Configuration design and kinematics analysis of foldable composite space capture mechanism

Abstract

To meet the requirements of high flexibility, high stiffness, and large working range of space debris cleaning technology, a foldable composite space capture mechanism is proposed based on the finite screw theory in this paper. The degree of freedom of the mechanism is analyzed by using the screw theory and its degree of freedom is obtained. The closed-loop vector method and DH method are used to analyze the kinematics of the mechanism, and the position solution of the mechanism is derived. The analytical expressions of velocity and acceleration solutions are obtained by differential calculation. According to the analysis of numerical example, the kinematic model of the mechanism is built in the simulation environment, and the simulation experiment is carried out for the motion trajectory of the mechanism. The correctness of theoretical analysis solution and method is proved by contrasting and analyzing motion trajectory, and it is verified that the space capture mechanism uses the foldable property to form large workspace to complete the capture of the target.

Introduction

With the development of aerospace industry, the amount of discarded space debris in space is increasing gradually. The space debris has the characteristics of irregular shape, excessive inertia, and unknown motion parameters, will cause impact threat to spacecraft in use. Therefore, it is very difficult and risky to capture space debris on orbit. ¹ The research and development of a space capture mechanism with low accuracy, high stiffness, high load capacity, and large workspace has important scientific significance and research value for enriching and improving space debris cleaning technology and improving innovative development in the aerospace field.

In view of this phenomenon, domestic and foreign scholars have carried out in-depth research on space debris cleaning technology. At present, there are three kinds of space debris cleaning technologies: rigid capture by space manipulator^2–4 (as shown in Figure 1), telescopic rod to capture engine nozzle ⁵ (as shown in Figure 2), and flexible capture by mechanisms such as flying nets and flying claws ⁶ (as shown in Figure 3). Most of the manipulators used for capturing are series manipulators, which have high dexterity and large workspace, but weak load capacity. In the work, the target will fail to be captured due to chattering or deformation. The parallel manipulator has high stiffness and strong load capacity, but small workspace. Using telescopic rod to capture engine nozzle requires high capture accuracy and difficult operation. Using flying net or flying claw capture mechanism has low stiffness, unable to carry out other operations on the target.

Figure 1.

European space manipulator ERA.

Figure 2.

SMART-OLEV.

Figure 3.

ROGER capture system: (a) fly net type and (b) fly claw type.

Aiming at the existing problems of the current space capture mechanism, this paper proposes a foldable composite space capture mechanism based on a single ring folding mechanism and a translation and rotation coupling mechanism. The single ring folding mechanism is a closed-loop link system composed of the link and the motion joint between the link, which has the advantages of large expansion ratio, high stiffness, and good flexibility, etc., and has been used in truss deployable antenna and other fields.^7,8 The translation and rotation coupling mechanism has large workspace, which can quickly locate the moving target to be captured and realize detumbling operation, so that its attitude and speed are synchronized with the target to be captured. The combined space capture mechanism of the two is easier to capture the target and can eliminate the impact and interference of space debris in the space station. Yao et al. ⁹ proposed a polyhedral network space capture mechanism based on three-RRS parallel mechanism. Liu et al. ¹⁰ completed the clamping and pushing of the docking target based on the scaling and flipping characteristics of the Bricard mechanism. However, the above research ignored the kinematic coupling between the composite mechanisms and only analyze the simplified model. Because of the variability of its structure and the complexity of its design problems, the analysis and research of this kind of mechanism has been a great challenge for a long time.

In view of the shortcomings of the current space debris cleaning technology, the configuration design and kinematics analysis of the foldable composite space capture mechanism are studied in this paper. The kinematic decoupling is realized by constructing the internal relation of the single ring folding mechanism and the translation and rotation coupling mechanism, to ensure that the composite space capture mechanism can complete the complex tasks such as detumbling operation and in-orbit maintenance. The specific research contents of this paper are as follows:

In section “Configuration design,” combined with the configuration characteristics of the space capture mechanism, the configuration characteristics of the composite mechanism are obtained by using the finite screw theory. In section “Degree of freedom analysis,” under the topological configuration of the mechanism, the screw theory is used to analyze the degree of freedom of the mechanism, to ensure that the foldable composite space capture mechanism can achieve the capture task. In section “Kinematic analysis,” the DH method is used to analyze the position solution of 8R mechanism. And then the closed-loop vector method is used to analyze the position solution of PRS-PSS-PRS-PSS mechanism. Finally, the velocity and acceleration solutions of the mechanism are derived by differential calculation. In section “Numerical examples,” based on the kinematics analysis of the mechanism, the numerical example analysis is carried out. By comparing the motion trajectory obtained by MATLAB software programming and SolidWorks software simulation, it is verified that the measured values are consistent with the theoretical values. And then the correctness and feasibility of the theoretical analysis method is verified.

Configuration design

Space debris mainly includes discarded satellites and rocket fragments, which are collectively referred to as space non-cooperative objects. Due to its large spin, ¹¹ it is convenient to capture it quickly and comprehensively. A foldable composite space capture mechanism is proposed in this paper, which the translation and rotation coupling mechanism is the main part and the single ring folding mechanism is the moving platform. The space capture mechanism should be defined with four degrees of freedom.

According to the different types of freedom, the 4-DOF (degree of freedom) parallel mechanism can be generally divided into 1T3R, 2T2R and 3T1R, where T represents the translation DOF, and R represents the rotation DOF. According to the shortcomings of the existing space capture mechanism, it is defined that the space capture mechanism should have 2T2R. Firstly, the configuration synthesis of the parallel mechanism is carried out. At present, some scholars have conducted configuration synthesis research on the 4-DOF parallel mechanism.^12,13 In this paper, based on the results of Xie’s configuration synthesis of the 2T2Rmechanism, ¹⁴ as shown in Table 1, the configuration design of the space capture mechanism is completed using the finite screw theory.

Table 1.

2T2Rmechanism configuration synthesis results.

Limb	Type of constraint	Limb configuration
Limb 1	One-dimensional couple constraint	PR(Pa)RR
Limb 2	One-dimensional force constraint	PRS
Limb 3	No constraint	PSS

Three limb configurations can be obtained by reasonable arrangement of all joints, as shown in Figure 4.

Figure 4.

Limb configuration: (a) PR(Pa)RR, (b) PRS, and (c) PSS.

Based on the finite screw theory, ¹⁵ the PR(Pa)RR limb configuration can be expressed as

$$\begin{array}{c}{\mathit{S}}_f=2\tan \frac{{\theta }_4}{2}\left(\begin{array}{c}{\mathit{s}}_4\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_4\times {\mathit{s}}_4}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\theta }_3}{2}\left(\begin{array}{c}{\mathit{s}}_3\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_3\times {\mathit{s}}_3}\end{array}\right)\hfill \\ \mathrm{\Delta }2\tan \frac{{\theta }_2}{2}\left(\begin{array}{c}{\mathit{s}}_2\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_2\times {\mathit{s}}_2}\end{array}\right)\mathrm{\Delta }{t}_1\left(\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{s}}_1}\end{array}\right)\hfill \end{array}$$

(1)

where, the symbol “Δ” represents the screw triangle product, $t_1\left(\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{s}}_1}\end{array}\right)$ for translation, $2\tan \frac{{\theta }_i}{2}\left(\begin{array}{c}{\mathit{s}}_i\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_i\times {\mathit{s}}_i}\end{array}\right)(i=2,3,4)$ for rotation.

