Kinematics Analysis and Singularity Avoidance of a Parallel Mechanism with Kinematic Redundancy

Abstract

The kinematic redundancy is considered as a way to improve the performance of the parallel mechanism. In this paper, the kinematics performance of a three degree-of-freedom parallel mechanism with kinematic redundancy (3-DOF PM-KR) and the influence of redundant parts on the PM-KR are analyzed. Firstly, the kinematics model of the PM-KR is established. The inverse solutions, the Jacobian matrix, and the workspace of the PM-KR are solved. Secondly, the influence of redundancy on the PM-KR is analyzed. Since there exists kinematic redundancy, the PM-KR possesses fault-tolerant performance. By locking one actuating joint or two actuating joints simultaneously, the fault-tolerant workspace is obtained. When the position of the redundant part is changed, the workspace and singularity will be changed. The results show that kinematic redundancy can be used to avoid singularity. Finally, the simulations are performed to prove the theoretical analysis.

1 Introduction

The parallel mechanism is defined as a closed-loop kinematic chain mechanism whose end-effector is linked to the fixed base by several independent kinematic chains [1]. It has some advantages of high stiffness, high accuracy, and small error accumulation. However, it also has some disadvantages, such as singularities in the workspace and difficulty to eliminate self-motion. The self-motion of the parallel mechanism means that when all its actuating joints are locked, the moving platform will produce a certain position or posture change, which proves that the kinematic performance of the mechanism is general. Many scholars have tried to improve the performance of the parallel mechanism by adding redundant actuators or redundant structures.

Scientist Merlet [2] pointed out that redundancy was a way to improve parallel mechanisms’ performance. For parallel mechanisms with redundant actuators, the number of actuators is greater than the number of the mobility of the parallel mechanism. Many researchers proved that redundant actuators could be used to solve the forward kinematics [3], avoid singularities [4, 5], and optimize the distribution of force torque [6,7,8]. However, the redundant actuators need strict movement coordination, which increases the control difficulty.

In order to overcome the shortcomings and realize the self-coordination between inputs and outputs, some scholars proposed to design parallel mechanisms with redundant kinematic structures. For the general parallel mechanism, the actuating joints, the mobility of the mechanism, and the DOFs of the moving platform are equal in number. However, for a parallel mechanism with kinematic redundancy, the number of mobility is equal to the number of actuators but it is greater than that of the DOF of the moving platform. Besides the earlier studies [9,10,11], many researchers have played great attention to the parallel mechanism with kinematic redundancy in recent years.

The Gosselin group conducted a lot of relevant analysis for the kinematically redundant parallel mechanism. Wen et al. [12, 13] proposed a (6+3) DOF kinematically redundant hybrid parallel robot and did a lot of kinematic analysis on it, and the singularities of this mechanism were proved avoidable, indicating that this mechanism had a large workspace. They introduced a new method for detecting mechanical interferences between two links that were not directly connected, proposed for evaluating the workspace of the kinematically redundant parallel mechanism. Landuré et al. [14] proposed a spherical parallel robot and proved that its workspace was huge. The analysis of the mechanism’s singularity showed that the kinematically redundant parallel mechanism's workspace had design advantages. Isaksson et al. [15] provided an analysis of a class of kinematically redundant parallel manipulators. They explored the singularity of this kind of parallel mechanism with kinematic redundancy by using the screw theory.

Many researchers explored the type synthesis of the parallel mechanism with kinematic redundancy. Qu et al. [16,17,18] discussed the difference between a parallel manipulator’s mobility and the relative degree of freedom (RDOF). Based on the proposed RDOF criterion, they performed the type synthesis of the parallel manipulator with open-loop limbs or closed-loop limbs. Besides, they synthesized a kind of kinematically redundant parallel manipulator based on the modified G-K formula and RDOF criterion. Li et al. [19] summarized several types of parallel mechanisms with 2R1T redundant parallel mechanisms and 2T1R redundant parallel mechanisms, and the results showed that all kinematic redundant mechanisms had a good performance. Wang et al. [20] proposed a hybrid strategy based on the combination of linear decoupling geometric analysis method and high-order convergence iteration method to solve the positive solution of parallel mechanism.

Studies show that the kinematics redundant parallel mechanism has dramatically improved its kinematic performance in its mechanism singularity and workspace [21,22,23]. Qu et al. [24] proposed a new 3-RRR parallel mechanism with kinematic redundancy and used three examples to illustrate that the workspace boundary and singular configuration could be changed by adjusting the kinematic redundant actuating parameter. Zhao et al. [25] proposed a 3-RPS/3-SPS parallel mechanism with redundant limbs and defined the evaluation index of the workspace. The results showed that the workspace and kinematics performance of the mechanism were improved. Jin et al. [26] designed a type of 2RIT kinematics redundant parallel mechanism, analyzed the inverse kinematics and workspace of this mechanism, and proved that it had a larger two-dimensional rotation angle of the workspace. Besides, the geometry of the kinematically redundant can be optimized based on the singularity and workspace [27].

For the kinematically redundant parallel mechanism, the influence of redundant kinematics components on the mechanism was explored by solving the statics and dynamics. Qu et al. [28] analyzed the statics of a planar parallel mechanism with kinematic redundancy, and its actuating torque was significantly lower than that of the 3-RRR parallel mechanism. This showed that the introduction of the kinematic redundant components could change the force transmission of the mechanism and improve the statics performance. Bahman et al. [29] took a 3-RPRR planar kinematics redundant mechanism as the research object and used the principle of virtual work to obtain the inverse dynamics of the mechanism. The comparison with the non-redundant mechanism showed that the mechanism could avoid the pass through the rotating joint by adding redundant components. Schreiber et al. [30] studied the dynamic trace planning for a 3-DOF space parallel robot and established its dynamic model. PM-KR has many application prospects, such as solar energy collection devices, 3D simulation equipment, and loading and unloading robots in manufacturing.

The above analysis proves that adding kinematic redundant components to the parallel mechanism has a positive effect on the optimization of mechanical structure and the improvement of mechanical performance. This paper takes a 3-DOF parallel mechanism with kinematic redundancy (PM-KR) as the research object. By building its structure model, the singular position and workspace of the mechanism can be solved. PM-KR can improve the fault tolerance of the mechanism while avoiding singular positions. By changing the height of the redundant component z_p, the workspace of the mechanism can be adjusted, and various singular positions of the mechanism can be avoided, which proves that it is a parallel mechanism with good kinematic performance. The kinematics simulation of the three singular positions of the mechanism is carried out, exploring the impact of changes in the structural parameters of redundant components on the performance of avoiding singular positions, which can prove that this kinematic redundant structure design of the mechanism has obvious advantages.

