Drive Layout Configuration Strategy and Scale Optimization Design of Multi-Drive Mode Parallel Mechanism

Han, Ming; Xiong, Bin; Liu, Jinyue; Yang, Dong; Li, Tiejun

doi:10.1115/1.4064237

Abstract

Aiming at the inherent defects of single-drive mode parallel mechanisms, such as small good workspace and many singular configurations, a new multi-drive mode parallel mechanism is proposed from the perspective of driving innovation. Taking the planar 6R parallel mechanism as an example, the driving layout configuration strategy and scale optimization design are studied. First, the potential driving layout of the mechanism is analyzed, and the inverse kinematics model of the mechanism is established. Then, based on the motion/force transmission index, the singular space and good transmission workspace of the mechanism are identified, and the mechanism-driven layout configuration strategy is formulated to complete the performance comparison analysis. Finally, based on the performance map method, the scale optimization design of the mechanism is carried out, the prototype is developed and manufactured, and the experimental verification is carried out. The research results show that the multi-drive mode parallel mechanism can improve its kinematic performance without changing the topological structure and scale parameters of the mechanism.

1 Introduction

Compared with the series mechanism, the parallel mechanism has obvious advantages in terms of overall stiffness [1], dynamic performance [2], and end bearing capacity [3], so it is often used as a main functional component in high precision and high bearing capacity. However, due to its own structural limitations, the single-drive mode parallel mechanism has inherent defects, such as a small good transmission workspace and many singular configurations, which has become a bottleneck problem that hinders its application and development. Therefore, it is urgent to innovate and design new types of mechanisms to provide new ideas for improving the performance of mechanisms in essence.

Any parallel mechanism has several reasonable driving layouts, and different driving layouts will have a great influence on the performance of the parallel mechanism [4]. For example, Liu and Liu [5] synthesized a parallel mechanism with 2R1T and 2T1R motion modes by using displacement manifold theory and analyzed the degree-of-freedom characteristics of the mechanism during the transition of motion modes by using screw theory, which verified the rationality of the driving layout under different motion modes. Cao et al. [6] analyzed the rationality of the driving layout based on the screw theory for the 3-PPRU parallel mechanism and analyzed the performance of the mechanism, such as speed, bearing capacity, and stiffness. However, the above research does not make use of the mechanism-driven layout from the global perspective, so it fails to exert the maximum potential of the mechanism. In recent years, multi-drive model parallel mechanisms have been proposed and used to improve the performance of the mechanism. Based on parallel mechanisms, this kind of mechanism can switch the corresponding rods back and forth between the active rods and the passive rods by adding clutch driving units or clutch driving branches, so as to achieve the purpose of switching the driving layout, avoiding singular configurations and expanding the good transmission workspace [7]. This kind of robot has a variety of available driving modes. By changing the driving layout, the kinematics performance can be changed and improved without redundancy and structural changes, which brings important research and application value to this kind of robot. For example, Wang et al. [8] proposed a 3-RPaS parallel mechanism with two rotational degrees-of-freedom and one translational degree-of-freedom. By analyzing the kinematic performance of the mechanism under different driving layouts, it was proved that a reasonable selection of driving layouts can effectively improve the comprehensive performance and eliminate singularity. Zhang et al. [9] designed a planar 3-RRR parallel robot, which can realize 27 kinds of driving layouts, defined the evaluation index of the kinematic performance of the mechanism by using the concept of matrix orthogonal basis, and analyzed the kinematic characteristics of the mechanism under different driving layouts.

In addition, scale optimization is also an important method to improve the performance of the mechanism, that is, by optimizing the geometric parameters of the mechanism, the maximum performance of the mechanism can be exerted. Generally speaking, scale optimization mainly involves two aspects: performance evaluation and optimization design. In the aspect of performance evaluation, the following three indicators are mainly used. The first one is based on the Jacobian matrix, and its condition number [10–13] and eigenvalue [14–16] are used to evaluate the different performances of parallel mechanisms. However, the dimensions of the Jacobian matrix of translational-translational coupling parallel mechanisms are inconsistent, and the evaluation indicators based on its mathematical characteristics will lose their physical significance. The second is to establish the performance evaluation index of the mechanism based on a matrix orthogonal basis. For example, Wang et al. [17] proposed a new planar 5R parallel mechanism with variable driving mode and used the transmission index based on a matrix orthogonal basis to optimize the scale of the mechanism. The third is to evaluate the kinematic performance of the mechanism by using the motion/force transmission index [18]. The research on this index can be traced back to the concept of imaginary coefficient put forward by Ball [19]. Later, based on the research of Yuan et al. [20], Tsai and Lee [21] improved the transmission index and proposed a method to calculate the maximum transmission efficiency. Chen and Angeles [22] put forward the global transmission index (GTI) and revised the maximum value of the effective coefficient. Wang et al. [23] systematically discussed the motion/force transmission characteristics of the input and output of the parallel mechanism, defined the corresponding input transmission index (ITI) and output transmission index (OTI), and also proposed the minimum method to define the local transmission index (LTI). The establishment of this index has nothing to do with the selection of a coordinate system and is dimensionless. In the aspect of mechanism scale optimization design, the objective function method and performance map method are mainly used. The objective function method is to set up an objective function according to a certain performance index and then use the numerical optimization algorithm to optimize the design parameters of the mechanism. This method has obvious shortcomings, and the uniqueness of the solution leads to the inability to reflect the relationship between the design parameters and the performance of the mechanism. The performance map method is intuitive, and designers can adjust the range of high-quality scale domains according to actual needs and select the mechanism parameters that meet the requirements, which can better reflect the relationship between the design parameters and the performance of the mechanism, so it is often used as an important tool for scale optimization [24–27] For example, Zhu et al. [28] used the performance map method to optimize the scale parameters of a redundant driven parallel mechanism and conducted in-depth research on its bearing capacity and workspace. Zhang et al. [25] put forward a new type of far-center parallel mechanism for minimally invasive surgery and analyzed the performance of the mechanism by using the motion/force transmission index and then optimized the scale of the mechanism based on the performance map method. Therefore, in this paper, the performance evaluation and scale optimization design of the multi-drive model parallel mechanism are carried out by using the motion/force transmission index and performance map method.

