Kinematic performance evaluation of 2-degree-of-freedom parallel mechanism applied in multilayer garage

Abstract

In this article, our recent work on a kind of 2-degree-of-freedom lower-mobility parallel mechanism, which has one rotation degree of freedom and one translational degree of freedom, used in multilayer garage is presented. It has the following characteristics: lower-mobility, non-symmetric structure but can realize symmetric movement and a good compatibility for different kinds of lifting work. Kinematic performance should be considered in the first of designing a new kind of mechanism, the optimal kinematic design and analysis of this lower-mobility parallel mechanism are primarily investigated. In process of study, the global conditioning index over workspace is adopted, we establish a new evaluation method for the lower-mobility parallel mechanism, called global symmetry index and simulation results are shown. In addition, the flexible workspace of this lower-mobility parallel mechanism is also proposed. The evaluation index can be also applied on other lower-mobility parallel mechanism, which needs steady and symmetric movement.

Keywords

2 degree-of-freedom lower-mobility parallel mechanism kinematic performance evaluation global symmetry index flexible workspace multilayer garage

Introduction

The wide usage of parallel mechanism has proven to be a very helpful way to realize steady heavy load lifting, precise manipulation, good isotropy, quick response and obtaining high stiffness. Meanwhile, compared with 6-degree-of-freedom (DOF) parallel mechanism, lower-mobility parallel mechanisms (LMPMs) which have 2–5 DOF have their advantages, such as simpler structure, easy to be controlled, and in specific situation, it does not need that many DOFs. For such reasons, more and more types of 6-DOF parallel mechanism and LMPMs are proposed for scientific research and production instrument. 6-DOF Stewart platform, DELTA mechanism, 3-UPU (U-universal joint, P-prismatic joint) mechanism and many other LMPMs are very successful examples applied in practical production.^1–5 However, optimal kinematic design for a good kinematic performance always comes first in mechanism design, and it is also one of the most challenging issues in the field.⁶ The kinematic performance maps the transmission property between kinematics and dynamics from joints to workspace. Based on isotropy theory, Gosselin established a dimension parameter relationship for a planar 3-DOF parallel mechanism and a spherical 3-DOF parallel mechanism, respectively.^7,8 Yoshikawa⁹ used determinant of Jacobin matrix as the index and, thereby, obtained the best posture of a 4-DOF finger. Gosselin and Angeles¹⁰ first proposed global conditioning index (GCI) to evaluate the kinematic performance of parallel mechanism. T Sun et al.^11–15 utilized multi-objective optimization method and artificial intelligence approach obtaining the dimensional synthesis results which bring best evaluation index results to different kinds of parallel mechanism, virtual power transmissibility index, kinetostatic performance index, position-orientation coupling (POC) subsystem kinematic performance index and GCI are included. Velocity amplification factor (VAF) was studied by F Majou et al.,¹⁶ and got a better useful workspace and Cartesian workspace of 2-DOF parallel mechanism. Meanwhile, Cinquemani et al.¹⁷ and G Legnani and collegues^18,19 studied the isotropy of different parallel mechanisms and proposed concept of ‘point of isotropy’ both for serial and parallel mechanism. Many indexes are proposed, such as transmission factor, generalized transmission index that can evaluate the motion/force transmissibility, local transmission index and homogeneously dimensionless local and global transmission indices.^20–24 Although different kinds of indexes are presented for parallel mechanism, an index which can reflect the degree of symmetry of kinematic performance of parallel mechanism is not presented. In this article, we propose an index called global symmetry index (GSI) to reflect the degree of symmetry of kinematic performance.

In this article, a kind of 2-DOF LMPM designed for multilayer garage is studied. This mechanism has two DOFs, one rotation DOF about the X axis of the global coordinate system and one translational DOF along the Z axis of the global coordinate system. LMPM is a very useful alternative to the multilayer garage. The main reason is that it can provide a reasonable rising movement range by the translational DOF and an initiative posture control by the rotation DOF to guarantee smooth and steady lifting work in multilayer garage. Multilayer garage is a helpful equipment to increase the amount of parking lot and improve people’s daily life. Take China as example, large population results in a large car ownership and the parking problem keeps growing every day. Different kinds of multilayer garage and mechanisms are proposed, such as up-down and translation equipment, vertical circulation equipment and laneway stacker.^25,26 These equipment are applied in many places, but there exists a big obstacle for further development, the safety problem. Domestic vehicles are expensive goods for people; however, many news about multilayer garage accidents are reported.^27,28 We apply parallel mechanism in multilayer garage, which can exert the advantages of parallel mechanism to improve the safety of multilayer garage.