2 Kinematics Analysis of PM-KR

2.1 Structure Description

The kinematically redundant parallel mechanism studied in this paper is shown in Figure 1. This parallel mechanism consists of a fixed base, an equilateral triangular moving platform, three kinematic limbs with the same structure and a lifting platform. Qu et al. [17] proved that the moving platform has three independent movements, and the mobility of the whole mechanism is equal to 4. The 4 prismatic joints connected to the base are selected as the actuating joints, respectively L₁, L₂, L₃, and L₄.

Figure 1

The model of 3-DOF PM-KR

One side of the limbs of the mechanism is connected to the base by a prismatic joint, and three prismatic joints are all actuating joints. The other side is connected to the moving platform through a spherical joint. The rotation joints of each limb are passive and are parallel to each other. A_i, C_i, D_i and E_i (i = 1, 2, 3) are the center points of the rotating joints in each limb; B_i is the center point of the spherical joint in each limb; P₄ is the center point of the lifting platform; the link B_iC_i connects to the link D_iE_i and passes through the rotating joint D_i, and D_i is a point of the link B_iC_i. Links A_iC_i, D_iE_i, and B_iC_i are denoted as link l_i1, link l_i2, and link l_i3 (i = 1, 2, 3), respectively. The coordinate systems O-xyz and P-uvw are establish. The z-axis is perpendicular to the base; the x-axis is towards the actuating joint P₁; the w-axis is perpendicular to the moving platform, and the u-axis is towards the joint B₁.

2.2 Inverse Kinematics Solution

The detailed process to solve the inverse kinematics solution of the mechanism is shown in Figure 2.

Figure 2

The inverse kinematics solution process of the mechanism

As shown in Figure 3, it is the geometric structure of the mechanism. The base coordinate system {O} is O-xyz with the y-axis pointing out of the paper and x-axis aligned with the axis of prismatic joint P_i. The moving platform coordinate system {P} is P-uvw with the w-axis perpendicular to the u-axis and the v-axis.

Figure 3

The geometric structure of the mechanism

The position of the center point P of the moving platform can be defined as shown in Eq. (1).

$${}^{O}{\varvec{P}} = \left[ {\begin{array}{*{20}c} {x_{p} } & {y_{p} } & {z_{p} } \\ \end{array} } \right]^{{\text{T}}} ,$$

(1)

Besides, the rotation matrix ^OR_P is used to define the position of the moving platform relative to the base coordinate system {O}, using the expression of pitch-roll-yaw, as shown in Eqs. (2) and (3).

$${}^{O}{\varvec{R}}_{P} = \left[ {\begin{array}{*{20}c} {u_{x} } & {v_{x} } & {w_{x} } \\ {u_{y} } & {v_{y} } & {w_{y} } \\ {u_{z} } & {v_{z} } & {w_{z} } \\ \end{array} } \right]{, }$$

(2)

$$\begin{gathered} {}^{O}{\varvec{R}}_{P} { = }{\varvec{R}}_{z} \left( \gamma \right){\varvec{R}}_{y} \left( \beta \right){\varvec{R}}_{x} \left( \alpha \right) = \left[ {\begin{array}{*{20}c} {c\beta c\gamma } & {c\gamma s\alpha s\beta - c\alpha s\gamma } & {c\alpha c\gamma s\beta + s\alpha s\gamma } \\ {c\beta s\gamma } & {s\alpha s\beta s\gamma + c\alpha c\gamma } & {c\alpha s\beta s\gamma - c\gamma s\alpha } \\ { - s\beta } & {c\beta s\alpha } & {c\alpha c\beta } \\ \end{array} } \right], \hfill \\ \hfill \\ \end{gathered}$$

(3)

where, α, β and γ are respectively expressed as Euler angles of the moving platform about the x, y and z axes. Therefore, according to Eq. (4), the midpoint’s representation on the moving platform in {O} can be obtained.

$${}^{O}P = {}^{O}{\varvec{R}}_{P} {}^{P}\varvec{P} + {\varvec{OP}}\user2{.}$$

(4)

The positions of the three spherical joints in the moving platform coordinate system {B} are expressed in Eq. (5).

$$\left\{ {\begin{array}{*{20}c} {{}^{P}\varvec{B}_{1} = \left[ {\begin{array}{*{20}c} r & 0 & 0 \\ \end{array} } \right]^{{\text{T}}} ,} \\ {{}^{P}\varvec{B}_{2} = \left[ {\begin{array}{*{20}c} { - r/2} & {\sqrt 3 r/2} & 0 \\ \end{array} } \right]^{{\text{T}}} ,} \\ {{}^{P}\varvec{B}_{3} = \left[ {\begin{array}{*{20}c} { - r/2} & { - \sqrt 3 r/2} & 0 \\ \end{array} } \right]^{{\text{T}}} .} \\ \end{array} } \right.$$

(5)

According to Eq. (5), the transformation formula of the three spherical joints can be obtained, as shown in Eq. (6).

$${}^{O}\varvec{B}_{i} = {}^{O}R_{P} {}^{P}\varvec{B}_{i} + {\varvec{OP}} .$$

(6)

Through calculation, the basic coordinate vector representation of B_i (i = 1,2,3) can be obtained, as shown in Eqs. (7–9).

$${}^{O}{\varvec{B}}_{1} = \left[ {\begin{array}{*{20}c} {x_{p} + ru_{x} } \\ {y_{p} + ru_{y} } \\ {z_{p} + ru_{z} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {x_{p} + r{\text{cos}}\beta {\text{cos}}\gamma } \\ {y_{p} + r{\text{cos}}\beta {\text{sin}}\gamma } \\ {z_{p} - r{\text{sin}}\beta } \\ \end{array} } \right],$$

(7)

$${}^{O}{\varvec{B}}_{2} { = }\left[ {\begin{array}{*{20}c} {x_{p} - \frac{1}{2}r\left( {\cos \beta \cos \gamma + \sqrt 3 ( - \cos\gamma \sin \alpha \sin \beta + \cos \alpha \sin \gamma )} \right)} \\ {y_{p} - \frac{1}{2}r\cos \beta \sin \gamma + \frac{\sqrt 3 }{2}r(\cos\alpha \cos \gamma + \sin \alpha \sin \beta \sin \gamma )} \\ {z_{p} + \frac{1}{2}r(\sqrt 3 \cos\beta \sin\alpha + \sin \beta )} \\ \end{array} } \right],$$