The rest of this paper is organized as follows: In Sec. 2, the potential driving layout of the planar 6R parallel mechanism is analyzed, and the inverse kinematics model of the mechanism is established. In Sec. 3, the local motion/force transmission index of the mechanism is established, and the identification rules of mechanism singular space and good transmission workspace are introduced. In Sec. 4, the mechanism-driven layout configuration strategy is formulated, and the effectiveness of this method is proved by analyzing the optimization effect of this strategy on the mechanism performance. In Sec. 5, the mechanism is optimized, and the prototype is manufactured by using the performance map method. In Sec. 6, experiments are carried out on the developed prototype, which proves that the multi-drive model parallel mechanism can significantly improve the performance of the mechanism. Finally, the research conclusion of this paper is given.

2 Kinematics Analysis of 6R Planar Parallel Mechanism

There are numerous ways to dynamically select the driving layout of the mechanism [7]. In a real process, the driving layout of the robot can be switched in real time according to the working conditions on the spot. Therefore, the mechanism can work in a non-redundant state in real time, and the best performance of the mechanism can be achieved. For example, when the end effector of a parallel mechanism approaches a singular trajectory, its driving layout can be changed according to predefined rules to avoid the singular trajectory and improve the motion performance of the robot. In the present study, the planar 6R parallel mechanism is considered as the research object to carry out the follow-up research.

2.1 Analysis of the Potential Driving Layout of the 6R Planar Mechanism.

Figure 1 shows that the planar 6R parallel mechanism consists of five connecting rods, which are connected end to end by six rotating pairs. Based on the screw theory [29], the mechanism has three degrees-of-freedom, including the movement along the x- and y-axes and the rotation around the z-axis. Therefore, this mechanism requires three drives to perform a certain movement.

Fig. 1

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Structural diagram of the 6R planar parallel mechanism

According to the type of the branched-chain drive, the mechanism has six potential drives, as shown in Fig. 2. The black arrow in the figure represents the driving joint. When the mechanism contains RRR or RRR branched chain, the driving coupling occurs, which significantly affects the kinematics performance of the mechanism. If the mechanism contains RRR branches, it will increase the inertia of the moving platform, which is not conducive to active control. In order not to increase the dynamics and control the complexity of the mechanism and, at the same time, ensure its kinematic performance, a combination of RRR and RRR or RRR and RRR branches can be dynamically selected for the driving layout. In this regard, four potential non-redundant driving layouts of the mechanism are presented in Table 1.

Fig. 2

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The potential driving forms of the branched chain: (a) RRR, (b) RRR, (c) RRR, (d) RRR, (e) RRR, and (f) RRR

Table 1

Four potential non-redundant driving layouts of the mechanism

Branched-chain combination	Driving layout	Driving joint
RRR and RRR	I	A₁B₁A₂
RRR and RRR	II	A₁B₁B₂
RRR and RRR	III	A₁A₂B₂
RRR and RRR	IV	B₁A₂B₂

2.2 Inverse Kinematics Analysis.

Figure 3 shows driving layout I as an example where joints A₁, B₁, and A₂ are active pairs, while joints B₂, C₁, and C₂ are passive pairs, and the rod C₁C₂ is the moving platform of the mechanism. The center point P is the reference point of the moving platform, and rod A₁A₂ is the frame, so a fixed reference coordinate system o-xy is established. The origin of the coordinates is located at the center of A₁A₂, and the x-axis is along the A₁A₂ direction, while the y-axis is perpendicular to A₁A₂. It is assumed that A₁B₁ = A₂B₂ = r₁, B₁C₁ = B₂C₂ = r₂, C₁C₂ = 2r₃, and A₁A₂ = 2r₄.

Fig. 3

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Schematic diagram of the 6R planar parallel mechanism

Assuming that the coordinate of point P is P = (x, y, φ), where φ is the rotation angle of the moving platform, the coordinates of each position point can be expressed as follows:

{\begin{matrix} a_{1} = (- r_{4}, 0)^{T} \\ b_{1} = (r_{1} \cos θ_{1} - r_{4}, r_{1} \sin θ_{1})^{T} \\ c_{1} = (x - r_{3} \cos φ, y - r_{3} \sin φ)^{T} \\ a_{2} = (r_{4}, 0)^{T} \\ b_{2} = (r_{1} \cos θ_{2} + r_{4}, r_{1} \sin θ_{2})^{T} \\ c_{2} = (x + r_{3} \cos φ, y + r_{3} \sin φ)^{T} \end{matrix}

(1)

where θ₁ is the angle between rods A₁A₂ and A₁B₁, and θ₂ is the angle between rods A₁A₂ and A₂B₂. According to the geometrical correlation between the length of rods

| b_{1} c_{1} |^{2} = r_{2}^{2}

(2)

Let

{\begin{matrix} A_{1} = (x - r_{3} \cos φ + r_{4})^{2} + (y - r_{3} \sin φ)^{2} + r_{1}^{2} - r_{2}^{2} \\ B_{1} = - 2 r_{1} (x - r_{3} \cos φ + r_{4}) \\ C_{1} = - 2 r_{1} (y - r_{3} \sin φ) \end{matrix}

(3)

Accordingly, Eq. can be rewritten in the form below

C_{1} \sin θ_{1} + B_{1} \cos θ_{1} + A_{1} = 0

(4)

Combined with the double-angle formula, let

{\begin{matrix} \sin θ_{1} = \frac{2 \tan (θ_{1} / 2)}{1 + \tan^{2} (θ_{1} / 2)} \\ \cos θ_{1} = \frac{1 - \tan^{2} (θ_{1} / 2)}{1 + \tan^{2} (θ_{1} / 2)} \end{matrix}

(5)

Simultaneous solution of Eqs. and yields the following results:

{\begin{matrix} θ_{1} = 2 \arctan \frac{- C_{1} \pm \sqrt{C_{1}^{2} + B_{1}^{2} - A_{1}^{2}}}{A_{1} - B_{1}} & A_{1} \neq B_{1} \\ θ_{1} = 2 \arctan \frac{- (A_{1} + B_{1})}{2 C_{1}} & A_{1} = B_{1} \end{matrix}