The PRS limb configuration can be expressed as

$$\begin{array}{c}{\mathit{S}}_f=2\tan \frac{{\theta }_5}{2}\left(\begin{array}{c}{\mathit{s}}_5\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_5\times {\mathit{s}}_5}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\theta }_4}{2}\left(\begin{array}{c}{\mathit{s}}_4\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_4\times {\mathit{s}}_4}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\theta }_3}{2}\left(\begin{array}{c}{\mathit{s}}_3\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_3\times {\mathit{s}}_3}\end{array}\right)\hfill \\ \mathrm{\Delta }2\tan \frac{{\theta }_2}{2}\left(\begin{array}{c}{\mathit{s}}_2\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_2\times {\mathit{s}}_2}\end{array}\right)\mathrm{\Delta }{t}_1\left(\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{s}}_1}\end{array}\right)\hfill \end{array}$$

(2)

The PSS limb configuration can be expressed as

$$\begin{array}{c}{\mathit{S}}_f=2\tan \frac{{\theta }_7}{2}\left(\begin{array}{c}{\mathit{s}}_7\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_7\times {\mathit{s}}_7}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\theta }_6}{2}\left(\begin{array}{c}{\mathit{s}}_6\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_6\times {\mathit{s}}_6}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\theta }_5}{2}\left(\begin{array}{c}{\mathit{s}}_5\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_5\times {\mathit{s}}_5}\end{array}\right)\hfill \\ \mathrm{\Delta }2\tan \frac{{\theta }_4}{2}\left(\begin{array}{c}{\mathit{s}}_4\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_4\times {\mathit{s}}_4}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\theta }_3}{2}\left(\begin{array}{c}{\mathit{s}}_3\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_3\times {\mathit{s}}_3}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\theta }_2}{2}\left(\begin{array}{c}{\mathit{s}}_2\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_2\times {\mathit{s}}_2}\end{array}\right)\mathrm{\Delta }{t}_1\left(\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{{\mathit{s}}_1}\end{array}\right)\hfill \end{array}$$

(3)

After the available limb configurations are obtained, the overall configuration of the mechanism is obtained through assembly conditions and actuation arrangement. Three configurations are listed here, as shown in Figure 5.

Figure 5.

Mechanism configuration: (a) PRS-PSS-PRS-PSS, (b) PRS-PRPaRR-2PSS, and (c) PRU-3PSS.

According to the application scenario, the configuration of PRS-PSS-PRS-PSS mechanism is selected, and its configuration characterization is shown in formula (4). Because its limbs are symmetrically distributed and the configuration is regular, and it can quickly locate the target to be captured. Thus, it is the main part of the foldable composite space capture mechanism in this paper.

$$\{\begin{array}{l}{\mathit{S}}_{f,i}=2\tan \frac{{\mathit{\theta }}_{i,5}}{2}\left(\begin{array}{c}{\mathit{s}}_{i,5}\\ {\mathit{r}}_{s,i}\times {\mathit{s}}_{i,5}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\mathit{\theta }}_{i,4}}{2}\left(\begin{array}{c}{\mathit{s}}_{i,4}\\ {\mathit{r}}_{s,i}\times {\mathit{s}}_{i,4}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\mathit{\theta }}_{i,3}}{2}\left(\begin{array}{c}{\mathit{s}}_{i,3}\\ {\mathit{r}}_{s,i}\times {\mathit{s}}_{i,3}\end{array}\right)\\ \mathrm{\Delta }2\tan \frac{{\mathit{\theta }}_{i,2}}{2}\left(\begin{array}{c}{\mathit{s}}_{i,2}\\ {\mathit{r}}_{r,i}\times {\mathit{s}}_{i,2}\end{array}\right)\mathrm{\Delta }{t}_{i,1}\left(\begin{array}{c}\mathit{0}\\ {\mathit{s}}_{i,1}\end{array}\right),i=1,3\\ {\mathit{S}}_{f,i}=2\tan \frac{{\mathit{\theta }}_{i,7}}{2}\left(\begin{array}{c}{\mathit{s}}_{i,7}\\ {\mathit{r}}_{i,7}\times {\mathit{s}}_{i,7}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\mathit{\theta }}_{i,6}}{2}\left(\begin{array}{c}{\mathit{s}}_{i,6}\\ {\mathit{r}}_{i,6}\times {\mathit{s}}_{i,6}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\mathit{\theta }}_{i,5}}{2}\left(\begin{array}{c}{\mathit{s}}_{i,5}\\ {\mathit{r}}_{i,5}\times {\mathit{s}}_{i,5}\end{array}\right)\\ \mathrm{\Delta }2\tan \frac{{\mathit{\theta }}_{i,4}}{2}\left(\begin{array}{c}{\mathit{s}}_{i,4}\\ {\mathit{r}}_{i,4}\times {\mathit{s}}_{i,4}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\mathit{\theta }}_{i,3}}{2}\left(\begin{array}{c}{\mathit{s}}_{i,3}\\ {\mathit{r}}_{i,2}\times {\mathit{s}}_{i,3}\end{array}\right)\mathrm{\Delta }2\tan \frac{{\mathit{\theta }}_{i,2}}{2}\left(\begin{array}{c}{\mathit{s}}_{i,2}\\ {\mathit{r}}_{i,2}\times {\mathit{s}}_{i,2}\end{array}\right)\mathrm{\Delta }{t}_{i,1}\left(\begin{array}{c}\mathit{0}\\ {\mathit{s}}_{i,1}\end{array}\right),i=2,4\end{array}$$

(4)

The rotation and workspace of the parallel mechanism are limited. To achieve flexible capture, the moving platform can adjust itself according to the target shape to find the most easily captured side of the target for capture. Therefore, the moving platform is a single ring folding mechanism, which has the function of folding and closing to pull the target inward and cooperate with the limb to clamp the target, and thus prevent the target from escaping. At present, the classic single ring folding mechanism includes four-links mechanism-Bennett mechanism, ¹⁶ six-links mechanism-Bricard mechanism, ¹⁷ and eight-links mechanism-8R mechanism. ¹⁸

The motion process of three single ring folding mechanisms is shown in Figures 6 to 8. Through comparative analysis, the touchpoints of Bennett mechanism and Bricard mechanism are 2, and the touchpoints of 8R mechanism are 4. Considering the symmetry of the folding mechanism, the 8R mechanism is selected as the single ring folding mechanism of the moving platform.

Figure 6.

Bennett mechanism: (a) fully expanded state, (b) intermediate state, and (c) fully folded state.

Figure 7.

Bricard mechanism: (a) fully expanded state, (b) intermediate state, and (c) fully folded state.

Figure 8.

8R mechanism: (a) fully expanded state, (b) intermediate state, and (c) fully folded state.

Based on the above analysis, this paper proposes a foldable composite space capture mechanism with PRS-PSS-PRS-PSS mechanism as the main part and 8R mechanism as the moving platform, as shown in Figure 9. The mechanism can meet the requirements of high flexibility, high stiffness, and large workspace for space debris cleaning technology.

Figure 9.

8R& PRS-PSS-PRS-PSS mechanism schematic diagram.

Degree of freedom analysis

As shown in Figure 10, PRS-PSS-PRS-PSS mechanism is composed of two PSS limbs and two PRS limbs. The PSS limb is shown in Figure 11, and the screw system of the limb can be expressed as

$${\$}_1=\{\begin{array}{c}{\$}_{11}=(\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & n_1\hfill \end{array})\hfill \\ \multicolumn{1}{c}{{\$}_{12}=(\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}0\hfill & a\hfill & 0\hfill \end{array})}\\ \multicolumn{1}{c}{{\$}_{13}=(\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array};\begin{array}{ccc}-a\hfill & 0\hfill & 0\hfill \end{array})}\\ \multicolumn{1}{c}{{\$}_{14}=(\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array})}\\ \multicolumn{1}{c}{\begin{array}{c}{\$}_{15}=(\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}0\hfill & d\hfill & -c\hfill \end{array})\hfill \\ \multicolumn{1}{c}{{\$}_{16}=(\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array};\begin{array}{ccc}-d\hfill & 0\hfill & b\hfill \end{array})}\\ \multicolumn{1}{c}{{\$}_{17}=(\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array};\begin{array}{ccc}c\hfill & -b\hfill & 0\hfill \end{array})}\end{array}}\end{array}$$

(5)

Figure 10.