(8)

$${}^{O}{\varvec{B}}_{3} = \left[ {\begin{array}{*{20}c} {x_{p} + \frac{1}{2}r\left( { - \cos \beta \cos \gamma + \sqrt 3 ( - \cos\gamma \sin \alpha \sin \beta + \cos \alpha \sin \gamma )} \right)} \\ {y_{p} - \frac{1}{2}r\left( {\sqrt 3 \cos \alpha \cos \gamma + (\cos\beta + \sqrt 3 \sin \alpha \sin \beta \sin \gamma )} \right)} \\ {z_{p} + \frac{1}{2}r( - \sqrt 3 \cos\beta \sin\alpha + \sin \beta )} \\ \end{array} } \right].$$

(9)

The base coordinate vector representation of the point A_i (i = 1,2,3) is shown in Eq. (10).

$$\left\{ {\begin{array}{*{20}c} {{}^{O}{\varvec{A}}_{1} = \left[ {\begin{array}{*{20}c} {L_{1} } & 0 & {d_{1} } \\ \end{array} } \right]^{{\text{T}}} ,} \\ {{}^{O}{\varvec{A}}_{2} = \left[ {\begin{array}{*{20}c} { - 1/2L_{2} } & {\sqrt 3 {/2}L_{2} } & {d_{1} } \\ \end{array} } \right]^{{\text{T}}} ,} \\ {{}^{O}{\varvec{A}}_{3} = \left[ {\begin{array}{*{20}c} { - 1/2L_{3} } & { - \sqrt 3 {/2}L_{3} } & {d_{1} } \\ \end{array} } \right]^{{\text{T}}} .} \\ \end{array} } \right.$$

(10)

According to the position coordinates of A_i and B_i, the distance between the joint A_i and the joint B_i can be calculated, as shown in Eq. (11).

$${\varvec{A}}_{i} {\varvec{B}}_{i} = {\varvec{B}}_{i} - {\varvec{A}}_{i} .$$

(11)

The vector s_i is set in three directions on the base, as shown in Eq. (12).

$$\left\{ {\begin{array}{*{20}c} {{}^{O}{\varvec{s}}_{1} = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 \\ \end{array} } \right]^{{\text{T}}} ,} \\ {{}^{O}{\varvec{s}}_{2} = \left[ {\begin{array}{*{20}c} {\sqrt 3 } & 1 & 0 \\ \end{array} } \right]^{{\text{T}}} ,} \\ {{}^{O}{\varvec{s}}_{3} = \left[ {\begin{array}{*{20}c} { - \sqrt 3 } & 1 & 0 \\ \end{array} } \right]^{{\text{T}}} .} \\ \end{array} } \right.$$

(12)

Because s_i is perpendicular to A_iB_i, the dot product of the two vectors is equal to 0, and the expressions of x_p, y_p, and γ can be obtained, as shown in Eqs. (13)–(15).

$$x_{p} = - \frac{{r\left( {\cos^{2} \alpha - \cos^{2} \beta + \sin^{2} \alpha \sin^{2} \beta } \right)}}{{2\sqrt {\left( {1 + \cos \alpha \cos \beta } \right)^{2} } }},$$

(13)

$$y_{p} = - \frac{r\cos \beta \sin \alpha \sin \beta }{{\sqrt {\left( {1 + \cos \alpha \cos \beta } \right)^{2} } }},$$

(14)

$$\gamma = \arctan \left[ {\frac{\cos \alpha + \cos \beta }{{\sqrt {\left( {1 + \cos \alpha \cos \beta } \right)^{2} } }},\frac{\sin \alpha \sin \beta }{{\sqrt {\left( {1 + \cos \alpha \cos \beta } \right)^{2} } }}} \right].$$

(15)

According to the above method, the coordinate representation of the remaining joints on the limb can be obtained sequentially.

The inverse solution of the mechanism is that the parameters α, β and z_p of the moving platform are known, and the actuating lengths L₁, L₂ and L₃ of the input prismatic joints are calculated. According to the vector closed-loop principle, a vector closed-loop equation is established in a limb of this mechanism, as shown in Figure 4. The four vector combinations OA₁, A₁C₁, C₁B₁, B₁O are selected as the closed-loop.

Figure 4

Diagram of closed-loop 1

According to Figure 4, the closed-loop equation 1 of limb 1 can be obtained, as shown in Eq. (16).

$${\varvec{OP}}_{1} + {\varvec{P}}_{1} {\varvec{A}}_{1} + {\varvec{A}}_{1} {\varvec{C}}_{1} + {\varvec{C}}_{1} {\varvec{B}}_{1} = {\varvec{OB}}_{1} .$$

(16)

Then Eq. (16) is expressed with the structural parameters of the mechanism, as shown in Eq. (17).

$$\left[ {\begin{array}{*{20}c} {x_{a1} } \\ 0 \\ {z_{a1} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {a_{1} \cos \theta_{11} } \\ 0 \\ {a_{1} \sin \theta_{11} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} { - \left( {a_{2} + a_{3} } \right)\cos \theta_{12} } \\ 0 \\ {\left( {a_{2} + a_{3} } \right)\sin \theta_{12} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {x_{b1} } \\ 0 \\ {z_{b1} } \\ \end{array} } \right].$$

(17)

Because the mechanism is a parallel mechanism with kinematic redundancy, its redundant structure is a lifting platform. The two closed-loop equations of the limb are solved simultaneously, as shown in Figures 5 and 6.

Figure 5

Diagram of closed-loop 2

Figure 6

Diagram of closed-loop 3

According to Figure 5, the closed-loop equation 2 of limb 1 can be obtained as Eq. (18).

$${\varvec{OP}}_{1} + {\varvec{P}}_{1} {\varvec{A}}_{1} + {\varvec{A}}_{1} {\varvec{C}}_{1} + {\varvec{C}}_{1} {\varvec{D}}_{1} + {\varvec{D}}_{1} {\varvec{E}}_{1} = {\varvec{OE}}_{1} .$$

(18)

Then Eq. (18) is expressed with the structural parameters of the mechanism, as shown in Eq. (19).

$$\left[ {\begin{array}{*{20}c} {x_{a1} } \\ 0 \\ {z_{a1} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {a_{1} \cos \theta_{11} } \\ 0 \\ {a_{1} \sin \theta_{11} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} { - a_{2} \cos \theta_{12} } \\ 0 \\ {a_{2} \sin \theta_{12} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} { - a_{4} \cos \theta_{13} } \\ 0 \\ { - a_{4} \sin \theta_{13} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {x_{e1} } \\ 0 \\ {z_{e1} } \\ \end{array} } \right].$$

(19)

