Motion Characteristics Analysis of a Novel Spherical Two-degree-of-freedom Parallel Mechanism

Abstract

Current research on spherical parallel mechanisms (SPMs) mainly focus on surgical robots, exoskeleton robots, entertainment equipment, and other fields. However, compared with the SPM, the structure types and research contents of the SPM are not abundant enough. In this paper, a novel two-degree-of-freedom (2DOF) SPM with symmetrical structure is proposed and analyzed. First, the models of forward kinematics and inverse kinematics are established based on D-H parameters, and the Jacobian matrix of the mechanism is obtained and verified. Second, the workspace of the mechanism is obtained according to inverse kinematics and link interference conditions. Next, rotational characteristics analysis shows that the end effector can achieve continuous rotation about an axis located in the mid-plane and passing through the rotation center of the mechanism. Moreover, the rotational characteristics of the mechanism are proved, and motion planning is carried out. A numerical example is given to verify the kinematics analysis and motion planning. Finally, some variant mechanisms can be synthesized. This work lays the foundation for the motion control and practical application of this 2DOF SPM.

1 Introduction

Spherical parallel mechanism (SPM) is a special spatial parallel mechanism. Its end effector can rotate freely around the point. The SPMs have important application value and have been widely used, such as the azimuth tracking system [1], the bionic robot [2], surgical robot [3], and the medical device [4]. The research about SPM mostly focuses on 2DOF SPM [5] and 3DOF SPM [6]. The theoretical research and practical application of 3DOF SPM are quite mature. For example, theoretical research about the typical 3-RRR 3DOF SPM has been studied in terms of its working space [7], singularity [8], dexterity [9], stiffness [10], dynamics [11]. In practical engineering applications, Gosselin et al. proposed the famous agile eye in 1994 [12], etc. In most cases, the 2DOF SPM can satisfy application requirements, such as pointing mechanisms [13] used in spherical engraving machines, azimuth tracking of satellite antennas, and automatic ground tracking equipment for various aircraft, etc., and some 2DOF artificial wrists sorted out by Bajaj et al. [14].

The representative 2DOF SPM is the spherical 5R mechanism. Ouerfelliz et al. [15] studied the direct and inverse kinematics, kinematic and dynamic optimization of a general spherical 5R linkage. Cervantes-Sanchez et al. [16] analyzed its workspace and singularity. Zhang et al. [17] had a further analysis of the workspace of spherical 5R mechanism and 2DOF SPM with actuation redundancy, as well as dynamic analysis [18, 19], trajectory planning [20], and parameter optimization [21]. Yu et al. [22] introduced a simple and visual graphic method for mobility analysis of parallel mechanisms and presented a novel 2DOF rotational parallel mechanism derived from well-known Omni Wrist III. Dong et al. [23] analyzed the kinematics, singularity, and workspace of a class of 2DOF rotational parallel manipulators in a geometric approach. Chen [24] proposed a new geometric kinematic modeling approach based on the concept of instantaneous single-rotation-angle and used for the 2DOF RPMs with symmetry in a homo-kinetic plane. Kim et al. [25] deformed the spherical 5R mechanism, designed the spatial self-adaptive finger clamp, and conducted constraint analysis, optimization design of the structure, and grasping experiment on it. Xu et al. [26] established a theory regarding the type synthesis of the two-rotational-degrees-of-freedom parallel mechanism with two continuous rotational axes systematically. Terence et al. [27] conducted the decoupling design of the 5R spherical mechanism and compared it with the traditional 5R spherical mechanism in motion characteristics and workspace. Cao et al. [28] obtained a three-rotation, one-translation (3R1T) manipulator for minimally invasive surgery by connecting the revolute pair and the prismatic pair to a 2DOF spherical mechanism, and analyzed its kinematics and singularity. Alamdar et al. [29] introduced a new non-symmetric 5R-SPM and developed a geometrical approach to analyze its configurations and singularities.

In this paper, a novel 2DOF SPM with symmetric structure and its variant Mechanisms are proposed. The paper is organized as follows: Section 2 gives the description of a SPM structure, analysis of its mobility, the models of forward kinematics and inverse kinematics are established, and the Jacobian matrix of the mechanism is obtained and verified. In Section 3, the workspace of the mechanism is obtained. The rotation characteristics of SMP are analyzed in Section 4. Section 5 describes variant mechanisms of the 2DOF SPM. Conclusions are presented in Section 6.

