Kinematic Solving and Stable Workspace Analysis of a Spatial Under-Constrained Cable-Driven Parallel Mechanism

Abstract

This study systematically investigates the kinematic characteristics and static stability of a spatial under-constrained four-cable-driven parallel mechanism, specifically designed for supporting aircraft models in wind tunnel tests. Addressing the inherent strong coupling between kinematics and statics in such systems, an integrated solution framework is proposed. Firstly, a hybrid intelligent algorithm integrating genetic algorithm, chaos optimization, and particle swarm optimization is introduced to efficiently solve the direct and inverse geometric-statics problems, ensuring the identification of physically feasible equilibrium configurations under constraints such as cable tension limits and mechanical interference. Subsequently, a stability evaluation method based on the eigenvalue analysis of the system’s total stiffness matrix is employed, establishing a criterion (minimum eigenvalue λ_min > 0) to identify statically stable equilibrium points. Finally, the static feasible workspace and the static stable workspace are systematically analyzed and quantified, providing practical operational limits for mechanism design and trajectory planning. The effectiveness of the proposed solution framework is validated through numerical computations, simulations, and experimental tests, demonstrating its superiority over benchmark methods. This study provides theoretical support for the design, analysis, and control of under-constrained four-cable-driven parallel mechanisms.

1. Introduction

Cable-driven parallel mechanisms (CDPMs) are a class of robotic manipulators based on parallel architectures, which utilize flexible cables to drive the end-effector along predefined spatial trajectories. These systems are characterized by their structural simplicity, high payload capacity, large workspace, superior stiffness, dynamic performance, and cost-effectiveness. In recent years, CDPMs have garnered significant attention from both researchers and industries, with emerging applications in diverse fields such as large-space three-dimensional (3D) printing [1], material handling [2], medical rehabilitation [3], radio telescopes [4], and wind tunnel tests of aircraft [5].

Generally, CDPMs can be classified into three configurations based on the relationship between the number of cables (δ) and the degrees of freedom (n) of the end-effector: under-constrained (δ < n + 1), fully constrained (δ = n + 1), and redundantly constrained (δ > n + 1). In addition, cable-driven suspension mechanisms with δ = n are also a type of fully constrained case, where gravity is often considered as an additional, constant force (analogous to an extra cable) acting downwards on the end-effector, effectively making the system behave like a fully constrained one (δ = n + 1) under certain conditions. There have been extensive studies on the redundantly and fully constrained cases. They cover topics ranging from geometric, kinematic, and static analysis, dynamics and control to other different architectures and applications [6]. Comparatively, less attention has been dedicated to the under-constrained cable-driven parallel mechanism, which employs fewer cables than the number of DOFs, thus allowing only a limited number of DOFs to be actively controlled. The motion of the end-effector becomes uncertain in the presence of external disturbances while the cable lengths remain unchanged. This offers a unique advantage for some special applications, such as active rehabilitation tasks and rescue operations, which benefit from lower dexterity and potentially simpler motion patterns (due to fewer actively controlled DOFs), allowing for reduced complexity, cost, and set-up time, etc. [7]. In recent years, under-constrained CDPMs have attracted attention due to their combination of under-constrained systems and cable robots [8,9].

The most challenging aspect for the study of the under-constrained CDPMs comes from the kinematic coupling of its geometry and statics, which makes it difficult to determine the effective equilibrium points and their stability. Several scholars and groups have already conducted some research on the questions mentioned above. For example, Carricato and Merlet expounded on the key problems of the under-constrained CDPMs’ kinematics and pointed out that the end-effector still preserved some certain DOFs even when the actuators were locked and the cable lengths were assigned. To tackle the direct and inverse geometric-static problems, a hybrid elimination procedure based on Groebner bases and Sylvester’s dialytic method was proposed to solve the high-degree polynomials (derived by eliminating cable tensions from the equilibrium equations), yielding all complex solutions [10]. However, the upper bound on the number of real roots was yet unknown, and it was further investigated using numerical approaches including Dietmaier’s algorithm and evolutionary techniques [11]. The methods aimed to maximize the number of equilibrium configurations but ignored cable tension feasibility and static stability requirements. The algorithms of interval analysis [12] and artificial neural network [13] were also utilized for the direct kinematic problem to search for real solutions with non-negative cable tensions, but constructing the training sets for the under-constrained cable-driven parallel mechanisms is rather difficult.

Kinematic workspace is also an important reference for the design of the under-constrained cable-driven parallel mechanisms. An analytical method for generating the boundaries of the wrench-feasible workspace for cable robots was presented in reference [14]. The method applies to both under-constrained and fully constrained cable-driven parallel mechanisms; complete analytical expressions for the boundaries are detailed for a planar and a spatial point-mass cable robot. When it relates to the spatial 6-DOF robot, the analytical method becomes quite complicated, and the expressions describing workspace boundaries cannot be obtained in closed forms for the under-constrained case. Duan et al. [15] proposed a consistent solution strategy for the static feasible workspace of different types of the under-constrained cable-driven parallel mechanisms and planar hybrid robots. Considering the constraint conditions of cable forces and taking the least squares error of the static equilibrium equations as the objective, the convex optimization was carried out to obtain the static feasible workspace of the under-constrained system. Jibi et al. [16] investigated the workspace of a four-cable under-constrained cable-driven parallel mechanism used in architectural applications, where the platform is not required to be in perfect static equilibrium. Instead, moments about two horizontal axes are constrained within predefined bounds to accommodate practical design requirements. Wang et al. [17] proposed to construct the wrench space of an under-constrained system by using the method of virtual cables. The workspace was calculated based on the generalized inverse matrix theory, which provides the optimal solution of the cable tensions that satisfies the motion requirements of the system. However, the effect of errors must be analyzed, especially at the boundaries of the calculated workspace.

Besides the kinematics study of the under-constrained CDPMs, the equilibrium stability assessment is another crucial issue. Bosscher et al. [18] introduced a slope-based measure of the stability of a pose of an under-constrained cable robot. Although this stability measure is physically meaningful, it is inefficient to calculate the overall stability in the task space. Behzadipour et al. [19] studied the stiffness of cable-based robots using an equivalent four-spring model, which was further used as the stability criterion to derive sufficient conditions for stability. In reference [20], an energetic approach by minimizing potential energy subject to complex algebraic constraints is applied, but the impact of cable tension constraints is ignored. Carricato et al. [21] investigated the stability of CDPMs under spatial constraints by proposing an efficient algorithm, which is rooted in the principle of virtual work and standard linear algebraic routines. This approach evaluates stability through the positive definiteness analysis of the Hessian matrix. Similarly, Surdilovic et al. [22] developed a method for calculating the eigenvalues of the Hessian matrix while incorporating pulley dynamic effects, thereby extending the applicability of this evaluation framework. Both approaches do not explore the sufficient conditions for the Hessian matrix to be positive definite, which are very meaningful and instructive for stability analysis. Zhu et al. [23] proposed a three-degree of freedom wire-driven parallel robot and used differential evolution algorithm to optimize the spring parameters installed on the telescopic rod by comprehensively considering acceleration and cable force. The minimum initial acceleration of the moving platform is used to characterize the dynamic stability of the robot, and dynamic stability analysis under impulsive disturbances is performed according to the Gauss principle of least constraint. Other research on the under-constrained CDPMs relates to the natural oscillations, tension optimization, oscillation reduction, and control [24,25,26]. From the aforementioned literature, it is evident that existing studies seldom provide comprehensive investigations into the kinematics/workspace and equilibrium stability of under-constrained systems, despite their critical importance for practical applications. In addressing the direct geometric-static problem of the under-constrained CDPMs, it is typically treated by eliminating cable tensions from equilibrium equations, and then solved by using the pseudo-inverse or least square solution method. For the stability analysis of a cable-based manipulator, it is deemed to be stable at a given pose when the manipulator is kinematically non-singular and the active stiffness matrix in the absence of any external load is positive definite [19].

In operational practice, the inverse kinematics problem with assigned platform poses and incorporating constraints on the cable tensions is of more concern. How to obtain feasible equilibrium solutions more efficiently when considering the constraints, and how to determine the stable workspace of the under-constrained CDPMs, are all problems that motivate us to carry out the research. Unlike the pursuit of general-purpose algorithms in most studies, the innovation of this paper is characterized as an “application-oriented integrated innovation.” Guided by the need to solve the specific problem of soft support for a full-scale aircraft model in wind tunnel flutter tests, this work, for the first time, systematically integrates the kinematic solving, static stability criterion, mechanical interference constraints, and workspace analysis for under-constrained CDPMs into a unified analytical framework. The ultimate goal of this framework is not to propose a universally applicable theory, but rather to provide a closed-loop solution directly applicable to experimental design. This ensures that the designed support system is not only kinematically feasible but also statically stable and safe. The specific methodological improvements described below are all organic components supporting this complete solution. The contributions presented in this paper are summarized as follows:

(1) An improved intelligent algorithm is proposed to deal with the geometric-static coupling problem and the constraints on cable tension, and equilibrium poses can be determined for the design of the under-constrained CDPMs.