In the following sections, kinematic analysis of the LMPM is presented in section ‘Kinematic analysis of 2-DOF LMPM’, where kinematic design, mobility analysis, forward kinematics analysis and inverse kinematics analysis are included. In section ‘Singularity and workspace analysis’, the singularity and workspace are studied, including the definition of flexible workspace of LMPM. In section ‘Kinematic performance evaluation using GSI’, GSI is proposed, which can reflect the symmetry of kinematic performance of LMPM and also can be used on other LMPMs that need steady and symmetric movement. In section ‘Prototype experiment’, we make a prototype which demonstrates that this kind of LMPM can be used for heavy load lifting.

Kinematic analysis of 2-DOF LMPM

Layout and mobility analysis of LMPM

In this section, kinematic analysis of LMPM is presented. Figure 1 shows the geometrical layout of LMPM.

Figure 1.

2-DOF lower-mobility parallel mechanism: (a) render graph of LMPM and (b) layout of LMPM.

The LMPM consists of a moving platform, whose width is $L_{41}$ and $L_{42}$ , and length is $L_{3}$ , and a fixed platform, whose width is $L_{21}$ and $L_{22}$ , and length is $L_{1}$ . Four prismatic pairs are used as driving units, $B_{1} M_{1}$ , $B_{2} M_{2}$ , $B_{3} M_{3}$ and $B_{4} M_{4}$ , the driving unit is connected with the fixed platform by a revolute joint and connected with the rod by a revolute joint, respectively. Four rods, $P_{1} N_{1}$ , $P_{2} N_{2}$ , $P_{3} N_{3}$ and $P_{4} N_{4}$ , which all the endpoints connected with the fixed platform and moving platform by revolute joints, meanwhile endpoints $N_{1}$ , $N_{2}$ , $N_{3}$ and $N_{4}$ can move along the lengthwise direction of the fixed platform and endpoints $P_{2}$ and $P_{3}$ can move along the lengthwise direction of the moving platform. Four assistive rods, $B_{1} Q_{1}$ , $B_{2} Q_{2}$ , $B_{3} Q_{3}$ and $B_{4} Q_{4}$ , connected with the fixed platform and rod by two revolute joints, respectively. Once the output of driving unit is confirmed, there comes out a triangle $Δ BMQ$ , and it results in a confirmed angle $∠ BMQ$ . Combining $BM$ , $∠ BMQ$ and $MN$ , a triangle $Δ BMN$ is obtained, then we can get the length of $BN$ . Therefore, the assistive rods can help LMPM keep locked at a specified position. Global coordinate system O-XYZ is fixed at the centre point of the fixed platform and local coordinate system $O' - X' Y' Z'$ is fixed at the centre point of the moving platform. For further explanation, in the LMPM, all the rotational axes of the revolute joints must be parallel to the X axis of the global coordinate. Therefore, each node of the LMPM has rotational ability which guarantees the LMPM running in smooth and steady status.

For LMPM is first proposed, so the mobility analysis is necessary. The LMPM can be simplified as shown in Figure 2. The number of component is expressed in Arabic numerals and the joint number is expressed in Roman numerals. The simplified configuration has eight components and 11 joints, according to the calculation formula of freedom of planar mechanism, $F = 3 n - 2 p_{L} - p_{H} = 24 - 22 = 2$ , which means that the simplified mechanism needs two independent driving units to ensure the mechanism has two freedoms. Component 6 and 7 are chosen to be the driving unit, when they move symmetrically about the origin point O, component 1 (moving platform) moves along Z axis and on the contrary the component 1 rotates about the X axis.

Figure 2.

Simplified configuration of LMPM.

The LMPM is symmetrical about the YOZ plane. Based on above mobility analysis, the outputs of $B_{1} M_{1}$ and $B_{4} M_{4}$ , and the outputs of $B_{2} M_{2}$ and $B_{3} M_{3}$ must be equal, respectively. For this reason, we can regard $B_{1} M_{1}$ and $B_{4} M_{4}$ as one group, and the same with $B_{2} M_{2}$ and $B_{3} M_{3}$ . When the output of $B_{1} M_{1}$ equals to the output of $B_{2} M_{2}$ , the moving platform can rise along Z axis. On the contrary, when the output are different, the moving platform can rotate about X axis.

Inverse kinematics analysis

For parallel mechanism, the inverse kinematics problem is easier to solve than the forward kinematics problem. Inverse kinematics analysis can be concluded as, knowing the position vector of moving platform and the dimension parameters to solve the output of the driving unit.^29,30

For LMPM, the position vector $P_{iO}$ of endpoints $P_{i} (i = 1, 2, 3, 4)$ of the rod in global coordinate system is

P_{iO} = r + R p_{i} = (\begin{matrix} x_{P_{iO}} & y_{P_{iO}} & z_{P_{iO}} \end{matrix})^{T}

(1)

where $r$ is the position vector of original point $O'$ of local coordinate system relative to original point $O$ of global coordinate system

r = (\begin{matrix} X_{P} & Y_{P} & Z_{P} \end{matrix})^{T}

(2)