(6)

Similarly, θ₂ can be obtained. Based on the geometry

θ_{3} = \arctan \frac{y - r_{3} \sin φ - r_{1} \sin θ_{1}}{x - r_{3} \cos φ + r_{4} - r_{1} \cos θ_{1}}

(7)

ρ₁ can be solved using the cosine theorem, where the obtained result is

\begin{aligned} ρ_{1} & = \arccos (\frac{r_{1}^{2} + r_{2}^{2} - {| a_{1} c_{1} |}^{2}}{2 r_{1} r_{2}}) \\ = \arccos (\frac{r_{1}^{2} + r_{2}^{2} - {(x - r_{3} \cos φ + r_{4})}^{2} - {(y - r_{3} \sin φ)}^{2}}{2 r_{1} r_{2}}) \end{aligned}

(8)

Similarly, the values of ρ₂, μ₁, and μ₂ can be obtained.

The performed analysis reveals that there are 2² sets of solutions to the inverse kinematics of this mechanism. When θ₁ is positive and θ₂ is negative, the mechanism is called the “+−” pattern. Figure 4 shows four inverse assembly models obtained by the same token. Because the multi-drive parallel robot developed in this paper is based on the application background of building construction, it is mainly used for the handling and assembly of building components such as glass curtain walls and gypsum boards, and the task occurs on one side of the robot, so the assembly configuration shown in Fig. 4(a) is excluded, and because the configuration shown in Fig. 4(d) has angle limit, it is impossible for them to intersect in the same plane, so the assembly configuration shown in Fig. 4(d) is also excluded. However, the two configurations shown in Figs. 4(b) and 4(c) both meet the requirements and are in a symmetrical relationship. Therefore, this paper only takes the assembly configuration shown in Fig. 4(b) as the research object and assumes that the motion range of the attitude angle φ (see Fig. 3) of the terminal moving platform is (− π, −π/2), so as to carry out subsequent innovative research in theory and technology.

Fig. 4

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Four sets of inverse assembly configurations: (a) “+ −,” (b) “+ +,” (c) “− −,” and (d) “− +”

3 Analysis of Motion/Force Transmission Index

3.1 Local Transmission Index.

In this paper, the LTI proposed by Liu Xinjun [23] is used to measure the motion performance of the mechanism. This index can be divided into two parts: ITI and OTI, which reflect the transmission efficiency of energy from the input end to the branch and from the branch to the output end, respectively. These indices can be mathematically expressed as follows:

γ_{I} = min_{i} {λ_{i}} = min_{i} {\frac{| $_{T i} \circ $_{I i} |}{{| $_{T i} \circ $_{I i} |}_{max}}}

(9)

and

γ_{O} = min_{i} {η_{i}} = min_{i} {\frac{| $_{T i} \circ $_{O i} |}{{| $_{T i} \circ $_{O i} |}_{max}}}

(10)

where γ_I and γ_O represent ITI and OTI, respectively. Moreover, $_Ii, $_Oi, and $_Ti are the input motion screw, output motion screw, and transmission force screw corresponding to the input joint i, respectively.

Equations and indicate that the values of γ_I and γ_O are in the range of [0,1]. In order for each torque to transmit the corresponding motion/force to the end effector well, the values of ITI and OTI should be as close to 1 as possible. In order for each of the transmission force screws to transfer the corresponding motion/force to the end effector, the values of ITI and OTI need to be as close to 1 as possible. Therefore, the LTI of the mechanism is defined as follows:

γ = min {γ_{I}, γ_{O}}

(11)

For the planar 6R parallel mechanism, the solution diagrams of the transmission force screw corresponding to each input joint are shown in Fig. 5.

Fig. 5

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Schematic diagram for solving the transmission force screw

The structure diagram of the RRR branched chain is shown in Fig. 5. There are three motion screws as follows:

{\begin{matrix} $_{1} = (\begin{matrix} 0, & 0, & 1; \begin{matrix} 0, & - r_{4}, & 0 \end{matrix} \end{matrix}) \\ $_{2} = (\begin{matrix} 0, & 0, & 1; \begin{matrix} r_{1} \sin θ_{2}, & - r_{4} - r_{1} \cos θ_{2}, & 0 \end{matrix} \end{matrix}) \\ $_{3} = (\begin{matrix} 0, & 0, & 1; \begin{matrix} r_{1} \sin θ_{2} + r_{2} \sin (ρ_{2} - θ_{2}), & r_{2} \cos (ρ_{2} - θ_{2}) - r_{4} - r_{1} \cos θ_{2}, & 0 \end{matrix} \end{matrix}) \end{matrix}

(12)

The constraint screws are

{\begin{matrix} $_{1}^{r} = $_{C 1} = (\begin{matrix} 0, & 0, & 1; & 0, & 0, & 0 \end{matrix}) \\ $_{2}^{r} = $_{C 2} = (\begin{matrix} 0, & 0, & 0; & 1, & 0, & 0 \end{matrix}) \\ $_{3}^{r} = $_{C 3} = (\begin{matrix} 0, & 0, & 0; & 0, & 1, & 0 \end{matrix}) \end{matrix}

(13)

Since the joint A₂ in the RRR branched chain is the driving pair, its corresponding motion screw $₁ is the input motion screw. When the input joint A₂ corresponding to the motion screw $₁ is locked, the screw $₁ will not belong to the motion screw system of the branched-chain RRR. In this case, the force screw, which is inversely related to other motion screws and linearly independent of the three constraint screws, is $_TA2. This can be expressed as follows:

$_{T A 2} = (- \cos (ρ_{2} - θ_{2}), \sin (ρ_{2} - θ_{2}), 0; 0, 0, r_{4} \sin (ρ_{2} - θ_{2}) + r_{1} \sin ρ_{2})

(14)

Similarly, the transmission force spinor $_TA1 corresponding to joint A₁ in the branched RRR and the transmission force spinor $_TB1 corresponding to joint B₁ in the branched RRR can be obtained.

