numerical solution. At this point, substitute the three equilibrium equations obtained from the static analysis into equations (15) and (16) to derive the values of T₁, T₂, F₁, F₂, and M.

It is important to note that when θ = 0°, the compression spring, as a linear spring, meets Hooke’s law. So, F₁ = 0 and M = 0. The three equilibrium equations developed by static analysis can be used to solve T₁ and T₂.

CDPSHJ variable stiffness analysis

Stiffness is an important feature of a robotic device. It is directly related to the structure and rigidity of the robotic device. Compliant design is needed to allow a WMRM to produce high stiffness when it is performing tasks and low stiffness during motions. In this article, the CDPSHJ has the feature with variable stiffness. The purpose of SSA of the CDPSHJ is as follows. First, understand the stiffness change in the mechanism to know whether the system has enough stiffness to withstand certain loads. Second, understand the stiffness change in the mechanism in the workspace to know how to adjust the system stiffness. Third, by studying the SSA, theoretical preparation is made for the design of mechanism control scheme and the realization of system.

Based on the static analysis, the Cartesian stiffness matrix is deduced by the Jacobian-based stiffness analysis method to analyze the CDPSHJ stiffness adjustability. As the CDPSHJ is to be applied to the WMRM that assists the disabled elderly people in performing ADLs, the elbow joint range of motion is 110°.⁴⁵ In conducting the variable stiffness analysis, cable is modeled as a massless spring, whose profile is a straight line. For CDPSHJ at static equilibrium, the incremental displacement $δ X$ is correlated with the incremental wrench $δ W$ by the Cartesian stiffness matrix K^33,34 as

δ W = K δ X

(17)

K = - [\begin{matrix} \frac{\partial S}{\partial y} T & \frac{\partial S}{\partial θ} T \end{matrix}] - S K_{l} J^{- 1}

(18)

where $K_{l} = [\begin{matrix} k_{1} & 0 \\ 0 & k_{2} \end{matrix}]$ and $k_{1}$ and $k_{2}$ are rigidity of cables 1and 2, respectively.

From equation (18), it can be observed that CDPSHJ stiffness actually comprised two types of stiffness in series—the cable elastic stiffness and the actuator stiffness arising from the closed-loop control system.^33,34 In general, the system stiffness is highly dependent upon the pose and actuation arrangement of the manipulator. Similarly, there are three categories for stiffness adjustment³⁴ of CDPSHJ. First, change the cable stiffness by varying the stiffness of each individual cable. Second, modulate the stiffness matrix through a reconfiguration of the system.⁴⁶ Third, choose the desired cable tension from the stiffness point of view, which is challenging. This article focuses on analyzing the latter two ways to achieve the variable stiffness adjustment of the mechanism.

Stiffness matrix homogenization

In order to evaluate and compare the different layouts of the CDPSHJ or the CDPSHJ in different poses in terms of stiffness, the authors need to develop stiffness indices based on the stiffness matrix. But the previous Cartesian stiffness matrix equation (18) is inhomogeneous, namely

\begin{matrix} {[\begin{matrix} F_{x} (N) \\ F_{y} (N) \\ M_{z} (N m) \end{matrix}]}_{3 \times 1} = {[\begin{matrix} {[K_{11} (N / m)]}_{2 \times 1} & {[K_{12} (N)]}_{2 \times 1} \\ {[K_{21} (N)]}_{1 \times 1} & {[K_{22} (N m)]}_{1 \times 1} \end{matrix}]}_{3 \times 2} \\ {[\begin{matrix} y (m) \\ θ (rad) \end{matrix}]}_{2 \times 1} \end{matrix}

(19)

The matrix needs to be homogenized first, as it is not possible to directly derive the stiffness indices. The authors adopt the method introduced in previous studies,^34,47 which partitions a unit-inconsistent matrix into homogeneous translational and rotational components.

Equation (19) can be decomposed as

F = K_{11} y + K_{12} θ = F_{y} + F_{θ}

(20)

M = K_{21} y + K_{22} θ = M_{y} + M_{θ}

(21)

