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High-order inverse dynamics of serial robots based on projective geometric algebra

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Abstract

The efficient algorithm for high-order inverse dynamics of serial robots is an essential need in design and model-based control of robots equipped with serial elastic joints. Although several efficient algorithms have been proposed, the introduction of new frameworks can lead to new understanding and further improvements. Based on the projective geometric algebra (PGA) for Euclidean geometry, this paper provides a recursive algorithm that is computationally efficient, intuitive, uniform, and coordinate invariant. In the PGA-based method, calculations of the exponential map and Lie brackets are simplified and rigid body motions are represented as vectors instead of matrices. All geometric elements used to model robots are represented as vectors uniformly, and all operations are modeled as algebraic operations with explicit geometric meaning. The validation of the algorithm is presented for the second-order inverse dynamics of the Franka Emika Panda using the algorithm based on PGA and the algorithm based on product of exponentials (POE) respectively. The proposed algorithm is 15% faster than the POE-based algorithm with correct results. It turns out that the kinematics part of the algorithm saves 69.82% multiplications and 73.58% additions than that in the POE-based algorithm. The relation between PGA and other popular concepts in robotics, such as dual quaternions, is also discussed.

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  1. The data file can be downloaded from https://ieeexplore.ieee.org/ielx7/7083369/9285111/9290369/lra-3044028-mm.zip?arnumber=9290369.

References

  1. Gattringer, H., Springer, K., Müller, A., Jörgl, M.: An efficient method for the dynamical modeling of serial elastic link/joint robots. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) Computer Aided Systems Theory – EUROCAST 2015. Lecture Notes in Computer Science, pp. 689–697. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-27340-2_85

    Chapter  Google Scholar 

  2. Müller, A.: An o(n)-algorithm for the higher-order kinematics and inverse dynamics of serial manipulators using spatial representation of twists. IEEE Robot. Autom. Lett. 6(2), 397–404 (2021). https://doi.org/10.1109/LRA.2020.3044028

    Article  Google Scholar 

  3. Oaki, J., Adachi, S.: Vibration suppression control of elastic-joint robot arm based on designless nonlinear state observer. IEEJ Trans. Ind. Appl. 135(5), 571–581 (2015). https://doi.org/10.1541/ieejias.135.571

    Article  Google Scholar 

  4. Bruzzone, L., Bozzini, G.: A statically balanced SCARA-like industrial manipulator with high energetic efficiency. Meccanica 46(4), 771–784 (2011). https://doi.org/10.1007/s11012-010-9336-6

    Article  MATH  Google Scholar 

  5. Feng, H., Xu, Y., Mao, D.: A novel adaptive balance-drive mechanism for industrial robots using a series elastic actuator. Int. J. Comput. Integr. Manuf. 33(10–11), 991–1005 (2020). https://doi.org/10.1080/0951192X.2020.1736715

    Article  Google Scholar 

  6. Denavit, J., Hartenberg, R.S.: A kinematic notation for lower-pair mechanisms based on matrices. J. Appl. Mech. 22(2), 215–221 (1955). https://doi.org/10.1115/1.4011045

    Article  MathSciNet  MATH  Google Scholar 

  7. Lynch, K.M., Park, F.C.: Modern Robotics. Cambridge University Press, Cambridge (2017)

    Google Scholar 

  8. Chen, G., Kong, L., Li, Q., Wang, H., Lin, Z.: Complete, minimal and continuous error models for the kinematic calibration of parallel manipulators based on POE formula. Mech. Mach. Theory 121, 844–856 (2018). https://doi.org/10.1016/j.mechmachtheory.2017.11.003

    Article  Google Scholar 

  9. Selig, J.M.: Geometrical Foundations of Robotics. Springer, New York (2005). https://doi.org/10.1007/b138859

    Book  MATH  Google Scholar 

  10. Müller, A., Kumar, S.: Closed-form time derivatives of the equations of motion of rigid body systems. Multibody Syst. Dyn. 53(3), 257–273 (2021). https://doi.org/10.1007/s11044-021-09796-8

    Article  MathSciNet  MATH  Google Scholar 

  11. Müller, A.: An overview of formulae for the higher-order kinematics of lower-pair chains with applications in robotics and mechanism theory. Mech. Mach. Theory 142, 103594 (2019). https://doi.org/10.1016/j.mechmachtheory.2019.103594

    Article  Google Scholar 

  12. Singh, S., Russell, R.P., Wensing, P.M.: Efficient analytical derivatives of rigid-body dynamics using spatial vector algebra. IEEE Robot. Autom. Lett. 7(2), 1776–1783 (2022). https://doi.org/10.1109/lra.2022.3141194

    Article  Google Scholar 

  13. Cibicik, A., Egeland, O.: Kinematics and dynamics of flexible robotic manipulators using dual screws. IEEE Trans. Robot. 37(1), 206–224 (2021). https://doi.org/10.1109/tro.2020.3014519

    Article  Google Scholar 

  14. Silva, F.F.A., Quiroz-Omaña, J.J., Adorno, B.V.: Dynamics of mobile manipulators using dual quaternion algebra. J. Mech. Robot. 14(6), 061005 (2022). https://doi.org/10.1115/1.4054320