$p_{i}$ is the position vector of $P_{i}$ in the local coordinate system

p_{i} = {\begin{matrix} p_{1} = {(\begin{matrix} \frac{L_{41}}{2} & - \frac{L_{3}}{2} & 0 \end{matrix})}^{T} \\ p_{2} = {(\begin{matrix} \frac{L_{42}}{2} & \frac{L_{3}}{2} + Δ l_{T} & 0 \end{matrix})}^{T} \\ p_{3} = {(- \begin{matrix} \frac{L_{42}}{2} & \frac{L_{3}}{2} + Δ l_{T} & 0 \end{matrix})}^{T} \\ p_{4} = {(- \begin{matrix} \frac{L_{41}}{2} & - \frac{L_{3}}{2} & 0 \end{matrix})}^{T} \end{matrix}

(3)

$Δ l_{T}$ is the displacement along $Y'$ axis of points $P_{2}$ and $P_{3}$ derived from the different outputs of $B_{1} M_{1}$ $(B_{4} M_{4})$ and $B_{2} M_{2}$ $(B_{3} M_{3})$ .

and $R$ is the rotation matrix from fixed platform to moving platform

R = (\begin{matrix} 1 & 0 & 0 \\ 0 & c_{φ} & - s_{φ} \\ 0 & s_{φ} & c_{φ} \end{matrix})

(4)

$φ$ is the rotational degree of moving platform and $c_{φ} = \cos φ$ , $s_{φ} = \sin φ$ .

When $r$ , $p_{i}$ , $Δ l_{T}$ and $φ$ are known, combined with mechanism dimension parameters, the outputs of $B_{1} M_{1}$ $(B_{4} M_{4})$ and $B_{2} M_{2}$ $(B_{3} M_{3})$ can be solved.

Relationship of output speed and input speed between the moving platform and driving unit is also about inverse kinematic analysis. Closed vector method is used to solve the problem

r = b_{i} + l_{i 1} \cdot q_{i 1} + l_{i 3} \cdot q_{i 3} - R \cdot p_{i}

(5)

where $b_{i}$ denotes the position vector of point $B_{i}$ in global coordinate system, $l_{i 1}$ and $l_{i 3}$ denote the length of $B_{i} M_{i}$ and $M_{i} P_{i}$ , respectively, and $q_{i 1}$ and $q_{i 3}$ denote the unit vector of $B_{i} M_{i}$ and $M_{i} P_{i}$ , respectively. Taking derivation of equation (5), the following equation is obtained

J \overset{\cdot}{w} = {\overset{\cdot}{l}}_{i 1}

(6)

where $J = (J_{i}) = {(\begin{matrix} q_{i 3}^{T} / (q_{i 1} \cdot q_{i 3} - {(q_{i 1} \times q_{i 3})}^{T} \cot θ_{i}) \\ {(R p_{i} \times q_{i 3})}^{T} / (q_{i 1} \cdot q_{i 3} - {(q_{i 1} \times q_{i 3})}^{T} \cot θ_{i}) \end{matrix})}^{T}$ , $\overset{\cdot}{w} = ({\overset{\cdot}{w}}_{i}) = (\begin{matrix} \overset{\cdot}{r} \\ \overset{\cdot}{φ} \end{matrix})$ , ${\overset{\cdot}{l}}_{i 1} = (\begin{matrix} {\overset{\cdot}{l}}_{11} \\ {\overset{\cdot}{l}}_{21} \\ {\overset{\cdot}{l}}_{31} \\ {\overset{\cdot}{l}}_{41} \end{matrix})$ , $θ_{i}$ denotes the angle between $B_{i} M_{i}$ and $M_{i} P_{i}$ . $J$ is the Jacobin matrix of LMPM, which can reflect the relationship between input speed and output speed of the LMPM.

Forward kinematics analysis

For LMPM, its forward kinematics analysis reflects the relationship of input from $B_{i} M_{i}$ and output $r$ and $φ$ .

Take driving unit $B_{1} M_{1}$ as example. $l_{11}$ is known, then $l_{15}$ , the length between $B_{1}$ and $N_{1}$ along Y axis, can be expressed as

l_{15} = \sqrt{| \frac{l_{12} (l_{14}^{2} - l_{11}^{2})}{l_{16}} + (l_{14}^{2} + l_{12} l_{16} + l_{12}^{2}) |}

(7)

where $l_{12}$ , $l_{14}$ and $l_{16}$ represents the length of $Q_{1} N_{1}$ , $B_{1} Q_{1}$ and $Q_{1} M_{1}$ , respectively. Then the position vector of $N_{1}$ in global coordinate system can be obtained

N_{1 O} = {(\begin{matrix} \frac{L_{41}}{2} & l_{15} - \frac{L_{1}}{2} & 0 \end{matrix})}^{T}

