1. Introduction
Cable-driven parallel robots (CDPRs) use flexible cables to control an end effector by winding and unwinding cables instead of using the rigid links of common parallel robots. The use of light cables with high load-withstanding ability as actuators can reduce the actuator weight as well as the inertia; thus, CDPRs have many typical advantages, such as high dynamics, low inertia, and large workspaces. Due to these advantages, CDPRs have been widely used in various fields, such as in radio telescopes [
1], multiple mobile cranes [
2], waist rehabilitation [
3], and construction cable robots [
4]. The CDPR system has also been studied by researchers from different perspectives. The control algorithm is an important factor affecting the accuracy and stability of the robot system. To ensure the stability of the end-effector during movement, the use of a hybrid position-force control algorithm as an asynchronous control of the cable force was proposed [
5]. A passivity-based control for a planar cable-driven manipulator system was also investigated. In this work, to simplify the modeling process of the system, a lumped-mass model is used, and each cable is modeled individually. Then, the derivation of Lagrange’s equation and the null-space method is used to derive the dynamics of the system [
6]. To dynamically control CDPRs in [
7], visual feedback with linearizing inner loop control was proposed, and this new approach allows for the control of CDPRs when using long cables or polymer cables. In addition, the stability of the dynamic control scheme was demonstrated using Tikhonov’s theorem. Characteristics of the cable used for CDPR that are related to the accuracy of the end-effector were also considered based on nonlinear viscoelastic properties, such as structural elongation and hysteresis. Therefore, to improve the accuracy, many studies have considered these attributes. Rather than the commonly used elastic cable approach, an approach based on the numerical analysis of cable elements was proposed in [
8], where a novel formulation for three-dimensional cables was developed. Besides that, the thermal-elastic of the proposed element was also analyzed. The proposed formulation leads to an explicit stiffness matrix this makes the new element more efficient in terms of the analysis time, and the research results in the literature indicate the advantages of the proposed scheme. Moreover, a simple model for cable elongation was incorporated to increase position accuracy for CDPR [
9]. A hysteresis model was developed based on the Bouc–Wen model, and dynamic behavior, such as static creep, hardening effects, long-term recovery, and hysteresis, was described based on a viscoelastic model with only an integrated dynamic model [
10,
11,
12]. However, to evaluate its performance, the workspace analysis of a CDPR, considering the condition of various accelerations, is an important issue.
The workspace of CDPR can be divided into two categories: dynamic and static [
13]. The static workspace can be further divided into a wrench-closure workspace. Most of the research on CDPR has only considered a static condition in the analysis of the workspace. A workspace analysis under equilibrium conditions to evaluate external force was presented in [
14]. Besides, a force-closure workspace analysis of CDPR was conducted based on a workspace with fully restrained positioning mechanisms. Because of the maintenance of positive cable tension, the general algorithm to generate a force-closure workspace was studied at various orientations of the end-effector [
15]. Moreover, a method that is seen as one of the most efficient interior-point methods was improved in [
16] to analyze the wrench-feasible workspace for general CDPR. This method uses the linear matrix inequalities and the projective method to obtain the wrench-feasible workspace. In [
17], to analyze the wrench-feasible printing workspace, a methodology that considers the impact of cable mass and end-effector orientation was proposed. To ignore the inertia effects of the cable, this proposed method assumes that the motion of the end effector is slow. The cable profile is also considered for the checking of collisions based on kinetostatic analyses of the cable robot. In addition, the limitations of planar cable-driven parallel robots can be improved so that they cause a zero force on the borders of such a static wrench. Two different CDPRs, with and without a counterbalancing mechanism, were proposed for study and comparison, and then the improving effects of the proposed mechanism on the static wrench (SW) and available wrench set (AW) were demonstrated [
18]. Similarly, two different CDPRs were used to analyze the workspace in [
19], and a novel workspace from the control was proposed, namely the Control Stability Workspace (CSW). This workspace can determine whether the controller will be able to guide the platform of the CDPR to the desired position. Moreover, to determine the wrench-feasible workspace (WFW) for CDPR when subjected to external force and high payload and improve the optimum designs of CDPR, the overall workspace was maximized. Based on the kinematic equilibrium, the wrench-feasible workspace and twist feasible workspace were obtained and presented based on these case studies [
20]. Despite much research already being conducted on the workspace analysis of general CDPR, a workspace analysis on musculoskeletal systems as cable-driven robots was proposed for the first time in [
21]; the research results show that the proposed approach can produce a realistic workspace.