Let $y = S_{y} P_{y}$ , $θ = S_{θ} P_{θ}$ , $y = H_{y} R_{y}$ , and $θ = H_{θ} R_{θ}$ . $P_{y}$ , $P_{θ}$ , $R_{y}$ , and $R_{θ}$ are all dimensionless intermediate variables. $S_{y}$ , $S_{θ}$ , $H_{y}$ , and $H_{θ}$ are the generalized orthogonal matrix, whose columns are the eigenvectors of $K_{11}^{T} K_{11}$ , $K_{12}^{T} K_{12}$ , $K_{21}^{T} K_{21}$ , and $K_{22}^{T} K_{22}$ respectively. For the stiffness matrix of this article, $K_{11}^{T} K_{11} K_{12}^{T} K_{12}$ , , $K_{21}^{T} K_{21}$ , and $K_{22}^{T} K_{22}$ are all scalars, whose eigenvalues are their scalar values and the eigenvector are all 1. Therefore, y and $θ$ are derived from the dimensionless parameters P and R by linear transformation. So, we have

F = K_{11} S_{y} P_{y} + K_{12} S_{θ} P_{θ} = G_{F} P

(22)

M = K_{21} H_{y} R_{y} + K_{22} H_{θ} R_{θ} = G_{M} R

(23)

where $P = [\begin{matrix} S_{y} P_{y} \\ S_{θ} P_{θ} \end{matrix}]$ , $R = [\begin{matrix} H_{y} R_{y} \\ H_{θ} R_{θ} \end{matrix}]$ , $S_{y}$ , $S_{θ}$ , $H_{y}$ , and $H_{θ}$ are all 1.

$G_{F} = {[\begin{matrix} K_{11} & K_{12} \end{matrix}]}_{2 \times 2}$ is a dimensionally homogeneous matrix, in N, which maps the dimensionless parameter vector $P$ to the force vector $F$ .

$G_{M} = {[\begin{matrix} K_{21} & K_{22} \end{matrix}]}_{1 \times 2}$ is a dimensionally homogeneous matrix, in N m, which maps the dimensionless parameter vector $R$ to the moment M.

$G_{F} G_{F}^{T}$ results in a 2 × 2 matrix whose eigenvectors and eigenvalues form an ellipse corresponding to the translational part of the stiffness. More specifically, the eigenvectors represent the directions of maximum and minimum translational stiffness, and the corresponding eigenvalues indicate the stiffness magnitudes in these directions.

$G_{M} G_{M}^{T}$ is reduced to a scalar value, which represents the rotational part of the stiffness.

λ_t ₁ and λ_t₂ are the eigenvectors of $G_{F} G_{F}^{T}$ and λ_r is $G_{M} G_{M}^{T}$ . Hence, the translational stiffness index is defined as $k_{t} = \min (\sqrt{λ_{t 1}}, \sqrt{λ_{t 2}})$ . Similarly, the rotational stiffness index can be defined as $k_{r} = \sqrt{λ_{r}}$ . The two performance indices $k_{t}$ and $k_{r}$ assess the stiffness of the joint, and a higher index indicates higher rigidity.⁴⁷ The index f is used to evaluate the CDPSHJ stiffness, so

f = k_{t} k_{r}

(24)

Numerical results of CDPSHJ stiffness adjustment

Simulation studies are performed with the software MATLAB 2016. Before illustrating the CDPSHJ stiffness adjustment, the following conditions should be satisfied.

Wrench-feasible condition

0 < T_{\min} \leq T_{i} \leq T_{\max}, i = 1, 2

(25)

Wrench-set condition

The cable mechanism is able to sustain a specified set of external wrenches. In this article, four different load cases, denoted by L1–L4 in Table 1, will be considered.

Table 1.

External wrench conditions.

	F_px (N)	F_py (N)	M_p (N m)
L1	0	0	0
L2	10	−20	−10
L3	10	−20	0
L4	0	−20	−10

Cable-length condition

0 < l_{\min} \leq l_{i}, i = 1, 2

(26)

Translational-constraint condition

It is known from equation (1) that the translation motion along the X-axis is limited by (y, θ). Meanwhile, structural size of CDPSHJ restricts the translation motion. Hence

x_{\min} \leq x \leq x_{\max}

(27)

In consideration of the CDPSHJ toward WMRM’s elbow joint, the values of the various maximum and minimum limits, in arbitrary units, corresponding to those required in inequalities equations (25)–(27) as used for all numerical examples are given in Table 2.

Table 2.

Numerical values of limits.

