Skip to main content
SpringerLink
Log in

Modularized Analysis of Kinematic and Mechanical Error for Planar Linkages Composed of Class 3 and Order 3 Assur Groups

  • Regular Paper
  • Published:
International Journal of Precision Engineering and Manufacturing Aims and scope Submit manuscript

Abstract

This paper presents a modular approach for analyzing kinematics and mechanical error for planar linkages or manipulators composed of class 3 and order 3 Assur groups (AGs). Since a planar multi-link mechanism or manipulator can be decomposed into several AGs, it is only necessary to individually analyze the modules that constitute the whole linkage or manipulator. Then, an overall analysis can be obtained by combining analyses of each module. To this end, analytical expressions for kinematic and mechanical error analysis of the class 3 and order 3 AGs kinematic chains are first derived. Derived algorithms can be programmed into user subroutines in advance. Established subroutines can be further used to cope with computations necessary for the analysis by substituting known parameters. This modular approach enables users to perform kinematic and mechanical error analysis by concentrating on topology decompositions of planar linkages or manipulators. The strength of the presented method is that there is no need to re-derive closed-loop equations and perform tedious solutions to the mechanism under investigation. This paper provides three numerical examples to demonstrate the presented algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Similar content being viewed by others

Abbreviations

\({\text{A(A}}_{{\text{x}}} {\text{, A}}_{{\text{y}}} {)}\) :

Positions of external joints A

\({\dot{\text{A}}}(\dot{{\text{A}}_{{\text{x}}} } , \dot{{\text{A}}_{{\text{y}}}})\) :

Velocities of external joints A

\({\ddot{\text{A}}}(\ddot{{\text{A}}_{{\text{x}}} },\ddot{{\text{A}}_{{\text{y}}} })\) :

Accelerations of external joints A

\({\text{A}}_{{1}} {\text{(x, y),B}}_{1} \left( {{\text{x}}_{{{\text{B1}}}} {\text{, y}}_{{{\text{B1}}}} } \right),{\text{C}}_{1} \left( {{\text{x}}_{{{\text{C1}}}} {\text{, y}}_{{{\text{C1}}}} } \right)\) :

Positions of the joint on ternary link

\({\text{B(B}}_{{\text{x}}} {\text{, B}}_{{\text{y}}} {)}\) :

Positions of external joints B

\({\text{B}}_{1} {\text{(B}}_{{{\text{1x}}}} {\text{, B}}_{{{\text{1y}}}} {)}\) :

Positions of external joints B1

\({\dot{\text{B}}} (\dot{{\text{B}}_{{\text{x}}}},\dot{{\text{B}}_{{\text{y}}}})\) :

Velocities of external joints B

\({\ddot{\text{B}}}(\ddot {{\text{B}}_{{\text{x}}} } ,\ddot{{\text{B}}_{{\text{y}}}})\) :

Accelerations of external joints B

\({\text{C(C}}_{{\text{x}}} {\text{, C}}_{{\text{y}}} {)}\) :

Positions of external joints C

\({\text{C}}_{1} {\text{(C}}_{{{\text{1x}}}} {\text{, C}}_{{{\text{1y}}}} {)}\) :

Positions of external joints C1

\({\dot{\text{C}}} (\dot {{\text{C}}_{{\text{x}}} } ,\dot{{\text{C}}_{{\text{y}}} } )\) :

Velocities of external joints C

\({\ddot{\text{C}(}}\ddot{{\text{C}}_{{\text{x}}} },\ddot{{\text{C}}_{{\text{y}}}})\) :

Accelerations of external joints C

\({\text{I}}\) :

Configuration of the AG

\({\text{L}}_{0}\) :

Length of the input link

\({\text{L}}_{1} ,{\text{L}}_{2} ,{\text{L}}_{3}\) :

Lengths of the binary links

\({\text{L}}_{11} ,{\text{L}}_{12} ,{\text{L}}_{21} ,{\text{L}}_{22} ,{\text{L}}_{31} ,{\text{L}}_{32}\) :

Lengths of the accessory triangle of 3RP AG

\({\text{l}}_{{{\text{AB}}}} ,{\text{l}}_{{{\text{BC}}}} ,{\text{l}}_{{{\text{AC}}}}\) :

Lengths of the ternary link

\({\text{O}}_{1} ,{\text{O}}_{2} ,{\text{O}}_{3}\) :

Offset distance of prismatic sliding path of 3PR AG

\({\text{S}}_{1} ,{\text{S}}_{2} ,{\text{S}}_{3}\) :

Strokes of the prismatic joints

\({\text{S}}_{1}^{{\prime}} ,{\text{S}}_{2}^{{\prime}} ,{\text{S}}_{3}^{{\prime}}\) :

Accessory strokes of 3PR AG

\(\dot {{\text{S}}_{{1}} } ,\dot{{\text{S}}_{2} } ,\dot {{\text{S}}_{3} }\) :

Linear velocities of the prismatic joints

\(\dot {{\text{S}}_{{1}}^{{\prime}} } ,\dot{{\text{S}}_{2}^{{\prime}}},\dot {{\text{S}}_{3}^{{\prime}} }\) :

Linear velocities of the accessory strokes of 3PR AG

\(\ddot{{\text{S}}_{1} },\ddot{{\text{S}}_{2} } ,\ddot {{\text{S}}_{3} }\) :

Linear acceleration of the prismatic joints

\(\ddot {{\text{S}}_{{1}}^{{\prime}} },\ddot {{\text{S}}_{2}^{{\prime}} },\ddot {{\text{S}}_{3}^{{\prime}} }\) :

Linear acceleration of the accessory strokes of 3PR AG

\(({\dot{\text{x}}}, {\dot{\text{y}}})\) :

3RR AG’s obtain velocity

\(({\ddot{\text{x}}},{\ddot{\text{y}}})\) :

3RR AG’s obtain acceleration

\(\upalpha ,\upalpha _{1} ,\upalpha _{2} ,\upalpha _{3}\) :

Angle of the ternary link

\(\upbeta _{1} ,\upbeta _{2} ,\upbeta _{3}\) :

Orientation of the prismatic sliding path of 3PR AG

\(\Delta {\text{A (}}\Delta {\text{A}}_{{\text{x}}} {,}\Delta {\text{A}}_{{\text{y}}} {),}\Delta {\text{B(}}\Delta {\text{B}}_{{\text{x}}} {,}\Delta {\text{B}}_{{\text{y}}} {),}\Delta {\text{C(}}\Delta {\text{C}}_{{\text{x}}} {,}\Delta {\text{C}}_{{\text{y}}} {)}\) :

Assembly errors of the external joints

\(\Delta {\text{D}}\) :

Translational error of the ternary link

\(\Delta {\text{L}}_{0}\) :

Manufacturing errors of the input link

\(\Delta {\text{L}}_{{1}} {,}\Delta {\text{L}}_{2} {,}\Delta {\text{L}}_{3}\) :

Manufacturing errors of the binary link length

\(\Delta {\text{l}}_{{{\text{AB}}}} {,}\Delta {\text{l}}_{{{\text{BC}}}} {,}\Delta {\text{l}}_{{{\text{AC}}}}\) :

Manufacturing errors of the ternary link length

\(\Delta {\text{O}}_{1} ,\Delta {\text{O}}_{2} ,\Delta {\text{O}}_{3}\) :

Manufacturing errors of the sliding offset of 3PR AG

\(\Delta {\text{S}}_{1} ,\Delta {\text{S}}_{2} ,\Delta {\text{S}}_{3}\) :

Stroke deviation of the prismatic joints

\(\Delta {\text{S}}_{1}^{{\prime}} ,\Delta {\text{S}}_{2}^{{\prime}} ,\Delta {\text{S}}_{3}^{{\prime}}\) :

Deviations of the accessory strokes of 3PR AG

\({(}\Delta {\text{x,}}\Delta {\text{y),}}\left( {\Delta {\text{x}}_{{{\text{B1}}}} {,}\Delta {\text{y}}_{{{\text{B1}}}} } \right),\left( {\Delta {\text{x}}_{{{\text{C1}}}} {,}\Delta {\text{y}}_{{{\text{C1}}}} } \right)\) :

Positional deviation of the ternary link

\(\Delta \upalpha _{1} ,\Delta \upalpha _{2} ,\Delta \upalpha _{3}\) :

Deviations of the interior angle of the ternary link

\(\Delta\upbeta _{1} ,\Delta\upbeta _{2} ,\Delta\upbeta _{3}\) :

Manufacturing errors of the sliding orientation of 3PR AG

\({\Delta }\upphi\) :

Orientation deviation of the ternary link

\(\Delta\uptheta _{1} ,\Delta\uptheta _{2} ,\Delta\uptheta _{3}\) :

Orientation errors of the external prismatic joint

\(\Delta\upgamma _{1} ,\Delta\upgamma _{2} ,\Delta\upgamma _{3}\) :

Deviations of the interior angle of the 3PR AG

\(\upphi\) :

Orientation of the ternary link

\({\dot{\upphi}}\) :

Angular velocity of the ternary link

\({\ddot{\upphi }}\) :

Angular acceleration of the ternary link

\(\uptheta _{1} ,\uptheta _{2} ,\uptheta _{3}\) :

Orientation of the prismatic pairs of 3PR AG

\(\uptheta _{{\text{A}}} ,\uptheta _{{\text{B}}} ,\uptheta _{{\text{C}}}\) :

Orientation of point A, B, and C

\(\dot {\uptheta _{{1}} },\dot {\uptheta _{2} },\dot {\uptheta _{3} }\) :

Angular velocity of the prismatic pairs of 3PR AG

\(\ddot {\uptheta _{{1}} },\ddot {\uptheta _{2} },\ddot {\uptheta _{3} }\) :

Angular acceleration of the prismatic pairs of 3PR AG

\(\upgamma _{1} ,\upgamma _{2} ,\upgamma _{3}\) :

Interior angle of the 3PR AG

References

  1. Gosselin, C. M. & Sefrioui, J. (1991). Polynomial solutions for the direct kinematic problem of planar three-degree-of-freedom parallel manipulators. In Proceedings of the IEEE international conference on robotics and automation (pp. 1124–1129).

  2. Gosselin, C. M., & Merlet, J. P. (1994). The direct kinematics of planar parallel manipulators: Special architectures and number of solutions. Mechanism and Machine Theory, 29(8), 1083–1097.

    Article  Google Scholar 

  3. Merlet, J. P. (1996). Direct kinematics of planar parallel manipulators. In Proceedings of the IEEE international conference on robotics and automation (pp. 3744–3749).

  4. Hayes, M. J. D., Zsombor-Murray, P. J., & Chen, C. (2004). Unified kinematics analysis of general planar parallel manipulators. Journal of Mechanical Design, 126, 866–874.

    Article  Google Scholar 

  5. Gosselin, C., & Angeles, J. (1988). The optimum kinematics design of a planar three-degree-of-freedom parallel manipulator. Journal of Mechanisms, Transmissions, and Automation in Design, 110, 35–41.

    Article  Google Scholar 

  6. Kumar, V. (1992). Characterization of workspaces of parallel manipulators. Journal of Mechanical Design, 114, 368–375.

    Article  Google Scholar 

  7. Merlet, J. P., Gosselin, C. M., & Mouly, N. (1998). Workspaces of planar manipulators. Mechanism and Machine Theory, 33(1–2), 7–20.

    Article  MathSciNet  MATH  Google Scholar 

  8. Voglewede, P., & Ebert-Uphoff, I. (2004). Application of workspace generation techniques to determine the unconstrained motion of parallel manipulators. Journal of Mechanical Design, 126, 283–290.

    Article  Google Scholar 

  9. Oetomo, D., Liaw, H. C., Alici, G., & Shirinzadeh, B. (2006). Direct kinematics and analytical solution to 3RRR parallel planar mechanisms. In IEEE international conference on control, automation, robotics and vision (pp. 2251–2256). Singapore.

  10. Staicu, S. (2008) Kinematics of the 3-RRR planar parallel robot. UPB Scientific Bulletin, Series D, 70(2).

  11. Zhang, X., Zhang, X., & Chen, Z. (2014). Dynamic analysis of a 3-RRR parallel mechanism with multiple clearance joints. Mechanism and Machine Theory, 78, 105–115.

    Article  Google Scholar 

  12. Zhang, X., Mills, J. K., & Cleghorn, W. L. (2007). Dynamic modeling and experimental validation of a 3-PRR parallel manipulator with flexible intermediate links. Journal of Intelligent and Robotic Systems, 50, 323–340.

    Article  MATH  Google Scholar 

  13. Ur-Rehman, R., Caro, S., Chablat, D., & Wenger, P. (2010). Multiobjective design optimization of 3-PRR planar parallel manipulators. In HAL 20th CIRP design conference (Nantes, France, April 2010).

  14. Gosselin, C. M., & Jean, M. (1996). Determination of the workspace of planar parallel manipulators with joint limits. Robotics and Autonomous Systems, 17, 129–138.

    Article  Google Scholar 

  15. Kong, X., & Gosselin, C. M. (2001). Forward displacement analysis of third-class analytic 3-RPR planar parallel manipulators. Mechanism and Machine Theory, 36, 1009–1018.

    Article  MATH  Google Scholar 

  16. Collins, C. L. (2002). Forward kinematics of planar parallel manipulators in the Clifford algebra of P2. Mechanism and Machine Theory, 37, 799–813.

    Article  MathSciNet  MATH  Google Scholar 

  17. Binaud, N., Caro, S., Bai, S., & Wenger, P. (2010). Comparison of 3-PPR parallel planar manipulators based on their sensitivity to joint clearances. In: IEEE/RSJ international conference on intelligent robots and systems, Taipei, Taiwan.

