Higher order kinematic formulas and its numerical computation employing dual numbers

References

• Abderazek, H., A. R. Yildiz, and S. Mirjalili. 2020. Comparison of recent optimization algorithms for design optimization of a cam-follower mechanism. Knowledge-Based Systems 191:105237. doi:10.1016/j.knosys.2019.105237.
   Web of Science ®Google Scholar
• Abreu, R., D. Stich, and J. Morales. 2013. On the generalization of the complex step method. Journal of Computational and Applied Mathematics 241:84–102. doi:10.1016/j.cam.2012.10.001.
   Web of Science ®Google Scholar
• Al-Mohy, A. H., and N. J. Higham. 2010. The complex step approximation to the Fréchet derivative of a matrix function. Numerical Algorithms 53 (1):133–48. doi:10.1007/s11075-009-9323-y.
   Web of Science ®Google Scholar
• Ball, R. S. 1900. A Treatise on the Theory of Screws. Cambridge: University of Cambridge.
   Google Scholar
• Barre, P.-J., R. Bearee, P. Borne, and E. Dumetz. 2005. Influence of a jerk controlled movement law on the vibratory behaviour of high-dynamics systems. Journal of Intelligent and Robotic Systems 42 (3):275–93. doi:10.1007/s10846-004-4002-7.
   Web of Science ®Google Scholar
• Baydin, A. G., B. A. Pearlmutter, A. Andreyevich Radul, and J. M. Siskind. 2018. Automatic differentiation in machine learning: Survey. Journal of Machine Learning Research 18 (153):1–43. http://jmlr.org/papers/v18/17-468.html.
   Google Scholar
• Bornemann, F. 2011. Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals. Foundations of Computational Mathematics 11 (1):1–63. doi:10.1007/s10208-010-9075-z.
   Web of Science ®Google Scholar
• Bücker, H. M., and G. F. Corliss. 2006. A bibliography of automatic differentiation. In Automatic Differentiation: Applications, Theory, and Implementations, edited by Martin Bücker, George Corliss, Uwe Naumann, Paul Hovland, and Boyana Norris, pp. 321–2. Berlin, Heidelberg: Springer.
   Google Scholar
• Caliò, F., M. Frontini, and G. V. Milovanović. 1991. Numerical differentiation of analytic functions using quadratures on the semicircle. Computers & Mathematics with Applications 22 (10):99–106. doi:10.1016/0898-1221(91)90196-B.
   Web of Science ®Google Scholar
• Cervantes-Sánchez, J. J., J. M. Rico-Martínez, and V. H. Pérez-Muñoz. 2014. On the angular velocity of a rigid body: Matrix and vector representations. European Journal of Mechanics - A/Solids 45:123–32. doi:10.1016/j.euromechsol.2013.11.016.
   Web of Science ®Google Scholar
• Chen, G., L. Zhang, C. Wang, H. Xiang, G. Tong, and D. Zhao. 2022. High-precision modeling of CNCs’ spatial errors based on screw theory. SN Applied Sciences 4 (2):45–59. doi:10.1007/s42452-021-04929-2.
   Web of Science ®Google Scholar
• Cheng, H. H. 1994. Programming with dual numbers and its applications in mechanisms design. Engineering with Computers 10 (4):212–29. doi:10.1007/BF01202367.
   Web of Science ®Google Scholar
• Cheng, Q., Q. Feng, Z. Liu, P. Gu, and G. Zhang. 2016. Sensitivity analysis of machining accuracy of multi-axis machine tool based on POE screw theory and Morris method. The International Journal of Advanced Manufacturing Technology 84 (9–12):2301–18. doi:10.1007/s00170-015-7791-x.
   Web of Science ®Google Scholar
• Clifford, W. K. 1873. Preliminary sketch of biquaternions. Proc. London Mathematical Society 1 (1–4):381–95.
   Google Scholar
• Cline, D. 2017. Variational principles in classical mechanics. Rochester, NY: University of Rochester River Campus Librarie.
   Google Scholar
• Dhingra, A. K., and N. K. Mani. 1993. Finitely and multiply separated synthesis of Link and Geared mechanisms using symbolic computing. Journal of Mechanical Design 115 (3):560–7. doi:10.1115/1.2919226.