As shown in Figure 6, the closed-loop equation 3 of limb 1 can be obtained, as shown in Eq. (20).

$${\varvec{E}}_{1} {\varvec{D}}_{1} + {\varvec{D}}_{1} {\varvec{B}}_{1} = {\varvec{E}}_{1} {\varvec{B}}_{1} .$$

(20)

Then Eq. (20) is expressed with the structural parameters of the mechanism, as shown in Eq. (21).

$$\left[ {\begin{array}{*{20}c} {a_{4} \cos \theta_{13} } \\ 0 \\ {a_{4} \sin \theta_{13} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} { - a_{3} \cos \theta_{12} } \\ 0 \\ {a_{3} \sin \theta_{12} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {x_{b1} - x_{e1} } \\ 0 \\ {z_{b1} - z_{e1} } \\ \end{array} } \right].$$

(21)

Eqs. (16)–(21) are expanded to get six equations, as shown in Eq. (22).

$$\left\{ {\begin{array}{*{20}c} {x_{a1} + a_{1} \cos \theta_{11} - \left( {a_{2} + a_{3} } \right)\cos \theta_{12} = x_{b1} , } \\ {z_{a1} + a_{1} \sin \theta_{11} + \left( {a_{2} + a_{3} } \right)\sin \theta_{12} = z_{b1} , } \\ {x_{a1} + a_{1} \cos \theta_{11} - a_{2} \cos \theta_{12} - a_{4} \cos \theta_{13} = x_{e1} , } \\ {z_{a1} + a_{1} \sin \theta_{11} + a_{2} \sin \theta_{12} - a_{4} \sin \theta_{13} = z_{e1} , } \\ {a_{4} \cos \theta_{13} - a_{3} \cos \theta_{12} = x_{b1} - x{}_{e1}, } \\ {a_{4} \sin \theta_{13} + a_{3} \sin \theta_{12} = z_{b1} - z_{e1} . } \\ \end{array} } \right.$$

(22)

According to the coordinate expressions of B₁ and E₁, D₁ is taken as the center of the circle, B₁D₁, and E₁D₁ as the radius. The equations of the two circles are listed respectively, and the coordinates of D₁ can be solved, as shown in the Eqs. (23) and (24).

$$\left( {x_{d1} - x_{b1} } \right)^{2} + \left( {z_{d1} - z_{b1} } \right)^{2} = a_{3}^{2} ,$$

(23)

$$\left( {x_{d1} - x_{e1} } \right)^{2} + \left( {z_{d1} - z_{e1} } \right)^{2} = a_{4}^{2} .$$

(24)

According to the coordinates of B₁ and D₁ and the tangent formula of the triangle, the size of θ₁₃ can be solved, as shown in Eq. (25).

$$\theta_{13} = \arctan \frac{{z_{d1} - z_{e1} }}{{x_{d1} - x_{e1} }}.$$

(25)

According to the closed-loop equation (22), the size of θ₁₂ can be solved, as shown in Eq. (26).

$$\theta_{12} = \arccos \frac{{ - x_{b1} + x_{e1} + a_{4} \cos \theta_{13} }}{{a_{3} }}.$$

(26)

Simultaneously eliminating θ₁₂ in Eqs. (22) and (27) can be obtained.

$$\begin{gathered} a_{1}^{2} + \left( {x_{a1} - x_{b1} } \right)^{2} + 2a_{1} \left( {x_{a1} - x_{b1} } \right)\cos \theta_{11} + \left( {z_{a1} - z_{b1} } \right)^{2} { + }2a_{1} \left( {z_{a1} - z_{b1} } \right)\sin \theta_{11} = \left( {a_{2} + a_{3} } \right)^{2} . \hfill \\ \hfill \\ \end{gathered}$$

(27)

According to Eq. (27), the expression of θ₁₁ can be obtained, as shown in Eq. (28).

$$\begin{gathered} \theta_{11} = \arctan \hfill \\ \left( {\frac{{ - a_{1} \left( {x_{a1} - x_{b1} } \right)\left( {a_{1}^{2} - \left( {a_{2} + a_{3} } \right)^{2} + \left( {x_{a1} - x_{b1} } \right)^{2} + \left( {z_{a1} - z_{b1} } \right)^{2} } \right)}}{{a_{1}^{2} \left( {\left( {x_{a1} - x_{b1} } \right)^{2} + \left( {z_{a1} - z_{b1} } \right)^{2} } \right)}}} \right. + \hfill \\ \left. \begin{gathered} \frac{{\sqrt { - a_{1}^{2} \left( {\left( { - a_{1} + a_{2} + a_{3} } \right)^{2} - \left( {x_{a1} - x_{b1} } \right)^{2} - \left( {z_{a1} - z_{b1} } \right)^{2} } \right)} }}{{a_{1}^{2} \left( {\left( {x_{a1} - x_{b1} } \right)^{2} + \left( {z_{a1} - z_{b1} } \right)^{2} } \right)}} \hfill \\ \frac{{\sqrt {\left( {\left( {a_{1} + a_{2} + a_{3} } \right)^{2} - \left( {x_{a1} - x_{b1} } \right)^{2} - \left( {z_{a1} - z_{b1} } \right)^{2} } \right)\left( {z_{a1} - z_{b1} } \right)^{2} } }}{{a_{1}^{2} \left( {\left( {x_{a1} - x_{b1} } \right)^{2} + \left( {z_{a1} - z_{b1} } \right)^{2} } \right)}} \hfill \\ \end{gathered} \right). \hfill \\ \end{gathered}$$

(28)

According to Eq. (22), only the actuating parameter L₁ is unknown in the equation. By solving the equation, the expression equation of L₁ can be obtained, as shown in Eq. (29).

$$L_{1} = x_{e1} - a_{1} \cos \theta_{11} - a_{2} \cos \theta_{12} - a_{4} \cos \theta_{13} .$$

(29)

In the same way, the expressions for actuating lengths L₂ and L₃ can be obtained by the closed-loop equations of limb 2 and limb 3, as shown in Eqs. (30) and (31).

$$L_{2} = - 2x_{e2} - a_{1} \cos \theta_{21} + a_{2} \cos \theta_{22} + a_{4} \cos \theta_{23} ,$$

(30)

$$L_{3} = - 2x_{e3} - a_{1} \cos \theta_{31} + a_{2} \cos \theta_{32} + a_{4} \cos \theta_{33} .$$

(31)

2.3 Singularity Analysis

The positive Jacobian matrix J_q is the relationship between the twist χ of the motion platform expressed by the six-dimensional vector and the rate of changing L_i of the actuating length, as shown in Eq. (32).