(8)

then according to the geometrical relationship, the position vector of $P_{1}$ can be obtained

\begin{matrix} P_{1 O} = (\begin{matrix} x_{P_{1 O}} & y_{P_{1 O}} & z_{P_{1 O}} \end{matrix})^{T} \\ = {(\begin{matrix} \frac{L_{41}}{2} & (l_{15} - \frac{L_{1}}{2}) - A_{1} l_{P_{1} N_{1}} & \sqrt{1 - A_{1}^{2}} \end{matrix} l_{P_{1} N_{1}})}^{T} \end{matrix}

(9)

where $A_{1}$ is cosine value of the angle between rod $P_{1} N_{1}$ and Y axis, and $l_{P_{1} N_{1}}$ is the length of the rod. Similarly, $P_{2 O}$ , $P_{3 O}$ and $P_{4 O}$ can be obtained, they can be expressed as

P_{iO} = (\begin{matrix} x_{P_{iO}} & y_{P_{iO}} & z_{P_{iO}} \end{matrix})^{T}

(10)

$φ$ can be expressed as

φ = \tan^{- 1} \frac{z_{P_{1 O}} - z_{P_{2 O}}}{y_{P_{2 O}} - y_{P_{1 O}}}

(11)

then $r$ can be obtained

r = {(\begin{matrix} 0 & (l_{15} - \frac{L_{1}}{2}) - A_{1} l_{P_{1} N_{1}} + c φ \frac{L_{3}}{2} & \sqrt{(1 - A_{1}^{2})} l_{P_{1} N_{1}} + s φ \frac{L_{3}}{2} \end{matrix})}^{T}

(12)

and $Δ l_{T}$ can be obtained as

Δ l_{T} = \frac{L_{1} - (l_{15} + l_{25}) + (A_{1} + A_{2}) l_{P_{1} N_{1}}}{c φ} - L_{3}

(13)

Singularity and workspace analysis

Singularity analysis

When parallel mechanism is at singularity configuration, it will lose DOF and results in mechanism being out of control. Therefore, the singularity configuration should be considered in designing process and avoided in practical application. Combining with above analysis and using Jacobin matrix method, we study the singularity of LMPM. According to the layout of LMPM, it can be easily seen that it needs the output of $B_{1} M_{1}$ be equal to $B_{4} M_{4}$ , and $B_{2} M_{2}$ be equal to $B_{3} M_{3}$ , respectively, otherwise, the LMPM will lose the reasonable DOF and run in error.

LMPM is symmetrical about YOZ plane, we take the part of $B_{1} M_{1}$ and $B_{2} M_{2}$ to explain the singularity analysis using Jacobin matrix method. The Jacobin matrix turns into

J' = (\begin{matrix} J_{1} \\ J_{2} \end{matrix}) = (\begin{matrix} q_{13}^{T} / (q_{11} \cdot q_{13} - {(q_{11} \times q_{13})}^{T} \cdot \cot θ_{1}) & (R p_{1} \times q_{13}) / (q_{11} \cdot q_{13} - {(q_{11} \times q_{13})}^{T} \cdot \cot θ_{1}) \\ q_{23}^{T} / (q_{21} \cdot q_{23} - {(q_{21} \times q_{23})}^{T} \cdot \cot θ_{2}) & (R p_{2} \times q_{23}) / (q_{21} \cdot q_{23} - {(q_{21} \times q_{23})}^{T} \cdot \cot θ_{2}) \end{matrix}) .

The determinant of $J'$ is

| J' | = \frac{q_{13}^{T} \cdot {(R p_{2} \times q_{23})}^{T} - q_{23}^{T} \cdot {(R p_{1} \times q_{13})}^{T}}{(q_{11} \cdot q_{13} - {(q_{11} \times q_{13})}^{T} \cdot \cot θ_{1}) \cdot (q_{21} \cdot q_{23} - {(q_{21} \times q_{23})}^{T} \cdot \cot θ_{2})}

(14)

When the determinant $| J' |$ equals 0, the LMPM will be in singularity configuration.²¹ Following conditions will make $| J' |$ equal 0:

$l_{13} = l_{23} = 0$ . This will lead $q_{13}$ and $q_{23}$ to be zero vector and then make $| J' | = 0$ . This condition is architecture singularity. Similarly, $l_{11} = l_{21} = 0$ will also make $| J' | = 0$ .

$q_{11} \times q_{13} = q_{21} \times q_{23} = 0$ . This condition means that the driving units are parallel to the rods, meanwhile the moving platform is also parallel to the fixed platform. The direction of driving force is horizontal so that the driving units are not able to drive LMPM. This condition is known as dead point singularity.

