The workspace of a PM is the set of all the working areas that the reference points on the moving platform can reach. It is an important parameter to measure the working performance of the PM. The shape and size of the working area determine the motion ability of the PM. Compared with serial mechanisms, the workspace of PMs is generally smaller and the shape is irregular, which limits the application range of PMs. Therefore, analysis of workspace is a key segment in the design of PMs.
Workspace analysis of PM is usually the process of solving its maximum workspace, the calculation process of the workspace of PM is generally very complex, the current solution methods include numerical solution and analytical solution. The geometric solution method proposed by Jo and developed by Gosselin is the most representative analytical solution [11, 19]. Based on CAD software, Arrouk [20] obtained the calculation of workspace of PM by calculating the intersection of simple 3D geometry. Xiong et al. [21] analyzed geometric isotropy indices for workspace. Fu et al. [22] and Antonov et al. [23] explore the dependence of the workspaces on the manipulator geometric parameters. The numerical solution of the workspace of PM is to use the inverse position solution to calculate the parameters such as the joint angle and the length of the link under the given posture. By comparing with the constraint conditions, it can judge whether the given posture can be reached, so as to determine the range of the workspace [24,25,26]. In this paper, the workspace of the designed PM is analyzed by using the numerical method of searching the workspace limit boundary.
3.1 Limitation of the Length of Links
There are many limiting factors of solving workspace. In which there are three main influencing factors: the limitation of the length of each driving link, the limitation of the rotation angle of each pair and the interference between the driving links.
According to Eq. (3), the length of each link li can be expressed as follows:
$$l_{i} = \left| {{\varvec{P}} + {\varvec{R}} \cdot {\varvec{a}}_{i} - {\varvec{b}}_{i} } \right|{, }\, i = 1 - 6.$$
(4)
If li min is the shortest length of the link li and li max is the longest length of the link i, the following constraints in the actual motion process can be expressed as follows:
$$l_{{i{\text{ min}}}} \le l_{i} \le l_{{i{\text{ max}}}} {, }\, i = 1 - 6.$$
(5)
Whenever the length of any link reaches its limit, the reference point on the moving platform reaches the limit boundary of its workspace.
3.2 Limitation of the Rotation Angle of Kinematic Pair
The equivalent spherical joint (U-R) is used between the moving platform and the driving link, and the Hooke joint is used between the base and the driving link. The equivalent spherical joint is three independent rotation pairs, and the diagram of the equivalent spherical joint is shown in Figure 8.
In the PM analyzed in this paper, the angles θ1, θ2 and θ3 are limited by the structural parameters and constraints of the PM, and range of their changes can be expressed as follows:
$$\theta_{i} \le \theta_{{i{\text{ max}}}} {, } \quad i = 1 - 3,$$
(6)
where θi max is the maximum limitation angle of each pair.
For the convenience of calculation, the three rotation angles of the equivalent spherical joint are simplified as the rotation angle θ of the spherical joint, which is shown in Figure 9. The rotation angle θ of the spherical pair is determined by the Z axis of the moving coordinate system and the vector u of the link, and its limitation value is the minimum value of θi max. Similarly, the rotation angles of Hooke joint also have a range.
Vector nai denotes the posture of the base of spherical pair on the moving platform in the moving coordinate system O-XYZ, and vector nbi denotes the posture of the base of the Hooke joint in the base coordinate system O0-X0Y0Z0. θai is the rotation angle of spherical pair, θbi is the rotation angle of Hooke joint, as shown in Figure 10.
Then the constraint condition of the spherical pair can be expressed as:
$$\theta_{ai} = \arccos \frac{{{\varvec{L}}_{i} \cdot \left( {{\varvec{Rn}}_{ai} } \right)}}{{\left| {{\varvec{L}}_{i} } \right|}} \le \theta_{{a{\text{ max}}}} {, } \quad i = 1 - 6,$$
(7)
where Li is the vector between two pairs of the ith link. The constraint condition of the Hooke joint can be expressed as:
$$\theta_{bi} = \arccos \frac{{{\varvec{L}}_{i} \cdot \left( {{\varvec{Rn}}_{bi} } \right)}}{{\left| {{\varvec{L}}_{i} } \right|}} \le \theta_{{b{\text{ max}}}} {, } \quad i = 1 - 6.$$
(8)
3.3 Interference of Link
The possible interference between the moving platform, the base and the driving link should be considered in the motion of PM due to the certain physical dimensions there all exist. Suppose that each link in the PM is a standard cylinder, of which D denotes the diameter of the link and Di denotes the distance between central axis of adjacent links. Therefore, the constraint condition of interference between links can be expressed as
$$D \le D_{i} {, } \quad i = 1 - 6,$$
(9)
where ni denotes the unit vector of the common normal between the adjacent link vectors Li and Li+1:
$${\varvec{n}}_{i} = \frac{{{\varvec{L}}_{i} \times {\varvec{L}}_{{i{ + 1}}} }}{{\left| {{\varvec{L}}_{i} \times {\varvec{L}}_{{i{ + 1}}} } \right|}},$$
(10)
where Δi denotes the minimum distance between the vectors Li and Li+1:
$$\Delta_{i} = \left| {{\varvec{n}}_{i} \cdot \left( {{\varvec{b}}_{{i{ + 1}}} - {\varvec{b}}_{i} } \right)} \right|.$$
(11)
The relationship between the minimum distance Δi and the distance Di depends on the positions of the common normal intersections Ci and Ci+1, The coordinates ci of intersection Ci can be calculated by Eq. (9):
$$\frac{{{\varvec{c}}_{i} - {\varvec{b}}_{i} }}{{{\varvec{a}}_{i} - {\varvec{b}}_{i} }} = \left| {\frac{{\left( {{\varvec{b}}_{{i{ + 1}}} - {\varvec{b}}_{i} } \right) \cdot {\varvec{m}}_{i} }}{{\left( {{\varvec{a}}_{i} - {\varvec{b}}_{i} } \right) \cdot {\varvec{m}}_{i} }}} \right|,$$
(12)
where ai denotes the coordinates of joints ai in base coordinate system O0-x0y0z0, mi can be expressed as:
$${\varvec{m}}_{i} = {\varvec{n}}_{i} \times \left( {{\varvec{a}}_{{i{ + 1}}} - {\varvec{b}}_{{i{ + 1}}} } \right).$$
(13)
Similarly, ci+1 can be calculated. According to the position of the intersection points Ci and Ci+1 on the link, there are three cases:
(1) The intersections Ci and Ci+1 are both on the link, shown as Figure 11(a). The interference condition of links under this circumstance can be expressed as Δi > D.
