Analysis of workspace boundary for multi-robot coordinated lifting system with rolling base

Abstract

An improved Monte-Carlo algorithm is proposed to address the problem of an unclear workspace boundary in a multi-robot coordinated lifting system. The spatial configuration of a multi-robot coordinated lifting system with rolling base is analyzed, and the kinematics and static workspace of the system are established. To solve the workspace boundary, first, the error introduced by the layers is reduced using an intra-layer thinning method. Second, each layer is divided simultaneously based on rows and columns, and the initial boundary points are extracted by searching for the best value. Third, random three-dimensional points are added in the neighborhood, and pseudo-boundary points are removed using three-dimensional local spherical coordinates to achieve a high-precision solution for the workspace boundary. Finally, the workspace volume is used to analyze the influence of structural parameters on the workspace boundary. The results show that the lifting system has limited carrying capacity and a data reference for selecting the structural parameters by analyzing the factors that affect the workspace. Findings provide a basis for further studies on the structural configuration and optimization of the lifting system.

1. Introduction

With the complexity and randomization of the lifting environment in the fields of road construction, cargo handling, rescue, and industrial production, as well as the limitations of a single lifting robot in the workspace, carrying capacity, and adaptability to the environment, the existing lifting equipment is challenging to complete the lifting task under heavy load and complex environment [1, 2]. Since the cable-driven parallel system has the advantages of a large workspace, and strong load capacity, replacing the boom of the traditional rigid parallel robot with the flexible cable and replacing the fixed base with the rolling base, which will form a rolling multi-robot coordinated lifting system. The rolling base and lifting robot can be used as a single unit or in combination with other vehicles and the system for lifting operations. The multi-robot coordinated towing system with rolling base belongs to the class of rope-driven parallel systems. It not only contains the advantages of the rope-driven parallel mechanism, such as strong flexibility and a large workspace, but also features the ability of the towing system to operate with high loads and a modular design. By adjusting the position of the rolling base, the end position of the towing robot, and the length of the cable, the position and posture of the suspended object can be changed [3, 4].

The workspace is an important index to measure the performance of a multi-robot coordinated lifting system with rolling base, and it is also the basis for parameter optimization and stability analysis of the lifting system. As a quantitative parameter describing the workspace, the workspace boundary can not only accurately describe the size and shape of the workspace but also accurately describe the system’s characteristics [5–7]. Therefore, the analysis of the workspace boundary has excellent theoretical and applied research value. There are many methods to solve the workspace, so we need to choose a solution method that best conforms to the characteristics of the system to get a correct and accurate workspace. At present, methods for solving the workspace of the parallel robot include analytical, geometric, and numerical methods [8]. Verhoeven et al. [9] estimated the workspace of the tendon platform by using algebraic sets. Bosscher et al. [10] used vector theory to solve the analytical solution of the workspace of the parallel system and discussed the possibility of solving the workspace boundary with vector theory. Diao et al. [11] solved the workspace of a parallel robot under completely constrained conditions based on the vector closure principle. Guan et al. [12] used the Monte-Carlo algorithm to analyze the reachable workspace of humanoid robots. For multiple degrees of freedom (DOF), robots or complex systems with multiple robots in parallel, analytical and geometric methods are not applicable, and this method cannot solve the three-dimensional (3D) workspace.

The numerical method is based on the forward kinematics to solve the workspace, which is simple and versatile, and is more suitable for solving the workspace of the lifting system. The Monte-Carlo algorithm, which takes into account both efficiency and accuracy, and is easy to program and has been widely used in robotic systems [13, 14]. However, the workspace solved by the numerical method is only an approximate representation of the actual workspace. To determine the ultimate position of the suspended object in the lifting space, it is very critical to accurately extract the boundary points of the workspace. At present, the commonly used methods are the traversal method and the extreme value method [15]. Cao et al. [16] extracted boundary points through the distribution of local coordinate quadrant points. This method has high accuracy but could be more computationally intensive. Yin et al. [17] first obtained the boundary points of the seed workspace and then increased the density of the boundary points with a Gaussian distribution. The proposed method improved the boundary accuracy, but it is not universal. Huang et al. [18] solved the workspace of redundant parallel mechanisms using the Monte-Carlo method based on boundary extremum point densification. Xu et al. [19] increased the boundary points of the workspace by adding random values in the joint angle neighborhood. Ji et al. [20] transformed the workspace boundary into an optimization problem and solved the workspace boundary by traversing the inverse solution of the parallel system. After layering the workspace, the boundary point extraction method is simple and easy to operate, but the boundary point extraction is uneven and has low accuracy. Considering that the boundary of the motion parameter determines the workspace boundary, if the joint angle is taken according to the β distribution, the point distribution density of the workspace boundary can be increased [21], but this method can only be used in systems with a unique inverse solution of the joint angle.