  18. Staicu, S. (2009). Dynamics of the 3-PRP planar parallel robot. In Revue Roumaine des Sciences Techniques – Serie de Mecanique Appliquee, Tome (vol. 54, No. 2, pp. 125–142). Bucarest.

  19. Mitsi, S., Bouzakis, K. D., Mansour, G., & Popescu, I. (2011). A method for forward displacement analysis of 3-RRP and 3-PRP planar parallel manipulators. In The Romanian Review Precision Mechanics, Optics and Mechantronics vol. 39.

  20. Zahedi, A., Behzadnia, H., Ghanbari, H., & Tabatabaei, S. H. (2016). Kinematic analysis of the triangle-star robot with telescope arm three kinematics chains as T-S robot (3-PRP). Recent Advances in Robotics Systems, Intech.

  21. Assur, L. V. (1952). Investigation of plane hinged mechanisms with lower pairs from the point of view of their structure and classification. In I. I. Artobolevskii (Ed.) Academic of Sciences, USSR.

  22. Hahn, E. & Shai, O. (2016). The unique engineering properties of AGs/Graph, Assur kinematic chains, Baranov trusses and parallel robots. In ASME IDETC/ CIE, Charlotte, North Carolina, USA.

  23. Mitsi, S., Bouzakis, K. D., Mansour, G., & Popescu, I. (2003). Position analysis in polynomial form of planar mechanisms with AGs of Class 3 including revolute and prismatic joints. Mechanism and Machine Theory, 38, 1325–1344.

    Article  MathSciNet  MATH  Google Scholar 

  24. Chung, W. Y. (2005). The position analysis of Assur kinematics chain with five links. Mechanism and Machine Theory, 40, 1015–1029.

    Article  MATH  Google Scholar 

  25. Chung, W. Y. (2007). Double configurations of five-link Assur kinematic chain and stationary configurations of Stephenson six-bar. Mechanism and Machine Theory, 42, 1653–1662.

    Article  MATH  Google Scholar 

  26. Mitsi, S., Bouzakis, K. D., Mansour, G., & Popescu, I. (2006). Position analysis in polynomial form of class 3 and order 3 with two or three prismatic joints. Journal Mechanisms and Manipulators, 5(2), 31–36.

    MATH  Google Scholar 

  27. Mitsi, S., Bouzakis, K. D., Mansour, G., & Popescu, I. (2008). Position analysis in polynomial form of class-three AGs with two or three prismatic joints. Mechanism and Machine Theory, 43, 1401–1415.

    Article  MATH  Google Scholar 

  28. Hartenberg, R., & Danavit, J. (1964). Kinematic synthesis of linkages. McGraw-Hill.

    Google Scholar 

  29. Garrett, R. E., & Hall, A. S. (1969). Effect of tolerance and clearance in linkage design. Journal of Engineering for Industry, 9(1), 198–202.

    Article  Google Scholar 

  30. Kolhatkar, S. A., & Yajnik, K. S. (1970). The effects of play in the joints of a function-generating mechanism. Journal of Mechanisms, 5(4), 521–532.

    Article  Google Scholar 

  31. Lakshminarayan, K., & Narayanamurthi, R. G. (1971). On the analysis of the effect of tolerances in linkages. Journal of Mechanisms, 6(1), 59–67.

    Article  Google Scholar 

  32. Chakraborty, J. (1975). Synthesis of mechanical error in linkages. Mechanism and Machine Theory, 10(2–3), 155–165.

    Article  Google Scholar 

  33. Choubey, M., & Rao, A. C. (1982). Synthesizing linkages with minimal structural and mechanical error based upon tolerance allocation. Mechanism and Machine Theory, 17(2), 91–97.

    Article  Google Scholar 

  34. Sharfi, O. M. A., & Smith, M. R. (1983). A simple method for the allocation of appropriate tolerances and clearances in linkage mechanisms. Mechanism and Machine Theory, 18(2), 123–129.

    Article  Google Scholar 

  35. Chen, F. C., & Huang, H. H. (2006). Taguchi-fuzzy-based approach for the sensitivity analysis of a four-bar function generator. Proceedings of the Institution of Mechanical Engineering, Part C, Journal of Mechanical Engineering Science, 220(9), 1413–1421.

    Article  Google Scholar 

  36. Chang, W. T., Lee, W. I., & Hsu, K. L. (2021). Analysis and experimental verification of mechanical errors in nine-link type double-toggle mold/die clamping mechanisms. Applied Sciences, 11(2), 832.

    Article  Google Scholar 

  37. Jawale, H. P., & Jaiswal, A. (2018). Investigation of mechanical error in four-bar mechanism under the effects of link tolerance. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40(383)

  38. Jaiswal, A., & Jawale, H. P. (2022). Influence of tolerances on error estimation in P3R and 4R planar mechanisms. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 44.

  39. Dhande, S. G., & Chakraborty, J. (1973). Analysis and synthesis of mechanical error in linkages—A stochastic approach. Journal of Engineering for Industry, 95(3), 672–676.

    Article  Google Scholar 

  40. Sukhija, R. P., & Rao, A. C. (1986). Mechanical error synthesis of path generating mechanism using reliability index. Transactions of the Canadian Society for Mechanical Engineering, 10(2), 85–90.

    Article  Google Scholar 

  41. Sukhija, R. P., & Rao, A. C. (1987). Synthesis of path-generating mechanisms and tolerance allocation using information theory. Journal of Mechanisms, Transmissions, and Automation in Design, 109(3), 366–371.

    Article  Google Scholar 

  42. Mallik, A. K., & Dhande, S. G. (1987). Analysis and synthesis of mechanical error in path-generating linkages using a stochastic approach. Mechanism and Machine Theory, 22(2), 115–123.

    Article  Google Scholar 

  43. Erkaya, S., & Uzmay, I. (2009). Determining link parameters using genetic algorithm in mechanisms with joint clearance. Mechanism and Machine Theory, 44(1), 222–234.

    Article  MATH  Google Scholar 

  44. Chatterjee, G. B., & Mallik, A. K. (1987). Mechanical error of a four-bar linkage coupler curve. Mechanism and Machine Theory, 22(1), 85–88.

    Article  Google Scholar 

  45. Mutawe, S., Al-Smadi, Y. M., & Sodhi, R. S. (2011). Planar four-bar path generation considering worst case joint tolerances. In World congress on engineering and computer science, San Francisco, United States.

  46. Chang, W. T., & Wu, L. I. (2013). Tolerance analysis and synthesis of cam-modulated linkages. Mathematical and Computer Modelling, 57(3–4), 641–660.

    Article  MathSciNet  MATH  Google Scholar 

  47. Ting, K. L., & Long, Y. F. (1996). Performance quality and tolerance sensitivity of mechanisms. Journal of Mechanical Design, 118(1), 144–150.

    Article  Google Scholar 

  48. Ch, J. U., Youn, I., Choi, K., & Lee, Y. J. (2011). Human-following robot using tether steering. International Journal of Precision Engineering and Manufacturing, 12(5), 899–906.

    Article  Google Scholar 

  49. Zhang, X., & Xu, Q. (2018). Design and testing of a new 3-DOF spatial flexure parallel micropositioning stage. International Journal of Precision Engineering and Manufacturing, 19(1), 109–118.

    Article  MathSciNet  Google Scholar 

  50. Wang, S. M., Lee, C. Y., Gunawan, H., & Yeh, C. C. (2022). On-line error-matching measurement and compensation method for a precision machining production line. International Journal of Precision Engineering and Manufacturing Green Technology, 9(2), 493–505.

    Article  Google Scholar 

  51. Hsu, K. L., & Chung, J. Y. (2021). A modular method for mechanical error analysis of planar linkages composed of class II AGs kinematic chain. Journal of Mechanisms and Robotics, 14(1).

Download references

Acknowledgements

The authors would like to acknowledge the support of the department of Mechanical Engineering at National Taiwan University. In addition, the author offers his grateful thanks to the dedicated referees of this work, whose collective comments form the fundamentals of the research. Most importantly, the Young Scholar Fellowship Program supported by the Ministry of Science and Technology of Taiwan (MOST 110-2636-E-002-023 and 111-2222-E-002-002-MY3) encourages the author to fearlessly devote to his research.

Funding

This work was supported by the Ministry of Science and Technology of Taiwan (MOST 110-2636-E-002-023 and MOST 111-2222-E-002-002-MY3).

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, National Taiwan University, Taipei, 10617, Taiwan

    Kuan-Lun Hsu & Ching-Hsiang Chang

Authors
  1. Kuan-Lun Hsu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ching-Hsiang Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by C-HC. The first draft of the manuscript was written by K-LH and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Kuan-Lun Hsu.

Ethics declarations

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

The following equations present coefficients defined in Eqs. (4)–(6) in Sect. 2.1.

$${\text{C}}_{2} = - 2{\text{A}}_{{\text{x}}}$$
(A1)
$${\text{C}}_{3} = - 2{\text{A}}_{{\text{y}}}$$
(A2)
$${\text{C}}_{4} = {\text{L}}_{1}^{2} - {\text{A}}_{{\text{x}}}^{2} - {\text{A}}_{{\text{y}}}^{2}$$
(A3)
$${\text{C}}_{5} = 2\left( {l_{{{\text{AB}}}} {\text{cos}}\upphi - {\text{B}}_{{\text{x}}} } \right)$$
(A4)
$${\text{C}}_{6} = 2\left( {l_{{{\text{AB}}}} {\text{sin}}\upphi - {\text{B}}_{{\text{y}}} } \right)$$
(A5)
$$\begin{aligned} {\text{C}}_{7} & = {\text{L}}_{2}^{2} - \left( {l_{{{\text{AB}}}} {\text{cos}}\upphi - {\text{B}}_{{\text{x}}} } \right)^{2} \\ & \quad - \left( {l_{{{\text{AB}}}} {\text{sin}}\upphi - {\text{B}}_{{\text{y}}} } \right)^{2} \\ \end{aligned}$$
(A6)
$${\text{C}}_{8} = 2\left[ {l_{{{\text{AC}}}} \cos \left( {\upphi + {\upalpha }} \right) - {\text{C}}_{{\text{x}}} } \right]$$
(A7)
$${\text{C}}_{9} = 2\left[ {l_{{{\text{AC}}}} \sin \left( {\upphi + {\upalpha }} \right) - {\text{C}}_{{\text{y}}} } \right]$$
(A8)
$$\begin{aligned} {\text{C}}_{10} & = {\text{L}}_{3}^{2} - \left[ {l_{{{\text{AC}}}} \cos \left( {\upphi + \upalpha } \right) - {\text{C}}_{{\text{x}}} } \right]^{2} \\ & \quad - \left[ {l_{{{\text{AC}}}} \sin \left( {\upphi + \upalpha } \right) - {\text{C}}_{{\text{y}}} } \right]^{2} \\ \end{aligned}$$
(A9)

The following equations present coefficients defined in Eq. (13) in Sect. 2.1.

$${\text{D}}_{1} = 2\left( {{\text{x}} - {\text{A}}_{{\text{x}}} } \right)$$
(A10)
$${\text{D}}_{2} = 2\left( {{\text{y}} - {\text{A}}_{{\text{y}}} } \right)$$
(A11)
$${\text{D}}_{3} = 0$$
(A12)
$${\text{D}}_{4} = {\text{D}}_{1} \dot {{\text{A}}_{{\text{x}}} } + {\text{D}}_{2} \dot {{\text{A}}_{{\text{y}}} }$$
(A13)
$${\text{D}}_{5} = 2\left( {{\text{x}} + l_{{{\text{AB}}}} {\text{cos}}\upphi - {\text{B}}_{{\text{x}}} } \right)$$
(A14)
$${\text{D}}_{6} = 2\left( {{\text{y}} + l_{{{\text{AB}}}} {\text{sin}}\upphi - {\text{B}}_{{\text{y}}} } \right)$$
(A15)
$${\text{D}}_{7} = - {\text{D}}_{5} l_{{{\text{AB}}}} {\text{sin}}\upphi + {\text{D}}_{6} l_{{{\text{AB}}}} {\text{cos}}\upphi$$
(A16)
$${\text{D}}_{8} = {\text{D}}_{5} \dot {{\text{B}}_{{\text{x}}} } + {\text{D}}_{6} \dot {{\text{B}}_{{\text{y}}} }$$
(A17)
$${\text{D}}_{9} = 2\left[ {{\text{x}} + l_{{{\text{AC}}}} \cos \left( {\upphi + \upalpha } \right) - {\text{C}}_{{\text{x}}} } \right]$$
(A18)
$${\text{D}}_{10} = 2\left[ {{\text{y}} + l_{{{\text{AC}}}} \sin \left( {\upphi + \upalpha } \right) - {\text{C}}_{{\text{y}}} } \right]$$
(A19)
$$\begin{aligned} {\text{D}}_{11} & = - {\text{D}}_{9} l_{{{\text{AC}}}} \sin \left( {\upphi + \upalpha } \right) \\ & \quad + {\text{D}}_{10} l_{{{\text{AC}}}} {\text{cos}}\left( {\upphi + \upalpha } \right) \\ \end{aligned}$$
(A20)
$${\text{D}}_{12} = {\text{D}}_{9} \dot {{\text{C}}_{{\text{x}}} } + {\text{D}}_{10} \dot {{\text{C}}_{{\text{y}}} }$$
(A21)

The following equations present coefficients defined in Eq. (14) in Sect. 2.1.

