# A New Kinematic Synthesis Model of Spatial Linkages for Designing Motion and Identifying the Actual Dimensions of a Double Ball Bar Test Based on the Data Measured

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Spatial Five-Bar Linkage of Double Ball Bar Tests of a Two-Axis Rotary Table

#### 2.1. Double Ball Bar Test of a Two-Axis Rotary Table

_{f},

**X**

_{f},

**Y**

_{f},

**Z**

_{f}} established on the machine tool frame, and each coordinate axis is along the corresponding machine tool linear axis direction. A moving coordinate system {O

_{m},

**x**

_{m},

**y**

_{m},

**z**

_{m}}, is set up on the worktable, and the axis

**z**is defined along the worktable rotation axis direction. A point P of the worktable can be expressed as a point vector

_{m}**R**in {O

_{Pf}_{f},

**X**

_{f},

**Y**

_{f},

**Z**

_{f}}:

**R**

_{Omf}is the position vector of O

_{m}in {O

_{f},

**X**

_{f},

**Y**

_{f},

**Z**

_{f}},

**r**

_{Pm}is the position vector of point P in {O

_{m},

**x**

_{m},

**y**

_{m},

**z**

_{m}}, and [

**M**

_{AC}] is the transformation matrix of two coordinate systems and expressed as follows:

_{1A}and θ

_{2C}are the machine tool A-axis and C-axis rotation angles, respectively.

_{1A}and θ

_{2C}of the two rotary tables. In the practical use of the two rotary tables, the C-axis will rotate full-circle while the A-axis does a non-circle. The motion ranges of θ

_{1A}and θ

_{2C}are defined as follows:

_{E},y

_{E},z

_{E}) and F (x

_{F},y

_{F},z

_{F}) of the DBB, is measured by the sensor, shown in Figure 1b. The vector connecting the two ball centers can be expressed by

**R**and

_{OE}**R**are the position vectors of the two points E and F, respectively.

_{OF}#### 2.2. The Mechanism Model for the DBB Test of a Two-Axis Rotary Table

_{0}, a

_{0}, a

_{1}, s

_{2}, a

_{2}, and α

_{12}, as shown in Figure 2. This spatial linkage is driven by two input variables or motors that control the rotation angles θ

_{1}and θ

_{2}of links 1 and 2, respectively, wherein the output parameter ‘d’ is an independent variable representing the distance between the centers of the two balls. Pay attention to the variation of the output parameter ‘d’, which is confined to a small range in millimeters as dictated by the DBB instrument.

**x**

_{0},

**y**

_{0},

**z**

_{0}} for frame 0, the coordinate system {C,

**x**

_{1},

**y**

_{1},

**z**

_{1}} for link 1, and the coordinate system {C′,

**x**

_{2},

**y**

_{2},

**z**

_{2}} for link 2. Two ball centers point F (a

_{2}cosθ

_{C}

_{0}, a

_{2}sinθ

_{C}

_{0}, s

_{2}) and E (s

_{0}, −a

_{0}sinθ

_{A}

_{0}, a

_{0}cosθ

_{A}

_{0}) are, respectively, expressed in terms of the parameters of the linkage. A closed-loop vector equation is formulated to articulate the geometric relationships within the spatial five-bar linkage.

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}, a

_{1}, α

_{12}, θ

_{1}, θ

_{2}, d), in which eight ones correspond to the constructure parameters of the two-axis rotary table, one is the output parameter, while two are the input parameters. The parameters for the coordinates of the two balls’ positions are (s

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}), and the constructure parameters of the A-axis and C-axis are (a

_{1}and α

_{12}), the distance and angular between the A-axis and C-axis. The input angles θ

_{1}and θ

_{2}of the spatial five-bar linkage are the independent variables, and the output displacement d of the RR-SPS linkage can be solved by Equation (9). For the given parameters (s

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}) of the coordinates of the two balls’ positions, the length d of the spatial five-bar linkage RR-SPS varies or outputs different values.