The multi-robot coordinated lifting system with rolling base has excellent potential in practical applications. These previous studies focused on the workspace of the fully constrained and over-constrained system from different perspectives. The workspace boundary of the under-constrained lifting system is rarely analyzed. Peng et al. [22] proposed an expression to describe the static equilibrium reachable workspace and its boundaries for a three-cable system, but the linearization process is relatively complex. Experiments are an essential way to verify the workspace of a lifting system, and theoretical analysis should be performed before experiments to verify the correctness of the method and the feasibility of the system under study. Taking the equilateral triangle configuration of the lifting system as an example, based on the workspace solved by the traditional Monte-Carlo method, a hierarchical strategy is proposed to extract the initial boundary points according to row and column partition. The precision of the workspace boundary is solved by adding 3D random points and removing the pseudo-boundary points by 3D local spherical coordinates. The influence of the structure parameters on the workspace volume is analyzed, and the boundary value of the structure parameters is determined.

The outline of the study is as follows. In Section 2, the configuration of the multi-robot coordinated lifting system with rolling base is introduced. In Section 3, the kinematic model of the lifting system is established. In Section 4, the static workspace of the lifting system is analyzed, which provides a reference for solving the workspace boundary. In Section 5, the workspace boundary of the lifting system is solved using the improved Monte-Carlo method. In Section 6, the influence of structural parameters on the workspace boundary is analyzed using the workspace volume. In the last section, some remarkable conclusions are given.

2. Analysis of the spatial configuration

The spatial configuration of a multi-robot coordinated lifting system with rolling base has various forms, the typical design is two-wheeled lifting robots arranged in a straight line, three-wheeled lifting robots arranged in a triangle, and four or more wheeled lifting robots arranged in a polygon. To make the research object more typical, the three-wheeled lifting robot system is studied, which not only meets the economic requirements in practical engineering but also can fully control the position of the suspended object in space. Fig. 1 shows the spatial configuration of the three-wheeled lifting robots in a triangular, $\{O\}$ is the global coordinate system, $\{O_{si}\}$ is the rolling-base coordinate system, and $\{O^{\prime}\}$ is the suspended object coordinate system. The rolling base is driven at a two-wheel differential speed. The barycenter $(x_{c},y_{c},z_{c})$ on the rolling base is equipped with a three-DOF joint robot, the link length of the robot is $(a_{i1},a_{i2},a_{i3})$ , and the joint angle of the robot is $(\theta _{i1},\theta _{i2},\theta _{i3})$ . Link 1 connected to the rolling base can be rotated around the $z$ axis, and links 2 and 3 can be rotated up and down, and both joints can only achieve rotation in the same vertical plane. The position vector of the cable is $\boldsymbol{L}_{i}$ , and the cable length is $P_{i}B_{i}$ .

Figure 1. Spatial configuration of the lifting system.

The spatial configuration of the multi-robot coordinated lifting system needs to be determined based on the actual task, and its spatial configuration is determined to be three-wheeled lifting robots coordinating to lift the same object. The rolling base is moved to the desired position, and the position of the rolling base is fixed before lifting the load. The suspended object can be changed by adjusting the length of the cable and the end position of the robot. When the lifting point of the object is arranged in a plane triangle, the spatial configuration of the three-wheeled lifting robots is shown in Fig. 2. The symmetry of the equilateral triangle is better and easier to calculate, so the position of the wheeled lifting robots is used as the equilateral triangle to analyze the kinematics, statics, and workspace of the lifting system.

Figure 2. Plane configuration of wheeled lifting robots.

3. Kinematic analysis

The end of the lifting robot can reach the desired position $(x_{Pi},y_{Pi},z_{Pi})$ by controlling the horizontal position $(x_{c},y_{c},z_{c})$ of the rolling base and the joint angles $(\theta _{i1},\theta _{i2},\theta _{i3})$ of the robot.