(2) Based on the dynamic equations of the under-constrained CDPMs, the first Lyapunov method from nonlinear system stability theory is employed to assess the equilibrium stability of the system using the eigenvalues of the system stiffness matrix, and sufficient stability conditions are derived.

(3) Taking the wind tunnel aeroelastic testing of a model (end-effector) suspension system as the engineering context, a comprehensive analysis is conducted to evaluate the under-constrained CDPMs in terms of their static feasible workspace and statically stable workspace.

The structure of the paper is as follows: Section 2 describes the geometric-static model of the under-constrained CDPMs. Section 3 introduces the improved algorithm for kinematics solving. Section 4 elaborates on the derivation of the system stiffness matrix and the stability evaluation method. Section 5 presents numerical simulation and analysis. Finally, Section 6 summarizes the main achievements of the paper. For clarity, the nomenclature is provided in Abbreviations.

2. Geometric-Static Model

A cable-driven parallel mechanism with δ cables (δ < 6) is typically categorized as under-constrained. In the context of flutter testing, this study employs a four-cable CDPM to suspend an aircraft model, as illustrated in Figure 1. Each cable i (where i = 1, 2, …, δ) extends from an anchor point $B_i$ on the fixed base frame to an attachment point $P_i$ on the end-effector (an aircraft model). The pose of the end-effector is controlled by adjusting the lengths of the cables. To simplify the kinematic modeling of the suspension system, the pulleys mounted on the fixed frame are idealized as frictionless revolute joints in a static reference frame, and their physical diameters are neglected. Furthermore, following the assumption commonly adopted in geometric statics analyses of CDPMs [23], the cables are modeled as massless and inextensible. This approximation is justified for systems with short cable spans and high elastic moduli, where elastic deformation has negligible impact on pose accuracy.

As shown in Figure 2, a fixed coordinate system $O_g\text{-}{X}_g{Y}_g{Z}_g$ is defined with respect to the base frame, and a body-fixed coordinate system $O_b\text{-}{X}_b{Y}_b{Z}_b$ is attached to the platform (aircraft model), thus enabling the establishment of the kinematic model of the system.

The cable length vector is:

$${\mathit{l}}_i={\mathit{{\rm B}}}_i-\left({\mathit{X}}_P+\mathit{R}{\mathit{x}}_{P_i}\right),i=1,2,\cdots ,\delta$$

(1)

Here, $B_i$ represents the coordinates of point $B_i$ in the fixed coordinate system, $X_P{\left(x^g,y^g,z^g\right)}^T$ is the coordinates of origin $O_b$ of the body-fixed coordinate system in the fixed coordinate system, $x_{P_i}{\left(x^b,y^b,z^b\right)}^T$ represents the coordinates of the point $P_i$ in the body-fixed coordinate system, $R=\left[\begin{array}{ccc}{R}_{11}& R_{12}& R_{13}\\ R_{21}& R_{22}& R_{23}\\ R_{31}& R_{32}& R_{33}\end{array}\right]$ is the rotation transformation matrix for coordinate conversion. Then the calculation of the i-th cable length can be expressed as Equation (2).

$$l_i=\sqrt{l_i^T{l}_i}$$

(2)

Let $l_i\left(i=1,2,\cdots ,\delta \right)$ be the scalar length of the i-th cable.

Since the platform has 6-DOF, its posture is ultimately determined by the equilibrium laws, namely:

$$J^{T}T+W=\mathbf{0}$$

(3)

Here $J$ is the mechanism’s Jacobian matrix, $J={\left[\begin{array}{c}{u}_1 \cdots u_{\delta }\\ r_1\times u_1\cdots r_{\delta }\times u_{\delta }\end{array}\right]}^T$, $T$ is the cable tension vector, $u_i$ is the i-th unit vector along the cable pointing toward the fixed base, and $r_i$ is the position vector of anchor point on the end-effector with respect to the center point $O_b$, $W$ is denoted as an external wrench, and in static equilibrium, only gravity exists.

Generally, the cable is only subjected to tension, and the tension on the cable is only effective when it is positive.

$$T_{max}\ge T_i\ge T_{min}>0\left(i=1,\dots ,\delta \right)$$

(4)

Here $T_{\min }$ and $T_{\max }$ are, respectively, the minimum and maximum tension values of the cable. The configuration change of the under-constrained CDPMs due to slack cable is not considered.

It is well-established that the kinematics of the under-constrained CDPMs is inherently coupled. Specifically, even when the cable lengths are prescribed, the end-effector may still undergo displacements induced by external forces or moments. Therefore, as shown in Equations (5) and (6), the posture of the moving platform is determined by both geometric and static relationships, given the prescribed cable lengths. Equations (5) and (6) amount to a total of (δ + 6) scalar relations that involve (2δ + 6) variables, namely, six components of the moving platform pose $X={\left(x^g,y^g,z^g,\varphi ,\theta ,\psi \right)}^T$, the cable lengths $l_i$ and the cable tensions $T_i$, where $\varphi$, $\theta$, and $\psi$ represent roll angle, pitch angle, and yaw angle, respectively. For the direct geometric-static problem, once δ variables concerning the cable lengths are assigned, the cable tensions and the platform pose could be solved. Similarly, as to the inverse geometric-static problem, δ variables concerning the platform pose need to be assigned to solve the cable lengths and tensions.

3. Intelligent Algorithm for Solving the Geometric-Static Problem

The kinematics problem of the under-constrained CDPMs involves dealing with the relationships among platform pose, cable lengths, and cable tensions, which must always be maintained in tension. Given the coupling of geometric-static equations, they need to be solved simultaneously. In essence, it is a problem of solving nonlinear multivariable algebraic equations with constraints.

As described in Section 1, most methods used in the literature focus on finding all possible kinematics solutions. Methods are further needed to identify the unique actual pose of the platform among all the real solutions. Numerical iterative methods are usually sensitive to the choice of initial values and the nature of constraint equations. Intelligent algorithms have the capabilities of global optimization and strong versatility, and they have been proposed to address the direct kinematics of parallel mechanisms. Particle swarm optimization (PSO) [27], a swarm intelligence-based global optimization algorithm, was first proposed by Kennedy and Eberhart in 1995. In this algorithm, the position of each particle represents a potential solution in the solution space, and particles gradually approach the global optimum by dynamically adjusting their velocities. With this in mind, and to avoid falling into a local optimum, an improved PSO algorithm is used here to solve the direct and inverse kinematics problems of the mechanism. On the one hand, the algorithm incorporates mechanisms inspired by the natural evolutionary process (e.g., from genetic algorithms (GA)) to enhance the search for the optimal individual (personal best). On the other hand, an improved method for calculating the inertia weight is proposed based on the current fitness.

When using the aforementioned intelligent optimization algorithms to solve the geometric-static equations, a transformation is required. Specifically, the cable length Equation (2) and the static equilibrium Equation (3) are, respectively, transformed into Equations (5) and (6).

$$\begin{array}{ll}{f}_i& ={l_i}^2-{\left(x_{B_i}^g-x^g-R_{11}{x}_i^b-R_{12}{y}_i^b-R_{13}{z}_i^b\right)}^2-{\left(y_{B_i}^g-y^g-R_{21}{x}_i^b-R_{22}{y}_i^b-R_{23}{z}_i^b\right)}^2-{\left(z_{B_i}^g-z^g-R_{31}{x}_i^b-R_{32}{y}_i^b-R_{33}{z}_i^b\right)}^2\\ & ={l_i}^2-A_{i1}^2\left(x^g,\varphi ,\theta ,\psi \right)-A_{i2}^2\left(y^g,\varphi ,\theta ,\psi \right)-A_{i3}^2\left(z^g,\varphi ,\theta ,\psi \right)=0\hspace{1em}\hspace{1em}\left(i=1,2,\cdots ,\delta \right)\end{array}$$