    Article  Google Scholar 

  15. Crowe, M.G., Williamson, R.E.: A History of Vector Analysis, vol. 37, p. 844 (1969). https://doi.org/10.1119/1.1975883

    Book  Google Scholar 

  16. Hestenes, D.: Spacetime physics with geometric algebra. Am. J. Phys. 71(7), 691–714 (2003). https://doi.org/10.1119/1.1571836

    Article  Google Scholar 

  17. Doran, C., Gullans, S.R., Lasenby, A., Lasenby, J., Fitzgerald, W.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003). https://doi.org/10.1017/CBO9780511807497

    Book  MATH  Google Scholar 

  18. Lopes, W.B., Lopes, C.G.: Geometric-algebra adaptive filters. IEEE Trans. Signal Process. 67(14), 3649–3662 (2019). https://doi.org/10.1109/TSP.2019.2916028

    Article  MathSciNet  MATH  Google Scholar 

  19. Bayro-Corrochano, E., Zamora-Esquivel, J.: Differential and inverse kinematics of robot devices using conformal geometric algebra. Robotica 25(1), 43–61 (2007). https://doi.org/10.1017/S0263574706002980

    Article  Google Scholar 

  20. Bayro-Corrochano, E., Reyes-Lozano, L., Zamora-Esquivel, J.: Conformal geometric algebra for robotic vision. J. Math. Imaging Vis. 24(1), 55–81 (2006). https://doi.org/10.1007/s10851-005-3615-1

    Article  MathSciNet  MATH  Google Scholar 

  21. Bayro-Corrochano, E.: Robot modeling and control using the motor algebra framework. In: 2019 12th International Workshop on Robot Motion and Control (RoMoCo), pp. 1–8 (2019). https://doi.org/10.1109/RoMoCo.2019.8787386

    Chapter  Google Scholar 

  22. Selig, J.M., Bayro-Corrochano, E.: Rigid body dynamics using Clifford algebra. Adv. Appl. Clifford Algebras 20(1), 141–154 (2008). https://doi.org/10.1007/s00006-008-0144-1

    Article  MathSciNet  MATH  Google Scholar 

  23. Hestenes, D., Fasse, E.D.: Homogeneous Rigid Body Mechanics with Elastic Coupling. Birkhäuser, Boston (2002). https://doi.org/10.1007/978-1-4612-0089-5_19.

    Book  MATH  Google Scholar 

  24. Gunn, C.G.: Doing Euclidean plane geometry using projective geometric algebra. Adv. Appl. Clifford Algebras 27(2), 1203–1232 (2017). https://doi.org/10.1007/s00006-016-0731-5

    Article  MathSciNet  MATH  Google Scholar 

  25. Gunn, C.: Geometric algebras for Euclidean geometry. Adv. Appl. Clifford Algebras 27(1), 185–208 (2017). https://doi.org/10.1007/s00006-016-0647-0

    Article  MathSciNet  MATH  Google Scholar 

  26. Gunn, C.: Geometry, kinematics, and rigid body mechanics in Cayley-Klein geometries. Doctoral thesis, Technische Universität Berlin, Fakultät II - Mathematik und Naturwissenschaften, Berlin (2011). https://doi.org/10.14279/depositonce-3058

  27. Hadfield, H., Lasenby, J.: Chap. 39. Constrained dynamics in conformal and projective geometric algebra. In: Computer Graphics International Conference. Lecture Notes in Computer Science, pp. 459–471 (2020). https://doi.org/10.1007/978-3-030-61864-3_39

    Chapter  Google Scholar 

  28. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60. Springer, New York (1978). https://doi.org/10.1007/978-1-4757-1693-1

    Book  Google Scholar 

  29. Pottmann, H., Wallner, J.: Computational Line Geometry. Mathematics and Visualization. Springer, Berlin (2001). https://doi.org/10.1007/978-3-642-04018-4

    Book  MATH  Google Scholar 

  30. Gunn, C.G.: Course notes geometric algebra for computer graphics, SIGGRAPH 2019. In: ACM SIGGRAPH 2019 Courses, pp. 1–140 (2019). https://doi.org/10.1145/3305366.3328099

    Chapter  Google Scholar 

  31. Gaz, C., Cognetti, M., Oliva, A., Robuffo Giordano, P., De Luca, A.: Dynamic identification of the Franka Emika Panda robot with retrieval of feasible parameters using penalty-based optimization. IEEE Robot. Autom. Lett. 4(4), 4147–4154 (2019). https://doi.org/10.1109/lra.2019.2931248

    Article  Google Scholar 

  32. Angeles, J.: Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms. Mechanical Engineering Series, vol. 124. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-01851-5

    Book  MATH  Google Scholar 

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Funding

This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 51935010, 51822506).

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  1. State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Guangzhen Sun & Ye Ding

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  1. Guangzhen Sun
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Sun, G., Ding, Y. High-order inverse dynamics of serial robots based on projective geometric algebra. Multibody Syst Dyn 59, 337–362 (2023). https://doi.org/10.1007/s11044-023-09915-7

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