$${\varvec{L}}_{i} { = }{\varvec{J}}_{q} {\varvec{\chi}}\user2{.}$$

(32)

The Jacobian matrix of the parallel mechanism with kinematic redundancy is the relationship between the angular velocity ω of the moving platform and the vector L₁, L₂, and L₃ of the actuating joint change rate, as shown in Eq. (33).

$${\varvec{L}} = {\varvec{J}} \cdot {\varvec{\omega}}\user2{.}$$

(33)

The relationship among the inverse Jacobian matrix J_x, the three-dimensional twist vector χ of the moving platform, and the angular velocity ω is shown in Eq. (34).

$${\varvec{\chi}}{ = }\left[ {\begin{array}{*{20}c} {{}^{A}{\varvec{v}}_{p} } \\ {\varvec{\omega}} \\ \end{array} } \right]{ = }{\varvec{J}}_{x} {\varvec{\omega}}\user2{.}$$

(34)

where χ represents the moving platform’s twist; ^Av_p is the speed of the moving platform, and ω is the angular velocity of the moving platform.

The closed-loop 2 of the limb1, 2, 3 is selected. According to the result of the inverse solution, the constraint equations of limbs 1, 2, and 3 are shown in Eqs. (35)–(37).

$$\begin{gathered} \left( {x_{a1} - x_{e1} - a_{2} \times \cos \theta_{12} - a_{4} \cos \theta_{13} } \right)^{2} + \left( {z_{a1} - z_{e1} + a_{2} \sin \theta_{12} - a_{4} \sin \theta_{13} } \right)^{2} = a_{1}^{2} , \hfill \\ \hfill \\ \end{gathered}$$

(35)

$$\begin{gathered} \left( {2x_{a2} - 2x_{e2} - a_{2} \cos \theta_{22} + a_{4} \cos \theta_{23} } \right)^{2} + \left( {z_{a2} - z_{e2} + a_{2} \sin \theta_{22} - a_{4} \sin \theta_{23} } \right)^{2} = a_{1}^{2} , \hfill \\ \hfill \\ \end{gathered}$$

(36)

$$\begin{gathered} \left( {2x_{a3} - 2x_{e3} + a_{2} \cos \theta_{32} + a_{4} \cos \theta_{33} } \right)^{2} + \left( {z_{a3} - z_{e3} + a_{2} \sin \theta_{32} - a_{4} \sin \theta_{33} } \right)^{2} = a_{1}^{2} . \hfill \\ \hfill \\ \end{gathered}$$

(37)

The partial derivative of the time t is calculated in the actuating parameters L₁, L₂, L_3, and the moving platform motion parameters α, β, z_p. After simplification, the Eqs. (38)–(40) can be obtained.

$$e_{1} = e_{11} \times \dot{\alpha } + e_{12} \times \dot{\beta } + e_{13} \times \dot{z}_{p} + e_{14} \times \dot{L}_{1} ,$$

(38)

$$e_{2} = e_{21} \times \dot{\alpha } + e_{22} \times \dot{\beta } + e_{23} \times \dot{z}_{p} + e_{24} \times \dot{L}_{2} ,$$

(39)

$$e_{3} = e_{31} \times \dot{\alpha } + e_{32} \times \dot{\beta } + e_{33} \times \dot{z}_{p} + e_{34} \times \dot{L}_{3} .$$

(40)

In Eqs. (38)–(40), e_ij (i = 1,2,3 and j = 1,2,3,4) respectively represent the result of the combination of the same type and the simplified expressions of the result in Eqs. (35)–(37). Because these expressions are too long, they are not shown in the paper.

Extracting the common parameter items in the above equations, the positive Jacobian matrix J_q and the inverse Jacobian matrix J_x can be obtained as Eqs. (41) and (42).

$${\varvec{J}}_{q} = \left[ {\begin{array}{*{20}c} {e_{14} } & 0 & 0 \\ 0 & {e_{24} } & 0 \\ 0 & 0 & {e_{34} } \\ \end{array} } \right],$$

(41)

$${\varvec{J}}_{x} = \left[ {\begin{array}{*{20}c} {e_{11} } & {e_{12} } & {e_{13} } \\ {e_{21} } & {e_{22} } & {e_{23} } \\ {e_{31} } & {e_{32} } & {e_{33} } \\ \end{array} } \right].$$

(42)

The singularity of the mechanism is solved. The structural parameters of the mechanism are set, d₁ = 30 mm, a₁ = 114 mm, a₂ = 92 mm, a₃ = 115 mm, a₄ = 132 mm, d₄ = 62.68 mm, r = 78 mm. The range of z_p is set as [150, 210] mm, and the range of α and β is set as [−2, 2] rad.

Let Det(J_x) = 0, the actuating parameter L₄ is taken as the structure parameter of the mechanism. When L₄  =  80 mm, the height of the lifting platform is located between the base and the moving platform, so that the mechanism as a whole maintains a normal working state. When the actuating joint changes, the height of the lifting platform will not have a significant impact on the position and posture of the mechanism. And the singularity diagram of the mechanism under L₄  =  80 mm is measured, as shown in Figure 7.

Figure 7

Singular diagram of mechanism when L₄ = 80 mm

Set z_p = 150/170/190/ 210 mm, respectively, and the positive motion singular images of the mechanism under different values of z_p are measured, as shown in Figure 8.

Figure 8

The singularity diagram of the mechanism when z_p takes different values

By analyzing the Jacobian matrix and singular image of the mechanism, the singular position of the mechanism can be obtained [31,32,33]. When θ_i1 = π/2 and θ_i2 = 0°of a limb, the mechanism will have a singular position 1, as shown in Figure 20a. When θ_i1 = θ_i2 = π/2 of the three limbs, the mechanism will have a singular position 2, as shown in Figure 24a. When θ_i1 = 0° of a limb, the mechanism produces a singular position 3, as shown in Figure 26a. The redundant components of the mechanism called lifting platform, its position will affect the singularity of the mechanism, which will be analyzed in the content.

2.4 Workspace Analysis

The structural parameters of the mechanism are set, d₁ = 30 mm, a₁ = 114 mm, a₂ = 92 mm, a₃ = 115 mm, a₄ = 132 mm, d₄ = 62.68 mm, r = 78 mm.

The workspace of the mechanism is analyzed by adopting the workspace search method. The length of the actuating joint L₄ is set as a fixed value, and the point set is obtained where the workspace satisfies the inverse solution of the mechanism. Finally, we can get the workspace diagram of the mechanism.