   Web of Science ®Google Scholar
• Diez-Martínez, C. R., J. M. Rico, J. J. Cervantes-Sánchez, and J. Gallardo. 2006. Mobility and connectivity in multiloop linkages. In Advances in Robot Kinematics, edited by Jadran Lennarčič and B. Roth, Dordrecht, pp. 455–64. Netherlands: Springer.
   Google Scholar
• Eager, D., A.-M. Pendrill, and N. Reistad. 2016. Beyond velocity and acceleration: Jerk, snap and higher derivatives. European Journal of Physics 37 (6):65008. doi:10.1088/0143-0807/37/6/065008.
   Web of Science ®Google Scholar
• Erkorkmaz, K., and Y. Altintas. 2001. High speed CNC system design. Part I: Jerk limited trajectory generation and quintic spline interpolation. International Journal of Machine Tools and Manufacture 41 (9):1323–45. doi:10.1016/S0890-6955(01)00002-5.
   Web of Science ®Google Scholar
• Fang, Y., J. Qi, J. Hu, W. Wang, and Y. Peng. 2020. An approach for jerk-continuous trajectory generation of robotic manipulators with kinematical constraints. Mechanism and Machine Theory 153:103957. doi:10.1016/j.mechmachtheory.2020.103957.
   Web of Science ®Google Scholar
• Fenton, R. G., and R. A. Willgoss. 1990. Comparison of methods for determining screw parameters of infinitesimal rigid body motion from position and velocity data. Journal of Dynamic Systems, Measurement, and Control 112 (4):711–6. doi:10.1115/1.2896199.
   Web of Science ®Google Scholar
• Figliolini, G., and C. Lanni. 2016. Jerk and jounce relevance for the kinematic performance of Long-Dwell mechanisms. Advances in Mechanism and Machine Science 73:219–28.
   Google Scholar
• Fike, J., and J. Alonso. 2011. The development of hyper-dual numbers for exact second-derivative calculations, AIAA2011-886. 49th AIAA Aerospace Sciences Meeting including the NewHorizons Forum and Aerospace Exposition. doi:10.2514/6.2011–886.
   Google Scholar
• Fike, J. A., and J. J. Alonso. 2012. Automatic differentiation through the use of hyper-dual numbers for second derivatives. In Recent Advances in Algorithmic Differentiation, edited by Shaun Forth, Paul Hovland, Eric Phipps, Jean Utke, and Andrea Walther, pp. 163–73. Berlin Heidelberg: Springer.
   Google Scholar
• Floudas, A., and P. M. Pardalos. 2009. Encyclopedia of Optimisation, 2nd Edition. Boston. MA: Springer-Verlag.
   Google Scholar
• Fornberg, B. 1981. Numerical differentiation of analytic functions. ACM Transactions on Mathematical Software 7 (4):512–26. doi:10.1145/355972.355979.
   Web of Science ®Google Scholar
• Gallardo-Alvarado, J. 2014. Hyper-Jerk analysis of robot manipulators. Journal of Intelligent & Robotic Systems 74 (3–4):625–41. doi:10.1007/s10846-013-9849-z.
   Web of Science ®Google Scholar
• Gasparetto, A., and V. Zanotto. 2008. A technique for time-jerk optimal planning of robot trajectories. Robotics and Computer-Integrated Manufacturing 24 (3):415–26. doi:10.1016/j.rcim.2007.04.001.
   Web of Science ®Google Scholar
• Giftthaler, M., M. Neunert, M. Stäuble, M. Frigerio, C. Semini, and J. Buchli. 2017. Automatic differentiation of rigid body dynamics for optimal control and estimation. Advanced Robotics 31 (22):1225–37. doi:10.1080/01691864.2017.1395361.
   Web of Science ®Google Scholar
• Gilat, A., and V. Subramaniam. 2008. Numerical methods for engineers and scientists: An introduction with applications using MATLAB. 1st ed. United States of America: John Wiley\& Sons.
   Google Scholar
• Griewank, A. 1989. Mathematical programming: Recent developments and applications. In On Automatic Differentiation, edited by M. Iri and K. Tanabe, pp. 83–108. Dordrecht: Kluwer Academic Publishers.