PRS-PSS-PRS-PSS mechanism schematic diagram..

Figure 11.

Screw analysis of PSS limb.

where, $n_1$ , $a$ , $b$ , $c$ , and $d$ are screw coordinates of motion joints determined by link length and position of motion joints. Coordinate of spherical joint B is $\left(\begin{array}{ccc}0\hfill & 0\hfill & a\hfill \end{array}\right)$ and coordinate of spherical joint C is $\left(\begin{array}{ccc}b\hfill & c\hfill & d\hfill \end{array}\right)$ . Through calculation, the screw system ${\$}_1$ has no reciprocal screw.

The PRS limb is shown in Figure 12, and the screw system of the limb can be expressed as

$${\$}_2=\{\begin{array}{c}{\$}_{21}=(\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & m_1\hfill \end{array})\hfill \\ \multicolumn{1}{c}{{\$}_{22}=(\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}0\hfill & e\hfill & 0\hfill \end{array})}\\ \multicolumn{1}{c}{{\$}_{23}=(\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}0\hfill & g\hfill & -f\hfill \end{array})}\\ \multicolumn{1}{c}{{\$}_{24}=(\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array};\begin{array}{ccc}-g\hfill & 0\hfill & 0\hfill \end{array})}\\ \multicolumn{1}{c}{{\$}_{25}=(\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array};\begin{array}{ccc}f\hfill & 0\hfill & 0\hfill \end{array})}\end{array}$$

(6)

where, $m_1$ , $e$ , $f$ , and $g$ are screw coordinates of motion joints determined by link length and position of motion joints. Coordinate of spherical joint B is $\left(\begin{array}{ccc}0\hfill & 0\hfill & e\hfill \end{array}\right)$ and coordinate of spherical joint C is $\left(\begin{array}{ccc}0\hfill & f\hfill & g\hfill \end{array}\right)$ . Through calculation, the reciprocal screw of the screw system ${\$}_2$ can be expressed as

$${\$}_2^r=(\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array};\begin{array}{ccc}0\hfill & g\hfill & -f\hfill \end{array})$$

(7)

Figure 12.

Screw analysis of PRS limb.

According to the screw analysis, the mechanism has two reciprocal screws, which can realize the 4-DOF motion with two directions of rotation and two directions of translation. Using the modified Grübler-Kutzbach formula, ¹⁹ it can be obtained

$$\begin{array}{c}\multicolumn{1}{c}{M_1=(6-\lambda )(n-g-1)+\sum _{i=1}^g{f}_i+\nu -\xi }\\ =(6-0)\times (10-12-1)+24+0-2=4\hfill \end{array}$$

(8)

where, $M_1$ represents the DOF; $\lambda$ is the number of common constraints; $n$ is the number of components; $g$ is the number of motion joints; $f_i$ represents the degree of freedom of the $i$ th motion joint; $v$ represents the number of parallel redundant constraints, and $\xi$ represents the number of partial freedom. ²⁰

As shown in Figure 13, the 8R mechanism is a closed ring composed of eight links and eight rotational joints, and the spatial angle of the axis of each rotational joint is 45°. In the general position, the mechanism has two planes of symmetry $V_1$ (green plane) and $V_2$ (red plane). The Cartesian coordinate system is established with the intersection point O of the axes of B, D, F, and H of the rotational joint as the origin. The $x$ axis in plane $V_1$ is parallel to DH, the $y$ axis in plane $V_2$ is parallel to BF, the $z$ axis is the intersection line between plane $V_1$ and plane $V_2$ , $\alpha$ is the angle between axis $y$ and axis B of the rotational joint, and $\beta$ is the angle between axis $x$ and axis H of the rotational joint.

Figure 13.

8R mechanism schematic diagram.

The 8R mechanism can be regarded as a parallel mechanism composed of two RRRR limbs. Let ${\$}_1=\left(\begin{array}{ccc}p& q& r\end{array}�\begin{array}{ccc}l& m& n\end{array}\right)$ , where $\mathit{s}=\left(\begin{array}{ccc}p\hfill & q\hfill & r\hfill \end{array}\right)$ is the axis direction of ${\$}_1$ , ${\mathit{s}}_0=\left(\begin{array}{ccc}l\hfill & m\hfill & n\hfill \end{array}\right)$ represents the axial formula of the joint, and ${\mathit{s}}_0=\mathit{r}\times \mathit{s}$ , $\mathit{r}=\left(\begin{array}{ccc}{x}_1\hfill & y_1\hfill & z_1\hfill \end{array}\right)$ are the distance vectors of the joint axis ${\$}_1$ in the coordinate system. Then the screw system of the first limb can be expressed as

$${\$}_{31}=\{\begin{array}{c}{\$}_1=(\begin{array}{ccc}p\hfill & q\hfill & r\hfill \end{array};\begin{array}{ccc}l\hfill & m\hfill & n\hfill \end{array})\hfill \\ \multicolumn{1}{c}{{\$}_2=(\begin{array}{ccc}0\hfill & \cos \alpha \hfill & \sin \alpha \hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array})}\\ \multicolumn{1}{c}{{\$}_3=(\begin{array}{ccc}-p\hfill & q\hfill & r\hfill \end{array};\begin{array}{ccc}l\hfill & -m\hfill & -n\hfill \end{array})}\\ \multicolumn{1}{c}{{\$}_4=(\begin{array}{ccc}-\cos \beta \hfill & 0\hfill & \sin \beta \hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array})}\end{array}$$

(9)

The screw system of the second limb can be expressed as

$${\$}_{32}=\{\begin{array}{c}{\$}_5=(\begin{array}{ccc}-p\hfill & -q\hfill & r\hfill \end{array};\begin{array}{ccc}-l\hfill & -m\hfill & n\hfill \end{array})\hfill \\ \multicolumn{1}{c}{{\$}_6=(\begin{array}{ccc}0\hfill & -\cos \alpha \hfill & \sin \alpha \hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array})}\\ \multicolumn{1}{c}{{\$}_7=(\begin{array}{ccc}p\hfill & -q\hfill & r\hfill \end{array};\begin{array}{ccc}-l\hfill & m\hfill & n\hfill \end{array})}\\ \multicolumn{1}{c}{{\$}_8=(\begin{array}{ccc}\cos \beta \hfill & 0\hfill & \sin \beta \hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array})}\end{array}$$

(10)

By calculating the reciprocal screw of the two limbs respectively, it can be obtained

$$\begin{array}{c}{\$}_{31}^r=\hfill \\ \{\begin{array}{c}{\$}_1^r=(\begin{array}{ccc}0\hfill & -\frac{n}{m}\hfill & 1\hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array})\hfill \\ \multicolumn{1}{c}{{\$}_2^r=(\begin{array}{ccc}-\frac{r\cos \alpha -q\sin \alpha }{l\cos \alpha }\hfill & -\frac{p\sin \beta }{m\cos \beta }\hfill & 0\hfill \end{array};\begin{array}{ccc}\frac{\sin \beta }{\cos \beta }\hfill & -\frac{\sin \alpha }{\cos \alpha }\hfill & 1\hfill \end{array})}\end{array}\hfill \end{array}$$