Analyzing limb 1, according to the Eqs. (25)–(28) of θ₁₁, θ₁₂, θ₁₃ obtained by the inverse solution, substituting the above three equations into Eq. (29), we can get L₁’s expression only have the moving platform kinematic parameters include α, β, and z_p. Similarly, we can get L₁, and L₂’s expression only has the moving platform kinematic parameters including α, β, and z_p.

The value range of L₁, L₂, and L₃ is set as [50, 200] mm; the range of z_p is set as [50, 350] mm, and the value range of α and β is set to [−π/2, π/2]. L₄ is used as a structural parameter, and the workspace of the mechanism can be calculated when L₄ = 90 mm.

The surface image [34] of the workspace of the analyzed mechanism can be obtained, as shown in Figure 9, by using the method of fitting a surface with a set of points and fitting the workspace point set of the mechanism.

Figure 9

Three-dimensional curve diagram of the workspace

Based on the obtained workspace in Figure 9, the workspace diagram of the mechanism with z_p = 230/250/270 /290 mm can be sketched, as shown in Figure 10. The moving platform parameters α and β are variables, and z_p is a fixed value.

Figure 10

Workspace of mechanism with different values of z_p

By analyzing the workspace of the mechanism, we can get the changes of the moving platform kinematics parameters α and β when the moving platform parameter z_p is a fixed value, as shown in Table 1.

Table 1 Changes of kinematic parameters α and β of the moving platform

Table 1 shows that when the moving platform is in working condition, the maximum angle α and β of its rotation around the x and y axes can reach 66°. The moving platform of this mechanism has the characteristics of a large rotation angle compared with the general parallel mechanism, which can meet the needs of specific working conditions.

3 The Influence of Redundant Components on Mechanism’s Kinematics

Adding a redundant part to a general parallel mechanism, the kinematic characteristics of the mechanism will be affected, such as fault tolerance, workspace, and singularity [35]. This section researches the impact of redundant components, analyzing the influence of the change of redundant part’s parameters on the mechanism’s kinematics.

3.1 Analysis of Mechanism Fault Tolerance

Fault tolerance refers to the change in the mechanism’s kinematic performance when a part of the mechanism fails. The better the fault-tolerant mechanism is, the smaller the difference in the mechanism’s kinematics when the components fail. For the mechanism with poor fault-tolerant performance, the components’ failure will significantly impact the overall kinematics of the mechanism.

When one of the actuating joints of the parallel mechanism fails, the overall kinematic performance of the mechanism will be affected. Compared with the normal mechanism, the DOF of the mechanism with failed actuating joints will be less or even zero, and the mechanism will not have any kinematic performance. A parallel mechanism with good kinematics has better fault tolerance, that is, when multiple actuating joints fail, it still has certain kinematic performance.

There are many indexes for measuring the fault tolerance performance of a mechanism, such as changes in the workspace, changes in actuating torque, and each joint’s integrity. In measuring the fault-tolerant performance of the 3-DOF PM-KR, the change of the workspace is selected as the performance index. When a specific actuating joint of the mechanism fails, the influence on the mechanism’s workspace will be very obvious. As a result, the change of the workspace of the mechanism is used to test the fault tolerance performance of the mechanism.

As shown in Figure 11, the image of the workspace of the mechanism when the machine is operating normally, the actuating joint P₁ fails, the actuating joints P₁ and P₂ all fail. In the image, the moving platform kinematic parameters z_p = 210–310 mm, the height of the lifting platform L₄ = 90 mm, the kinematics parameters of the moving platform α, β∈[−90°, 90°], the actuating parameters L₁, L₂, L₃∈[50, 200] mm.

Figure 11

Images of the workspace when the L₄ = 90 mm

The point search method is adopted to analyze the fault-tolerant performance of the mechanism, and the distance between the points is 3 mm. First, on the condition that the mechanism is normal, the number of points in the workspace is used as the base. Then the effective proportion of the workspace is analyzed when one actuating joint fails and when two actuating joints fail simultaneously.

As shown in Table 2, it can be found that when the kinematic parameters of the moving platform z_p = 210/230/ 250/310 mm, the workspace when the actuating joint P₁ fails is about 60% of the normal workspace. When the actuating joint P₁ and P₂ fail simultaneously, the workspace is about 30% of the normal workspace, indicating that the mechanism has better performance in this moving platform height.

Table 2 Changes in the workspace of the mechanism

When the moving platform’s kinematic parameters are z_p  =  270/290 mm, the workspace when the actuating joint P₁ fails is about 30% of the normal workspace. When the actuating joint P₁ and P₂ fail simultaneously, the workspace is about 10% of the normal workspace. Compared with the fault tolerance of the mechanism obtained by the previous parameters, the mechanism's fault tolerance performance decreases.

Moreover, the position where the actuating joint fails has an impact on the fault tolerance of the mechanism. Measure the influence of the position where the actuating joint fails on the workspace. When the actuating joint P₁ fails, P₁ and P₂ fail simultaneously, the influence of the failure position of the actuating joint on the workspace is shown in Table 3.

Table 3 The influence of the failure position of the actuating joint on the workspace

Through the analysis of Table 3, we can find that when the actuating joint fails at [50, 100] mm, the workspace when the actuating joint P₁ fails is 10.91% of the normal workspace, and the workspace when the actuating joints P₁ and P₂ fail simultaneously is 1.9% of the normal workspace. Because this position of the actuating joint is very close to the center of the whole mechanism, indicating that the actuating joint has a fault at this position, the fault tolerance of the mechanism is very poor. When the fault position of the actuating joint is [100, 150] mm, the fault tolerance of the mechanism is improved, the workspace when the actuating joint P₁ fails is 29.68% of the normal workspace, and the workspace when the actuating joints P₁ and P₂ fail simultaneously is 11.86% of the normal workspace. When the actuating joint’s fault position is [150, 200] mm, the fault tolerance of the mechanism is excellent, and the position of the actuating joint is the farthest from the center of the mechanism.

In summary, the 3-DOF PM-KR has good fault tolerance. By analyzing the different height values z_p of the moving platform, the fault tolerance of the mechanism improves, the maximum workspace when the actuating joint P₁ fails is 60.78% of the normal workspace, and the maximum workspace when the actuating joint P₁ and P₂ fail simultaneously is 32.35% of the normal workspace. At the same time, the position where the actuating joint fails also affects the fault tolerance of the mechanism. When the actuating joint fails farther away from the center, the better the fault tolerance of the mechanism; otherwise, the worse the fault tolerance of the mechanism.