$q_{13}^{T} \cdot (R p_{2} \times q_{23})^{T} = q_{23}^{T} \cdot (R p_{1} \times q_{13})^{T}$ . If the equation is satisfied, $| J' | = 0$ . It means that the two parts $B_{1} N_{1} P_{1}$ and $B_{2} N_{2} P_{2}$ are identical one part and results in architecture singularity.

Workspace analysis

Different from workspace analysis of other parallel mechanism, they consider all positions that the moving platform can reach as the workspace of parallel mechanism.³¹ LMPM is designed for lifting vehicle in multilayer garage, we pay attention on the smooth and steady operation, therefore relationship between the centroid of the moving platform and the centroid of the vehicle should be controlled. For this reason, the workspace of LMPM is defined as all of the points that centroid of the moving platform $O'$ can reach when the outputs of driving units change from maximum to minimum.

First, we assign certain values to the components of LMPM. Maximum value of driving units $l_{i 1} = 1655 mm$ , and the minimum value is $l_{i 1} = 1190 mm$ , $l_{P_{i} N_{i}} = 3500 mm$ , sub item $l_{i 2} = 1750 mm$ , $l_{i 2} = 1185 mm$ , $l_{i 6} = 565 mm$ , length of the assistive rods $l_{i 4}$ keep equal to rod, $L_{1} = 4040 mm$ , $L_{21} = 1015 mm$ , $L_{22} = 1015 mm$ , $L_{41} = 1460 mm$ , $L_{42} = 1240 mm$ , and the initial value of $L_{3}$ is 4040 mm. This will construct a specific LMPM, which can be applied in multilayer garage. The following will combine the forward kinematics analysis and give the simulation results for the workspace of LMPM according to the selected parameters.

In Figure 3, it shows that the workspace of LMPM generated by 16 iterations, namely 256 points totally. It can be seen that the points that centroid $O'$ can reach have following characteristics:

On X axis, the centroid does not move;

On Y axis, the centroid moves only in the negative direction, maximum value is 0 mm and minimum value is −170.4 mm;

On Z axis, the centroid moves only in the positive direction, maximum value is 2000 mm and minimum value is 227 mm.

Figure 3.

Workspace of LMPM generated by 16 iterations.

Limited by iteration times, the points showed in Figure 3 appear sparse, so we increase iteration times to 31 and 39, respectively, as shown in Figures 4 and 5. It can be seen that the workspace generated by 39 iterations is more intensive. Meanwhile, most of the new appeared points locate at the right side of the workspace, and according to this phenomenon, we propose the following Table 1.

Figure 4.

Workspace of LMPM generated by 31 iterations.

Figure 5.

Workspace of LMPM generated by 39 iterations.

Table 1.

Distribution of points that $O'$ can reach (X = 0, −180 < Y < 0, 0 < Z < 2200 mm).

Iteration times	Total number of points	–10 < Y < 0	–20 < Y < –10	–30 < Y < –20	–180 < Y < –30
39	1521	43.98%	14.46%	9.34%	32.22%
31	961	43.81%	14.36%	9.78%	32.05%
16	256	42.19%	14.06%	10.16%	33.59%

In Table 1, following aspects can be obtained, in the range from −20 to 0, the points locate in this range occupy more than 60% of the total points, and the proportion of points that locate in each specified range almost keeps invariant, and the proportion in each range decrease progressively from ‘–10 to 0’ to ‘–180 to −170’. It illustrates that the LMPM has a more flexible attitude adjustment selection in the range −20 to 0. Therefore, we divide the workspace of LMPM along Y axis direction into several subareas by 10 mm. From 0 to −180, there are 18 subareas, and we define the two subareas that contain the largest proportion of workspace points as the flexible workspace of LMPM. Taking the above dimension parameters as example, the range X = 0, −20 < Y < 0, 0 < Z < 2200, is the flexible workspace of LMPM. This flexible workspace reflects that LMPM has more attitude adjustment selections and more smooth movement in these subareas. Moreover, the flexible workspace represents that the LMPM can realize fine and continuous posture adjustment in a small range, and it just meets the requirement that it does not need the instrument to make remarkable adjustment in multilayer garage.

In addition, once the dimension parameter is changed, the workspace and flexible workspace will change too. The change of flexible workspace will be illustrated in the following section.

Kinematic performance evaluation using GSI

In this section, according to the characteristics of LMPM, we propose a GSI based on the GCI to evaluate the kinematic performance of LMPM, and it can reflect the kinematic performance of the mechanism designed for realizing symmetry movement but driven by mutual independent units. The flexible workspace is also studied combined with the GCI. In the end, we developed a simulation software based on MATLAB guide, which can intuitively show the kinematic performance of LMPM.