For the planar 3-DOF 6R parallel mechanism, it is necessary to determine its input and output motion screws. Since the three input joints of the parallel mechanism are single-degree-of-freedom pairs, the input motion screw is their corresponding motion screw. Those can be expressed as follows:

{\begin{matrix} $_{IA 1} = (\begin{matrix} 0, & 0, & 1; \begin{matrix} 0, & r_{4}, & 0 \end{matrix} \end{matrix}) \\ $_{IB 1} = (\begin{matrix} 0, & 0, & 1; \begin{matrix} r_{1} \sin θ_{1}, & r_{4} - r_{1} \cos θ_{1}, & 0 \end{matrix} \end{matrix}) \\ $_{I A 2} = (\begin{matrix} 0, & 0, & 1; \begin{matrix} 0, & - r_{4}, & 0 \end{matrix} \end{matrix}) \end{matrix}

(15)

The performed analysis indicates that the input joints A₁, B₁, and A₂ of the mechanism have certain effects on the output motion/force of the end-moving platform. In the single-drive branched RRR, if input joints A₁ and B₁ are “locked while joint A₂ is driven, only the motion screw $_IA2 corresponding to joint A₂ is transmitted to the end-moving platform under the action of the transmission force screw $_TA2. At this time, the original planar 3DOF 6R mechanism has become a single DOF mechanism. In order to express the unit instantaneous motion of the moving platform at the end of the mechanism, the unit output motion screw $_OA2 is used in this study. Accordingly, it can be expressed as follows:

\begin{matrix} $_{T A 1} \circ $_{OA 1} = 0 \\ $_{TB 1} \circ $_{OB 1} = 0 \end{matrix}

(16)

and

$_{O A 2} \circ $_{C i} = 0, (i = 1, 2, 3)

(17)

Applying the augmented matrix method to solve one-dimensional null space yields the following results:

$_{O A 2} = (0, 0, 1; P_{OA 2}, Q_{OA 2}, 0)

(18)

where

P_{OA 2} = r_{1} \sin θ_{2} + r_{2} \sin (ρ_{2} - θ_{2}) + 2 r_{3} \sin (θ_{2} - ρ_{2} - μ_{2})

and

Q_{OA 2} = r_{2} \cos (ρ_{2} - θ_{2}) - r_{4} - r_{1} \cos θ_{2} - 2 r_{3} \cos (θ_{2} - ρ_{2} - μ_{2})

⁠.

Similarly, the output motion screw $_OA2 corresponding to the input joint A₂ in the RRR branch chain and the output motion screw $_OB1 corresponding to the joint B₁ in the RRR branch chain can be obtained.

Combining Eqs. –, the LTI of the planar 3-DOF 6R mechanism can be obtained in the form below:

\begin{aligned} γ & = min {γ_{I}, γ_{O}} = min {λ_{1}, λ_{2}, λ_{3}, η_{1}, η_{2}, η_{3}} \\ = min {| \sin ρ_{2} |, | \frac{r_{1} \sin ρ_{1}}{\sqrt{r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos ρ_{1}}} |, | \sin μ_{2} |} \end{aligned}

(19)

Similarly, the solution diagrams and final results of the motion/force transmission indices of driving layouts II–IV can be calculated, and the obtained results are presented in Table 2.

Table 2

LTI of driving layouts II–IV

Driving layout	Solution diagram	LTI
II		$LT I_{A_{1} B_{1} B_{2}} = min {1, 1, \| \sin ρ_{2} \|, \| \frac{r_{1}}{A_{1} C_{1}} \sin ρ_{1} \|, \| \frac{r_{1}}{A_{1} C_{1}} \sin ρ_{1} \|, \| \sin α_{1} \|}$
III		$LT I_{A_{1} A_{2} B_{2}} = min {\| \sin ρ_{1} \|, 1, 1, \| \sin μ_{1} \|, \| \frac{r_{1}}{A_{2} C_{2}} \sin ρ_{2} \|, \| \frac{r_{1}}{A_{2} C_{2}} \sin ρ_{2} \|}$
IV		$LT I_{B_{1} A_{2} B_{2}} = min {\| \sin ρ_{1} \|, 1, 1, \| \sin α_{1} \|, \| \frac{r_{1}}{A_{2} C_{2}} \sin ρ_{2} \|, \| \frac{r_{1}}{A_{2} C_{2}} \sin ρ_{2} \|}$

The posture of the moving platform of a planar 3-DOF 6R parallel mechanism is defined as the angle between the rod C₁C₂ and the x-axis. For a constant posture at the end of the robot, the motion range of the reference point is called the fixed-pose workspace of the robot. Assuming that the dimensionless parameters of the parallel mechanism are r₁ = r₂ = 1.5, r₃ = 0.4, and r₄ = 0.6, the distribution diagram of LTI in the fixed-pose workspace and a moving platform of −π/2 is shown in Fig. 6.

Fig. 6

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The distribution diagram of LTI in the fixed-pose workspace of the mechanism: (a) driving layout I, (b) driving layout II, (c) driving layout III, and (d) driving layout IV

When the posture angle of the platform changes continuously in the range of (−π, −π/2), the movement range that the reference point of the end-moving platform can reach an arbitrary posture angle is called the dexterous workspace of the mechanism. The LTI distribution diagram of the mechanism in the dexterous workspace is shown in Fig. 7.

Fig. 7

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Distribution diagram of LTI in the dexterous workspace of the mechanism: (a) driving layout I, (b) driving layout II, (c) driving layout III, and (d) driving layout IV

The results can be used to develop a theoretical foundation for solving the performance indices of the parallel mechanism, such as transmission singularity workspace and good transmission workspace.

3.2 Singularity Analysis.

The 6R planar parallel mechanism transmits the motion/force of the input joint to the output end by transmission force screw to realize a motion with three degrees-of-freedom. When the robot cannot support a 3-DOF movement of the end effector or cannot offset the external forces, transmission singularity occurs. It is worth noting that there is no constraint singularity in the planar 3-DOF 6R parallel mechanism, and only the transmission singularity should be considered in calculations.