$${\text{E}}_{1} = 2\left( {{\text{x}} - {\text{A}}_{{\text{x}}} } \right)$$
(A22)
$${\text{E}}_{2} = 2\left( {{\text{y}} - {\text{A}}_{{\text{y}}} } \right)$$
(A23)
$${\text{E}}_{3} = 0$$
(A24)
$$\begin{aligned} {\text{E}}_{4} & = {\text{E}}_{1} \dot {{\text{A}}_{{\text{x}}} } + {\text{E}}_{2} \dot {{\text{A}}_{{\text{y}}} } \\ & \quad - 2\left( {{\dot{\text{x}}} - \dot {{\text{A}}_{{\text{x}}} } } \right)^{2} - 2\left( {{\dot{\text{y}}} - \dot {{\text{A}}_{{\text{y}}} } } \right)^{2} \\ \end{aligned}$$
(A25)
$${\text{E}}_{5} = 2\left( {{\text{x}} + l_{{{\text{AB}}}} {\text{cos}}\upphi - {\text{B}}_{{\text{x}}} } \right)$$
(A26)
$${\text{E}}_{6} = 2\left( {{\text{y}} + l_{{{\text{AB}}}} {\text{sin}}\upphi - {\text{B}}_{{\text{y}}} } \right)$$
(A27)
$${\text{E}}_{7} = - {\text{E}}_{5} l_{{{\text{AB}}}} {\text{sin}}\upphi + {\text{E}}_{6} l_{{{\text{AB}}}} {\text{cos}}\upphi$$
(A28)
$$\begin{aligned} {\text{E}}_{8} & = {\text{E}}_{5} {\dot{\upphi}} ^{2} l_{{{\text{AB}}}} \cos\upphi + {\text{E}}_{4} \dot {{\text{B}}_{{\text{x}}} } \\ & \quad + {\text{E}}_{6} {\dot{\upphi}} ^{2} l_{{{\text{AB}}}} \sin\upphi + {\text{E}}_{5} \dot {{\text{B}}_{{\text{y}}} } \\ & \quad - 2\left( {{\dot{\text{x}}} - {\dot{\upphi}} l_{{{\text{AB}}}} \sin\upphi - \dot {{\text{B}}_{{\text{x}}} } } \right)^{2} \\ & \quad - 2\left( {{\dot{\text{y}}} - {\dot{\upphi}} {\text{l}}_{{{\text{AB}}}} \cos\upphi - \dot {{\text{B}}_{{\text{y}}} } } \right)^{2} \\ \end{aligned}$$
(A29)
$${\text{E}}_{9} = 2\left[ {{\text{x}} + l_{{{\text{AC}}}} \cos \left( {\upphi + \upalpha } \right) - {\text{C}}_{{\text{x}}} } \right]$$
(A30)
$${\text{E}}_{10} = 2\left[ {{\text{y}} + l_{{{\text{AC}}}} \sin \left( {\upphi + \upalpha } \right) - {\text{C}}_{{\text{y}}} } \right]$$
(A31)
$$\begin{aligned} {\text{E}}_{11} & = - {\text{E}}_{9} l_{{{\text{AC}}}} \sin \left( {\upphi + \upalpha } \right) \\ & \quad + {\text{E}}_{10} l_{{{\text{AC}}}} {\text{cos}}\left( {\upphi + \upalpha } \right) \\ \end{aligned}$$
(A32)
$$\begin{aligned} {\text{E}}_{12} & = {\text{E}}_{9} {\dot{\upphi}} ^{2} l_{{{\text{AC}}}} \cos \left( {\upphi + \upalpha } \right) + {\text{E}}_{8} \dot {{\text{C}}_{{\text{x}}} } \\ & \quad + {\text{E}}_{10} {\dot{\upphi}} ^{2} l_{{{\text{AC}}}} \sin \left( {\upphi + \upalpha } \right) + {\text{E}}_{9} \dot {{\text{C}}_{{\text{y}}} } \\ & \quad - 2\left[ {{\dot{\text{x}}} - {\dot{\upphi}} {\text{l}}_{{{\text{AC}}}} \sin \left( {\upphi + \upalpha } \right) - \dot {{\text{C}}_{{\text{x}}} } } \right]^{2} \\ & \quad - 2\left[ {{\dot{\text{y}}} - {\dot{\upphi}} {\text{l}}_{{{\text{AC}}}} \cos \left( {\upphi + \upalpha } \right) - \dot {{\text{C}}_{{\text{y}}} } } \right]^{2} \\ \end{aligned}$$
(A33)

Appendix B

The following equations present coefficients defined in Eqs. (22)–(24) in Sect. 2.2.

$$\begin{aligned} {\text{F}}_{1} & = - 2[\left( {{\text{B}}_{{\text{x}}} - {\text{A}}_{{\text{x}}} - {\text{L}}_{2} \sin\uptheta _{2} - {\text{L}}_{1} \sin\uptheta _{1} } \right)\cos\uptheta _{1} \\ & \quad + \left( {{\text{B}}_{{\text{y}}} - {\text{A}}_{{\text{y}}} + {\text{L}}_{2} \cos\uptheta _{2} + {\text{L}}_{1} \cos\uptheta _{1} } \right)\sin\uptheta _{1} ] \\ \end{aligned}$$
(B1)
$$\begin{aligned} {\text{F}}_{2} & = 2[({\text{B}}_{{\text{x}}} - {\text{A}}_{{\text{x}}} - {\text{L}}_{2} \sin\uptheta _{2} - {\text{L}}_{1} \sin\uptheta _{1} )\cos\uptheta _{2} \\ & \quad + ({\text{B}}_{{\text{y}}} - {\text{A}}_{{\text{y}}} + {\text{L}}_{2} \cos\uptheta _{2} + {\text{L}}_{1} \cos\uptheta _{1} )\sin\uptheta _{2} ] \\ \end{aligned}$$
(B2)
$${\text{F}}_{3} = - 2{\text{cos}}\left( {\uptheta _{1} +\uptheta _{2} } \right)$$
(B3)
$$\begin{aligned} {\text{F}}_{4} & = \left( {{\text{B}}_{{\text{x}}} - {\text{A}}_{{\text{x}}} - {\text{L}}_{2} \sin\uptheta _{2} - {\text{L}}_{1} \sin\uptheta _{1} } \right)^{2} \\ & \quad + \left( {{\text{B}}_{{\text{y}}} - {\text{A}}_{{\text{y}}} + {\text{L}}_{2} \cos\uptheta _{2} + {\text{L}}_{1} \cos\uptheta _{1} } \right)^{2} \\ & \quad - l_{{{\text{AB}}}}^{2} \\ \end{aligned}$$
(B4)
$$\begin{aligned} {\text{F}}_{5} & = - 2[({\text{C}}_{{\text{x}}} - {\text{B}}_{{\text{x}}} - {\text{L}}_{3} \sin\uptheta _{3} + {\text{L}}_{2} \sin\uptheta _{2} )\cos\uptheta _{2} \\ & \quad + ({\text{C}}_{{\text{y}}} - {\text{B}}_{{\text{y}}} + {\text{L}}_{3} \cos\uptheta _{3} - {\text{L}}_{2} \cos\uptheta _{2} )\sin\uptheta _{2} ] \\ \end{aligned}$$
(B5)
$$\begin{aligned} {\text{F}}_{6} & = 2[({\text{C}}_{{\text{x}}} - {\text{B}}_{{\text{x}}} - {\text{L}}_{3} \sin\uptheta _{3} + {\text{L}}_{2} \sin\uptheta _{2} )\cos\uptheta _{3} \\ & \quad + ({\text{C}}_{{\text{y}}} - {\text{B}}_{{\text{y}}} + {\text{L}}_{3} \cos\uptheta _{3} - {\text{L}}_{2} \cos\uptheta _{2} )\sin\uptheta _{3} ] \\ \end{aligned}$$
(B6)
$${\text{F}}_{7} = - 2{\text{cos}}\left( {{\uptheta }_{2} - {\uptheta }_{3} } \right)$$
(B7)
$$\begin{aligned} {\text{F}}_{8} & = \left( {{\text{C}}_{{\text{x}}} - {\text{B}}_{{\text{x}}} - {\text{L}}_{3} \sin\uptheta _{3} + {\text{L}}_{2} \sin\uptheta _{2} } \right)^{2} \\ & \quad + \left( {{\text{C}}_{{\text{y}}} - {\text{B}}_{{\text{y}}} + {\text{L}}_{3} \cos\uptheta _{3} - {\text{L}}_{2} \cos\uptheta _{2} } \right)^{2} \\ & \quad - l_{{{\text{BC}}}}^{2} \\ \end{aligned}$$
(B8)
$$\begin{aligned} {\text{F}}_{9} & = 2[({\text{A}}_{{\text{x}}} - {\text{C}}_{{\text{x}}} + {\text{L}}_{1} \sin\uptheta _{1} + {\text{L}}_{3} \sin\uptheta _{3} )\cos\uptheta _{1} \\ & \quad + ({\text{A}}_{{\text{y}}} - {\text{C}}_{{\text{y}}} - {\text{L}}_{1} \cos\uptheta _{1} - {\text{L}}_{3} \cos\uptheta _{3} )\sin\uptheta _{1} ] \\ \end{aligned}$$
(B9)
$$\begin{aligned} {\text{F}}_{10} & = - 2[({\text{A}}_{{\text{x}}} - {\text{C}}_{{\text{x}}} + {\text{L}}_{1} \sin\uptheta _{1} + {\text{L}}_{3} \sin\uptheta _{3} )\cos\uptheta _{3} \\ & \quad + ({\text{A}}_{{\text{y}}} - {\text{C}}_{{\text{y}}} - {\text{L}}_{1} \cos\uptheta _{1} - {\text{L}}_{3} \cos\uptheta _{3} )\sin\uptheta _{3} ] \\ \end{aligned}$$
(B10)
$${\text{F}}_{11} = - 2{\text{cos}}\left( {\uptheta _{1} -\uptheta _{3} } \right)$$
(B11)
$$\begin{aligned} {\text{F}}_{12} & = \left( {{\text{A}}_{{\text{x}}} - {\text{C}}_{{\text{x}}} + {\text{L}}_{1} \sin\uptheta _{1} + {\text{L}}_{3} \sin\uptheta _{3} } \right)^{2} \\ & \quad + \left( {{\text{A}}_{{\text{y}}} - {\text{C}}_{{\text{y}}} - {\text{L}}_{1} \cos\uptheta _{1} - {\text{L}}_{3} \cos\uptheta _{3} } \right)^{2} \\ & \quad - l_{{{\text{AC}}}}^{2} \\ \end{aligned}$$
(B12)

The following equations present coefficients defined in Eq. (30) in Sect. 2.2.