(11)

$$\begin{array}{c}\multicolumn{1}{c}{{\$}_{32}^r=\{\begin{array}{c}{\$}_3^r=(\begin{array}{ccc}\frac{n}{l}\hfill & 0\hfill & 1\hfill \end{array};\begin{array}{ccc}0\hfill & 0\hfill & 0\hfill \end{array})\hfill \\ \multicolumn{1}{c}{{\$}_4^r=(\begin{array}{ccc}\frac{r\cos \alpha -q\sin \alpha }{l\cos \alpha }\hfill & \frac{p\sin \beta }{m\cos \beta }\hfill & 0\hfill \end{array};\begin{array}{ccc}-\frac{\sin \beta }{\cos \beta }\hfill & \frac{\sin \alpha }{\cos \alpha }\hfill & 1\hfill \end{array})}\end{array}}\end{array}$$

(12)

Finally, the 8R mechanism has four reciprocal screws, so the degree of freedom of the mechanism is 2. Using the modified G-K formula can be obtained

$$\begin{array}{c}\multicolumn{1}{c}{M_2=(6-\lambda )(n-g-1)+\sum _{i=1}^g{f}_i+\nu }\\ =6\times (8-8-1)+8+0=2\hfill \end{array}$$

(13)

where, $M_2$ represents the DOF; $\lambda$ is the number of common constraints; $n$ is the number of components; $g$ is the number of motion joints; $f_i$ represents the degree of freedom of the $i$ th motion joint; $v$ represents the number of parallel redundant constraints.

Based on the above analysis, according to the mutual constraints between the limbs of the 8R mechanism and the PRS-PSS-PRS-PSS mechanism, the 8R& PRS-PSS-PRS-PSS composite space capture mechanism can realize the 4-DOF motion of two-direction rotation and two-direction translation. It can meet the requirements of quick location, detumbling operation, synchronous capture, large workspace, and high capture efficiency.

Kinematic analysis

Kinematics analysis is the premise of the research mechanism, and its main content is to analyze the position, speed, acceleration, and other aspects of the mechanism. The key to the kinematic analysis of the composite space capture mechanism lies in the kinematic transformation between the 8R&PRS-PSS-PRS-PSS mechanism and the PRS-PSS-PRS-PSS mechanism.

In this section, the DH method is firstly used to establish the link coordinate system at each joint of the 8R mechanism, and then the position analysis of PRS-PSS-PRS-PSS mechanism is carried out through the closed-loop vector method to reveal the relationship between the actuation parameters of the mechanism and the end pose. Finally, the velocity and acceleration solutions of the mechanism are derived through differential calculation to complete the kinematic analysis of the mechanism. As shown in Figure 14.

Figure 14.

Kinematic analysis flow diagram.

When 8R&PRS-PSS-PRS-PSS mechanism performs the capture task, any captured object can be converted into an envelope sphere. Therefore, after the outer contour spherical radius of the captured object is known, the length of the opposite side BF and DH of the 8R mechanism, can be determined. The positions of the end points $C_1$ , $C_2$ , $C_3$ , and $C_4$ of the PRS-PSS-PRS-PSS mechanism in the unified coordinate system can be determined. Finally, the analysis of 8R&PRS-PSS-PRS-PSS mechanism can be converted into the analysis of PRS-PSS-PRS-PSS mechanism, to complete the kinematics analysis of the mechanism. As shown in Figure 15. Taking the capture of the blue sphere as an example, the radius of the sphere is known to be 40 mm, and the reference coordinate system O-xyz is established with the center of the sphere as the origin. The coordinate system is the same as the O-xyz coordinate mentioned below. Then the length of BF and DH can be obtained as 80 mm, $C_1=\left(\begin{array}{cc}0\hfill & -40\hfill \end{array}\right)$ , $C_2\left(\begin{array}{cc}40\hfill & 0\hfill \end{array}\right)$ , $C_3\left(\begin{array}{cc}0\hfill & 40\hfill \end{array}\right)$ , and $C_4\left(\begin{array}{cc}-40\hfill & 0\hfill \end{array}\right)$ can be obtained. Thus, the analysis of 8R&PRS-PSS-PRS-PSS mechanism is transformed into the analysis of PRS-PSS-PRS-PSS mechanism.

Figure 15.

xOy cross section illustration.

Position analysis of 8R mechanism

As shown in Figure 16, the link coordinate system is established at each rotational joint of the mechanism, and the coordinate axis $z_i$ of the $i$ th rotational joint is specified to be along the rotation axis direction of the joint, and the coordinate axis $x_i$ is along the common normal direction of coordinate axis $z_i$ and $z_{i+1}$ , and points from $z_i$ to $z_{i+1}$ .

Figure 16.

8R mechanism structure diagram.

The coordinate system is established according to the DH method, and the corresponding parameters of the connecting link are shown in Table 2.

Table 2.

DH parameters of 8R mechanism.

$i$	$a_{i-1}$	${\alpha }_{i-1}$	$d_i$	${\theta }_i$
1	$L$	45°	0	${\theta }_1$
2	$L$	−45°	0	${\theta }_2$
3	$L$	45°	0	${\theta }_3$
4	$L$	−45°	0	${\theta }_4$
5	$L$	45°	0	${\theta }_5$
6	$L$	−45°	0	${\theta }_6$
7	$L$	45°	0	${\theta }_7$
8	$L$	−45°	0	${\theta }_8$

For the established coordinate system, the general formula of transformation matrix from the $i-1$ th coordinate system to the $i$ th coordinate system is

$$\begin{array}{c}\multicolumn{1}{c}{{}_i^{i-1}\mathit{T}=\left[\begin{array}{cccc}\mathrm{c}{\theta }_i\hfill & -\mathrm{s}{\theta }_i\hfill & 0\hfill & a_{i-1}\hfill \\ \multicolumn{1}{c}{\mathrm{c}{\alpha }_{i-1}\mathrm{s}{\theta }_i}& \mathrm{c}{\alpha }_{i-1}\mathrm{c}{\theta }_i\hfill & -\mathrm{s}{\alpha }_{i-1}\hfill & -d_i\mathrm{s}{\alpha }_{i-1}\hfill \\ \multicolumn{1}{c}{\mathrm{s}{\alpha }_{i-1}\mathrm{s}{\theta }_i}& \mathrm{s}{\alpha }_{i-1}\mathrm{c}{\theta }_i\hfill & \mathrm{c}{\alpha }_{i-1}\hfill & d_i\mathrm{c}{\alpha }_{i-1}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]}\end{array}$$

(14)

Where, $c{\theta }_i$ and $\mathrm{s}{\theta }_i$ respectively represent $\cos {\theta }_i$ and $\sin {\theta }_i$ ; the first three columns of the matrix respectively represent the direction cosine of axis $x_i$ , axis $y_i$ and axis $z_i$ of the $i$ th coordinate system in the $i-1$ th coordinate system; the fourth column represents the position of the origin of the coordinates of the $i$ th coordinate system in the $i-1$ th coordinate system.

It is known that the 8R mechanism has two degrees of freedom, the actuated angle $\theta$ and $\phi$ with the slave angle satisfy the relation: ${\theta }_2={\theta }_4={\theta }_6={\theta }_8=\theta$ and ${\theta }_1={\theta }_3={\theta }_5={\theta }_7=\phi$ .