Assuming that the three-wheeled lifting robots have the same structure, the end position of the lifting robot in the $\{O_{si}\}$ is

(1) \begin{align} \left\{\begin{array}{l} x_{Pi}^{\ast }=a_{i1}\cos \theta _{i1}\cos \theta _{i2}+a_{i2}\cos \theta _{i1}\cos \theta _{i3}\\[3pt] y_{Pi}^{\ast }=a_{i2}\sin \theta _{i1}\cos \theta _{i2}+a_{i3}\sin \theta _{i1}\cos \theta _{i3}\\[3pt] z_{Pi}^{\ast }=a_{i1}+a_{i2}\sin \theta _{i2}+a_{i3}\sin \theta _{i3} \end{array}\right. \end{align}

From the structure of the lifting robot and the related theory of robot kinematics, the end position of the lifting robot in the $\{O\}$ is

(2) \begin{align} \left\{\begin{array}{l} x_{Pi}=x_{C}+a_{i2}\cos \theta _{i2}\cos \left(\phi +\theta _{i1}\right)+a_{i3}\cos \left(\theta _{i2}+\theta _{i3}\right)\cos \left(\phi +\theta _{i1}\right)\\[3pt] y_{Pi}=y_{C}+a_{i2}\cos \theta _{i2}\sin \left(\phi +\theta _{i1}\right)+a_{i3}\cos \left(\theta _{i2}+\theta _{i3}\right)\sin \left(\phi +\theta _{i1}\right)\\[3pt] z_{Pi}=z_{C}+a_{i1}+a_{i2}\sin \theta _{i2}+a_{i3}\sin \left(\theta _{i2}+\theta _{i3}\right) \end{array}\right. \end{align}

where $\varphi$ is the angle between the forward direction of the rolling base and $x$ axis of the coordinate system $\{O\}$ .

The kinematic relationship between the end of the lifting robot and the wheeled rolling base can be obtained by Equations (1) and (2). $\square$

The cable-driven parallel lifting system consists of three cables and a three-DOF suspended object. The spatial configuration of the lifting system is shown in Fig. 3.

Figure 3. Cable-driven parallel lifting system.

Depending on the geometrical relations, the position vector of the cable can be obtained by the closed-loop vector method;

(3) \begin{align} \boldsymbol{L}_{i}= \boldsymbol{P}_{i}- \boldsymbol{B}_{i}= \boldsymbol{P}_{i}- \left(\boldsymbol{RB}_{i}^{\boldsymbol{\prime}} + \boldsymbol{r} \right)= \boldsymbol{P}_{i}- \boldsymbol{RB}_{i} ^{\boldsymbol{\prime}} - \boldsymbol{r} \end{align}

where ${\boldsymbol{P}}_{i}$ is the end position of the lifting robot in the $\{O\}, {\boldsymbol{B}}_{i}$ is the connection points between the suspended object and the cables in the $\{O\}, {\boldsymbol{B}}_{i} ^{\boldsymbol{\prime}}$  is the connection points between the suspended object and the cables in the $\{O^{\prime}\}, \boldsymbol{R}$ is the transformation matrix of the $\{O^{\prime}\}$ with respect to the $\{O\}, \boldsymbol{r}=[\begin{array}{ll} \begin{array}{ll} x & y \end{array} & z \end{array}]^{\mathrm{T}}$ is the position of the suspended object in the $\{O\}$ .

As the cable was in a tensioned state at the time of lifting, the mass of the cable was neglected [23]. The cable length can be obtained from the coordinates of $\boldsymbol{P}_{i}$ and $\boldsymbol{B}_{i}$ :

(4) \begin{align} L_{i}=\sqrt{\left(x_{Pi}-x_{Bi}\right)^{2}+\left(y_{Pi}-y_{Bi}\right)^{2}+\left(z_{Pi}-z_{Bi}\right)^{2}} \end{align}

According to the motion law of the wheeled rolling base and the lifting robot, the end coordinates of the lifting robot, the cable length, and the position of the suspended object can be obtained by connecting Equations (1) to (4).

In contrast to a single robot, the static workspace of a multi-robot system needs to consider the collaborative lifting among multiple robots. To obtain the workspace of the suspended object, the geometric constraints of the lifting system should be taken into account during the kinematic solution:

(1) The position of the suspended object is always in the triangular prism formed by the endpoints of the three robots and their projection points on the horizontal plane.

(2) To avoid the unbalanced distribution of the cable tension, the posture of the suspended object must be less than a certain angle.

(3) The cable tension must be between the minimum preload force and the maximum allowable tension to avoid the virtual strain and fracture of the cable.