_{1}and θ

_{2}. In particular, d is a constant in two cases: (1) both the A-axis and the C-axis are the ideal orthogonal intersected axes, and two balls are specially located on the theoretical positions, and (2) the motion of the two input angles θ

_{1}and θ

_{2}has to be planned as a theoretical function relationship. Generally, case 2 is easy to be realized by designating the two input parameters θ

_{1}and θ

_{2}, which can be solved by the kinematics of a spatial four-bar RRSS linkage degenerated by RRSPS.

## 3. Motion Design of Spatial Four-Bar Linkage for the DBB Test of a Two-Axis Rotary Table

_{1}and θ

_{2}must be designed for the DBB test, and the dimensions of the links must also be determined. In fact, the motion plan of the RRSPS is to determine the relationship between the two input parameters θ

_{1}and θ

_{2}. The design of the dimensions for the RRSPS involves determining the installation positions of both precision balls in order to measure the full workspace of the two rotary axes. Specifically, the A-axis has its own workspace with two extreme positions, while the C-axis can achieve a full rotation.

_{2}, while link 1 is a rocker with an input angle θ

_{1}corresponding to its two extreme positions. The variation in displacement d during a DBB test remains zero in cases where there is an ideal motion for RR when both input angles θ

_{1}and θ

_{2}follow a theoretical function.

#### 3.1. Motion Function Design for the DBB Test

_{1}and θ

_{2}have a theoretical function, and the output length d of the DBB test remains constant when the A and C axes have ideal motion.

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}, a

_{1}, α

_{12}, θ

_{1}, θ

_{2}, d), which are the same as those of RRSPS, shown in Figure 3. The parameters of two balls’ positions coordinates are (s

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}), and the constructure parameters of the A-axis and C-axis are (a

_{1}and α

_{12}). The length d of link 3, the distance between two balls, is a constant. The rotational angle θ

_{1}of link 1 is defined as the independent variable, or the input parameter, while the output angle θ

_{2}of link 2 is the dependent variable or a function of angle θ

_{1}. Based on Figure 3 and Equation (8), the motion relationship between the dependent variable θ

_{2}and the independent variable θ

_{1}can be derived as follows:

_{1}is expressed as the dependent variable while θ

_{2}is designated as the independent variable.

_{1}and the output parameter θ

_{2}of the spatial four-bar linkage RRSS. The rotational angles θ

_{1}of link 1 and θ

_{2}of link 2 are expressed in the coordinate system of RRSS, which can be converted into the coordinate system of the machine tool, and the angle θ

_{1A}of the A-axis and the orientation angle θ

_{2C}of the C-axis of a two-axis rotary table can be expressed, respectively,

_{A}

_{0}and θ

_{C}

_{0}have the geometrical meanings of the installation initial angles of the spherical joints S

_{E}and S

_{F}in the coordinate system of the machine tool.

_{1A}and θ

_{2C}of the two-axis rotary table for the DBB test can be calculated. For a general RRSS linkage, the orientation angles θ

_{1A}and θ

_{2C}only belong to a part of the workspace of the two-axis rotary table. It is necessary to design both the reasonable installation parameters of the DBB and the corresponding functions of θ

_{1A}and θ

_{2C}, that is to say, for the length of the DBB test to have a full work space, a crank, and a rocker.

#### 3.2. Motion Range of the DBB Test of an Ideal Two-Axis Rotary Table

_{2}(0,360) and link 0 occupies its extreme positions, or θ

_{1}(θ

_{1min}and θ

_{1max}). According to Section 3.1, the output parameter θ

_{1}and the input parameter θ

_{2}of the four-bar linkage RRSS correspond to their extreme values or locations. Based on the relationship between θ

_{1}and θ

_{2}, Equation (10) can be rewritten as

_{2}is expressed as

_{1}relative to θ

_{2}is zero, expressed by

_{1}when rocker 0 is at the extreme position.

**z**

_{1}of the spatial four-bar linkage RRSS is discussed when rocker 0 is located at the extreme position. That is, three vectors, direction vector

**x**

_{1}of crank 2, direction vector

**z**

_{1}of its rotation axis, and vector R

_{EF}of link 2, will be coplanar, as shown in Figure 4, and their inner product is zero:

_{1}is zero instantaneously.

