Design and kinetostatic analysis of a tip-stiffness improved compliant continuum robot using anti-buckling universal joints

Abstract

Cable-driven compliant continuum robots (CCRs) can reach target areas in constrained spaces due to their elastic bodies being controlled remotely. They have been widely employed to inspect, maintain, or repair industrial machines such as gas turbine engines and oil pipes. However, their performances are usually limited by the low tip stiffness and the stiffness usually decreases with the increase of cable pulling forces. This paper aims to address the above problems by presenting a tip stiffness improved CCR formed by compliant anti-buckling universal joints (ACCR). The normalized nonlinear spatial models of the one-segment CCR and three-segment CCR are proposed and comprehensively verified using commercial nonlinear finite element analysis software. Given cable forces, prescribed cable displacements, tip loads, and gravity, the motion of any points on the CCRs can be analytically obtained. The performance characteristics of the one-segment CCRs and three-segment CCRs are extensively studied under different loading conditions, including the maximum CCR deformation, tip-location accuracy, shape dexterity, and tip stiffness. The results show that the tip stiffness of the ACCR is always much higher than that of the counterpart CCR under the same loading conditions. For example, the one-segment and three-segment ACCRs both have a high in-plane tip stiffness under in-plane actuation, which can increase by 49.0% and 31.3%, respectively; and they both have a high transverse tip stiffness under spatial actuation, which can increase by 48.9% and 31.2%, respectively. It is also confirmed that the ACCRs can increase tip stiffness by increasing cable actuation. Several preliminary planar experimental tests are carried out to validate the fabrication feasibility of the prototype, the accuracy of the analytical model, and/or the above-mentioned stiffness improvement.

1. Introduction

There has been a rapid increase in research on compliant continuum robots (CCRs) because they can bring potential advantages to many applications (S. Li and Hao, 2021). A CCR usually has a slender, elastic, and continuous shape, which is composed of multiple segments (Webster and Jones, 2010). It can enable access to confined spaces and ensures safe physical interactions with the surrounding environment, thanks to its exceptional flexibility. Dong et al. introduced a slender CCR for on-wing inspection or repair of gas turbine engines. Simaan and Taylor (2004) designed a CCR using multiple Nickel-titanium (NiTi) tubes with three distal dexterity units for laryngeal surgery. Amanov et al. (2021) designed an extensible CCR with path-following capabilities for minimally invasive surgery.

The performance of the majority of existing cable-driven CCRs is limited due to their low tip stiffness (Dong et al., 2022; Ji et al., 2020; Oliver-Butler et al., 2019; Ouabi et al., 2022; Smoljkic et al., 2014; Wang et al., 2018). The low tip stiffness of the CCRs driven by cables is mainly attributed to two coupled factors: the inherent small twisting stiffness (Dong et al., 2015) and negative load-dependent effects (associated with bending stiffness that decreases with the increase of cable forces) of constitutive joints/units. The following explanations show how low twisting stiffness in the structure reduces stiffness at the tip and how negative load-dependent effects contribute to low tip stiffness. The bending and twisting are depicted in Figure 1(f). Because the body’s centerline/backbone in deformation is not in line with that of the tip part, the twisting moment at the tip can bend partial non-proximal units of the CCR. Similarly, when a bending load acts at the tip of an actuated CCR, it can twist partial non-proximal units of the CCR. The smaller the twisting stiffness of the CCR, the lower the tip stiffness achieved. Most cable-driven CCRs are generally formed with modular compliant revolute/universal joints consisting of compressive flexures (Kutzer et al., 2011; Thomas et al., 2020; Wu et al., 2017; Zhang et al., 2021; Y. Zhao et al., 2024), where the cable forces only exert compression forces on these joints. The rotational/bending stiffness of these modular units usually decreases with the increase of cable forces, which is called the negative beam load-dependent effect (S. Li et al., 2022). Buckling may occur in CCRs if the cable force exceeds the critical load.

Figure 1.

Description of the ACCR: (a) modular unit Type 1, (b) modular unit Type 2, (c) exploded view of Type 1, (d) exploded view of Type 2, (e) a rotated inversion-based sheet under a compressive force, (f) a rotated non-inversion-based sheet under a compressive force, (g) simplified equivalent pseudo-rigid-body model for calculating the rotational stiffness of the deformed inversion-based sheet, (h) cable-loading positions, (i) spatial display of the ACCR, and (j) mimicking elephant’s trunk design-ACCR.

Many researchers are motivated to increase the stiffness of CCRs at different design levels. For example, at the bending segment design level, Langer et al. applied a reversible stiffening sheath for each bending segment by using dimension jamming (Clark and Rojas, 2019; Langer et al., 2018); Xing et al. employed low-melting-point alloy to obtain a solid phase for each segment (Xing et al., 2021). At the modular unit design level, Yang et al. added SMA alloys into each modular unit to lock robot movements (Yang et al., 2020); At the actuation level, Shiva et al. varied CCRs’ stiffness by antagonistically implementing two types of actuation (Hassan et al., 2017; Shiva et al., 2016; Stilli et al., 2017).

This paper focuses on improving the tip stiffness of cable-driven CCRs, whose elastic bodies can be controlled remotely. This improvement is achieved by incorporating anti-buckling universal joints (S. Li and Hao, 2021). CCRs using universal joints are quite common to have a much smoother curvature (in comparison with the revolute joints based CCR designs (Dong et al., 2015) with improved twisting stiffenss (compared with the wire-beam-backbone-based CCR designs (K. Xu et al., 2015). Many researchers designed CCRs using rigid universal joints combined with springs and multi-tendons (Hannan and Walker, 2000, 2001; Y. Liu et al., 2019b; Rone et al., 2018; Tang et al., 2017; Anderson, 1970; K. Xu et al., 2014; W. Xu et al., 2017; Yeshmukhametov et al., 2018; Yigit and Boyraz, 2017). In particular, K. Xu et al. developed a design specifically tailored for minimally invasive surgery (K. Xu et al., 2014). By incorporating rigid universal joints, they achieved a significant increase in twisting stiffness. Their work encourages us to design a CCR with compliant universal joints. To the best of our knowledge, no studies have reported the development of a CCR using a compliant universal joint with crossing spring pivots, which allows for a larger range of motion compared to existing compliant universal joints consisting of short sheets (Palmieri et al., 2012). The special advantage of the compliant universal joints is the intrinsic high rotational compliance in the degrees of freedom (DoF) direction and significantly high stiffness in the DoC (degrees of constraint) directions (S. Li and Hao, 2022). By introducing an inverse arrangement of the compliant universal joint, we can obtain an anti-buckling compliant universal joint, where its rotational (i.e., bending) stiffness can increase or remain constant by increasing cable forces (S. Li et al., 2022b). The compliant continuum robot utilizing anti-buckling compliant universal joints (ACCR) benefits from the distinct advantages of these joints. In other words, the tip stiffness of the ACCR can increase by increasing cable forces.

In the modeling method of the ACCR, we introduce a nonlinear spatial method utilizing the free-body diagram to derive a more accurate analytical model. There are many existing modeling methods for modeling the CCR, but they have some limitations as briefly listed as follows. (1) The center shifts of each anti-buckling universal joint should not be ignored, so the kinematic methods are not applied, such as DH method (Dong et al., 2015), constant curvature assumption (Goldman et al., 2014; Z. Li and Du, 2013; Nuelle et al., 2020; Webster and Jones, 2010), pseudo-rigid-body model (Venkiteswaran et al., 2019), and Bézier curve fitting (Yuan et al., 2017). In addition, these methods cannot effectively capture the nonlinearity of the ACCR; (2) The ACCR is not only limited to a very small ratio of cross-sectional dimension to its length, so its whole body cannot be simplified as a slender elastic rod. Therefore, Cosserat rod theory (Alqumsan et al., 2019; Caleb Rucker and Webster, 2014; Chikhaoui et al., 2019; Till et al., 2019, 2020; Till and Rucker, 2017) is not sufficient for capturing shear and twisting deformations of the ACCR. The thorough comparison between the proposed analytical model and existing analytical models for CCRs is summarized in Section 6.5.

While the anti-buckling universal joint itself is novel, its true significance lies in how it enables the construction of a new type of CCR. Specifically, the advantage of stiffness increase — a known bottleneck in most state-of-the-art CCRs— is discovered, analyzed, and validated through both finite element analysis (FEA) and experimental results in our work. We hope this can provide key insights to the community and enable a new generation of CCRs. The benefits and advantages of this CCR, made possible by the novel joint, extend far beyond the behavior of the joint itself, representing a broader scientific contribution. The primary contributions of this paper are summarized as follows:

(1) Two types of CCRs have been proposed: the ACCR and its counterpart CCR (as a benchmark design), both utilizing universal joints with crossing spring pivots. Each type is represented by one segment featuring three universal joints or by three segments totaling nine universal joints. The CCR that uses anti-buckling universal joints with inversion-based sheets (ACCR) has a much higher stiffness compared to the CCR that uses universal joints with non-inversion-based sheets (the counterpart CCR).

(2) The nonlinear spatial forward kinetostatic model of a CCR has been first derived, the system modeling framework in which can be used to model other CCRs with different compliant joints. Given any actuation and external loads (including cable forces, prescribed cable displacements, tip loads, and gravity), spatial displacements and rotations of any points on the stiffness improved CCR can be analytically obtained.

(3) The performance characteristics including the tip stiffness of the ACCR and the counterpart CCR are thoroughly analyzed and compared under different geometric parameters and loading conditions, using both analytical and FEA models.

(4) One-segment and three-segment prototypes are manufactured and experimentally tested. The modeling results have been rigorously verified through the utilization of nonlinear FEA and experiments. It has been experimentally confirmed that the tip stiffness of the one-segment ACCR can increase by increasing cable forces, and it is much larger than that of the one-segment counterpart CCR.

This paper is organized as follows. Section 2 describes the design of the CCR incorporating a series of anti-buckling universal joints. In Section 3, the normalized nonlinear spatial models of one-segment and three-segment CCRs are derived. The maximum CCR deformation, tip-location accuracy, shape dexterity, and tip stiffness improvement of the one-segment and three-segment CCRs are studied and verified with FEA and experiments in Sections 4 and 5, respectively. Section 6 discusses the effects of gravity, unaccounted considerations, scalability, reliability, and compares the three-segment ACCR with existing representative work. Conclusions and future work are finally drawn in Section 7.

2. Design description of the ACCR

We have presented two types of anti-buckling universal joints for the ACCR, as shown in Figure 1(a) and (b). Figure 1(c) and (d) are their exploded views, respectively. Each type of modular unit consists of four parts: a rigid base, a rigid middle loop, four elastic tensile sheets, and a rigid motion stage. The three rigid parts are not directly connected. The motion stage (red) and middle loop (green) are connected by a crossing spring pivot. The middle loop (green) and the base (blue) are connected by the other crossing spring pivot. Therefore, the three rigid parts are connected in series, which forms the anti-buckling universal joint. The only difference between the two types of modular units is the position of the middle loop. Type 1 has an internal middle loop, and Type 2 has an external middle loop. Type 1 and Type 2 joints have the same analytical model and experimental results in principle. More details on the design of these joints can be found in (S. Li et al., 2020b) and (S. Li and Hao, 2022).

Our design leverages the load-dependent effect of the compliant mechanism to increase ACCR stiffness by increasing cable actuation. This method provides an intrinsic way to enhance the ACCR’s stiffness. We use an elastic sheet as an example to demonstrate how load-dependent effects influence the stiffness of compliant mechanisms under applied loads. The load-dependent effects are classified in two types: structure load-dependent effects and beam load-dependent effects (S. Li et al., 2022). The structure load-dependent effect refers to how different loading positions can influence the stiffness of compliant mechanisms. The effective rotational stiffness of the deformed elastic sheet, denoted by K_eff, is quite different when the loading position of the compressive force is near to or far from its free end, as shown in Figure 1(e) and (g). Small angle assumption (i.e., sinθ ≈ θ) is used for calculating the rotational stiffness associated with the structure load-dependent effect. The beam load-dependent effect refers to how a compressive or tensile axial force influences the stiffness of compliant mechanisms. The rotational stiffness of the deformed elastic sheet increases with the increase of tensile axial forces and decreases with the increase of compressive axial forces. We denote this variation in rotational stiffness caused by beam load-dependent effects as ΔK, where ΔK >0 when the elastic sheet is under a tensile axial force, otherwise ΔK <0. By comparing Figure 1(e) and (f), we can observe that a compressive force in the non-inversion-based sheet is converted into a tensile force in the inversion-based sheet. Therefore, by an inversion-based arrangement of elastic sheets, compressive forces exerted on the motion stage of the anti-buckling universal joint lead to tensile axial forces exerted on elastic sheets. The anti-buckling universal joints have the following features. (1) The four elastic sheets thereof are long compared to the ones with short sheets in (S. Li et al., 2020b), which leads to a large deformation universal joint rather than some existing compliant universal joints with a smaller range of motion (Dong et al., 2014; Palmieri et al., 2012). (2) Its rotational stiffness is customizable (i.e., the stiffness can be increased or kept constant) with the increase of cable forces by regulating its geometric parameters and loading positions, as verified in (S. Li et al., 2022; S. Li and Hao, 2022). (3) Its stiffness along any of the constraint/bearing directions (axial motion along the Y_J-axis and twisting/rotational motion about the Y_J-axis) is as high as desired, as detailed in (S. Li and Hao, 2022).

A hybrid active and passive cable-driven method is employed for actuating each segment, which implies that each segment is driven by four cables. Compared with the active cable-driven method (i.e., each universal joint is driven by four cables), the hybrid active and passive cable-driven method enables a lighter mechanism with fewer motors and reduced complexity for the control (T. Liu et al., 2018). The cable-loading positions of an anti-buckling universal joint are shown in Figure 1(h). ACCR should consist of at least two segments to obtain an “S” shape under a hybrid active and passive actuation. The assembly of three segments is considered to form the ACCR, as shown in Figure 1(i). The cables of segment 3 pass through all the motion stages of the universal joints in segments 2 and 1, and the cables of segment 2 pass through all the motion stages in segment 1 (Z. Li et al., 2017; T. Liu et al., 2019a; W. Xu et al., 2018), as shown in Figure 1(j). As each segment of the three-segment ACCR is driven by four cables, each segment obtains two degrees of freedom (DoFs), and the three-segment ACCR produces six DoFs in total.

From a bio-inspired perspective, the ACCR is designed with a gradually decreasing rotational stiffness along segments toward the tip, mimicking the biomechanical properties of an elephant’s trunk. In nature, this progressive stiffness variation allows an elephant’s trunk to maintain both strength at the base for load-bearing and flexibility at the tip for precise manipulation. Similarly, in our design as presented in Figure 1(j), the sheet width (or thickness) of the gray anti-buckling universal joint (proximal) is larger than that of the red one (middle), and the sheet width (or thickness) of the red one is larger than that of the orange one (distal). This stiffness variation allows the ACCR to achieve smooth and continuous deformations. Each segment consists of three anti-buckling universal joints with identical stiffness, ensuring uniform mechanical properties within each segment while maintaining the overall stiffness gradient across the segments. Additionally, segment 2 is rotated by −π/2 about the ACCR central axis (Y_T) in its undeformed state, in order to increase twisting stiffness about the Y_T-axis. The ACCR’s stiffness is increased by increasing cable actuation, aligning with the adaptive mechanics of an elephant’s trunk, as elephants can actively increase stiffness through muscle contraction.

3. Normalized nonlinear spatial models

Throughout this paper, we use the right-handed coordinate system and right-handed rule; use capital symbols to denote dimensional parameters, and use lower-case symbols to denote normalized (dimensionless) ones if not specified otherwise. All the load-equilibrium conditions are derived in a deformed condition, which means all the translational matrices and cable-force components are derived in a deformed condition. All the subscripts for nomenclatures of the normalized coordinates, translational matrices, and loads include their relative coordinate systems, for instance, D_{p_J1} and D_{p_J2} denote the normalized translational matrices with respect to O_J1-X_J1Y_J1Z_J1 and O_J2-X_J2Y_J2Z_J2, respectively. All the normalized coordinates with ^* means that the normalized coordinates are the results with respect to the global fixed coordinate system in a deformed condition, such as Q₁^* denotes the normalized coordinates of cable-loading position Q₁ in a deformed condition with respect to O_G-X_GY_GZ_G.

3.1. Kinetostatic model of a one-segment ACCR

In this paper, the modeling procedure remains consistent regardless of the type of modular unit, with Type 1 serving as an illustrative example. As depicted in Figure 2(a), the one-segment CCR is composed of three anti-buckling universal joints (i.e., six inversion-based symmetric cross spring pivots (IS-CSP)) mounted in a serial arrangement. It is actuated with four cables. O_G-X_GY_GZ_G denotes the global fixed coordinate system of the one-segment ACCR, which is located at the rotational center of the non-deformed anti-buckling universal joint 1, that is, O_J1. Two types of local mobile coordinate systems of the anti-buckling universal joint are denoted by O_JΩ-X_JΩY_JΩZ_JΩ and O_JΩ(M)-X_JΩ(M)Y_JΩ(M)Z_JΩ(M) (Ω = 1, 2, or 3). In the O_JΩ-X_JΩY_JΩZ_JΩ system definition, O_J1-X_J1Y_J1Z_J1 and O_G-X_GY_GZ_G always overlap as shown in Figure 2(b), and the other two always accompany with the motion of the prior adjacent joint. In other words, O_J2-X_J2Y_J2Z_J2 is located at O_J2 after the motions of the anti-buckling universal joint 1, and O_J3-X_J3Y_J3Z_J3 is located at O_J3 after the motions of the anti-buckling universal joints 1 and 2. O_JΩ(M)-X_JΩ(M)Y_JΩ(M)Z_JΩ(M) denotes the new local mobile coordinate system with regard to the corresponding O_JΩ-X_JΩY_JΩZ_JΩ after rotations and translations of the universal joint-Ω (subscript (M) denotes mobile). The relationship between the O_JΩ-X_JΩY_JΩZ_JΩ (Ω = 1, 2, or 3) and O_JΩ(M)-X_JΩ(M)Y_JΩ(M)Z_JΩ(M) is formulated in Appendix A.