$$\begin{aligned} {\text{G}}_{1} & = - 2\left( {{\text{B}}_{{\text{x}}} - {\text{A}}_{{\text{x}}} + {\text{S}}_{2} \cos\uptheta _{2} - {\text{S}}_{1} \cos\uptheta _{1} - {\text{L}}_{2} \sin\uptheta _{2} - {\text{L}}_{1} \sin\uptheta _{1} } \right)\cos\uptheta _{1} \\ & \quad - 2\left( {{\text{B}}_{{\text{y}}} - {\text{A}}_{{\text{y}}} + {\text{S}}_{2} \sin\uptheta _{2} - {\text{S}}_{1} \sin\uptheta _{1} + {\text{L}}_{2} \cos\uptheta _{2} + {\text{L}}_{1} \cos\uptheta _{1} } \right)\sin\uptheta _{1} \\ \end{aligned}$$
(B13)
$$\begin{aligned} {\text{G}}_{2} & = 2\left( {{\text{B}}_{{\text{x}}} - {\text{A}}_{{\text{x}}} + {\text{S}}_{2} \cos\uptheta _{2} - {\text{S}}_{1} \cos\uptheta _{1} - {\text{L}}_{2} \sin\uptheta _{2} - {\text{L}}_{1} \sin\uptheta _{1} } \right)\cos\uptheta _{2} \\ & \quad + 2\left( {{\text{B}}_{{\text{y}}} - {\text{A}}_{{\text{y}}} + {\text{S}}_{2} \sin\uptheta _{2} - {\text{S}}_{1} \sin\uptheta _{1} + {\text{L}}_{2} \cos\uptheta _{2} + {\text{L}}_{1} \cos\uptheta _{1} } \right)\sin\uptheta _{2} \\ \end{aligned}$$
(B14)
$${\text{G}}_{3} = 0$$
(B15)
$$\begin{aligned} {\text{G}}_{4} & = - 2\left( {{\text{B}}_{{\text{x}}} - {\text{A}}_{{\text{x}}} + {\text{S}}_{2} \cos\uptheta _{2} - {\text{S}}_{1} \cos\uptheta _{1} - {\text{L}}_{2} \sin\uptheta _{2} - {\text{L}}_{1} \sin\uptheta _{1} } \right) \\ & \quad \left( {\dot {{\text{B}}_{{\text{x}}} } - \dot {{\text{A}}_{{\text{x}}} } - {\text{S}}_{2} \dot {\uptheta _{2} } \sin\uptheta _{2} + {\text{S}}_{1} \dot {\uptheta _{1} } \sin\uptheta _{1} - {\text{L}}_{2} \dot {\uptheta _{2} } \cos\uptheta _{2} - {\text{L}}_{1} \dot {\uptheta _{1} } \cos\uptheta _{1} } \right) \\ & \quad - 2\left( {{\text{B}}_{{\text{y}}} - {\text{A}}_{{\text{y}}} + {\text{S}}_{2} \sin\uptheta _{2} - {\text{S}}_{1} \sin\uptheta _{1} + {\text{L}}_{2} \cos\uptheta _{2} + {\text{L}}_{1} \cos\uptheta _{1} } \right) \\ & \quad \left( {\dot {{\text{B}}_{{\text{y}}} } - \dot {{\text{A}}_{{\text{y}}} } + {\text{S}}_{2} \dot {\uptheta _{2} } \cos\uptheta _{2} - {\text{S}}_{1} \dot {\uptheta _{1} } \cos\uptheta _{1} - {\text{L}}_{2} \dot {\uptheta _{2} } \sin\uptheta _{2} - {\text{L}}_{1} \dot {\uptheta _{1} } \sin\uptheta _{1} } \right) \\ \end{aligned}$$
(B16)
$${\text{G}}_{5} = 0$$
(B17)
$$\begin{aligned} {\text{G}}_{6} & = - 2\left( {{\text{C}}_{{\text{x}}} - {\text{B}}_{{\text{x}}} + {\text{S}}_{3} \cos\uptheta _{3} - {\text{S}}_{2} \cos\uptheta _{2} - {\text{L}}_{3} \sin\uptheta _{3} + {\text{L}}_{2} \sin\uptheta _{2} } \right)\cos\uptheta _{2} \\ & \quad - 2\left( {{\text{C}}_{{\text{y}}} - {\text{B}}_{{\text{y}}} + {\text{S}}_{3} \sin\uptheta _{3} - {\text{S}}_{2} \sin\uptheta _{2} + {\text{L}}_{3} \cos\uptheta _{3} - {\text{L}}_{2} \cos\uptheta _{2} } \right)\sin\uptheta _{2} \\ \end{aligned}$$
(B18)
$$\begin{aligned} {\text{G}}_{7} & = 2\left( {{\text{C}}_{{\text{x}}} - {\text{B}}_{{\text{x}}} + {\text{S}}_{3} \cos\uptheta _{3} - {\text{S}}_{2} \cos\uptheta _{2} - {\text{L}}_{3} \sin\uptheta _{3} + {\text{L}}_{2} \sin\uptheta _{2} } \right)\cos\uptheta _{3} \\ & \quad + 2\left( {{\text{C}}_{{\text{y}}} - {\text{B}}_{{\text{y}}} + {\text{S}}_{3} \sin\uptheta _{3} - {\text{S}}_{2} \sin\uptheta _{2} + {\text{L}}_{3} \cos\uptheta _{3} - {\text{L}}_{2} \cos\uptheta _{2} } \right)\sin\uptheta _{3} \\ \end{aligned}$$
(B19)
$$\begin{aligned} {\text{G}}_{8} & \quad = - 2\left( {{\text{C}}_{{\text{x}}} - {\text{B}}_{{\text{x}}} + {\text{S}}_{3} \cos\uptheta _{3} - {\text{S}}_{2} \cos\uptheta _{2} - {\text{L}}_{3} \sin\uptheta _{3} + {\text{L}}_{2} \sin\uptheta _{2} } \right) \\ & \quad \left( {\dot {{\text{C}}_{{\text{x}}} } - \dot {{\text{B}}_{{\text{x}}} } - {\text{S}}_{3} \dot {\uptheta _{3} } \sin\uptheta _{3} + {\text{S}}_{2} \dot {\uptheta _{2} } \sin\uptheta _{2} - {\text{L}}_{3} \dot {\uptheta _{3} } \cos\uptheta _{3} + {\text{L}}_{2} \dot {\uptheta _{2} } \cos\uptheta _{2} } \right) \\ & \quad - 2\left( {{\text{C}}_{{\text{y}}} - {\text{B}}_{{\text{y}}} + {\text{S}}_{3} \sin\uptheta _{3} - {\text{S}}_{2} \sin\uptheta _{2} + {\text{L}}_{3} \cos\uptheta _{3} - {\text{L}}_{2} \cos\uptheta _{2} } \right) \\ & \quad \left( {\dot {{\text{C}}_{{\text{y}}} } - \dot {{\text{B}}_{{\text{y}}} } + {\text{S}}_{3} \dot {\uptheta _{3} } \cos\uptheta _{3} - {\text{S}}_{2} \dot {\uptheta _{2} } \cos\uptheta _{2} - {\text{L}}_{3} \dot {\uptheta _{3} } \sin\uptheta _{3} + {\text{L}}_{2} \dot {\uptheta _{2} } \sin\uptheta _{2} } \right) \\ \end{aligned}$$
(B20)
$$\begin{aligned} {\text{G}}_{9} & = 2\left( {{\text{A}}_{{\text{x}}} - {\text{C}}_{{\text{x}}} + {\text{S}}_{1} \cos\uptheta _{1} - {\text{S}}_{3} \cos\uptheta _{3} + {\text{L}}_{1} \sin\uptheta _{1} + {\text{L}}_{3} \sin\uptheta _{3} } \right)\cos\uptheta _{1} \\ & \quad + 2\left( {{\text{A}}_{{\text{y}}} - {\text{C}}_{{\text{y}}} + {\text{S}}_{1} \sin\uptheta _{1} - {\text{S}}_{3} \sin\uptheta _{3} - {\text{L}}_{1} \cos\uptheta _{1} - {\text{L}}_{3} \cos\uptheta _{3} } \right)\sin\uptheta _{1} \\ \end{aligned}$$
(B21)
$${\text{G}}_{10} = 0$$
(B22)
$$\begin{aligned} {\text{G}}_{11} & = - 2\left( {{\text{A}}_{{\text{x}}} - {\text{C}}_{{\text{x}}} + {\text{S}}_{1} \cos\uptheta _{1} - {\text{S}}_{3} \cos\uptheta _{3} + {\text{L}}_{1} \sin\uptheta _{1} + {\text{L}}_{3} \sin\uptheta _{3} } \right)\cos\uptheta _{3} \\ & \quad - 2\left( {{\text{A}}_{{\text{y}}} - {\text{C}}_{{\text{y}}} + {\text{S}}_{1} \sin\uptheta _{1} - {\text{S}}_{3} \sin\uptheta _{3} - {\text{L}}_{1} \cos\uptheta _{1} - {\text{L}}_{3} \cos\uptheta _{3} } \right)\sin\uptheta _{3} \\ \end{aligned}$$
(B23)
$$\begin{aligned} {\text{G}}_{12} & = - 2\left( {{\text{A}}_{{\text{x}}} - {\text{C}}_{{\text{x}}} + {\text{S}}_{1} \cos\uptheta _{1} - {\text{S}}_{3} \cos\uptheta _{3} + {\text{L}}_{1} \sin\uptheta _{1} + {\text{L}}_{3} \sin\uptheta _{3} } \right) \\ & \quad \left( {\dot {{\text{A}}_{{\text{x}}} } - \dot {{\text{C}}_{{\text{x}}} } - {\text{S}}_{1} \dot {{\uptheta }_{1} } \sin\uptheta _{1} + {\text{S}}_{3} \dot {\uptheta _{3} } \sin\uptheta _{3} + {\text{L}}_{1} \dot {\uptheta _{1} } \cos\uptheta _{1} + {\text{L}}_{3} \dot {\uptheta _{3} } \cos\uptheta _{3} } \right) \\ & \quad - 2\left( {{\text{A}}_{{\text{y}}} - {\text{C}}_{{\text{y}}} + {\text{S}}_{1} \sin\uptheta _{1} - {\text{S}}_{3} \sin\uptheta _{3} - {\text{L}}_{1} \cos\uptheta _{1} - {\text{L}}_{3} \cos\uptheta _{3} } \right) \\ & \quad \left( {\dot {{\text{A}}_{{\text{y}}} } - \dot {{\text{C}}_{{\text{y}}} } + {\text{S}}_{1} \dot {{\uptheta }_{1} } \cos\uptheta _{1} - {\text{S}}_{3} \dot {\uptheta _{3} } \cos\uptheta _{3} + {\text{L}}_{1} \dot {\uptheta _{1} } \sin\uptheta _{1} + {\text{L}}_{3} \dot {\uptheta _{3} } \sin\uptheta _{3} } \right) \\ \end{aligned}$$
(B24)

The following equations present coefficients defined in Eq. (33) in Sect. 2.2.