Based on the foregoing analysis, when ITI is equal to 0, the motion/force of the input joint cannot be transmitted. In this case, it is called input transmission singularity, which is defined as follows:

ITS = min_{i} {| $_{T i} \circ $_{I i} |} \to 0, (i = 1, 2, 3)

(20)

Similarly, when OTI is equal to 0, the parallel mechanism has an output transmission singularity (OTS), which is defined as follows:

OTS = min_{i} {| $_{T i} \circ $_{O i} |} \to 0, (i = 1, 2, 3)

(21)

Because the high-performance parallel robot developed in this paper is mainly used in the handling and loading and unloading of large-mass components, the task occurs on one side of the robot. Therefore, as shown in Fig. 8, we have studied the influence of different driving layouts on the mechanism performance when the end moving platform rotates at point (0.5,2).

Fig. 8

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Variations of ITI and OTI of the mechanism with the end posture angle under different driving layouts

Figure 8 reveals that when the driving layouts III and IV are adopted, ITI is equal to 0, and the mechanism is in a configuration of input transmission singularity. When the driving layouts I, III, and IV are adopted, and the corresponding end posture angle φ is set to $- 132 \deg$ ⁠, $- 153 \deg$ ⁠, and $- 160 \deg$ ⁠, respectively, OTI is equal to 0, and the mechanism is in a configuration of output transmission singularity. The farther the mechanism is from the singular configuration, the better its motion/force transmission indices.

When the posture angle of the moving platform of the mechanism changes continuously in the range of (−π, −π/2), the set of all fixed-pose workspaces is called the reachable workspace of the mechanism. The transmission singular workspace of the mechanism in the reachable workspace is shown in Fig. 9.

Fig. 9

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Transmission singular workspace of the mechanism in the reachable workspace: (a) driving layout I, (b) driving layout II, (c) driving layout III, and (d) driving layout IV

The dotted area in Fig. 9 represents the transmission singular workspace of the mechanism. In other words, when the moving platform operates at the end of the mechanism, a singular configuration will appear at a certain posture angle. Therefore, it is necessary to avoid working in the area near the transmission singular workspace.

3.3 Identification of Good Transmission Workspace.

When the posture of the robot end moving platform is given, the set of all position points with LTI ≥ 0.7 is defined as the good transmission position workspace (GTPW) of the mechanism.

For the same dimensionless parameters and assuming that the attitude angle of the moving platform is −π/2, the distributions of GTPW for different driving layouts are shown in Fig. 10. The region with LTI ≥ 0.7 represents the GTPW of the mechanism, and S represents its area size. The discussion of the mechanism here is conducted under dimensionless dimensions, so the area of high-quality workspace is dimensionless. It is observed that the GTPW area of the planar 6R parallel mechanism for different driving layouts is 0.70, 0.68, 4.41, and 0.53, respectively.

Fig. 10

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Distribution of GTPW: (a) driving layout I, (b) driving layout II, (c) driving layout III, and (d) driving layout IV

In order to further explore the rotational ability of the mechanism at a given position (x,y), the set of all φ’s satisfying LTI ≥ 0.7 at a given position is defined as good transmission position space (GTPS). For the same dimensionless parameters, when the reference point of the moving platform at the end of the mechanism is (0,1.5), variations of the GTPS of the mechanism for different driving layouts are shown in Fig. 11.

Fig. 11

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Variations of the GTPS mechanism with different driving layouts

It is observed that when the mechanism is driven by the driving layouts I and III, the angle range of GTPS is $(- 180 \deg, - 155 \deg)$ and $(- 116 \deg, - 90 \deg)$ ⁠, respectively. When the mechanism is driven by the driving layouts II and IV, there is no GTPS.

In order to ensure the overall performance of the mechanism, the area with LTI ≥ 0.7 is defined as the good transmission workspace (GTW), and the GTW distribution in the flexible workspace is shown in Fig. 12. It is observed that the GTW areas of the planar 6R parallel mechanism in the driving layouts I to IV are 0.04, 0.01, 0.21, and 0.49, respectively.

Fig. 12

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GTW distribution diagrams of mechanisms: (a) driving layout I, (b) driving layout II, (c) driving layout III, and (d) driving layout IV

4 Drive Layout Configuration

The performed analyses in the proceeding sections demonstrate that the planar 3-DOF 6R parallel mechanism under four non-redundant driving layouts has a large singular space, a small good workspace, and poor motion/force transmission performance. A common solution to resolve this problem is to introduce redundant drives into the parallel mechanism. However, the introduction of a redundant drive unavoidably increases the complexity of the control system and results in serious challenges in internal force coupling, thereby affecting investigations to improve dynamic performance. Drive layout configuration is a potential way to improve the kinematic performance of parallel mechanisms without changing the topological structure and parameters. It should be indicated that the difference and complementarity of motion/force transmission performance under different driving layouts provide new ideas to improve the kinematic performance of parallel mechanisms.

4.1 Drive Layout Configuration Strategy.

According to the optimal principle, the drive layout configuration strategy can be formulated as follows:

{\begin{matrix} Driving layout I : LT I_{A_{1} B_{1} A_{2}} \geq max {LT I_{A_{1} B_{1} B_{2}}, LT I_{A_{1} A_{2} B_{2}}, LT I_{B_{1} A_{2} B_{2}}} \\ Driving layout II : LT I_{A_{1} B_{1} B_{2}} \geq max {LT I_{A_{1} B_{1} A_{2}}, LT I_{A_{1} A_{2} B_{2}}, LT I_{B_{1} A_{2} B_{2}}} \\ Driving layout III : LT I_{A_{1} A_{2} B_{2}} \geq max {LT I_{A_{1} B_{1} A_{2}}, LT I_{A_{1} B_{1} B_{2}}, LT I_{B_{1} A_{2} B_{2}}} \\ Driving layout IV : LT I_{B_{1} A_{2} B_{2}} \geq max {LT I_{A_{1} B_{1} A_{2}}, LT I_{A_{1} B_{1} B_{2}}, LT I_{A_{1} A_{2} B_{2}}} \end{matrix}

(22)

The subscripts of LTI in Eq. (22) represent different driving layouts; for example, $LT I_{A_{1} B_{1} A_{2}}$ represents the LTI of the mechanism when the mechanism is driven by joint A₁B₁A₂. For the same scale parameters, the drive layout configuration strategy in a fixed attitude workspace is shown in Fig. 13.