Recently, several studies on CDPRs focused on workspace analysis, but most of this research only considered static equilibrium conditions or gravity to determine the workspace. In their study, Heo et al. presented a wrench-feasible workspace [
22]. Their study not only considered dynamic conditions based on the effect of various accelerations but also took into account the pulley-bearing friction factor. However, this factor analysis is only suitable for CDPR systems where the design of the drive uses multiple pulleys in series and is not applicable to other designs that only use a single pulley at anchor points of CDPR systems. As previously mentioned, cable tension is quite important. In a real CDPR system, the predicted cable tensions cannot easily improve the features of the robot due to the nonlinear characteristics of the cable as well as the nonlinear tension that occurs during motion. These characteristics affect the dynamic stability of CDPRs. In addition, the high acceleration generated on the wrench-feasible workspace affected the CDPR system significantly. Therefore, the workspace analysis of a CDPR with consideration of the nonlinear cable tension is an important issue in evaluating its performance. In this study, we focused on the nonlinear cable tension effect because tension varies when cables are actuated in dynamic conditions. Cable nonlinear tension modeling cannot be neglected in situations where high tension is imposed under varying acceleration. The methodologies to determine the feasible workspace of CDPR have been presented in many studies. However, for CDPR using polymer cables, the linear tension model can be used to determine the cable tension, but this model cannot be used to determine the feasible workspace of the CDPR system because it does not satisfy the nonlinear elastic properties of the polymer cable. In addition, the operation of the robot is also affected by dynamic phenomena associated with high cable length variations. The main contributions of this paper can be summarized as follows:
The dynamics of the redundant cable-driven parallel robot (CDPR) and the influence of pulley kinematics are presented.
A methodology for determining the wrench-feasible workspace (WFW) of a cable-driven parallel robot (CDPR) is proposed based on a nonlinear cable tension model.
The dynamic elastic modulus of the polymer cable for the nonlinear tension model is obtained based on the dynamic mechanical analysis (DMA) method by changing the frequencies of the applied force on the cable and is presented for the first time in this article.
An approach to obtaining the redundant tensions and ensuring that the cable tensions are positive, using a nonlinear constraint for optimal function, is proposed.
The structure of the article is organized as follows. First, the kinematics equation and dynamics of CDPRs were derived. Next, a nonlinear tension formulation of cables was constructed, and experiments were carried out at different frequencies based on the dynamic mechanical analysis (DMA) method, from which the dynamic modulus of elasticity coefficients was determined. Third, cable nonlinear tensions were considered. Subsequently, optimization was conducted considering these nonlinear tensions, and a wrench-feasible workspace analysis was also conducted. Then, the simulation results are discussed, with the main contributions presented in the Conclusions section.
2. Modeling of the CDPR System
The configuration of the designed CDPR is shown in
Figure 1. The cube frame dimension of the CDPR was 1 m × 1 m × 1 m (L × W × H). The pulleys were located asymmetrically to prevent intersections with each other during operation (
Figure 1b). As shown in
Figure 1b, four pulleys (P1, P2, P3, and P4) were upper pulleys. In contrast, four pulleys (P5, P6, P7, and P8) were lower pulleys. Additionally, the coordinates of the pulley location are listed in
Table 1.
A flexible cable with high-strength Dynamica rope was used in this study. These cables started from the winches and then passed through the pulleys to the end-effector and were divided into four upper cables and lower cables. The position of the end-effector was controlled by these cables. In
Figure 1a, it can be seen that, to reduce the moment caused by the end-effector during movement, the upper cables were connected to the bottom anchor points of the end-effector, and in contrast, the bottom cables were connected to the upper anchor points of the end-effector as described earlier [
14]. Here, we used an end-effector size of 0.26 m × 0.26 m × 0.26 m, with a mass of 1 kg.
2.1. Kinematics of the Redundant-Actuated CDPR System
Briefly reviewing the kinematic model, the geometry of the CDPR robot is described by its proximal anchor points on the robot frame and the distal anchor points on the end-effector, which were defined by the vector.
Figure 2 shows that B was the origin of the global coordinate, and the vector
was described from the frame B to anchor point A of the frame. The vector denoted as
represented the vector from B to the center of the end-effector fixed frame E, the vector
was connected from the end-effector fixed frame E to the anchor point C of the end-effector, and it described the position vector of the anchor point from E. In addition, the vector of cable length denoted as
was derived from the given vector. The cable length of the cables could be obtained by solving the inverse position kinematics (IPK). By applying a vector loop, the cable vector
was established as follows:
where the index
i denotes the cable number.