_{2}, calculated by Equation (15), that is

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}, a

_{1}, and α

_{12}) of a spatial four-bar linkage RRSS, the motion range of the angles θ

_{1}and θ

_{2}can be calculated.

#### 3.3. Kinematic Synthesis of a Spatial Four-Bar Linkage for the DBB Test

_{1}

^{(i)}, θ

_{2}

^{(i)}, i = 3, ……, n.

_{Fm}, y

_{Fm}, z

_{Fm}) of link 2 and the fixed sphere point E (x

_{Ef}, y

_{Ef}, z

_{Ef}) of the frame link 0, respectively.

_{Fm}, y

_{Fm}, z

_{Fm}) of link 2 traces a spatial trajectory Γ

_{F}: R

_{F}(i), i = 1, ……, n in the frame link 0, as link 1 and link 2 rotate θ

_{1}

^{(i)}, θ

_{2}

^{(i)}, which can be expressed as follows:

_{Fm}, y

_{Fm}, z

_{Fm}), whose trajectory Γ

_{F}is the closest spherical curve.

**R**

_{F}

^{(i)}}, a spherical surface adaptively determined by letting the maximum fitting error be minimum, whose spherical surface is the saddle spherical surface and the error is defined as the saddle spherical surface error. Based on the definition, we set up the mathematics model of a saddle spherical surface fitting as

^{(i)}(

**x**)} is the normal fitting error of the discrete point set {

**R**

_{F}

^{(i)}} and a fitting spherical surface, the optimization variables

**x**= (x

_{Ef}, y

_{Ef}, z

_{Ef}, d) are the saddle sphere center’s coordinates (x

_{Ef}, y

_{Ef}, z

_{Ef}) of link 0 and its diameter d, respectively; n is the number of discrete points {

**R**

_{F}

^{(i)}} and Δ

_{ss}is the saddle spherical surface fitting error for {

**R**

_{F}

^{(i)}},which is called the first saddle program.

_{Fm}, y

_{Fm}, z

_{Fm}) of link 2 is firstly chosen in link 2, which corresponds to a saddle sphere with its center’s coordinates (x

_{Ef}, y

_{Ef}, z

_{Ef}) and diameter d. For the moving link 2 with given discrete parameters (x

^{(i)}

_{omf}, y

^{(i)}

_{omf}, z

^{(i)}

_{omf}, θ

_{1}

^{(i)}, θ

_{2}

^{(i)}), there certainly exists a point F (x

_{Fm}, y

_{Fm}, z

_{Fm}) whose trajectory corresponds to a saddle spherical surface error, which achieves a minimum value with respect to the other points in its neighborhood, which is called the saddle sphere point of link 2. On the other hand, the spatial RRSS linkage has to meet not only the kinematics but also non-geometrical inference, etc. We present the optimization mathematic model of a saddle synthesis of RRSS linkage for the DBB test as follows:

_{ss}(

**Z**) is the objective function or the saddle spherical surface error for any point trajectory {

**R**

_{F}

^{(i)}}, obtained by Equation (17),

**Z**= (x

_{Fm}, y

_{Fm}, z

_{Fm}) are the optimization variables. δ

_{ss}is the saddle sphere point error. It is called the second saddle program. The constraint equations are given as follows according to the DBB test device and process.

**r**

_{LEF}is the vector function of the ball bar

**r**

_{S}is the edge vector function of the rotatory bale and can be written as

**R**

_{T}is the center position vector of the rotatory bale; r

_{t}is the radius of the table;

**e**

_{z}

_{2}(φ

_{s}) is the unit circle vector; φ

_{s}is the rotational angles.

_{1}, d

_{2}…… are the dimensional series of the ball bar.