(5)

$$\left[\begin{array}{c}{f}_{\delta +1}\\ f_{\delta +2}\\ f_{\delta +3}\\ f_{\delta +4}\\ f_{\delta +5}\\ f_{\delta +6}\end{array}\right]=\left[\begin{array}{c}{\displaystyle \sum _{i=1}^4{A}_{i1}\left(x^g,\varphi ,\theta ,\psi \right)\frac{T_i}{l_i}}=l_{1x^g}{\displaystyle \frac{T_1}{l_1}}+l_{2x^g}{\displaystyle \frac{T_2}{l_2}}+l_{3x^g}{\displaystyle \frac{T_3}{l_3}}+l_{4x^g}{\displaystyle \frac{T_4}{l_4}}=0\\ {\displaystyle \sum _{i=1}^4{A}_{i2}\left(y^g,\varphi ,\theta ,\psi \right)\frac{T_i}{l_i}}=l_{1y^g}{\displaystyle \frac{T_1}{l_1}}+l_{2y^g}{\displaystyle \frac{T_2}{l_2}}+l_{3y^g}{\displaystyle \frac{T_3}{l_3}}+l_{4y^g}{\displaystyle \frac{T_4}{l_4}}=0\\ {\displaystyle \sum _{i=1}^4{A}_{i3}\left(z^g,\varphi ,\theta ,\psi \right)\frac{T_i}{l_i}}=l_{1z^g}{\displaystyle \frac{T_1}{l_1}}+l_{2z^g}{\displaystyle \frac{T_2}{l_2}}+l_{3z^g}{\displaystyle \frac{T_3}{l_3}}+l_{4z^g}{\displaystyle \frac{T_4}{l_4}}-mg=0\\ {\displaystyle \sum _{i=1}^4{H}_{i1}\left(y^g,z^g,\varphi ,\theta ,\psi \right)\frac{T_i}{l_i}}=r_{1x^g}\times l_{1x^g}{\displaystyle \frac{T_1}{l_1}}+r_{2x^g}\times l_{2x^g}{\displaystyle \frac{T_2}{l_2}}+r_{3x^g}\times l_{3x^g}{\displaystyle \frac{T_3}{l_3}}+r_{4x^g}\times l_{4x^g}{\displaystyle \frac{T_4}{l_4}}=0\\ {\displaystyle \sum _{i=1}^4{H}_{i2}\left(x^g,z^g,\varphi ,\theta ,\psi \right)\frac{T_i}{l_i}}=r_{1y^g}\times l_{1y^g}{\displaystyle \frac{T_1}{l_1}}+r_{2y^g}\times l_{2y^g}{\displaystyle \frac{T_2}{l_2}}+r_{3y^g}\times l_{3y^g}{\displaystyle \frac{T_3}{l_3}}+r_{4y^g}\times l_{4y^g}{\displaystyle \frac{T_4}{l_4}}=0\\ {\displaystyle \sum _{i=1}^4{H}_{i3}\left(x^g,y^g,\varphi ,\theta ,\psi \right)\frac{T_i}{l_i}}=r_{1z^g}\times l_{1z^g}{\displaystyle \frac{T_1}{l_1}}+r_{2z^g}\times l_{2z^g}{\displaystyle \frac{T_2}{l_2}}+r_{3z^g}\times l_{3z^g}{\displaystyle \frac{T_3}{l_3}}+r_{4z^g}\times l_{4z^g}{\displaystyle \frac{T_4}{l_4}}=0\end{array}\right]=\left[\begin{array}{c}{\displaystyle \sum _{i=1}^4{A}_{i1}\left(x^g,\varphi ,\theta ,\psi \right)\frac{T_i}{l_i}}=0\\ {\displaystyle \sum _{i=1}^4{A}_{i2}\left(y^g,\varphi ,\theta ,\psi \right)\frac{T_i}{l_i}}=0\\ {\displaystyle \sum _{i=1}^4{A}_{i3}\left(z^g,\varphi ,\theta ,\psi \right)\frac{T_i}{l_i}}=0\\ {\displaystyle \sum _{i=1}^4{H}_{i1}\left(y^g,z^g,\varphi ,\theta ,\psi \right)\frac{T_i}{l_i}}=0\\ {\displaystyle \sum _{i=1}^4{H}_{i2}\left(x^g,z^g,\varphi ,\theta ,\psi \right)\frac{T_i}{l_i}}=0\\ {\displaystyle \sum _{i=1}^4{H}_{i3}\left(x^g,y^g,\varphi ,\theta ,\psi \right)\frac{T_i}{l_i}}=0\end{array}\right]$$

(6)

Here all coefficients $A_{i1}$, $A_{i2}$, $A_{i3}$ and $H_{i1}$, $H_{i2}$, $H_{i3}$ are functions of $\left(x^g,y^g,z^g,\varphi ,\theta ,\psi \right)$. We define the fitness function as Equation (7).

$$\min Q\left(x\right)={\omega }_l{\displaystyle \sum _{i=1}^{\delta }{\left(\frac{\sqrt{f_i}}{l_0}\right)}^2+{\omega }_f{{\displaystyle \sum _{i=\delta +1}^{\delta +3}{\left(\frac{f_i}{G}\right)}^2+{\omega }_{\tau }{\displaystyle \sum _{i=\delta +4}^{\delta +6}\left(\frac{f_i}{G{l}_0}\right)}}}^2+\mu {\displaystyle \sum _{j=1}^{\delta }{\left(\frac{\min \left(0,T_j\right)}{G}\right)}^2}}$$

(7)

Here, $l_0$ is the maximum cable length, $G$ is the platform’s weight, $\mu$ is a Lagrange factor, ${\omega }_l$, ${\omega }_f$ and ${\omega }_{\tau }$ represent the weight coefficients for each part, used to adjust the importance of various types of errors. When $Q$ meets the requirement of residual error, the optimal solution is obtained.

Figure 3 shows the flowchart of geometric-static problems solving, and the key is to transform it into an optimization problem.

In an N-dimension space, the swarm is represented as $S=\left\{X_1,X_2,\cdots ,X_{M^{\prime }}\right\}$ with $M^{\prime }$ particles, where $X_{\eta }=\left(x_{\eta 1},x_{\eta 2},\cdots ,x_{\eta \mathrm{N}}\right)$ and $V_{\eta }=\left(v_{\eta 1},v_{\eta 2},\cdots ,v_{\eta \mathrm{N}}\right)$ are, respectively, recorded as the position and velocity of the η-th particle. The best solution found by the current particle is denoted as $P_{best}$, and the global best solution found by the entire swarm is denoted as $G_{best}$. For each generation, the velocity and the position of η-th particle will be updated by the following equations:

$$V_{\eta }(k+1)=\omega V_{\eta }(k)+c_1{\delta }_1[P_{\mathit{best}}(k)-P_{\eta }(k)]+c_2{\delta }_2[G_{\mathit{best}}(k)-P_{\eta }(k)]$$

(8)

$$P_{\eta }(k+1)=P_{\eta }(k)+V_{\eta }(k+1)$$

(9)

Here, $V_{\eta }$ represents the velocity of the η-th particle at generation $k$, ${\delta }_1$ and ${\delta }_2$ are random numbers in the interval (0, 1), and $P_{\eta }$ represents the position of the η-th particle at generation $k$. On the right-hand side of Equation (8), the first part represents the momentum term, where $\omega$ is the inertia weight coefficient, indicating the extent to which the previous velocity of the particle affects its current velocity. The second part represents the cognitive component, reflecting the particle’s self-cognition, with $c_1$ being the self-learning factor that determines the difference between the individual best position and the current position to guide the direction of particle movement. The third part represents the social component, where $c_2$ is the social-learning factor, indicating that particles move towards the global best position $G_{best}$ through information sharing among them. Typically, $c_1$ and $c_2$ take values between 0 and 2.

For the velocity update formula in Equation (8), a large $\omega$ leads to strong global optimization ability, while a small $\omega$ leads to strong local search capability. Thus, the inertia weight coefficients become the key parameters to balance global and local search in the PSO algorithm. The nonlinear dynamic inertia weight coefficients used are expressed in Equation (10).

$$\omega =\left\{\begin{array}{c}{\omega }_{\mathit{min}}+{\displaystyle \frac{({\omega }_{\mathit{max}}-{\omega }_{\mathit{min}}\left)\right(Q-Q_{\mathit{min}})}{Q_{\mathit{avg}}-Q_{\mathit{min}}}}, Q<Q_{\mathit{avg}}\\ {\omega }_{\mathit{max}},\hspace{1em}\hspace{1em}Q>Q_{\mathit{avg}}\end{array}\right.$$

(10)

Here, ${\omega }_{\max }$ and ${\omega }_{\min }$ are the maximum and minimum values of $\omega$, respectively; $Q$ is the fitness value of the function, $Q_{\min }$ represents its minimum value, and $Q_{avg}=\left({\displaystyle \sum _{i=1}^{M^{\prime }}{Q}_{\eta }}\right){\displaystyle /}{M}^{\prime }$ represents the average fitness value.