T_min (N)	T_max (N)	l_min (m)	x_min (m)	x_max (m)	θ_min (rad)	θ_max (rad)	y_min (m)	y_max (m)
5	300	0.025	−0.08	0.08	−1.57	1.57	0.05	0.095

Before illustrating the effects of the CDPSHJ variable stiffness adjustment, cable and spring parameters must be specified. For the cable actuators, 6 × 7 wire rope is considered. According to the method utilized by Anson et al.,³⁴ the stiffness of the ith cable is formulated as

k_{i} = \frac{E_{i} A_{Ci}}{l_{i} + l_{cw}}

(28)

where the modulus of elasticity of the cable E is 68 GPa, the cross-sectional area of the cable A_C is 131.88 mm², and the length of the actuating winch l_cw is 30 mm, which is assumed to be constant. Diameter of steel wire d_w is 2 mm. The spring is made of carbon steel. Elastic modulus of spring E_s is 196 GPa, Young’s modulus G is 78.5 GPa, diameter of spring wire d_s is 5 mm, pitch diameter of the spring D is 40 mm, initial height of the compression spring l₀ is 105 mm, active coil number of the compression spring n₀ is 8, and spring constant K is 23,840 N/m. The spring bending stiffness β satisfies the following relationship⁴¹

β = β_{0} \frac{l}{l_{0}}

(29)

where l is the compressed length of the spring and β₀ is the initial bending stiffness of the spring. Since the spring is treated as an elastic beam of variable length and has large deformation, it is assumed that $l = \sqrt{x^{2} + y^{2}} = ‖ \vec{O_{1} O_{2}} ‖$ . Let I be the inertia moment of the cross section of the spring, namely, $I = 3.068 \times 10^{- 11}$ . Then, $β_{0} = 4 E_{s} G I l_{0} / (π d_{s} n_{0} (E_{s} + 2 G)) =$ $0.559$ .⁴¹

Translation motion and stiffness adjustment

As shown in Figure 2(a), rigid shaft 1 passes through the upper platform, forming the series part, so the CDPSHJ stiffness is adjusted through the translation motion. Once the cable is selected, according to equation (18), the stiffness adjustment mainly involves adjusting the tension of the driving cable T or the structural matrix $S$ or the velocity Jacobian matrix $J$ . There are many factors that affect T, $S$ , and $J$ , in which the effect of cable length is mainly analyzed. When CDPSHJ is at an angle, according to equation (2), the cable length is affected by y. That is to say, when θ is fixed, through adjusting the y, CDPSHJ stiffness is adjusted. The adjustable range of y should make sure $y - d > 0$ , and the set of poses meet inequalities (25), (26), and (27). Figure 5 shows CDPSHJ stiffness at θ = −58°, a = 0.05 m, b = 0.05 m, and d = 0.04 m.

Figure 5.

CDPSHJ variable stiffness at θ = −58°, a = b = 0.05 m, d = 0.04 m: (a) k_t, k_r at L1; (b) f, T₁, T₂ at L1, (c) k_t, k_r at L2, (d) f, T₁, T₂ at L2, (e) k_t, k_r at L3, (f) f, T₁, T₂ at L3, (g) k_t, k_r at L4, and (h) f, T₁, T₂ at L4.

From Figure 5, the joint translational stiffness is always smaller than rotational stiffness. The joint stiffness adjustment range and cable tensions T₁ and T₂ are affected by external loads. But when the CDPSHJ pose is the same at L1, L2, L3, and L4, the stiffness values are all the same and not affected by external loads. All the maximum values are at y = 0.072 m. In other words, at L2, L3, and L4, respectively, the stiffness range is the subset of the stiffness set at L1.

Structural parameters (a, b, d) on stiffness adjustment

The last section indicates that once structural parameters are decided, the joint stiffness is adjusted by translation motion when the joint is at an angle, but the stiffness values are all the same and not affected by external loads. However, if structural parameters (a, b, and d) change, the stiffness values change. This can be proved by equation (18). That is to say, in order to maximize CDPSHJ stiffness, reconfigure the system. Namely

max (min (f (a, b, d)))

subject to 0.035 m ≤ a ≤0.08 m, 0.035 m ≤ b ≤ 0.08 m, 0.025 m ≤ d ≤ 0.055 m.

Other parameters are listed in Tables 1 and 2. Due to the space limitation, this section only analyzes the effects of structural parameters on stiffness adjustment under L1 and L2. Here, a × b × d is discretized into n × n × n points. Traverse n × n points of (a, b) to figure out the f (a, b, d) corresponding to d (n points), as shown in Figure 6. The numerical analysis is performed with MATLAB at n = 5. Figure 6 shows f (a, b, d) when a × b × d is traversed at L1 and L2. There should be 25 curves in Figure 6(a) and (b). The maximum target value falls on curves 1 and 2. At L1 in Figure 6(a), f (a, b, d) is the maximum at d = 0.055 m, a = 0.035, and b = 0.08 m. At L2 from Figure 6(b), f (a, b, d) is the maximum at d = 0.0448 m, a = 0.0575, and b = 0.035 m. In future work, the optimization design of CDPSHJ should be multi-objective, taking f (a, b, d) into account in addition to the effects of workspace.