3.2 The Impact of Redundant Components on Workspace

Adding redundant components to the parallel mechanism will have an impact on the entire mechanism's workspace [35, 36]. In the 3-DOF PM-KR, the lifting platform is connected with link l_i3 through three redundant link l_i2 (i = 1, 2, 3) at rotating joints. The position and posture of each link in the limb change with the height of the lifting platform, which will affect the workspace of the mechanism.

The kinematic parameter z_p = 250 mm of the moving platform is set as a fixed value, and the influence of height variation of lifting platform, L₄, on the motion parameters α and β of the moving platform can be obtained. As shown in Figure 12, it is the changing image of the workspace of the mechanism when L₄ = 60/65/70/75 mm.

Figure 12

L₄ = 60–75 mm workspace images of the mechanism

By observing the image of the workspace, it can be concluded that with the increase of the value of L₄, the workspace of the mechanism also becomes larger, indicating that the workspace of the mechanism in this posture increases with the structural parameters of redundant components.

As shown in Figure 13, it is an image of the workspace when L₄ = 80/85/90/95 mm. In this posture, when the value of L₄ increases from 80 mm to 90 mm, the workspace continues to increase. But when the value of L₄ continues to increase, the lifting platform is located in the center of the entire mechanism, its position has little influence on the posture of the limb, so the change of the workspace of the mechanism is not apparent.

Figure 13

L₄ = 80–95 mm workspace images of the mechanism

Continue to increase the value of L₄, and the changing image of the workspace is taken when L₄ = 100/105/110/115 mm, as shown in Figure 14. By observing the posture change of the workspace, we can explain that as the value of L₄ increases in this range, the mechanism's workspace is continuously reduced. Since the kinematic parameter z_p of the moving platform is set to a fixed value, in this posture, the lifting platform's height is constantly rising close to the moving platform. At this time, the mechanism's posture inhibits the movement of the limb, which makes the workspace of the mechanism continuously reduced.

Figure 14

L₄ = 100–115 mm workspace images of the mechanism

Analyzing the mechanism’s workspace shows that the change of the kinematic redundant part's position significantly impacts the workspace. When the lifting platform is at the starting position, as its height value increases, the mechanism's workspace also becomes larger. Because each mechanism’s workspace is fixed, when the position of the lifting platform reaches a specific height value, the change of the workspace of the mechanism is not obvious at this time. As the lifting platform's height continues to increase, its position slowly approaches the moving platform of the mechanism, and the redundant part has a significant inhibitory effect on the movement of the limb, which causes the workspace to reduce significantly.

3.3 The Effect of Redundant Components on Singularity

The structural parameter’s change of the redundant part will impact the mechanism’s kinematics, the most significant of which is to change the mechanism to avoid the singularity. Kinematics analysis obtains the Jacobian matrix of the mechanism and then solves the mechanism’s singularity. Finally, we can obtain the singular position of the motion posture of the mechanism. In this section, the kinematics parameter z_p of the moving platform changes with the values of the redundant part L₄, and different L₄ corresponds to different singular configurations of the mechanism. Therefore, the influence of the redundant components on the singularity avoidance of the mechanism is analyzed.

As shown in Figure 15, it is an image of the singularity of the mechanism when the moving platform’s kinematics parameter is z_p = 150 mm. When the value of L₄ is increased from 60 to 100 mm, the singular position’s image is significantly reduced. Indicating that at this position, as the lifting platform's height value increases, the point set of the singular position of the mechanism continues to shrink.

Figure 15

The image of the singular position of the mechanism when z_p = 150 mm

When the value of actuating joint L₄ becomes larger, the change of the singular position’s range of the mechanism is observed. As shown in Figure 16, it is a singular image when the kinematics parameter z_p of the moving platform of the mechanism is 170 mm, and the value range of L₄ is [60,100] mm. Compared with the singular images, when the height of the moving platform z_p is equal to 150 mm, the point set of the mechanism’s singular position at this height is increased. However, with the increase of the lifting platform’s height, the point set of the mechanism's singular position decreases significantly.

Figure 16

The image of the singular position of the mechanism when z_p = 170 mm

Figures 17 and 18 show the singular image when the kinematic parameters of the moving platform z_p are equal to 190 mm and 210 mm, and the value range of L₄ is [60,100] mm. Research shows that when L₄ is a fixed value, as the value of the kinematic parameter z_p of the moving platform increases, the mechanism's singular position will continue to increase. Similarly, the change of the lifting platform's height value will affect the number of point sets of the singular positions.

Figure 17

The image of the singular position of the mechanism when z_p = 190 mm

Figure 18

The image of the singular position of the mechanism when z_p = 210 mm

Research shows that the change of the kinematic parameters of redundant components has an obvious influence on the mechanism's singular performance. When the moving platform is at a certain height, the lifting platform's height change will reduce the singular position of the mechanism. It also shows that in this 3-DOF parallel mechanism, the addition of redundant components will positively affect the kinematics of the mechanism. When the mechanism is in a singular position, the mechanism's posture can be changed by changing the height of the lifting platform so as to avoid singularity.

4 Simulation Analysis

4.1 Simulations of Mechanism Avoiding Singular Position Under Three Actions

In the previous section, it has been found that the 3-DOF PM-KR has the characteristics of avoiding singularity. When the mechanism is in singular positions, it can avoid the singular positions by changing the lifting platform’s height. In this section, the Adams software is used to simulate the three singular positions of the mechanism. By changing the height of the lifting platform, the singular position of the mechanism can be avoided, which can verify the kinematic characteristics of the mechanism.

The mechanism’s initial position is shown in Figure 19. The height of the moving platform z_p = 170 mm, the deflection angle α = β = 0°, the height of the lifting platform L₄ = 60 mm, and the three actuating parameters on the base L₁ = L₂ = L₃ = 135 mm.

Figure 19

The initial position of the mechanism simulation

As shown in Figure 20a, take limb 1 as an example, it is the first singular position of the mechanism, and the actuating joint’s positions are L₁ = 215 mm, L₂ = L₃ = 100 mm, and L₄ = 60 mm. At this time, θ₁₁ is approximately equal to 90°, the link $A_{1} C_{1}$ of limb 1 is approximately perpendicular to the base, the link $C_{1} B_{1}$ is approximately parallel to the base, and the moving platform occurs a large angle. To avoid this singular position, we can increase the height of the lifting platform. When L₄ is equal to 100 mm, the position and posture of the mechanism change, so as to the mechanism can avoid the singular position, as shown in Figure 20b. Similarly, when such singularity occurs in limb 2 or limb 3, it can also be avoided by adjusting the height of the lifting platform.