2 Kinematics Analysis of the 2DOF SPM

2.1 Mobility Analysis

The schematic diagram of the 2DOF SPM is shown in Figure 1, all the revolute axes intersect at one-point O, called the rotation center of the mechanism. The base is connected with the end effector by three spherical serial 3R sub-chains: B₁B₂B₃, B₄B₅B₆, and B₇B₈B₉. There is a special spherical sub-chain consisting of link 9, link 10, and component 11 and connected by two arc prismatic pairs, limiting the revolute axes OB₂, OB₅, and OB₈ on a plane, which is defined as the mid-plane of the mechanism. And the spherical 3R sub-chains B₇B₈B₉ and component 11 forming a symmetric double arc slider-rocker mechanism aims at keeping the mid-plane always coplanar with the angular bisector of spherical angle ∠B₁B₂B₃ [30], ensuring the base and the end effector are symmetric concerning the mid-plane during the movement of the mechanism.

Figure 1

Schematic diagram of the 2-DOF SPM

The DOF of the parallel mechanism can be calculated by using the G-K formula:

$$M = d(n - g - 1) + \sum\limits_{i = 1}^{g} {f_{i} } ,$$

(1)

where d is the order of a mechanism (for the spherical mechanism d = 3), n is the number of components including the base, g is the number of kinematic pairs, f_i is the freedom of the ith kinematic pair. For this mechanism n = 11, g = 14, and ∑f_i = 14. Therefore, the degree of freedom of this mechanism is two.

2.2 Inverse Kinematics of the SPM

2.2.1 Establishment of the Coordinate Systems

As shown in Figure 2, a global coordinate system O-x₀y₀z₀ is located at the rotation center O with the x₀-axis passing through point Q, the midpoint of arc link B₁B₂, the z₀-axis is perpendicular to the plane where the arc link B₁B₂ lies on, and y₀-axis is defined by right-hand rule. The parameter θ_ij, where ij = 21, 32, 43, 54, 65, 61, 74, 87, 81, represents the angle between the two planes that the two adjacent links lying on. Looking at the rotation center along the revolute axis, the positive direction is counterclockwise.

Figure 2

Kinematic model and parameter representation of the 2-DOF SPM

Due to the characteristics of the SPM that each revolute axis intersects at the rotation center O, the parameters α_i and d_ij equal zero, where ij = 21, 32, 43, 54, 65, 61, 74, 87, 81. The ith local coordinate systems are also located at the rotation center O. The x_i-axis along with each revolute axis, where x₁ coincides with x₉, x₂ coincides with x₁₀, x₃ coincides with x₇, x₅ coincides with x₁₁, and x₈ coincides with x₁₂. The z_i-axis is perpendicular to the plane where the ith link is located and the y_i-axis is defined by the right-hand rule.

Because this SPM has two DOFs, the configuration can be represented by two angles φ and γ, where φ represents the angle between the OP and x₀-axis, and γ represents the angle between the mid-perpendicular plane of the end effector and the plane O-x₀z₀. Designate point P as the output reference point of the mechanism, and the driving parameters of the mechanism are θ₂₁ and θ₆₁.

In the inverse kinematics, the driving parameters θ₂₁ and θ₆₁ can be solved when the configuration parameters φ and γ of the end effector are given.

2.2.2 Description of the Configuration

Suppose each link moves on a spherical surface with a radius R, and the position of outputs reference point P can be described by angle φ and ω:

$$P = \left[ {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right] = R\left[ {\begin{array}{*{20}c} {\cos \varphi } \\ {\sin \varphi \sin \omega } \\ {\sin \varphi \cos \omega } \\ \end{array} } \right],$$

(2)

where ω is the angle between the plane OPQ and the plane O-x₀z₀, which also represents the angle between the projection of OP on the plane O-y₀z₀ and the positive direction of the z₀-axis. The relationship between γ and ω can be derived from the spherical triangle PQM and MNQ. According to the characteristics of the spherical mechanism and the knowledge of spherical trigonometry [31], the relevant parameters are expressed in Figure 3 for clear observation.

Figure 3

Schematic of spherical triangle PQM (a) and MNQ (b)

The point M in Figure 3(a) is the intersection point of the arc MQ (intersecting line of the mid-perpendicular plane of the end effector and spherical surface) and arc MP (intersecting line of the plane O-x₀z₀ and spherical surface), and the point M' in Figure 3(b) is the same point with M for convenient description. The point N is the midpoint of the arc PQ (intersecting line of the plane passing through the two lines OP and OQ and spherical surface), that is, the arc MN is the intersecting line of the midplane and spherical surface.