To enhance the randomness and flexibility of the described algorithm and to overcome local optima, a combination of chaotic behavior and GA crossover mutation is introduced into the PSO algorithm after each iteration. The chaotic process of the particle swarm is implemented using the Tent equation [28]. This involves normalizing the global best position to the domain [0, 1] of the Tent equation, then generating chaotic sequences for each particle’s position independently. These chaotic sequences are subsequently mapped back to the original solution space to obtain the post-chaos particle positions, and the fitness of these particles is calculated. Finally, the best particle after chaos is randomly selected to replace one of the particles in the original swarm, introducing chaotic factors into the particle swarm and increasing the diversity of the search within the solution space. The normalization formula and the Tent equation are shown in Equations (11) and (12), where $\epsilon$ represents the breakpoint.

$${\mathrm{\Lambda }}_{norm}={\displaystyle \frac{\mathrm{\Lambda }-{\mathrm{\Lambda }}_{min}}{{\mathrm{\Lambda }}_{max}-{\mathrm{\Lambda }}_{min}}}$$

(11)

$${\kappa }_{t+1}=\left\{\begin{array}{l}{\displaystyle \frac{{\kappa }_t}{\epsilon }},\hspace{1em}0⩽{\kappa }_t<\epsilon ;\\ {\displaystyle \frac{1-{\kappa }_t}{1-\epsilon }},\hspace{1em}\epsilon ⩽{\kappa }_t⩽1.\end{array}\right.$$

(12)

Here, $\mathrm{\Lambda }$ represents the original data points, ${\mathrm{\Lambda }}_{\min }$ and ${\mathrm{\Lambda }}_{\max }$ are the minimum and maximum values in the dataset, respectively, and ${\mathrm{\Lambda }}_{norm}$ is the normalized data point, ${\kappa }_t$ denotes the normalized coordinates of the particle.

The flowchart of the improved algorithm is shown in Figure 4.

4. Equilibrium Stability Analysis

For stability evaluation, it is essential to determine the equilibrium configurations corresponding to known cable lengths and static external wrenches, which aligns with the solutions to the direct geometric-static problem. Since the under-constrained CDPMs are more easily disturbed than the redundant case, whether the equilibrium is stable or has the tendency to return to the original pose when subjected to external disturbances needs to be further discussed in detail. Therefore, it is of great importance to address the stability analysis to avoid undesirable situations. To this end, a stability measure based on Lyapunov theory is proposed for the under-constrained CDPMs, and the sufficient conditions will also be presented in the following section.

The equilibrium stability analyzed above is actually static stability, which is concerned with the system’s behavior at the equilibrium point when disturbed, and whether the system automatically produces forces and moments which tend to reduce the disturbance.

The dynamic equations for the under-constrained CDPMs can be expressed in matrix form based on the Newton-Euler method.

$$M\ddot{X}+N\left(\dot{X}\right)-W_g-W_e=-J^{T}T$$

(13)

Here, $M$ represents the mass matrix of the end-effector, which is assumed to be constant in this analysis. $N\left(\dot{X}\right)$ denotes the nonlinear Coriolis and centrifugal force vector, $W_g$ is the gravity vector of the end-effector, and $W_e$ represents the external wrench applied to the end-effector. $X={\left(x^g,y^g,z^g,\varphi ,\theta ,\psi \right)}^T$ is the set of poses of the end-effector, let $x_1=X$, $x_2=\dot{X}$. Then, the state-space equations can be expressed as Equation (14).

$$\dot{x}=f=\left[\begin{array}{l}{\mathbf{0}}_{6\times 6}\hspace{1em}{E}_{6\times 6}\\ {\mathbf{0}}_{6\times 6}\hspace{1em}{\mathbf{0}}_{6\times 6}\end{array}\right]x+\left[\begin{array}{l}{\mathbf{0}}_{6\times 1}\\ U_{6\times 1}\end{array}\right]$$

(14)

Here, $x={\left[x_1^T,x_2^T\right]}^T$ and $U_{6\times 1}=M^{-1}\left(-J^{T}T-N+W_g+W_e\right)$, $E_{6\times 6}$ is the identity matrix. $N$ has the general form $\omega \times \left({\rm I}\omega \right)$, where $\omega$ represents the angular velocity, and ${\rm I}$ is the inertia matrix.

Then, the nonlinear system’s Jacobian matrix could be expressed as Equation (15).

$${\displaystyle \frac{\partial f}{\partial x}}=\left[\begin{array}{l}{\mathbf{0}}_{6\times 6}\hspace{1em}{E}_{6\times 6}\\ {\mathbf{0}}_{6\times 6}\hspace{1em}{\mathbf{0}}_{6\times 6}\end{array}\right]+\left[\begin{array}{l}{\mathbf{0}}_{6\times 12}\\ {\displaystyle \frac{\partial U}{\partial x}}\end{array}\right]$$

(15)

The term of ${\displaystyle \frac{\partial U}{\partial x}}$ could be further divided into two parts:

$${\displaystyle \frac{\partial U}{\partial x}}=\left[{\displaystyle \frac{\partial U}{\partial x_1}}\hspace{1em}{\displaystyle \frac{\partial U}{\partial x_2}}\right]$$

(16)

$${\displaystyle \frac{\partial U}{\partial x_1}}=M^{-1}\left[-{\displaystyle \frac{\partial J^{\mathrm{T}}T}{\partial X}}-{\displaystyle \frac{\partial N}{\partial X}}+{\displaystyle \frac{\partial W_g}{\partial X}}+{\displaystyle \frac{\partial W_e}{\partial X}}\right]$$

(17)

$${\displaystyle \frac{\partial U}{\partial x_2}}=M^{-1}\left[-{\displaystyle \frac{\partial N}{\partial \dot{X}}}+{\displaystyle \frac{\partial W_e}{\partial \dot{X}}}\right]$$

(18)

Here, ${\displaystyle \frac{\partial W_g}{\partial X}}$, ${\displaystyle \frac{\partial N}{\partial X}}$, ${\displaystyle \frac{\partial N}{\partial \dot{X}}}$ are always zero matrix at the equilibrium state, and the derivative of external forces ${\displaystyle \frac{\partial W_e}{\partial X}}$ and ${\displaystyle \frac{\partial W_e}{\partial \dot{X}}}$ are not considered, the nonlinear system’s Jacobian matrix becomes ${\displaystyle \frac{\partial f}{\partial x}}=\left[\begin{array}{cc}{\mathbf{0}}_{6\times 6}& E_{6\times 6}\\ -M^{-1}{\displaystyle \frac{\partial J^{T}T}{\partial X}}& -{\displaystyle \frac{\partial N}{\partial \dot{X}}}\end{array}\right]$.

This study analyzes the stability of the flying model (moving platform) under the assumption of zero pose and symmetric mechanism configuration. Under these conditions, the body-fixed coordinate frame aligns with the principal axes of inertia of the platform, resulting in a diagonal inertia matrix. The time derivative of the angular momentum, expressed as ${\displaystyle \frac{\partial N}{\partial \omega }}=\left[\begin{array}{ccc}{I}_x& {\omega }_z\left(I_z-I_y\right)& {\omega }_y\left(I_z-I_y\right)\\ {\omega }_z\left(I_x-I_z\right)& I_y& {\omega }_x\left(I_x-I_z\right)\\ {\omega }_y\left(I_y-I_x\right)& {\omega }_x\left(I_y-I_x\right)& I_z\end{array}\right]$, where $I_x$, $I_y$, and $I_z$ denote the moments of inertia about the O_bX_b-axis, O_bY_b-axis, and O_bZ_b-axis, respectively, is obviously positive-definite at the equilibrium state ($\omega =0$.) According to the first Lyapunov method of stability theory for the nonlinear system, the system is stable if and only if the real parts of the eigenvalues of matrix ${\displaystyle \frac{\partial f}{\partial x}}$ are negative. This condition implies that the system stiffness matrix ${\displaystyle \frac{\partial J^{T}T}{\partial X}}$ must be positive-definite. This conclusion is consistent with the definiteness requirements on the matrix in reference [21], that is, the stability of the under-constrained CDPM requires the system stiffness matrix $K$ to be a positive definite matrix [19]. It is known that ${\displaystyle \frac{\partial J^{T}T}{\partial X}}$ can be written in the form of Equation (19).

$$K={\displaystyle \frac{\partial J^{\mathrm{T}}T}{\partial X}}=J^T{K}_{c}J+K_s$$

(19)