Based on the condition number $κ$ of $J'$ , the transmission property between joints and the workspace can be connected by its distribution in workspace. $K = \int κ dV$ , V denotes the volume of workspace. For LMPM, $K$ is a square matrix whose dimension is determined by the iteration times ‘n’ of driving unit $B_{1} M_{1}$ $(B_{4} M_{4})$ and $B_{2} M_{2}$ $(B_{3} M_{3})$ . In $K$ , each element represents the condition number at each specific position of LMPM. Meanwhile, a smaller $κ$ represents a better transmission property.

To obtain a good kinematic performance, dimension parameter of LMPM should be adjusted. The dimension parameters in section ‘Workspace analysis’ are unchanged, except $l_{i 2}$ and $l_{i 6}$ . We denote $0.3 \leq λ_{l_{i 2}} = l_{i 2} / l_{P_{i} N_{i}} \leq 0.5$ and $0.3 \leq λ_{l_{i 6}} = l_{i 6} / (l_{i 3} + l_{i 6}) \leq 0.5$ , and then use monotonicity analysis method to find a group of $λ_{l_{i 2}}$ and $λ_{l_{i 6}}$ which can lead to a best kinematic performance.

In Figures 6 –9, X label represents iterations of driving unit $B_{1} M_{1}$ $(B_{4} M_{4})$ , Y label represents iterations of driving unit $B_{2} M_{2}$ $(B_{3} M_{3})$ , Z label represents condition number and the colour band represents the position that point $O'$ reaches along Z axis of the global coordinate system. The condition number distributions when $λ_{l_{i 2}} = 0.33$ and 0.36 are omitted for the distributions are similar with $λ_{l_{i 2}} = 0.30$ and $λ_{l_{i 2}} = 0.39$ . In Table 2, ‘R’ is the specific value of the maximum value and the minimum value that $O'$ can reach along Z axis. ‘scn’ and ‘ecn’ are the value of condition number at starting point and ending point on axle wire, respectively. Axle wire is shown in red dashed line from Figures 6 –8. Considering from workspace, the points on axle wire represent that the output of $B_{1} M_{1}$ $(B_{4} M_{4})$ and $B_{2} M_{2}$ $(B_{3} M_{3})$ are equal. Comparing the 64 groups, they have the same characteristic, and the condition number decreases monotonously from the starting point to the ending point. From Figures 6 –9, it can be seen that the distribution becomes more symmetrical when $λ_{l_{i 2}}$ increases. Combining with Table 2, when $λ_{l_{i 2}} = 0.50$ and $λ_{l_{i 6}} = 0.33$ , the LMPM gets a biggest $R = 2015.9 mm / 373.8 mm = 5.393 mm$ , which means that LMPM can get a big folding and unfolding space. Meanwhile, the smallest ‘ecn’ and the biggest ‘scn’ are obtained, which represent that the kinematic performance of LMPM gets the biggest increasing range and reaches the best kinematic performance of 3.364 among the 64 groups. Therefore, we preliminarily take $λ_{l_{i 2}} = 0.50$ and $λ_{l_{i 6}} = 0.33$ .

Figure 6.

Condition number distribution: $λ_{l_{i 2}} = 0.30$ and $λ_{l_{i 2}} = 0.39$ , $λ_{l_{i 6}} = 0.30 ~ 0.50$ .

Figure 7.

Condition number distribution: $λ_{l_{i 2}} = 0.41$ and $λ_{l_{i 2}} = 0.44$ , $λ_{l_{i 6}} = 0.30 ~ 0.50$ .

Figure 8.

Condition number distribution: $λ_{l_{i 2}} = 0.47$ , $λ_{l_{i 6}} = 0.30 ~ 0.50$ and $λ_{l_{i 2}} = 0.50$ , $λ_{l_{i 6}} = 0.36, 0.39, 0.41, 0.44, 0.47, 0.50$ .

Figure 9.

Condition number distribution: $λ_{l_{i 2}} = 0.50$ , $λ_{l_{i 6}} = 0.33$ .

Table 2.

Characteristics of condition number distribution with different $λ_{l_{i 2}}$ and $λ_{l_{i 6}}$ .