^{(i)}

_{omf}, y

^{(i)}

_{omf}, z

^{(i)}

_{omf}, θ

_{1}

^{(i)}, θ

_{2}

^{(i)}, i = 1, ……, n) into Equation (19), we can obtain the parameters (x

_{Ef}, y

_{Ef}, z

_{Ef}, x

_{Fm}, y

_{Fm}, z

_{Fm}, d) of RRSS. For installing two balls on the table and the spindle, the parameters of RRSS have to be transformed into the coordinate system of the machine tool, that is

## 4. Actual Parameter Identification of Spatial Five-Bar Linkage for the DBB Test

_{1}and θ

_{2}are independently driven by two motors as theoretical input values for the RRSS linkage. In practice, both the two-axis rotary table and the ball positions may have errors, such as geometric errors and elastic deformations. These errors are revealed by the output values of the DBB test, specifically by variations in the length d* of the RRSPS linkage, as shown in Figure 6.

_{i}

^{*}and the theoretical calculating values d

_{i}be minimum during a period of DBB test, that is

_{MA}(

**x**)} is the corresponding error between the measured value d

_{i}

^{*}of the DBB test and the calculated output value d

_{i}of the RR-SPS linkage for all instances i = 1,2,…n, the optimization variables

**x**= (s

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2},θ

_{C}

_{0}, a

_{1}, α

_{12})

^{T}are the eight parameters of RR-SPS linkage, respectively; n is the number of discrete points {d

_{i}

^{*}} measured by the DBB test, and Δ

_{MA}is the saddle optimization error for {d

_{i}

^{*}},which is the unidentified errors caused by manufacturing, such as the kinematic pairs errors.

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}, a

_{1}, α

_{12}) of the mechanism. Subsequently, we can determine the installation error of the ball bar and the structural error of the machine tool under testing.

## 5. Experiments of the Double Ball Bar Test of the Two-Axis Rotary Table

#### 5.1. Measurement Motion Function and Range of the DBB Test

_{1}= 0, α

_{12}= 270°) of the A and C two-axis rotary table are known. The rotational angle θ

_{1}of the A-axis has bounds of ±110°, and the rotational angle θ

_{2}of the C-axis has bounds of 360°, as shown in Table 1.

_{1}(0–360°) while link 0 has values θ

_{1}(−110°, 110°).

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}) for the two precision balls of the RRSS linkage, as shown in Table 2, and the length d of the DBB is determined to be a nominal length of d = 300 mm.

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}) of the synthesized RRSS linkage in Table 2, we precisely locate the positions of two balls on the spindle and the worktable, respectively. And θ

_{2}(0, 360°) is the independent variable, and θ

_{1}(−110°,110°) is the dependent variable calculated by Equation (10).

_{1}and θ

_{2}into Equation (7), we can determine the position of the fixed ball point E of the DBB and the spatial trajectory formed by the moving ball point F in the fixed coordinate system, as shown in Figure 8. The blue grid represents a spherical surface with E as the center and the nominal length d of the DBB as the radius. The trajectory of F is a spatial curve on this spherical surface. The red line represents the line connecting the moving ball points and the fixed ball point of the DBB at each moment in time.

_{2}(0–360°) and the range of the A-axis θ

_{1}(−110°,110°). These results provide evidence that the novel optimization synthesis model of RRSS proposed in this paper is perfectly suitable for designing the measurement motion of DBB tests for a two-axis rotary table.

#### 5.2. The Actual Parameter Identification of the DBB Test

_{1}= 0, α

_{12}= 270°, θ

_{1}= −110~110°, θ

_{2}= 0~360°) of the A and C two-axis rotary table are given in Table 1. The mounting position coordinates (s

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}) of two precision balls are listed in Table 2.

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}) of two balls are located as per Table 2.

_{0}

^{(2)}is assigned to be s

_{0}+ 0.2 mm. This is achieved by operating the linear axis-X by CNC, while other linear axes maintain the same fixed values as in case M1.

_{0}

^{(3)}is designated to be s

_{0}− 0.2 mm. This is also achieved by operating the linear axis-X by CNC, with other linear axes maintaining the same fixed values as in case M1.