$$\begin{aligned} {\text{H}}_{1} & = - 2\left( {{\text{B}}_{{\text{x}}} - {\text{A}}_{{\text{x}}} + {\text{S}}_{2} \cos\uptheta _{2} - {\text{S}}_{1} \cos\uptheta _{1} - {\text{L}}_{2} \sin\uptheta _{2} - {\text{L}}_{1} \sin\uptheta _{1} } \right)\cos\uptheta _{1} \\ & \quad - 2\left( {{\text{B}}_{{\text{y}}} - {\text{A}}_{{\text{y}}} + {\text{S}}_{2} \sin\uptheta _{2} - {\text{S}}_{1} \sin\uptheta _{1} + {\text{L}}_{2} \cos\uptheta _{2} + {\text{L}}_{1} \cos\uptheta _{1} } \right)\sin\uptheta _{1} \\ \end{aligned}$$
(B25)
$$\begin{aligned} {\text{H}}_{2} & = 2\left( {{\text{B}}_{{\text{x}}} - {\text{A}}_{{\text{x}}} + {\text{S}}_{2} \cos\uptheta _{2} - {\text{S}}_{1} \cos\uptheta _{1} - {\text{L}}_{2} \sin\uptheta _{2} - {\text{L}}_{1} \sin\uptheta _{1} } \right)\cos\uptheta _{2} \\ & \quad + 2\left( {{\text{B}}_{{\text{y}}} - {\text{A}}_{{\text{y}}} + {\text{S}}_{2} \sin\uptheta _{2} - {\text{S}}_{1} \sin\uptheta _{1} + {\text{L}}_{2} \cos\uptheta _{2} + {\text{L}}_{1} \cos\uptheta _{1} } \right)\sin\uptheta _{2} \\ \end{aligned}$$
(B26)
$${\text{H}}_{3} = 0$$
(B27)
$$\begin{aligned} {\text{H}}_{4} & = - 2(\dot {{\text{B}}_{{\text{x}}} } - \dot {{\text{A}}_{{\text{x}}} } + \dot {{\text{S}}_{2} } \cos\uptheta _{2} - {\text{S}}_{2} \dot {\uptheta _{2} } \sin\uptheta _{2} \\ & \quad - \dot {{\text{S}}_{1} } \cos\uptheta _{1} + {\text{S}}_{1} \dot {\uptheta _{1} } \sin\uptheta _{1} - {\text{L}}_{2} \dot {\uptheta _{2} } \cos\uptheta _{2} \\ & \quad - {\text{L}}_{1} \dot {\uptheta _{1} } \cos\uptheta _{1} )^{2} \\ & \quad - 2(\dot {{\text{B}}_{{\text{y}}} } - \dot {{\text{A}}_{{\text{y}}} } + \dot {{\text{S}}_{2} } \sin\uptheta _{2} + {\text{S}}_{2} \dot {\uptheta _{2} } \cos\uptheta _{2} \\ & \quad - \dot {{\text{S}}_{1} } \sin\uptheta _{1} - {\text{S}}_{1} \dot {\uptheta _{1} } \cos\uptheta _{1} - {\text{L}}_{2} \dot {\uptheta _{2} } \sin\uptheta _{2} \\ & \quad - {\text{L}}_{1} \dot {{\uptheta }_{1} } \sin\uptheta _{1} )^{2} \\ & \quad - 2({\text{B}}_{{\text{x}}} - {\text{A}}_{{\text{x}}} + {\text{S}}_{2} \cos\uptheta _{2} - {\text{S}}_{1} \cos\uptheta _{1} \\ & \quad - {\text{L}}_{2} \sin\uptheta _{2} - {\text{L}}_{1} \sin\uptheta _{1} ) \\ & \quad (\dot {{\text{B}}_{{\text{x}}} } - \dot {{\text{A}}_{{\text{x}}} } - 2\dot {{\text{S}}_{2} } \dot {\uptheta _{2} } \sin\uptheta _{2} - {\text{S}}_{2} \dot {\uptheta _{2} } \sin\uptheta _{2} \\ & \quad - {\text{S}}_{2} {\dot {\uptheta _{2} }}^{2} \cos\uptheta _{2} + 2\dot {{\text{S}}_{1} } \dot {\uptheta _{1} } \sin\uptheta _{1} + {\text{S}}_{1} \dot {\uptheta _{1} } \sin\uptheta _{1} \\ & \quad + {\text{S}}_{1} {\dot {\uptheta _{1} }}^{2} \cos\uptheta _{1} - {\text{L}}_{2} \dot {\uptheta _{2} } \cos\uptheta _{2} + {\text{L}}_{2} {\dot {\uptheta _{2} }}^{2} \sin\uptheta _{2} \\ & \quad - {\text{L}}_{1} \dot {\uptheta _{1} } \cos\uptheta _{1} + {\text{L}}_{1} {\dot {\uptheta _{1} }}^{2} \sin\uptheta _{1} ) \\ & \quad - 2({\text{B}}_{{\text{y}}} - {\text{A}}_{{\text{y}}} + {\text{S}}_{2} \sin\uptheta _{2} - {\text{S}}_{1} \sin\uptheta _{1} \\ & \quad + {\text{L}}_{2} \cos\uptheta _{2} + {\text{L}}_{1} \cos\uptheta _{1} ) \\ & \quad (\dot {{\text{B}}_{{\text{y}}} } - \dot {{\text{A}}_{{\text{y}}} } + 2\dot {{\text{S}}_{2} } \dot {\uptheta _{2} } \cos\uptheta _{2} + {\text{S}}_{2} \dot {\uptheta _{2} } \cos\uptheta _{2} \\ & \quad - {\text{S}}_{2} {\dot {\uptheta _{2} }}^{2} \sin\uptheta _{2} - 2\dot {{\text{S}}_{1} } \dot {\uptheta _{1} } \cos\uptheta _{1} - {\text{S}}_{1} \dot {\uptheta _{1} } \cos\uptheta _{1} \\ & \quad + {\text{S}}_{1} {\dot {\uptheta _{1} }}^{2} \sin\uptheta _{1} - {\text{L}}_{2} \dot {{\uptheta }_{2} } \sin\uptheta _{2} - {\text{L}}_{2} {\dot {\uptheta _{2} }}^{2} \cos\uptheta _{2} \\ & \quad - {\text{L}}_{1} \dot {\uptheta _{1} } \sin\uptheta _{1} - {\text{L}}_{1} {\dot {\uptheta _{1} }}^{2} \cos\uptheta _{1} ) \\ \end{aligned}$$
(B28)
$${\text{H}}_{5} = 0$$
(B29)
$$\begin{aligned} {\text{H}}_{6} & = - 2\left( {{\text{C}}_{{\text{x}}} - {\text{B}}_{{\text{x}}} + {\text{S}}_{3} \cos\uptheta _{3} - {\text{S}}_{2} \cos\uptheta _{2} - {\text{L}}_{3} \sin\uptheta _{3} + {\text{L}}_{2} \sin\uptheta _{2} } \right)\cos\uptheta _{2} \\ & \quad - 2\left( {{\text{C}}_{{\text{y}}} - {\text{B}}_{{\text{y}}} + {\text{S}}_{3} \sin\uptheta _{3} - {\text{S}}_{2} \sin\uptheta _{2} + {\text{L}}_{3} \cos\uptheta _{3} - {\text{L}}_{2} \cos\uptheta _{2} } \right)\sin\uptheta _{2} \\ \end{aligned}$$
(B30)
$$\begin{aligned} {\text{H}}_{7} & = 2\left( {{\text{C}}_{{\text{x}}} - {\text{B}}_{{\text{x}}} + {\text{S}}_{3} \cos\uptheta _{3} - {\text{S}}_{2} \cos\uptheta _{2} - {\text{L}}_{3} \sin\uptheta _{3} + {\text{L}}_{2} \sin\uptheta _{2} } \right)\cos\uptheta _{3} \\ & \quad + 2\left( {{\text{C}}_{{\text{y}}} - {\text{B}}_{{\text{y}}} + {\text{S}}_{3} \sin\uptheta _{3} - {\text{S}}_{2} \sin\uptheta _{2} + {\text{L}}_{3} \cos\uptheta _{3} - {\text{L}}_{2} \cos\uptheta _{2} } \right)\sin\uptheta _{3} \\ \end{aligned}$$
(B31)
$$\begin{aligned} {\text{H}}_{8} & = - 2(\dot {{\text{C}}_{{\text{x}}} } - \dot {{\text{B}}_{{\text{x}}} } + \dot {{\text{S}}_{3} } \cos\uptheta _{3} - {\text{S}}_{3} \dot {\uptheta _{3} } \sin\uptheta _{3} \\ & \quad - \dot {{\text{S}}_{2} } \cos\uptheta _{2} + {\text{S}}_{2} \dot {{\uptheta }_{2} } \sin\uptheta _{2} - {\text{L}}_{3} \dot {\uptheta _{3} } \cos\uptheta _{3} \\ & \quad + {\text{L}}_{2} \dot {\uptheta _{2} } \cos\uptheta _{2} )^{2} \\ & \quad - 2(\dot {{\text{C}}_{{\text{y}}} } - \dot {{\text{B}}_{{\text{y}}} } + \dot {{\text{S}}_{3} } \sin\uptheta _{3} + {\text{S}}_{3} \dot {\uptheta _{3} } \cos\uptheta _{3} \\ & \quad - \dot {{\text{S}}_{2} } \sin\uptheta _{2} - {\text{S}}_{2} \dot {\uptheta _{2} } \cos\uptheta _{2} - {\text{L}}_{3} \dot {\uptheta _{3} } \sin\uptheta _{3} \\ & \quad + {\text{L}}_{2} \dot {\uptheta _{2} } \sin\uptheta _{2} )^{2} \\ & \quad - 2({\text{C}}_{{\text{x}}} - {\text{B}}_{{\text{x}}} + {\text{S}}_{3} \cos\uptheta _{3} - {\text{S}}_{2} \cos\uptheta _{2} \\ & \quad - {\text{L}}_{3} \sin\uptheta _{3} + {\text{L}}_{2} \sin\uptheta _{2} ) \\ & \quad (\dot {{\text{C}}_{{\text{x}}} } - \dot {{\text{B}}_{{\text{x}}} } - 2\dot {{\text{S}}_{3} } \dot {{\uptheta }_{3} } \sin\uptheta _{3} - {\text{S}}_{3} \dot {\uptheta _{3} } \sin\uptheta _{3} \\ & \quad - {\text{S}}_{3} {\dot {\uptheta _{3} }}^{2} \cos\uptheta _{3} + 2\dot {{\text{S}}_{2} } \dot {\uptheta _{2} } \sin\uptheta _{2} + {\text{S}}_{2} \dot {\uptheta _{2} } \sin\uptheta _{2} \\ & \quad + {\text{S}}_{2} {\dot {\uptheta _{2} }}^{2} \cos\uptheta _{2} - {\text{L}}_{3} \dot {\uptheta _{3} } \cos\uptheta _{3} + {\text{L}}_{3} {\dot {\uptheta _{3} }}^{2} \sin\uptheta _{3} \\ & \quad + {\text{L}}_{2} \dot {\uptheta _{2} } \cos\uptheta _{2} - {\text{L}}_{2} {\dot {\uptheta _{2} }}^{2} \sin\uptheta _{2} ) \\ & \quad - 2({\text{C}}_{{\text{y}}} - {\text{B}}_{{\text{y}}} + {\text{S}}_{3} \sin\uptheta _{3} - {\text{S}}_{2} \sin\uptheta _{2} \\ & \quad + {\text{L}}_{3} \cos\uptheta _{3} - {\text{L}}_{2} \cos\uptheta _{2} ) \\ & \quad (\dot {{\text{C}}_{{\text{y}}} } - \dot {{\text{B}}_{{\text{y}}} } + 2\dot {{\text{S}}_{3} } \dot {\uptheta _{3} } \cos\uptheta _{3} + {\text{S}}_{3} \dot {\uptheta _{3} } \cos\uptheta _{3} \\ & \quad - {\text{S}}_{3} {\dot {\uptheta _{3} }}^{2} \sin\uptheta _{3} - 2\dot {{\text{S}}_{2} } \dot {\uptheta _{2} } \cos\uptheta _{2} - {\text{S}}_{2} \dot {\uptheta _{2} } \cos\uptheta _{2} \\ & \quad + {\text{S}}_{2} {\dot {\uptheta _{2} }}^{2} \sin\uptheta _{2} - {\text{L}}_{3} \dot {\uptheta _{3} } \sin\uptheta _{3} - {\text{L}}_{3} {\dot {\uptheta _{3} }}^{2} \cos\uptheta _{3} \\ & \quad + {\text{L}}_{2} \dot {\uptheta _{2} } \sin\uptheta _{2} + {\text{L}}_{2} {\dot {\uptheta _{2} }}^{2} \cos\uptheta _{2} ) \\ \end{aligned}$$
(B32)
$$\begin{aligned} {\text{H}}_{9} & = 2\left( {{\text{A}}_{{\text{x}}} - {\text{C}}_{{\text{x}}} + {\text{S}}_{1} \cos\uptheta _{1} - {\text{S}}_{3} \cos\uptheta _{3} + {\text{L}}_{1} \sin\uptheta _{1} + {\text{L}}_{3} \sin\uptheta _{3} } \right)\cos\uptheta _{1} \\ & \quad + 2\left( {{\text{A}}_{{\text{y}}} - {\text{C}}_{{\text{y}}} + {\text{S}}_{1} \sin\uptheta _{1} - {\text{S}}_{3} \sin\uptheta _{3} - {\text{L}}_{1} \cos\uptheta _{1} - {\text{L}}_{3} \cos\uptheta _{3} } \right)\sin\uptheta _{1} \\ \end{aligned}$$
(B33)
$${\text{H}}_{10} = 0$$
(B34)
$$\begin{aligned} {\text{H}}_{11} & = - 2\left( {{\text{A}}_{{\text{x}}} - {\text{C}}_{{\text{x}}} + {\text{S}}_{1} \cos\uptheta _{1} - {\text{S}}_{3} \cos\uptheta _{3} + {\text{L}}_{1} \sin\uptheta _{1} + {\text{L}}_{3} \sin\uptheta _{3} } \right)\cos\uptheta _{3} \\ & \quad - 2\left( {{\text{A}}_{{\text{y}}} - {\text{C}}_{{\text{y}}} + {\text{S}}_{1} \sin\uptheta _{1} - {\text{S}}_{3} \sin\uptheta _{3} - {\text{L}}_{1} \cos\uptheta _{1} - {\text{L}}_{3} \cos\uptheta _{3} } \right)\sin\uptheta _{3} \\ \end{aligned}$$
(B35)
$$\begin{aligned} {\text{H}}_{12} & = - 2(\dot {{\text{A}}_{{\text{x}}} } - \dot {{\text{C}}_{{\text{x}}} } + \dot {{\text{S}}_{1} } \cos\uptheta _{1} - {\text{S}}_{1} \dot {\uptheta _{1} } \sin\uptheta _{1} \\ & \quad - \dot {{\text{S}}_{3} } \cos\uptheta _{3} + {\text{S}}_{3} \dot {\uptheta _{3} } \sin\uptheta _{3} + {\text{L}}_{1} \dot {\uptheta _{1} } \cos\uptheta _{1} \\ & \quad + {\text{L}}_{3} \dot {\uptheta _{3} } \cos\uptheta _{3} )^{2} \\ & \quad - 2(\dot {{\text{A}}_{{\text{y}}} } - \dot {{\text{C}}_{{\text{y}}} } + \dot {{\text{S}}_{1} } \sin\uptheta _{1} + {\text{S}}_{1} \dot {\uptheta _{1} } \cos\uptheta _{1} \\ & \quad - \dot {{\text{S}}_{3} } \sin\uptheta _{3} - {\text{S}}_{3} \dot {\uptheta _{3} } \cos\uptheta _{3} + {\text{L}}_{1} \dot {\uptheta _{1} } \sin\uptheta _{1} \\ & \quad + {\text{L}}_{3} \dot {\uptheta _{3} } \sin\uptheta _{3} )^{2} \\ & \quad + 2({\text{A}}_{{\text{x}}} - {\text{C}}_{{\text{x}}} + {\text{S}}_{1} \cos\uptheta _{1} - {\text{S}}_{3} \cos\uptheta _{3} \\ & \quad + {\text{L}}_{1} \sin\uptheta _{1} + {\text{L}}_{3} \sin\uptheta _{3} ) \\ & \quad (\dot {{\text{A}}_{{\text{x}}} } - \dot {{\text{C}}_{{\text{x}}} } - 2\dot {{\text{S}}_{1} } \dot {{\uptheta }_{1} } \sin\uptheta _{1} - {\text{S}}_{1} \dot {\uptheta _{1} } \sin\uptheta _{1} \\ & \quad - {\text{S}}_{1} {\dot {\uptheta _{1} }}^{2} \cos\uptheta _{1} + 2\dot {{\text{S}}_{3} } \dot {\uptheta _{3} } \sin\uptheta _{3} + {\text{S}}_{3} \dot {\uptheta _{3} } \sin\uptheta _{3} \\ & \quad + {\text{S}}_{3} {\dot {\uptheta _{3} }}^{2} \cos\uptheta _{3} + {\text{L}}_{1} \dot {\uptheta _{1} } \cos\uptheta _{1} - {\text{L}}_{1} {\dot {\uptheta _{1} }}^{2} \sin\uptheta _{1} \\ & \quad + {\text{L}}_{3} \dot {\uptheta _{3} } \cos\uptheta _{3} - {\text{L}}_{3} {\dot {\uptheta _{3} }}^{2} \sin\uptheta _{3} ) \\ & \quad + 2({\text{A}}_{{\text{y}}} - {\text{C}}_{{\text{y}}} + {\text{S}}_{1} \sin\uptheta _{2} - {\text{S}}_{3} \sin\uptheta _{3} \\ & \quad - {\text{L}}_{1} \cos\uptheta _{1} - {\text{L}}_{3} \cos\uptheta _{3} ) \\ & \quad (\dot {{\text{A}}_{{\text{y}}} } - \dot {{\text{C}}_{{\text{y}}} } + 2\dot {{\text{S}}_{1} } \dot {\uptheta _{1} } \cos\uptheta _{1} + {\text{S}}_{1} \dot {\uptheta _{1} } \cos\uptheta _{1} \\ & \quad - {\text{S}}_{1} {\dot {\uptheta _{1} }}^{2} \sin\uptheta _{1} - 2\dot {{\text{S}}_{3} } \dot {\uptheta _{3} } \cos\uptheta _{3} - {\text{S}}_{3} \dot {\uptheta _{3} } \cos\uptheta _{3} \\ & \quad + {\text{S}}_{3} {\dot {\uptheta _{3} }}^{2} \sin\uptheta _{3} + {\text{L}}_{1} \dot {\uptheta _{1} } \sin\uptheta _{1} + {\text{L}}_{1} {\dot {\uptheta _{1} }}^{2} \cos\uptheta _{1} \\ & \quad + {\text{L}}_{3} \dot {\uptheta _{3} } \sin\uptheta _{3} + {\text{L}}_{3} {\dot {\uptheta _{3} }}^{2} \cos\uptheta _{3} ) \\ \end{aligned}$$
(B36)

Appendix C

The following equations present coefficients defined in Eqs. (51)–(53) in Sect. 2.3.