In a typical CDPR robot system, the pulleys have an important role, and they are used to guide the cable at point A towards the end-effector; the proximal anchor points of the robots are simplified to be ideal points. Thus, the starting point of the cable described by the position vector is not a constant but depends on the current pose of the robot and the wrapping angle of the cable around the pulley is a variable. Moreover, in a CDPR system, the real cables achieve a reasonable lifetime only when a minimum bending radius is exceeded [
23]. Thus, determining the starting point of the cable is essential to improve the accuracy of its position and the effect of pulley kinematics in this system. A model of the pulley kinematics is presented in
Figure 3, and the kinematic equation is given as follows:
where
Lcomp is a cable length compensation,
Lc is the free cable length from point A to point B,
r is the pulley radius and
θ is the wrapping angle of the cable. For the two right-angled triangles ADO and AOE shown in
Figure 3, OA is the common side. Therefore, based on the Pythagorean theorem for the two right-angled triangles, the following can be obtained:
Considering the tetragon EODA, there are two angles that are right-angles. Thus, the enclosed angle
θ1 + θ2 at point A equals the sought complementary angle
θ at point O [
5,
23,
24]. Using the elementary trigonometric function, the cable wrapping angle around the pulley is given by Equation (4).
where
. Given the coordinates of the point at pulley O whose coordinates are O(
x2,
y2,
x2), at the anchor point of end-effector A(
x1,
y1,
z1), and at the center of end-effector G(
x0,
y0,
z0), then the values
dx,
dy, and
dz are coordinates in the
x,
y,
z directions, which are calculated by the equation:
L(
x,
y,
z) = O − A − G.
dz represents coordinates of point E with respect to the frame point
B coordinates. Finally, the equation for determining the corrected cable length, which takes into account pulley kinematics to solve the inverse kinematics, can be transformed as follows:
2.2. Dynamic Modeling for the CDPR System
To derive the dynamics equation for the CDPR system, first, a static equilibrium model for the system should be established by applying the forces and moments to the end-effector, which consists of internal forces and external forces. These forces and moments are from different parts, with one part from the eight cables and the other from the environment. Considering the static equilibrium, the total forces and moments exerted on the end effector should be zero, and the equilibrium equations can be obtained as follows:
where
Ti denotes the tension vector for the
ith cable,
m is the mass of the end-effector,
g is the gravitational acceleration, and
Fr and
Mr indicate the resultant forces and moment vector, respectively. For convenience, we assumed that the origin geometrical center was located at the center of gravity. Therefore, the moment caused by gravity could be ignored. Combining
Fr and
Mr together, we obtained the wrench vector:
From Equations (6)–(8), the static equilibrium equation can be described as follows:
where
J represents the cable tension Jacobian matrix, [
J]
T represents the transposed matrix of
J, and
and
denote the unit cable length vector.
T is the vector of cable tensions:
To consider the effects of external force caused by the acceleration of the end-effector, the dynamics equation was derived as shown in Equation (12).
where
Fe and
Me denote the external force and moment, respectively; matrices m and
I are a [3 × 3] diagonal matrix of the end-effector mass and tensor of inertia, respectively;
is the acceleration of the end-effector; and ω denotes the angular velocity vector. These are defined as follows:
In this work, the CDPR system was based on translation-only motion with a nearly constant orientation. Thus, the angular velocity ω was small enough and the last term, including the angular velocity on the right side of Equation (13), was negligible in this research.
4. Workspace Analysis
In this article, simulations were performed for the proposed CDPR system considering nonlinear cable tension. The wrench-feasible workspace (WFW) was analyzed for a given motion trajectory of the end-effector and to solve Equation (20). The wrench-feasible workspace of a CDPR is defined as the set of positions for which a given set of wrenches, called the required net wrench set, can be balanced by the set of all wrenches, which the CDPR cables can generate in a determined position of the workspace [
18,
30,
31]. To determine the wrench-feasible workspace, simulations were carried out for the end-effector moves in the range of X, Y, and Z directions. The moving ranges were limited in direction by the
X-axis and the
Y-axis from −0.3 to 0.3 m and limited in the
Z-axis from 0.1 to 0.7 m. Additionally, the simulations were implemented at various accelerations at 10 and 100 m/s
2, and a geometric model was built by MATLAB simulation. The weight of the end-effector is 10 N and the maximum loading limit of the cable is 4000 N for a Dynamica cable with a diameter of 6 mm (breaking strength certificate). The step size for the MATLAB WFW simulation is 100 mm. According to the definition of feasible workspace, all of the cable tension at each position needed to be constrained to maintain positive tension and not exceed the maximum loading of the cable. Besides, each box was denoted with a color in the diagram; each color represented the maximum cable tension value at each acting position. Moreover, when the calculated cable tension values exceeded the limited value, the end-effector at these positions was eliminated from the workspace domain and appeared vacant in the figures.