Figure 2.

Description of the one-segment ACCR: (a) cable-loading positions and the two overlapping coordinate systems: the global coordinate system O_G-X_GY_GZ_G and local coordinate system O_J1-X_J1Y_J1Z_J1, (b) two types of local coordinate systems of the three anti-buckling universal joints: O_JΩ-X_JΩY_JΩZ_JΩ and O_J(M)Ω-X_J(M)ΩY_J(M)ΩZ_J(M)Ω (Ω = 1, 2, or 3), respectively, and (c) the derivation of the cable-force components due to the cable force f_c1.

We use f_xTipG, f_yTipG, f_zTipG, m_xTipG, m_yTipG, and m_zTipG to denote the tip loads with respect to O_G-X_GY_GZ_G loading at tip O_TipG. The directions for external forces or moments are based on the global fixed coordinate system throughout the motions of the one-segment CCR, as seen in Figure 2(a) and (b) for the tip loads. We use f_cn (n = 1, 2, 3, or 4) to denote the normalized cable forces of cable-n, which is a positive constant along the cable-n; use A _n (n = 1, 2, 3, or 4) and Q _ε (ε = 1, 2, …, or 12) to denote the pulley positions and cable-loading positions of the one-segment ACCR, respectively, where A₃ is in the front of A₄, and two corresponding cables are not indicated in Figure 2(a). The four cables are fixed at Q₉ to Q₁₂ and pass through Q₁ to Q₈.

L_d denotes the characteristic length for normalization, which is equal to the diagonal length of the ACCR (cf. Figure 2(a)). All translational displacements and length parameters are divided by the footprint L_d. l_a, u, and t denote the normalized length, width, and thickness of each sheet, respectively. α, λ, h, l_J, and l_G denote sheet tilt angle, crossing position, normalized loading position, the normalized diameter, and normalized height of the anti-buckling universal joint, respectively. They are the geometric parameters of the one-segment ACCR. Forces and moments are divided by EI_z/L_d² and EI_z/L_d, respectively, where I_z denotes the cross-section moment and is expressed as UT³/12; E is Young’s modulus of the material. (For more details on the geometric parameters, the third figure in (S. Li and Hao, 2022) can be visited).

According to our previous work of modeling the IS-CSP (S. Li and Hao, 2022), given loading inputs (tip loads and cable forces) acting at the motion stage of the IS-CSP, translational displacements and rotations of the IS-CSP can be solved (six unknowns). We take each IS-CSP as a modular unit to model a one-segment ACCR, composed of six IS-CSPs, which makes a total of thirty-six unknowns. When geometric parameters (l_a, u, t, α, λ, h, l_J, and l_G) and loading inputs of the one-segment ACCR with respect to O_G-X_GY_GZ_G are given, the performances of the one-segment ACCR can be obtained by solving the load-equilibrium condition and compatibility condition (thirty-six equations in total) with an initial guess, which are summarized in Figure 3 and detailed below.

Step 1 : The compatibility conditions of the one-segment ACCR.

Figure 3.

Forward kinetostatic model of the one-segment ACCR.

We use R_s, R_J, and R_G to denote the rotational matrices of an IS-CSP, an anti-buckling universal joint, and the one-segment ACCR with respect to their global coordinate systems, respectively. Their rotational sequences are described by the Euler angles using a ZXY convention as derived in equation (1).

$\mathbf{R}={\mathbf{R}}_{\mathrm{x}}\left({\theta }_{\mathrm{x}}\right)\hspace{0.17em}{\mathbf{R}}_{\mathrm{z}}\left({\theta }_{\mathrm{z}}\right)\hspace{0.17em}{\mathbf{R}}_{\mathrm{y}}\left({\theta }_{\mathrm{y}}\right)$

(1)

where R_x, R_y, and R_z denote the rotational matrices about the X, Y, and Z axes, respectively, which are shown as equation (2).

${\mathbf{R}}_{\mathrm{x}}\left({\theta }_{\mathrm{x}}\right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \hspace{0.17em}{\theta }_{\mathrm{x}}& -\sin \hspace{0.17em}{\theta }_{\mathrm{x}}\\ 0& \sin \hspace{0.17em}{\theta }_{\mathrm{x}}& \cos \hspace{0.17em}{\theta }_{\mathrm{x}}\end{array}\right];\hspace{0.17em}{\mathbf{R}}_{\mathrm{y}}\left({\theta }_{\mathrm{y}}\right)=\left[\begin{array}{ccc}\cos \hspace{0.17em}{\theta }_{\mathrm{y}}& 0& \sin \hspace{0.17em}{\theta }_{\mathrm{y}}\\ 0& 1& 0\\ -\sin \hspace{0.17em}{\theta }_{\mathrm{y}}& 0& \cos \hspace{0.17em}{\theta }_{\mathrm{y}}\end{array}\right];\hspace{0.17em}{\mathbf{R}}_{\mathrm{z}}\left({\theta }_{\mathrm{z}}\right)=\left[\begin{array}{ccc}\cos \hspace{0.17em}{\theta }_{\mathrm{z}}& -\sin \hspace{0.17em}{\theta }_{\mathrm{z}}& 0\\ \sin \hspace{0.17em}{\theta }_{\mathrm{z}}& \cos \hspace{0.17em}{\theta }_{\mathrm{z}}& 0\\ 0& 0& 1\end{array}\right]$

(2)

R_JΩ (Ω = 1, 2, or 3) denotes a rotational matrix of the anti-buckling universal joint-Ω with respect to O_JΩ-X_JΩY_JΩZ_JΩ, which can be derived as equations (3) to (5). (For more details on R_JΩ, equation (21) in (S. Li and Hao, 2022) can be visited).

${\mathbf{R}}_{\mathrm{J}1}={\mathbf{R}}_{\mathrm{s}1}{\mathbf{R}}_{\mathrm{y}}\left(\pi /2\right){\mathbf{R}}_{\mathrm{s}2}{\mathbf{R}}_{\mathrm{y}}\left(-\pi /2\right)$

(3)

${\mathbf{R}}_{\mathrm{J}2}={\mathbf{R}}_{\mathrm{s}3}{\mathbf{R}}_{\mathrm{y}}\left(\pi /2\right){\mathbf{R}}_{\mathrm{s}4}{\mathbf{R}}_{\mathrm{y}}\left(-\pi /2\right)$

(4)

${\mathbf{R}}_{\mathrm{J}3}={\mathbf{R}}_{\mathrm{s}5}{\mathbf{R}}_{\mathrm{y}}\left(\pi /2\right){\mathbf{R}}_{\mathrm{s}6}{\mathbf{R}}_{\mathrm{y}}\left(-\pi /2\right)$

(5)

where R_sw (w = 1, 2, …, or 6) denotes a rotational matrix of the IS-CSP-w with respect to its local coordinate system. The numbering of IS-CSP-w (w = 1, 2, …, or 6) starts from the IS-CSP located at the base.

The normalized center shift of the anti-buckling universal joint-Ω (Ω = 1, 2, or 3) with respect to O_JΩ-X_JΩY_JΩZ_JΩ is denoted by d_cJΩ = [d_xcJΩ, d_ycJΩ, d_zcJΩ]^T, as given in equation (6).

${\mathbf{d}}_{\text{cJ}\mathrm{\Omega }}=\left[\begin{array}{c}{d}_{\text{xc}\left(2\mathrm{\Omega }-1\right)}\\ d_{\text{yc}\left(2\mathrm{\Omega }-1\right)}\\ d_{\text{zc}\left(2\mathrm{\Omega }-1\right)}\end{array}\right]+{\mathbf{R}}_{\mathrm{s}\left(2\mathrm{\Omega }-1\right)}{\mathbf{R}}_{\mathrm{y}}\left(\frac{\mathrm{\pi }}{2}\right)\left[\begin{array}{c}{d}_{\text{xc}\left(2\mathrm{\Omega }\right)}\\ d_{\text{yc}\left(2\mathrm{\Omega }\right)}\\ d_{\text{zc}\left(2\mathrm{\Omega }\right)}\end{array}\right]$

(6)

where [d_xcw, d_ycw, d_zcw]^T (w = 2Ω−1 or 2 Ω and Ω = 1, 2, or 3) denotes the normalized center shifts of the IS-CSP-w with respect to its local coordinate system, which is given as equation (7). (For more details on [d_xcw, d_ycw, d_zcw]^T, equation (31) in (S. Li and Hao, 2022) can be visited).

$\left[\begin{array}{c}{d}_{\text{xc}w}\\ d_{\text{yc}w}\\ d_{\text{zc}w}\end{array}\right]={\mathbf{R}}_{\mathrm{z}1}^{-1}\left({\beta }_1\right)\left[\begin{array}{c}{d}_{\mathrm{x}\left(2w-1\right)}\\ d_{\mathrm{y}\left(2w-1\right)}\\ d_{\mathrm{z}\left(2w-1\right)}\end{array}\right]-\left({\mathbf{R}}_{\mathrm{s}\left(2w-1\right)}\left[\begin{array}{c}\lambda l_{\mathrm{a}}\hspace{0.17em}\sin \hspace{0.17em}\alpha \\ -\lambda l_{\mathrm{a}}\hspace{0.17em}\cos \hspace{0.17em}\alpha \\ l_{\mathrm{s}}\end{array}\right]-\left[\begin{array}{c}\lambda l_{\mathrm{a}}\hspace{0.17em}\sin \hspace{0.17em}\alpha \\ -\lambda l_{\mathrm{a}}\hspace{0.17em}\cos \hspace{0.17em}\alpha \\ l_{\mathrm{s}}\end{array}\right]\right)$

(7)

• Step 2 : The load-equilibrium conditions of the one-segment ACCR.

All cable-force components due to f_c1 through f_c4 acting on the cable-loading positions in a deformed condition with respect to O_G-X_GY_GZ_G should be obtained first. They are denoted by f_{NetQε_G}^* (ε = 1, 2, …, 12), as shown in Figure 2(c). We applied the Euler-Eytelwein formula to obtain the cable-force components, and the cable-force components perpendicular to the cable sliding plane are neglected in (S. Li and Hao, 2022). However, the cable-force components are calculated by a vector method and all cable-force components are considered in this paper. Let us take f_c1 as an example to calculate the cable-force components with respect to O_G-X_GY_GZ_G acting at loading positions Q₁^*, Q₅^*, and Q₉^*, as derived in equations (8)–(10). Q₁^*, Q₅^*, and Q₉^* denote the points of Q₁, Q₅, and Q₉ in a deformed condition.

${{\mathbf{f}}_{\text{NetQ}1\_\mathrm{G}}}^{*}=f_{\mathrm{c}1}\frac{{\mathbf{A}}_1-{{\mathbf{Q}}_1}^{*}}{\Vert {\mathbf{A}}_1-{{\mathbf{Q}}_1}^{*}\Vert }+f_{\mathrm{c}1}\frac{{{\mathbf{Q}}_5}^{*}-{{\mathbf{Q}}_1}^{*}}{\Vert {{\mathbf{Q}}_5}^{*}-{{\mathbf{Q}}_1}^{*}\Vert }$

(8)

${{\mathbf{f}}_{\text{NetQ}5\_\mathrm{G}}}^{*}=f_{\mathrm{c}1}\frac{{{\mathbf{Q}}_9}^{*}-{{\mathbf{Q}}_5}^{*}}{\Vert {{\mathbf{Q}}_9}^{*}-{{\mathbf{Q}}_5}^{*}\Vert }+f_{\mathrm{c}1}\frac{{{\mathbf{Q}}_1}^{*}-{{\mathbf{Q}}_5}^{*}}{\Vert {{\mathbf{Q}}_1}^{*}-{{\mathbf{Q}}_5}^{*}\Vert }$

(9)

${{\mathbf{f}}_{\text{NetQ}9\_\mathrm{G}}}^{*}=f_{\mathrm{c}1}\frac{{{\mathbf{Q}}_5}^{*}-{{\mathbf{Q}}_9}^{*}}{\Vert {{\mathbf{Q}}_5}^{*}-{{\mathbf{Q}}_9}^{*}\Vert }$

(10)

where A₁, Q₁^*, Q₅^*, and Q₉^*denote the normalized coordinates of A₁, Q₁^*, Q₅^*, and Q₉^*with respect to O_G-X_GY_GZ_G;

Q₁^* is derived as equation (11) due to the rotations and translations of the anti-buckling universal joint 1; Q₅^* is calculated as equation (12) due to the rotations and translations of the anti-buckling universal joints 1 and 2; Q₉^* is written as equation (13) due to the rotations and translations of the three anti-buckling universal joints. Q_{1_JΩ}, Q_{5_JΩ}, Q_{9_JΩ} (Ω = 1, 2, or 3) denote the normalized coordinates of points Q₁, Q₅, and Q₉ with respect to O_JΩ-X_JΩY_JΩZ_JΩ in a non-deformed condition, so Q_{1_JΩ(M)} = Q_{1_JΩ}, Q_{5_JΩ(M)} = Q_{5_JΩ}, Q_{9_JΩ(M)} = Q_{9_JΩ} (Ω = 1, 2, or 3).

$\left(\begin{array}{c}{{\mathbf{Q}}_1}^{*}\\ 1\end{array}\right)={}^{\mathrm{G}}\mathbf{T}_{\mathrm{J}1\left(\mathrm{M}\right)}\left(\begin{array}{c}{\mathbf{Q}}_{1\_\mathrm{J}1 \left(\mathrm{M}\right)}\\ 1\end{array}\right)$

(11)

$\left(\begin{array}{c}{{\mathbf{Q}}_5}^{*}\\ 1\end{array}\right)={}^{\mathrm{G}}\mathbf{T}_{\mathrm{J}1\left(\mathrm{M}\right)}{}^{\mathrm{J}1\left(\mathrm{M}\right)}\mathbf{T}_{\mathrm{J}2}{}^{\mathrm{J}2}\mathbf{T}_{\mathrm{J}2\left(\mathrm{M}\right)}\left(\begin{array}{c}{\mathbf{Q}}_{5\_\mathrm{J}2\left(\mathrm{M}\right)}\\ 1\end{array}\right)$

(12)

$\left(\begin{array}{c}{{\mathbf{Q}}_9}^{*}\\ 1\end{array}\right)={}^{\mathrm{G}}\mathbf{T}_{\mathrm{J}1\left(\mathrm{M}\right)}{}^{\mathrm{J}1\left(\mathrm{M}\right)}\mathbf{T}_{\mathrm{J}2}{}^{\mathrm{J}2}\mathbf{T}_{\mathrm{J}2\left(\mathrm{M}\right)}{}^{\mathrm{J}2\left(\mathrm{M}\right)}\mathbf{T}_{\mathrm{J}3}{}^{\mathrm{J}3}\mathbf{T}_{\mathrm{J}3\left(\mathrm{M}\right)}\left(\begin{array}{c}{\mathbf{Q}}_{9\_\mathrm{J}3\left(\mathrm{M}\right)}\\ 1\end{array}\right)$

(13)

where Q_{1_J1(M)} denotes the normalized coordinates of Q₁ with respect to O_J1(M)-X_J1(M)Y_J1(M)Z_J1(M), as shown in Figure 2(b); ^GT_J1(M) denotes the homogeneous transformation matrix, which is used for expressing the coordinates with respect to O_J1(M)-X_J1(M)Y_J1(M)Z_J1(M) into the coordinates with respect to O_G-X_GY_GZ_G, and their results are shown in Appendix A; The nomenclature of other transformation matrices can be understand similarly.

Therefore, the cable-force components loading at Q₁^*, Q₅^*,^, and Q₉^* with respect to O_J1-X_J1Y_J1Z_J1, O_J2-X_J2Y_J2Z_J2, and O_J3-X_J3Y_J3Z_J3, respectively, are formulated in equation (14). Other cable-force components due to f_c2 through f_Sc4 can be derived similarly.

${{\mathbf{f}}_{\text{NetQ}1\_\mathrm{J}1}}^{*}={{\mathbf{f}}_{\text{NetQ}1\_\mathrm{G}}}^{*},{{\mathbf{f}}_{\text{NetQ}5\_\mathrm{J}2}}^{*}={{\mathbf{R}}_{\mathrm{J}1}}^{-1}{{\mathbf{f}}_{\text{NetQ}5\_\mathrm{G}}}^{*},{{\mathbf{f}}_{\text{NetQ}9\_\mathrm{J}3}}^{*}={\left({\mathbf{R}}_{\mathrm{J}1}{\mathbf{R}}_{\mathrm{J}2}\right)}^{-1}{{\mathbf{f}}_{\text{NetQ}9\_\mathrm{G}}}^{*}$

(14)

where f_{NetQ1_J1}^*, f_{NetQ5_J2}^*, and f_{NetQ9_J3}^* denote cable-force components acting at Q₁^*, Q₅^*,^, and Q₉^* with respect to O_J1-X_J1Y_J1Z_J1, O_J2-X_J2Y_J2Z_J2, and O_J3-X_J3Y_J3Z_J3, respectively.

Then all the loading inputs, including tip loads and cable-force components with respect to O_G-X_GY_GZ_G, should be equivalently converted to the loads with respect to O_JΩ-X_JΩY_JΩZ_JΩ (Ω = 1, 2, or 3) for each anti-buckling universal joint in a deformed condition. The cable-force components loading at Q₁^*–Q₄^* have been considered in the analytical modeling of the anti-buckling universal joint 1, similarly to the cable-force components loading at Q₅^*–Q₈^*, and Q₉^*–Q₁₂^* have been considered in the analytical modeling of an anti-buckling universal joints 2 and 3. The load-equilibrium condition for each single IS-CSP is completed. The load-equilibrium condition between different universal joints should be further derived because the loads loading at the top joint influence the motions of its underneath joints. For instance, the cable-force components loading at the anti-buckling universal joint 3 and tip loads with respect to O_G-X_GY_GZ_G, lead to moments acting at the anti-buckling universal joint 2. We use f_xJΩ, f_yJΩ, f_zJΩ, m_xJΩ, m_yJΩ, and m_zJΩ, to denote normalized loads of the anti-buckling universal joint-Ω acting at O_JΩ with respect to O_JΩ-X_JΩY_JΩZ_JΩ, respectively. The load-equilibrium condition of the one-segment ACCR is formulated as equations (15), (20), and (25).