(2) If the intersection point Ci or Ci+1 is not on the link, as shown in Figure 11(b) and (c), Di is calculated according to the position of Ci or Ci+1. If Ci+1 is on link ai+1bi+1, Di is the distance from ai to link ai+1bi+1:
$$D_{i} = \left| {\frac{{({\varvec{a}}_{i} - {\varvec{b}}_{{i{ + 1}}} ) \times {\varvec{L}}_{{i{ + 1}}} }}{{{\varvec{L}}_{{i{ + 1}}} }}} \right|.$$
(14)
If Ci+1 is on link aibi, as shown in Figure 11(c), Di is the distance from ai+1 to link aibi:
$$D_{i} = \left| {\frac{{({\varvec{a}}_{{i{ + 1}}} - {\varvec{b}}_{i} ) \times {\varvec{L}}_{i} }}{{{\varvec{L}}_{i} }}} \right|,$$
(15)
(3) If the intersection points Ci or Ci+1 are not on the link, as shown in Figure 11(d)‒(f), Di depends on the positions of Mi and Mi+1, where, Mi is the intersection of the line passing through the joint point ai+1 and perpendicular to Li and Li+1, and Mi+1 is the intersection of the line passing through the joint point and perpendicular to Li and Li+1. Under this condition, there are three possibilities as follows:
As shown in Figure 11(d), when Mi is outside the link aibi and M1 is on the link ai+1bi+1, Di can be obtained by Eq. (14); As shown in Figure 11(e), when Mi is on the link aibi and M1 is outside the link ai+1bi+1, Di can be obtained by Eq. (15); As shown in Figure 11(f), when Mi and M1 are both outside the links, Di is the distance from ai to link ai+1bi+1.
3.4 Calculation on Workspace
3.4.1 Spherical Coordinate Searching Method
The spherical coordinate searching method is to express any point P = [Xp Yp Zp]T in space by establishing spherical coordinate system. The radial distance ρ represents the distance from the target point P to the origin O of the coordinate system, the zenith angle φ represents the angle between the line OP and the positive direction of the Z axis, and the azimuth Φ represents the deflection angle of the line OP relative to the XZ plane, as shown in Figure 12.
The specific steps of calculation on workspace boundary are shown in Figure 13. Firstly, the initial pose and structural parameters of the PM are acquired according to the requirements of this subject. The components XP, YP and ZP of point P are expressed in spherical coordinates. Maintain the zenith angle φ and increase the azimuth Φ from 0 to 2 π, the maximum radial distance Pmax is found according to the kinematic theory and constraint conditions of PM. Gradually, increase the Zenith angle from 0 to π, according to the previous steps to find out the maximum radial distance Pmax.
3.4.2 Influence of Structural Parameters on Workspace
The workspace of the PM can be obtained by the spherical coordinate searching method. The structure parameters of PM are related to radius R, circle angle α, radius R0 and circle angle β. And in former analysis we have α=β and R=R0. In order to analyze the influence of the structure parameters on the workspace, the influence of the four parameters on the workspace is analyzed.
Figure 14(a) shows the workspaces of PM with the radius R (R=R0) of 50 mm, 55 mm and 60 mm. It can be obtained that when the angle α and β and other constraints remain unchanged, the workspace size within a certain range decreases with the increase of radius R and R0. It can be obtained from Figure 14(b) that under the condition that radius R (R0) is 50 mm and other constraints remain unchanged, with the increase of circumference angle α and β with the increase of the value, the workspace of the PM is increasing.
3.4.3 Presentation of Workspace
According to the spherical coordinate searching method, the workspace of the PM is searched and drawn by using MATLAB software. What in Figure 15 signifies the workspace of the PM. Figure 15(a) is a 3D view of the workspace of the PM. Figure 15(b) is the projection of the workspace in XY plane. Figure 15(c) is the projection of the workspace in XZ plane.
It can be acquired from Figure 15 that the workspace of the PM basically presents a relatively regular shape, and the working range of the reference point of the PM in the X axis direction is within ±30 mm. The working range of the reference point in the Y axis direction is about −35 mm and +30 mm, and the working range of the reference point in the Z axis direction is about 120 mm and 150 mm, which fully meets the design requirements.
In order to calculate the obliquity of the PM in X, Y, Z directions, which also means to solve the maximum and minimum values of the three pose angles φ, θ, γ in the workspace, the curve of the pose angle is shown in Figure 16. As shown in Figure 16, the inclination angle range in X direction is from −27° to 27.79°, the inclination angle range in Y direction is from −8.76° to 8.76°, the inclination angle range in Z direction is from −10.37° to 8.43°.