According to the above angle relationship, the corresponding transformation matrix can be written as

$${}_2^1\mathit{T}=\left[\begin{array}{cccc}{n}_{x_2}\hfill & o_{x_2}\hfill & a_{x_2}\hfill & p_{x_2}\hfill \\ \multicolumn{1}{c}{n_{y_2}}& o_{y_2}\hfill & a_{y_2}\hfill & p_{y_2}\hfill \\ \multicolumn{1}{c}{n_{z_2}}& o_{z_2}\hfill & a_{z_2}\hfill & p_{z_2}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(15)

$${}_4^1\mathit{T}={}_2^1\mathit{T}{}_3^2\mathit{T}{}_4^3\mathit{T}=\left[\begin{array}{cccc}{n}_{x_4}\hfill & o_{x_4}\hfill & a_{x_4}\hfill & p_{x_4}\hfill \\ \multicolumn{1}{c}{n_{y_4}}& o_{y_4}\hfill & a_{y_4}\hfill & p_{y_4}\hfill \\ \multicolumn{1}{c}{n_{z_4}}& o_{z_4}\hfill & a_{z_4}\hfill & p_{z_4}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(16)

$${}_6^1\mathit{T}={}_2^1\mathit{T}{}_3^2\mathit{T}{}_4^3\mathit{T}{}_5^4\mathit{T}{}_6^5\mathit{T}=\left[\begin{array}{cccc}{n}_{x_6}\hfill & o_{x_6}\hfill & a_{x_6}\hfill & p_{x_6}\hfill \\ \multicolumn{1}{c}{n_{y_6}}& o_{y_6}\hfill & a_{y_6}\hfill & p_{y_6}\hfill \\ \multicolumn{1}{c}{n_{z_6}}& o_{z_6}\hfill & a_{z_6}\hfill & p_{z_6}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(17)

$${}_8^1\mathit{T}={}_2^1\mathit{T}{}_3^2\mathit{T}{}_4^3\mathit{T}{}_5^4\mathit{T}{}_6^5\mathit{T}{}_7^6\mathit{T}{}_8^7\mathit{T}=\left[\begin{array}{cccc}{n}_{x_8}\hfill & o_{x_8}\hfill & a_{x_8}\hfill & p_{x_8}\hfill \\ \multicolumn{1}{c}{n_{y_8}}& o_{y_8}\hfill & a_{y_8}\hfill & p_{y_8}\hfill \\ \multicolumn{1}{c}{n_{z_8}}& o_{z_8}\hfill & a_{z_8}\hfill & p_{z_8}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(18)

The coordinates of connection points B, D, F, and H of 8R mechanism and PRS-PSS-PRS-PSS mechanism in the reference coordinate system $O\mathrm{-}\mathrm{xyz}$ are: $B(\begin{array}{ccc}{P}_{x_2}\hfill & P_{y_2}\hfill & P_{z_2}\hfill \end{array})$ , $D(\begin{array}{ccc}{P}_{x_4}\hfill & P_{y_4}\hfill & P_{z_4}\hfill \end{array})$ , $F(\begin{array}{ccc}{P}_{x_6}\hfill & P_{y_6}\hfill & P_{z_6}\hfill \end{array})$ , $H(\begin{array}{ccc}{P}_{x_8}\hfill & P_{y_8}\hfill & P_{z_8}\hfill \end{array})$ .

Therefore, the relationship between moving platform dimensions $\left|\mathit{BF}\right|$ , $\left|\mathit{DH}\right|$ , and actuated angle can be expressed as

$$\left|\mathit{BF}\right|=2b=\sqrt{{(p_{x_6}-p_{x_2})}^2+{(p_{y_6}-p_{y_2})}^2+{(p_{z_6}-p_{z_2})}^2}$$

(19)

$$\left|\mathit{DH}\right|=2b\prime =\sqrt{{(p_{x_8}-p_{x_4})}^2+{(p_{y_8}-p_{y_4})}^2+{(p_{z_8}-p_{z_4})}^2}$$

(20)

Inverse position analysis of PRS-PSS-PRS-PSS mechanism

As shown in Figure 17, a fixed coordinate system $O\mathrm{-}\mathrm{xyz}$ is established. This coordinate system is the same as the reference coordinate system $O\mathrm{-}\mathrm{xyz}$ of the analysis connection points of 8R mechanism and PRS-PSS-PRS-PSS mechanism. The origin O is located at the geometric center of the fixed platform, the $x$ axis points to the P joint point $A_3$ on the PRS limb, the $y$ axis points to the P joint point $A_2$ on the PSS limb, and the z axis points parallel to the normal direction of the fixed platform; The moving coordinate system $P\mathrm{-}\mathrm{x}\prime \mathrm{y}\prime \mathrm{z}\prime$ is established. The origin P is located at the geometric center of the moving platform. The $x\prime$ axis points to the S joint point $C_3$ on the PRS limb, the $y\prime$ axis points to the S joint point $C_2$ of the PSS limb, and the $z\prime$ axis points parallel to the normal of the moving platform. $A_i(i=1,2,3,4)$ is the base point of the translational joints of each limb, $B_i(i=1,2,3,4)$ is the center of the middle spherical joint and the rotational joint, and $C_i(i=1,2,3,4)$ is the center of the spherical joint on the moving platform.

Figure 17.

PRS-PSS-PRS-PSS mechanism structure diagram.

For the PRS-PSS-PRS-PSS mechanism, any vector $\mathit{R}\prime$ in the moving coordinate system can be converted to the fixed coordinate system by using the coordinate transformation method, as shown in the following formula

$$\mathit{R}=\left[T\right]\mathit{R}\prime +\mathit{P}$$

(21)

where,

$$\begin{array}{c}\multicolumn{1}{c}{\left[T\right]=\left[\begin{array}{ccc}{d}_{11}\hfill & d_{12}\hfill & d_{13}\hfill \\ \multicolumn{1}{c}{d_{21}}& d_{22}\hfill & d_{23}\hfill \\ \multicolumn{1}{c}{d_{31}}& d_{32}\hfill & d_{33}\hfill \end{array}\right]}\\ =\left[\begin{array}{ccc}\cos \gamma \hfill & 0\hfill & \sin \gamma \hfill \\ \multicolumn{1}{c}{\sin \gamma \sin \phi }& \cos \phi \hfill & -\sin \phi \cos \gamma \hfill \\ \multicolumn{1}{c}{-\sin \gamma \cos \phi }& \sin \phi \hfill & \cos \gamma \cos \phi \hfill \end{array}\right],\hfill \\ \mathit{P}={\left(\begin{array}{ccc}{X}_P\hfill & Y_P\hfill & Z_P\hfill \end{array}\right)}^{\mathrm{T}}\hfill \end{array}$$

where, $\left[T\right]$ is the direction cosine matrix of the attitude of the moving platform, in which three columns are the direction cosines of $x\prime$ , $y\prime$ , and $z\prime$ of the moving coordinate system in the fixed coordinate system, and $\mathit{P}$ is the coordinates of the origin $P$ of the moving coordinate system in the fixed coordinate system.

Take one PSS limb as the research object, as shown in Figure 18. Given $\mathit{OP}=P$ , $\mathit{P}{\mathit{C}}_2$ is parallel to the $x\prime$ direction of the moving coordinate system. And $\left|\mathit{P}{\mathit{C}}_2\right|$ is determined by the size of the moving platform, which is expressed as $\mathit{P}{\mathit{C}}_2={\left(\begin{array}{ccc}b\hfill & 0\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}$ . The coordinates of the spherical joint at the end of the PSS limb in the fixed coordinate system can be obtained by substituting formula (21)

$$\begin{array}{c}\mathit{O}{\mathit{C}}_2=\left[T\right]\mathit{P}{\mathit{C}}_2+\mathit{OP}\hfill \\ =\left[\begin{array}{ccc}{d}_{11}\hfill & d_{12}\hfill & d_{13}\hfill \\ \multicolumn{1}{c}{d_{21}}& d_{22}\hfill & d_{23}\hfill \\ \multicolumn{1}{c}{d_{31}}& d_{32}\hfill & d_{33}\hfill \end{array}\right]{\left(\begin{array}{ccc}b\hfill & 0\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}+{\left(\begin{array}{ccc}0\hfill & Y_P\hfill & Z_P\hfill \end{array}\right)}^{\mathrm{T}}\hfill \\ =\left(\begin{array}{c}b{d}_{11}\hfill \\ \multicolumn{1}{c}{b{d}_{21}+Y_P}\\ \multicolumn{1}{c}{b{d}_{31}+Z_P}\end{array}\right)\hfill \end{array}$$

(22)

Figure 18.