We assume each anti-buckling universal joint has the gravity located at its rotational center. Equation (15) shows the load-equilibrium condition of the anti-buckling universal joint 3 with respect to O_J3-X_J3Y_J3Z_J3, which is formulated due to the tip loads and the gravity of the anti-buckling universal joint 3.

$\left[\begin{array}{c}{\mathbf{f}}_{\mathrm{J}3}\\ {\mathbf{m}}_{\mathrm{J}3}\end{array}\right]={{\mathbf{D}}_{\text{pOTipG}\_\mathrm{J}3}^{\mathrm{T}}}^{*}{\left({\mathbf{R}}_{\text{JJ}1}{\mathbf{R}}_{\text{JJ}2}\right)}^{-1}\left[\begin{array}{c}{\mathbf{f}}_{\text{TipG}}\\ {\mathbf{m}}_{\text{TipG}}\end{array}\right]+{{\mathbf{D}}_{\text{pOmaJ}3\_\mathrm{J}3}^{\mathrm{T}}}^{*}{\left({\mathbf{R}}_{\text{JJ}1}{\mathbf{R}}_{\text{JJ}2}\right)}^{-1}\left[\begin{array}{c}0\\ f_{\text{maJ}3\_\mathrm{G}}\\ 0\\ {\mathbf{0}}_{3\times 1}\end{array}\right]$

(15)

where f_J3 = [f_xJ3, f_yJ3, f_zJ3]^T, m_J3 = [m_xJ3, m_yJ3, m_zJ3]^T, f_TipG = [f_xTipG, f_yTipG, f_zTipG]^T, and m_TipG = [m_xTipG, m_yTipG, m_zTipG]^T. D_{pOTipG_J3}^* and D_{pOmaJ3_J3}^* denote normalized translation matrices of points O_{TipG_J3}^* and O_{maJ3_J3}^*, which are given as equation (16); a ₀ represents either O_{TipG_J3}^* or O_{maJ3_J3}^* here. O_{TipG_J3}^* and O_{maJ3_J3}^* denote the normalized coordinate of the tip O_TipG and the mass center of the anti-buckling universal joint 3 O_maJ3 with respect to O_J3-X_J3Y_J3Z_J3 in a deformed condition, as calculated in equations (17) and (18), respectively; f_{maJ3_G} denotes the gravity of the anti-buckling universal joint 3 with respect to O_G-X_GY_GZ_G;

${\mathbf{D}}_{\text{pa}0}=\left[\begin{array}{cc}{\mathbf{I}}_{3\times 3}& \left[\begin{array}{ccc}0& {\mathbf{a}}_{\mathbf{0}}\left(3,1\right)& -{\mathbf{a}}_{\mathbf{0}}\left(2,1\right)\\ -{\mathbf{a}}_{\mathbf{0}}\left(3,1\right)& 0& {\mathbf{a}}_{\mathbf{0}}\left(1,1\right)\\ {\mathbf{a}}_{\mathbf{0}}\left(2,1\right)& -{\mathbf{a}}_{\mathbf{0}}\left(1,1\right)& 0\end{array}\right]\\ {\mathbf{0}}_{3\times 3}& {\mathbf{I}}_{3\times 3}\end{array}\right]$

(16)

$\left[\begin{array}{c}{{\mathbf{O}}_{\text{TipG}\_\mathrm{J}3}}^{*}\\ 1\end{array}\right]={}^{\mathrm{J}3}\mathrm{T}_{\mathrm{J}3\left(\mathrm{M}\right)}\left[\begin{array}{c}{\mathbf{O}}_{\text{TipG}\_\mathrm{J}3\left(\mathrm{M}\right)}\\ 1\end{array}\right]$

(17)

$\left[\begin{array}{c}{{\mathbf{O}}_{\text{maJ}3\_\mathrm{J}3}}^{*}\\ 1\end{array}\right]={}^{\mathrm{J}3}\mathrm{T}_{\mathrm{J}3\left(\mathrm{M}\right)}\left[\begin{array}{c}{\mathbf{O}}_{\text{maJ}3\_\mathrm{J}3\left(\mathrm{M}\right)}\\ 1\end{array}\right]$

(18)

where ^J3T_J3(M) can be found in Appendix A; d_cJ3 = [d_xcJ3, d_ycJ3, d_zcJ3]^T is the center shift of the anti-buckling universal joint 3, as derived in equation (6); I_3×3 denotes an identity matrix;

R_JJ1 and R_JJ2 in equation (15) denote 6 × 6 rotational matrices, as derived in equation (19).

${\mathbf{R}}_{\text{JJ}1}=\left[\begin{array}{cc}{\mathbf{R}}_{\mathrm{J}1}& {\mathbf{0}}_{3\times 3}\\ {\mathbf{0}}_{3\times 3}& {\mathbf{R}}_{\mathrm{J}1}\end{array}\right]\hspace{1em}\mathrm{and}\hspace{1em}{\mathbf{R}}_{\text{JJ}2}=\left[\begin{array}{cc}{\mathbf{R}}_{\mathrm{J}2}& {\mathbf{0}}_{3\times 3}\\ {\mathbf{0}}_{3\times 3}& {\mathbf{R}}_{\mathrm{J}2}\end{array}\right]$

(19)

Equation (20) shows the load-equilibrium condition of the anti-buckling universal joint 2 with respect to O_J2-X_J2Y_J2Z_J2 due to the effects of tip loads, gravity, and cable-force components loading at anti-buckling universal joint 3, and gravity of the anti-buckling universal joint 2.

$\left[\begin{array}{c}{\mathbf{f}}_{\mathrm{J}2}\\ {\mathbf{m}}_{\mathrm{J}2}\end{array}\right]={{\mathbf{D}}_{\text{pOTipG}\_\mathrm{J}2}^{\mathrm{T}}}^{*}{\mathbf{R}}_{\text{JJ}1}^{-1}\left[\begin{array}{c}{\mathbf{f}}_{\text{TipG}}\\ {\mathbf{m}}_{\text{TipG}}\end{array}\right]+{{\mathbf{D}}_{\text{pOmaJ}3\_\mathrm{J}2}^{\mathrm{T}}}^{*}{\mathbf{R}}_{\text{JJ}1}^{-1}\left[\begin{array}{c}0\\ f_{\text{maJ}3\_\mathrm{G}}\\ 0\\ {\mathbf{0}}_{3\times 1}\end{array}\right]+{\displaystyle \sum }_{k=9}^{12}{{\mathbf{D}}_{\text{pQ}k\_\mathrm{J}2}^{\mathrm{T}}}^{*}{\mathbf{R}}_{\text{JJ}1}^{-1}\left[\begin{array}{c}{{\mathbf{f}}_{\text{NetQ}k\_\mathrm{G}}}^{*}\\ {\mathbf{0}}_{3\times 1}\end{array}\right]+{{\mathbf{D}}_{\text{pOmaJ}2\_\mathrm{J}2}^{\mathrm{T}}}^{*}{\mathbf{R}}_{\text{JJ}1}^{-1}\left[\begin{array}{c}0\\ f_{\text{maJ}2\_\mathrm{G}}\\ 0\\ {\mathbf{0}}_{3\times 1}\end{array}\right]$

(20)

where f_J2 = [f_xJ2, f_yJ2, f_zJ2]^T, m_J2 = [m_xJ2, m_yJ2, m_zJ2]^T , f_{maJ2_G} denotes the gravity of the anti-buckling universal joint 2 with respect to O_G-X_GY_GZ_G; D_{pTipG_J2}^*, D_{pmaJ3_J2}^*, D_{pQk_J2}^* (k = 9, 10, 11, or 12), and D_{pmaJ2_J2}^* denote the normalized translational matrices of points O_{TipG_J2}^*, O_{maJ3_J2}^*, Q_{k_J2}^*, and O_{maJ2_J2}^*, as derived in equation (16). a ₀ represents O_{TipG_J2}^*, Q_{k_J2} ^* , O_{maJ3_J2}^*, or O_{maJ2_J2}^* here and denotes the normalized coordinates of the tip O_TipG, center of mass O_maJ3, loading positions Q _k or center of mass of the anti-buckling universal joint 2 O_maJ2 with respect to O_J2-X_J2Y_J2Z_J2 in a deformed condition;

$\left(\begin{array}{c}{{\mathbf{O}}_{\text{TipG}\_\mathrm{J}2}}^{*}\\ 1\end{array}\right)={\mathbf{T}}_{\text{Unknown}1}\left(\begin{array}{c}{\mathbf{O}}_{\text{TipG}\_\mathrm{J}3\left(\mathrm{M}\right)}\\ 1\end{array}\right)$

(21)

$\left(\begin{array}{c}{{\mathbf{Q}}_{k\_\mathrm{J}2}}^{*}\\ 1\end{array}\right)={\mathbf{T}}_{\text{Unknown}1}\left(\begin{array}{c}{\mathbf{Q}}_{k\_\mathrm{J}3\left(\mathrm{M}\right)}\\ 1\end{array}\right)$

(22)

$\left(\begin{array}{c}{{\mathbf{O}}_{\text{maJ}3\_\mathrm{J}2}}^{*}\\ 1\end{array}\right)={\mathbf{T}}_{\text{Unknown}1}\left(\begin{array}{c}{\mathbf{O}}_{\text{maJ}3\_\mathrm{J}3\left(\mathrm{M}\right)}\\ 1\end{array}\right)$

(23)

$\left(\begin{array}{c}{{\mathbf{O}}_{\text{maJ}2\_\mathrm{J}2}}^{*}\\ 1\end{array}\right)={}^{\mathrm{J}2}\mathbf{T}_{\mathrm{J}2\left(\mathrm{M}\right)}\left(\begin{array}{c}{\mathbf{O}}_{\text{maJ}2\_\mathrm{J}2\left(\mathrm{M}\right)}\\ 1\end{array}\right)$

(24)

where T_Unknown1 = ^J2T_J2(M) ^J2(M)T_J3 ^J3T_J3(M) and the transformation matrices can be found in Appendix A.

Equation (25) formulates the load equilibrium of the anti-buckling universal joint 1 with respect to O_J1-X_J1Y_J1Z_J1 due to the effects of tip loads, gravity, cable-force components loading at anti-buckling universal joint 3; gravity and cable-force components loading at anti-buckling universal joint 2; and gravity of the anti-buckling universal joint 1.

$\left[\begin{array}{c}{\mathbf{f}}_{\mathrm{J}1}\\ {\mathbf{m}}_{\mathrm{J}1}\end{array}\right]={{\mathbf{D}}_{\text{pOTipG}\_\mathrm{J}1}^{\mathrm{T}}}^{*}\left[\begin{array}{c}{\mathbf{f}}_{\text{TipG}}\\ {\mathbf{m}}_{\text{TipG}}\end{array}\right]+{\displaystyle \sum }_{W=1}^3{{\mathbf{D}}_{\text{pOmaJ}W\_\mathrm{J}1}^{\mathrm{T}}}^{*}\left[\begin{array}{c}0\\ f_{\text{maJ}W\_\mathrm{G}}\\ 0\\ {\mathbf{0}}_{3\times 1}\end{array}\right]+{\displaystyle \sum }_{k=9}^{12}{{\mathbf{D}}_{\text{pQ}k\_\mathrm{J}1}^{\mathrm{T}}}^{*}\left[\begin{array}{c}{{\mathbf{f}}_{\text{NetQ}k\_\mathrm{G}}}^{*}\\ {\mathbf{0}}_{3\times 1}\end{array}\right]+{\displaystyle \sum }_{\varsigma =5}^8{{\mathbf{D}}_{\text{pQ}\varsigma \_\mathrm{J}1}^{\mathrm{T}}}^{*}\left[\begin{array}{c}{{\mathbf{f}}_{\text{NetQ}\varsigma \_\mathrm{G}}}^{*}\\ {\mathbf{0}}_{3\times 1}\end{array}\right]$

(25)

where f_J1 = [f_xJ1, f_yJ1, f_zJ1]^T; m_J1 = [m_xJ1, m_yJ1, m_zJ1]^T; f_{maJ1_G} denotes the gravity of the anti-buckling universal joint 1 with respect to O_G-X_GY_GZ_G; D_{pOTipG_J1}^*, D_{pOmaJ3_J1}^*, D_{pQk_J1}^* (k = 9, 10, 11, or 12), D_{pOmaJ2_J1}^*, D_{pQς_J1}^* (ς = 5, 6, 7, or 8) and D_{pOmaJ1_J1}^* denote the normalized translational matrices of O_{TipG_J1}^*, O_{maJ3_J1}^*, Q_{k_J1}^*, O_{maJ2_J1}^*, Q_{ς_J1}^*, and O_{maJ1_J1}^*, as given in equation (16); a ₀ represents O_{TipG_J1}^*, O_{maJ3_J1}^*, Q_{k_J1}^*, O_{maJ2_J1}^*, Q_{ς_J1}^*, or O_{maJ1_J1}^* here and denotes the normalized coordinate of tip O_TipG, center of mass O_maJ3, cable-loading position Q _k , center of mass O_maJ2, cable-loading position Q _ς , or center of mass of the anti-buckling universal joint 1 O_maJ1 with respect to O_G-X_GY_GZ_G in a deformed condition, as written in equations (26)–(30); d_cJ1 = [d_xcJ1, d_ycJ1, d_zcJ1]^T is the center shift of the anti-buckling universal joint 1, as derived in equation (6);

$\left(\begin{array}{c}{{\mathbf{O}}_{\text{TipG}\_\mathrm{J}1}}^{*}\\ 1\end{array}\right)={\mathbf{T}}_{\text{Unknown}2}\left(\begin{array}{c}{\mathbf{O}}_{\text{TipG}\_\mathrm{J}3\left(\mathrm{M}\right)}\\ 1\end{array}\right)$

(26)

$\left[\begin{array}{c}{{\mathbf{Q}}_{k\_\mathrm{J}1}}^{*}\\ 1\end{array}\right]={\mathbf{T}}_{\text{Unknown}2}\left[\begin{array}{c}{\mathbf{Q}}_{k\_\mathrm{J}3\left(\mathrm{M}\right)}\\ 1\end{array}\right]$

(27)

$\left[\begin{array}{c}{{\mathbf{Q}}_{\varsigma \_\mathrm{J}1}}^{*}\\ 1\end{array}\right]={\mathbf{T}}_{\text{Unknown}2}\left[\begin{array}{c}{\mathbf{Q}}_{\varsigma \_\mathrm{J}3\left(\mathrm{M}\right)}\\ 1\end{array}\right]$

(28)

$\left[\begin{array}{c}{{\mathbf{O}}_{\text{maJ}i\_\mathrm{J}1}}^{*}\\ 1\end{array}\right]={\mathbf{T}}_{\text{Unknown}2}\left[\begin{array}{c}{\mathbf{O}}_{\text{maJ}i\_\mathrm{J}3\left(\mathrm{M}\right)}\\ 1\end{array}\right]\hspace{0.17em}(i=1\hspace{0.17em}\mathrm{or}\hspace{0.17em}2)$

(29)

$\left[\begin{array}{c}{{\mathbf{O}}_{\text{maJ}1\_\mathrm{J}1}}^{*}\\ 1\end{array}\right]={}^{\mathrm{G}}\mathbf{T}_{\mathrm{J}1\mathrm{M}}\left[\begin{array}{c}{\mathbf{O}}_{\text{maJ}1\_\mathrm{J}1\left(\mathrm{M}\right)}\\ 1\end{array}\right]$

(30)

where T_Unknown2 = ^GT_J1(M) ^J1(M)T_J2 ^J2T_J2(M) ^J2(M)T_J3 ^J3T_J3(M) and the transformation matrices can be found in Appendix A.

• Step 3 : Calculation on the motions and cable length after solving the performance of the one-segment ACCR. Motions of other points on the ACCR can be solved similarly.

The normalized tip displacements are denoted by d_TipG = [d_xTipG, d_yTipG, d_zTipG]^T with respect to O_G-X_GY_GZ_G are formulated in equation (31), which resultes from the rotational and translational motions of the three anti-buckling universal joints.

${\mathbf{d}}_{\text{TipG}}={\mathbf{R}}_{\mathrm{J}1}{\mathbf{O}}_{\text{TipG}-\mathrm{J}1}-{\mathbf{O}}_{\text{TipG}-\mathrm{J}1}+{\mathbf{R}}_{\mathrm{J}1}\left({\mathbf{R}}_{\mathrm{J}2}{\mathbf{O}}_{\text{TipG}-\mathrm{J}2}-{\mathbf{O}}_{\text{TipG}-\mathrm{J}2}\right)+{\mathbf{R}}_{\mathrm{J}1}{\mathbf{R}}_{\mathrm{J}2}\left({\mathbf{R}}_{\mathrm{J}3}{\mathbf{O}}_{\text{TipG}-\mathrm{J}3}-{\mathbf{O}}_{\text{TipG}-\mathrm{J}3}\right)+{\mathbf{d}}_{\text{cG}}$

(31)

where O_TipG-JΩ (Ω = 1, 2, or 3) denotes the normalized coordinates of the tip O_TipG in a non-deformed condition with respect to O_JΩ-X_JΩY_JΩZ_JΩ; d_cG = [d_xcG, d_ycG, d_zcG]^T denotes the normalized center shift of the one-segment ACCR with respect to O_G-X_GY_GZ_G as derived in equation (32); d_cJΩ = [d_xcJΩ, d_ycJΩ, d_zcJΩ]^T (Ω = 1, 2, or 3) denotes the normalized center shift of the anti-buckling universal joint- Ω with respect to O_JΩ-X_JΩY_JΩZ_JΩ.

${\mathbf{d}}_{\text{cG}}={\mathbf{d}}_{\text{cJ}1}+{\mathbf{R}}_{\mathrm{J}1}{\mathbf{d}}_{\text{cJ}2}+{\mathbf{R}}_{\mathrm{J}1}{\mathbf{R}}_{\mathrm{J}2}{\mathbf{d}}_{\text{cJ}3}$

(32)

R_G denotes the tip rotational matrix of the one-segment ACCR with respect to O_G-X_GY_GZ_G, which is derived in equation (33).

