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Contact constraints-based dynamic manipulation control of the multi-fingered hand robot: a force sensorless approach

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Abstract

A novel analytical modeling and dynamic manipulation control method of the multi-fingered hand robot has been proposed in the paper. Based on the contact constraints, the explicit dynamics modeling of the hand robot manipulating an object is hierarchically established by Udwadia–Kalaba equation with no auxiliary variable (e.g., Lagrange multipliers or quasi-generalized variables). Through the second order of the contact constraints, the grasping forces of the hand robot in the manipulation work space are derived explicitly and decoupled with the control torques of the finger joints. Consider the hand robot and the object as an entire system in the control design. Motivated by Udwadia’s work, the manipulation task of the grasped object is novelly used to formulate a set of servo constraints. In virtue of following the servo constraints, the hand robot can manipulate the object to accomplish the desired task. With the formulated contact forces model, a model-based dynamic control method is proposed for the hand robot to handle the object, which does not depend on the force feedback from the fingertips sensors (force sensorless). The system performance under the proposed control can be guaranteed by the theoretical proofs and demonstrated by the simulation of a three-fingered hand robot in the three-dimensional work space.

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References

  1. Devi, M.A., Udupa, G., Sreedharan, P.: A novel underactuated multi-fingered soft robotic hand for prosthetic application. Robot. Auton. Syst. 100, 267–277 (2018)

    Article  Google Scholar 

  2. Zhang, T., Jiang, L., Liu, H.: Design and functional evaluation of a dexterous myoelectric hand prosthesis with biomimetic tactile sensor. IEEE Trans. Neural Syst. Rehabil. Eng. 26(7), 1391–1399 (2018)

    Article  Google Scholar 

  3. Zhao, Z., Li, D., Gao, S., et al.: Development of a dexterous hand for space service. In: 2016 IEEE International Conference on Robotics and Biomimetics (ROBIO), pp. 408–412. Qingdao, China (2016)

  4. Jin, X., Fang, Y., Zhang, D., et al.: Synthesis of 3-[P][S] parallel mechanism-inspired multimode dexterous hands with parallel finger structure. J. Mech. Des. 142(8), 1–39 (2020)

    Article  Google Scholar 

  5. Rojas, N., Ma, R.R., Dollar, A.M.: The GR2 gripper: an underactuated hand for open-loop in-hand planar manipulation. IEEE Trans. Rob. 32(3), 763–770 (2016)

    Article  Google Scholar 

  6. Jin, X., Fang, Y., Zhang, D., et al.: Design of dexterous hands based on parallel finger structures. Mech. Mach. Theory 152, 103952 (2020)

    Article  Google Scholar 

  7. Honarpardaz, M., Alvander, J., Tarkian, M.: Fast finger design automation for industrial robots. Robot. Auton. Syst. 113, 120–131 (2019)

    Article  Google Scholar 

  8. Lippiello, V., Ruggiero, F., Villani, L.: Inverse kinematics for object manipulation with redundant multi-fingered robotic hands. Robot Motion and Control 2009, pp. 255–264. Springer, London (2009)

  9. Cui, L., Sun, J., Dai, J.: In-hand forward and inverse kinematics with rolling contact. Robotica 35(12), 2381 (2017)

    Article  Google Scholar 

  10. Sahbani, A., Khoury, S.E., Bidaud, P.: An overview of 3D object grasp synthesis algorithms. Robot. Auton. Syst. 60(3), 326–336 (2012)

    Article  Google Scholar 

  11. Sundaralingam, B., Hermans, T.: Relaxed-rigidity constraints: kinematic trajectory optimization and collision avoidance for in-grasp manipulation. Auton. Robot. 43(2), 469–483 (2019)

    Article  Google Scholar 

  12. Reis, M.F., Leite, A.C., Lizarralde, F., et al.: Kinematic modeling and control design of a multifingered robot hand. In: 2015 IEEE 24th International Symposium on Industrial Electronics (ISIE). Buzios, Brazil (2015)

  13. Droukas, L., Doulgeri, Z.: Rolling contact motion generation and control of robotic fingers. J. Intell. Robot. Syst. 82(1), 21–38 (2016)

    Article  Google Scholar 

  14. Reis, M.F., Leite, A.C., From, P.J., et al.: Visual servoing for object manipulation with a multifingered robot hand. IFAC-PapersOnLine. 48(19), 1–6 (2015)

    Article  Google Scholar 

  15. Sanchez, J., Corrales, J.A., Bouzgarrou, B.C., et al.: Robotic manipulation and sensing of deformable objects in domestic and industrial applications: a survey. Int. J. Robot. Res. 37(7), 688–716 (2018)

    Article  Google Scholar 

  16. Zaidi, L., Corrales, J.A., Bouzgarrou, B.C., et al.: Model-based strategy for grasping 3D deformable objects using a multi-fingered robotic hand. Robot. Auton. Syst. 95, 196–206 (2017)

    Article  Google Scholar 

  17. Boughdiri, R., Nasser, H., Bezine, H., et al.: Lagrange dynamic modeling of a multi-fingered robot hand in free motion considering the coupling dynamics. Adv. Mater. Res. 588, 1659–1663 (2012)

    Article  Google Scholar 

  18. Deng, H., Zhong, G., Li, X., et al.: Slippage and deformation preventive control of bionic prosthetic hands. IEEE/ASME Trans. Mechatron. 22(2), 888–897 (2016)

    Article  Google Scholar 

  19. Costanzo, M., De Maria, G., Natale, C.: Two-fingered in-hand object handling based on force/tactile feedback. IEEE Trans. Rob. 36(1), 157–173 (2019)

    Article  Google Scholar 

  20. Zhang, T., Jiang, L., Fan, S., et al.: Development and experimental evaluation of multi-fingered robot hand with adaptive impedance control for unknown environment grasping. Robotica 34(5), 1168–1185 (2016)