(4) The wheels of the rolling base are always in contact with the ground to ensure that the rolling base will not overturn.

4. Analysis of the static workspace

The balance equation of the suspended object is

(5) \begin{align} \boldsymbol{J}^{\mathrm{T}}\boldsymbol{T}=\boldsymbol{F} \end{align}

where $\boldsymbol{T} = [\begin{array}{lll} T_{1} & T_{2} & T_{3} \end{array}]^{\mathrm{T}}, 0\lt T_{\min }^{s}\leq T_{i}\leq T_{\max }^{s}$ . $\boldsymbol{F}$ is the external force matrix of the suspended object:

(6) \begin{align} \boldsymbol{F}=\left[\left(m\frac{\mathrm{d}\boldsymbol{v}}{\mathrm{d}t}\right)^{\mathrm{T}} \left(\left[\boldsymbol{I}_{x}\ \boldsymbol{I}_{y}\ \boldsymbol{I}_{z}\right]\!\frac{\mathrm{d}\boldsymbol{\omega }}{\mathrm{d}t}\right)^{\mathrm{T}}\right]^{\mathrm{T}} \end{align}

where $\boldsymbol{I}_{x},\ \boldsymbol{I}_{y},\ \boldsymbol{I}_{z}$ are the moment of inertia of the suspended object, respectively, and $\boldsymbol{\nu}$ and $\boldsymbol{\omega }$ are the linear and angular velocity of the suspended object, respectively.

In Equation (5), $\boldsymbol{J}^{\mathrm{T}}=[\begin{array}{ll} \begin{array}{ll} \boldsymbol{J}_{1} & \boldsymbol{J}_{2} \end{array} & \boldsymbol{J}_{3} \end{array}]$ is the structure matrix of the lifting system:

(7) \begin{align} \boldsymbol{J}_\boldsymbol{i} = \left[ \begin{array}{c} \boldsymbol{e}_{i} \\ \left(\boldsymbol{RB}^{\boldsymbol{\prime}}_{i}\right)\times \boldsymbol{e}_{i} \end{array}\right]\left(i=1,2,3\right) \end{align}

where $\boldsymbol{e}_{i}=\boldsymbol{L}_{i}/\| \boldsymbol{L}_{i}\|$ is the unit vector of the cable length.

When the position of the suspended object load is known, Equation (5) can be equivalent to inhomogeneous linear equations for solving the cable tension, and the condition $\mathrm{rank}(\boldsymbol{J})\leq \min (g,h)$ is satisfied. The static workspace of the lifting system is solved in the following steps [24]:

(1) The end position of the robot and the position of the suspended object are randomly generated by the Monte-Carlo algorithm. The $\mathrm{rank}(\boldsymbol{J})$ and $\mathrm{rank}(\boldsymbol{J},\boldsymbol{F})$ are separately calculated.

(2) Determining the $\mathrm{rank}(\boldsymbol{J})=\mathrm{rank}(\boldsymbol{J},\boldsymbol{F})=g$ whether it is true, if it is true, $\boldsymbol{T}=\boldsymbol{J}^{-1}\boldsymbol{F}$ , and the program enter step (4), if it is not true, and enter step (3).

(3) Determining the $\mathrm{rank}(\boldsymbol{J})=\mathrm{rank}(\boldsymbol{J},\boldsymbol{F})\lt g$ whether it is true, if it is true, $\boldsymbol{T}^{*}=\boldsymbol{J}^{\mathrm{T}}\left(\boldsymbol{J}\right.\left.\boldsymbol{J}^{\mathrm{T}}\right)^{-1}\boldsymbol{F}$ , and the program enter step (4), if it is not true, the program return to step (1).

(4) Determining whether the cable tension $0\lt T_{\min }^{s}\leq T_{i}\leq T_{\max }^{s}$ is satisfied, if so, the position of the point is recorded, and the program enter step (5), if not, directly return to step (1).

(5) Steps (1) to (4) were repeated until the cycle ends.

5. Analysis of the workspace boundary

The solution of the workspace boundary is the basis of the workspace analysis, and the workspace boundary is also an index to evaluate whether the lifting system can work stably. The boundary analysis of the workspace to the lifting system is known the range of the structural parameters and motion parameters of the robot, and the ultimate position of the suspended object is solved. Compared with a single robot of the workspace, the solution of the workspace of a multi-robot coordinated lifting system is more complicated, and the main difficulty is the quantification of the workspace. A low-precision workspace can only describe one index of the system characteristics qualitatively, while a high-precision workspace can describe the parameters of the system characteristics quantitatively [25].