According to the spherical triangular sine theorem, from the spherical triangle M'NQ shown in Figure 3(b) it can be derived that:

$$\frac{\sin \angle M^{\prime}}{{\sin \angle m^{\prime}}} = \frac{\sin \angle N}{{\sin \angle n}}.$$

(3)

Similarly, it is available in a spherical triangle PQM shown in Figure 3(a):

$$\frac{\sin \angle Q}{{\sin \angle q}} = \frac{\sin \angle M}{{\sin \angle m}}.$$

(4)

In Eqs. (3) and (4), ∠M=180°−γ, ∠M'=∠M/2, ∠m=φ, ∠m'=∠m/2, ∠q=∠n, and ∠N=90°.

It can be derived from Eqs. (3) and (4) that:

$$\omega = \arcsin \frac{{\sin (180 - \gamma )\sin \frac{\varphi }{2}}}{{\sin \left( {\frac{180 - \gamma }{2}} \right)\sin \varphi }}.$$

(5)

2.2.3 Solutions of Coordinates with Configuration Parameters

As shown in Figure 3, the circle where arc B₂B₅ is located in the large circle corresponding to the middle plane of the mechanism, so the equation of the circle where arc B₂B₅ is located in the base coordinate system O-x₀y₀z₀, can be expressed as

$$\left\{ {\begin{array}{*{20}l} {x^{2} + y^{2} + z^{2} = R^{2} ,} \hfill \\ {(\cos \varphi - 1) \cdot x + \sin \varphi \sin \omega \cdot y + \sin \varphi \cos \omega \cdot z = 0.} \hfill \\ \end{array} } \right.$$

(6)

The trajectory of point B₂ in the global coordinate system O-x₀y₀z₀ is determined by a spherical surface and a plane. As shown in Figure 4, the radius of the spherical surface is OB₁ and the center is O. The plane is vertical to OB₁ and passing through the line B₂B₂'.

Figure 4

Front view of link B₁B₂

The trajectory equation is:

$$\left\{ {\begin{array}{*{20}l} {x^{2} + y^{2} + z^{2} = R^{2} ,} \hfill \\ {\cos \frac{{\alpha_{1} }}{2} \cdot x - \sin \frac{{\alpha_{1} }}{2} \cdot y = R\cos \alpha_{2} .} \hfill \\ \end{array} } \right.$$

(7)

Therefore, the coordinate of B₂ = [x₂ y₂ z₂]^T in the global coordinate system O-x₀y₀z₀, can be obtained by Eqs. (6) and (7). And the coordinate of B₅ = [x₅ y₅ z₅]^T in the global coordinate system O-x₀y₀z₀, can be obtained similarly.

2.2.4 Solutions of Coordinates with Driving Parameters

The coordinates of B₂ and B₅ can also be derived by D-H link parameters.

${}_{i}^{i - 1} {\varvec{T}}$ is a forward transformation matrix [32] between the adjacent local ith and (i−1)th coordinate system, which is the coordinate transformation from ith link to (i−1)th link, it can be obtained by the following equation:

$${}_{i}^{i - 1} {\varvec{T}} = {\text{Rot}}(z,\alpha_{i} ){\text{Trans}}(0,0,a_{i} ){\text{Trans}}(\alpha_{ij} ,0,0){\text{Rot}}(x,\theta_{ij} ).$$

(8)

${}_{{i{ - }1}}^{i} {\varvec{T}}$ is an inverse transformation matrix between the adjacent local ith and (i−1)th coordinate system, which is the coordinate transformation from (i−1)th link to ith link, and is the transpose matrix of ${}_{i}^{i - 1} {\varvec{T}}$. Then, it can be derived that:

$${}_{i - 1}^{i} {\varvec{T}} = {}_{i}^{i - 1} {\varvec{T}}^{ - 1} = {}_{i}^{i - 1} {\varvec{T}}^{{\text{T}}} .$$

(9)

The coordinates of revolute pairs B₂ and B₅ in the global coordinate system O-x₀y₀z₀ can be obtained from Eqs. (8) and (9):

$${\varvec{b}}_{{{\mathbf{20}}}} = {}_{0}^{1} {\varvec{T}} \cdot {}_{1}^{2} {\varvec{T}} \cdot {\varvec{b}}_{{{\mathbf{22}}}} = \left[ {\begin{array}{*{20}c} {x_{2} } & {y_{2} } & {z_{2} } \\ \end{array} } \right]^{{\text{T}}} ,$$