    Article  Google Scholar 

  21. Grossard, M., Mathieu, J., da Cruz Pacheco, G.F.: Control-oriented design and robust decentralized control of the CEA dexterous robot hand. IEEE/ASME Trans. Mechatron. 20(4), 1809–1821 (2014)

    Article  Google Scholar 

  22. Wen, R., Yuan, K., Wang, Q., et al.: Force-guided high-precision grasping control of fragile and deformable objects using sEMG-based force prediction. IEEE Robot. Autom. Lett. 5(2), 2762–2769 (2020)

    Article  Google Scholar 

  23. Udwadia, F.E., Kalaba, R.E.: Analytical Dynamics: A New Approach. Cambridge University Press, New York (2007)

    MATH  Google Scholar 

  24. Udwadia, F.E., Kalaba, R.E.: A new perspective on constrained motion. Proc. R. Soc. A Math. Phys. Eng. Sci. 439(1906), 407–410 (1992)

    MathSciNet  MATH  Google Scholar 

  25. Udwadia, F.E., Kalaba, R.E.: What is the general form of the explicit equations of motion for constrained mechanical systems? J. Appl. Mech. 69(3), 335–339 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Udwadia, F.E., Schutte, A.D.: Equations of motion for general constrained systems in Lagrangian mechanics. Acta Mech. 213(1–2), 111–129 (2010)

    Article  MATH  Google Scholar 

  27. Schutte, A., Udwadia, F.: New approach to the modeling of complex multibody dynamical systems. J. Appl. Mech. 78(2), 021018 (2011)

    Article  Google Scholar 

  28. Udwadia, F. E., Wanichanon, T.: Explicit equation of motion of constrained systems. Nonlinear Approaches in Engineering Applications. pp. 315–347. Springer, New York (2012)

  29. Udwadia, F.E., Wanichanon, T.: On general nonlinear constrained mechanical systems. Numer. Algebra Control Optim. 3(3), 425–443 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Cabannes, H.: General Mechanics, \(2^{nd}\) Edition (in English, translated from the original French). Blaisdell/Ginn, Waltham, MA (1968)

    Google Scholar 

  31. Papastavridis, J.G.: Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems; for Engineers, Physicists, and Mathematicians. Oxford University Press, New York (2002)

    MATH  Google Scholar 

  32. Udwadia, F.E.: A new perspective on the tracking control of nonlinear structural and mechanical systems. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 459(2035), 1783–1800 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Udwadia, F.E., Kalaba, R.E.: New directions in the control of nonlinear mechanical systems. In: Guttalu, R.S. (ed.) Mechanics and Control. Springer, Boston (1994)

    MATH  Google Scholar 

  34. Udwadia, F.E.: Optimal tracking control of nonlinear dynamical systems. Proc. R. Soc. A Math. Phys. Eng. Sci. 464(2097), 2341–2363 (2008)

    MathSciNet  MATH  Google Scholar 

  35. Udwadia, F. E., Wanichanon, T.: A new approach to the tracking control of uncertain nonlinear multi-body mechanical systems. Nonlinear Approaches in Engineering Applications 2. Springer, New York (2014)

  36. Udwadia, F. E., Wanichanon, T.: Control of uncertain nonlinear multibody mechanical systems. J. Appl. Mech. 81(4) (2014)

  37. Udwadia, F.E., Schutte, A.D.: A unified approach to rigid body rotational dynamics and control. Proc. R. Soc. A Math. Phys. Eng. Sci. 468(2138), 395–414 (2012)

    MathSciNet  MATH  Google Scholar 

  38. Udwadia, F.E., Koganti, P.B., Wanichanon, T., Stipanovia, D.M.: Decentralised control of nonlinear dynamical systems. Int. J. Control 87(4), 827–843 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Udwadia, F.E.: A new approach to stable optimal control of complex nonlinear dynamical systems. J. Appl. Mech. 81(3), 031001 (2014)

    Article  Google Scholar 

  40. Koganti, P.B., Udwadia, F.E.: Dynamics and precision control of tumbling multibody systems. J. Guid. Control. Dyn. 40(3), 584–602 (2017)

    Article  Google Scholar 

  41. Koganti, P.B., Udwadia, F.E.: Unified approach to modeling and control of rigid multibody systems. J. Guid. Control. Dyn. 39(12), 2683–2698 (2016)

    Article  Google Scholar 

  42. Cho, H., Udwadia, F.E.: Explicit solution to the full nonlinear problem for satellite formation-keeping. Acta Astronaut. 67(3–4), 369–387 (2010)

    Article  Google Scholar 

  43. Cho, H., Udwadia, F.E.: Explicit control force and torque determination for satellite formation-keeping with attitude requirements. J. Guid. Control. Dyn. 36(2), 589–605 (2013)

  44. Zhao, R., Chen, Y.H., Jiao, S., et al.: A constraint-following control for uncertain mechanical systems: given force coupled with constraint force. Nonlinear Dyn. 93(3), 1201–1217 (2018)

    Article  MATH  Google Scholar 

  45. Zhao, R., Li, M., Niu, Q., et al.: Udwadia–Kalaba constraint-based tracking control for artificial swarm mechanical systems: dynamic approach. Nonlinear Dyn. 100, 2381–2399 (2020)

    Article  Google Scholar 

  46. Leopold, A.P.: A Treatise on Analytical Dynamics. OX Bow Press, New York (1979)

    Google Scholar 

  47. Rosenberg, R.: Analytical Dynamics of Discrete Systems. Plenum Press, New York (1977)

    Book  MATH  Google Scholar 

  48. Udwadia, F.E., Kalaba, R.E., Eun, H.C.: Equations of motion for constrained mechanical systems and the extended d’Alembert’s principle. Q. Appl. Math. 55(2), 321–331 (1997)