The simulation results of the wrench-feasible workspace considering nonlinear tension are shown in
Figure 7 and
Figure 8. In the case of the end-effector moves in the X, Y, and Z directions, the figures show the maximum cable tension distribution of the eight cables when applying various accelerations for a given position motion of the end-effector. As shown in these figures, the average cable tension was ∼100 N at an acceleration of 10 m/s
2 and ~200 N at an acceleration of 100 m/s
2. Moreover, the total maximum tensions increased significantly from 600 N to 1000 N when the acceleration varied from 10 m/s
2 to 100 m/s
2. In addition, maximum tension values appeared at the vertices and edges of the workspace. The negative cable tension disappeared in the moving direction due to the constraint of the optimal method. Considering the case of X-direction acceleration (
Figure 7a,b, and
Figure 8a,b), the same directions between the front view and back view surface were compared, and the tension distribution of the back view surface was slightly higher than that in the opposite surface. The tension region had a C shape, with the edges created by the contiguous faces, and the surrounding area had a higher tension distribution than that of the front view surface. This tension distribution was similar for both cases of X-direction acceleration. In particular, the vertical edge was the edge created between the back view surface of the X-direction and the front view surface of the Y-direction, which had the highest tension distribution in this position because of the optimal method and the variable cable length; thus, the cable was expected to maintain a positive tension for the acceleration in the X-direction—the greater the acceleration, the greater the tension distribution in this region. This finding means that the CDPR system is restricted from working in this area and the tension must be decreased so that the wrench-feasible workspace can be improved.
Similarly, the range of acceleration in the Y-direction (
Figure 7c,d, and
Figure 8c,d) and the surface of the front view of the Y-direction showed a higher tension distribution compared to the opposite surface in the same direction of movement. Although the applied acceleration was different, the tension distribution on the faces in the same direction was similar. In addition, the tension distribution in the vicinities of the edges on these surfaces had high tension values because the cable maintained a positive tension for Y-direction acceleration. In particular, a high-tension distribution phenomenon occurred at the bottom edges on these surfaces. Here, as the acceleration increased, the tension on the line increased. In particular, in the case of 100 m/s
2, the tension distribution reached the maximum value; therefore, the CDPR system should be restricted from working in this region. Overall, neither tension distribution case exceeded the allowable tension of 4000 N.
Figure 9a,b show the quarter section of the wrench-feasible workspace in the simulation results, and they provide detailed information on the internal workspace and show that both quarter sections were filled with blue boxes. The cable tension distribution in the interior of the workspace is shown to be relatively low, with an average value of approximately 100 N. Therefore, these results were desirable. Additionally,
Figure 9c,d show the workspace results at Z = 0.2 m and Z = 0.4 m, respectively. It is obvious that the blue boxes comprised the bulk of the internal workspace and the tension was similar in these two diagrams. However, the high-tension distribution appeared at the corner of its slice and the vicinities of this position.
Figure 10a–d show the wrench-feasible workspace of CDPR when extending the moving ranges in the X and Y directions from 0.6 m to 0.8 m and the
Z-axis from 0.1 m to 0.8 m. The applied acceleration was 100 m/s
2. In these figures, it can be seen that both faces of the back view in the X and Y directions had the highest cable tension and appeared in regions that were eliminated from the workspace because they exceeded the constraint in the range of 0 ≤ Ti ≤ 4000 N. This shows that the simulation results were suitable for the wrench-feasible workspace of the proposed CDPR system as well as the real system due to the necessity of having a safe working space to prevent collisions between the end-effector and the frame during work. Therefore, extensions of the moving range directions and the high-tension distribution, which occurred at the vertical edges of the workspace, should be considered. In addition, compared with results from previous related studies in [
4,
22,
32], the research results of the paper show that under the same conditions of motion acceleration, the end-effector range of motion is smaller but the wrench-feasible workspace analysis results are significantly improved. This indicates the effectiveness of the proposed method.