Here, the first term is the passive stiffness, which is always a positive definite matrix, where $K_c$ denotes the cable stiffness; the second term is the active stiffness $K_s$. Further clarification is as follows [19]:

$$\left\{\begin{array}{l}{\mathit{K}}_c=diag\left(v\right)\\ {\mathit{K}}_s={\mathit{K}}_1+{\mathit{K}}_2\\ {\mathit{K}}_1={\displaystyle \sum _{i=1}^{\delta }\frac{T_i}{L_{\mathrm{i}}}\left[\begin{array}{cc}{\mathit{I}}_{3\times 3}-{\mathit{u}}_i{\mathit{u}}_i^T& {[{\mathit{r}}_i\times ]}^T-{\mathit{u}}_i{\mathit{u}}_i^T{[{\mathit{r}}_i\times ]}^T\\ [{\mathit{r}}_i\times ]-[{\mathit{r}}_i\times ]{\mathit{u}}_i{\mathit{u}}_i^T& [{\mathit{r}}_i\times ]{[{\mathit{r}}_i\times ]}^T-[{\mathit{r}}_i\times ]{\mathit{u}}_i{u}_i^T{[{\mathit{r}}_i\times ]}^T\end{array}\right]}\\ {\mathit{K}}_2=-{\displaystyle \sum _{i=1}^{\delta }{T}_i}\left[\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& [{\mathit{u}}_i\times ][{\mathit{r}}_i\times ]\end{array}\right]\end{array}\right.$$

(20)

Here, $v=\left[\begin{array}{cccc}{\displaystyle \frac{k_c}{l_1}}& {\displaystyle \frac{k_c}{l_2}}& \dots & {\displaystyle \frac{k_c}{l_{\delta }}}\end{array}\right]$, where $k_c$ is the unit stiffness of the cable and $l_i\left(i=1,\cdots ,\delta \right)$ is the length of the i-th cable. The skew-symmetric matrices $\left[u_i\times \right]$ and $\left[r_i\times \right]$ denote the cross-product operators associated with vectors $u_i$ and $r_i$, respectively. The first term $K_1$ is symmetric and positive definite if all the cable tensions are taut. Generally, if external loads are applied to the end-effector, the second term $K_2$ becomes asymmetric, and its eigenvalues cannot be determined. However, $K_2$ is symmetric only when the external torque acting on the end-effector (platform) is zero, since $\left[{\displaystyle \sum _{i=1}^{\delta }{T}_i\left(u_i\times r_i\right)}\right]=\mathbf{0}$, and:

$${\mathit{K}}_2-{\mathit{K}}_2^{\mathrm{T}}=\left[\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \left[{\displaystyle \sum _{i=1}^{\delta }(T_i{\mathit{u}}_i\times {\mathit{r}}_i)}\right]\times \end{array}\right]=\mathbf{0}$$

(21)

If the stiffness matrix is asymmetric, the judgment of the equilibrium stability can be based on its eigenvalues, which can be easily calculated.

It is worth mentioning that when the external wrench is nonzero, $K_2$ and $K$ are both asymmetric; the only criterion for determining stability is then based on the eigenvalues of the complete stiffness matrix.

5. Numerical Examples and Analysis

For the kinematic solution and stability analysis of the workspace, a special spatial under-constrained CDPM with four cables is considered, as shown in Figure 5. An aircraft model serves as the moving platform. This mechanism could be used to support the aircraft model for wind tunnel tests, such as the full-model flutter test, which needs a soft suspension with the natural frequency as small as possible to determine the model’s critical flutter velocity. The geometric parameters of the mechanism are given in

Table 1. Coordinates of cable connections. 

Bi	Coordinates	Pi	Coordinates
B1	(−0.710, 0.000, −0.985) m	P1	(−0.113, 0.000, −0.020) m
B2	(−0.710, 0.000, −0.385) m	P2	(−0.113, 0.000, 0.020) m
B3	(0.660, −0.500, −0.735) m	P3	(0.050, −0.022, 0.000) m
B4	(0.660, 0.500, −0.735) m	P4	(0.050, 0.022, 0.000) m

. In the base frame, the initial position of the aircraft model is (0, 0, −0.685) m.

5.1. Geometric-Static Solution and Analysis

In order to validate the effectiveness of the proposed method, both inverse and direct kinematics problems were solved. An example of a four-cable mechanism is presented below.

To satisfy the compatibility between the number of equations and unknown parameters, four of the variables were specified, including the pitch angle and three coordinates of the center of mass, so that the corresponding equilibrium points could be found. In solving an inverse geometric-static problem, the initial orientation of the moving platform was set to (0°, 0°, 0°), and the coordinates of the center of mass were set to (0, 0, −0.685) m. A continuous motion solution was then obtained for the pitch direction ranging from 1° to 10°. Subsequently, using the improved PSO algorithm, the position parameters along the other two coordinate axes, the lengths of the four cables, and the cable tensions were calculated. The relevant parameters were as follows: the mass of the moving platform was 1.06 kg, with its center of mass located at the origin of the body-fixed coordinate system; the principal moments of inertia were I_x = 0.000684 kg·m², I_y = 0.000852 kg·m² and I_z = 0.000846 kg·m²; the learning factors $c_1$ and $c_2$ were both set to 2; the numbers of particles and iterations were 500 and 2000, respectively; and the inertial weights were ${\omega }_{\max }=0.8$ and ${\omega }_{\min }=0.4$. As shown in

Table 2. Solutions for inverse kinematic problem. 

$\mathit{X}/\mathbf{m}$	$\mathit{Y}/\mathbf{m}$	${\mathit{l}}_1/\mathbf{m}$	${\mathit{l}}_2/\mathbf{m}$	${\mathit{l}}_3/\mathbf{m}$	${\mathit{l}}_4/\mathbf{m}$	${\mathit{T}}_1/\mathbf{N}$	${\mathit{T}}_2/\mathbf{N}$	${\mathit{T}}_3/\mathbf{N}$	${\mathit{T}}_4/\mathbf{N}$	$\mathit{Q}$
0	0	0.7651	0.7670	0.7766	0.7766	104.0	83.9	93.2	93.0	1.4 × 10−5
0	0	0.7642	0.7680	0.7767	0.7767	88.2	64.3	75.7	75.7	2.4 × 10−6
0	0	0.7633	0.7691	0.7768	0.7768	77.3	50.9	63.7	63.7	3.8 × 10−8
0	0	0.7625	0.7702	0.7769	0.7769	69.4	41.1	54.9	54.9	5.3 × 10−5
0	0	0.7617	0.7713	0.7770	0.7770	63.4	33.7	48.3	48.3	4.8 × 10−6
0	0	0.7609	0.7724	0.7772	0.7772	58.7	27.8	43.1	43.1	9.8 × 10−7
0	0	0.7602	0.7736	0.7773	0.7773	54.9	23.1	38.9	38.9	2.2 × 10−7
0	0	0.7595	0.7748	0.7774	0.7774	51.7	19.2	35.4	35.4	4.7 × 10−8
0	0	0.7588	0.7761	0.7776	0.7776	49.0	15.9	32.5	32.5	3.3 × 10−5
0	0	0.7582	0.7773	0.7778	0.7778	46.5	13.1	30.0	30.0	1.1 × 10−5

, the inverse kinematics solutions of 10 different equilibrium configurations were listed, all of which satisfied x = y = 0 m and had sufficiently small fitness values, thus demonstrating the feasibility of the proposed algorithm.

After determining the cable lengths, the proposed optimization method can also be employed to solve the direct geometric-static problem.

Table 3. Solutions for direct kinematic problem. 

$\mathit{R}\mathit{o}\mathit{l}\mathit{l}/°$	$\mathit{P}\mathit{i}\mathit{t}\mathit{c}\mathit{h}/°$	$\mathit{Y}\mathit{a}\mathit{w}/°$	$\mathit{X}/\mathbf{m}$	$\mathit{Y}/\mathbf{m}$	$\mathit{Z}/\mathbf{m}$	${\mathit{T}}_1/\mathbf{N}$	${\mathit{T}}_2/\mathbf{N}$	${\mathit{T}}_3/\mathbf{N}$	${\mathit{T}}_4/\mathbf{N}$	$\mathit{Q}$
1 × 10−12	1	1 × 10−12	−3 × 10−11	−2 × 10−21	−0.685	104.0	83.9	93.2	93.0	1.4 × 10−6
1 × 10−12	2	1 × 10−12	−5 × 10−12	−3 × 10−17	−0.685	88.2	64.3	75.7	75.7	2.1 × 10−7
4 × 10−10	3	5 × 10−9	7 × 10−17	1 × 10−15	−0.685	77.3	50.9	63.7	63.7	3.7 × 10−9
1 × 10−12	4	1 × 10−12	−2 × 10−10	−7 × 10−18	−0.685	69.4	41.1	54.9	54.9	5.0 × 10−6
1 × 10−12	5	1 × 10−12	−1 × 10−11	−3 × 10−18	−0.685	63.4	33.7	48.3	48.3	3.4 × 10−7
1 × 10−12	6	1 × 10−12	−7 × 10−13	−4 × 10−17	−0.685	58.7	27.8	43.1	43.1	2.1 × 10−8
4 × 10−11	7	3 × 10−10	−6 × 10−14	6 × 10−17	−0.685	54.9	23.1	38.9	38.9	2.2 × 10−8
1 × 10−12	8	1 × 10−12	−4 × 10−16	3 × 10−19	−0.685	51.7	19.2	35.4	35.4	2.3 × 10−11
1 × 10−12	9	2 × 10−12	−1 × 10−13	2 × 10−19	−0.685	49.0	15.9	32.5	32.5	3.3 × 10−6
1 × 10−12	10	1 × 10−12	8 × 10−13	3 × 10−17	−0.685	46.5	13.1	30.0	30.0	3.7 × 10−8

shows the results of the direct kinematics problem using the cable lengths from Table 2. As shown in Table 3, the results are in good agreement with the given initial parameters in inverse kinematic problem.