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}) of the spatial five-bar linkage RRSPS synthesized in Table 2, the installation positions of the moving and fixed ball points are calculated. The machine tool spindle’s X, Y, and Z linear axes are controlled to install the ball seat at the corresponding positions. The fixed ball point is installed on the spindle, and the moving ball point is installed on the worktable. The machine tool is controlled to move according to the parameters designed by Formulas (10) and (11), realizing the DBB test of the two-axis turntable. All these d

_{1}

^{*}values are taken as the data for case M1, shown as an orange * line in Figure 11.

_{0}

^{(2)}is adjusted to s

_{0}+ 0.2 mm. This means that while the position of the moving ball point relative to the machine tool worktable remains unchanged, only X

_{Ef}

^{(2)}of the fixed ball point’s three linear axes moves 0.2 mm in the negative direction relative to X

_{Ef}

^{(1)}. The DBB test of the two-axis rotary table is synchronously measured, and the output data d

_{2}

^{*}of the DBB test is recorded as the data for case M2, shown as a green line in Figure 11.

_{0}

^{(3)}is adjusted to s

_{0}− 0.2 mm. This means that while the position of the moving ball point relative to the machine tool worktable remains unchanged, only X

_{Ef}

^{(3)}of the fixed ball point’s three linear axes moves 0.2 mm in the positive direction relative to X

_{Ef}

^{(1)}. The DBB test of the two-axis rotary table is synchronously measured, and the output data d

_{3}

^{*}of the DBB test is taken as data for case M3, shown as a blue dashed line in Figure 11.

_{1}

^{*}}, {d

_{2}

^{*}}, and {d

_{3}

^{*}} from all three cases are substituted into mathematical model (26). These equations are solved with identical initial values for the optimization variables to obtain identification results. The actual eight parameters (s

_{0}, a

_{0}, θ

_{A}

_{0}, s

_{2}, a

_{2}, θ

_{C}

_{0}, a

_{1}, α

_{12}) of the five-bar linkage RRSPS for all three cases are completely identified by Equation (26) based on measured data case M1-M3 and are listed in Table 3.

_{0}that shows a significant difference across the three cases, that is

_{0}, s

_{0}− 0.2 s

_{0}+ 0.2. The other seven parameters (a

_{0}, θ

_{A0}, s

_{2}, a

_{2}, θ

_{C0}, a

_{1}, α

_{12}) show only slight variations. This suggests that the identification process is accurate and reliable for these parameters.

_{0}, a

_{0}, θ

_{A0}, s

_{2}, a

_{2}, θ

_{C0}, a

_{1}, α

_{12}) identified in Table 3 and the mechanism input angle parameters θ

_{1}and θ

_{2}into Equation (8), we can calculate the output d of the mechanism under the identified parameters, which is the ideal length of the DBB. A comparison of this calculated output with actual test results provides valuable insights into the accuracy of our parameter identification process and the effectiveness of our computational model. As shown in Figure 12, test results are represented by colored * marks, while the corresponding mechanism output is depicted as a black solid line.