$${\text{I}}_{1} = - 2\left( {{\text{L}}_{1} + {\text{O}}_{2} \sin\upgamma _{1} } \right)$$
(C1)
$$\begin{aligned} {\text{I}}_{2} & = 2[\left( {{\text{O}}_{1} + {\text{O}}_{2} \cos\upgamma _{1} } \right)\sin\upgamma _{1} \\ & \quad - \left( {{\text{L}}_{1} + {\text{O}}_{2} \sin\upgamma _{1} } \right)\cos\upgamma _{1} ] \\ \end{aligned}$$
(C2)
$${\text{I}}_{3} = 2\cos\upgamma _{1}$$
(C3)
$$\begin{aligned} {\text{I}}_{4} & = \left( {{\text{O}}_{1} + {\text{O}}_{2} \cos\upgamma _{1} } \right)^{2} + \left( {{\text{L}}_{1} + {\text{O}}_{2} \sin\upgamma _{1} } \right)^{2} \\ & \quad - \left( {{\text{B}}_{{\text{x}}} - {\text{A}}_{{\text{x}}} } \right)^{2} - \left( {{\text{B}}_{{\text{y}}} - {\text{A}}_{{\text{y}}} } \right)^{2} \\ \end{aligned}$$
(C4)
$${\text{I}}_{5} = - 2\left( {{\text{L}}_{2} + {\text{O}}_{3} \sin\upgamma _{2} } \right)$$
(C5)
$$\begin{aligned} {\text{I}}_{6} & = 2[\left( {{\text{O}}_{2} + {\text{O}}_{3} \cos\upgamma _{2} } \right)\sin\upgamma _{2} \\ & \quad - \left( {{\text{L}}_{2} + {\text{O}}_{3} \sin\upgamma _{2} } \right)\cos\upgamma _{2} ] \\ \end{aligned}$$
(C6)
$${\text{I}}_{7} = 2\cos\upgamma _{2}$$
(C7)
$$\begin{aligned} {\text{I}}_{8} & = \left( {{\text{O}}_{2} + {\text{O}}_{3} \cos\upgamma _{2} } \right)^{2} + \left( {{\text{L}}_{2} + {\text{O}}_{3} \sin\upgamma _{2} } \right)^{2} \\ & \quad - \left( {{\text{C}}_{{\text{x}}} - {\text{B}}_{{\text{x}}} } \right)^{2} - \left( {{\text{C}}_{{\text{y}}} - {\text{B}}_{{\text{y}}} } \right)^{2} \\ \end{aligned}$$
(C8)
$$\begin{gathered} {\text{I}}_{9} = 2[\left( {{\text{O}}_{3} + {\text{O}}_{1} \cos\upgamma _{3} } \right)\sin\upgamma _{3} \hfill \\ - \left( {{\text{L}}_{3} + {\text{O}}_{1} \sin\upgamma _{3} } \right)\cos\upgamma _{3} ] \hfill \\ \end{gathered}$$
(C9)
$${\text{I}}_{10} = - 2\left( {{\text{L}}_{3} + {\text{O}}_{1} \sin\upgamma _{3} } \right)$$
(C10)
$${\text{I}}_{11} = 2\cos\upgamma _{3}$$
(C11)
$$\begin{aligned} {\text{I}}_{12} & = \left( {{\text{O}}_{3} + {\text{O}}_{1} \cos\upgamma _{3} } \right)^{2} + \left( {{\text{L}}_{3} + {\text{O}}_{1} \sin\upgamma _{3} } \right)^{2} \\ & \quad - \left( {{\text{A}}_{{\text{x}}} - {\text{C}}_{{\text{x}}} } \right)^{2} - \left( {{\text{A}}_{{\text{y}}} - {\text{C}}_{{\text{y}}} } \right)^{2} \\ \end{aligned}$$
(C12)

The following equations present coefficients defined in Eqs. (54) in Sect. 2.3.

$$\begin{aligned} {\text{I}}_{13} & = {\text{O}}_{1} + \left( {{\text{L}}_{3} - {\text{S}}_{3}^{{\prime}} } \right)\sin\upgamma _{3} \\ & \quad + {\text{O}}_{3} \cos\upgamma _{3} \\ \end{aligned}$$
(C13)
$$\begin{aligned} {\text{I}}_{14} & = - {\text{S}}_{1}^{{\prime}} + \left( {{\text{L}}_{3} - {\text{S}}_{3}^{{\prime}} } \right)\cos\upgamma _{3} \\ & \quad - {\text{O}}_{3} \sin\upgamma _{3} \\ \end{aligned}$$
(C14)

The following equations present coefficients defined in Eq. (63) in Sect. 2.3.

$${\text{J}}_{1} = 2\left( {{\text{S}}_{1}^{{\prime}} - {\text{L}}_{1} + {\text{S}}_{2}^{{\prime}} \cos\upgamma _{1} - {\text{O}}_{2} \sin\upgamma _{1} } \right)$$
(C15)
$${\text{J}}_{2} = 2\left( {{\text{S}}_{2}^{{\prime}} + {\text{O}}_{1} \sin\upgamma _{1} + {\text{S}}_{1}^{{\prime}} \cos\upgamma _{1} - {\text{L}}_{1} \cos\upgamma _{1} } \right)$$
(C16)
$${\text{J}}_{3} = 0$$
(C17)
$$\begin{aligned} {\text{J}}_{4} & = 2[\left( {{\text{B}}_{{\text{x}}} - {\text{A}}_{{\text{x}}} } \right)\left( {\dot {{\text{B}}_{{\text{x}}} } - \dot {{\text{A}}_{{\text{x}}} } } \right) \\ & \quad + \left( {{\text{B}}_{{\text{y}}} - {\text{A}}_{{\text{y}}} } \right)\left( {\dot {{\text{B}}_{{\text{y}}} } - \dot {{\text{A}}_{{\text{y}}} } } \right)] \\ \end{aligned}$$
(C18)
$${\text{J}}_{5} = 0$$
(C19)
$${\text{J}}_{6} = 2\left( {{\text{S}}_{2}^{{\prime}} - {\text{L}}_{2} + {\text{S}}_{3}^{{\prime}} \cos\upgamma _{2} - {\text{O}}_{3} \sin\upgamma _{2} } \right)$$
(C20)
$${\text{J}}_{7} = 2\left( {{\text{S}}_{3}^{{\prime}} + {\text{O}}_{2} \sin\upgamma _{2} + {\text{S}}_{2}^{{\prime}} \cos\upgamma _{2} - {\text{L}}_{2} \cos\upgamma _{2} } \right)$$
(C21)
$$\begin{aligned} {\text{J}}_{8} & \quad = 2[\left( {{\text{C}}_{{\text{x}}} - {\text{B}}_{{\text{x}}} } \right)\left( {\dot {{\text{C}}_{{\text{x}}} } - \dot {{\text{B}}_{{\text{x}}} } } \right) \\ & \quad + \left( {{\text{C}}_{{\text{y}}} - {\text{B}}_{{\text{y}}} } \right)\left( {\dot {{\text{C}}_{{\text{y}}} } - \dot {{\text{B}}_{{\text{y}}} } } \right)] \\ \end{aligned}$$
(C22)
$${\text{J}}_{9} = 2\left( {{\text{S}}_{1}^{{\prime}} + {\text{O}}_{3} \sin\upgamma _{3} + {\text{S}}_{3}^{{\prime}} \cos\upgamma _{3} - {\text{L}}_{3} \cos\upgamma _{3} } \right)$$
(C23)
$${\text{J}}_{10} = 0$$
(C24)
$${\text{J}}_{11} = 2\left( {{\text{S}}_{3}^{{\prime}} - {\text{L}}_{3} + {\text{S}}_{1}^{{\prime}} \cos\upgamma _{3} - {\text{O}}_{1} \sin\upgamma _{3} } \right)$$
(C25)
$$\begin{aligned} {\text{J}}_{12} & = 2[\left( {{\text{A}}_{{\text{x}}} - {\text{C}}_{{\text{x}}} } \right)\left( {\dot {{\text{A}}_{{\text{x}}} } - \dot {{\text{C}}_{{\text{x}}} } } \right) \\ & \quad + \left( {{\text{A}}_{{\text{y}}} - {\text{C}}_{{\text{y}}} } \right)\left( {\dot {{\text{A}}_{{\text{y}}} } - \dot {{\text{C}}_{{\text{y}}} } } \right)] \\ \end{aligned}$$
(C26)

The following equations present coefficients defined in Eq. (66) in Sect. 2.3.

$${\text{K}}_{1} = 2\left( {{\text{S}}_{1}^{{\prime}} - {\text{L}}_{1} + {\text{S}}_{2}^{{\prime}} \cos\upgamma _{1} - {\text{O}}_{2} \sin\upgamma _{1} } \right)$$
(C27)
$${\text{K}}_{2} = 2\left( {{\text{S}}_{2}^{{\prime}} + {\text{O}}_{1} \sin\upgamma _{1} + {\text{S}}_{1}^{{\prime}} \cos\upgamma _{1} - {\text{L}}_{1} \cos\upgamma _{1} } \right)$$
(C28)
$${\text{K}}_{3} = 0$$
(C29)
$$\begin{aligned} {\text{K}}_{4} & = - 2\left( {{\dot {{\text{S}}_{1}^{{\prime}} }}^{2} + {\dot {{\text{S}}_{2}^{{\prime}} }}^{2} + 2\dot {{\text{S}}_{1}^{{\prime}} } \dot {{\text{S}}_{2}^{{\prime}} } \cos\upgamma _{1} } \right) + 2\left( {\dot {{\text{B}}_{{\text{x}}} } - \dot {{\text{A}}_{{\text{x}}} } } \right)^{2} \\ & \quad + 2\left( {{\text{B}}_{{\text{x}}} - {\text{A}}_{{\text{x}}} } \right)\left( {\dot {{\text{B}}_{{\text{x}}} } - \dot {{\text{A}}_{{\text{x}}} } } \right) + 2\left( {\dot {{\text{B}}_{{\text{y}}} } - \dot {{\text{A}}_{{\text{y}}} } } \right)^{2} \\ & \quad + 2\left( {{\text{B}}_{{\text{y}}} - {\text{A}}_{{\text{y}}} } \right)\left( {\dot {{\text{B}}_{{\text{y}}} } - \dot {{\text{A}}_{{\text{y}}} } } \right) \\ \end{aligned}$$
(C30)
$${\text{K}}_{5} = 0$$
(C31)
$${\text{K}}_{6} = 2\left( {{\text{S}}_{2}^{{\prime}} - {\text{L}}_{2} + {\text{S}}_{3}^{{\prime}} \cos\upgamma _{2} - {\text{O}}_{3} \sin\upgamma _{2} } \right)$$
(C32)
$${\text{K}}_{7} = 2\left( {{\text{S}}_{3}^{{\prime}} + {\text{O}}_{2} \sin\upgamma _{2} + {\text{S}}_{2}^{{\prime}} \cos\upgamma _{2} - {\text{L}}_{2} \cos\upgamma _{2} } \right)$$
(C33)
$$\begin{aligned} {\text{K}}_{8} & = - 2\left( {{\dot {{\text{S}}_{2}^{{\prime}} }}^{2} + {\dot {{\text{S}}_{3}^{{\prime}} }}^{2} + 2\dot {{\text{S}}_{2}^{{\prime}} } \dot {{\text{S}}_{3}^{{\prime}} } \cos\upgamma _{2} } \right) + 2\left( {\dot {{\text{C}}_{{\text{x}}} } - \dot {{\text{B}}_{{\text{x}}} } } \right)^{2} \\ & \quad + 2\left( {{\text{C}}_{{\text{x}}} - {\text{B}}_{{\text{x}}} } \right)\left( {\dot {{\text{C}}_{{\text{x}}} } - \dot {{\text{B}}_{{\text{x}}} } } \right) + 2\left( {\dot {{\text{C}}_{{\text{y}}} } - \dot {{\text{B}}_{{\text{y}}} } } \right)^{2} \\ & \quad + 2\left( {{\text{C}}_{{\text{y}}} - {\text{B}}_{{\text{y}}} } \right)\left( {\dot {{\text{C}}_{{\text{y}}} } - \dot {{\text{B}}_{{\text{y}}} } } \right) \\ \end{aligned}$$
(C34)
$${\text{K}}_{9} = 2\left( {{\text{S}}_{1}^{{\prime}} + {\text{O}}_{3} \sin\upgamma _{3} + {\text{S}}_{3}^{{\prime}} \cos\upgamma _{3} - {\text{L}}_{3} \cos\upgamma _{3} } \right)$$
(C35)
$${\text{K}}_{10} = 0$$
(C36)
$${\text{K}}_{11} = 2\left( {{\text{S}}_{3}^{{\prime}} - {\text{L}}_{3} + {\text{S}}_{1}^{{\prime}} \cos\upgamma _{3} - {\text{O}}_{1} \sin\upgamma _{3} } \right)$$
(C37)
$$\begin{aligned} {\text{K}}_{12} & = - 2\left( {{\dot {{\text{S}}_{1}^{{\prime}} }}^{2} + {\dot {{\text{S}}_{3}^{{\prime}} }}^{2} + 2\dot {{\text{S}}_{1}^{{\prime}} } \dot {{\text{S}}_{3}^{{\prime}} } \cos\upgamma _{3} } \right) + 2\left( {\dot {{\text{A}}_{{\text{x}}} } - \dot {{\text{C}}_{{\text{x}}} } } \right)^{2} \\ & \quad + 2\left( {{\text{A}}_{{\text{x}}} - {\text{C}}_{{\text{x}}} } \right)\left( {\dot {{\text{A}}_{{\text{x}}} } - \dot {{\text{C}}_{{\text{x}}} } } \right) + 2\left( {\dot {{\text{A}}_{{\text{y}}} } - \dot {{\text{C}}_{{\text{y}}} } } \right)^{2} \\ & \quad + 2\left( {{\text{A}}_{{\text{y}}} - {\text{C}}_{{\text{y}}} } \right)\left( {\dot {{\text{A}}_{{\text{y}}} } - \dot {{\text{C}}_{{\text{y}}} } } \right) \\ \end{aligned}$$
(C38)