   Google Scholar
• Huang, J., P. Hu, K. Wu, and M. Zeng. 2018. Optimal time-jerk trajectory planning for industrial robots. Mechanism and Machine Theory 121:530–44. doi:10.1016/j.mechmachtheory.2017.11.006.
   Web of Science ®Google Scholar
• Rico, J. M., and A. J. Gallardo. 1996. Acceleration analysis, via screw theory and characterization of singularities of closed chains, pp. 139–48. Dordrecht: Springer Netherlands.
   Google Scholar
• Kiran, R., and K. Khandelwal. 2014. Numerically approximated Cauchy integral (NACI) for implementation of constitutive models. Finite Elements in Analysis and Design 89:33–51. doi:10.1016/j.finel.2014.05.016.
   Web of Science ®Google Scholar
• Lai, K.-L., and J. L. Crassidis. 2008. Extensions of the first and second complex-step derivative approximations. Journal of Computational and Applied Mathematics 219 (1):276–93. doi:10.1016/j.cam.2007.07.026.
   Web of Science ®Google Scholar
• Lerbet, J. 1998. Analytic geometry and singularities of mechanisms. ZAMM 78 (10):687–94. doi:10.1002/(SICI)1521-4001(199810)78:10<687::AID-ZAMM687>3.0.CO;2-T.
   Web of Science ®Google Scholar
• López-Custodio, P. C., J. M. Rico, J. J. Cervantes-Sánchez, G. I. Pérez-Soto, and C. R. Díez-Martínez. 2017. Verification of the higher order kinematic analyses equations. European Journal of Mechanics—A/Solids 61 (1):198–215. doi:10.1016/j.euromechsol.2016.09.010.
   Google Scholar
• Martins, J. R. R. A., P. Sturdza, and J. J. Alonso. 2003. The complex-step derivative approximation. ACM Transactions on Mathematical Software 29 (3):245–62. doi:10.1145/838250.838251.
   Web of Science ®Google Scholar
• Martins, J. R. R. A., and J. T. Hwang. 2013. Review and unification of methods for computing derivatives of multidisciplinary computational models. AIAA Journal 51 (11):2582–99. doi:10.2514/1.J052184.
   Web of Science ®Google Scholar
• Müller, A. 2014. Higher derivatives of the kinematic mapping and some applications. Mechanism and Machine Theory 76:70–85. doi:10.1016/j.mechmachtheory.2014.01.007.
   Web of Science ®Google Scholar
• Müller, A. 2018. Higher-order analysis of kinematic singularities of lower pair linkages and serial manipulators. Journal of Mechanisms and Robotics 10 (1) doi:10.1115/1.4038528.
   Google Scholar
• Müller, A. 2019. An overview of formulae for the higher-order kinematics of lower-pair chains with applications in robotics and mechanism theory. Mechanism and Machine Theory 142 (12):103594. doi:10.1016/j.mechmachtheory.2019.103594.
   Google Scholar
• Müller, A., and S. Kumar. 2021. Closed-form time derivatives of the equations of motion of rigid body systems. Multibody System Dynamics 53 (3):257–73. doi:10.1007/s11044-021-09796-8.
   Web of Science ®Google Scholar
• Nam, S.-H., and Y. Min-Yang. 2004. A study on a generalized parametric interpolator with real-time jerk-limited acceleration. Computer-Aided Design 36 (1):27–36. doi:10.1016/S0010-4485(03)00066-6.
   Web of Science ®Google Scholar
• Peñuñuri, F., O. Carvente, M. A. Zambrano-Arjona, R. Peón, and C. A. Cruz-Villar. 2019. Dual numbers for algorithmic differentiation. Ingeniría–Revista Académica de la Facultad de Ingeniría, Universidad Autónoma de Yucatán 23 (3):71–81.
   Google Scholar
• Peñuñuri, F., R. Peón, D. González-Sánchez, and M. A. Escalante-Soberanis. 2020. Dual numbers and automatic differentiation to efficiently compute velocities and accelerations. Acta Applicandae Mathematicae 170 (1):649–59. doi:10.1007/s10440-020-00351-9.