PSS limb position analysis.

Given that $\mathit{O}{\mathit{A}}_2$ is parallel to the $x$ direction of the fixed coordinate system, $\left|\mathit{O}{\mathit{A}}_2\right|$ is determined by the size of the fixed platform. Assuming $\mathit{O}{\mathit{A}}_2={\left(\begin{array}{ccc}a\hfill & 0\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}$ , the vector of ${\mathit{A}}_2{\mathit{C}}_2$ in the fixed coordinate system can be expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathit{A}}_2{\mathit{C}}_2=\mathit{O}{\mathit{C}}_2-\mathit{O}{\mathit{A}}_2}& =\left(\begin{array}{c}b{d}_{11}\hfill \\ \multicolumn{1}{c}{b{d}_{21}+Y_P}\\ \multicolumn{1}{c}{b{d}_{31}+Z_P}\end{array}\right)-{\left(\begin{array}{ccc}a\hfill & 0\hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}\hfill \\ =\left(\begin{array}{c}b{d}_{11}-a\hfill \\ \multicolumn{1}{c}{b{d}_{21}+Y_P}\\ \multicolumn{1}{c}{b{d}_{31}+Z_P}\end{array}\right)\hfill \end{array}$$

(23)

Since ${\mathit{A}}_2{\mathit{B}}_2$ is parallel to the z direction of the fixed coordinate system. Assuming ${\mathit{A}}_2{\mathit{B}}_2={\left(\begin{array}{ccc}0\hfill & 0\hfill & c_1\hfill \end{array}\right)}^{\mathrm{T}}$ , the angle between ${\mathit{A}}_4{\mathit{C}}_4$ and ${\mathit{A}}_4{\mathit{B}}_4$ is

$$\begin{array}{c}\multicolumn{1}{c}{\cos \theta =\frac{{\mathit{A}}_2{\mathit{C}}_2·{\mathit{A}}_2{\mathit{B}}_2}{\left|{\mathit{A}}_2{\mathit{C}}_2\right|\left|{\mathit{A}}_2{\mathit{B}}_2\right|}}\\ =\frac{b{d}_{31}+Z_P}{\sqrt{{(b{d}_{11}-a)}^2+{(b{d}_{21}+Y_P)}^2+{(b{d}_{31}+Z_P)}^2}}\hfill \end{array}$$

(24)

$\left|{\mathit{B}}_i{\mathit{C}}_i\right|$ is the length $l$ of the $B_i{C}_i$ link, which can be obtained according to the law of cosines

$$\begin{array}{c}{\left|{\mathit{B}}_2{\mathit{C}}_2\right|}^2={\left|{\mathit{A}}_2{\mathit{B}}_2\right|}^2+{\left|{\mathit{A}}_2{\mathit{C}}_2\right|}^2-2\left|{\mathit{A}}_2\mathit{B}{}_2\right|\left|{\mathit{A}}_2{\mathit{C}}_2\right|\cos \theta \hfill \\ l^2={c_1}^2+{(b{d}_{11}-a)}^2+{(b{d}_{21}+Y_P)}^2+{(b{d}_{31}+Z_P)}^2\hfill \\ -2c_1(b{d}_{31}+Z_P)\hfill \end{array}$$

(25)

If there is but one unknown $c_1$ in formula (25), the actuated distance $\left|{\mathit{A}}_2{\mathit{B}}_2\right|$ of the translational joint of the PSS limb can be obtained as

$$\begin{array}{l}\left|{\mathit{A}}_2{\mathit{B}}_2\right|=c_1=Z_p+b{d}_{32}\\ -\sqrt{-a^2-b^2{d}_{11}{}^2-b^2{d}_{21}{}^2+2b{d}_{21}{Y}_P-2ab{d}_{11}+l^2-Y_p{}^2}\end{array}$$

(26)

Similarly, the actuated distance of the translational joint of another PSS limb can be expressed as

$$\begin{array}{c}\left|{\mathit{A}}_4{\mathit{B}}_4\right|=c_3=Z_p-b{d}_{32}\hfill \\ -\sqrt{-a^2-b^2{d_{11}}^2-b^2{d_{21}}^2+2b{d}_{21}{Y}_P+2\mathrm{ab}{d}_{11}+l^2-{Y_p}^2}\hfill \end{array}$$

(27)

Take one PRS limb as the research object, as shown in Figure 19. Given $\mathit{OP}=P$ , $\mathit{P}{\mathit{C}}_1$ is parallel to the $y\prime$ direction of the moving coordinate system. And $\left|\mathit{P}{\mathit{C}}_1\right|$ is determined by the size of the moving platform, which is expressed as $\mathit{P}{\mathit{C}}_1={\left(\begin{array}{ccc}0\hfill & b\prime \hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}$ . The coordinates of the spherical joint at the end of the PRS limb in the fixed coordinate system can be obtained by substituting formula (21)

$$\begin{array}{c}\mathit{O}{\mathit{C}}_1=\left[T\right]\mathit{P}{\mathit{C}}_1+\mathit{P}\hfill \\ =\left[\begin{array}{ccc}{d}_{11}\hfill & d_{12}\hfill & d_{13}\hfill \\ \multicolumn{1}{c}{d_{21}}& d_{22}\hfill & d_{23}\hfill \\ \multicolumn{1}{c}{d_{31}}& d_{32}\hfill & d_{33}\hfill \end{array}\right]{\left(\begin{array}{ccc}0\hfill & b\prime \hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}+{\left(\begin{array}{ccc}0\hfill & Y_P\hfill & Z_P\hfill \end{array}\right)}^{\mathrm{T}}\hfill \\ =\left(\begin{array}{c}b\prime d_{12}\hfill \\ \multicolumn{1}{c}{b\prime d_{22}+Y_P}\\ \multicolumn{1}{c}{b\prime d_{32}+Z_P}\end{array}\right)\hfill \end{array}$$

(28)

Figure 19.

PRS limb position analysis.

Given that $\mathit{O}{\mathit{A}}_1$ is parallel to the $y$ direction of the fixed coordinate system, $\left|\mathit{O}{\mathit{A}}_1\right|$ is determined by the size of the fixed platform. Assuming $\mathit{O}{\mathit{A}}_1={\left(\begin{array}{ccc}0\hfill & a\prime \hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}$ , the vector of ${\mathit{A}}_1{\mathit{C}}_1$ in the fixed coordinate system can be expressed as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{A}}_1{\mathit{C}}_1=\mathit{O}{\mathit{C}}_1-\mathit{O}{\mathit{A}}_1=\left(\begin{array}{c}b\prime d_{12}\hfill \\ \multicolumn{1}{c}{b\prime d_{22}+Y_P}\\ \multicolumn{1}{c}{b\prime d_{32}+Z_P}\end{array}\right)-{\left(\begin{array}{ccc}0\hfill & a\prime \hfill & 0\hfill \end{array}\right)}^{\mathrm{T}}}\\ =\left(\begin{array}{c}b\prime d_{12}\hfill \\ \multicolumn{1}{c}{b\prime d_{22}-a\prime +Y_P}\\ \multicolumn{1}{c}{b\prime d_{32}+Z_P}\end{array}\right)\hfill \end{array}$$

(29)