$λ_{l_{i 6}}$	$λ_{l_{i 2}}$
	0.30			0.33			0.36			0.39
0.30	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.30	1.407	5.188	4.132	1.434	4.978	4.191	1.479	4.759	4.157	1.549	4.541	4.058
0.33	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.33	1.396	5.298	4.381	1.418	5.075	4.360	1.453	4.850	4.276	1.510	4.631	4.146
0.36	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.36	1.389	5.422	4.593	1.405	5.189	4.514	1.433	4.960	4.393	1.479	4.737	4.238
0.39	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.39	1.384	5.566	4.786	1.395	5.325	4.664	1.417	5.090	4.512	1.455	4.865	4.337
0.41	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.41	1.383	5.678	4.911	1.390	5.431	4.765	1.409	5.192	4.596	1.441	4.963	4.409
0.44	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.44	1.382	5.874	5.101	1.385	5.618	4.924	1.398	5.371	4.733	1.424	5.136	4.530
0.47	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.47	1.384	6.116	5.302	1.383	5.846	5.099	1.391	5.589	4.888	1.411	5.345	4.670
0.50	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.50	1.389	6.417	5.525	1.382	6.129	5.297	1.386	5.855	5.067	1.401	5.597	4.835
$λ_{l_{i 6}}$	$λ_{l_{i 2}}$
	0.41			0.44			0.47			0.50
0.30	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.30	1.670	4.343	3.912	1.909	4.219	3.731	2.614	4.449	3.526	n/a	n/a	n/a
0.33	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.33	1.604	4.431	3.980	1.778	4.288	3.789	2.192	4.347	3.579	5.393	7.310	3.364
0.36	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.36	1.554	4.535	4.056	1.686	4.379	3.856	1.965	4.359	3.642	2.963	5.104	3.428
0.39	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.39	1.515	4.658	4.142	1.619	4.492	3.934	1.821	4.428	3.717	2.367	4.766	3.505
0.41	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.41	1.494	4.754	4.207	1.584	4.581	3.993	1.751	4.497	3.775	2.156	4.715	3.564
0.44	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.44	1.468	4.920	4.317	1.541	4.738	4.097	1.671	4.631	3.876	1.952	4.749	3.667
0.47	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.47	1.446	5.120	4.446	1.506	4.927	4.220	1.611	4.804	3.997	1.819	4.866	3.791
0.50	R	scn	ecn	R	scn	ecn	R	scn	ecn	R	scn	ecn
0.50	1.429	5.360	4.601	1.479	5.156	4.368	1.564	5.018	4.142	1.725	5.047	3.939

Degree of symmetry of the condition number distribution in workspace should be concerned in LMPM’s kinematic performance evaluation. LMPM is designed for realizing steady lifting and posture adjustment, so it is expected that the driving units $l_{B_{1} M_{1}}$ $(l_{B_{4} M_{4}})$ and $l_{B_{2} M_{2}}$ $(l_{B_{3} M_{3}})$ play the same role in operation. According to the distribution condition from Figures 6 –9, we define a GSI for LMPM.

GSI is defined based on GCI. The condition number distribution K is

\begin{matrix} K = [\begin{matrix} κ_{11} & \dots & κ_{1 n} \\ ⋮ & ⋱ & ⋮ \\ κ_{n 1} & \dots & κ_{nn} \end{matrix}], K_{U} = [\begin{matrix} κ_{11} & \dots & κ_{1 n} \\ 0 & ⋱ & ⋮ \\ 0 & 0 & κ_{nn} \end{matrix}], \\ K_{D} = [\begin{matrix} κ_{11} & 0 & 0 \\ ⋮ & ⋱ & 0 \\ κ_{n 1} & \dots & κ_{nn} \end{matrix}] \\ K_{S} = (K_{U})^{T} - K_{D} = [\begin{matrix} K_{S_{11}} & K_{S_{12}} & \dots & K_{S_{1 n}} \\ K_{S_{21}} & K_{S_{22}} & \dots & K_{S_{2 n}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ K_{S_{n 1}} & K_{S_{n 2}} & \dots & K_{S_{nn}} \end{matrix}] \\ = [\begin{matrix} 0 & 0 & \dots & 0 \\ κ_{12} - κ_{21} & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ κ_{1 n} - κ_{n 1} & \dots & κ_{(n - 1) n} - κ_{n (n - 1)} & 0 \end{matrix}] \end{matrix}

We denote $K_{GSI}$ as the value of GSI, and $K_{GSI}$ is expressed as

K_{GSI} = max (| K_{S_{uv}} |)

(15)

where u = 1, 2,…, n and v = 1, 2,…, n.

The value of $K_{GSI}$ presents the degree of symmetry of kinematic performance of LMPM. When $K_{GSI}$ is bigger, LMPM has a more asymmetric condition number distribution, and the kinematic performance of LMPM is more asymmetric. On the contrary, the two groups of driving unit bring similar kinematic performance, and it is also helpful to realize smooth and steady lifting movement and posture adjustment.

In order to intuitively illustrate the symmetry degree of kinematic performance of LMPM, $K_{SG}$ is defined as $K_{SG} = | K_{S} | + | K_{S}^{T} |$ , its distribution in workspace is shown in Figure 10. $K_{SG}$ is symmetry about the axle wire. It is obviously that on axle wire, the symmetry of kinematic performance gets its best value zero. Take axle wire as boundary, the left side and right side is symmetrical. Both sides can reflect the symmetry degree of kinematic performance of LMPM.

Figure 10.

Symmetry degree of kinematic performance when $l_{i 6} = 569 mm$ .