  49. Spong, M.W., Seth, H., Mathukumalli, V.: Robot Modeling and Control. Wiley, New York (2006)

    Google Scholar 

  50. Craig, J.J.: Introduction to Robotics Mechanics and Control. Pearson Education, London (2009)

    Google Scholar 

  51. Slotine, J.J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, New Jersey (1991)

    MATH  Google Scholar 

  52. Bae, J.H., Arimoto, S.: Important role of force/velocity characteristics in sensory-motor coordination for control design of object manipulation by a multi-fingered robot hand. Robotica 22(5), 479–491 (2004)

    Article  Google Scholar 

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Acknowledgements

This study was funded by China Postdoctoral Natural Science Foundation (No. 2021T140585), Fundamental Research Funds for Chinese Central Universities (No. 300102259306), Shaanxi Province Natural Science Foundation (No. 2020JM-240) and Key Research and Development Program of Shaanxi (No. 2021ZDLGY09-02).

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Authors and Affiliations

  1. National Engineering Laboratory for Highway Maintenance Equipment, Chang’an University, Xi’an, 710065, Shaanxi, People’s Republic of China

    Ruiying Zhao, Jin Yu & Hao Yang

  2. The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Ye-Hwa Chen

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  1. Ruiying Zhao
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Appendices

Appendix A

The details of the dynamics modeling for the hand robot in contact with an object in (4.1) are as follows:

$$\begin{aligned} M_{fi_{11}}&=m_{i1}l^{2}_{ci1}c^{2}_{i2}\nonumber \\&\quad +m_{i2}\big (l^{2}_{i1}c^{2}_{i2}+l^{2}_{ci2}c^{2}_{i23}+l_{i1}l_{ci2}(c_{i3}+c1_{i23})\big )\nonumber \\&\quad +m_{i3}\big (l^{2}_{i1}c^{2}_{i2}+l^{2}_{i2}c^{2}_{i23}+l^{2}_{ci3}c^{2}_{i234}\nonumber \\&\quad +l_{i1}l_{i2}c_{i3}+l_{i1}l_{i2}c1_{i23}+l_{i1}l_{ci3}(c_{i34}+c1_{i234})\nonumber \\&\quad +l_{i2}l_{ci3}(c_{i4}+c2_{i234})\big )+I_{i1x}s^{2}_{i2}+I_{i1y}c^{2}_{i2}\nonumber \\&\quad +I_{i2x}s^{2}_{i23}+I_{i2y}c^{2}_{i23}+I_{i3x}s^{2}_{i234}+I_{i3y}c^{2}_{i234}, \end{aligned}$$
(A.1)
$$\begin{aligned} M_{fi_{22}}&=m_{i1}l^{2}_{ci1}+m_{i2}(l^{2}_{i1}+l^{2}_{ci2}+2l_{i1}l_{ci2}c_{i3})\nonumber \\&\quad +m_{i3}(l^{2}_{i1}+l^{2}_{i2}+l^{2}_{ci3}+2l_{i1}l_{i2}c_{i3}\nonumber \\&\quad +2l_{i1}l_{ci3}c_{i34}+2l_{i2} l_{ci3}c_{i4})+I_{i1z}+I_{i2z}\nonumber \\&\quad +I_{i3z}, M_{fi_{34}}=M_{i_{43}}=m_{i3}(l^{2}_{ci3}\nonumber \\&\quad +l_{i2}l_{ci3}c_{i4})+I_{i3z}, \end{aligned}$$
(A.2)
$$\begin{aligned}&M_{fi_{23}}=M_{fi_{32}}=m_{i2}(l^{2}_{ci2}+l_{i1}l_{ci2}c_{i3})\nonumber \\&\quad +m_{i3}(l^{2}_{i2}+l^{2}_{ci3}+l_{i1}l_{i2}c_{i3}+l_{i1}l_{ci3}c_{i34}\nonumber \\&\quad +2l_{i2}l_{ci3}c_{i4})+I_{i2z}+I_{i3z},\nonumber \\&M_{fi_{24}}=M_{fi_{42}}=m_{i3}(l^{2}_{ci3}+l_{i2}l_{ci3}c_{i4}\nonumber \\&\quad +l_{i1}l_{ci3}c_{i34})+I_{i3z}, \end{aligned}$$
(A.3)
$$\begin{aligned}&M_{fi_{33}}=m_{i2}l^{2}_{ci2}+m_{i3}(l^{2}_{i2}+l^{2}_{ci3}+2l_{i2}l_{ci3}c_{i4})\nonumber \\&\quad +I_{i2z}+I_{i3z}, M_{fi_{44}}=m_{i3}l^{2}_{ci3}+I_{i3z}, \end{aligned}$$
(A.4)
$$\begin{aligned}&Q_{fi_{1}}=\bigg \{2m_{i1}l^{2}_{ci1}s_{i2}c_{i2}{\dot{q}}_{fi_{2}}\nonumber \\&\quad +m_{i2}\Big [2\big (l^{2}_{i1}s_{i2}c_{i2}+l^{2}_{ci2}s_{i23}c_{i23}+l_{i1}l_{ci2}s1_{i23}\big )\nonumber \\&\quad \times {\dot{q}}_{fi_{2}}+\big (2l^{2}_{ci2}s_{i23}c_{i23}+l_{i1}\nonumber \\&\quad \times l_{ci2}(s_{i3}+s1_{i23})\big ){\dot{q}}_{fi_{3}}\Big ]\nonumber \\&\quad +m_{i3}\Big [2\big (l^{2}_{i1}s_{i2}c_{i2}+l^{2}_{i2}s_{i23}c_{i23}+l^{2}_{ci3}s_{i234}c_{i234}\nonumber \\&\quad +l_{i1}l_{i2}s1_{i23}+l_{i1}l_{ci3}\nonumber \\&\quad \times s1_{i234}+l_{i2}l_{ci3}s2_{i234}\big ){\dot{q}}_{fi_{2}}\nonumber \\&\quad +\big (2l^{2}_{i2}s_{i23}c_{i23}+2l^{2}_{ci3}s_{i234}c_{i234}\nonumber \\&\quad +l_{i1}l_{i2}(s_{i3}+s1_{i23})+l_{i1}l_{ci3}\nonumber \\&\quad \times (s_{i34}+s1_{i234})+2l_{i2}l_{ci3}s2_{i234}\big ){\dot{q}}_{fi_{3}}\nonumber \\&\quad +\big (2l^{2}_{ci3}s_{i234}c_{i234}+l_{i2}l_{ci3}(s_{i4}+s2_{i234})\nonumber \\&\quad +l_{i1}l_{ci3}(s_{i34}+s1_{i234})\big ){\dot{q}}_{fi_{4}}\Big ]\nonumber \\&\quad +2\Big [\big (I_{i1y}-I_{i1x}\big )s_{i2}c_{i2}+\big (I_{i2y}-I_{i2x}\big )s_{i23}c_{i23}\nonumber \\&\quad +\big (I_{i3y}-I_{i3x}\big )s_{i234}c_{i234}\Big ]{\dot{q}}_{fi_{2}}\nonumber \\&\quad +2\Big [\big (I_{i2y}-I_{i2x}\big )s_{i23}c_{i23}+\big (I_{i3y}-I_{i3x}\big )\nonumber \\&\quad \times s_{i234}c_{i234}\Big ]{\dot{q}}_{fi_{3}}+2\big (I_{i3y}-I_{i3x}\big )\nonumber \\&\quad \times s_{i234}c_{i234}{\dot{q}}_{fi_{4}}\bigg \}{\dot{q}}_{fi_{1}}, \end{aligned}$$