(10)

$${\varvec{b}}_{{{\mathbf{50}}}} = {}_{0}^{6} {\varvec{T}} \cdot {}_{6}^{5} {\varvec{T}} \cdot {\varvec{b}}_{{{\mathbf{55}}}} = \left[ {\begin{array}{*{20}c} {x_{5} } & {y_{5} } & {z_{5} } \\ \end{array} } \right]^{{\text{T}}} ,$$

(11)

where, ${\varvec{b}}_{{{\mathbf{55}}}} = \left[ {\begin{array}{*{20}c} R & 0 & 0 \\ \end{array} } \right]^{{\text{T}}}$ are the coordinates of revolute pairs B₂ and B₅ in the local coordinate system O-x₂y₂z₂ respectively.

Derived from the coordinate of B₂=[x₂ y₂ z₂]^T and Eq. (10):

$$\theta_{21} { = }\arcsin \frac{{z_{2} }}{{R\sin \alpha_{2} }}.$$

(12)

Derived from the coordinate of B₅=[x₅ y₅ z₅]^T and Eq. (11):

$$\theta_{61} { = }\arcsin \frac{{z_{5} }}{{R\sin \alpha_{6} }}.$$

(13)

In Eqs. (12) and (13), z₂ and z₅ both have two solutions (z₂ < π/2, z₂ > π/2, z₅ < π/2 and z₅ > π/2), which means one position corresponds to four sets of solutions. The four initial configurations with different arrangements of the drive links are shown in Figure 5. Meanwhile, the initial configurations in Figure 5(a) were selected to analyze the kinematics characteristic of the spherical mechanism.

Figure 5

Four initial configurations with different arrangements of the drive links

2.3 Forward Kinematics of the SPM

Given the driving parameters θ₂₁ and θ₆₁, the solution of the configuration parameters φ and γ can be figured out, that is the forward kinematics of the spherical mechanism. And the normal vector of the mid-plane is obtained by Eqs. (10) and (11):

$${\varvec{b}}_{{{\mathbf{20}}}} \times {\varvec{b}}_{{{\mathbf{50}}}} { = }\left[ {\begin{array}{*{20}c} {r \cdot {\varvec{i}}} & {t \cdot {\varvec{j}}} & {s \cdot {\varvec{k}}} \\ \end{array} } \right]^{{\text{T}}} ,$$

(14)

where,

$$r = R^{2} \sin \alpha_{6} \sin \theta_{61} (\sin \frac{{\alpha_{1} }}{2}\cos \alpha_{2} + \cos \frac{{\alpha_{1} }}{2}\sin \alpha_{2} \cos \theta_{21} )- R^{2} \sin \alpha_{2} \sin \theta_{21} (\sin \frac{{\alpha_{1} }}{2}\cos \alpha_{6} + \cos \frac{{\alpha_{1} }}{2}\sin \alpha_{6} \cos \theta_{61} ),$$

$$s = R^{2} \sin \alpha_{2} \sin \theta_{21} (\cos \frac{{\alpha_{1} }}{2}\cos \alpha_{6} - \sin \frac{{\alpha_{1} }}{2}\sin \alpha_{6} \cos \theta_{61} )- R^{2} \sin \alpha_{6} \sin \theta_{61} (\cos \frac{{\alpha_{1} }}{2}\cos \alpha_{2} + \sin \frac{{\alpha_{1} }}{2}\sin \alpha_{2} \cos \theta_{21} )- R^{2} \sin \alpha_{6} \sin \theta_{61} (\cos \frac{{\alpha_{1} }}{2}\cos \alpha_{2} + \sin \frac{{\alpha_{1} }}{2}\sin \alpha_{2} \cos \theta_{21} ),$$

$$t = - R^{2} (\cos \alpha_{6} \sin \alpha_{2} \cos \theta_{21} - \cos \alpha_{2} \sin \alpha_{6} \cos \theta_{61} ).$$

The equation of the mid-plane can be described as:

$$r \cdot x + s \cdot y + t \cdot z = 0.$$

(15)

The midpoint of the fixed link Q = [R 0 0]^T and point P = [x y z]^T are symmetric with respect to the mid-plane. The intersection point of the line PQ and the mid-plane is H = [x_h y_h z_h]^T.