Since ${\mathit{A}}_1{\mathit{B}}_1$ is parallel to the z direction of the fixed coordinate system. Assuming ${\mathit{A}}_1{\mathit{B}}_1={\left(\begin{array}{ccc}0\hfill & 0\hfill & c_2\hfill \end{array}\right)}^{\mathrm{T}}$ , the angle between ${\mathit{A}}_1{\mathit{C}}_1$ and ${\mathit{A}}_1{\mathit{B}}_1$ is

$$\begin{array}{c}\multicolumn{1}{c}{\cos \theta =\frac{{\mathit{A}}_1{\mathit{C}}_1·{\mathit{A}}_1{\mathit{B}}_1}{\left|{\mathit{A}}_1{\mathit{C}}_1\right|\left|{\mathit{A}}_1{\mathit{B}}_1\right|}}\\ =\frac{b\prime d_{32}+Z_P}{\sqrt{{(b\prime d_{12})}^2+{(b\prime d_{22}-a\prime +Y_P)}^2+{(b\prime d_{32}+Z_P)}^2}}\hfill \end{array}$$

(30)

$\left|{\mathit{B}}_i{\mathit{C}}_i\right|$ is the length $h$ of the $B_i{C}_i$ link, which can be obtained according to the law of cosines

$$\begin{array}{c}{\left|{\mathit{B}}_1{\mathit{C}}_1\right|}^2={\left|{\mathit{A}}_1{\mathit{B}}_1\right|}^2+{\left|{\mathit{A}}_1{\mathit{C}}_1\right|}^2-2\left|{\mathit{A}}_1{\mathit{B}}_1\right|\left|{\mathit{A}}_1{\mathit{C}}_1\right|\cos \theta \hfill \\ h^2={c_2}^2+{(b\prime d_{12})}^2+{(b\prime d_{22}-a\prime +Y_P)}^2+{(b\prime d_{32}+Z_P)}^2\hfill \\ -2c_2(b\prime d_{32}+Z_P)\hfill \end{array}$$

(31)

If there is but one unknown $c_2$ in formula (31), the actuated distance $\left|{\mathit{A}}_1{\mathit{B}}_1\right|$ of the translational joint of the PRS limb can be obtained as

$$\begin{array}{c}\left|{\mathit{A}}_1{\mathit{B}}_1\right|=c_2=Z_p+b\prime d_{32}\hfill \\ -\sqrt{-{b\prime }^2{d_{12}}^2-{b\prime }^2{d_{22}}^2+2a\prime b\prime d_{22}-2b\prime d_{22}{Y}_P-{a\prime }^2+2a\prime Y_P+h^2-{Y_P}^2}\hfill \end{array}$$

(32)

Similarly, the actuated distance of the translational joint of another PRS limb can be expressed as

$$\begin{array}{c}\left|{\mathit{A}}_3{\mathit{B}}_3\right|=c_4=Z_p-b\prime d_{32}\hfill \\ -\sqrt{-{b\prime }^2{d_{12}}^2-{b\prime }^2{d_{22}}^2+2a\prime b\prime d_{22}+2b\prime d_{22}{Y}_P-{a\prime }^2-2a\prime Y_P+h^2-{Y_P}^2}\hfill \end{array}$$

(33)

Velocity and acceleration analysis

Based on the above position analysis, the vector relationship between the limbs can be obtained according to Figure 20(a) and (b), and the closed-loop vector equation can be expressed as

$$\mathit{OP}+\mathit{P}{\mathit{C}}_i=\mathit{O}{\mathit{A}}_i+{\mathit{A}}_i{\mathit{B}}_i+{\mathit{B}}_i{\mathit{C}}_i$$

(34)

Figure 20.

Velocity analysis of the PRS-PSS-PRS-PSS mechanism: (a) $\mathrm{yOz}$ plane view and (b) $\mathrm{xOz}$ plane view.

If the unit vector of ${\mathit{A}}_i{\mathit{B}}_i$ is ${\mathit{n}}_i$ and the unit vector of ${\mathit{B}}_i{\mathit{C}}_i$ is ${\mathit{m}}_i$ , then

$$\mathit{OP}+\mathit{P}{\mathit{C}}_i=\mathit{O}{\mathit{A}}_i+l_i{\mathit{n}}_i+h_i{\mathit{m}}_i$$

(35)

Formula (35) through differential calculation can be obtained as

$${\mathit{v}}_P+\mathit{\omega }\times \mathit{P}{\mathit{C}}_i={\stackrel{·}{l}}_i{\mathit{n}}_i+h_i({\mathit{\omega }}_i\times {\mathbf{m}}_i)$$

(36)

where, ${\mathit{v}}_P$ is the velocity vector of the origin P of the moving coordinate system relative to the fixed coordinate system, ${\mathit{v}}_P=\left(\begin{array}{ccc}0\hfill & {\stackrel{·}{P}}_y\hfill & {\stackrel{·}{P}}_z\hfill \end{array}\right)$ is the first order differential of $\mathit{OP}=\left(\begin{array}{ccc}0\hfill & P_y\hfill & P_z\hfill \end{array}\right)$ over time, $\mathit{\omega }$ is the angular velocity vector of the moving platform, $\mathit{\omega }=\left(\begin{array}{ccc}\stackrel{·}{\phi }\hfill & \stackrel{·}{\gamma }\hfill & 0\hfill \end{array}\right)$ , ${\mathit{\omega }}_i$ is the angular velocity vector of the link RS and SS of the $i$ th limb, $h_i$ is the length of the link RS and SS of the $i$ th limb. ${\stackrel{·}{l}}_i$ is the first order differential decomposition of the actuated distance function of the P joint of the $i$ th limb over time. ²¹

To facilitate calculation, multiply the left and right ends of formula (36) by ${\mathit{m}}_i$ can be obtained as

$$({\mathit{v}}_P+\mathit{\omega }\times \mathit{P}{\mathit{C}}_i)·{\mathit{m}}_i={\stackrel{·}{l}}_i{\mathit{n}}_i·{\mathit{m}}_i+h_i({\mathit{\omega }}_i\times {\mathit{m}}_i)·{\mathit{m}}_i$$

(37)

Since the direction of ${\mathit{\omega }}_i\times {\mathit{m}}_i$ is perpendicular to ${\mathit{m}}_i$ , so $h_i({\mathit{\omega }}_i\times {\mathit{m}}_i)·{\mathit{m}}_i=0$ , then

$${\stackrel{·}{l}}_i{\mathit{n}}_i·{\mathit{m}}_i=({\mathit{v}}_P+\mathit{\omega }\times \mathit{P}{\mathit{C}}_i)·{\mathit{m}}_i,i=1,2,3,4$$

(38)

If the actuation equation of the actuated distance $l_i$ and the vectors ${\mathit{n}}_i$ , ${\mathit{m}}_i$ , and $\mathit{P}{\mathit{C}}_i$ in fixed coordinates are known, then there are only four unknowns ( ${\stackrel{·}{P}}_y$ , ${\stackrel{·}{P}}_z$ , $\stackrel{·}{\phi }$ , and $\stackrel{·}{\gamma }$ ), then the velocity vector ${\mathit{v}}_P$ and the angular velocity vector $\mathit{\omega }$ can be obtained to complete the velocity analysis of the PRS-PSS-PRS-PSS mechanism.