Fig. 13

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Drive layout configuration strategy for different attitude angles: (a) $φ = - π / 2$ ⁠, (b) $φ = - 2 π / 3$ ⁠, (c) $φ = - 5 π / 6$ ⁠, and (d) $φ = - π$

The drive layout configuration strategy in a flexible workspace is shown in Fig. 14.

Fig. 14

View large Download slide

Drive layout configuration strategy in a flexible workspace

4.2 Performance Analysis Before and After Drive Layout Configuration.

To evaluate the comprehensive local transmission index (CLTI) of the mechanism after the drive layout configuration, the distribution of the motion/force transmission index in the workspace is defined as the CLTI of the mechanism after adopting the drive layout configuration strategy. Figure 15 shows GTPW in the fixed posture after adopting the drive layout configuration strategy.

Fig. 15

View large Download slide

Distribution diagram of GTPW at different posture angles: (a) $φ = - π / 2$ ⁠, (b) $φ = - 2 π / 3$ ⁠, (c) $φ = - 5 π / 6$ ⁠, and (d) $φ = - π$

As can be seen from Fig. 15, compared with the mechanism driven by joint A₁B₂A₁ in Fig. 10, the GTPW of the mechanism in the fixed posture workspace is greatly increased by the drive layout configuration, indicating that the motion/force transmission performance can be improved after adopting drive layout configuration strategy in this paper. It is observed that compared with the GTPW area of the driving layout I, the area for posture angles −π/2, −2π/3, −5π/6, and −π increased by 4.66, 2.83, 1.41, and 0.71, and the increased amplitude was 6.7 times, 1.4 times, 0.5 times, and 0.2 times, respectively. It is concluded that the improvement effect was significant.

Figure 16 shows the distribution diagram of GTPS after adopting the drive layout configuration strategy when the reference point of the moving platform at the end of the mechanism rotates at a fixed point (01.5).

Fig. 16

View large Download slide

Distribution diagram of GTPS of end-moving platform at a specific position

Figure 16 indicates that when the moving platform at the end of the mechanism rotates from $- 180 \deg$ to $- 139 \deg$ ⁠, the mechanism is driven by the driving layout I, and when the rotation angle varies from $- 139 \deg$ to $- 90 \deg$ ⁠, the mechanism is driven by the driving layout III. It should be indicated that the angle range of the GTPS after adopting the drive layout configuration strategy is $(- 180 \deg, - 155 \deg)$ and $(- 116 \deg, - 90 \deg)$ ⁠, which is 1.04 times larger than that of the driving layout I alone. Accordingly, it is concluded that this strategy greatly improves the performance of the mechanism and effectively eliminates singularity.

Figure 17 illustrates the GTW distribution diagram in the flexible workspace after adopting the mechanism-driven layout configuration strategy.

Fig. 17

View large Download slide

GTW distribution diagram of mechanism in flexible workspace

Compared with Fig. 12 with a single driving layout, the GTW of the mechanism in the flexible workspace with a drive layout configuration strategy is greatly increased in a specific corner. Accordingly, it is inferred that the drive layout configuration strategy can improve the motion/force transmission performance of the mechanism. Compared with the case with the driving layout I, the area of GTW increases by 0.58, the amplitude increases by 14.5 times, and the improvement is significant.

4.3 Scale Optimization of the Parallel Mechanism.

Determining the dimensions of the mechanism is one of the main challenges in optimal design. It is worth noting that the appropriate selection of parameters will directly affect the performance of the mechanism. In this regard, it is intended to use the space model theory to establish the dimensionless processing method and determine the design parameters of the planar 6R mechanism. Then, the mapping relationship between the good workspace area of the mechanism and the motion/force transmission performance index is established, and the performance atlas is drawn. Finally, the design of the mechanism is optimized from global aspects.

4.4 Dimensionless Method and Parameter Design Space.

There are four dimensionless scale variables in the planar 6R parallel mechanism. Suppose that

{\begin{matrix} r_{i} = R_{i} / L \\ L = \frac{1}{4} \sum_{i = 1}^{4} R_{i} \end{matrix}, (i = 1, 2, 3, 4)

(23)

where R_i and r_i represent the actual length and dimensionless parameters of the bar, respectively. Let

r_{1} + r_{2} + r_{3} + r_{4} = 4

(24)

Due to the structural characteristics and assembly constraints of the robot, the lengths should meet the following constraints

{\begin{matrix} 0 < r_{1}, r_{2} < 4 \\ 0 < r_{3}, r_{4} < 2 \end{matrix}

(25)

The parameter design space of the mechanism can be obtained as shown in Fig. 18.

Fig. 18

View large Download slide

Parameter design space of the parallel mechanism: (a) space model and (b) plan of the space model

From the above analysis, it can be seen that the dimensionless dimensions of the planar 3-DOF 6R parallel mechanism are within the space model ABCDEFG, shown in Fig. 18(a) (excluding the boundary).

In the spatial coordinate system o − r₁r₂r₃, the coordinates of each point of the polyhedron ABCDEFG are

{\begin{matrix} A = (2, 0, 2) \\ B = (4, 0, 0) \\ C = (0, 4, 0) \\ D = (0, 2, 2) \\ E = (0, 0, 2) \\ F = (2, 0, 0) \\ G = (0, 2, 0) \end{matrix}

(26)

In the spatial coordinate system, when r₄ changes between (0,2), the coordinates of each point of trapezoidal A′B′C′D′ are

{\begin{matrix} A^{'} = (\frac{4 - r_{4}}{2}, 0, 2) \\ B^{'} = (4 - r_{4}, 0, 0) \\ C^{'} = (0, 4 - r_{4}, 0) \\ D^{'} = (0, \frac{4 - r_{4}}{2}, 2) \end{matrix}

(27)

For convenience, we are used to using a plane coordinate system instead of the dimensionless coordinate system, so we can get a plan of the space model, as shown in Fig. 18(b).