(A.5)
$$\begin{aligned}&Q_{fi_{2}}=-m_{i1}\big (l^{2}_{ci1}c_{i2}s_{i2}{\dot{q}}^{2}_{fi_{1}}+l_{ci1}c_{i2}g\big )\nonumber \\&\quad +m_{i2}\Big [-\big (l^{2}_{i1}s_{i2}c_{i2}+l^{2}_{ci2}s_{i23}c_{i23}\nonumber \\&\quad +l_{i1}l_{ci2}s1_{i23}\big ){\dot{q}}^{2}_{fi_{1}}+l_{i1}l_{ci2}\nonumber \\&\quad \times s_{i3}(2{\dot{q}}_{fi_{2}}{\dot{q}}_{fi_{3}}+{\dot{q}}^{2}_{fi_{3}})-l_{i1}c_{i2}g-l_{ci2}c_{i23}g\Big ]\nonumber \\&\quad +m_{i3}\Big [-\big (l^{2}_{i1}s_{i2}c_{i2}+l^{2}_{i2}s_{i23}c_{i23}+l^{2}_{ci3}\nonumber \\&\quad \times s_{i234}c_{i234}+l_{i1}l_{i2}s1_{i23}+l_{i2}l_{ci3}s2_{i234}+l_{i1}\nonumber \\&\quad \times l_{ci3}s1_{i234}\big ){\dot{q}}^{2}_{fi_{1}}+\big (l_{i1}l_{i2}s_{i3}+l_{i1}l_{ci3}s_{i34}\big ){\dot{q}}^{2}_{fi_{3}}\nonumber \\&\quad +\big (l_{i1}l_{ci3}s_{i34} l_{ci3}s_{i4}\big ){\dot{q}}^{2}_{fi_{4}}\nonumber \\&\quad +2\big (l_{i1}l_{i2}s_{i3}+l_{i1}l_{ci3}s_{i34}\big ){\dot{q}}_{fi_{2}}{\dot{q}}_{fi_{3}}\nonumber \\&\quad +2\big (l_{i1}l_{ci3}s_{i34}+l_{i2}l_{ci3}s_{i4}\big )\nonumber \\&\quad \times ({\dot{q}}_{fi_{2}}{\dot{q}}_{fi_{4}}+{\dot{q}}_{fi_{3}}{\dot{q}}_{fi_{4}})-l_{i1}c_{i2}g-l_{i2}c_{i23}g \nonumber \\&\quad -l_{ci3}c_{i234}g\Big ]+\Big [\big (I_{i1x}-I_{i1y}\big )s_{i2}c_{i2}\nonumber \\&\quad +\big (I_{i2x}-I_{i2y}\big )s_{i23}c_{i23}+\big (I_{i3x}-I_{i3y}\big )\nonumber \\&\qquad s_{i234}c_{i234}\Big ]{\dot{q}}^{2}_{fi_{1}}, \end{aligned}$$
(A.6)
$$\begin{aligned} Q_{fi_{3}}&=-m_{i2}\Big [\big (l^{2}_{ci2}s_{i23}c_{i23}+\frac{1}{2}l_{i1}l_{ci2}(s_{i3}+s1_{i23})\big )\nonumber \\&\quad \times {\dot{q}}^{2}_{fi_{1}}+l_{i1}l_{ci2}s_{i3}{\dot{q}}^{2}_{fi_{2}}+l_{ci2}c_{i23}g\Big ]\nonumber \\&\quad +m_{i3}\Big [l_{i2}l_{ci3}s_{i4} {\dot{q}}^{2}_{fi_{4}}-\big (l^{2}_{i2}s_{i23}c_{i23}\nonumber \\&\quad +l^{2}_{ci3}s_{i234}c_{i234}+\frac{1}{2}l_{i1}l_{i2}(s_{i3}+s1_{i23})\nonumber \\&\quad +\frac{1}{2}l_{i1}l_{ci3}(s_{i34}+s1_{i234})+l_{i2}l_{ci3}\nonumber \\&\quad \times s2_{i234}\big ){\dot{q}}^{2}_{fi_{1}}-\big (l_{i1}l_{i2}s_{i3}+l_{i1}l_{ci3}s_{i34}\big ){\dot{q}}^{2}_{fi_{2}}\nonumber \\&\quad +2l_{i2}l_{ci3}s_{i4}({\dot{q}}_{fi_{2}}+{\dot{q}}_{fi_{3}}){\dot{q}}_{fi_{4}}-l_{i2}c_{i23}g\nonumber \\&\quad -l_{ci3}c_{i234} g\Big ]+\Big [\big (I_{i2x}-I_{i2y}\big )s_{i23}c_{i23}\nonumber \\&\quad +\big (I_{i3x}-I_{i3y}\big )s_{i234}c_{i234}\Big ]{\dot{q}}^{2}_{fi_{1}}, \end{aligned}$$
(A.7)
$$\begin{aligned} Q_{fi_{4}}&=-\Big [m_{i3}\big (l^{2}_{ci3}s_{i234}c_{i234}+\frac{1}{2}l_{i1}l_{ci3}(s_{i34}+s1_{i234})\nonumber \\&\quad +\frac{1}{2}l_{i2}l_{ci3}(s_{i4}+s2_{i234})\big )+\big (I_{i3y}-I_{i3x}\big )s_{i234}\nonumber \\&\quad \times c_{i234}\Big ]{\dot{q}}^{2}_{fi_{1}}-m_{i3}\big (l_{i1}l_{ci3}s_{i34}+l_{i2}l_{ci3}s_{i4}\big ){\dot{q}}^{2}_{fi_{2}}\nonumber \\&\quad -m_{i3}l_{i2}l_{ci3}s_{i4}\big (2{\dot{q}}_{fi_{2}}\nonumber \\&\quad +{\dot{q}}_{fi_{3}}\big ){\dot{q}}_{fi_{3}}-m_{i3}gl_{ci3}\times c_{i234}. \end{aligned}$$
(A.8)