The following equations present coefficients defined in Eqs. (68) in Sect. 2.3.

$${\text{K}}_{13} = {\text{O}}_{1} + \left( {{\text{L}}_{3} - {\text{S}}_{3}^{{\prime}} } \right)\sin\upgamma _{3} + {\text{O}}_{3} \cos\upgamma _{3}$$
(C39)
$${\text{K}}_{14} = - {\text{S}}_{1}^{{\prime}} + \left( {{\text{L}}_{3} - {\text{S}}_{3}^{{\prime}} } \right)\cos\upgamma _{3} - {\text{O}}_{3} \sin\upgamma _{3}$$
(C40)

Appendix D

The following equation presents α+Δα defined in Eq. (72) in Sect. 3.1.

$${\upalpha } + {{\Delta \upalpha }} = \cos^{ - 1} \left[ {\frac{{\left( {l_{{{\text{AB}}}} + {\Delta }l_{{{\text{AB}}}} } \right)^{2} + \left( {l_{{{\text{AC}}}} + {\Delta }l_{{{\text{AC}}}} } \right)^{2} - \left( {l_{{{\text{BC}}}} + {\Delta }l_{{{\text{BC}}}} } \right)^{2} }}{{2\left( {l_{{{\text{AB}}}} + {\Delta }l_{{{\text{AB}}}} } \right)\left( {l_{{{\text{AC}}}} + {\Delta }l_{{{\text{AC}}}} } \right)}}} \right]$$
(D1)

The following equations present coefficients defined in Eq. (73) in Sect. 3.1.

$${\text{M}}_{1} = 2\left( {{\text{x}} - {\text{A}}_{{\text{x}}} - {\Delta A}_{{\text{x}}} } \right)$$
(D2)
$${\text{M}}_{2} = 2\left( {{\text{y}} - {\text{A}}_{{\text{y}}} - {\Delta A}_{{\text{y}}} } \right)$$
(D3)
$${\text{M}}_{3} = 0$$
(D4)
$$\begin{aligned} {\text{M}}_{4} & = - { }\left( {{\text{A}}_{{\text{x}}} + {\Delta A}_{{\text{x}}} } \right)^{2} - \left( {{\text{x}} - 2{\text{A}}_{{\text{x}}} - 2{\Delta A}_{{\text{x}}} } \right){\text{x}} \\ & \quad - \left( {{\text{A}}_{{\text{y}}} + {\Delta A}_{{\text{y}}} } \right)^{2} - \left( {{\text{y}} - 2{\text{A}}_{{\text{y}}} - 2{\Delta A}_{{\text{y}}} } \right){\text{y}} \\ & \quad + \left( {{\text{L}}_{1} + {\Delta L}_{1} } \right)^{2} \\ \end{aligned}$$
(D5)
$$\begin{aligned} {\text{M}}_{5} & = 2\left( {{\text{x}} - {\text{B}}_{{\text{x}}} - {\Delta B}_{{\text{x}}} } \right) \\ & \quad + 2\left( {l_{{{\text{AB}}}} + {\Delta }l_{{{\text{AB}}}} } \right){\text{cos}}\upphi \\ \end{aligned}$$
(D6)
$$\begin{aligned} {\text{M}}_{6} & = 2\left( {{\text{y}} - {\text{B}}_{{\text{y}}} - {\Delta B}_{{\text{y}}} } \right) \\ & \quad + 2\left( {l_{{{\text{AB}}}} + {\Delta }l_{{{\text{AB}}}} } \right){\text{sin}}\upphi \\ \end{aligned}$$
(D7)
$$\begin{aligned} {\text{M}}_{7} & = - { }2\left( {l_{{{\text{AB}}}} + {\Delta }l_{{{\text{AB}}}} } \right)\left( {{\text{x}} - {\text{B}}_{{\text{x}}} - {\Delta B}_{{\text{x}}} } \right){\text{sin}}\upphi \\ & \quad + 2\left( {l_{{{\text{AB}}}} + {\Delta }l_{{{\text{AB}}}} } \right)\left( {{\text{y}} - {\text{B}}_{{\text{y}}} - {\Delta B}_{{\text{y}}} } \right){\text{cos}}\upphi \\ \end{aligned}$$
(D8)
$$\begin{aligned} {\text{M}}_{8} & = - { }2\left( {l_{{{\text{AB}}}} + {\Delta }l_{{{\text{AB}}}} } \right)\left( {{\text{x}} - {\text{B}}_{{\text{x}}} - {\Delta B}_{{\text{x}}} } \right){\text{sin}}\upphi \\ & \quad - 2\left( {l_{{{\text{AB}}}} + {\Delta }l_{{{\text{AB}}}} } \right)\left( {{\text{y}} - {\text{B}}_{{\text{y}}} - {\Delta B}_{{\text{y}}} } \right){\text{cos}}\upphi \\ & \quad - \left( {{\text{B}}_{{\text{x}}} + {\Delta B}_{{\text{x}}} } \right)^{2} - \left( {{\text{x}} - 2{\text{B}}_{{\text{x}}} - 2{\Delta B}_{{\text{x}}} } \right){\text{x}} \\ & \quad - \left( {{\text{B}}_{{\text{y}}} + {\Delta B}_{{\text{y}}} } \right)^{2} - \left( {{\text{y}} - 2{\text{B}}_{{\text{y}}} - 2{\Delta B}_{{\text{y}}} } \right){\text{y}} \\ & \quad - \left( {l_{{{\text{AB}}}} + {\Delta }l_{{{\text{AB}}}} } \right)^{2} + \left( {{\text{L}}_{2} + {\Delta L}_{2} } \right)^{2} \\ \end{aligned}$$
(D9)
$$\begin{aligned} {\text{M}}_{9} & = 2\left( {{\text{x}} - {\text{C}}_{{\text{x}}} - {\Delta C}_{{\text{x}}} } \right) \\ & \quad + 2\left( {l_{{{\text{AC}}}} + {\Delta }l_{{{\text{AC}}}} } \right){\text{cos}}\left( {\upphi + \upalpha + \Delta \upalpha } \right) \\ \end{aligned}$$
(D10)
$$\begin{aligned} {\text{M}}_{10} & = 2\left( {{\text{y}} - {\text{C}}_{{\text{y}}} - \Delta {\text{C}}_{{\text{y}}} } \right) \\ & \quad + 2\left( {l_{{{\text{AC}}}} + \Delta l_{{{\text{AC}}}} } \right){\text{sin}}\left( {\upphi + \upalpha + \Delta \upalpha } \right) \\ \end{aligned}$$
(D11)
$$\begin{aligned} {\text{M}}_{11} & = - { }2\left( {l_{{{\text{AC}}}} + \Delta l_{{{\text{AC}}}} } \right)\left( {{\text{x}} - {\text{C}}_{{\text{x}}} - \Delta {\text{C}}_{{\text{x}}} } \right)\sin \left( {\upphi + \upalpha + \Delta \upalpha } \right) \\ & \quad + 2\left( {l_{{{\text{AC}}}} + \Delta l_{{{\text{AC}}}} } \right)\left( {{\text{y}} - {\text{C}}_{{\text{y}}} - \Delta {\text{C}}_{{\text{y}}} } \right)\cos \left( {\upphi + \upalpha + \Delta \upalpha } \right) \\ \end{aligned}$$
(D12)
$$\begin{aligned} {\text{M}}_{12} & = - { }2\left( {l_{{{\text{AC}}}} + \Delta l_{{{\text{AC}}}} } \right)\left( {{\text{x}} - {\text{C}}_{{\text{x}}} - \Delta {\text{C}}_{{\text{x}}} } \right)\cos \left( {\upphi + \upalpha + \Delta \upalpha } \right) \\ & \quad - 2\left( {l_{{{\text{AC}}}} + {\Delta }l_{{{\text{AC}}}} } \right)\left( {{\text{y}} - {\text{C}}_{{\text{y}}} - {\Delta C}_{{\text{y}}} } \right){\text{sin}}\left( {\upphi + \upalpha + \Delta \upalpha } \right) \\ & \quad - \left( {{\text{C}}_{{\text{x}}} + \Delta {\text{C}}_{{\text{x}}} } \right)^{2} - \left( {{\text{x}} - 2{\text{C}}_{{\text{x}}} - 2\Delta {\text{C}}_{{\text{x}}} } \right){\text{x}} - \left( {{\text{C}}_{{\text{y}}} + \Delta {\text{C}}_{{\text{y}}} } \right)^{2} \\ & \quad - \left( {{\text{y}} - 2{\text{C}}_{{\text{y}}} - 2\Delta {\text{C}}_{{\text{y}}} } \right){\text{y}} - \left( {l_{{{\text{AC}}}} + \Delta l_{{{\text{AC}}}} } \right)^{2} + \left( {{\text{L}}_{3} + \Delta {\text{L}}_{3} } \right)^{2} \\ \end{aligned}$$
(D13)

Appendix E

The following equations present coefficients defined in Eq. (84) in Sect. 3.2.

$$\begin{aligned} {\text{N}}_{1} & = 2{\text{S}}_{1} - 2{\text{N}}_{13} \cos \left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right) \\ & \quad - 2{\text{N}}_{14} \sin \left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right) \\ & \quad - 2{\text{S}}_{2} \cos \left[ {\left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) - \left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right)} \right] \\ \end{aligned}$$
(E1)
$$\begin{aligned} {\text{N}}_{2} & = 2{\text{S}}_{2} + 2{\text{N}}_{13} \cos \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) \\ & \quad + 2{\text{N}}_{14} \sin \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) \\ & \quad - 2{\text{S}}_{1} \cos \left[ {\left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) - \left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right)} \right] \\ \end{aligned}$$
(E2)
$${\text{N}}_{3} = 0$$
(E3)
$$\begin{aligned} {\text{N}}_{4} & = - {\text{S}}_{1}^{2} - {\text{S}}_{2}^{2} - {\text{N}}_{13}^{2} - {\text{N}}_{14}^{2} + \left( {l_{{{\text{AB}}}} + {\Delta }l_{{{\text{AB}}}} } \right)^{2} \\ & \quad - 2\left[ {{\text{N}}_{13} \cos \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) + {\text{N}}_{14} \sin \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right)} \right]{\text{S}}_{2} \\ & \quad + 2\left[ {{\text{N}}_{13} \cos \left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right) + {\text{N}}_{14} \sin \left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right)} \right]{\text{S}}_{1} \\ & \quad + 2{\text{S}}_{1} {\text{S}}_{2} \cos \left[ {\left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) - \left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right)} \right] \\ \end{aligned}$$
(E4)
$${\text{N}}_{5} = 0$$
(E5)
$$\begin{aligned} {\text{N}}_{6} & = 2{\text{S}}_{2} - 2{\text{N}}_{15} \cos \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) \\ & \quad - 2{\text{N}}_{16} \sin \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) \\ & \quad - {\text{S}}_{3} {\text{cos}}\left[ {\left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) - \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right)} \right] \\ \end{aligned}$$
(E6)
$$\begin{aligned} {\text{N}}_{7} & = 2{\text{S}}_{3} + 2{\text{N}}_{15} \cos \left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) \\ & \quad + 2{\text{N}}_{16} \sin \left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) \\ & \quad - 2{\text{S}}_{2} {\text{cos}}\left[ {\left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) - \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right)} \right] \\ \end{aligned}$$
(E7)
$$\begin{aligned} {\text{N}}_{8} & = - {\text{S}}_{2}^{2} - {\text{S}}_{3}^{2} - {\text{N}}_{15}^{2} - {\text{N}}_{16}^{2} + \left( {{\text{l}}_{{{\text{BC}}}} + {\Delta l}_{{{\text{BC}}}} } \right)^{2} \\ & \quad - 2\left[ {{\text{N}}_{15} \cos \left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) + {\text{N}}_{16} \sin \left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right)} \right]{\text{S}}_{3} \\ & \quad + 2\left[ {{\text{N}}_{15} \cos \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) + {\text{N}}_{16} \sin \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right)} \right]{\text{S}}_{2} \\ & \quad + {\text{S}}_{2} {\text{S}}_{3} {\text{cos}}\left[ {\left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) - \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right)} \right] \\ \end{aligned}$$
(E9)
$${\text{N}}_{10} = 0$$
(E10)
$$\begin{aligned} {\text{N}}_{11} & = 2{\text{S}}_{3} - 2{\text{N}}_{17} {\text{cos}}\left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) \\ & \quad - 2{\text{N}}_{18} {\text{sin}}\left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) \\ & \quad - 2{\text{S}}_{1} {\text{cos}}\left[ {\left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right) - \left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right)} \right] \\ \end{aligned}$$
(E11)
$$\begin{aligned} {\text{N}}_{12} & = - {\text{S}}_{1}^{2} - {\text{S}}_{3}^{2} - {\text{N}}_{17}^{2} - {\text{N}}_{18}^{2} + \left( {{\text{l}}_{{{\text{AC}}}} + {\Delta l}_{{{\text{AC}}}} } \right)^{2} \\ & \quad - 2\left[ {{\text{N}}_{17} \cos \left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right) + {\text{N}}_{18} \sin \left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right)} \right]{\text{S}}_{1} \\ & \quad + 2\left[ {{\text{N}}_{17} \cos \left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) + {\text{N}}_{18} \sin \left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right)} \right]{\text{S}}_{3} \\ & \quad + 2{\text{S}}_{1} {\text{S}}_{3} {\text{cos}}\left[ {\left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right) - \left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right)} \right] \\ \end{aligned}$$
(E12)
$$\begin{aligned} {\text{N}}_{13} & = \left( {{\text{B}}_{{\text{x}}} + {\Delta B}_{{\text{x}}} } \right) - \left( {{\text{A}}_{{\text{x}}} + {\Delta A}_{{\text{x}}} } \right) \\ & \quad - \left( {{\text{L}}_{2} + {\Delta L}_{2} } \right)\sin \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) \\ & \quad - \left( {{\text{L}}_{1} + {\Delta L}_{1} } \right){\text{sin}}\left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right) \\ \end{aligned}$$
(E13)
$$\begin{aligned} {\text{N}}_{14} & = \left( {{\text{B}}_{{\text{y}}} + {\Delta B}_{{\text{y}}} } \right) - \left( {{\text{A}}_{{\text{y}}} + {\Delta A}_{{\text{y}}} } \right) \\ & \quad + \left( {{\text{L}}_{2} + {\Delta L}_{2} } \right)\cos \left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) \\ & \quad + \left( {{\text{L}}_{1} + {\Delta L}_{1} } \right){\text{cos}}\left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right) \\ \end{aligned}$$
(E14)
$$\begin{aligned} {\text{N}}_{15} & = \left( {{\text{C}}_{{\text{x}}} + {\Delta C}_{{\text{x}}} } \right) - \left( {{\text{B}}_{{\text{x}}} + {\Delta B}_{{\text{x}}} } \right) \\ & \quad - \left( {{\text{L}}_{3} + {\Delta L}_{3} } \right)\sin \left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) \\ & \quad + \left( {{\text{L}}_{2} + {\Delta L}_{2} } \right){\text{sin}}\left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) \\ \end{aligned}$$
(E15)
$$\begin{aligned} {\text{N}}_{16} & = \left( {{\text{C}}_{{\text{y}}} + {\Delta C}_{{\text{y}}} } \right) - \left( {{\text{B}}_{{\text{y}}} + {\Delta B}_{{\text{y}}} } \right) \\ & \quad + \left( {{\text{L}}_{3} + {\Delta L}_{3} } \right)\cos \left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) \\ & \quad - \left( {{\text{L}}_{2} + {\Delta L}_{2} } \right){\text{cos}}\left( {{\uptheta }_{2} + {{\Delta \uptheta }}_{2} } \right) \\ \end{aligned}$$
(E16)
$$\begin{aligned} {\text{N}}_{17} & = \left( {{\text{A}}_{{\text{x}}} + {\Delta A}_{{\text{x}}} } \right) - \left( {{\text{C}}_{{\text{x}}} + {\Delta C}_{{\text{x}}} } \right) \\ & \quad + \left( {{\text{L}}_{1} + {\Delta L}_{1} } \right)\sin \left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right) \\ & \quad + \left( {{\text{L}}_{3} + {\Delta L}_{3} } \right){\text{sin}}\left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) \\ \end{aligned}$$
(E17)
$$\begin{aligned} {\text{N}}_{18} & = \left( {{\text{A}}_{{\text{y}}} + {\Delta A}_{{\text{y}}} } \right) - \left( {{\text{C}}_{{\text{y}}} + {\Delta C}_{{\text{y}}} } \right) \\ & \quad - \left( {{\text{L}}_{1} + {\Delta L}_{1} } \right)\cos \left( {{\uptheta }_{1} + {{\Delta \uptheta }}_{1} } \right) \\ & \quad - \left( {{\text{L}}_{3} + {\Delta L}_{3} } \right){\text{cos}}\left( {{\uptheta }_{3} + {{\Delta \uptheta }}_{3} } \right) \\ \end{aligned}$$
(E18)