Figure 6.

f (a, b, d) at traversing a × b × d at L1, L2: (a) f (a, b, d) at L1 and (b) f (a, b, d) at L2.

Bending and compression spring on stiffness adjustment

As the spring is an elastic beam, the effects of axial elongation and shear are negligible. This section discusses that the spring flexural rigidity is on CDPSHJ stiffness due to bending and compression. According to equation (29),⁴¹ once the spring is selected, the spring bending stiffness β is only relevant for the spring compressed length l.

In this article, it is assumed that $l (y, θ) =$ $\sqrt{x^{2} + y^{2}} = ‖ \vec{O_{1} O_{2}} ‖$ . Hence, Figure 7 shows β approaching (y, θ) limit under the four load conditions at a = b = 0.05 m and d = 0.04 m. According to Figure 7, obviously β increases with the increase in y, which means the spring gets softer with the compression. Meanwhile, β increases with the increase in θ. So, the spring is softened for the bending motion under the compression effect and gets stiffer for rotation. Figure 8 shows β and CDPSHJ stiffness f in the four load cases at θ = −58° and y = 0.085 m, respectively, when a = b = 0.05 m and d = 0.04 m. According to Figure 8, the spring gets softer under the compression effect and gets stiffer for rotation, like in Figure 7. When the amount of compression is constant, the CDPSHJ stiffness f is consistent with the spring bending stiffness β. The linear helical compression spring provides the nonlinear stiffness feature under the bending and compression effects. The low-cost CDPSHJ stiffness adjustment can be achieved either by translational motion of the upper plate or adjustment of the bending angle. But the CDPSHJ stiffness adjustment amplitude through translational motion of the upper plate is smaller than that through adjustment of the bending angle. Meanwhile, because the amplitude is too large, it shows CDPSHJ stiffness f approaching 0 in Figure 8(b), (d), (f), and (h).

Figure 7.

β approaching (y, θ) limit at a = b = 0.05 m and d = 0.04 m: (a) β at L1, (b) β at L2, (c) β at L3, and (d) β at L4.

Figure 8.

β and CDPSHJ stiffness f at a = b = 0.05 m and d = 0.04 m: (a) β, f at θ = −58° at L1; (b) β, f at y = 0.085 m at L1; (c) β, f at θ = −58° at L2; (d) β, f at y = 0.085 m at L2; (e) β, f at θ = −58° at L3; (f) β, f at y = 0.085 m at L3, (g) β, f at θ = −58° at L4; and (h) β, f at y = 0.085 m at L4.

Conclusion

This article proposes the CDPSHJ for WMAMs, aiming to analyze the stiffness adjustability. After introducing the elbow joint concept, kinematic analysis of the joint is carried out to calculate the velocity Jacobian matrix. Then, the cable tension is analyzed through static modeling. Finally, stiffness adjustment analysis is carried out in four different load cases with MATLAB. The linear helical compression spring provides the nonlinear stiffness feature under the bending and compression effects. The CDPSHJ stiffness adjustment can be achieved either by translational motion of the upper plate or adjustment of the bending angle, and the stiffness values are not affected by external loads, but affected by structural parameters (a, b, and d). Featured with stiffness adjustability, large workspace, and light structure, the proposed CDPSHJ proves to be of great potential use for WMRMs. The future work will address the effects of different parameters of the spring on the CDPSHJ workspace and stiffness, develop the shoulder and wrist joints (three or four cables driven) of the anthropopathic manipulator, and make experiments on joints, including elbow, shoulder and wrist joints, and the manipulator.

Footnotes

Handling Editor: Roslinda Nazar

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation (NSF) of China (51275152) and NSF of Hebei Province (2018202114).

ORCID iD

Dongxing Cao

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Analysis on variable stiffness of a cable-driven parallel–series hybrid joint toward wheelchair-mounted robotic manipulator

Abstract

Keywords

Introduction

CDPSHJ description

Velocity Jacobian matrix

Tension analysis

Static analysis of lateral bending and compression spring

CDPSHJ variable stiffness analysis

Stiffness matrix homogenization

Numerical results of CDPSHJ stiffness adjustment

Wrench-feasible condition

Wrench-set condition

Cable-length condition

Translational-constraint condition

Translation motion and stiffness adjustment

Structural parameters (a, b, d) on stiffness adjustment

Bending and compression spring on stiffness adjustment

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References