Formula (36) through differential calculation can be obtained as

$$\begin{array}{c}{\mathit{a}}_P+(\mathit{\epsilon }\times \mathit{P}{\mathit{C}}_i)+\mathit{\omega }\times (\mathit{\omega }\times \mathit{P}{\mathit{C}}_i)={\stackrel{··}{l}}_i{\mathit{n}}_i+h_i{\mathit{\epsilon }}_i\times {\mathit{m}}_i\hfill \\ +[h_i{\mathit{\omega }}_i\times ({\mathit{\omega }}_i\times {\mathit{m}}_i)]\hfill \end{array}$$

(39)

where, ${\mathit{v}}_P$ is the acceleration vector of the origin P of the moving coordinate system relative to the fixed coordinate system, ${\mathit{a}}_P=\left(\begin{array}{ccc}0\hfill & {\stackrel{··}{P}}_y\hfill & {\stackrel{··}{P}}_z\hfill \end{array}\right)$ is the second order differential of $\mathit{OP}=\left(\begin{array}{ccc}0\hfill & P_y\hfill & P_z\hfill \end{array}\right)$ over time, $\mathit{\epsilon }=\left(\begin{array}{ccc}\stackrel{··}{\phi }\hfill & \stackrel{··}{\gamma }\hfill & 0\hfill \end{array}\right)$ is the rotational angular acceleration vector of the moving platform, ${\mathit{\omega }}_i$ is the angular velocity vector of the link RS and SS of the $i$ th limb, ${\mathit{\epsilon }}_i$ is the angular acceleration vector of the link RS and SS of the $i$ th limb, $h_i$ is the length of the link RS and SS of the $i$ th limb. ${\stackrel{··}{l}}_i$ is the second order differential decomposition of the actuated distance function of the P joint of the $i$ th limb over time.

To facilitate calculation, multiply the left and right ends of formula (39) by ${\mathit{m}}_i$ can be obtained as

$$\begin{array}{c}{\mathit{a}}_P{\mathit{m}}_i+(\mathit{\epsilon }\times \mathit{P}{\mathit{C}}_i){\mathit{m}}_i+\mathit{\omega }\times (\mathit{\omega }\times \mathit{P}{\mathit{C}}_i){\mathit{m}}_i={\stackrel{··}{l}}_i{\mathit{n}}_i{\mathit{m}}_i\hfill \\ +(h_i{\mathit{\epsilon }}_i\times {\mathit{m}}_i){\mathit{m}}_i+[h_i{\mathit{\omega }}_i\times ({\mathit{\omega }}_i\times {\mathit{m}}_i)]{\mathit{m}}_i\hfill \end{array}$$

(40)

Since the direction of $h_i{\mathit{\epsilon }}_i\times {\mathit{m}}_i$ is perpendicular to ${\mathit{m}}_i$ , so $(h_i{\mathit{\epsilon }}_i\times {\mathit{m}}_i){\mathit{m}}_i$ , then

$$\begin{array}{c}{\mathit{a}}_P{\mathit{m}}_i+(\mathit{\epsilon }\times \mathit{P}{\mathit{C}}_i){\mathit{m}}_i+\mathit{\omega }\times (\mathit{\omega }\times \mathit{P}{\mathit{C}}_i){\mathit{m}}_i={\stackrel{··}{l}}_i{\mathit{n}}_i{\mathit{m}}_i\hfill \\ +[h_i{\mathit{\omega }}_i\times ({\mathit{\omega }}_i\times {\mathit{m}}_i)]{\mathit{m}}_i\hfill \end{array}$$

(41)

If the actuation equation of the actuated distance $l_i$ and the vectors ${\mathit{n}}_i$ , ${\mathit{m}}_i$ , $\mathit{P}{\mathit{C}}_i$ and the angular velocity vector $\mathit{\omega }$ in fixed coordinates are known, then there are only four unknowns ( ${\stackrel{··}{P}}_y$ , ${\stackrel{··}{P}}_z$ , $\stackrel{··}{\phi }$ , and $\stackrel{··}{\gamma }$ ), then the acceleration vector ${\mathit{a}}_P$ and the angular acceleration vector $\epsilon$ can be obtained to complete the acceleration analysis of the PRS-PSS-PRS-PSS mechanism.

Numerical examples

The 8R&PRS-PSS-PRS-PSS mechanism analysis has been converted to the PRS-PSS-PRS-PSS mechanism analysis in Section “Degree of freedom analysis.” And the PRS-PSS-PRS-PSS mechanism is the main body of the foldable composite space capture mechanism, so the simulation of its motion analysis is carried out to verify the correctness and feasibility of its theoretical derivation. Fixed platform radius $a=a\prime =110\mathrm{mm}$ , moving platform radius $b=b\prime =73.6\mathrm{mm}$ , SS limb and RS limb length $l=h=93\mathrm{mm}$ . As shown in Figure 21, the moving platform of the mechanism is set to carry out 6 s four degrees of freedom motion, that is, all the motions of rotating around the x and y axes and moving along the y and z axes are completed at the same time.

Figure 21.

Motion trajectory of the center point of the moving platform.

The position vector $\mathit{P}={\left[\begin{array}{ccc}0\hfill & P_y\hfill & P_z\hfill \end{array}\right]}^{\mathrm{T}}$ of the center point of the moving platform relative to the center point of the static platform is defined as the function of $P_y\left(t\right)$ and $P_z\left(t\right)$ with respect to time $t(s)$ . In the transformation matrix $\left[T\right]$ of the moving coordinate system relative to the fixed coordinate system, the function of transformation angle $\gamma (°)$ and $\phi (°)$ with respect to time $t(s)$ is $\gamma \left(t\right)$ and $\phi \left(t\right)$ . The translation along the y axis is represented by the parametric equation $P_y\left(t\right)$ of actuated 1. The rotation around the x axis is represented by the parametric equation $\gamma \left(t\right)$ of actuated 2. The rotation around the y axis is represented by the parametric equation $\phi \left(t\right)$ of actuated 3. The translation along the z axis is represented by the parametric equation $P_z\left(t\right)$ of actuated 4, as shown in formula (42).

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{P}_y\left(t\right)=-0.07t^3+0.6t^2-0.5t+1.52\hfill \\ \multicolumn{1}{c}{\gamma \left(t\right)=-0.27t^3+2.4t^2-1.65t+1.82}\\ \multicolumn{1}{c}{\phi \left(t\right)=-0.67t^3+6t^2-4t+2.72}\\ \multicolumn{1}{c}{P_z\left(t\right)=-0.4t^3+3.6t^2-2.5t+1.8}\end{array}\begin{array}{c}0\le t\le 6\hfill \end{array}}\end{array}$$

(42)

The software MATLAB and SolidWorks are used to analyze the motion laws in the above process, and the results are shown in Figures 22 and 23. In Figure 22, the four lines are the change curves of actuated parameter of limb 1, 2, 3, and 4 obtained by MATLAB analysis respectively. In Figure 23, the curves represented by $x,y,z$ and $x\prime ,y\prime ,z\prime$ are the coordinate values of the center point P of the moving platform obtained by MATLAB and SolidWorks in the fixed coordinate system, and the analysis shows that the two results are completely consistent.

Figure 22.

The change curves of actuated parameter.

Figure 23.

Position change of point P in the fixed coordinate system.

Conclusion

To solve the existing acquisition problems of space debris cleaning technology, a composite space capture mechanism with variable platform is proposed in this paper, which has important scientific significance and research value for enriching and improving space debris cleaning technology and improving innovative development in the aerospace field. The main conclusions are as follows

• (1) Based on the finite screw theory, the configuration design of the 8R&PRS-PSS-PRS-PSS composite space capture mechanism is completed. The mechanism has the advantages of high flexibility, high stiffness, and large spread to harvest ratio.

• (2) The screw theory is used to complete the analysis of the degree of freedom of the mechanism. The DH method and the closed-loop vector method are used to solve the analytical expression of the position of the 8R mechanism and PRS-PSS-PRS-PSS mechanism. The kinematic theory analysis of the mechanism is completed through the differential operation.

• (3) Through transforming 8R&PRS-PSS-PRS-PSS mechanism analysis into PRS-PSS-PRS-PSS mechanism analysis. The correctness and feasibility of the theoretical analysis method of PRS-PSS-PRS-PSS mechanism are verified by numerical examples.

• (4) In the future research, according to the application scenarios, the shape and size of the captured target should be comprehensively considered, and the size and section parameters of the mechanism should be optimized to promote its engineering application scenarios.