Appendix F

The following equations present coefficients defined in Eq. (99) in Sect. 3.3.

$${\text{Q}}_{1} = 2{\text{Q}}_{14}$$
(F1)
$$\begin{aligned} {\text{Q}}_{2} & = 2[{\text{Q}}_{13} \sin \left( {\upgamma _{1} + \Delta\upgamma _{1} } \right) \\ + {\text{Q}}_{14} {\text{cos}}\left( {\upgamma _{1} + \Delta\upgamma _{1} } \right)] \\ \end{aligned}$$
(F2)
$${\text{Q}}_{3} = 0$$
(F3)
$$\begin{aligned} {\text{Q}}_{4} & = \left[ {\left( {{\text{B}}_{{\text{x}}} + \Delta {\text{B}}_{{\text{x}}} } \right) - \left( {{\text{A}}_{{\text{x}}} + \Delta {\text{A}}_{{\text{x}}} } \right)} \right]^{2} \\ & \quad + \left[ {\left( {{\text{B}}_{{\text{y}}} + \Delta {\text{B}}_{{\text{y}}} } \right) - \left( {{\text{A}}_{{\text{y}}} + \Delta {\text{A}}_{{\text{y}}} } \right)} \right]^{2} \\ & \quad - {\text{Q}}_{13}^{2} - {\text{Q}}_{14}^{2} \\ \end{aligned}$$
(F4)
$${\text{Q}}_{5} = 0$$
(F5)
$${\text{Q}}_{6} = 2{\text{Q}}_{16}$$
(F6)
$$\begin{aligned} {\text{Q}}_{7} & = 2[{\text{Q}}_{15} \sin \left( {\upgamma _{2} + \Delta\upgamma _{2} } \right) \\ & \quad + {\text{Q}}_{16} {\text{cos}}\left( {\upgamma _{2} + \Delta\upgamma _{2} } \right)] \\ \end{aligned}$$
(F7)
$$\begin{aligned} {\text{Q}}_{8} & = \left[ {\left( {{\text{C}}_{{\text{x}}} + \Delta {\text{C}}_{{\text{x}}} } \right) - \left( {{\text{B}}_{{\text{x}}} + \Delta {\text{B}}_{{\text{x}}} } \right)} \right]^{2} \\ & \quad + \left[ {\left( {{\text{C}}_{{\text{y}}} + \Delta {\text{C}}_{{\text{y}}} } \right) - \left( {{\text{B}}_{{\text{y}}} + \Delta {\text{B}}_{{\text{y}}} } \right)} \right]^{2} \\ & \quad - {\text{Q}}_{15}^{2} - {\text{Q}}_{16}^{2} \\ \end{aligned}$$
(F8)
$$\begin{aligned} {\text{Q}}_{9} & = 2[{\text{Q}}_{17} \sin \left( {\upgamma _{3} + \Delta\upgamma _{3} } \right) \\ & \quad + {\text{Q}}_{18} {\text{cos}}\left( {\upgamma _{3} + \Delta\upgamma _{3} } \right)] \\ \end{aligned}$$
(F9)
$${\text{Q}}_{10} = 0$$
(F10)
$${\text{Q}}_{11} = 2{\text{Q}}_{18}$$
(F11)
$$\begin{aligned} {\text{Q}}_{12} & = \left[ {\left( {{\text{A}}_{{\text{x}}} + \Delta {\text{A}}_{{\text{x}}} } \right) - \left( {{\text{C}}_{{\text{x}}} + \Delta {\text{C}}_{{\text{x}}} } \right)} \right]^{2} \\ & \quad + \left[ {\left( {{\text{A}}_{{\text{y}}} + \Delta {\text{A}}_{{\text{y}}} } \right) - \left( {{\text{C}}_{{\text{y}}} + \Delta {\text{C}}_{{\text{y}}} } \right)} \right]^{2} \\ & \quad - {\text{Q}}_{17}^{2} - {\text{Q}}_{18}^{2} \\ \end{aligned}$$
(F12)
$$\begin{aligned} {\text{Q}}_{13} & = \left( {{\text{O}}_{1} + \Delta {\text{O}}_{1} } \right) + {\text{S}}_{2}^{{\prime}} \sin \left( {\upgamma _{1} + \Delta\upgamma _{1} } \right) \\ & \quad + \left( {{\text{O}}_{2} + \Delta {\text{O}}_{2} } \right){\text{cos}}\left( {\upgamma _{1} + \Delta\upgamma _{1} } \right) \\ \end{aligned}$$
(F13)
$$\begin{aligned} {\text{Q}}_{14} & = {\text{S}}_{1}^{{\prime}} - \left( {{\text{L}}_{1} + \Delta {\text{L}}_{1} } \right) + {\text{S}}_{2}^{{\prime}} \cos \left( {\upgamma _{1} + \Delta\upgamma _{1} } \right) \\ & \quad - \left( {{\text{O}}_{2} + \Delta {\text{O}}_{2} } \right){\text{sin}}\left( {\upgamma _{1} + \Delta\upgamma _{1} } \right) \\ \end{aligned}$$
(F14)
$$\begin{aligned} {\text{Q}}_{15} & = \left( {{\text{O}}_{2} + \Delta {\text{O}}_{2} } \right) + {\text{S}}_{3}^{{\prime}} \sin \left( {\upgamma _{2} + \Delta\upgamma _{2} } \right) \\ & \quad + \left( {{\text{O}}_{3} + \Delta {\text{O}}_{3} } \right){\text{cos}}\left( {\upgamma _{2} + \Delta\upgamma _{2} } \right) \\ \end{aligned}$$
(F15)
$$\begin{aligned} {\text{Q}}_{16} & = {\text{S}}_{2}^{{\prime}} - \left( {{\text{L}}_{2} + \Delta {\text{L}}_{2} } \right) + {\text{S}}_{3}^{{\prime}} \cos \left( {\upgamma _{2} + \Delta\upgamma _{2} } \right) \\ & \quad - \left( {{\text{O}}_{3} + \Delta {\text{O}}_{3} } \right){\text{sin}}\left( {\upgamma _{2} + \Delta\upgamma _{2} } \right) \\ \end{aligned}$$
(F16)
$$\begin{aligned} {\text{Q}}_{17} & = \left( {{\text{O}}_{3} + \Delta {\text{O}}_{3} } \right) + {\text{S}}_{1}^{{\prime}} \sin \left( {\upgamma _{3} + \Delta\upgamma _{3} } \right) \\ & \quad + \left( {{\text{O}}_{1} + \Delta {\text{O}}_{1} } \right){\text{cos}}\left( {\upgamma _{3} + \Delta\upgamma _{3} } \right) \\ \end{aligned}$$
(F17)
$$\begin{aligned} {\text{Q}}_{18} & = {\text{S}}_{3}^{{\prime}} - \left( {{\text{L}}_{3} + \Delta {\text{L}}_{3} } \right) + {\text{S}}_{1}^{{\prime}} \cos \left( {\upgamma _{3} + \Delta\upgamma _{3} } \right) \\ & \quad - \left( {{\text{O}}_{1} + \Delta {\text{O}}_{1} } \right){\text{sin}}\left( {\upgamma _{3} + \Delta\upgamma _{3} } \right) \\ \end{aligned}$$
(F18)

The following equations present coefficients defined in Eqs. (100) in Sect. 3.3.

$$\begin{aligned} {\text{Q}}_{19} & = \left( {{\text{O}}_{1} + \Delta {\text{O}}_{1} } \right) \\ & \quad + \left[ {\left( {{\text{L}}_{3} + \Delta {\text{L}}_{3} } \right) - \left( {{\text{S}}_{3}^{{\prime}} + \Delta {\text{S}}_{3}^{{\prime}} } \right)} \right] \\ & \quad \sin \left( {\upgamma _{3} + \Delta\upgamma _{3} } \right) + \left( {{\text{O}}_{3} + {\Delta O}_{3} } \right) \\ & \quad {\text{cos}}\left( {\upgamma _{3} + \Delta\upgamma _{3} } \right) \\ \end{aligned}$$
(F19)
$$\begin{aligned} {\text{Q}}_{20} & = - \left( {{\text{S}}_{1}^{{\prime}} + \Delta {\text{S}}_{1}^{{\prime}} } \right) \\ & \quad + \left[ {\left( {{\text{L}}_{3} + \Delta {\text{L}}_{3} } \right) - \left( {{\text{S}}_{3}^{{\prime}} + \Delta {\text{S}}_{3}^{{\prime}} } \right)} \right] \\ & \quad \cos \left( {\upgamma _{3} + \Delta\upgamma _{3} } \right) - \left( {{\text{O}}_{3} + \Delta {\text{O}}_{3} } \right) \\ & \quad {\text{sin}}\left( {\upgamma _{3} + \Delta\upgamma _{3} } \right) \\ \end{aligned}$$
(F20)

The following equation presents S +ΔS defined in Eqs. (102)-(103) in Sect. 3.3.

$$\begin{aligned} \left( {{\text{S}}_{1} + \Delta {\text{S}}_{1} } \right) & = \sin \left( {\upbeta _{3} + \Delta\upbeta _{3} } \right)\frac{{\left( {l_{{{\text{AC}}}} + \Delta l_{{{\text{AC}}}} } \right)}}{{\sin \left( {\upgamma _{3} + \Delta \upgamma \upgamma _{3} } \right)}} \\ & \quad - \left( {{\text{S}}_{1}^{{\prime}} + \Delta {\text{S}}_{1}^{{\prime}} } \right) \\ \end{aligned}$$
(F21)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hsu, KL., Chang, CH. Modularized Analysis of Kinematic and Mechanical Error for Planar Linkages Composed of Class 3 and Order 3 Assur Groups. Int. J. Precis. Eng. Manuf. 24, 1161–1190 (2023). https://doi.org/10.1007/s12541-023-00804-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12541-023-00804-9

Keywords

Search

Navigation