Assuming that $\frac{{x_{h} - R}}{r} = \frac{{y_{h} }}{s} = \frac{{z_{h} }}{t} = k$, the coordinate of the outputs reference point P can be obtained from the symmetrical characteristic of the mechanism:

$$\left\{ {\begin{array}{*{20}l} {x = 2x_{h} - R,} \hfill \\ {y = 2y_{h} ,} \hfill \\ {z = 2z_{h} .} \hfill \\ \end{array} } \right.$$

(16)

It can be obtained by the spherical triangular cosine theorem from the spherical triangle M'NQ in Figure 3(b) that:

$$\cos \angle M^{\prime} = - \cos \angle N\cos \angle Q + \sin \angle N\sin \angle Q\cos \angle m^{\prime}.$$

(17)

According to Figure 3(a), the configuration parameters can be obtained by Eqs. (2), (16), and (17):

$$\left\{ {\begin{array}{*{20}l} {\varphi = \arccos \frac{x}{R},} \hfill \\ {\gamma { = }180^\circ - 2\arccos (\sin (\arctan \frac{y}{z})\cos (\frac{1}{2}\arccos \frac{x}{R})).} \hfill \\ \end{array} } \right.$$

(18)

2.4 Jacobian Matrix Analysis

By taking the derivative of Eq. (2) with respect to time, the following equation can be obtained:

$$\dot{\user2{P}} = \left[ {\begin{array}{*{20}c} {\dot{x}} \\ {\dot{y}} \\ {\dot{z}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - R\sin \varphi } \\ {R\cos \varphi \sin \omega } \\ {R\cos \varphi \cos \omega } \\ \end{array} } & {\begin{array}{*{20}c} 0 \\ {R\sin \varphi \cos \omega } \\ { - R\sin \varphi \sin \omega } \\ \end{array} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\dot{\varphi }} \\ {\dot{\omega }} \\ \end{array} } \right].$$

(19)

From the symmetrical characteristic of the mechanism, we can know that:

$$\left\{ {\begin{array}{*{20}c} {{\varvec{OB_{2} }} \cdot {\varvec{OP}}\ = {\varvec{b}}_{20} \cdot {\varvec{P}} = R^{2} \cos \angle QOB_{2} ,} \\ {{\varvec{OB_{5} }} \cdot {\varvec{OP}} = {\varvec{b}}_{50} \cdot {\varvec{P}} = R^{2} \cos \angle QOB_{5} .} \\ \end{array} } \right.$$

(20)

Take the derivative each side of Eq. (20) with respect to time, the following equation can be obtained:

$$\left\{ {\begin{array}{*{20}c} {\dot{\user2{b}}_{20} \cdot {\varvec{P}} + {\varvec{b}}_{20} \cdot \dot{\user2{P}} = R^{2} \sin \alpha_{2} \sin \frac{{\alpha_{1} }}{2}\sin \theta_{21} \cdot \dot{\theta }_{21} ,} \\ {\dot{\user2{b}}_{50} \cdot {\varvec{P}} + {\varvec{b}}_{50} \cdot \dot{\user2{P}} = R^{2} \sin \alpha_{6} \sin \frac{{\alpha_{1} }}{2}\sin \theta_{61} \cdot \dot{\theta }_{61} ,} \\ \end{array} } \right.$$

(21)

where $\dot{\user2{b}}_{20} = \left[ {\begin{array}{*{20}c} {\dot{x}_{2} } & {\dot{y}_{2} } & {\dot{z}_{2} } \\ \end{array} } \right]^{{\text{T}}}$, $\dot{\user2{b}}_{50} = \left[ {\begin{array}{*{20}c} {\dot{x}_{2} } & {\dot{y}_{2} } & {\dot{z}_{2} } \\ \end{array} } \right]^{{\text{T}}}$.