Due to the complex structure of the unconstrained multi-robot coordinated lifting system, the solution of its workspace is solved by numerical method. The point set of the traditional Monte-Carlo method for solving the workspace boundary is sparse, so it is impossible to extract the high-precision workspace boundary. By improving the Monte-Carlo algorithm, the workspace boundary is more accurate. Based on the existing boundary points extraction algorithms, the hierarchical method, partitioning method, adding random points in the neighborhood, and local coordinate method are modified and optimized to extract the workspace boundary of the multi-robot coordinated lifting system using the improved Monte-Carlo algorithm.

5.1. Solving the initial workspace

Taking the wheeled lifting robot as an equilateral triangle configuration, the reachable workspace of the lifting system is solved. It is known that the mass of the suspended object is 10 kg, the allowable tension of the cable is 100 N, the connecting points between the suspended object and the cables is an equilateral triangle, the distance between the connecting point and the barycenter of the suspended object is 0.2 m. The barycenter of the three lifting robots is an equilateral triangle with an outer circle with a radius of 2 m, and the link length of the robot is $a_{i1}=1\,\mathrm{m}, a_{\mathrm{i}2}=0.5\,\mathrm{m}, a_{i3}=0.5\,\mathrm{m}$ , respectively.

The workspace of the suspended object solved according to the traditional Monte-Carlo method is shown in Fig. 4, with a total of 50,000 points satisfying the conditions. Fig. 4(b), 4(c), and 4(d) are the 3D projections of the workspace.

Figure 4. Workspace of the suspended object.

The overall workspace is a triangular prism in Fig. 4, and the distribution of points in the workspace is gradually sparse from the inside to the boundary. As the number of cycles increases, the number of points in the workspace will increase, but the workspace boundary remains unclear. At the same time, a small number of points are found to exceed the overall workspace, which should be eliminated when solving the workspace boundary, and the distribution and density of boundary points should be improved. The wheeled robots are positioned before lifting, so their positions are fixed. The boundary points of the workspace form an equilateral triangle with a roughly symmetric distribution. Therefore, the vertices of the triangle can roughly reflect the relative distribution of the positions of the wheeled robots.

5.2. Layering of the workspace

The boundary points of the workspace are the limited position of the suspended object in the lifting space, and the characteristics of the lifting system can be expressed through the quantification of the workspace. For the workspace of the unconstrained multi-robot coordinated lifting system, the boundary is extracted by layering, and the 3D random points in each layer are projected onto a plane of the layer, and the workspace is transformed from 3D to 2D. Although the boundary of the projection plane becomes clearer, errors are introduced, as shown in Fig. 5, where the boundary below the projection plane increases after the downward projection, and similarly, the upward projection causes the boundary above the projection plane to decrease. This error can be neglected when the layer spacing is small enough, but an increase in the number of layers will increase the calculation.

Figure 5. Diagram of layering error.

The error is reduced by the intra-layer refinement, that is, a point within a thinner sub-layer is taken from the center of the large layer of the initial layering and projected onto the central plane of the large layer, effectively reducing the projection error caused by layering and reducing the useless points projected onto the plane. The workspace shown in Fig. 4 is divided into 50 large layers averaged along the Z-axis, and then each large layer is divided into 10 small layers averaged along the Z-axis. The points of a small layer in the middle of each large layer are taken and projected onto the central plane of the large layer as the boundary points of the whole layer. According to the above method, after layering workspace and projections are shown in Fig. 6.

Figure 6. Workspace after layering.

5.3. Extracting the boundary points

It is difficult to extract the boundary points of the workspace due to the uneven distribution of the points. The workspace points of each layer of the projection plane were divided into 50 columns along the X-axis, the maximum and minimum value points of the Y coordinate in each column were found, these two points were taken as the outer boundary points of the column, and all the obtained boundary points were defined as the column boundary points. Similarly, the row boundary points of the workspace can be obtained by dividing the workspace points into 50 rows along the Y-axis. The row boundary points and column boundary points extracted at the 20th layer according to the above method are shown in Fig. 7.

Figure 7. Boundary points of the workspace.