_{MA}in the identification model (26). Its values are shown in Table 4.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Hunt, K.H. Kinematic Geometry of Mechanisms; Oxford University Press: Oxford, UK, 1978. [Google Scholar]
- McCarthy, J.M.; Soh, C.S. Geometric Design of Linkages, 2nd ed.; Interdisciplinary Applied Mathematics 11; Springer: New York, NY, USA, 2010. [Google Scholar]
- Bryan, J. A simple method for testing measuring machines and machine tools Part 1: Principles and applications. Precis. Eng.
**1982**, 4, 61–69. [Google Scholar] [CrossRef] - Bryan, J. A simple method for testing measuring machines and machine tools Part 2: Construction details. Precis. Eng.
**1982**, 4, 125–138. [Google Scholar] [CrossRef] - ISO 230-4; Test Code for Machine Tools-Part 4: Circular Tests for Numerically Controlled Machine Tools. ISO: London, UK, 2005.
- ASME B5.54; Methods for Performance Evaluation of Computer Numerically Controlled Machining Centers. American National Standards Institute: Washington, DC, USA, 2005.
- Huang, Y.B.; Fan, K.C.; Lou, Z.F.; Sun, W. A Novel Modeling of Volumetric Errors of Three-Axis Machine Tools Based on Abbe and Bryan Principles. Int. J. Mach. Tools Manuf.
**2020**, 151, 103527. [Google Scholar] [CrossRef] - Lee, K.I.; Yang, S.H. Accuracy Evaluation of Machine Tools by Modeling Spherical Deviation Based on Double Ball-Bar Measurements. Int. J. Mach. Tools Manuf.
**2013**, 75, 46–54. [Google Scholar] [CrossRef] - Lei, W.T.; Sung, M.P.; Liu, W.L. Double Ballbar Test for the Rotary Axes of Five-Axis CNC Machine Tools. Int. J. Mach. Tools Manuf.
**2007**, 47, 273–285. [Google Scholar] [CrossRef] - Lasemi, A.; Xue, D.; Gu, P. Accurate Identification and Compensation of Geometric Errors of 5-Axis CNC Machine Tools Using Double Ball Bar. Meas. Sci. Technol.
**2016**, 27, 055004. [Google Scholar] [CrossRef] - Chen, J.; Lin, S.; He, B. Geometric Error Measurement and Identification for Rotary Table of Multi-Axis Machine Tool Using Double Ballbar. Int. J. Mach. Tools Manuf.
**2014**, 77, 47–55. [Google Scholar] [CrossRef] - Zhu, S.; Ding, G.; Qin, S. Integrated Geometric Error Modeling, Identification and Compensation of CNC Machine Tools. Int. J. Mach. Tools Manuf.
**2012**, 52, 24–29. [Google Scholar] [CrossRef] - Pahk, H.J.; Kim, Y.S.; Moon, J.H. A new technique for volumetric error assessment of CNC machine tools incorporating ball bar measurement and 3D volumetric error model. Int. J. Mach. Tools Manuf.
**1997**, 37, 1583–1596. [Google Scholar] [CrossRef] - Zhong, L.; Bi, Q.; Wang, Y. Volumetric accuracy evaluation for five-axis machine tools by modeling spherical deviation based on double ball-bar kinematic test. Int. J. Mach. Tools Manuf.
**2017**, 122, 106–119. [Google Scholar] [CrossRef] - Xu, K.; Li, G.; He, K.; Tao, X. Identification of Position-Dependent Geometric Errors with Non-Integer Exponents for Linear Axis Using Double Ball Bar. Int. J. Mech. Sci.
**2020**, 170, 105326. [Google Scholar] [CrossRef] - Xia, C.; Wang, S.; Wang, S.; Ma, C.; Xu, K. Geometric error identification and compensation for rotary worktable of gear profile grinding machines based on single-axis motion measurement and actual inverse kinematic model. Mech. Mach. Theor.
**2021**, 155, 104042. [Google Scholar] [CrossRef] - Xia, H.-J.; Peng, W.-C.; Ouyang, X.-B.; Chen, X.-D.; Wang, S.-J. Identification of geometric errors of rotary axis on multi-axis machine tool based on kinematic analysis method using double ball bar. Int. J. Mach. Tools Manuf.
**2017**, 122, 161–175. [Google Scholar] [CrossRef] - Chai, X.; Zhang, N.; He, L.; Li, Q.; Ye, W. Kinematic Sensitivity Analysis and Dimensional Synthesis of a Redundantly Actuated Parallel Robot for Friction Stir Welding. Chin. J. Mech. Eng.
**2020**, 33, 1–10. [Google Scholar] [CrossRef] - Bai, S.; Li, Z.; Angeles, J. Exact Path Synthesis of RCCC Linkages for a Maximum of Nine Prescribed Positions. ASME J. Mech. Robot.
**2022**, 14, 021011. [Google Scholar] [CrossRef] - Chen, C.; Angeles, J. A novel family of linkages for advanced motion synthesis. Mech. Mach. Theory
**2008**, 43, 882–890. [Google Scholar] [CrossRef] - Zhao, C.; Guo, W. Inverted Modelling: An Effective Way to Support Motion Planning of Legged Mobile Robots. Chin. J. Mech. Eng.
**2023**, 36, 19. [Google Scholar] [CrossRef] - Huang, X.; Liu, C.; Xu, H.; Li, Q. Displacement Analysis of Spatial Linkage Mechanisms Based on Conformal Geometric Algebra. J. Mech. Eng.
**2021**, 57, 39–50. [Google Scholar] - Denavit, J.; Hartenberg, R.S. A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices. J. Appl. Mech.
**1955**, 22, 215–221. [Google Scholar] [CrossRef] - Wang, D.; Wang, W. Kinematic Differential Geometry and Saddle Synthesis of Linkages; John Wiley & Sons: Singapore, 2015. [Google Scholar]
- Wang, D.; Wang, Z.; Wu, Y.; Dong, H.; Yu, S. Invariant errors of discrete motion constrained by actual kinematic pairs. Mech. Mach. Theory
**2018**, 119, 74–90. [Google Scholar] [CrossRef]