${\mathbf{R}}_{\mathrm{G}}={\mathbf{R}}_{\mathrm{J}1}{\mathbf{R}}_{\mathrm{J}2}{\mathbf{R}}_{\mathrm{J}3}$

(33)

where R_JΩ (Ω = 1, 2, or 3) denotes the rotational matrix of the anti-buckling universal joint-Ω with respect to O_JΩ-X_JΩY_JΩZ_JΩ, which can be derived as R_J1 = R_s1R_y(π/2)R_s2R_y(− π/2), R_J2 and R_J3 can be derived similarly; R_sw (w = 1, 2, …, or 6) denotes the rotational matrix of IS-CSP-w with respect to its local coordinate system.

As we determine the rotational sequence of all the rotational matrices in equation (1), the three rotations at the tip O_TipG (denoted by θ_xTipG, θ_yTipG, and θ_zTipG) are derived as (34).

${\theta }_{\text{xTipG}}=\mathrm{atan}\hspace{0.17em}2\left({\mathbf{R}}_{\mathrm{G}}\left(3,2\right),{\mathbf{R}}_{\mathrm{G}}\left(2,2\right)\right);\hspace{0.17em}{\theta }_{\text{yTipG}}=\mathrm{atan}\hspace{0.17em}2\left({\mathbf{R}}_{\mathrm{G}}\left(1,3\right),{\mathbf{R}}_{\mathrm{G}}\left(1,1\right)\right);\hspace{0.17em}{\theta }_{\text{zTipG}}=\mathrm{atan}\hspace{0.17em}2\left(-{\mathbf{R}}_{\mathrm{G}}\left(1,2\right),\sqrt{{\mathbf{R}}_{\mathrm{G}}{\left(1,1\right)}^2+{\mathbf{R}}_{\mathrm{G}}{\left(1,3\right)}^2}\right)$

(34)

where atan2 is a computing function in MATLAB.

The cable displacement is the result of the cable length in a non-deformed condition minus the cable length in a deformed condition. The normalized cable displacements of the four cables are then found as equation (35) through (38). When ∆l_cn > 0, the cable is pulled down, otherwise, the cable is elongated.

$\mathrm{\Delta }{l}_{\mathrm{c}1}=\left(\overline{{\mathbf{Q}}_9{\mathbf{Q}}_5}+\overline{{\mathbf{Q}}_5{\mathbf{Q}}_1}+\overline{{\mathbf{Q}}_1{\mathbf{A}}_1}\right)-\left(\overline{{{\mathbf{Q}}_9}^{*}{{\mathbf{Q}}_5}^{*}}+\overline{{{\mathbf{Q}}_5}^{*}{{\mathbf{Q}}_1}^{*}}+\overline{{{\mathbf{Q}}_1}^{*}{\mathbf{A}}_1}\right)$

(35)

$\mathrm{\Delta }{l}_{\mathrm{c}2}=\overline{{\mathbf{Q}}_{10}{\mathbf{Q}}_6}+\overline{{\mathbf{Q}}_6{\mathbf{Q}}_2}+\overline{{\mathbf{Q}}_2{\mathbf{A}}_2}-\left(\overline{{{\mathbf{Q}}_{10}}^{*}{{\mathbf{Q}}_6}^{*}}+\overline{{{\mathbf{Q}}_6}^{*}{{\mathbf{Q}}_2}^{*}}+\overline{{{\mathbf{Q}}_2}^{*}{\mathbf{A}}_2}\right)$

(36)

$\mathrm{\Delta }{l}_{\mathrm{c}3}=\overline{{\mathbf{Q}}_{11}{\mathbf{Q}}_7}+\overline{{\mathbf{Q}}_7{\mathbf{Q}}_3}+\overline{{\mathbf{Q}}_3{\mathbf{A}}_3}-\left(\overline{{{\mathbf{Q}}_{11}}^{*}{{\mathbf{Q}}_7}^{*}}+\overline{{{\mathbf{Q}}_7}^{*}{{\mathbf{Q}}_3}^{*}}+\overline{{{\mathbf{Q}}_3}^{*}{\mathbf{A}}_3}\right)$

(37)

$\mathrm{\Delta }{l}_{\mathrm{c}4}=\overline{{\mathbf{Q}}_{12}{\mathbf{Q}}_8}+\overline{{\mathbf{Q}}_8{\mathbf{Q}}_4}+\overline{{\mathbf{Q}}_4{\mathbf{A}}_4}-\left(\overline{{{\mathbf{Q}}_{12}}^{*}{{\mathbf{Q}}_8}^{*}}+\overline{{{\mathbf{Q}}_8}^{*}{{\mathbf{Q}}_4}^{*}}+\overline{{{\mathbf{Q}}_4}^{*}{\mathbf{A}}_4}\right)$

(38)

where ∆l_cn and ∆L_cn denote the normalized and non-normalized cable displacement of cable-n in the one-segment ACCR, respectively, and ∆l_cn = ∆L_cn /L_d. $\overline{{{\mathbf{Q}}_9}^{*}{{\mathbf{Q}}_5}^{*}}$ denotes the normalized distance between Q₉^* and Q₅^*; $\overline{{\mathbf{Q}}_9{\mathbf{Q}}_5}$ denotes the normalized distance between Q₉ and Q₅; others are the same.

3.2. Kinetostatic model of a three-segment ACCR

Figure 4(a) depicts the front view of the three-segment ACCR, including the cable-loading positions and coordinate systems. Each segment has three anti-buckling universal joints and is actuated by four cables. The cables of segment 3 go through the motion stages of segments 2 and 1. The cables of segment 2 go through the motion stages of segment 1. We use O_T-X_TY_TZ_T to denote the global coordinate system of the three-segment ACCR; use O_GΩ-X_GΩY_GΩZ_GΩ (Ω = 1, 2, or 3) to denote the local coordinate systems of segment Ω in Figure 4(b); use f_SE1-cn, f_SE2-cn, and f_SE3-cn (n = 1, 2, 3, or 4) to denote the positive tensions along cable-n of segments 1, 2, and 3, respectively, as shown in Figure 4(c); use f_xTip, f_yTip, f_zTip, m_xTip, m_yTip, and m_zTip to denote the tip loads with respect to O_T-X_TY_TZ_T loading at tip O_Tip; use Q_wSE1, Q_wS2S1, and Q_wS3S1 (w = 1, 2, …,12) to denote the cable-loading positions acting at segment 1 due to the cables of segments 1, 2, and 3, respectively; use Q_wSE2 and Q_wS3S2 to denote the cable-loading positions acting at segment 2 due to cables of segments 2 and 3, respectively; use Q_wSE3 to denote the cable-loading positions acting at segment 3 due to cables of segment 3.

Figure 4.

Description of the three-segment ACCR: (a) section view from XY plane of the cable-force loading positions, (b) global coordinate system O_T-X_TY_TZ_T and local coordinate systems O_GΩ-X_GΩY_GΩZ_GΩ (Ω = 1, 2, or 3), and (c) top view of the cables in each segment.

We take each anti-buckling universal joint as a modular unit to model a three-segment ACCR. Given geometric parameters, tip loads, and cable forces of the three-segment ACCR, all loading inputs with respect to O_T-X_TY_TZ_T should be equivalently converted to the loads with respect to O_JΩ-X_JΩY_JΩZ_JΩ (Ω = 1, 2, or 3) for each anti-buckling universal joint. The displacements and rotations of the nine anti-buckling universal joints can be obtained by solving the nonlinear spatial models of the anti-buckling universal joints and load-equilibrium condition of a three-segment ACCR in a deformed condition. Then any points on the three-segment ACCR can be obtained by solving the compatibility condition in a deformed condition, such as tip-location and shape dexterity.

Step 1 : The load-equilibrium conditions of the three-segment ACCR.

The normalized cable-force components with respect to O_T-X_TY_TZ_T acting at each segment are derived firstly using the same method in Section 3.1. We use f_xJΩSE1, f_yJΩSE1, f_zJΩSE1, m_xJΩSE1, m_yJΩSE1, and m_zJΩSE1 to denote normalized loads of the anti-buckling universal joint-Ω acting at its rotational center, O_JΩSE1, with respect to O_JΩSE1-X_JΩSE1Y_JΩSE1Z_JΩSE1 in segment 1, respectively (segments 2 and 3 have similar nomenclatures).

The load-equilibrium condition of the three-segment ACCR includes the load-equilibrium conditions between each anti-buckling universal joint. The load-equilibrium condition of the anti-buckling universal joints in segment 3 is the same as those of Section 3.1. However, more cable-force components and gravities should be considered for those of segments 1 and 2, in terms of two aspects: (1) The cable-force components acting at the motion stages due to other segments’ cables should be added for the modeling of anti-buckling universal joints. (2) Gravities and cable-force components acting at the top anti-buckling universal joints affect the motions of bottom anti-buckling universal joints, which should be added for the load-equilibrium condition between different anti-buckling universal joints.

We take the anti-buckling universal joint 3 in segment 2 as an example. In the modeling of this modular unit, the load-equilibrium condition should include cable-force components due to other segments’ cables, that is, the cable-force components loading at Q_kS3S2 (k = 9, 10, 11, or 12). Reference (S. Li and Hao, 2022) gives a generalized analytical model for the modeling of an anti-buckling universal joint.

Equation (39) shows the load-equilibrium condition of the anti-buckling universal joint 3 in segment 2 with respect to its local coordinate system, O_J3SE2-X_J3SE2Y_J3SE2Z_J3SE2, due to effects of loads that act at other anti-buckling universal joints and its gravity. In the right side of equation (39), the items, in turn, are formulated due to the effects of tip loads, cable-force components loading at the motion stages of the three anti-buckling universal joints in segment 3, gravities of the three anti-buckling universal joints in segment 3, and the gravity of the anti-buckling universal joint 3 in segment 2. Therefore, the load-equilibrium condition of other anti-buckling universal joints in segments 1 and 2 can be similarly derived.

$\left[\begin{array}{c}{\mathbf{f}}_{\mathrm{J}3\text{SE}2}\\ {\mathbf{m}}_{\mathrm{J}3\text{SE}2}\end{array}\right]={{\mathbf{D}}_{\text{pOTip}\_\mathrm{J}3\text{SE}2}^{\mathrm{T}}}^{*}\mathbf{R}\left[\begin{array}{c}{\mathbf{f}}_{\text{Tip}}\\ {\mathbf{m}}_{\text{Tip}}\end{array}\right]+{\displaystyle \sum }_{\epsilon =1}^{12}{{\mathbf{D}}_{\text{pQ}\epsilon \text{SE}3\_\mathrm{J}3\text{SE}2}^{\mathrm{T}}}^{*}\mathbf{R}\left[\begin{array}{c}{{\mathbf{f}}_{\text{NetQ}\epsilon \text{SE}3\_\mathrm{T}}}^{*}\\ {\mathbf{0}}_{3\times 1}\end{array}\right]+{\displaystyle \sum }_{\mathrm{\Omega }=1}^3{{\mathbf{D}}_{\text{pOmaJ}\mathrm{\Omega }\text{SE}3\_\mathrm{J}3\text{SE}2}^{\mathrm{T}}}^{*}\mathbf{R}\left[\begin{array}{c}\begin{array}{c}0\\ f_{\text{maJ}\mathrm{\Omega }\text{SE}3\_\mathrm{T}}\\ 0\end{array}\\ {\mathbf{0}}_{3\times 1}\end{array}\right]+{{\mathbf{D}}_{\text{pOmaJ}3\text{SE}2\_\mathrm{J}3\text{SE}2}^{\mathrm{T}}}^{*}\mathbf{R}\left[\begin{array}{c}\begin{array}{c}0\\ f_{\text{maJ}3\text{SE}2\_\mathrm{T}}\\ 0\end{array}\\ {\mathbf{0}}_{3\times 1}\end{array}\right]$

(39)

where f_J3SE2 = [f_xJ3SE2, f_yJ3SE2, f_zJ3SE2]^T, m_J3SE2 = [m_xJ3SE2, m_yJ3SE2, m_zJ3SE2]^T, f_{NetQεSE3_T} denotes the cable-force components loading at Q_εSE3 of the three anti-buckling universal joints in segment 3 with respect to O_T-X_TY_TZ_T; f_{maJΩSE3_T} denotes the gravity loading at the rotational centers of the three anti-buckling universal joints in segment 3 with respect to O_T-X_TY_TZ_T; f_{maJ3SE2_T} denotes the gravity loading at the rotational center of the anti-buckling universal joint 3 in segment 2 with respect to O_T-X_TY_TZ_T; D_{pOTip_J3SE2}^*, D_{pQεSE3_J3SE2}^* (ε = 1, 2, …, or 12), D_{pOmaJΩSE2_J3SE2}^* (Ω = 1, 2, or 3) and D_{pOmaJ3SE2_J3SE2}^* denote the normalized translational matrices of the points O_{Tip_J3SE2}^*, Q_{εSE3_J3SE2}^*, O_{maJΩSE2_J3SE2}^*, and O_{maJ3SE2_J3SE2}^*, respectively, as written in equation (16). a ₀ represents O_{Tip_J3SE2}^*, Q_{εSE3_J3SE2}^*, O_{maJΩSE2_J3SE2}^*, or O_{maJ3SE2_J3SE2}^* and denotes the normalized coordinate of tip O_Tip, cable-loading positions of segment 3 Q_εSE3, mass centers of segment 3 O_maJΩSE3, or the mass center of itself O_maJ3SE2 with respect to O_J3SE2-X_J3SE2Y_J3SE2Z_J3SE2 in a deformed condition. Each of them is the result of the displacements and rotations of the anti-buckling universal joint 3 in segment 2 and three anti-buckling universal joints in segment 3, which can be derived similarly as we did in Section 3. 1. R is a rotational matrix, which is utilized to rotate the loads with respect to O_T-X_TY_TZ_T to those with respect to O_J3SE2-X_J3SE2Y_J3SE2Z_J3SE2, as expressed in equation (40);

$\mathbf{R}={\left({\mathbf{R}}_{\text{JJ}1\text{SE}2}{\mathbf{R}}_{\text{JJ}2\text{SE}2}\right)}^{-1}{\left[{\mathbf{R}}_{\text{GG}1}{\mathbf{R}}_{\text{yy}}\left(-\mathrm{\pi }/2\right)\right]}^{-1}$

(40)

R_JJ1SE2, R_JJ2SE2, R_GG1, and R_yy(−π/2) are 6 × 6 rotational matrices, as shown in equation (41) below:

${\mathbf{R}}_{\text{JJ}1\text{SE}2}=\left[\begin{array}{cc}{\mathbf{R}}_{\mathrm{J}1\text{SE}2}& {\mathbf{I}}_{3\times 3}\\ {\mathbf{I}}_{3\times 3}& {\mathbf{R}}_{\mathrm{J}1\text{SE}2}\end{array}\right]; {\mathbf{R}}_{\text{JJ}2\text{SE}2}=\left[\begin{array}{cc}{\mathbf{R}}_{\mathrm{J}2\text{SE}2}& {\mathbf{I}}_{3\times 3}\\ {\mathbf{I}}_{3\times 3}& {\mathbf{R}}_{\mathrm{J}2\text{SE}2}\end{array}\right]; {\mathbf{R}}_{\text{GG}1}=\left[\begin{array}{cc}{\mathbf{R}}_{\mathrm{G}1}& {\mathbf{I}}_{3\times 3}\\ {\mathbf{I}}_{3\times 3}& {\mathbf{R}}_{\mathrm{G}1}\end{array}\right]; {\mathbf{R}}_{\text{yy}}\left(-\mathrm{\pi }/2\right)=\left[\begin{array}{cc}{\mathbf{R}}_{\mathrm{y}}\left(-\mathrm{\pi }/2\right)& {\mathbf{I}}_{3\times 3}\\ {\mathbf{I}}_{3\times 3}& {\mathbf{R}}_{\mathrm{y}}\left(-\mathrm{\pi }/2\right)\end{array}\right]$

(41)

• Step 2: The compatibility conditions of the three-segment ACCR.

The rotational matrix for segment Ω with respect to O_GΩ-X_GΩY_GΩZ_GΩ (Ω = 1, 2, or 3) is written in equation (42).

${\mathbf{R}}_{\mathrm{G}\mathrm{\Omega }}={\mathbf{R}}_{\mathrm{J}1\text{SE}\mathrm{\Omega }}{\mathbf{R}}_{\mathrm{J}2\text{SE}\mathrm{\Omega }}{\mathbf{R}}_{\mathrm{J}3\text{SE}\mathrm{\Omega }}$

(42)

where R_GΩ denotes the rotational matrix of segment Ω; R_J1SEΩ denotes the rotational matrix of the anti-buckling universal joint 1 in segment Ω, similar to R_J1SEΩ and R_J1SEΩ.

As segment 2 rotates by −π/2 about both segments 1 and 3, the normalized displacements of segment 2 with respect to O_G2-X_G2Y_G2Z_G2 can be described by those with respect to O_G1-X_G1Y_G1Z_G1, as shown in equation (43).

$\left[\begin{array}{c}{d}_{\text{xG}1}\\ d_{\text{yG}1}\\ d_{\text{zG}1}\end{array}\right]={\mathbf{R}}_{\mathrm{G}1}{\mathbf{R}}_{\mathrm{y}}\left(-\mathrm{\pi }/2\right)\left[\begin{array}{c}{d}_{\text{xG}2}\\ d_{\text{yG}2}\\ d_{\text{zG}2}\end{array}\right]$

(43)

The normalized displacements of segment 3 with respect to O_G3-X_G3Y_G3Z_G3 can be described by those with respect to O_G2-X_G2Y_G2Z_G2, as shown in equation (44).

$\left[\begin{array}{c}{d}_{\text{xG}2}\\ d_{\text{yG}2}\\ d_{\text{zG}2}\end{array}\right]={\mathbf{R}}_{\mathrm{G}2}{\mathbf{R}}_{\mathrm{y}}\left(\mathrm{\pi }/2\right)\left[\begin{array}{c}{d}_{\text{xG}3}\\ d_{\text{yG}3}\\ d_{\text{zG}3}\end{array}\right]$

(44)

Combining equations (43) and (44), the normalized displacements of segment 3 with respect to O_G3-X_G3Y_G3Z_G3 can be described by those with respect to O_G1-X_G1Y_G1Z_G1, as shown in equation (45).