Due to the influence of divided accuracy and the layered shape, there is a blank distribution of boundary points in the workspace, and it is difficult to extract the boundary points of the upper corner and the left and right corners in Fig. 7. Considering that there are no voids in the workspace of the suspended object, to avoid the problems of low precision and discontinuity of the boundary due to the uniaxial division, a superposition of row boundary points and column boundary points is the most reasonable way to extract the boundary points.

The boundary points of the workspace all satisfy the dynamic balance conditions of the suspended object and the geometric constraints of the lifting system. The boundary points are all within the workspace of the suspended object, so these points are reachable. Taking the 20th layer as an example, the boundary points initially extracted by the superposition method are shown in Fig. 8, with 132 points in total. The distribution of boundary points is sparse, and the boundary points are all inside the workspace. The cross-sectional shape of the workspace is approximately triangular. The extracted points provide the basis for solving the workspace boundary.

Figure 8. Initial extracted boundary points.

5.4. Solving the workspace boundary

Since the inverse solution of the lifting system is not unique, it can not form a single mapping relationship with the joint angles of the robot. Therefore, random points are added in the neighborhood of the initial extracted boundary points to improve the sparse distribution of boundary points. First, random points are generated in the neighborhood of the extracted boundary points, and then the workspace points satisfying the conditions are extracted from the rank of the structure matrix and the tension of the cables. Since the neighborhood points are a set of reachable points, there may be pseudo-boundary points, so it is necessary to screen the generated new points and eliminate useless points.

The pseudo-boundary points in the workspace boundary can be eliminated using the local coordinate method. First, a local coordinate system is established with point b as the target point in Fig. 9, and all the points satisfying the constraints $\left\{a\!\left(x_{i},y_{i}\right)\right.\left.\!\left| \left(x_{i}-x_{b}\right)^{2}+\left(y_{i}-y_{b}\right)^{2}\lt r^{2}\right.\right\}$ are searched. As long as there is no point in one quadrant, point b is determined as the boundary point, and the selection of the boundary points depends on the searched radius [26].

Figure 9. Schematic diagram of local coordinate method.

According to the revelations of the 2D local coordinate method, a 3D local spherical coordinate method is proposed to extract the boundary points. Whether the target point is a boundary point is judged by the positive or negative difference between the origin and the spherical coordinate. For the boundary points extracted from the column maxima, a point is a boundary point if the difference between the coordinates of the target point and the Y-axis in the neighborhood of the column boundary point is positive. For a boundary point extracted from a column minimum, if the difference is negative, then the point is a boundary point. Similarly, for the boundary points extracted from row maxima and row minima, the boundary points are extracted by comparing the coordinates of the X-axis.

Since the random points generated by the 3D local spherical coordinate method are in space, the errors introduced by layering can be eliminated after many cycles, and the final workspace boundary is more precise after removing the pseudo-boundary points. The steps of the improved Monte-Carlo method to solve the workspace of the lifting system are as follows:

(1) The initial workspace is obtained by the traditional Monte-Carlo algorithm.

(2) The maximum and minimum values of the Z-axis are found in the workspace, and it is divided into 50 large layers evenly at first, and then each large layer is divided into 10 small layers. The boundary points in the middle of the small layer are taken as the boundary points in the large layer.

(3) Each layer of the workspace is divided into 50 rows and 50 columns along the Y-axis and X-axis, and the points of the maximum and minimum values in each row and column are taken as the initial extracted boundary points.

(4) Based on the initial extracted boundary points of each layer, 10 random points are generated in its neighborhood $[-0.05,0.05]$ , and the points satisfying the constraint conditions are added to the boundary points obtained in step (3).

(5) The added points are searched and screened by using the local coordinate method. The radius of the circle or ball in the local coordinate method is 0.05 m, and the boundary points satisfying the conditions are saved.

(6) Cycling steps (4) and (5), until the number of cycles is reached, and finally, all the extracted boundary points are obtained.

Taking the 20th layer as an example, the workspace boundary after adding 10 random points in the neighborhood of the initially extracted boundary points is shown in Fig. 10, the red points are the boundary points of the initial extraction, and the blue points are the newly added points. Fig. 10(a) and (c), respectively, show the workspace and projection after adding 2D random points, and Fig. 10(b) and (d), respectively, show the workspace and projection after adding 3D random points.

Figure 10. Workspace after adding random points.

Fig. 10 shows that the boundary points obtained by adding 3D random points are evenly distributed and sparse compared with those obtained by adding 2D random points; the boundary points obtained by adding 2D random points are distributed in a neighborhood of few points and some points are beyond the neighborhood; the boundary points obtained by adding 3D random points are evenly distributed in the neighborhood with more points.