**Figure 1.**A and C two-axis rotary table and double ball bar: (

**a**) A-C two-axis rotary table; (

**b**) the double ball bar.

**Figure 8.**Calculation trajectory of the moving ball point and the fixed ball point; (

**a**) angle measurement range as −110°~0°; (

**b**) angle measurement range as 0°~+110°.

**Figure 9.**The workspace of two-axis A and C calculated; (

**a**) angle measurement range as −110°~0°; (

**b**) angle measurement range as 0°~+110°.

Technical Parameters | Value | Unit |
---|---|---|

X/Y/Z travel | 650/650/450 | mm |

A-axis motion range | −130~+130 | ° |

C-axis motion range | 0~360 | ° |

Table diameter | Φ650 | mm |

Distance from spindle end face to table | 90~540 | mm |

Perpendicular distance of AC axis line (a1) | 0 | mm |

Included angle of AC axis line (α12) | 270 | ° |

s_{0}/mm | a_{0}/mm | θ_{A0/}° | s_{2}/mm | a_{2}/mm | θ_{C0}/° | |
---|---|---|---|---|---|---|

angle measurement range as −110°~0° | 96.310 | 352.114 | −55.000 | 80.000 | 30.000 | −161.592 |

angle measurement range as 0°~+110° | 96.310 | 352.114 | 55.000 | 80.000 | 30.000 | −161.592 |

s_{0}/mm | a_{0}/mm | θ_{A0}/° | s_{2}/mm | a_{2}/mm | θ_{C0}/° | a_{1}/mm | α_{12}/° | |
---|---|---|---|---|---|---|---|---|

Case M1 | 96.865 | 351.891 | −54.995 | 79.871 | 30.063 | −161.531 | −0.023 | 270.093 |

Case M2 | 97.124 | 351.900 | −54.996 | 79.904 | 30.080 | −161.541 | −0.021 | 270.102 |

Case M3 | 96.682 | 351.951 | −54.996 | 79.934 | 30.111 | −161.559 | −0.019 | 270.094 |

Design value | 96.310 | 352.114 | −55.000 | 80.000 | 30.000 | −161.592 | 0 | 270.000 |

Case M1 | Case M2 | Case M3 | |
---|---|---|---|

Δ_{MA}/μm | 1.587 | 1.607 | 1.674 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liao, Z.; Tang, S.; Wang, D.
A New Kinematic Synthesis Model of Spatial Linkages for Designing Motion and Identifying the Actual Dimensions of a Double Ball Bar Test Based on the Data Measured. *Machines* **2023**, *11*, 919.
https://doi.org/10.3390/machines11090919

**AMA Style**

Liao Z, Tang S, Wang D.
A New Kinematic Synthesis Model of Spatial Linkages for Designing Motion and Identifying the Actual Dimensions of a Double Ball Bar Test Based on the Data Measured. *Machines*. 2023; 11(9):919.
https://doi.org/10.3390/machines11090919

**Chicago/Turabian Style**

Liao, Zuping, Shouchen Tang, and Delun Wang.
2023. "A New Kinematic Synthesis Model of Spatial Linkages for Designing Motion and Identifying the Actual Dimensions of a Double Ball Bar Test Based on the Data Measured" *Machines* 11, no. 9: 919.
https://doi.org/10.3390/machines11090919