$\left[\begin{array}{c}{d}_{\text{xG}1}\\ d_{\text{yG}1}\\ d_{\text{zG}1}\end{array}\right]={\mathbf{R}}_{\mathrm{G}1}{\mathbf{R}}_{\mathrm{y}}\left(-\mathrm{\pi }/2\right){\mathbf{R}}_{\mathrm{G}2}{\mathbf{R}}_{\mathrm{y}}\left(\mathrm{\pi }/2\right)\left[\begin{array}{c}{d}_{\text{xG}3}\\ d_{\text{yG}3}\\ d_{\text{zG}3}\end{array}\right]$

(45)

Therefore, the rotational compatibility condition of the three-segment ACCR is derived in equation (46).

${\mathbf{R}}_{\mathrm{T}}={\mathbf{R}}_{\mathrm{G}1}{\mathbf{R}}_{\mathrm{y}}\left(-\mathrm{\pi }/2\right){\mathbf{R}}_{\mathrm{G}2}{\mathbf{R}}_{\mathrm{y}}\left(\mathrm{\pi }/2\right){\mathbf{R}}_{\mathrm{G}3}$

(46)

If we determine the rotational sequence is the same as equation (1), the three rotations of the three-segment ACCR (denoted by θ_xTip, θ_yTip, and θ_zTip) can be calculated as equation (47).

${\theta }_{\text{xTip}}=\mathrm{atan}\hspace{0.17em}2\left({\mathbf{R}}_{\mathrm{T}}\left(3,2\right),{\mathbf{R}}_{\mathrm{T}}\left(2,2\right)\right);\hspace{0.17em}{\theta }_{\text{yTip}}=\mathrm{atan}\hspace{0.17em}2\left({\mathbf{R}}_{\mathrm{T}}\left(1,3\right),{\mathbf{R}}_{\mathrm{T}}\left(1,1\right)\right);\hspace{0.17em}{\theta }_{\text{zTip}}=\mathrm{atan}\hspace{0.17em}2\left(-{\mathbf{R}}_{\mathrm{T}}\left(1,2\right),\sqrt{{\mathbf{R}}_{\mathrm{T}}{\left(1,1\right)}^2+{\mathbf{R}}_{\mathrm{T}}{\left(1,3\right)}^2}\right)$

(47)

• Step 3 : Calculation on the tip motions and cable length after solving the performance of the three-segment ACCR. Motions of other points on the ACCR can be solved similarly.

The normalized coordinate of a point on the ACCR in a deformed condition is derived depending on its position. Equation (48) gives the normalized coordinates of the tip O_Tip with respect to O_G-X_GY_GZ_G in a deformed condition. The normalized coordinates of other points on the three-segment ACCR with respect to O_G-X_GY_GZ_G in a deformed condition can be similarly derived.

$\left(\begin{array}{c}{{\mathbf{O}}_{\text{Tip}\_\mathrm{T}}}^{*}\\ 1\end{array}\right)={\mathbf{T}}_{\text{SE}1}{\mathbf{T}}_{\text{SE}2}{\mathbf{T}}_{\text{SE}3}\left(\begin{array}{c}{\mathbf{O}}_{\text{Tip}\_\mathrm{T}}\\ 1\end{array}\right)$

(48)

where T_SE1 = ^GT_J1(M)SE1 ^J1(M)SE1T_J2SE1 ^J2SE1T_J2(M)SE1 ^J2(M)SE1T_J3SE1 ^J3SE1T_SE2 = T_J3(M)SE1; ^J3(M)SE1T_J1SE2 ^J1SE2T_J1(M)SE2 ^J1(M)SE2T_J2SE2 ^J2SE2T_J2(M)SE2 ^J2(M)SE2T_J3SE2 ^J3SE2T_J3(M)SE2;

T_SE3 = ^J3(M)SE2T_J1SE3 ^J1SE3T_J1(M)SE3 ^J1(M)SE3T_J2SE3 ^J2SE3T_J2(M)SE3 ^J2(M)SE3T_J3SE3 ^J3SE3T_J3(M)SE3 ^J3(M)SE3T_T; Their explanations are detailed in Appendix A; O_{Tip_T} denotes the normalized coordinates of the tip O_Tip in a non-deformed condition with respect to O_T-X_TY_TZ_T.

The cable displacements of the four cables for each segment can be derived in a similar way to what we did in Section 3.1. The normalized cable displacement of cable-1 in segment 3 is taken as an example, as shown in equation (49).

$\begin{array}{c}\mathrm{\Delta }{l}_{\text{SE}3-\mathrm{c}1}=\overline{{\mathbf{Q}}_{9\text{SE}3}{\mathbf{Q}}_{5\text{SE}3}}+\overline{{\mathbf{Q}}_{5\text{SE}3}{\mathbf{Q}}_{1\text{SE}3}}+\overline{{\mathbf{Q}}_{1\text{SE}3}{\mathbf{Q}}_{9\mathrm{S}3\mathrm{S}2}}-\left(\overline{{{\mathbf{Q}}_{9\text{SE}3}}^{*}{{\mathbf{Q}}_{5\text{SE}3}}^{*}}+\overline{{{\mathbf{Q}}_{5\text{SE}3}}^{*}{{\mathbf{Q}}_{1\text{SE}3}}^{*}}+\overline{{{\mathbf{Q}}_{1\text{SE}3}}^{*}{{\mathbf{Q}}_{9\mathrm{S}3\mathrm{S}2}}^{*}}\right)\\ +\overline{{\mathbf{Q}}_{9\mathrm{S}3\mathrm{S}2}{\mathbf{Q}}_{5\mathrm{S}3\mathrm{S}2}}+\overline{{\mathbf{Q}}_{5\mathrm{S}3\mathrm{S}2}{\mathbf{Q}}_{1\mathrm{S}3\mathrm{S}2}}+\overline{{\mathbf{Q}}_{1\mathrm{S}3\mathrm{S}2}{\mathbf{Q}}_{9\mathrm{S}3\mathrm{S}1}}-\left(\overline{{{\mathbf{Q}}_{9\mathrm{S}3\mathrm{S}2}}^{*}{{\mathbf{Q}}_{5\mathrm{S}3\mathrm{S}2}}^{*}}+\overline{{{\mathbf{Q}}_{5\mathrm{S}3\mathrm{S}2}}^{*}{{\mathbf{Q}}_{1\mathrm{S}3\mathrm{S}2}}^{*}}+\overline{{{\mathbf{Q}}_{1\mathrm{S}3\mathrm{S}2}}^{*}{{\mathbf{Q}}_{9\mathrm{S}3\mathrm{S}1}}^{*}}\right)\\ +\overline{{\mathbf{Q}}_{9\mathrm{S}3\mathrm{S}1}{\mathbf{Q}}_{5\mathrm{S}3\mathrm{S}1}}+\overline{{\mathbf{Q}}_{5\mathrm{S}3\mathrm{S}1}{\mathbf{Q}}_{1\mathrm{S}3\mathrm{S}1}}+\overline{{\mathbf{Q}}_{1\mathrm{S}3\mathrm{S}1}{\mathbf{A}}_1}-\left(\hspace{0.5em}\overline{{{\mathbf{Q}}_{9\mathrm{S}3\mathrm{S}1}}^{*}{{\mathbf{Q}}_{5\mathrm{S}3\mathrm{S}1}}^{*}}+\overline{{{\mathbf{Q}}_{5\mathrm{S}3\mathrm{S}1}}^{*}{{\mathbf{Q}}_{1\mathrm{S}3\mathrm{S}1}}^{*}}+\overline{{{\mathbf{Q}}_{1\mathrm{S}3\mathrm{S}1}}^{*}{{\mathbf{A}}_1}^{*}}\right)\end{array}$

(49)

4. Spatial Analysis of the one-segment ACCR

In this section, the accuracy of the kinetostatic model of the one-segment ACCR is assessed by using the nonlinear FEA model, which is built in COMSOL 5.0. In the FEA model, the mesh size of the parts excluding elastic sheets is not required but they should be set as “rigid domain” in COMSOL 5.0. The scale factor of the total displacement figures given by the COMSOL 5.0 is set to 1, which means the deformation is not being enlarged or reduced. The details of the FEA process are shown in Appendix B.

Without loss of generality, the following parameters are assigned for the analytical analysis and FEA verifications. The geometric parameters, cable-loading positions, and four pulley positions of the one-segment ACCR are set as those listed in Tables 1 and 2, respectively. The elastic sheets are made of Mn 65 spring steel. The Young’s modulus and yield stress of the Mn 65 spring steel take 198 × 10⁹ (Pa) and 7.8 ×10⁸ (Pa), respectively. We set the gravity to be 0 and use α = 45° to evaluate the tip-location accuracy and shape dexterity of the one-segment ACCR; We then consider α = 15°, 30°, and 45° to evaluate the tip stiffness between the one-segment ACCR and the counterpart CCR. Note that λ is constant at 0.5 in Section 4.

Table 1.

The geometric parameters of the one-segment ACCR (length unit: mm).

Parameters	Values	Parameters	Values
La sheet length	50	LG height of an anti-buckling universal joint	118
T sheet thickness	1	Length of the one-segment ACCR	3LG = 354
U sheet width	12	HJ mid-loop height	88
LJ ACCR diameter	100	Ld diagonal length of the CCR	(9LG2 + LJ2)1/2

Table 2.

The normalized coordinates of the cable-loading positions and pulley positions of the one-segment ACCR with respect to O_G-X_GY_GZ_G, where x_Q1 = 60 (mm)/L_d; y_Q1 = −λl_acosα −t/2sinα; x_A = 30 (mm)/L_d; and y_A = −100 (mm)/L_d − λl_acosα−t/2sinα.

Type of points	Points	Normalized coordinates
Cable-loading positions	Q1	[−xQ1, yQ1, 0]T
	Q2	[xQ1, yQ1, 0]T
	Q3	[0, yQ1, xQ1]T
	Q4	[0, yQ1, − xQ1]T
	Qn+4 (n = 1, 2, 3, or 4)	Q n + [0, lG, 0]T
	Qn+8 (n = 1, 2, 3, or 4)	Q n + [0, 2lG, 0]T
Pulley positions	A1	[ −xA, yA, 0 ]T
	A2	[ xA, yA, 0 ]T
	A3	[ 0, yA, xA ]T
	A4	[ 0, yA, −xA ]T

The analytical results show that the ACCR is not evenly bending under the hybrid active and passive cable-driven method. When a tip moment, a tip force, or a cable force loading at the one-segment ACCR without considering the gravity effect, the rotations of the three anti-buckling universal joints with respect to their local coordinate systems are shown in Table 3 (α = 45°). The one-segment ACCR cannot be a constant curvature except for the pure tip moment.

Table 3.

The rotations of the three anti-buckling universal joints with respect to their local coordinate systems (unit: rad).

Loading conditions	θzJ1 (OJ1-XJ1YJ1ZJ1)	θzJ2 (OJ2-XJ2YJ2ZJ2)	θzJ3 (OJ3-XJ3YJ3ZJ3)
Tip moment: MzTipG = 0.7 (N·m)	0.089	0.089	0.089
Tip bending force: FxTipG = − 2 (N)	0.075	0.045	0.015
Cable force: Fc1 = 9 (N)	0.073	0.068	0.068

4.1. Maximum ACCR deformation

Figure 5(a) and (b) show the tip motions of analytical and FEA results of the one-segment ACCR under tip moments and tip forces. Three tip forces remain constant (F_xTipG = F_zTipG = 5 (N), F_yTipG = − 5 (N)) and two tip moments are equal to each other (M_xTipG = M_zTipG) ranging from 0 to 0.5 (N·m) with a step of 0.1 (N·m). The average errors between the analytical and FEA results of D_xTipG, D_yTipG, D_zTipG, θ_xTipG, θ_yTipG, and θ_zTipG, are 6.4%, 7.3%, 4.5%, 5.2%, 7.4%, and 6.8%, respectively. The maximum deformation of the one-segment ACCR is shown in Figure 5(c) for M_xTipG = M_zTipG = 0.5 (N·m) and the maximum Von Mises stress is 7.65 ×10⁸ (Pa).

Figure 5.

Maximum deformation evaluation under two moments and three forces acting at the tip of the one-segment ACCR: (a) tip displacements, (b) tip rotations, and (c) total displacements under three constant tip forces and two equal moments.

The corresponding tip displacements and rotations are D_xTipG = 0.046 (m), D_yTipG = − 0.035 (m), D_zTipG = 0.125 (m), θ_xTipG = 0.621 (rad), θ_yTipG = − 0.095 (rad), and θ_zTipG = − 0.166 (rad), that is, 35.6°, 5.45°, and 9.52°, respectively. Note that throughout the paper, the rotational angles are the Euler angles, not the roll-pitch-yaw angles. Therefore, the tip rotations are not the ACCRs’ bending or twisting angles.

4.2. Tip-location accuracy and shape dexterity

The tip-location accuracy and shape dexterity of the one-segment ACCR are studied when α = 45° and the ACCR is driven by cable forces. The cable forces: F_c1 ranges from 1 (N) to 9 (N) with a step of 2 (N), F_c3 = 5 (N), and other cable forces are equal to 0. From the analytical model, all cable-force components in a deformed condition can be calculated, which are used to be point loads loading at the cable-loading positions Q₁ to Q₁₂ in the FEA model. Then the displacements of any point with respect to O_G-X_GY_GZ_G in the FEA model can be evaluated by using “Point Evaluation” in COMSOL 5.0. R_G of the FEA model can be solved by using the coordinate transformation method (For more details on R_G, equation (33) in (S. Li and Hao, 2022) can be visited), which is substituted into equation (34) to obtain the FEA rotations. Tip displacements and rotations of the FEA and nonlinear kinetostatic models are compared in Figure 6(a) and (b). We use the coordinates of points Q₁^*, Q₅^*, and Q₉^* with respect to O_G-X_GY_GZ_G to describe the shape of the one-segment ACCR, which is depicted in Figure 6(c) and (d).

Figure 6.

Shape dexterity of the one-segment ACCR with respect to O_G-X_GY_GZ_G: (a) tip displacements, (b) tip rotations, (c) shape dexterity with the increase of F_c1, (d) total displacement under F_c1 = 1 (N), (e) total displacement under F_c1 = 3 (N), and (f) total displacement under F_c1 = 9 (N). (Average differences between the analytical results and their corresponding FEA results in Figure 6(a) and (b): D_xTipG: 5.8%, D_yTipG: 8.5%, D_zTipG: 5.2%, θ_xTipG: 5.6%, θ_yTipG: 7.8%, and θ_zTipG: 3.3%. The average differences of the coordinates between the analytical and FEA ACCR shapes in Figure 6(c) with respect to X_G, Y_G, and Z_G-axes are 2.9%, 2.3%, and 2.3%, respectively).

4.3. Tip-stiffness improvement

We first study the in-plane and the out-of-plane tip stiffness of the one-segment ACCR and counterpart CCR under two different in-plane actuations. Then the transverse tip stiffness of the two CCRs under a spatial actuation is investigated.

A counterpart CCR composed of non-inversion-based crossing spring pivots is built and plays as a comparative group. The counterpart CCR is representative of the stiffness characteristics of CCR designs reported in the literature. The tip stiffness of the counterpart CCR decreases with the increasing cable actuation. As we stated in the Introduction, the performance of many existing cable-driven CCRs is limited due to their low tip stiffness. The counterpart CCR is the most appropriate choice as a comparative design for the following reasons. The primary factors influencing the performance characteristics of different CCRs are the cable actuation, joints, and structural features. In both the ACCR and the counterpart CCR, the cable-loading positions are located at the free-end planes of each elastic sheet, with equal actuation forces applied. Both designs utilize the same compliant universal joints, with the only difference coming from the arrangement of joints in terms of an inversion-based configuration or non-inversion-based configuration. Thereby, we can reasonably compare the stiffness of the two designs while eliminating the influence of the CCR’s structural features on the results.

We require all the geometric parameters of ACCR and the counterpart CCR to be the same, and cable-force loading positions to be located at the level of the sheets’ free ends for each universal joint of the two CCRs, as shown in Figure 7. K_∆Fx-∆Dx denotes the in-plane tip stiffness along the X_G axis, which can calculated as equation (50) (Mahvash and Dupont, 2011; Yang et al., 2020). We use K_∆Fz-∆Dz to denote the out-of-plane tip stiffness along the Z_G axis, which can be computed similarly.

$K_{\mathrm{\Delta }F\mathrm{x}-\mathrm{\Delta }D\mathrm{x}}=\frac{\partial F_{\text{xTip}}}{\partial D_{\text{xTip}}}$

(50)

where ∂F_xTip and ∂D_xTip denote the adjacent differences of a series of tip forces and tip displacements (at least two forces) along the X_G axis, respectively;

Figure 7.

Description of the two CCRs: (a) ACCR and (b) the counterpart CCR (α = 45° here and cable-4 is not drawn).

The improvement ratio of the in-plane tip stiffness is given as equation (51). The improvement ratio of the other stiffness can be solved similarly.

$\frac{{K_{\mathrm{\Delta }F\mathrm{x}-\mathrm{\Delta }D\mathrm{x}}}_{\_\text{ACCR} }-K_{\mathrm{\Delta }F\mathrm{x}-\mathrm{\Delta }D\mathrm{x}\_\text{coun}}}{{K_{\mathrm{\Delta }F\mathrm{x}-\mathrm{\Delta }D\mathrm{x}\_}}_{\text{coun} }}\times 100\%$

(51)

where K_{∆Fx-∆Dx_ACCR} and K_{∆Fx-∆Dx_Coun} denote the in-plane tip stiffness of the ACCR and the counterpart CCR, respectively.

λ takes 0.5, α = 15°, 30°, or 45°, and the loading condition is an increasing tip force F_xTipG with a constant cable-displacement actuation ∆L_c1. ∆L_c1 is fixed at 4 (mm) and other cables are free. These loading conditions are held in the following two situations.