Here, \(s1_{i23}=\sin {(2q_{fi_{2}}+q_{fi_{3}})}\), \(s1_{i234}=\sin {(2q_{fi_{2}}}{+q_{fi_{3}}+q_{fi_{4}})}\), \(c1_{i234}=\cos {(2q_{fi_{2}}+q_{fi_{3}}+q_{fi_{4}})}\), \(s2_{i234}=\sin {(2q_{fi_{2}}+2q_{fi_{3}}+q_{fi_{4}})}\), \(c2_{i234}=\cos {(2q_{fi_{2}}+2q_{fi_{3}}+q_{fi_{4}})}\). The unlisted elements of matrix \(M_{fi}\) and \(Q_{fi}\) are zeroes.

Appendix B

The detailed expressions of matrices \(A_{fi}\), \(A_{oi}\) in (4.9) and \(b_{di}\) in (4.10) are

$$\begin{aligned} A_{fi_{11}}&=s_{i1}(l_{i1}c_{i2}+l_{i2}c_{i23}+l_{i3}c_{i234}\nonumber \\&\quad +x_{fci}c_{i234})-y_{fci}s_{i234}s_{i1}-z_{fci}c_{i1}, \end{aligned}$$
(B.1)
$$\begin{aligned} A_{fi_{12}}&=c_{i1}(l_{i1}s_{i2}+l_{i2}s_{i23}+l_{i3}s_{i234}+x_{fci}s_{i234}\nonumber \\&\quad +y_{fci}c_{i234}), \end{aligned}$$
(B.2)
$$\begin{aligned} A_{fi_{13}}&=c_{i1}(l_{i2}s_{i23}+l_{i3}s_{i234}+x_{fci}s_{i234}+y_{fci}c_{i234}), \end{aligned}$$
(B.3)
$$\begin{aligned} A_{fi_{14}}&=c_{i1}\Big [(l_{i3}+x_{fci})s_{i234}+y_{fci}c_{i234}\Big ], A_{fi_{24}}\nonumber \\&=s_{i1}\Big [(l_{i3}+x_{fci})s_{i234}+y_{fci}c_{i234}\Big ],\end{aligned}$$
(B.4)
$$\begin{aligned} A_{fi_{21}}&=-c_{i1}(l_{i1}c_{i2}+l_{i2}c_{i23}+l_{i3}c_{i234}\nonumber \\&\quad +x_{fci}c_{i234}-y_{fci}s_{i234})-z_{fci}s_{i1}, A_{fi_{31}}=0, \end{aligned}$$
(B.5)
$$\begin{aligned} A_{fi_{22}}&=s_{i1}(l_{i1}s_{i2}+l_{i2}s_{i23}+l_{i3}s_{i234}\nonumber \\&\quad +x_{fci}s_{i234}+y_{fci}c_{i234}),\end{aligned}$$
(B.6)
$$\begin{aligned} A_{fi_{23}}&=s_{i1}(l_{i2}s_{i23}+l_{i3}s_{i234}+x_{fci}s_{i234}+y_{fci}c_{i234}),\end{aligned}$$
(B.7)
$$\begin{aligned} A_{fi_{32}}&=y_{fci}s_{i234}-l_{i1}c_{i2}-l_{i2}c_{i23}-l_{i3}c_{i234}\nonumber \\&\quad -x_{fci}c_{i234},\end{aligned}$$
(B.8)
$$\begin{aligned} A_{fi_{33}}&=y_{fci}s_{i234}-l_{i2}c_{i23}-(l_{i3}+x_{fci})c_{i234},\nonumber \\ A_{fi_{34}}&=y_{fci}s_{i234}-(l_{i3}+x_{fci})c_{i234},\end{aligned}$$
(B.9)
$$\begin{aligned} A_{oi_{14}}&=-x_{oci}(s_{\phi }c_{\theta }c_{\psi }+c_{\phi }s_{\psi })+y_{oci}(s_{\phi }c_{\theta }s_{\psi }\nonumber \\&\quad -c_{\phi }c_{\psi })-z_{oci}s_{\phi }s_{\theta }, \end{aligned}$$
(B.10)
$$\begin{aligned} A_{oi_{15}}&=-x_{oci}c_{\phi }s_{\theta }c_{\psi }+y_{oci}c_{\phi }s_{\theta }s_{\psi }+z_{oci}c_{\phi }c_{\theta },\nonumber \\ A_{oi_{11}}&=A_{oi_{22}}=A_{oi_{33}}=1,\end{aligned}$$
(B.11)
$$\begin{aligned} A_{oi_{16}}&=-x_{oci}(c_{\phi }c_{\theta }s_{\psi }+s_{\phi }c_{\psi })-y_{oci}\nonumber \\&\quad \times (c_{\phi }c_{\theta }c_{\psi }-s_{\phi }s_{\psi }), \end{aligned}$$
(B.12)
$$\begin{aligned} A_{oi_{24}}&=x_{oci}(c_{\phi }c_{\theta }c_{\psi }-s_{\phi }s_{\psi })-y_{oci}(c_{\phi }c_{\theta }s_{\psi }\nonumber \\&\quad +s_{\phi }c_{\psi })+z_{oci}c_{\phi }s_{\theta }, \end{aligned}$$
(B.13)
$$\begin{aligned} A_{oi_{25}}&=y_{oci}s_{\phi }s_{\theta }s_{\psi }+z_{oci}s_{\phi }c_{\theta }-x_{oci}s_{\phi }s_{\theta }c_{\psi },\end{aligned}$$
(B.14)
$$\begin{aligned} A_{oi_{26}}&=-x_{oci}(s_{\phi }c_{\theta }s_{\psi }-c_{\phi }c_{\psi })-y_{oci}\nonumber \\&\quad \times (s_{\phi }c_{\theta }c_{\psi }+c_{\phi }s_{\psi }), \end{aligned}$$
(B.15)
$$\begin{aligned} A_{oi_{35}}&=-x_{oci}c_{\theta }c_{\psi }+y_{oci}c_{\theta }s_{\psi }-z_{oci}s_{\theta },\nonumber \\ A_{oi_{36}}&=x_{oci}s_{\theta }s_{\psi }+y_{oci}s_{\theta }c_{\psi },\end{aligned}$$
(B.16)
$$\begin{aligned} b_{\phi _{i1}}&=\Big [x_{oci}\big (c_{\phi }c_{\theta }c_{\psi }-s_{\phi }s_{\psi }\big )+y_{oci}\big (-c_{\phi }c_{\theta }s_{\psi }\nonumber \\&\quad -s_{\phi }c_{\psi }\big )+z_{oci}c_{\phi }s_{\theta }\Big ]{\dot{\phi }}\nonumber \\&\quad +s_{\phi }\big (-x_{oci}s_{\theta }c_{\psi }+y_{oci}s_{\theta }s_{\psi }\nonumber \\&\quad +z_{oci}c_{\theta }\big ){\dot{\theta }}+\Big [x_{oci}\big (-s_{\phi }c_{\theta }s_{\psi }\nonumber \\&\quad +c_{\phi }c_{\psi }\big )-y_{oci}\big (s_{\phi }c_{\theta }c_{\psi }+c_{\phi }s_{\psi }\big )\Big ]{\dot{\psi }},\end{aligned}$$
(B.17)