The results in each row of Table 3 correspond to the cable length inputs listed in the same row of Table 2, and the pitch angle is consistent with the preset equilibrium configuration, verifying the correctness of the solver.

It is worth noting that the cable tension constraints have a major impact on the solutions to the direct and inverse kinematics problems. For the cases shown above, the cable tensions are all constrained to be positive ($T_i>0$), but when they are confined to the range $\left[T_{\min },T_{\max }\right]$, the number of feasible solutions decreases due to the strong constraints. Furthermore, if the range of the cable tension constraints is too restrictive (e.g., the upper limit is too low or the lower limit is too high relative to the required forces), there will be no feasible solution.

To quantitatively evaluate the effectiveness of the proposed hybrid GA + PSO + CHAOs algorithm, a comparative study was conducted against the standard PSO algorithm [27]. All parameter settings relevant to the inverse kinematics solution were kept identical between the two approaches, except that the inertia weight in the standard PSO was fixed at a constant value. For each algorithm, independent runs were performed using randomly initialized populations. The fitness value Q was adopted as the metric to assess solution accuracy. As shown in

Table 4. Performance Comparison of the Algorithm. 

Pitch Angle (°)	Q (Standard PSO)	Q (GA + PSO + CHAOs)
1	2.85 × 10−4	1.4 × 10−5
2	2.37 × 10−4	2.4 × 10−6
3	2.01 × 10−6	3.8 × 10−8
4	1.05 × 10−4	5.3 × 10−5
5	1.72 × 10−5	4.8 × 10−6
6	1.63 × 10−5	9.8 × 10−7
7	6.99 × 10−6	2.2 × 10−7
8	3.37 × 10−6	4.7 × 10−8
9	3.09 × 10−4	3.3 × 10−5
10	7.17 × 10−4	1.1 × 10−5

, the results demonstrate that the hybrid GA + PSO + CHAOs algorithm exhibits superior robustness and global search capability when addressing the highly nonlinear and constraint-coupled geometric-static problems inherent in under-constrained cable-driven parallel manipulators. This provides a research foundation for optimizing complex, high-dimensional, and dynamic research objects, such as redundant constraint models, in the future.

Furthermore, to further validate the superiority of the proposed hybrid GA-PSO-CHAOs algorithm in solving such strongly coupled, multi-constrained geometric-static problems, a quantitative comparison with a currently advanced metaheuristic algorithm, the gray wolf optimizer (GWO) [29], has been added. Under identical parameter settings (population size, number of iterations, objective function), independent solutions were sought for the 10 pitch angle equilibrium configurations listed in Table 2. The statistical results indicate that, when converging to the same accuracy (Q < 1.0 × 10⁻⁵), the average number of iterations required by the algorithm presented in this paper is reduced by approximately 33% compared to GWO. Moreover, the proportion of successful feasible solutions found within 20 independent runs reached 98%, whereas the GWO algorithm has an approximate 17% probability of getting trapped in local optima. This quantitative comparison demonstrates that the algorithm presented in this paper is superior to the referenced advanced algorithm in terms of convergence speed, solution accuracy, and robustness, making it particularly suitable for the complex solution space search associated with under-constrained cable-driven parallel mechanisms.

5.2. Workspace and Static Stability Analysis

In this subsection, we perform workspace simulations and system static stability analysis for a special spatial under-constrained support configuration with four cables (CDPM-4). The wrench feasible workspace (WFW) is the most fundamental workspace for cable-driven parallel mechanisms, defined as a set of poses of the end-effector, where the mechanism can apply any external wrench while keeping the cables taut and avoiding interference between the cables and the end-effector. The static feasible workspace (SFW) is a special subset of the WFW, in which the wrench induced solely by the weight of the end-effector must be balanced by the counteracting wrenches generated by the cable tensions.

5.2.1. Set Initial Conditions

The support layout and model parameters were designed according to the experimental requirements, with the following specific parameter settings: The aircraft model (see Figure 5) has a mass of 1.06 kg, a wing reference area of 0.0199 m², an average aerodynamic chord length of 0.092 m, and moments of inertia about the center of mass of [0.000684, 0.000852, 0.000846] kg·m². According to Table 1, the initial cable lengths are [0.6594, 0.6594, 0.7766, 0.7766] m, and the initial pretension forces are [108.4802, 84.5109, 111.2222, 111.2222] N.

According to reference [30], after the aircraft model in a four-cable suspension system is subjected to aerodynamic forces, the displacement changes in the three translational directions are small. In contrast, the changes in the attitude angles are significant. Therefore, it is crucial to analyze the adjustment range of the support system’s attitude angle workspace. For the attitude angle workspace, the necessary condition for the cable tensions to have a solution must first be satisfied, i.e., when rank (J) = rank ([J W]), there exists a unique positive solution for the cable tensions. However, for the under-constrained parallel support mechanism shown in Figure 5, most poses do not meet this condition. Considering that active motion control of the aircraft model is not required during flutter testing, only the constraint that the cable tensions must be greater than zero needs to be satisfied. Thus, simulations are conducted based on the coordinates of points $B_i$ and $P_i$ in Table 1.

5.2.2. Interference Analysis

The workspace of the cable-driven parallel support mechanism is constrained by potential interferences among cables, between cables and the aircraft model, and between cables and the surrounding support structures, which limits the expansion of the workspace. Based on the mechanism configuration shown in Figure 5 and the cable anchor point coordinates listed in Table 1, interferences among cables and between cables and the surrounding support structures can be neglected. Therefore, the primary focus is on analyzing the potential interference between the cables and the aircraft model itself. Considering the external shape of the aircraft model used in this paper and the distribution of the cables, the interference between the cables and the aircraft model can be categorized into two main types: interference between the rear cables and the wings (as illustrated in Figure 6a), and interference between the front cables and the wings (as illustrated in Figure 6b).

The interference between the cable and the wing requires evaluating the angle between the cable and the plane in which the wing resides. This can be approximated by the angle α between the cable and the fuselage axis. Taking the rear cable $P_i{B}_i$ as an example, when interference occurs between the cable and the wing, the cable will move to a new position $P_i{B}_i^{\prime }$. At this moment, as shown in Figure 6a, the angle α between the vectors $\overrightarrow{P_i{B}_i}$ and $\overrightarrow{P_i{B}_i^{\prime }}$, representing the cable length, becomes zero. The mathematical expression for this condition is given by:

$$\alpha =\arccos \left({\displaystyle \frac{\overrightarrow{P_i{B}_i}\cdot \overrightarrow{P_i{B}_i^{\prime }}}{\left|\overrightarrow{P_i{B}_i}\right|\left|\overrightarrow{P_i{B}_i^{\prime }}\right|}}\right)=0$$

(22)

Similarly, when interference occurs between the front cable and the wing, as illustrated in Figure 6b, the angle β between the vectors $\overrightarrow{P_j{B}_j}$ and $\overrightarrow{P_j{B}_j^{\prime }}$, representing the cable length, also becomes zero. This condition is expressed as:

$$\beta =\arccos \left({\displaystyle \frac{\overrightarrow{P_j{B}_j}\cdot \overrightarrow{P_j{B}_j^{\prime }}}{\left|\overrightarrow{P_j{B}_j}\right|\left|\overrightarrow{P_j{B}_j^{\prime }}\right|}}\right)=0$$

(23)

This method can be employed to determine whether interference occurs at certain extreme positions within the workspace. By converting the interference judgment criterion into a computational algorithm and analyzing whether the sign of the positional relationship between the cable and the wing plane changes before and after an attitude variation, it can be determined whether the given attitude belongs to the feasible workspace of the mechanism.