Based on the workspace after adding random points in Fig. 10, the workspace boundary after removing pseudo-boundary points by using the 2D local coordinate method and 3D local spherical coordinate method are shown in Fig. 11(a) and 11(b). It can be found that under the same number of cycles, the workspace boundary of removing pseudo-boundary points by using the 3D local spherical coordinate method is clearer.

Figure 11. Workspace boundary with pseudo-boundary points removed.

According to the above solved method, after adding random points and removing pseudo-boundary points, the workspace boundary obtained after 10 cycles are shown in Fig. 12, in which Figures (a), (c), (e), and (g) are the boundary and projections of the workspace after adding 2D random points and removing pseudo-boundary points, respectively. There are 116 boundary points in each layer. The error in the X-axis is $[-0.20,0.22]$ , and the error in the Y-axis is $[-0.08,0.15]$ . Figures (b), (d), (f), and (h) are the boundary and projections of the workspace after adding 3D random points and removing pseudo-boundary points, respectively. There are 128 boundary points in each layer. The error in the X-axis is $[-0.13,0.12]$ , and the error in the Y-axis is $[-0.05,0.06]$ .

Figure 12. Workspace boundary.

It should be noted that the workspace boundaries are obtained regardless of whether it is a 2D or 3D workspace, without considering the stability and other constraints of the lifting system. Compared to the workspace boundary of the initial solution, the improved workspace boundary is obtained by superposing the central plane boundary points of each layer. As can be seen from Fig. 12, the workspace boundary after adding 2D random points and removing pseudo-boundary points has a larger error between layers and a smaller number of boundary points, and the maximum error is 0.22 m on the X-axis and 0.15 m on the Y-axis. The maximum error is 0.13 m on the X-axis and 0.06 m on the Y-axis when the workspace boundary is added with 3D random points and the pseudo-boundary points are removed. For the extensive multi-robot coordinated lifting system with heavy load and low speed, the error of the workspace boundary obtained by the former method is too large, and the error mainly comes from the layering of the workspace. The error in the workspace boundary obtained by the latter method can meet the accuracy of the lifting system. Compared to the multi-robot coordinated lifting system with fixed base, the literature [27] conducted a detailed analysis of the workspace of a lifting system using the principle of least squares combined with the Monte-Carlo algorithm. This method can quickly calculate the static balance workspace of a multi-robot coordinated lifting system with different spatial configurations, but there is a problem of unclear workspace boundaries, which makes it difficult to accurately quantify for guiding practical lifting tasks. The improved Monte-Carlo algorithm increases the number of workspace points by about 30.96% compared to the traditional algorithm, which proves that the improved Monte-Carlo algorithm obtains a more complete workspace and clearer workspace boundaries, as shown in Fig. 12, which not only improves the problem of uneven distribution of random points in the Monte-Carlo algorithm but also avoids the waste of encrypted points, which proves the effectiveness of the proposed method for obtaining workspace boundaries.

6. Analysis of the structural parameters on the workspace boundary

The workspace is the basis for parameter optimization and stability analysis in the multi-robot coordinated lifting system. The workspace of the multi-robot coordinated lifting system is jointly determined by the structure of the single-wheeled lifting robot and the cable-driven parallel lifting system. Different lifting tasks can be met by changing the structural parameters, which will inevitably affect the workspace boundary. Therefore, it is necessary to analyze the influence of critical structural parameters on the workspace. The main structural parameters that affect the workspace of the multi-robot coordinated lifting system are the mass of the suspended object, the link length of the robot, the distance between the wheeled lifting robots, and the distance between the end of the robot and the lifting point of the suspended object. The workspace volume describes the size of the workspace. It is necessary to solve the workspace volume based on the workspace boundary and then analyze the influence of the structural parameters on the workspace [28].

6.1. Workspace volume

The methods to solve the workspace volume include the voxel method and the integral method. The voxel method is to approximate the volume of the space with many voxels [20]. When the voxel is small, the volume is incomplete due to the sparse distribution of points on the workspace boundary, and when the voxel is large, the error increases. The integral method is fast and easy to calculate. The main error of the integral method comes from the accuracy of the boundary points in the layer plane and the boundary errors between layers. The previous section solved the relatively high-precision workspace boundary, and since the workspace of the suspended object is a closed triangular prism with no internal void, the integral method was chosen to solve the workspace volume.