(i) To compare in-plane tip stiffness K_∆Fx-∆Dx of the ACCR and counterpart CCR, the in-plane tip load F_xTipG ranges from 0.2 (N) to 1.6 (N) with a step of 0.2 (N). Their results are compared in Figure 8(a). The maximum difference between the analytical and FEA results is 3.9%. K_{∆Fx-∆Dx_ACCR} has high values for α = 15°. However, K_{∆Fx-∆Dx_coun} has low values for the same α. K_{∆Fx-∆Dx_ACCR} is larger than K_{∆Fx-∆Dx_coun}, and K_{∆Fx-∆Dx_ACCR} increases by 44.1%, 38.8%, and 30.1% under α = 15°, 30°, or 45°, respectively.

(ii) To compare the out-of-plane tip stiffness K_∆Fz-∆Dz of the ACCR and counterpart CCR, the out-of-plane tip load F_zTipG ranges from 0.2 (N) to 1.6 (N) with a step of 0.2 (N). Their results are as shown in Figure 8(b). The maximum difference between the analytical and FEA results is 5.4%. K_∆Fz-∆Dz of the two CCRs have high values for α = 15°. K_∆Fz-∆Dz of the two CCRs are close but K_{∆Fz-∆Dz_ACCR} is slightly higher than K_{∆Fz-∆Dz_coun}, where K_{∆Fz-∆Dz_ACCR} and K_{∆Fz-∆Dz_coun} denote the out-of-plane tip stiffness of the ACCR and counterpart CCR, respectively. K_{∆Fz-∆Dz_ACCR} increases by 1.2%, 0.9%, and 0.6% under α = 15°, 30°, or 45°, respectively.

Figure 8.

The in-plane and the out-of-plane tip stiffness comparison between the one-segment ACCR and counterpart CCR under a constant cable-displacement actuation and increasing tip forces: (a) the in-plane tip stiffness K_∆Fx-∆Dx under a series of increasing F_xTipG and (b) the out-of-plane tip stiffness K_∆Fz-∆Dz under a series of increasing F_zTipG.

Similarly, the tip stiffness study of the two one-segment CCRs under the increasing cable actuation is studied in Appendix C. The tip stiffness comparison between the ACCR and counterpart CCR under in-plane actuation is summarized in Table 4. The in-plane tip stiffness of the ACCR increases significantly under the increasing in-plane actuation. Under a small α, the tip stiffness of the two CCRs varies significantly, and ACCR has higher K_∆Fx-∆Dx and K_{∆Fz_∆Dz} than those of the counterpart CCR. Many improved ratios are calculated using equation (51) under the increasing tip loads or cable actuation, and the maximum improved ratio is the highest value among them.

Table 4.

The summary of tip stiffness evaluation between the one-segment ACCR and counterpart CCR under in-plane actuation.

CCRs types	Loads: Constant cable actuation and increasing tip loads (see Section 4.3)		Loads: Increasing cable actuation (see Appendix C)
	In-plane tip stiffness K∆Fx-∆Dx	Out-of-plane tip stiffness K∆Fz-∆Dz	In-plane tip stiffness K∆Fx-∆Dx	Out-of-plane tip stiffness K∆Fz-∆Dz
ACCR	Slightly increase	Both slightly increase	Increase over the range of 307 ∼ 319 N/m	Slightly increase
Counterpart CCR	Slightly decrease		Decrease over the range of 218 ∼ 214 N/m	Slightly decrease
	Max improved ratio 44.1%	Max improved ratio 1.2%	Max improved ratio 49.0%	Max improved ratio 5.6%

The transverse tip stiffness of the two CCRs under the increasing spatial actuation is studied in Appendix D. Their results are summarized in Table 5. The transverse tip stiffness of the ACCR increases significantly under the increasing spatial actuation. Therefore, when the one-segment ACCR and the counterpart CCR are under the same loading conditions and geometric parameters, the results show that the ACCR has a slenderer shape (i.e., small α), a higher tip stiffness, and a larger in-plane tip stiffness range.

Table 5.

The summary of the transverse tip stiffness evaluation between the one-segment ACCR and counterpart CCR under spatial actuation.

CCRs types	Loads: Increasing spatial actuation (see Appendix D)
	Transverse tip stiffness in the XG direction K∆Fx-∆Dx	Transverse tip stiffness in the ZG direction K∆Fz-∆Dz
ACCR	Increase over the range of 307.3 ∼ 319.8 N/m	Increase over the range of 308.4 ∼ 318.7 N/m
Counterpart CCR	Decrease over the range of 217.8 ∼ 213.4 N/m	Decrease over the range of 218.3 ∼ 214.3 N/m
	Max improved ratio 49.8%	Max improved ratio 48.7%

4.4. Experiment

The deformation, shape dexterity and tip stiffness have been thoroughly analyzed in Sections 4.2 using analytical models and FEA. In this section, the experimental comparison between the ACCR and counterpart CCR is investigated. Each fabricated prototype is shown for verifying design and modeling but is not optimized for specified size and performance. We use Type 1 joints to construct the ACCR instead of Type 2 joints, primarily due to its ease of fabrication.

Appendix E details two fabrication methods for the proposed robot, resorting to 3D printed parts and their assembly. The use of a 3D printer can lead to a low-cost and lightweight prototype over a short lead period. Therefore, the material for elastic sheets and the design parameters of the CCR are different from those mentioned earlier in the analytical and FEA studies. All 3D printing material is selected to be PLA where its Young’s modulus is 3150 (MPa) and Poisson’s ratio is 0.35. The fill density was set to be 50%. These parts are connected with each other by using Grease Welding Flux, which is a high-strength glue for professional industrial applications. Each anti-buckling universal joint has a weight of close to 22 (g). The length of the tip marker is 14 (mm) and the mass of the tip marker is 1 (g). The cable is made of strong nylon with a transparent color, which has trivial elongation. We use different weights to apply cable forces and tip forces.

The prototypes of the two one-segment CCRs using fabrication method 1 of Appendix E are shown in Figure 9(a) and (b). The geometric parameters are shown in Table 6. We compare the in-plane tip stiffness due to the significant improvement observed in Section 4.3. We designed a limiter for the CCR, which means the rigid parts of the CCR will contact to avoid excessive loads, as shown in Figure 9(c). Therefore, the maximum cable force that can be applied to the one-segment CCR is 50 (g) and the corresponding maximum cable displacement calculated by the analytical model is 4 (mm).

Figure 9.

Fabrication parts for the two one-segment CCRs: (a) parts for fabrication of a counterpart universal joint, (b) parts for fabrication of an anti-buckling universal joint, and (c) prototypes of the two joints.

Table 6.

Geometric parameters of the one-segment CCRs using fabrication method 1.

Parameters	Values (length unit: mm)	Parameters	Values
L a	15.1	λ	0.5
T	0.7	L d	(9LG2 + LJ2)1/2
U	5.86	x A1	−25/Ld
L G	37	y A1	−40/Ld
L J	40	x A2	25/Ld
Length of the one-segment ACCR	111	y A2	−40/Ld
α	45°	|x Qε |	30.3/Ld

Two tests are carried out. Each prototype is fixed on the working bench, as depicted in Figure 10(a). Test 1 is designed to investigate the tip stiffness of the ACCR and counterpart CCR under conditions of an increasing tip force and a constant cable displacement, as shown in Figure 10(b). The tip load can be calculated using equation (52). Test 2 is designed to investigate the tip stiffness under conditions of an increasing cable force and a constant cable displacement, as shown in Figure 10(c).

$F_{\text{xTipG}}=\frac{G}{2\hspace{0.17em}\cos \hspace{0.17em}\theta }$

(52)

where G denotes the mass weight for providing the tip force; θ denotes the angle for calculating the tip force.

Figure 10.

Test setup for the tip stiffness evaluation of the ACCR and the counterpart CCR: (a) prototype of the ACCR is unloaded, (b) prototype of the ACCR in Test 1, and (c) prototype of the counterpart CCR in Test 2.

We use 1/8-inch graph paper and an HD camera to calculate the tip motions of the CCR, as shown in Figure 11(a). The HD camera should face directly opposite the tip and photograph a series of figures when the loads change. The tip displacements can be calculated by counting the number of squares in the graph paper. In Test 1, the mass for providing tip force ranges from 0 (g) to 50 (g) with a step of 10 (g), and the cable displacement ΔL_c1 is constant at 4 (mm), so that the stiffness results for the ACCR and the counterpart CCR are compared in Figure 11(b). The analytical tip stiffness of the ACCR and that of the counterpart CCR are 32.9 (N/m) and 27.3 (N/m), respectively, when the weight of F_xTipG is 50 (g). Therefore, the maximum improved ratio is 20.3%. In Test 2, the mass for providing increasing cable force ranges from 10 (g) to 50 (g) with a step of 10 (g), and the cable displacement ΔL_c1 is constant at 4 (mm). The stiffness results for the ACCR and the counterpart CCR are compared in Figure 11(c). The analytical tip stiffness of the ACCR and the counterpart CCR is 30.5 (N/m) and 26.1 (N/m), respectively, when the weight of F_c2 is 50 (g). Thus, the maximum improved ratio is 17.1%. The differences between the analytical and experimental results of the ACCR (or the counterpart CCR) are smaller than 8% as shown in Figure 11(d). These experimental results confirm that the tip stiffness of the one-segment ACCR always increases with the increase of the tip force FxTipG (or cable force Fc₂).

Figure 11.

Results comparison of the two CCRs: (a) the image processing methods, (b) results of Test 1, (c) results of Test 2, and (d) the differences between the analytical and experimental results of the ACCR (or the counterpart CCR).

5. Spatial analysis of the three-segment ACCR

In this section, we use the nonlinear FEA model to confirm the accuracy of the kinetostatic model of the three-segment ACCR. The FEA settings are the same as those in Section 4. Most geometric parameters of the three segments are the same as those of Table 1, except for the following parameters. λ is assigned to 0.5 only for analyzing the maximum deformation, tip-location accuracy, and shape dexterity. The elastic sheet widths of segments 1, 2, and 3 are 12 (mm), 11 (mm), and 10 (mm), respectively. The radii of the cable-loading positions of segments 1, 2, and 3 are 70 (mm), 65 (mm), and 60 (mm), respectively.

5.1. Maximum ACCR deformation

Figure 12(a) and (b) show the tip motions of the analytical and FEA results of the three-segment ACCR under a tip moment and tip forces. Three constant tip forces (F_xTip = 0, F_yTip = − 1 (N), F_zTip = 1 (N)), and a tip moment M_zTip ranging from 0 to 1 (N·m) with a step of 0.2 (N·m) are applied at the ACCR tip. The average errors between the analytical and FEA results of D_xTip, D_yTip, D_zTip, θ_xTip, θ_yTip, and θ_zTip are 5.3%, 3.6%, 7.9%, 5.2%, 4.4%, and 6.5%, respectively. The maximum deformation of the three-segment ACCR occurs at M_zTip = 1.8 (N·m) given by the FEA simulation in Figure 12(c), and the maximum Von Mises stress is 7.76 ×10⁸ (Pa). The corresponding tip displacements and rotations are D_xTip = − 0.710 (m), D_yTip = − 0.839 (m), D_zTip = 0.300 (m), θ_xTip = − 1.05 (rad), θ_yTip = − 0.651 (rad), θ_zTip = 1.07 (rad), that is, −60.2°, 37.3, and 61.3°, respectively.

Figure 12.

Maximum deformation evaluation under one tip moment and two tip forces acting at the three-segment ACCR: (a) tip displacements, (b) tip rotations, and (c) total displacements under M_zTip = 1.8 (N·m).

5.2. Tip-location accuracy and shape dexterity

We vary the cable forces to evaluate the “L” and “S” shapes of the three-segment ACCR.

(1) “L” shape

The cable forces F_SE1-c1 = 5 (N), F_SE1-c3 = 3 (N), and F_SE3-c1 range from 0.2 (N) to 1.4 (N) with a step of 0.2 (N), and other cable forces are equal to 0, as shown in Figure 13(d). The cable-force components loading at Q_1SE1, Q_5SE1, and Q_9SE1 due to F_SE1-c1, cable-force components loading at Q_3SE1, Q_7SE1, and Q_11SE1 due to F_SE1-c3, and cable-force components loading at Q_1SE3, Q_5SE3, Q_9SE3, Q_1S3S2, Q_5S3S2, Q_9S3S2, Q_1S3S1, Q_5S3S1, and Q_9S3S1 due to F_SE3-c1 can be solved in the analytical model (as seen in Figure 4), which are substituted into the FEA model. The FEA R_T can be solved using the coordinate transformation method (similar to Section 4.2), and tip rotations of the FEA model with respect to O_T-X_TY_TZ_T can be solved by substituting the FEA R_T into equation (47). Tip displacements and rotations resulting from the analytical and FEA models are compared in Figure 13(a) and (b). We use a series of points in a deformed condition with respect to O_T-X_TY_TZ_T to describe the shape of the three-segment ACCR, including Q_1SE1^*, Q_5SE1^*, Q_9SE1^*, Q_3SE2-sp^*, Q_7SE2-sp^*, Q_11SE2-sp^*, Q_1SE3-sp^*, Q_5SE3-sp^*, and Q_9SE3-sp^*, as shown in Figure 13(e). The “L shape” is shown in Figure 13(c) when F_SE3-c1 is varied, and the total displacements of FEA simulations are given as Figure 13(d)–(f).

(2) “S” shape

Figure 13.

“L” shape of the three-segment ACCR with respect to O_T-X_TY_TZ_T: (a) tip displacements, (b) tip rotations, (c) shape dexterity with the increase of F_SE3-c1, (d) the total displacement under F_SE3-c1 = 0.2 (N), (e) the total displacement under F_SE3-c1 = 0.8 (N), and (f) the total displacement under F_SE3-c1 = 1.4 (N). The average differences between the analytical results and their corresponding FEA results: D_xTip: 5.5%, D_yTip: 8.3%, D_zTip: 5.9%, θ_xTip: 6.0%, θ_yTip: 8.5%, and θ_zTip: 5.3%. The average differences of the coordinates between the analytical and FEA ACCR shapes in Figure 13(c) with respect to X_T, Y_T, and Z_T axes are 2.2%, 0.1%, and 6.2%, respectively).

The cable forces: F_SE1-c1 ranges from 0 (N) to 5 (N) with a step of 1 (N), F_SE1-c3 = 2 (N), F_SE3-c2 = 1.2 (N), an d other cable forces are equal to 0. The cable-force components loading at Q_1SE1, Q_5SE1, and Q_9SE1 due to F_SE1-c1, the cable-force components loading at Q_3SE1, Q_7SE1, and Q_11SE1 due to F_SE1-c3, and the cable-force components loading at Q_2SE3, Q_6SE3, Q_10SE3, Q_2S3S2, Q_6S3S2, Q_10S3S2, Q_2S3S1, Q_6S3S1, and Q_10S3S1 due to F_SE3-c2 can be solved in the analytical model, which are substituted into the FEA model. Tip displacements and rotations of the FEA and nonlinear kinetostatic models are compared in Figure 14(a) and (b). “S” shape with the increasing of F_SE1-c1 is shown in Figure 14(c)–(j).

Figure 14.

“S” shape of the three-segment ACCR with respect to O_T-X_TY_TZ_T: (a) tip displacements, (b) tip rotations, (c) shape dexterity with the increase of F_SE1-c1, (d) total displacement when F_SE1-c1 = 0 (N), and (e) total displacement when F_SE1-c1 = 5 (N). The average differences between the analytical results and their corresponding FEA results: D_xTip: 5.1%, D_yTip: 8.1%, D_zTip: 5.4%, θ_xTip: 6.0%, θ_yTip: 7.9%, and θ_zTip: 2.7%. The average differences of the coordinates between the analytical and FEA ACCR shapes in Figure 14(c) with respect to X_T, Y_T, and Z_T axes are 0.52%, 0.05%, and 6.3%, respectively).

5.3. Tip-stiffness improvement

Similar to Section 4.3, the tip stiffness of the three-segment ACCR and the counterpart CCR are compared. The two three-segment CCRs have the same geometric parameters and cables are located at the level of the sheets’ free ends. λ takes 0.5, α = 15°, 30°, or 45°, and the loading condition is an increasing tip load with constant cable-displacement actuation. ∆L_SE1-c1 = 6 (mm) and ∆L_SE3-c2 = − 2 (mm), which means that L_SE1-c1 is pulled down by 6 (mm) and L_SE3-c2 is elongated by 2 (mm). Other cables are free. These loading conditions are held in the following three situations.

(i) To compare the in-plane tip stiffness K_∆Fx-∆Dx of the ACCR and counterpart CCR, the in-plane tip load F_xTip ranges from 0.02 to 0.08 (N) with a step of 0.02 (N). Their results are shown in Figure 15(a). The maximum difference between the analytical and FEA results is 5.2%. K_{∆Fx-∆Dx_ACCR} has high values under a small α, which is the highest under α = 15°. However, K_{∆Fx-∆Dx_coun} has low values under a small α, which is the smallest under α = 15°. K_{∆Fx-∆Dx_ACCR} is larger than K_{∆Fx-∆Dx_coun}, and K_{∆Fx-∆Dx_ACCR} under α = 15°, 30° or 45° increases by 30.5%, 26.8%, and 21.1%, respectively.

(ii) To compare the out-of-plane tip stiffness K_∆Fz-∆Dz of the ACCR and counterpart CCR, the out-of-plane tip load F_zTip ranges from 0.02 to 0.08 (N) with a step of 0.02 (N). Their results are shown in Figure 15(b). The maximum difference between the analytical and FEA results is 6.5%. K_∆Fz-∆Dz of the two CCRs both have high values under a small α. K_∆Fz-∆Dz of the two CCRs are close but K_{∆Fz-∆Dz_ACCR} is slightly larger than K_{∆Fz-∆Dz_coun} under the same geometric parameters and loading conditions. K_{∆Fz-∆Dz_ACCR} under α = 15°, 30° or 45° increases by 1.4%, 1.1%, and 0.7%, respectively.