5.2.3. ADAMS Workspace Simulation Analysis

Based on the initial conditions set above, the four-cable support system configuration shown in Figure 5 and the coordinate values of points $B_i$ and $P_i$ listed in Table 1, an ADAMS simulation model was created. When the aircraft model is subjected to aerodynamic forces, the displacements in the three translational directions are relatively small, while the displacements in the three rotational directions are larger (than those in the translational directions), indicating a significant change in attitude angles. Therefore, this section focuses on analyzing the adjustment range of the aircraft model’s attitude angles. According to the kinematic solution method presented in Section 2, the relationship between the required cable lengths and the attitude angles was calculated based on the static feasible workspace for attitude angles, resulting in curve functions (e.g., $\dot{\mathit{l}}=\mathit{J}{\dot{\mathit{X}}}_{\omega }$). These functions were then imported into ADAMS for motion control simulation.

$$\left\{\begin{array}{l}\dot{\mathit{l}}=\mathit{J}{\dot{\mathit{X}}}_{\omega }\\ {\dot{\mathit{X}}}_{\omega }={\left({\dot{x}}^g,{\dot{y}}^g,{\dot{z}}^g,{\omega }_x,{\omega }_y,{\omega }_z\right)}^T\end{array}\right.$$

(24)

Here, ${\left({\omega }_x,{\omega }_y,{\omega }_z\right)}^T$ is the angular velocity vector, which has the following relationship with the attitude angular velocity vector ${\left(\dot{\varphi },\dot{\theta },\dot{\psi }\right)}^T$:

$$\left(\begin{array}{c}{\omega }_x\\ {\omega }_y\\ {\omega }_z\end{array}\right)=\left[\begin{array}{ccc}\cos \theta \cos \psi & -\sin \psi & 0\\ \cos \theta \sin \psi & \cos \psi & 0\\ -\sin \theta & 0& 1\end{array}\right]\left[\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]$$

(25)

The simulation results illustrating the adjustment of the support system’s attitude angles are shown in Figure 7. Based on the geometric relationships and the calculated cable lengths, the results indicate that the four-cable suspension system designed in this study can achieve a maximum pitch angle of 23°, a maximum yaw angle of 22°, and a roll angle adjustment range exceeding 49°. Given the application of flutter testing, this attitude angle adjustment range can meet the attitude angle requirements for wind tunnel testing of special under constrained support models for space applications.

5.2.4. Static Feasible Workspace Analysis

Considering the experimental requirement for the mechanism to have a certain degree of attitude angle adjustment capability while avoiding interference between the cables and the aircraft model structure, the search range for attitude angles is set to [−50°, 50°]. Additionally, the cable tensions serve to balance external loads. To prevent excessive external loads that could cause cable breakage, deformation, or damage to the aircraft model, upper and lower limits for cable tensions are specified by setting the tension range to (0, 200) N, which maintains cable tension and prevents excessive loads. By performing exhaustive calculations on the discrete points with a step size of 1°, the attitude adjustment range is determined. The static feasible workspace for the attitude angles is obtained by substituting all parameters into the MATLAB R2024b program (which includes the interference judgment method from Section 5.2.2) for simulation analysis, as shown in Figure 8.

From the static reachable workspace shown in Figure 8 (defined by its boundary point coordinates), it can be seen that, under conditions avoiding structural component interference or excessive cable tension, the roll angle of the aircraft model can be safely adjusted within the range of [−50°, 50°], while the pitch and yaw angles have adjustable ranges of [−18°, 25°] and [−18°, 21°], respectively. This indicates that although the roll attitude shows a larger operational range, its rotational authority—defined as the ability to generate controlled moments about the roll axis—is limited due to the symmetric cable configuration and the absence of dedicated moment arms in the roll direction. As a result, fine control over roll motion may require higher precision in cable length adjustments, which could affect dynamic responsiveness. The mechanism layout is physically constrained in pitch and yaw directions, limiting their operational space. However, by adjusting key design parameters (e.g., the position of anchor point $B_i$), a broader adjustment range for the aircraft model’s attitude angles can be achieved. These findings are consistent with the ADAMS simulation results.

5.2.5. System Stiffness and Eigenvalue Analysis

The stable workspace is defined as the set of discrete points that can balance all external loads applied to the aircraft model under the given constraints of cable tension, while also satisfying the stability conditions at the current position and attitude. Based on the system equilibrium stability analysis in Section 4, the eigenvalues of the system’s total stiffness matrix are calculated according to the eigenvalue-based stability criterion. That is, when all eigenvalues of the stiffness matrix are positive, the system has no negative stiffness modes and can resist small disturbances, and is therefore considered stable. According to Equation (3), the initial pretension forces of the four cables in the initial equilibrium state are as follows:

$$T=\left[\begin{array}{cccc}108.4802& 84.5109& 111.2222& 111.2222\end{array}\right]\mathrm{N}$$

To obtain the total stiffness matrix $\mathit{K}$ of the system, the cable tension values calculated from the aforementioned conditions are substituted into the corresponding equations in Equation (20) to solve for ${\mathit{K}}_c$ and ${\mathit{K}}_s$. These are then substituted into Equation (19) to yield the total stiffness matrix $\mathit{K}$, as follows:

$$\mathit{K}=\left[\begin{array}{cccccc}35,289& -0& -1122& 0& 54& -0\\ -0& 8881& 0& -18& -0& 163\\ -1122& 0& 5331& 0& 342& 0\\ 0& -18& 0& 5& -0& -1\\ 54& -0& 342& -0& 58& 0\\ -0& 163& 0& -1& -0& 40\end{array}\right]$$

The six elements on the main diagonal of the stiffness matrix represent the translational stiffness of the aircraft model along the O_gX_g, O_gY_g, and O_gZ_g axes in the fixed coordinate system (measured in N/m), and the rotational stiffness around the O_gX_g, O_gY_g, and O_gZ_g axes (measured in N·m/rad). The off-diagonal elements form the coupling stiffness submatrices. For the convenience of discussion in the rest of this work, the six values on the main diagonal are selected to form a diagonal matrix:

$$diag(\mathit{K})=\left[\begin{array}{cccccc}35,289& 8881& 5331& 5& 58& 40\end{array}\right]$$

(26)

In this section, simulations are conducted for a special spatial under-constrained CDPM-4 configuration with four cables. The pitch angle range is set from −25° to 25°, and calculations are performed at intervals of 5°. The system stiffness values resulting from these calculations are summarized in

Table 5. Simulation results of total stiffness of the CDPM-4 system under varying pitch angles. 

	Stiffness	Translational Stiffness(N/m)			Rotational Stiffness(N·m/rad)
Pitch		OgXg	OgYg	OgZg	OgXg	OgYg	OgZg
−25°		34,743	8286	4819	5	103	5
−20°		34,875	8351	4859	4	79	7
−15°		34,987	8413	4896	3	59	9
−10°		35,079	8478	4938	2	45	12
−5°		35,158	8571	5013	2	40	17
0°		35,289	8881	5331	5	58	40
5°		34,896	7735	4025	−7	−24	−48
10°		35,038	8212	4534	−2	21	−11
15°		35,016	8278	4585	−0	42	−5
20°		34,952	8285	4575	1	64	0
25°		34,858	8267	4539	3	90	−1

.

Based on the above simulation results, it can be seen that for the CDPM-4, the translational stiffness of the aircraft model along the O_gX_g axis is clearly one order of magnitude higher than that along the other two axes (O_gY_g and O_gZ_g). The rotational stiffness about the O_gX_g axis is relatively weak because the cables providing this stiffness are located close to the fuselage axis, resulting in a minimum absolute value of rotational stiffness about the O_gX_g axis of 0 N·m/rad. This is consistent with the experimental requirement for the support configuration to constrain the model’s motion degrees of freedom, specifically, the configuration is designed to constrain five degrees of freedom of the model except for the incoming flow direction (opposite to the positive O_bX_b axis), while ensuring high translational stiffness along the O_gX_g axis to maintain the stability of the aircraft model during testing. The results are consistent with the ADAMS simulation results.

Furthermore, the values of the eigenvalue $\lambda$ of the system’s total stiffness matrix can be obtained as it varies with the pitch angle $\theta$, as shown in Figure 9. From the trend of the curve, it can be seen that within the pitch angle range of −10° to 10°, the value of $\lambda$ first increases, then decreases, and increases again. This is due to the aircraft model transitioning from a nonzero attitude back to a zero attitude and then moving away from it. Meanwhile, the traction cables go through cycles of slackening and tightening, causing continuous changes in the direction of the tensile force applied by the cables. As analyzed in Section 4 regarding the composition of the system’s total stiffness (Equation (20)), changes in the tensile force of the cables lead to fluctuations in the overall system stiffness. This is considered a normal phenomenon and conforms to the basic laws of motion.