It can be derived by Eqs. (10), (11), and (19) that:

$$\left[ {\begin{array}{*{20}c} {\dot{\theta }_{21} } & {\dot{\theta }_{61} } \\ \end{array} } \right]^{{\text{T}}} = {\varvec{J}}\left[ {\begin{array}{*{20}c} {\dot{\varphi }} & {\dot{\omega }} \\ \end{array} } \right]^{{\text{T}}} .$$

(22)

J in Eq. (22) is the inverse kinematics Jacobian matrix.

$${\varvec{J}} = \left[ {\begin{array}{*{20}c} {e_{2} /d_{2} } & {f_{2} /d_{2} } \\ {e_{6} /d_{6} } & {f_{6} /d_{6} } \\ \end{array} } \right],$$

(23)

where

$$d_{i} = D_{i} (x - R) + E_{i} y + F_{i} z,$$

$$e_{i} = - R( - x_{j} \sin \varphi + y_{j} \cos \varphi \sin \omega + z_{j} \cos \varphi \cos \omega ),$$

$$f_{i} = - R(y_{j} \sin \varphi \cos \omega - z_{j} \sin \varphi \sin \omega ),$$

$D_{i} = R\sin \alpha_{i} \sin \frac{{\alpha_{1} }}{2}\sin \theta_{i1} ,$ $E_{i} = R\sin \alpha_{i} \sin \frac{{\alpha_{1} }}{2}\cos \theta_{i1} ,$

$F_{i} = R\sin \alpha_{i} \cos \theta_{i1} ,$ (i = 2, 6, when i = 2 and 6, j = 2 and 5 respectively).

2.5 Verification of Kinematic Analysis

When two tiny values are given as inputs, the correctness of the Jacobian matrix and the forward kinematics are verified by comparing the numerical solution of Eqs. (22) and (23) and with the measurements of the 3D model [33].

Four sets of data under two general configurations are given, as shown in Table 1. Then, the correctness of the inverse kinematic model is verified in the same way, which means the correctness of the kinematics analysis of the 2DOF SPM.

Table 1 Verification of the Jacobian matrix

3 Workspace Analysis

Due to the interference of the mechanism, the reference point P of the end effector can’t reach every point on the spherical surface. As shown in Figure 6, suppose the width of each link of the mechanism is 8 mm, the effective radius is R = 200 mm, that is, OP = OQ = 200 mm, α₁ = α₄ = 60°, α₂ = α₃ = α₅ = α₆ = 40°, and α₇ = α₈ = 50°.

Figure 6

Workspace of the spherical mechanism

To avoid interference, considering the width of the links, assume that the angle between the rotation axes OB₁ and OB₃ and the angle between OB₄ and OB₆ is not less than 10°. The workspace of the mechanism in Figure 6 can be obtained according to the inverse kinematics and the interference condition. The specific limited configuration and corresponding position parameters of the mechanism are shown in Table 2.

Table 2 Limited configuration parameters of the spherical mechanism

4 Equivalent Rotation Characteristics of the Mechanism

4.1 Equivalent Rotation Characteristics

The end effector of the 2DOF SPM can realize continuous rotation around the axis that passes through the rotation center and lies on the mid-plane during the moving process. Moreover, the 2DOF SPM also has the following motion properties: Given the initial position and the end position, the end link can realize the pose transformation through a rotation around a fixed axis, which is called the equivalent rotation of the mechanism.

As the simplified motion model shown in Figure 7(a), the end effector moves from position I to position II, and the mid-planes at the initial and final positions are s₁ and s₂, respectively. The symmetric points of Q about the mid-plane are P₁ and P₂, respectively. The line l is the intersection line of the two mid-planes, and the axis of the equivalent rotation [34]. For a clear obversion, a plane s₃ is set, which passes through line OP₁ and is perpendicular to line l, as shown in Figure 7(b). S is the intersection point of the line l and the plane s₃. K is the intersection point of line QP₁ and plane s₁. J is the intersection point of line QP₂ and plane s₂.

Figure 7

Schematic diagram of the initial and final configuration (a) and front view of plane s₃ (b)

4.2 How to Realize the Equivalent Rotation

As shown in Figure 7(a), the two parameters φ₁ and γ₁ of the initial configuration of the mechanism and the two parameters φ₂ and γ₂ of the final configuration are given. The coordinates of output reference point can be obtained by Eq. (2). The equation of axis l, which is the intersection line of the two mid-planes, can be obtained by Eq. (15). The equation of plane s₃, which is passing through lines QP₁ and QP₂, can be obtained according to the structural characteristics. And the coordinates of the point S can be obtained by the equations of axis l and plane s₃.

Then the rotated angle of the output reference point P can be derived that:

$$\theta = \arccos \left(\frac{{{\varvec{SP}}_{{\mathbf{1}}} \cdot {\varvec{SP}}_{{\mathbf{2}}} }}{{\left| {{\varvec{SP}}_{{\mathbf{1}}} } \right| \cdot \left| {{\varvec{SP}}_{{\mathbf{2}}} } \right|}}\right),$$

(24)

where SP₁ = P₁−S and SP₂ = P₂−S.