$$\begin{aligned} b_{\theta _{i1}}&=s_{\phi }\big (-x_{oci}s_{\theta }c_{\psi }+y_{oci}s_{\theta }s_{\psi }+z_{oci}c_{\theta }\big ){\dot{\phi }}\nonumber \\&\quad +c_{\phi }\big (x_{oci}c_{\theta }c_{\psi }-y_{oci}c_{\theta }s_{\psi }\nonumber \\&\quad +z_{oci}s_{\theta }\big ){\dot{\theta }}-c_{\phi }s_{\theta }\big (x_{oci}s_{\psi }\nonumber \\&\quad +y_{oci}c_{\psi }\big ){\dot{\psi }},\end{aligned}$$
(B.18)
$$\begin{aligned} b_{\psi _{i1}}&=\Big [x_{oci}\big (-s_{\phi }c_{\theta }s_{\psi }+c_{\phi }c_{\psi }\big )-y_{oci}\big (s_{\phi }c_{\theta }c_{\psi }\nonumber \\&\quad +c_{\phi }s_{\psi }\big )\Big ]{\dot{\phi }}-c_{\phi }s_{\theta }\big (x_{oci}s_{\psi }\nonumber \\&\quad +y_{oci}c_{\psi }\big ){\dot{\theta }}+\Big [x_{oci}\big (c_{\phi }c_{\theta }c_{\psi }\nonumber \\&\quad -s_{\phi }s_{\psi }\big )-y_{oci}\big (c_{\phi }c_{\theta }s_{\psi }+s_{\phi }c_{\psi }\big )\Big ]{\dot{\psi }},\end{aligned}$$
(B.19)
$$\begin{aligned} b_{q_{i11}}&=\Big [-c_{i1}\big (l_{i1}c_{i2}+l_{i2}c_{i23}+l_{i3}c_{i234}+x_{fci}c_{i234}\nonumber \\&\quad -y_{fci}s_{i234}\big )-z_{fci}s_{i1}\Big ]{\dot{q}}_{fi_{1}}+s_{i1}\big (l_{i1}s_{i2}\nonumber \\&\quad +l_{i2}s_{i23}+l_{i3}s_{i234}+x_{fci}s_{i234}+z_{fci}c_{i234}\big )\nonumber \\&\quad \times {\dot{q}}_{fi_{2}}+s_{i1}\big (l_{i2}s_{i23}+l_{i3}s_{i234}+x_{fci}s_{i234}\nonumber \\&\quad +z_{fci}c_{i234}\big ) {\dot{q}}_{fi_{3}}+s_{i1}\big (l_{i3}s_{i234}+x_{fci}s_{i234}\nonumber \\&\quad +z_{fci}c_{i234}\big ){\dot{q}}_{fi_{4}},\end{aligned}$$
(B.20)
$$\begin{aligned} b_{q_{i21}}&=s_{i1}\big (l_{i1}s_{i2}+l_{i2}s_{i23}+(l_{i3}+x_{fci})s_{i234}\nonumber \\&\quad +y_{fci}c_{i234}\big ){\dot{q}}_{fi_{1}}\nonumber \\&\quad -c_{i1}\big (l_{i1}c_{i2}+l_{i2}c_{i23}+l_{i3}c_{i234}+x_{fci}\nonumber \\&\quad \times c_{i234}-y_{fci}s_{i234}\big ){\dot{q}}_{fi_{2}}-c_{i1}\big (l_{i2}c_{i23}\nonumber \\&\quad +l_{i3}c_{i234}+x_{fci}c_{i234}-y_{fci}s_{i234}\big ){\dot{q}}_{fi_{3}}\nonumber \\&\quad -c_{i1}\big (l_{i3}c_{i234}+x_{fci}c_{i234}-y_{fci}s_{i234}\big ){\dot{q}}_{fi_{4}},\end{aligned}$$
(B.21)
$$\begin{aligned} b_{q_{i31}}&=s_{i1}\big (l_{i2}s_{i23}+(l_{i3}+x_{fci})s_{i234}+y_{fci}c_{i234}\big ){\dot{q}}_{fi_{1}}\nonumber \\&\quad -c_{i1}\big (l_{i2}c_{i23}+(l_{i3}+x_{fci})c_{i234}-y_{fci}s_{i234}\big )\nonumber \\&\quad \times ({\dot{q}}_{fi_{2}}+{\dot{q}}_{fi_{3}})-c_{i1}\big ((l_{i3}\nonumber \\&\quad +x_{fci})c_{i234}-y_{fci}s_{i234}\big ){\dot{q}}_{fi_{4}},\end{aligned}$$
(B.22)
$$\begin{aligned} b_{q_{i41}}&=s_{i1}\big ((l_{i3}+x_{fci})s_{i234}+y_{fci}c_{i234}\big ){\dot{q}}_{fi_{1}}\nonumber \\&\quad -c_{i1}\big (l_{i3}c_{i234}\nonumber \\&\quad +x_{fci}c_{i234}-y_{fci}s_{i234}\big )({\dot{q}}_{fi_{2}}+{\dot{q}}_{fi_{3}}+{\dot{q}}_{fi_{4}}\big ),\end{aligned}$$
(B.23)
$$\begin{aligned} b_{\phi _{i2}}&=\Big [x_{oci}\big (s_{\phi }c_{\theta }c_{\psi }+c_{\phi }s_{\psi }\big )+y_{oci}\big (c_{\phi }c_{\psi }\nonumber \\&\quad -s_{\phi }c_{\theta }s_{\psi }\big )+z_{oci}s_{\phi }s_{\theta }\Big ]{\dot{\phi }}\nonumber \\&\quad +c_{\phi }\big (x_{oci}s_{\theta }c_{\psi }-y_{oci}s_{\theta }s_{\psi }-z_{oci}\nonumber \\&\quad \times c_{\theta }\big ){\dot{\theta }}+\Big [x_{oci}\big (c_{\phi }c_{\theta }s_{\psi }\nonumber \\&\quad +s_{\phi }c_{\psi }\big )+y_{oci}\big (c_{\phi }c_{\theta }c_{\psi }-s_{\phi }s_{\psi }\big )\Big ]{\dot{\psi }},\end{aligned}$$
(B.24)
$$\begin{aligned} b_{\theta _{i2}}&=c_{\phi }\big (x_{oci}s_{\theta }c_{\psi }-y_{oci}s_{\theta }s_{\psi }-z_{oci}c_{\theta }\big ){\dot{\phi }}\nonumber \\&\quad +s_{\phi }\big (x_{oci}c_{\theta }c_{\psi }-y_{oci}c_{\theta }s_{\psi }+z_{oci}s_{\theta }\big ){\dot{\theta }}\nonumber \\&\quad -s_{\phi }s_{\theta }\big (x_{oci}s_{\psi }+y_{oci}\nonumber \\&\quad \times c_{\psi }\big ){\dot{\psi }},\end{aligned}$$
(B.25)