(iii) To compare the out-of-plane tip stiffness K_∆My-∆θy of the ACCR and counterpart CCR, the tip twisting moment M_yTip ranges from 0.02 to 0.08 (N) with a step of 0.02 (N). Their results are shown in Figure 15(c). The maximum difference between the analytical and FEA results is 7.2%. K_∆My-∆θy of the two CCRs have high values under a big α, which means K_∆My-∆θy is the largest under α = 45°. K_{∆My-∆θy_ACCR} is slightly higher than K_{∆My-∆θy_coun} under the same geometric parameters, where K_{∆My-∆θy_ACCR} and K_{∆My-∆θy_coun} denote the K_∆My-∆θy of the ACCR and counterpart CCR, respectively. K_{∆My-∆θy_ACCR} under α = 15°, 30° or 45° increases by 0.5%, 1.1%, and 1.7%, respectively. The effects of α and tip loads on the tip stiffness of the three-segment CCRs are in line with those of the one-segment CCRs.

Figure 15.

The in-plane and out-of-plane tip stiffness comparison between the three-segment ACCR and the counterpart CCR under a constant cable-displacement actuation: (a) the in-plane tip stiffness K_∆Fx-∆Dx under a series of increasing F_xTip, (b) the out-of-plane tip stiffness K_∆Fz-∆Dz under a series of increasing F_zTip, and (c) the out-of-plane tip stiffness K_∆My-∆θy under a series of increasing M_yTip.

Similarly, the tip stiffness study of the two three-segment CCRs under increasing cable actuation is studied in Appendix F. The tip stiffness comparison between the two three-segment CCRs under in-plane actuation is summarized in Table 7. The in-plane tip stiffness of the three-segment ACCR increases significantly under the increasing in-plane actuation.

Table 7.

The summary of tip stiffness evaluation between the three-segment ACCR and the counterpart CCR under in-plane actuation.

CCRs types	Loads: Constant cable actuation and increasing tip loads (see Section 5.3)			Loads: Increasing cable actuation (see Appendix F)
	In-plane tip stiffness K∆Fx-∆Dx	Out-of-plane tip stiffness K∆Fz-∆Dz	Out-of-plane tip stiffness K∆My-∆θy	In-plane tip stiffness K∆Fx-∆Dx	Out-of-plane tip stiffness K∆Fz-∆Dz	Out-of-plane tip stiffness K∆My-∆θy
ACCR	Both constant	Both slightly increase	Both constant	Slightly increase over the range of 21.78 ∼ 21.92 N/m	Slightly increase	Decrease with ∆LSE1-c1 increases	Slightly increase with FSE1-c2 increases
Counterpart CCR				Slightly decrease over the range of 16.76 ∼ 16.69 N/m	Slightly decrease
	Max improved ratio 30.5%	Max improved ratio 1.4%	Max improved ratio 1.7%	Max improved ratio 31.3%	Max improved ratio 2.6%	Max improved ratio 1.6%

The transverse tip stiffness of the two three-segment CCRs under increasing spatial actuation is further studied in Appendix D. Their results are summarized in Table 8. The transverse tip stiffness of the ACCR increases significantly under the increasing spatial actuation. When the three-segment ACCR and the counterpart CCR are under the same loading conditions and geometric parameters, their results show that the ACCR has a higher tip stiffness with a slenderer shape. In addition, the effects of various λ and α on the tip stiffness are also studied in Appendix F.

Table 8.

The summary of the transverse tip stiffness evaluation between the three-segment ACCR and the counterpart CCR under spatial actuation.

CCRs types	Loads: Increasing spatial actuation (see Appendix D)
	Transverse tip stiffness in the XT direction K∆Fx-∆Dx	Transverse tip stiffness in the ZT direction K∆Fz-∆Dz
ACCR	Slightly increase over the range of 21.58 ∼ 21.70 N/m	Slightly increase over the range of 21.75 ∼ 21.77 N/m
Counterpart CCR	Slightly decrease over the range of 16.59 ∼ 16.55 N/m	Slightly decrease over the range of 16.65 ∼ 16.60 N/m
	Max improved ratio 31.1%	Max improved ratio 31.2%

5.4. Experiments

We experimentally test a three-segment ACCR prototype under planar actuation, aiming to assess its manufacturability and validate the proposed analytical model across various scenarios. In Figure 16, the prototype of an ACCR using fabrication method 1. (For more details on fabrication, Appendix E can be visited). The corresponding parameters of the prototype are shown in Table 9. All modular units have identical rigid structures, differing only in sheet width. Each anti-buckling universal joint has a weight of close to 22 (g).

Figure 16.

Fabricated modular unit and the ACCR: (a) modular unit, (b) modular units of segments 1, 2, and 3, (c) three segments, and (d) ACCR prototype with cables.

Table 9.

Geometric parameters of the three-segment ACCR using fabrication method 1.

Parameters	Values (length unit: mm)	Parameters	Values
L a	14.5	λ	0.5
T	0.7	L d	(81LG2 + LJ2)1/2
USE1 (sheet width for segment 1)	5.8	x A1	−200/Ld
USE2 (sheet width for segment 2)	5.3	y A1	−30/Ld
USE3 (sheet width for segment 3)	4.8	x A2	200/Ld
L G	37	y A2	−30/Ld
L J	40	|xQεSE1|	30.3/Ld
Length of the three-segment ACCR	333	|xQεSE2|	28.8/Ld
α	45°	|xQεSE3|	25.3/Ld

During the experiments, we observed that segment 1 helps to resist unwanted deformations when large forces are applied at the tip. The more flexible segment 3 allows for finer, more dexterous movements, making it ideal for precise positioning. The gradual transition in stiffness creates smoother flexibility variations, enhancing the ACCR’s overall performance in kinetostatic tasks. Three performance indicators, tip displacements, tip trajectory, and tip stiffness are used to evaluate the prototype. The test setup is shown in Figure 17(a). The base of the prototype is fixed on the working bench, and the three-segment ACCR is placed in an upside down position to eliminate the gravity effect in the non-deformed shape. We use different weights to provide cable forces, as shown in Figure 17(b) and (c). The tip displacements can be calculated by counting the number of squares in the graph paper, as shown in Figure 17(d).

Figure 17.

Experiments of the three-segment ACCR: (a) no-loads acting on the ACCR, (b) the weight of F_SE1-c2 is 40 g, (c) the weight of F_SE1-c2 is 40 g and the weight of F_SE3-c2 is 100 g, and (d) the double exposure of Figs. 17(b) and (c) for calculating tip displacements. (Note that “double exposure” typically refers to a photographic technique where two images are combined into a single frame, creating a layered or blended effect).

The L shape of the three-segment ACCR is tested first, as shown in Figure 18(a). The weight that provides the cable force F_SE1-c2 is constant at 40 (g). The weights that provide the cable force F_SE3-c2 range from 0 (g) to 100 (g) with a step of 20 (g). The analytical and experimental results are compared in Figure 18(b) and (c), and the average differences of D_xTip and D_yTip are 4.5% and 5.5%, respectively.

Figure 18.

L shape test of the three-segment ACCR: (a) tip displacement with loads, (b) tip displacements D_xTip and D_yTip, and (c) tip trajectory of the tip marker.

The S shape of the three-segment ACCR is tested as shown in Figure 19(a). The weight that provides the cable force F_SE1-c2 is constant at 100 (g). The weights that give the cable force F_SE3-c1 range from 0 (g) to 100 (g) with a step of 20 (g). The analytical and experimental results are compared in Figure 19(b) and (c), and the average differences of D_xTip and D_yTip are 5.3% and 4.6%, respectively.

Figure 19.

S shape test of the three-segment ACCR: (a) loading conditions, (b) tip displacements D_xTip and D_yTip, and (c) tip trajectory of the tip marker.

The tip stiffness is also evaluated as shown in Figure 20. Two cable displacements are constant, and other cables are free. ΔL_SE1-c1 = 6 (mm) and ΔL_SE3-c2 = 4 (mm). The cable displacement is adjusted by a 3D printing cylinder. The cylinder diameter is 1/π. Each time the cable wraps around the cylinder, and it pulls by 1 mm. The weight that provided tip force along the X_T-axis increases from 0 to 70 (g) with a step of 10 (g). The analytical and experimental results are compared in Figure 20(b)–(d), and the average differences of D_xTip, D_yTip, and K_ΔFx-Δx are 5.5%, 4.6%, and 5.07%, respectively. K_ΔFx-Δx increases significantly with the increase of F_xTip, as described in Figure 20(d). This is because the minimum available weight in the test is 10 (g), causing a much larger increment over the increase of F_xTip.

Figure 20.

The tip stiffness test of the three-segment ACCR: (a) loading conditions, (b) tip displacements D_xTip and D_yTip, (c) the tip trajectory of the tip marker, and (d) the tip stiffness increases with F_xTip under constant cable displacements.

The limitation of using an HD camera plus a 1/8-inch graph paper for shape measurement lies in its inability to simultaneously align directly opposite to all test points. Therefore, the shape of the ACCR prototype was not measured during the above experimental tests. We have used a laser sensor to evaluate another fabricated ACCR (prototype 2) in terms of tip displacements and shape measurement, as explained in Appendix G. However, the maximum difference for prototype 2 between the analytical and experimental results is much larger, about 10.6%. The comparison between the two prototypes is shown in Table 10. It is noted that both of the two displacement measurement methods are only suitable for planar testing rather than spatial testing.

Table 10.

The comparison between the two prototypes.

Prototypes	Fabrication method	Materials	Displacement measurement	Level of difficulty	The max average difference between the analytical and experimental results	Test conducted in plane
						Shape	Tip deformation	Tip stiffness
Prototype 1 in Figure 16	Method 1: 3D printing using Ultimaker 3D printer, with printing elastic sheets (method 1 in Appendix E)	PLA	1/8-inch graph paper and an HD camera (Section 5.4)	Simpler	5.5%	No	Yes	Yes
Prototype 2 in Fig. G1 (Appendix G)	Method 2: 3D printing using Ultimaker 3D printer, without printing elastic sheets (method 2 in Appendix E)	C17200 beryllium copper, and others are PLA	Laser (Appendix G)	This requires more time in terms of aligning each test point with the laser point	10.6%	Yes	Yes	Yes

Although experimental results of the three-segment CCRs under increasing cable actuation were not obtained, the analytical results in both FEA and analytical modeling show that the tip stiffness can be increased by increasing the cable force or displacements for the ACCR, which can be seen in Figure 2(c) and (d) of Appendix F.

6. Remarks of the ACCR

Any dominant loads/displacements acting on the CCR have been considered in the proposed kinetostatic model (such as cable forces, prescribed cable actuation displacements, tip loads, and gravity), and their effects on the CCR’s performances have been evaluated in Sections 4 and 5. In this section, we will discuss many aspects of the analytical model and the ACCR including the gravity effect, unaccounted considerations, fabrication scalability, reliability, and the comparison with other existing designs.

6.1. Gravity effect

The gravity effect has been included in the analytical, experimental, and FEA models. The gravity effect on the ACCR depends on the application environment and ACCR’s placements. For example, the gravity of the ACCR has less influence on its deformation when it is utilized in underwater or space. This section discusses how gravity influences the shape dexterity of the three-segment ACCR, when it is placed upside down, upright, or horizontally under a constant cable-force actuation and a tip load. F_SE1-c1 = 3 (N), F_SE3-c2 = 0.3 (N) and F_zTip = 0.1 (N). We assume that the mass of each anti-buckling is 20 (g), and the gravity of each anti-buckling universal joint is located at its rotational center. λ = 0.5 and α = 15° here and other geometric parameters are the same as those of Section 5.1.

Figure 21(a) describes the FEA model of the three-segment ACCR under λ = 0.5 and α = 15°. Figure 21(b) and (c) show the shape dexterity through X_TY_T and Y_TZ_T planes, respectively. The maximum difference between the analytical and FEA results is 3.4%. When the ACCR is placed upside down and upright, the gravity of each anti-buckling universal joint is along the positive and negative direction of the Y_T-axis, respectively, as given in Figure 21(e) and (f). The analytical model in Section 3.2 can be used directly. When the ACCR is placed horizontally, the gravity of each anti-buckling universal joint is along the positive direction of the X_T-axis, as shown in Figure 21(g). The gravity-related equation should be revised, such as equation (39), [0, f_{maJΩSE3_T}, 0, 0_3×1]^T should be revised as [f_{maJΩSE3_T}, 0, 0, 0_3×1]^T. In the XY plane, the gravity influences ACCR’s shape significantly when the ACCR is placed horizontally. In the YZ plane, the gravity influences ACCR’s shape greatly when the ACCR is placed upright.

Figure 21.

Gravity effect on the three-segment ACCR that is placed upside down, upright, or placed horizontally under the same actuation conditions. (a) Three-segment ACCR under λ = 0.5 and α = 15°, (b) shape dexterity through the XY plane, (c) shape dexterity through the YZ plane; the FEA total displacements when the ACCR is (d) no gravity, (e) placed upside down, (f) placed upright, and (g) placed horizontally.

6.2. Unaccounted considerations in modeling of the ACCR

Friction (Roy et al., 2017), elongation and creep of cables (J. Li et al., 2020a), slack and hysteresis of cables, hysteresis of ACCR’s material, temperature (Kim et al., 2017), and dynamics (Z. Liu et al., 2021), are common factors influencing the performance of robot. In this section, their effects on the ACCR’s performance are discussed.

(1) Friction

Compliant joints do not introduce any friction. The source of friction in the continuum robot system is mainly due to the contact between the cables and the holes, which can increase the actuation forces slightly (i.e., actuation stiffness increase). Note we use Clear Nylon Fishing Wire (Eruinfang) as the actuation cable for the experiment, which is very smooth for making trivial contribution to influence our analytical model. The experimental displacement results are often slightly larger than the analytical models (i.e., actuation compliance increase) due to two reasons: i) the deformation of rigid parts is neglected in the proposed kinetostatic model and ii) the inevitable stiffness loss due to assembly. The above two effects can usually cancel out each other to make the experimental model agree with the analytical model.

(2) Elongation and creep of cables

We did not consider cable elongation in the proposed analytical model. Dyneema thread is an ideal choice for a cable-driven ACCR. The cable break load and the Young’s modulus of a Փ0.4 (mm) Dyneema thread is 50 (kg) and at least 109 (GPa) (Sanborn et al., 2015), respectively. The Dyneema thread is of high strength, low density, and low elongation at break, making it highly suitable for the ACCR that demands substantial actuation forces. However, in our experiment, we used an alternative Փ0.4 (mm) Clear Nylon Fishing Wire (Eruinfang) as the actuation cable. The Young’s modulus of nylon is at least 2.7 (GPa). Our maximum actuation force is less than 0.1 (kg) in the experiment (due to the low stiffness of 3D printed joints), far below the cable break load (12.97 (kg)). So the elongation is not significant to affect the agreement between the analytical models and the experimental results as seen in Figures 18–20.

(3) Slack and hysteresis of cables

The effects of cable slack and hysteresis on the ACCR’s repeatability are not studied in this paper. These effects will be investigated in our future experiments including cyclic loading tests, drift analysis over extended operation, feedback control, or tension monitoring systems. We will develop a more robust control strategy that compensates for these effects in real-time.

(4) Hysteresis of ACCR’s material

The anti-buckling universal joints are fabricated from PLA. While polymers do exhibit some degree of creep under sustained stress, the impact on the ACCR depends on factors such as load magnitude, duration, and temperature. Creep in PLA is likely to occur when the material is under continuous mechanical stress, especially at temperatures approaching or exceeding its glass transition temperature (∼55 °C–65 °C). When we did the experiment, the ACCR was operated in a room temperature at about (22°C–23°C) during a short time. We also designed a limiter for the ACCR, which limits further deformation under heavy loads, providing additional protection.

We are planning further experiments to quantify the long-term impact of material creep on joint performance and to explore possible mitigation strategies, such as optimizing joint design. This will help us to ensure reliable performance in practical applications. Overall, while the material’s hysteresis and creep effects are present, our prototype exhibits acceptable for the ACCR’s intended tasks.

(5) Temperature variations

In this paper, we did not investigate the effect of temperature variations on the performance of the ACCR. In our design and experiment, we did not apply materials that are sensitive to temperature variations, such as shape memory alloys, and we conducted our experiment in room temperature. Thus, the temperature variations on the performance of the proposed robot are not considered in this paper. However, if the ACCR is used in extreme temperature environments, such temperature effects should be modeled and tested, which is out of the scope of this paper.

(6) Dynamics

Continuum robots in real-world applications are often used for slow movements as shown in (Chen et al., 2021; Dong et al., 2015, 2019; Kang et al., 2017; K. Xu et al., 2015; K. Xu and Simaan, 2008), which has motivated us focus on the kinetostatic modeling and analysis as presented in this paper. We plan to develop a more comprehensive control system and method based on the proposed kinetostatic modeling approach. In addition, even without considering the velocity and acceleration contributions, a slow control (Abdelaziz et al., 2011) can be conducted where inertia and damping are not considered.

6.3. Scalability

Two methods using a low-cost Ultimaker 3D printer have been proposed in this paper for fabricating two assembled prototypes. A small-scale modular unit with a diameter of 40 mm and a height of 37 mm has been shown in Figure 16. As a comparison, a large-scale anti-buckling universal joint, with a diameter of 110 mm and a height 87 mm, is also presented in (S. Li and Hao, 2022).

We believe that the limiting factors to a miniaturized CCR would be the capability of the used 3D printer and/or the assembly of the small parts. The biggest hurdle to further increase the size of the robot system would be the significant gravity effect due to the heavy mass introduced, which can increase the actuation effort and control complexity (due to cable pretension). The mass of the CRR can be considered an important parameter in our analytical model, as described in Section 6.1. The scale (such as how large it is possible) of the robot system is also dependent on an actual application (with desired specific performances). For example, for the human-machine collaboration, a large-scale ACCR alike a human arm would be desired.

Thus, mass, assembly methods, segment numbers, cable distributions/numbers, and geometric parameters are all influencing factors that contribute to both the scale and the performances of the ACCR.