In the vicinity of $λ_{l_{i 6}} = 0.33$ , we change $l_{i 6}$ and obtain different characteristics of the distribution of $κ$ , as shown in Table 3. Through comprehensive analysis and key-factor analysis, it is obtained that when $l_{i 2} = 1750 mm$ and $l_{i 6} = 569 mm$ , LMPM gets a good kinematic performance, reasonable workspace, good GCI and small $K_{GSI}$ .

Table 3.

Characteristics of distribution of $κ$ with different $l_{i 6}$ .

$l_{i 6}$	scn	ecn	$K_{GSI}$
575 mm	7.69	3.361	1.2163
570 mm	8.81	3.355	1.1461
565 mm	11.07	3.350	1.1732
560 mm	19.82	3.344	1.1603
569 mm	9.127	3.354	1.1231

In section ‘Singularity and workspace analysis’, the flexible workspace of LMPM is defined. Along with the change of dimension parameters, the workspace and flexible workspace are also changed, in the following table, the related characteristics are illustrated.

In Table 4, $Z_{max}$ and $Z_{min}$ , that is, the maximum and minimum value along Z axis that $O'$ can reach in flexible workspace, total amount of the points that $O'$ can reach in flexible workspace and $K_{GSI}$ in flexible workspace with different $l_{i 6}$ are included.

Table 4.

Characteristics of flexible workspace with different $l_{i 6}$ .

$l_{i 6}$	$Z_{\min}$ and $Z_{\max}$ (mm)		Points in flexible workspace	$K_{GSI}$
560	122	2015	875	1.1603
561	149	2015	875	1.1727
562	172	2015	877	1.1795
563	192	2015	881	1.1815
564	210	2015	889	1.1792
565	227	2015	889	1.1732
566	242	2015	891	1.1642
567	256	2015	891	1.1525
568	270	2015	891	1.1387
569	283	2015	897	1.1231
570	295	2015	901	1.1461
571	307	2015	905	1.1727
572	318	2015	905	1.1924
573	329	2015	909	1.2058
574	339	2015	909	1.2136
575	349	2015	915	1.2163

$Z_{max}$ keeps almost invariant and $Z_{min}$ increases with $l_{i 6}$ . The total amount of the points that $O'$ can reach increases with $l_{i 6}$ as well. The $K_{GSI}$ in flexible workspace equals to the $K_{GSI}$ in global workspace. The main kinematic performance index in flexible workspace and global workspace are almost identical, so it can be considered that the flexible workspace is the main workspace of LMPM, and LMPM can realize its best performance in flexible workspace.

Based on the above analysis, we developed a simulation software based on MATLAB guide to show the kinematic performance of LMPM intuitively, as shown in Figure 11. In the software, we can change the dimension parameter and choose different kinematic performance index to see the differences of each index under different dimension parameters.

Figure 11.

Kinematic performance simulation software of LMPM.

Prototype experiment

Combining the above study about the 2-DOF LMPM applied in multilayer garage, we made a prototype to verify the feasibility of this project.

The battery pack consists of six lead storage batteries in series connection, its nominal voltage is 12 V and it powers the LMPM. The DC motor, whose power is 2 KW, pumps oil from an 8 L hydraulic oil tank through switch valves to the driving units. Figure 12(a) shows LMPM located in prone position, Figure 12(b) shows LMPM in rising process and Figure 12(c) shows LMPM lifting the carrying board whose mass is about 300 kg. Proof by facts, the design project is reasonable and feasible.

Figure 12.

Prototype of LMPM: (a) prone position, (b) rising process, and (c) lifting carrying board.

Conclusion

In this article, we have shown our study and research about a new kind of lifting instrument applied in multilayer garage. The kinematic analysis and kinematic performance evaluation are presented, the unique kinematic characteristics of LMPM have been obtained meanwhile we establish two kinds of evaluation standard: flexible workspace and GCI; which can also be applied in kinematic analysis of mechanisms like LMPM which is expected to realize steady and symmetry movement. In kinematic level, we have established a good model for this kind of mechanism.

In future work, to get a better stability and symmetrical movement, following aspect will be studied: the influence that existing clearances in the joint brings to LMPM. Clearances in mechanism are unavoidable due to assemblage, manufacturing errors and wear. Moreover, the clearance occurs in each joint with the movement of the mechanism. The mechanism takes risk of losing movement accuracy, stability and the reliability. Meanwhile kinetostatic optimization should also be taken into consideration, combining dynamic analysis and kinetostatic analysis, the reliability, economical and practical factors are satisfied as well as the LMPM can get a good dynamic response.

Footnotes

Handling Editor: Jose Ramon Serrano

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was supported by the Heilongjiang Province Education Department aids project, Study of Omni-bearing Transporting Robot used in Intelligent Multilayer Garage (GC13A410).

ORCID iD

Tianhua He

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