$$\begin{aligned} b_{\psi _{i2}}&=\Big [x_{oci}\big (c_{\phi }c_{\theta }s_{\psi }+s_{\phi }c_{\psi }\big )\nonumber \\&\quad +y_{oci}\big (c_{\phi }c_{\theta }c_{\psi }\nonumber \\&\quad -s_{\phi }s_{\psi }\big )\Big ]{\dot{\phi }}-s_{\phi }s_{\theta }\big (x_{oci}s_{\psi }\nonumber \\&\quad +y_{oci}c_{\psi }\big ){\dot{\theta }}+\Big [x_{oci}\big (s_{\phi }c_{\theta }c_{\psi }\nonumber \\&\quad +c_{\phi }s_{\psi }\big )+y_{oci}\big (-s_{\phi }c_{\theta }s_{\psi }+c_{\phi }c_{\psi }\big )\Big ]{\dot{\psi }},\end{aligned}$$
(B.26)
$$\begin{aligned} b_{q_{i12}}&=-s_{i1}\big (l_{i1}c_{i2}+l_{i2}c_{i23}+(l_{i3}+x_{fci})c_{i234}\nonumber \\&\quad -y_{fci}s_{i234}){\dot{q}}_{fi_{1}}\nonumber \\&\quad -c_{i1}\big (l_{i1}s_{i2}+l_{i2}s_{i23}+(l_{i3}+x_{fci})\nonumber \\&\quad \times s_{i234}+y_{fci}c_{i234}\big ){\dot{q}}_{fi_{2}}\nonumber \\&\quad -c_{i1}\big (l_{i2}s_{i23}+(l_{i3}+x_{fci})s_{i234}\nonumber \\&\quad +y_{fci}c_{i234}\big ){\dot{q}}_{fi_{3}}-c_{i1}\Big [(l_{i3}+x_{fci})\nonumber \\&\quad \times s_{i234}+z_{fci}c_{i234}\Big ]{\dot{q}}_{fi_{4}},\end{aligned}$$
(B.27)
$$\begin{aligned} b_{q_{i22}}&=-c_{i1}\big (l_{i1}s_{i2}+l_{i2}s_{i23}+(l_{i3}+x_{fci})s_{i234}\nonumber \\&\quad +y_{fci}c_{i234}\big ){\dot{q}}_{fi_{1}}\nonumber \\&\quad -s_{i1}\big (l_{i1}c_{i2}+l_{i2}c_{i23}+(l_{i3}+x_{fci})\nonumber \\&\quad \times c_{i234}-y_{fci}s_{i234}\big ){\dot{q}}_{fi_{2}}-s_{i1}\big (l_{i2}c_{i23}+(l_{i3}\nonumber \\&\quad +x_{fci})c_{i234}-y_{fci}s_{i234}\big ){\dot{q}}_{fi_{3}} \nonumber \\&\quad -s_{i1}\Big [(l_{i3}+x_{fci}) c_{i234}-y_{fci}s_{i234}\Big ]{\dot{q}}_{fi_{4}},\end{aligned}$$
(B.28)
$$\begin{aligned} b_{q_{i32}}&=-c_{i1}\big (l_{i2}s_{i23}+(l_{i3}+x_{fci})s_{i234}+y_{fci}c_{i234}\big )\nonumber \\&\quad \times {\dot{q}}_{fi_{1}}-s_{i1}\big (l_{i2}c_{i23}+(l_{i3}+x_{fci})\nonumber \\&\quad \times c_{i234}-y_{fci}s_{i234}\big )({\dot{q}}_{fi_{2}}+{\dot{q}}_{fi_{3}}\big )\nonumber \\&\quad -s_{i1}\big ((l_{i3}+x_{fci})c_{i234}-y_{fci}s_{i234}\big )\nonumber \\&\quad \times {\dot{q}}_{fi_{4}}, b_{\phi _{i3}}=0,\end{aligned}$$
(B.29)
$$\begin{aligned} b_{q_{i42}}&=-c_{i1}\big ((l_{i3}+x_{fci})s_{i234}+y_{fci}c_{i234}\big ){\dot{q}}_{fi_{1}}\nonumber \\&\quad -s_{i1}\big ((l_{ci3}+x_{fci})c_{i234}-y_{fci}s_{i234}\big )({\dot{q}}_{fi_{2}}\nonumber \\&\quad +{\dot{q}}_{fi_{3}}+{\dot{q}}_{fi_{4}}), b_{\theta _{i3}}=\big (z_{oci}c_{\theta }-x_{oci}s_{\theta }c_{\psi }\nonumber \\&\quad -y_{oci}s_{\theta }s_{\psi }\big ){\dot{\theta }}-c_{\theta }\big (x_{oci}s_{\psi }+y_{oci}c_{\psi }\big ){\dot{\psi }},\end{aligned}$$
(B.30)
$$\begin{aligned} b_{\psi _{i3}}&=-\big (x_{oci}c_{\theta }s_{\psi }+y_{oci}c_{\theta }c_{\psi }\big ){\dot{\theta }}\nonumber \\&\quad -c_{\psi }\big (x_{oci}s_{\theta }+y_{oci}c_{\theta }\big ){\dot{\psi }}, b_{q_{i13}}=0,\end{aligned}$$
(B.31)
$$\begin{aligned} b_{q_{i23}}&=-\big (l_{i1}s_{i2}+l_{i2}s_{i23}+(l_{i3}+x_{fci})s_{i234}\nonumber \\&\quad +y_{fci}c_{i234}\big ){\dot{q}}_{fi_{2}}\nonumber \\&\quad -\big (l_{i2}s_{i23}+(l_{i3}+x_{fci})s_{i234}+y_{fci}\nonumber \\&\quad \times c_{i234}\big ){\dot{q}}_{fi_{3}}-\big ((l_{i3}+x_{fci})s_{i234}+y_{fci}c_{i234}\big )\nonumber \\&\quad \times {\dot{q}}_{fi_{4}},\end{aligned}$$
(B.32)
$$\begin{aligned} b_{q_{i33}}&=-\big (l_{i2}s_{i23}+(l_{i3}+x_{fci})s_{i234}\nonumber \\&\quad +y_{fci}c_{i234}\big )\big ({\dot{q}}_{fi_{2}}+{\dot{q}}_{fi_{3}}\big )-\big ((l_{i3}\nonumber \\&\quad +x_{fci})s_{i234}+y_{fci}c_{i234}\big ){\dot{q}}_{fi_{4}},\end{aligned}$$
(B.33)
$$\begin{aligned} b_{q_{i43}}&=-\big (l_{i3}s_{i234}+x_{fci}s_{i234}+y_{fci}c_{i234}\big )\big ({\dot{q}}_{fi_{2}}\nonumber \\&\quad +{\dot{q}}_{fi_{3}}+{\dot{q}}_{fi_{4}}\big ). \end{aligned}$$
(B.34)

The zero entries of the above matrices are not listed.

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Zhao, R., Yu, J., Yang, H. et al. Contact constraints-based dynamic manipulation control of the multi-fingered hand robot: a force sensorless approach. Nonlinear Dyn 107, 1081–1105 (2022). https://doi.org/10.1007/s11071-021-07044-4

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