Differential equations of motion in matrix form of multibody systems driven by electric motors
Keywords:multibody system, electromechanical system, equations of motion, dynamic models, underactuated system
This paper presents the dynamic model of multibody systems driven by electric motors, the so-called electromechanical systems. The mechanical systems considered in this study include an open loop and/or a closed loop, a full-actuated and an under-actuated one. The dynamic model of this electromechanical systems is established in matrix form by applying the Lagrangian equation with and without multipliers and substructure method. With this approach it is easy to obtain the differential equation of motion of the electro-mechanical systems based on the corresponding differential equations of the purely available mechanical system. These obtained equations describe the electromechanical systems in engineering better in case the systems are purely described by mechanical equations. The differential equations of serial and parallel manipulators, slider-crank mechanism, and overhead crane driven by electric motors are established as illustrated examples. In addition, a simplified dynamic model obtained by neglecting of current variation is also validated by numerical simulations.
A. A. Shabana. Dynamics of multibody systems. Cambridge University Press, 3rd edition, (2005).
T. R. Kane and D. A. Levinson. Dynamics: Theory and applications. McGraw-Hill, New York, (1985).
J. Wittenburg. Dynamics of multibody systems. Springer, (2007).
F. Amirouche. Fundamentals of multibody dynamics - Theory and applications. Birkhaeuser Boston, (2006).
R. M. Murray, Z. Li, and S. S. Sastry. A mathematical introduction to robotic manipulation. CRC Press, (2017).
R. Von Schwerin. Multibody system simulation: Numerical methods, algorithms, and software. Springer-Verlag Berlin Heidelberg, (1999).
F. C. Moon. Applied dynamics with applications to multibody and mechatronic systems. JohnWiley & Sons, (1998).
R. N. Jazar. Advanced dynamics: Rigid body, multibody, and aerospace applications. John Wiley & Sons, (2011).
N. V. Khang. Dynamics of multibody systems. Science and Technology Publishing House, Hanoi, (2007). (in Vietnamese).
L.-W. Tsai. Robot analysis: the mechanics of serial and parallel manipulators. John Wiley & Sons, (1999).
J. Angeles and J. Angeles. Fundamentals of robotic mechanical systems, Vol. 2. Springer, (2002).
J. G. De Jalon and E. Bayo. Kinematic and dynamic simulation of multibody systems: the real-time challenge. Springer-Verlag New York Inc, (1994).
M. Ceccarelli. Fundamentals of mechanics of robotic manipulation, Vol. 27. Springer Science & Business Media, (2004).
L. Sciavicco and B. Siciliano. Modelling and control of robot manipulators. Springer Science & Business Media, (2012).
M. W. Spong, S. Hutchinson, and M. Vidyasagar. Robot modeling and control, Vol. 3. Wiley, New York, (2006).
J.-P. Merlet. Parallel robots, Vol. 208. Springer Science & Business Media, (2006).
N. V. Khang. Kronecker product and a new matrix form of Lagrangian equations with multipliers for constrained multibody systems. Mechanics Research Communications, 38, (4), (2011), pp. 294–299. https://doi.org/10.1016/j.mechrescom.2011.04.004.
N. V. Khang and D. T. Tung. A contribution to the dynamic simulation of robot manipulator with the software RobotDyn. Vietnam Journal of Mechanics, 26, (4), (2004), pp. 215–225. https://doi.org/10.15625/0866-7136/26/4/5705.
N. V. Khang. Partial derivative of matrix functions with respect to a vector variable. Vietnam Journal of Mechanics, 30, (4), (2008), pp. 269–279. https://doi.org/10.15625/0866-7136/30/4/5632.
C. A. My and V. M. Hoan. Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms. Vietnam Journal of Mechanics, 41, (2), (2019), pp. 141–155. https://doi.org/10.15625/0866-7136/13073.
L. Sass, J. McPhee, C. Schmitke, P. Fisette, and D. Grenier. A comparison of different methods for modelling electromechanical multibody systems. Multibody System Dynamics, 12, (3), (2004), pp. 209–250. https://doi.org/10.1023/B:MUBO.0000049196.78726.da.
M. Scherrer and J. McPhee. Dynamic modelling of electromechanical multibody systems. Multibody System Dynamics, 9, (1), (2003), pp. 87–115. https://doi.org/10.1023/A:1021675422011.
M. Kim, W. Moon, D. Bae, and I. Park. Dynamic simulations of electromechanical robotic systems driven by DC motors. Robotica, 22, (5), (2004), pp. 523–531. https://doi.org/10.1017/S0263574704000177.
J. McPhee, C. Schmitke, and S. Redmond. Dynamic modelling of mechatronic multibody systems with symbolic computing and linear graph theory. Mathematical and Computer Modelling, 10, (1), (2004), pp. 1–23. https://doi.org/10.1080/13873950412331318044.
J. Moreno-Valenzuela, R. Campa, and V. Santibanez. Model-based control of a class of voltage-driven robot manipulators with non-passive dynamics. Computers & Electrical Engineering, 39, (7), (2013), pp. 2086–2099. https://doi.org/10.1016/j.compeleceng.2013.06.006.
M. H. Hiller and K. Hirsch. Multibody system dynamics and mechatronics. Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik, 86, (2), (2006), pp. 87–109. https://doi.org/10.1002/zamm.200510253.
F. L. Lewis, D. M. Dawson, and C. T. Abdallah. Robot manipulator control: Theory and practice. CRC Press, (2003).
Y. Yu and Z. Mi. Dynamic modeling and control of electromechanical coupling for mechanical elastic energy storage system. Journal of Applied Mathematics, 2013, (2013). https://doi.org/10.1155/2013/603063.
J. Baumgarte. Stabilization of constraints and integrals of motion in dynamical systems. Computer Methods in Applied Mechanics and Engineering, 1, (1), (1972), pp. 1–16. https://doi.org/10.1016/0045-7825(72)90018-7.