From the above analysis, it is easy to know that the transformation relationship of the coordinate system in Figs. 18(a) and 18(b) is as follows:

{\begin{matrix} x = \frac{2 \sqrt{3}}{3} r_{1} + \frac{\sqrt{3}}{3} r_{3} \\ y = r_{3} \end{matrix}

(28)

4.5 Scale Optimization of the Parallel Mechanism Based on Performance Atlas.

The scale of the parallel mechanism should be optimized based on reasonable and effective design indices. Since the performance of the mechanism is different in different poses of the same scale, and there is no constraint performance evaluation problem in the planar 3-DOF 6R parallel mechanism, let

Λ = min {γ} = min {γ_{I}, γ_{O}}

(29)

where γ_I, γ_O, and γ represent ITI, OTI, and CLTI, respectively. Λ represents the local design index (LDI), which is actually equivalent to CLTI. The set of all pose points with LDI ≥ 0.7 is defined as the good transmission workspace (GTW) of the parallel mechanism, and the obtained area is called a global index of the scale optimization design of the parallel mechanism. Then, when r₄ = 0.3, the mapping relationship between the workspace area and the scale of the planar 6R parallel mechanism is shown in Fig. 19(a).

Fig. 19

View large Download slide

Mechanism performance map: (a) good transmission workspace area, (b) GTI, and (c) high-quality scale domain

One feasible way to make the global transmission index closer to the real value is to increase the density of discrete points. In this regard, an index is defined as follows:

Γ = \frac{\int_{W} Λ d w}{\int_{W} d w}

(30)

where Γ represents the global transmission index (GTI), and w represents the good transmission workspace (GTW). This expression reflects the average value of the LDI in the good transmission workspace. Applying this method to solve the global transmission index, the mapping relationship between GTI and dimensions of the parallel mechanism can be obtained, as shown in Fig. 19(b). Combined with Fig. 19(a), the high-quality scale domain of the parallel mechanism in the parameter design space is shown in Fig. 19(c). In the same way, the performance map of the mechanism, when r₄ takes other values, can be obtained, which can guide the mechanism to carry out dimensional optimization design.

Considering the interference and assembly constraints of components and the results of scale optimization, the scale parameters selected in the optimal region are shown in Table 3.

Table 3

Scale parameters of the 6R mechanism

R₁/m	R₂/m	R₃/m	R₄/m
1.2	1.2	0.15	0.33

Then, the design processing assembly and debugging of the prototype parallel robot with a multi-drive layout are completed. Figure 20 illustrates the configuration of the assembled robot.

Fig. 20

View large Download slide

Prototype of the parallel robot with multiple driving modes

Further experimental investigations are carried out to verify the feasibility of the drive layout and analyze its influence on the kinematic performance of the mechanism.

5 Experiment

Professor Liu Xinjun of Tsinghua University revealed the influence mechanism of mechanism motion/force transmission index on accuracy and proved that the higher the mechanism output transmission index, the better the accuracy performance [30,31]. Therefore, this section verifies the effectiveness of the driving layout configuration strategy through the precision performance experiments of the parallel robot before and after the driving layout configuration. During the experiment, the self-developed multi-drive robot prototype is taken as the experimental object, and the position of the robot end is measured by API-Radian laser tracker produced by American Automated Precision Incorporated; the relevant data are collected and processed by a PC. Select three measuring points in the flexible workspace of the robot, which are marked as P₁, P₂, and P₃, respectively, and the coordinates in the space are shown in Table 4. By editing the end trajectory in the software of the upper computer, the robot passes through these three points in turn by adopting the drive mode I in the course of the movement, repeats the process 30 times, calculates the average measurement value. Next, the layout is transformed according to the driving layout configuration strategy shown in Fig. 15. That is, when the robot end passes through P₁, P₂, and P₃ in turn, the driving modes II, III, and IV are adopted, respectively, and then repeated for 30 times, and data are recorded. Figure 21 shows the inspection site photos of the parallel robot at point P₁ under different driving layouts, where the black arrow represents the position of the driving joint.

Fig. 21

View large Download slide

Precision performance experiment of driving layout before and after configuration: (a) driving layout I and (b) driving layout II

Table 4

Coordinate of measuring points before and after driving layout configuration

P₁/mm	P₂/mm	P₃/mm
(1005,1311)	(505,2200)	(1005,1105)

For the processing of measurement data, because the laser tracker measures the same measurement point many times, the measured coordinate points form a set of measured position points, and a circumscribed ball can be constructed that envelopes the three-dimensional coordinates of all measured points. The position accuracy of the robot refers to the distance between the center of the circumscribed ball formed by the set of commanded position points and measured position points, and the radius RP_l of the circumscribed ball is the position repeatability of the robot end effector, which is also called the position repeatability of the robot. The calculation method is as follows:

R P_{l} = \bar{l} + 3 S_{t}

(31)

where

{\begin{matrix} \bar{l} = \frac{1}{n} \sum_{j = 1}^{n} l_{j} \\ l_{j} = \sqrt{{(x_{j} - \bar{x})}^{2} + {(y_{j} - \bar{y})}^{2}} \\ S_{t} = \sqrt{\frac{\sum_{j = 1}^{n} (l_{j} - \bar{l})}{n - 1}} \\ \bar{x} = \frac{1}{n} \sum_{j = 1}^{n} x_{j} \\ \bar{y} = \frac{1}{n} \sum_{j = 1}^{n} y_{j} \end{matrix}

(32)

The position accuracy is recorded as AP_P, and the calculation process is as follows:

A P_{P} = \sqrt{{(x - \bar{x})}^{2} + {(y - \bar{y})}^{2}}

(33)

where

{\begin{matrix} \bar{x} = \frac{1}{n} \sum_{j = 1}^{n} x_{j} \\ \bar{y} = \frac{1}{n} \sum_{j = 1}^{n} y_{j} \end{matrix}

(34)

In Eqs. (32) and (34), n is the number of times the same measuring point is executed, generally taken as 30, which is the actual measured value obtained by executing the same measuring point for the jth time and is the average value of the actual measured value obtained after repeatedly executing the same measuring point for n times. The position accuracy parameters shown in Table 5 can be obtained by data processing the measurement results of the redundant driven parallel robot according to Eq. (31).