The direction vector l = [l_x l_y l_z]^T and the rotation angle θ of the end effector rotating around the axis l are already obtained, and the rotation matrix R_(θ) can be expressed by:

$${\varvec{R}}_{(\theta )} = \left[ {\begin{array}{*{20}c} {l_{x} l_{x} \xi + \cos \theta } & {l_{y} l_{x} \xi - l_{z} \sin \theta } & {l_{z} l_{x} \xi + l_{y} \sin \theta } \\ {l_{x} l_{y} \xi + l_{z} \sin \theta } & {l_{y} l_{y} \xi + \cos \theta } & {l_{z} l_{y} \xi - l_{x} \sin \theta } \\ {l_{x} l_{z} \xi - l_{y} \sin \theta } & {l_{y} l_{z} \xi + l_{x} \sin \theta } & {l_{z} l_{z} \xi + \cos \theta } \\ \end{array} } \right],$$

(25)

where $\xi$ = $(1 - \cos \theta )$.

The vector QP₂ can be expressed as:

$${\varvec{QP}}_{{\mathbf{2}}} = {\varvec{R}}_{(\theta )} {\varvec{SP}}_{1} + {\varvec{QS}}.$$

(26)

The coordinates of point P₂ can be obtained by Eq. (26), and the other parameters of the mechanism can be obtained by the inverse kinematics described in Section 2.2. Thereby, the driving parameters θ₂₁ and θ₆₁ of the rotation process can be obtained. It provides the basis for the motion planning of the spherical mechanism.

4.3 Motion Planning of the Equivalent Rotation

As shown in Figure 2, suppose the effective radius is R = 200 mm, that is, OP = OQ = 200 mm, α₁ = α₄ = 60°, α₂ = α₃ = α₅ = α₆ = 40°, and α₇ = α₈ = 50°. The parameters of the initial position are φ₁ = 75°, γ₁ = − 20°, and the parameters of final position are φ₂ = 70°, γ₂ = 20°. The four configurations of the mechanism from the initial position to the final position are shown in Figure 8(a)–(d), respectively. The detailed parameters of each configuration are listed in Table 3.

Figure 8

Numerical example of the equivalent rotation

Table 3 Numerical calculation example of the equivalent rotation

5 Variant Mechanisms of the 2DOF SPM

Based on the 3DOF planar sub-chain, a group of variant 2DOF SPMs with the same characteristics are synthesized, providing more potential possibilities for practical application.

In the middle of this mechanism, there are two arc prismatic pairs connecting links 9, 10, and 11, which function to keep the lines OB₂, OB_5, and OB₈ on the same mid-plane. According to the mechanism theory, the 3DOF planar sub-chain can restrict the revolute to the middle plane of the mechanism, ensuring that the relative motion between each motion pair is only planar. Therefore, the 3DOF planar sub-chains are used to replace the spherical links to provide the same constraints. By this method, a set of 2DOF SPMs without arc prismatic pairs can be obtained.

There are seven different configurations of the 3DOF planar sub-chain can be obtained: [RRR], [RPR], [PRR], [RRP], [PPR], [PRP], [RPP], in which R represents revolute pair and P represents prismatic pair [34]. Based on the 3DOF constrained planar sub-chain, seven kinds of equivalent 2DOF SPMs can be obtained, four of which are shown in Figure 9.

Figure 9

Several 2DOF SPMs based on 3DOF planar sub-chain

6 Conclusions

A novel 2DOF Spherical Parallel Mechanism (SPM) is proposed. The SPM can realize continuous rotation around any line on the mid-plane which passes through the rotation center of the spherical mechanism, and the rotational axis can be fixed during the rotation process, which means any form of motion of the mechanism can be transformed into a rotation with a fixed axis.

The forward and inverse kinematics of the mechanism are solved based on D-H parameters and analytical geometry. The inverse Jacobian matrix of the 2DOF SPM is obtained by taking the derivative of the constraint equation, and its workspace is analyzed by considering the interference condition of the links. The correctness of the kinematics and motion planning of the mechanism is verified by the motion examples presented.

A group of variant 2DOF SPMs are constructed based on the different 3DOF planar sub-chain that can provide more possibilities for practical application.