The workspace volume in Figs. 12 (a) and (b) is 6.498 m³ and 7.399 m³, respectively, by using the integral method. Under the same number of cycles, the workspace volume obtained by adding 3D random points is significantly larger, which verifies the result of extracting the workspace boundary in the previous section.

6.2. Factors affecting the workspace

After solving the workspace volume, a one-way function mapping relationship between the structural parameters and the workspace volume can be established. In the lifting condition with variable cable length (the end position of the robot is unchanged), several critical structural parameters affecting the workspace volume are the mass m of the suspended object, the link length L of the robot, the diameter D of the outer circle of the triangle where the wheeled lifting robot is located, and the horizontal distance d between the end of the robot and the lifting point of the suspended object. A specific analysis will be provided below:

(1) At any position in the workspace, the mass of the suspended object is proportional to the cable tension. To ensure the stability and safety of the lifting, it is necessary to limit the mass of the suspended object.

(2) The link length of the robot is unified into a structural parameter L, which is the distance from the end of the robot to the barycenter of the rolling base. When the mass of the suspended object is known, an increase in L can increase the workspace volume, but this will lead to an increase in the torque on the rolling base, resulting in a decrease in the carrying capacity.

(3) The horizontal distance between the lifting robots will affect on the position of the workspace. After the configuration of the lifting system is determined as a triangle, the distance between the lifting robots is determined by the diameter D of the outer circle of the triangle. On the premise that the lifting does not fail, the larger the D, the larger the workspace volume.

(4) To avoid interference between the cables during lifting, the horizontal distance between the robot end and the lifting point of the suspended object should be reasonable. The size of d affects the inclination angle of the cable and directly affects the workspace volume of the suspended object.

The influence of structural parameters on the volume of the workspace is simulated using the control variable method and forward search method. Due to the coupling relationship between D, L, and d, the influence of these three parameters on the workspace volume is analyzed. Figure 13 show the changes in workspace volume when d = 0.4 m, 0.6 m, 0.8 m, and 1.0 m, respectively. The following conclusions can be obtained: (1) with the increase of d, the workspace volume decreases, and the maximum volume decreases from 11.118 m³ (Figure a) to 7.216 m³ (Figure d); (2) when L is determined, the maximum workspace volume first increases and then decreases with the increase of D; (3) when D is determined, the maximum workspace volume first increases and then decreases with the increase of L.

Figure 13. Influence of D, L, and d on the workspace volume.

When D is determined, the waterfall diagram of L and d on the workspace volume is shown in Fig. 14. The workspace volume decreases with the increase of d, and the workspace volume increases first and then decreases with the increase of L. When d increases to a specific value, the workspace volume is discontinuous, and the maximum value of d is 0.6 m. For L = 1.86 m, the workspace volume is the maximum at different d, and then the workspace volume gradually decreases. Therefore, L = 1.86 m is the boundary value, which also indicates the maximum workspace.

Figure 14. Influence of L and d on the workspace volume.

When d and L are determined, the effect of D on the workspace volume is shown in Fig. 15. The workspace volume first increases and then decreases with the increase of D. For D = 4.46 m, the workspace volume is the largest, and the workspace has the largest boundary.

Figure 15. Influence of D on the workspace volume.

When D, L, and d are determined, the influence of the mass m of the suspended object on the workspace volume is shown in Fig. 16. The workspace volume decreases with the increase of m. For m = 200 kg, there are too few points that can reach the workspace, the workspace boundary cannot be solved, and it is considered that the system cannot perform the specified lifting task. If the minimum lifting requirement is that the workspace volume is equal to 0.5 m³, the maximum value of m is 185 kg.

Figure 16. Influence of m on the workspace volume.

7. Conclusion

Based on the analysis of the spatial configuration of a multi-robot coordinated lifting system with rolling base, the kinematic equations and static balance equations of the lifting system are established. To address the problem of an unclear workspace boundary of the suspended object using the traditional Monte-Carlo algorithm, a hierarchical strategy is proposed, the initial boundary points are extracted through row and column division, and the workspace boundary is solved by adding 3D random points and removing pseudo-boundary points using 3D local spherical coordinates. The effectiveness of the proposed method is verified through simulations. As a new research direction for a multi-robot coordinated lifting system, the workspace boundary has been the focus of the research. The analysis of the factors affecting the workspace provides a valuable data reference for the choice of structural parameters and contributes to the practical application of the lifting system.