6.4. Reliability

A reliable CCR can perform its tasks consistently, safely, and efficiently, while avoiding breakdowns, errors, and failures. The reliability of the ACCR is discussed in light of breakdown possibility and motion accuracy.

(1) Breakdown possibility

The breakdown possibility of the ACCR mainly depends on the properties of elastic sheets. The ratio of flexure strength to Young’s modulus of the elastic sheets is large, which decreases the possibility of breakdown. What’s more, the ACCR is designed for the use in a low-cycle application to avoid fatigue. Throughout the deformation evaluation in this paper, the max von Mise stress obtained by FEA of the ACCR is always far less than the yield stress.

(2) Motion accuracy

The proposed robot is a distributed-compliance mechanism by using long elastic sheets. The motion accuracy is not very sensitive to the geometrical errors of the elastic sheets so the ACCR can accommodate small manufacturing or fabrication errors. We have compared two three-segment ACCRs, in terms of tip displacements and tip rotations, with one slightly different geometrical parameter of the sheet while the remaining other sheet parameters are the same. The remaining parameters in the ACCR are the same as those explained in Section 5.2, and the loading condition is the same as the L shape test in Section 5.2 with F_SE3-c1 = 1.4 (N). In Table 11, the deformation differences between the ACCR with determined sheet parameters and the ACCR with sheet fabrication errors are calculated. We can observe that the tip motion is insensitive to the small change of sheet parameters that reflects the manufacturing/assembly errors.

Table 11.

Validation of the three-segment ACCR in accommodating manufacturing or fabrication errors.

Sheet geometric parameters (mm)		Tip displacements (m)						Tip rotations (rad)
		D xTip		D yTip		D zTip		θ xTip		θ yTip		θ zTip
Determined sheet	L = 50, USE1 = 12, T = 1, α = 45°	−0.172		−0.019		0.071		0.082		0.0091		0.239
1) Fabrication error in sheet length	L = 52.5, USE1 = 12, T = 1, α = 45°	−0.184	Diff. 6.7%	−0.022	Diff. 11.2%	0.076	Diff. 6.8%	0.086	Diff. 5.0%	0.010	Diff. 9.6%	0.252	Diff. 4.8%
2) A fabrication error in sheet width	L = 50, U SE1 = 11, T = 1, α = 45°	−0.187	Diff. 8.4%	−0.023	Diff. 16.3%	0.077	Diff. 8.3%	0.089	Diff. 8.7%	0.011	Diff. 16.5%	0.263	Diff. 8.7%
3) A fabrication error in sheet thickness	L = 50, USE1 = 12, T = 0.98, α = 45°	−0.182	Diff. 5.7%	−0.022	Diff. 11.1%	0.075	Diff. 5.7%	0.087	Diff. 6.0%	0.010	Diff. 11.5%	0.254	Diff. 5.8%

Note: U_SE2 and U_SE3 are equal to U_SE1−0.5 mm, U_SE1−1 mm, respectively. Diff. denotes the difference percentage of tip deformation between the ACCR with determined sheets and the ACCR with fabrication error in sheet.

6.5. Comparison with existing representative work

In this section, we summarize the comparisons between the ACCR and existing CCRs from three aspects, including working principle of increasing stiffness, analytical model, and other characteristics.

The working principle of increasing the stiffness of the ACCR is to utilize the load-dependent effects of compliant mechanisms. Table 12 shows the advantages and shortcomings of different methods to increase CCRs’ stiffness.

Table 12.

The advantages and shortcomings of different methods to increase CCRs’ stiffness.

CCRs designed by	The methods for increasing robot stiffness	Advantages	Shortcomings
Anti-buckling universal joints (this paper)	Regulate cable actuation, based on the working principle of load-dependent effect in compliant mechanisms	① High twisting stiffness and anti-buckling	Complex joint structures, leading to great fabrication challenges at a very small scale such as a diameter of 10 mm
		② Stiffness can be regulated more accurately
		③ No delay associated with material phase transitions (e.g., LMT or SMA)
		④ Not sensitive to temperature
Elastic tube (Clark and Rojas, 2019)	Regulate friction between granules	① Can afford very large tip load, such as 29.7 N with 90° bending	① High weight, such as 438 g
		② Not sensitive to temperature	② Long activation time, such as 3.69 s
		③ Suitable for large-size CCR, for example, a diameter of 48 mm	③ Not accurate in regulating the stiffness
			④ Nonlinear stiffness regulation
Elastic rods (Langer et al., 2018)	Regulate friction between sheaths	The smallest design among existing stiffening sheath CCRs, with a diameter of 13 mm	① Low twisting stiffness under lateral forces
			② In the presence of buckling
			③ Not accurate in regulating the stiffness
			④ Nonlinear stiffness regulation
Elastic tube (Xing et al., 2021)	Regulate temperature of low-melting-point alloy (LMT)	The copper tube not only acts as a structural skeleton but also provides efficient heat transfer	① It takes 15s from soft to stiffening state
			② Variations in environmental temperature affect the phase transition speed and the resulting stiffness
Elastic rods (Yang et al., 2020)	Regulate temperature of shape memory alloys (SMA)	Localized Stiffness Control: using SMA springs to locally adjust friction between drive rods and constraint disks	① Twisting stiffness can be limited due to the use of elastic rods
			② Long cooling time: 22∼29s
			③ Friction dependency: wear and tear could affect the stiffness
			④ Potential nonlinearity will increase the difficulty in control
Elastic rods (B. Zhao et al., 2020)	Regulate the length of an insertable rigid rod	① Minimal structural changes	① Disturbance during stiffness adjustment
		② High stiffness variation ratio	② Complexity in multi-segment designs
			③ Dependence on position and pose
Elastic rods (Shiva et al., 2016; Stilli et al., 2017)	Antagonistically implement two types of actuation modes: pneumatic and tendon-driven actuations	① Can support high loads without excessive deformation	① Complexity in actuation
		② If one actuation mode fails, another one can still provide stiffness	② Sealing wear and tear of pneumatic actuators
			③ High energy consumption

The proposed analytical model is based on the spatial nonlinear beam constraint model (BCM) and free-body diagram. Table 13 summarizes the advantages and shortcomings of the proposed analytical model and existing analytical models. The proposed analytical model has three key advantages over existing analytical models.

(1) The proposed analytical model accounts for localized, discrete, and distributed loads, whereas other analytical models have limitations in handling these types of loads.

Table 13.

The advantages and shortcomings of the proposed analytical model and existing analytical models.

Analytical models used in CCRs	Advantages	Shortcomings
Spatial nonlinear BCM based on free-body diagram for the ACCR (this paper)	① An explicit physics-based free-body diagram method to model nonlinear large deformation of CCR, retaining a balance between mechanical rigor and practical adaptability	① Limited to the CCR with elastic rods or sheets
	② Accounting for localized, discrete, and distributed loads along the CCR	② Motion of each joint is limited to medium range due to the limitation of spatial BCM
	③ Highly extensible to new designs by modeling other compliant mechanisms constructed from elastic rods or sheets, not limited solely to CCRs	③ Only limited to kinetostatic modeling
	④ Facilitating optimization of the CCR by considering all geometric parameters, including the elastic sheets and rigid components
	⑤ Friction can be included (our future work)
Cosserat rod theory (Alqumsan et al., 2019; Caleb Rucker and Webster, 2014; Chikhaoui et al., 2019; Till et al., 2019, 2020; Till and Rucker, 2017)	① Suitable for very large spatial deformation	① Limited to a slender CCR (i.e., 1D element): cross-sectional dimension should be significantly smaller than its length
	② Capture nonlinear deformation	② Not accounting for friction
		③ Implicit method
		④ Not accurate to deal with discrete or highly localized loads
Constant curvature assumption (Goldman et al., 2014; Z. Li and Du, 2013; Nuelle et al., 2020; Webster and Jones, 2010)	① Facilitating closed-form solutions	① Limited to CCRs with small length
	② Simple and computationally efficient	② Unable to address complex deformation problems, such as those involving torsion, shear, or non-circular bending
		③ Not accurate to deal with discrete or highly localized loads
		④ Not accounting for material properties
DH method (Dong et al., 2015; W. Xu et al., 2018)	① Kinematic method, easy to use	① Assuming that all joints are ideal kinematic joints
	② Computationally efficient	② Not suitable for large deformation
Pseudo rigid body model (Venkiteswaran et al., 2019)	① Simplified modeling	① High complexity in fully capturing the nonlinearity and large deformation
	② Quick computation	② Not suitable for spatial deformation
	③ Taking advantage of the knowledge body in kinematics
Bézier curve fitting (Guo et al., 2023; Yuan et al., 2017)	① Bézier curves are inherently smooth and continuous to achieve modeling of complex shapes	① Not explicitly accounting for friction or external forces acting along the length of the robot
	② The method is mathematically simple because it relies on polynomial interpolation	② Limited accuracy for large deflections

Many practical engineering applications, especially those involving tendon-driven robots, do not fit neatly into a purely distributed-loading framework. Instead, localized or discrete loads often occur, such as contact forces and friction at cable-routing holes or pivot joints. The proposed analytical model directly captures localized interactions without forcing them into a distributed framework. Each contact interface (e.g., a cable-routing hole) can be treated as an individual node. This preserves the true mechanical behavior at each interface and allows any force or moment to be precisely calculated from fundamental laws of equilibrium. For example, cables pass through all the holes along the ACCR, and the forces applied by the cables at each hole are considered individually.

Constant curvature or polynomial-based models are widely used for CCRs primarily due to their simplicity in describing geometry. However, they often approximate distributed loads when non-tip loads act on CCRs, which can cause inaccuracies for discrete or highly localized loads. Cosserat Rod Theory is among the most comprehensive mathematical formalisms for describing continuum structures because it naturally captures distributed loads via continuous functions (e.g., f(s) for forces and m(s) for moments along the elastic rod, where s is the arc length). However, external forces and moments are usually treated as continuous distributions. When forces arise from discrete contacts, such as friction at a small interface, representing them as f(s) may involve artificial “spread-out” distributions or Dirac delta functions.

(2) Shear and twisting deformations can be derived accurately by using the proposed analytical model, whereas other analytical models do not provide this level of accuracy or are limited to a stronger assumption, such as being slender CCRs for the Cosserat Rod Theory.

CCRs are designed to be highly flexible and deformable, often undergoing complex spatial motions that include shear, twisting, and bending. We use spatial BCM (Bai et al., 2021) to derive the proposed analytical model of the ACCR, which accounts for shear, twisting and bending deformations for each elastic sheet. Cosserat Rod Theory treats the entire CCR as multiple one-dimensional (1D) elements, so that the shear, twisting, and bending deformations along the CCR can be derived accurately. However, it is not ideal for capturing the shear and twisting of a CCR when its cross-sectional dimension is not significantly smaller than the CCR’s length.

(3) The proposed analytical model makes the overall model highly extensible to new designs and optimizations, which is suitable for other compliant mechanisms constructed from elastic rods or sheets, not limited solely to CCRs.

Researchers can introduce or modify friction interfaces, cable-routing hole geometries, or internal mechanisms by simply incorporating them into the local free-body diagrams. In contrast, adapting Cosserat Rod Theory to capture such detailed physics might require substantial reformulation or extensive approximations. As such, the proposed model strikes a balance between mechanical rigor and ease of customization, aligning well with the design-oriented needs of advanced CCRs. The analytical model accounts for all design parameters of the CCR, including elastic sheets, rigid components, and cable-loading positions, along with material properties. These factors influence key performance characteristics of CCRs, such as tip stiffness, tip position, and shape dexterity. By optimizing these design parameters, the CCR can be tailored to meet specific application requirements.

In this paper, the three-segment ACCR is physically developed in a medium size, targeting industrial applications. Its diameter and length are 40 (mm) and 333 (mm), respectively. The potential applications include industrial inspection and repairing (Dong et al., 2017) as well as human-machine integrated robotic systems (W. Liu et al., 2022). We applied the normalized tip loads to compare the tip load across scales. Normalized loads are dimensionless, which can account for material’s properties such as Young’s modulus and CCRs’ dimension, representing an appropriate normalized metric for cross-scale comparisons. We assume the CCR is an equivalent elastic beam and therefore, the maximum normalized tip load can be derived by using equation (53). The maximum tip load is defined based on the maximum deformation, just before buckling occurs.

$f_{\max }=\frac{F_{\max }{L_{\mathrm{d}}}^2}{E{I}_{\text{CCR}}}$

(53)

where F_max denotes the maximum tip load; f_max denotes the corresponding normalized tip load; I_CCR = πD²/64 with D being the diameter of the CCR; E is the Young’s modulus of the elastic parts in the CCR.

The characteristic comparison of the ACCR with the existing cable-driven CCRs is shown in Table 14. The tip stiffness of the existing reprehensive cable-driven CCRs decreases with cable actuation, while the tip stiffness of the ACCR increases with cable actuation. Our design achieves the highest approximate normalized maximum tip load, indicating an improvement in payload ability.

Table 14.

The comparison of the ACCR and other cable-driven CCRs.

CCR types	Experimental overall dimension (mm)		Tip stiffness against cable actuation	Joint type/compositional unit	Elastic flexure	Cable types	Approx. max. applied tip load (N)	Approx. max. normalized applied tip load	Applications	Analytical models
	Length	Diameter
ACCR (Section 5.4 in this paper)	333	40	Increase	Universal joint	PLA sheet	Nylon	0.98	4.456 ×10−7	Potential industrial inspection or repairing, etc.	Spatial BCM
Backbone CCRs (K. Xu et al., 2014)	33	7	Decrease	Wire beam	NiTi long rod	NiTi	2	1.262 ×10−8	Surgical robot	Constant curvature assumption
Compliant-joint CCRs (Dong et al., 2017)	1270	40	Decrease	Revolute joint	Steel short beam	Steel	1.96	2.014 ×10−7	Industrial inspection or repairing	DH method
Spring CCRs (Li et al., 2018)	170	20	Decrease	Wire beam	Steel Spring	Polyethylene	NA	NA	Potential industrial inspection	Piecewise constant curvature

7. Conclusions and future work

A tip stiffness improved cable-driven compliant continuum robot (ACCR) is proposed in this paper, which consists of multiple anti-buckling universal joints. The nonlinear spatial analytical models (i.e., forward kinetostatic models) of the ACCR have been derived under spatial loading conditions including cable-force actuation, cable-displacement actuation, tip loads, and gravity. One potential advantage of the proposed nonlinear analytical models is that any loads acting on the ACCR can be considered, and loads due to environmental constraints can be potentially considered in the analytical models if necessary. The proposed system modeling framework can be used to model other CCRs with different compliant joints. The maximum deformation, tip-location accuracy, shape dexterity, and tip stiffness of the ACCR have been comprehensively studied. The tip stiffness comparison between the ACCR and the counterpart CCR has been particularly investigated under two in-plane loading conditions, including (a) constant cable actuation, and (b) increasing cable actuation. The effect of geometric parameters, cable displacements, and cable forces on tip stiffness has been analyzed for the ACCR and the counterpart CCR. The tip stiffness of the one-segment ACCR and that of the counterpart CCR have been compared in experimental tests. The tip displacement, tip trajectory and tip stiffness of the three-segment ACCR have been experimentally evaluated.

The main results are summarized as follows.

(1) The proposed nonlinear analytical models have a high accuracy compared with FEA models when each anti-buckling universal joint has an intermediate range of motion. The maximum difference between the analytical and FEA results is less than 7% except for those results of very small axial displacements and twisting rotations. The maximum difference between the analytical and experimental results is less than 6%.

(2) Our design can produce high bending curvatures and variable shapes, whose motions in the DoC directions are well-constrained, such as twisting motions.

(3) The tip stiffness of the ACCR is always higher than that of the counterpart CCR under α ≤ 45° with any λ, when they are under the same geometric parameters and loading conditions. Therefore, ACCR enables a slenderer shape and higher tip stiffness than the counterpart CCR.

(4) Effects of geometric parameters and loading conditions on the tip stiffness of the one-segment and three-segment CCRs under increasing in-plane actuation have been summarized in Table 4 and Table 7, respectively, which are also briefly presented below.

(a) The in-plane tip stiffness of the ACCR increases with the increase of cable forces or displacements, and the tip stiffness change range is larger. However, the in-plane tip stiffness of the counterpart CCR decreases with the increase of cable forces or displacements.

(b) The out-of-plane tip stiffness (due to the out-of-plane tip forces) of the ACCR slightly increases with the increase of cable forces or displacements. However, the out-of-plane tip stiffness of the counterpart CCR has the opposite trend.

(c) The out-of-plane tip stiffness (due to the twisting tip moments) of the ACCR and that of the counterpart CCR all decreases with the increase of cable displacements while slightly increasing with the increase of cable forces.

(5) When the ACCR and the counterpart CCR are all under increasing spatial actuation, all the transverse tip stiffness of the ACCR can increase. Their results are summarized in Table 5 and Table 8 in Appendix D.

(6) The one-segment CCR experimental results validate that a) ACCR’s stiffness is much larger than that of the counterpart CCR; b) the ACCR tip stiffness can increase by increasing actuation forces. The three-segment CCR experimental results validate that a) the proposed analytical model has a high accuracy for predicting tip trajectory and various body shapes; b) the tip stiffness increases with the increase of tip forces.

In the future, the design parameters of the ACCR will be optimized so that the tip stiffness of the ACCR can increase more significantly by increasing the cable actuation. We will build a comprehensive control system for the ACCR. The actuation, sensing, and motion/force control of the ACCR will be investigated by developing both hardware and associated software/algorithms. Leveraging the advantages of the kinetostatic modeling method, we will explore the crucial intrinsic ability to detect contact locations and interaction forces between the ACCR and its environment. In addition, we will carry out a series of experiments on the unaccounted considerations in modeling of the ACCR. across various scenarios.

Supplemental Material

Supplemental Material - Design and kinetostatic analysis of a tip-stiffness improved compliant continuum robot using anti-buckling universal joints

Supplemental Material for Design and kinetostatic analysis of a tip-stiffness improved compliant continuum robot using anti-buckling universal joints by Shiyao Li, Salih Abdelaziz, Long Wang, and Guangbo Hao in The International Journal of Robotics Research