In fact, the positive definiteness of the system’s total stiffness matrix indicates that the system is stable. However, for a better qualitative analysis and considering that active motion control of the aircraft model is not required during flutter testing, we select the eigenvalues at a pitch angle of 0° for detailed analysis. The eigenvalues are: $\lambda =\left[\begin{array}{cccccc}\mathrm{e}{\mathrm{v}}_1& \mathrm{e}{\mathrm{v}}_2& \mathrm{e}{\mathrm{v}}_3& \mathrm{e}{\mathrm{v}}_4& \mathrm{e}{\mathrm{v}}_5& \mathrm{e}{\mathrm{v}}_6\end{array}\right]=\left[\begin{array}{cccccc}35,331& 8884& 5312& 37& 35& 5\end{array}\right]\mathrm{N}/\mathrm{m}$, with the smallest eigenvalue being $\mathrm{e}{\mathrm{v}}_6=5\mathrm{N}/\mathrm{m}$. This means that the eigenvalues of the total stiffness matrix of the system are all greater than zero, and it can be concluded that the system’s total stiffness matrix is positive-definite, indicating that the system is stable. Therefore, the specific spatial under-constrained support configuration CDPM-4 with four traction cables considered in this paper meets the stability requirements.

To evaluate the numerical robustness of the system, we compute the condition number of the total stiffness matrix $\mathit{K}$. Based on the provided matrix and eigenvalue analysis, the maximum eigenvalue is λ_max = 35,331 N/m and the minimum eigenvalue is λ_min = 5 N/m, yielding a condition number $\kappa \left(\mathit{K}\right)$ ≈ 7.1 × 10³. The results indicate that $\kappa \left(\mathit{K}\right)$ is less than 10⁴ in most regions, indicating good numerical conditions.

5.2.6. Geometric Parameter Sensitivity and Robustness Analysis

The descriptions and analyses provided in the preceding subsections were based on ideal geometric parameters. To evaluate the impact of manufacturing and installation errors on the stability of the system’s equilibrium pose, i.e., the system’s robustness, a preliminary geometric error sensitivity analysis was performed.

Based on the Monte Carlo principle [31], random perturbations were applied to key geometric parameters, and the changes in system outputs (e.g., pose, tension) were statistically analyzed. The anchor points $B_i$ and $P_i$ (coordinate values given in Table 1), which are most sensitive to the system’s statics, were selected as the sources of error. It is assumed that the error in each coordinate component is an independent and identically distributed random variable, following a normal distribution with a mean of zero and a standard deviation of σ = 1 mm (ΔB_i,x, ΔB_i,y, … ~ N(0, σ²)). N = 100 sets of geometric parameter samples with random errors were generated. For each error sample, the actual equilibrium poses $X^{\prime }$ and cable tensions $T^{\prime }$ of the platform were obtained by solving the static equations (Equations (5) and (6)) using the hybrid intelligent algorithm described in Section 3, under the initial equilibrium pose (zero pose).

The distributions of pose errors ($\Delta X=X^{\prime }-X$) and relative tension changes ($\Delta T_i/T_i$) were statistically analyzed. The analysis reveals that the maximum values of the attitude angle errors for the platform’s equilibrium pose do not exceed 0.3°, the maximum relative change in cable tension is less than 5%, and the system’s total stiffness matrix $\mathit{K}$ remains positive definite for all sample sets. This indicates that the four-cable support system designed in this paper is insensitive to geometric errors and exhibits good robustness in terms of its static equilibrium pose.

5.2.7. Analysis of Stable Workspace

To better demonstrate that the specific spatial under-constrained support structure using four cables considered in this paper can meet specific experimental balance stability requirements, additional conditions for judging system balance stability were added based on the settings for analyzing the static feasible attitude workspace. The simulation of the static stable attitude workspace was performed using MATLAB software, with the results shown in Figure 10.

Clearly, based on the system equilibrium stability conditions, the static stable orientation workspace boundary shrinks to varying degrees compared to the static feasible orientation workspace. The achievable ranges of roll, pitch, and yaw angles are [−50°, 50°], [−13°, 18°], and [−16°, 13°], respectively. Among them, the pitch and yaw angles show moderate shrinkage, while the roll angle remains unchanged. This is because, as shown in the mechanism configuration in Figure 5, even with the added stability constraints, the control and constraint capability for the aircraft model’s rolling motion remains relatively weak, whereas the constraints on pitch and yaw motions are further enhanced. As can be seen from Figure 10, the closer the aircraft model is to its zero-pose state, the stronger the mechanism’s capability to control lateral tilt and deviation becomes. This result is consistent with the analysis of static stability based on the system’s total stiffness matrix. Since active motion control of the aircraft model is not required during flutter testing, the above analysis of the static stable orientation workspace indicates that the specially designed spatial under-constrained support configuration with four cable-driven actuators satisfies the stability requirements and meets experimental needs, specifically, when the aircraft model experiences oscillations due to incoming airflow, the system possesses the capability to restore the model to the desired equilibrium state required by the test.

The comprehensive analysis presented above demonstrates that the system’s stability is determined by the positive definiteness of the total stiffness matrix $\mathit{K}$. To gain a deeper understanding of the stability behavior at the workspace boundaries, we further clarify that the criterion for the loss of system stability is the result of the combined effect of cable tensions approaching their prescribed minimum value (T_i → T_min) and the stiffness matrix approaching singularity (minimum eigenvalue λ_min → 0). In the boundary regions of the workspace, this often corresponds to the tension in one or more cables approaching its lower limit T_min. At this point, as indicated by Equation (20), the tension term providing the active stiffness ${\mathit{K}}_s$ is significantly reduced, leading to the minimum eigenvalue λ_min of the system’s total stiffness matrix $\mathit{K}$ approaching zero, placing the system in a “quasi-singular” state. In this state, the mechanism’s ability to resist disturbances in specific directions deteriorates sharply, defining the stability boundary. Therefore, the static stable workspace depicted in this paper (as shown in Figure 10) is precisely defined by the two conditions “T_i > T_min “and “λ_min > 0” acting together. This analysis links tension feasibility with the singularity of the stiffness matrix, deepening the understanding of the stability mechanism of under-constrained cable-driven parallel mechanisms and providing a more complete theoretical basis for accurately defining their workspace.

5.2.8. Discussion

This paper proposes an integrated solution framework specifically for the “under-constrained cable-driven mechanism” paradigm and the particular scenario of “wind tunnel model support.” Its innovation lies in the systematic integration of kinematic solving, a static stability criterion based on the eigenvalues of the stiffness matrix (serving as an effective tool for singularity/stability discrimination), mechanical interference constraints, and static stable workspace analysis. This framework complements modern research methodologies, such as reachable set computation based on interval analysis [12] and singularity discrimination based on Lie algebra derivatives [32]. Unlike works primarily focusing on fully constrained or redundantly constrained systems, this framework specifically addresses the core issues of under-constrained systems near their equilibrium points: their “weak stability” and the close relationship between the workspace boundaries and both the lower limit of cable tension and stiffness matrix singularity. It thus provides a more targeted design and analysis tool for the safe and stable operation of such mechanisms.

6. Experimental Verification

To validate the effectiveness of the workspace predicted by our theoretical calculations and simulations, we constructed a CDPM-4 prototype, which includes a support frame, transmission systems, an Attitude and Heading Reference System (AHRS), an industrial computer, and an aircraft model, as shown in Figure 11a. Experiments were conducted to determine the reachable attitude workspace of the aircraft model under physical structure limitations and potential interferences between the cables and the aircraft model. The positive limit values for each attitude angle were determined: roll angle $\varphi =49.7°$ (Figure 11b), pitch angle $\theta =24.8°$ (Figure 11c), and yaw angle $\psi =21.2°$ (Figure 11d). These results are largely consistent with those from theoretical calculations, ADAMS simulations, and MATLAB simulations. This indicates that the proposed special spatial under-constrained support configuration driven by four cables is feasible and can meet experimental requirements.

7. Conclusions

This paper investigates a special spatial under-constrained cable-driven parallel mechanism (UCDPM) with four cables, presenting a systematic study on its kinematic modeling and stability analysis. To address the strong coupling between kinematics and statics inherent in such systems, a collaborative solution framework is proposed.

First, the governing force and moment equilibrium equations are formulated. A hybrid intelligent optimization algorithm—integrating genetic algorithm, chaos optimization, and particle swarm optimization—is employed to efficiently solve both forward and inverse kinematic problems, enabling the identification of physically feasible equilibrium configurations that satisfy all system constraints.

Second, the static stability of these equilibrium configurations is evaluated through eigenvalue analysis of the system’s total stiffness matrix. A stability criterion based on the minimum eigenvalue (λ_min > 0) is established to delineate stable regions within the workspace.

Third, two distinct workspace definitions are introduced: the statically feasible workspace, which accounts for tension limits and interference constraints, and the statically stable workspace, which further incorporates the aforementioned stability criterion. These definitions provide more practical and reliable boundaries for mechanism layout design and trajectory planning.

The effectiveness of the proposed framework is validated through numerical case studies, demonstrating superior performance compared to benchmark algorithms. However, the current work focuses primarily on static equilibrium. Future research will extend the analysis to dynamic behavior and include comprehensive experimental validation using a physical prototype to confirm the applicability and robustness of the proposed framework.