   Web of Science ®Google Scholar
• Peñuñuri, F., R. Peón-Escalante, and A. Espinosa-Romero. 2022. Data availability: Higher order kinematic formulas and its numerical computation employing dual numbers. doi:10.5281/zenodo.6615524.
   Google Scholar
• Peón-Escalante, R., F. Cuenca Jiménez, M. A. Escalante Soberanis, and F. Peñuñuri. 2020. Path generation with dwells in the optimum dimensional synthesis of Stephenson III six-bar mechanisms. Mechanism and Machine Theory 144:103650. doi:10.1016/j.mechmachtheory.2019.103650.
   Web of Science ®Google Scholar
• Rico, J. M., and J. Duffy. 1996. An application of screw algebra to the acceleration analysis of serial chains. Mechanism and Machine Theory 31 (4):445–57. doi:10.1016/0094-114X(95)00089-H.
   Web of Science ®Google Scholar
• Rico, J. M., J. Gallardo, and J. Duffy. 1999. Screw theory and higher order kinematic analysis of open serial and closed chains. Mechanism and Machine Theory 34 (4):559–86. doi:10.1016/S0094-114X(98)00029-9.
   Web of Science ®Google Scholar
• Sandor, G. N., R. E. Kaufman, A. G. Erdman, T. J. Foster, J. P. Sadler, and T. N. Kershaw. 1970. Kinematic synthesis of geared linkages. Journal of Mechanisms 5 (1):59–87. doi:10.1016/0022-2569(70)90052-2.
   Google Scholar
• Sateesh, N., C. S. Rao, and T. A. Janardhan Reddy. 2009. Optimisation of cam-follower motion using B-splines. International Journal of Computer Integrated Manufacturing 22 (6):515–23. doi:10.1080/09511920802546814.
   Web of Science ®Google Scholar
• Schot, S. H. 1978. Jerk: The time rate of change of acceleration. American Journal of Physics 46 (11):1090–4. doi:10.1119/1.11504.
   Web of Science ®Google Scholar
• Squire, W., and G. Trapp. 1998. Using complex variables to estimate derivatives of real functions. SIAM Review 40 (1):110–2. doi:10.1137/S003614459631241X.
   Web of Science ®Google Scholar
• Sun, T., B. Lian, S. Yang, and Y. Song. 2020. Kinematic calibration of serial and parallel robots based on finite and instantaneous screw theory. IEEE Transactions on Robotics 36 (3):816–34. doi:10.1109/TRO.2020.2969028.
   Web of Science ®Google Scholar
• Szirmay-Kalos, L. 2020. Higher order automatic differentiation with dual numbers. Periodica Polytechnica Electrical Engineering and Computer Science 65 (1):1–10. doi:10.3311/PPee.16341.
   Google Scholar
• Valderrama-Rodríguez, J. I., J. M. Rico, and J. J. Cervantes-Sánchez. 2022. A screw theory approach to compute instantaneous rotation axes of indeterminate spherical linkages. Mechanics Based Design of Structures and Machines 50 (8):2836–76. doi:10.1080/15397734.2020.1787841.
   Web of Science ®Google Scholar
• Verulkar, A., C. Sandu, D. Dopico, and A. Sandu. 2022. Computation of direct sensitivities of spatial multibody systems with joint friction. Journal of Computational and Nonlinear Dynamics 17 (7):071006. doi:10.1115/1.4054110.
   Web of Science ®Google Scholar
• Xianwen, K., H. Yongguo, and Y. Tingli. 1993. Kinematic error analysis of robots. Robot 15 (3):7.
   Google Scholar
• Yin, F. W., W. J. Tian, H. T. Liu, T. Huang, and D. G. Chetwynd. 2020. A screw theory based approach to determining the identifiable parameters for calibration of parallel manipulators. Mechanism and Machine Theory 145:103665. doi:10.1016/j.mechmachtheory.2019.103665.
   Web of Science ®Google Scholar
• Yu, W., and M. Blair. 2013. DNAD, a simple tool for automatic differentiation of Fortran codes using dual numbers. Computer Physics Communications 184 (5):1446–52. doi:10.1016/j.cpc.2012.12.025.
   Web of Science ®Google Scholar