Table 5

Accuracy performance before and after driving layout configuration

Measuring point/mm	Actuated layout	Repeatability of position/mm	Position accuracy/mm
P₁	I	0.052	0.512
P₁	II	0.045	0.465
P₂	I	0.042	0.525
P₂	III	0.035	0.453
P₃	I	0.045	0.563
P₃	IV	0.039	0.496

By analyzing the maximum position repeatability and position accuracy of each point, it is known that the accuracy of the mechanism has been greatly improved after adopting the driving layout configuration strategy, which shows that the precision performance of the mechanism can be fully improved by adopting the driving layout configuration strategy. Moreover, by comparison, it can be seen that when the rotation angle range of the moving platform at the end of the mechanism is (−π, −π/2), its position repetition accuracy and position accuracy are 0.045 and 0.496, respectively, which are 0.007 and 0.067 higher than those driven by A₁B₂A₁ joint, and the increased amplitude is 13% and 12%, respectively, and the improvement effect is obvious.

6 Conclusion

Aiming at optimizing the kinematics performance of parallel mechanisms, the drive layout configuration for a planar 6R parallel mechanism is studied, and the dimensions of the mechanism are optimized based on the performance atlas method. Finally, a prototype is designed and fabricated. The main achievements of this article can be summarized as follows:

The potential driving forms of the branches in the planar 6R parallel mechanism are analyzed, four potential non-redundant driving layouts are screened out, the inverse kinematics model of the mechanism is developed, the performance index of the mechanism’s motion/force transmission is established, and the singularity of the mechanism and the good workspace are identified. The obtained results show that the GTPW area for four driving layouts is 0.70, 0.68, 4.41, and 0.53, and the corresponding area of GTW is 0.04, 0.01, 0.21, and 0.49, respectively. When the mechanism is driven by the driving layouts I and III, the GTPS angle varies in the range of $(- 180 \deg, - 155 \deg)$ and $(- 116 \deg, - 90 \deg)$ ⁠, respectively. Moreover, there is no GTPS when the mechanism is driven by driving layouts II and IV.
The simulation results show that compared with the driving layout I, the GTPW area for drive layout configuration increased by 4.66, 2.83, 1.41, and 0.71, respectively, and the increased amplitude was 6.7 times, 1.4 times, 0.5 times, and 0.2 times, respectively. The GTPS angle varies in the range of $(- 180 \deg, - 155 \deg)$ and $(- 116 \deg, - 90 \deg)$ ⁠, with an increase of 1.04 times. Furthermore, the GTW area increases by 0.58, the amplitude increases by 14.5 times, and the improvement is significant.
The experimental results show that compared with the A₁B₂A₁ joint drive, the position repetition accuracy and position accuracy are 0.045 and 0.496, respectively, and the increased amplitude is 13% and 12%, respectively, which fully proves that the multi-drive mode parallel mechanism can significantly improve the performance of the mechanism.

The main contribution of this paper is to propose a new type of multi-drive mode parallel mechanism from the perspective of drive innovation and to carry out research on drive layout configuration strategy and scale optimization. The research results show that the multi-drive mode parallel mechanism can improve its kinematics performance without changing the topological structure and scale parameters of the mechanism and avoid or even eliminate the inherent defects of the single-drive model parallel mechanism, such as small good workspace and many singular configurations. In addition, although there is a lot of research on the motion/force transmission index in previous papers, this paper applies this index to the performance evaluation of multi-drive mode parallel mechanisms for the first time and expands the definition of comprehensive local transmission index (CLTI) suitable for this kind of mechanisms, enriching and expanding the evaluation system of motion/force transmission index.

Funding Data

This work was supported by the National Natural Science Foundation of China (Grant No. U20A20283) and the Project supported by Hebei postdoctoral program (Grant No. B2023005005).

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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Drive Layout Configuration Strategy and Scale Optimization Design of Multi-Drive Mode Parallel Mechanism

Abstract

1 Introduction

2 Kinematics Analysis of 6R Planar Parallel Mechanism

2.1 Analysis of the Potential Driving Layout of the 6R Planar Mechanism.

2.2 Inverse Kinematics Analysis.

3 Analysis of Motion/Force Transmission Index

3.1 Local Transmission Index.

3.2 Singularity Analysis.

3.3 Identification of Good Transmission Workspace.

4 Drive Layout Configuration

4.1 Drive Layout Configuration Strategy.

4.2 Performance Analysis Before and After Drive Layout Configuration.

4.3 Scale Optimization of the Parallel Mechanism.

4.4 Dimensionless Method and Parameter Design Space.

4.5 Scale Optimization of the Parallel Mechanism Based on Performance Atlas.

5 Experiment

6 Conclusion

Funding Data

Conflict of Interest

Data Availability Statement

References

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Drive Layout Configuration Strategy and Scale Optimization Design of Multi-Drive Mode Parallel Mechanism

Abstract

1 Introduction

2 Kinematics Analysis of 6R Planar Parallel Mechanism

2.1 Analysis of the Potential Driving Layout of the 6R Planar Mechanism.

2.2 Inverse Kinematics Analysis.

3 Analysis of Motion/Force Transmission Index

3.1 Local Transmission Index.

3.2 Singularity Analysis.

3.3 Identification of Good Transmission Workspace.

4 Drive Layout Configuration

4.1 Drive Layout Configuration Strategy.

4.2 Performance Analysis Before and After Drive Layout Configuration.

4.3 Scale Optimization of the Parallel Mechanism.

4.4 Dimensionless Method and Parameter Design Space.

4.5 Scale Optimization of the Parallel Mechanism Based on Performance Atlas.

5 Experiment

6 Conclusion

Funding Data

Conflict of Interest

Data Availability Statement

References

Get Email Alerts

Cited By

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