Figure 20

Increasing the height of the lifting platform to avoid singularity at the singular position 1

In the first stage of the simulation, the mechanism moves from the initial position to the singular position 1, the simulation time is 0–5 s; the second stage is the process of avoiding the singular positions of the mechanism, and the simulation time is 5–8 s. In the simulation using Adams software, the rotation angle of the moving platform, revolving joint and spherical joint are measured to judge whether the mechanism completes the action of avoiding singular positions.

As shown in Figure 21, it is the simulation value of ∠A₁C₁B₁ of the mechanism. At the 5th second, indicate that link A₁C₁ and link C₁B₁ are approximately perpendicular at this time. At this time, the mechanism is close to the singular position, and link A₁C₁ and link C₁B₁ are close to vertical. If the position of the lifting platform does not change, the mechanism will be singular. By changing the height of the lifting platform, the mechanism can avoid the singular position. When the position of the lifting platform increases, the angle between the two links changes. ∠A₁C₁B₁ is greater than 90° and keeps increasing, which shows that the result of avoiding singular positions is obvious.

Figure 21

The simulation value of ∠A₁C₁ B₁ when the mechanism avoids singular position 1

The rotation angle of the spherical joint B₁ in the xoz plane is set as θ₁₄. Figures 22 and 23 show the variation of ∠A₁C₁B₁ and θ₁₄ respectively. At the 5th second, the variation of ∠A₁C₁B₁ is approximately 31.15°. After the 5th second, as the lifting platform's height increases, the variation of ∠A₁C₁B₁ gradually increases. At the 8th second, the amount is 61.26°. The variation of θ₁₄ gradually increases until the mechanism reaches the singular position. Then it decreases as the lifting platform's height increases, which indicates that the mechanism avoids singularity and returned to its normal position and posture.

Figure 22

The variation of ∠A₁C₁B₁ when the mechanism avoids singular position 1 (The symbol “$\Delta$” represents the variation of the angle)

Figure 23

The variation of θ₁₄ when the mechanism avoids singular position 1

When θ_i1 and θ_i2 of the mechanism are approximately equal to 90°, it is the singular position 2 of the mechanism. The actuating joint's positions are L₁ = L₂ = L₃ = 100 mm and L₄ = 100 mm, the link l_i1 and link l_i3 of the three limbs are approximately perpendicular to the base, as shown in Figure 24a. By decreasing the height of the lifting platform, the mechanism can avoid singularity. When L₄ = 60 mm, the posture of the limb changes and the height of the moving platform decreases simultaneously, and the mechanism returns to its normal posture, as shown in Figure 24b.

Figure 24

Decreasing the height of the lifting platform to avoid singularity at the singular position 2

As shown in Figure 25, it is the variation of θ₁₁ and ∠A₁C₁B₁ when the mechanism avoids singular position 2. At the 5th second, the variation of θ₁₁ and ∠A₁C₁B₁ reach their maximum values of 68.34° and 126.45°. If the three actuating joints of the mechanism continue to move forward, θ₁₁ will continue to increase, and ∠A₁C₁B₁ will continue to decrease, so the mechanism will reach the singular position. By decreasing the height of the lifting platform, link l_i2 can change the position of link l_i1 and link l_i3, which can change the overall posture of the mechanism. If θ_i1 and θ_i2 are less than 90° and keep decreasing, it is proved that the mechanism can avoid singular position 2.

Figure 25

The variation of θ₁₁ and ∠A₁C₁B₁ when the mechanism avoids singular position 2

After 5 s, the lifting platform’s height decreases, and the variation of θ₁₁ and ∠A₁C₁B₁ are reduced to 21.28° and 58.27° respectively, which indicates that the mechanism has avoided the singular positions.

When the angle θ_i1 is approximately equal to 0°, the third singular position of the mechanism occurs. The actuating joint’s positions are L₁ = L₂ = L₃ = 143 mm and L₄ = 60 mm, and the link l_i1 of the three limbs is approximately parallel to the base, as follows, as shown in Figure 26a. By increasing the height of the lifting platform, the mechanism can avoid singularity. When L₄ = 100 mm, the moving platform’s height is increased, and the overall posture of the mechanism changes.

Figure 26

Increasing the height of the lifting platform to avoid singularity at the singular position 3

Figures 27 and 28 show the simulation values of θ₁₁ and ∠C₁D₁E₁ when the mechanism avoids singular position 3. At the 5th second, the value of θ₁₁ is 1.66°, close to 0°, which indicates that link l_i1 is approximately parallel to the base at this position. If the actuating joint continues to move, the mechanism will be singular. At this time, by increasing the height of the lifting platform, the position of the link l_i2 will move and the position of the link l_i1 and the link l_i3 will be changed. If θ₁₁ increases, ∠C₁D₁E₁ will decrease, proving that the mechanism can avoid singular positions. After 5 s, increasing the height of the lifting platform, θ₁₁ significantly increases to 40.68°, and ∠C₁D₁E₁ also changes significantly, which can indicate that the height of the lifting platform makes the mechanism successfully avoid the singularity.

Figure 27

The simulation value of θ₁₁ when the mechanism avoids singular position 3

Figure 28

The simulation value of ∠C₁D₁E₁ when the mechanism avoids singular position 3

4.2 Discussion

It is proved that the singularities can be avoided by changing the lifting platform’s height when the mechanism is near or already in the singular positions. It also shows that the redundant kinematics part significantly improves the mechanism's kinematics performance and proves that the 3-DOF PM-KR has the advantages of structural design.

However, there are some problems in the simulation research. For example, the simulation of the mechanism is too ideal and doesn’t fully consider the influence of the errors of the mechanism’s materials and joints on the simulation results; when the mechanism reaches different singular positions, the force and deformation between joints and links will affect the overall mechanism. The parameters changing of the mechanism will also affect the effect of avoiding singular positions. In the follow-up research, we should solve these known problems as much as possible.

5 Conclusions

In this paper, the kinematics performance of a 3-DOF PM-KR and the influence of redundant parts on the mechanism are analyzed. The conclusions are drawn as follows:

• (1) Through the analysis of the inverse kinematics of the PM-KR, it can be concluded that different actuating parameters of kinematic redundancy lead to different solutions. Since there are infinite solutions for the inverse kinematics of PM-KR, the actuating parameter of kinematic redundancy should be set as a constant firstly.

• (2) When the parameter of the redundant part is increased, the singularity of the mechanism can be significantly reduced. This proves that the kinematic redundancy affects the workspace and singularity of the 3-DOF PM-KR.

• (3) Due to the existed kinematic redundancy, the PM-KR possesses the characteristic of fault tolerance. When the position of the failed actuating joint is far away from the center, the fault-tolerant performance of the PM-KR is better.