Abstract
In this paper, we develop a new method of analytical design and control of the spatial motion of a rigid body (in particular, a spacecraft considered as a rigid body) in a nonlinear dynamic formulation using dual quaternions (Clifford biquaternions). The control provides the asymptotic stability in general of any selected programmed motion in the inertial coordinate system and the desired dynamics of the controlled motion of the body. To build control laws, new biquaternion differential equations of perturbed spatial motion of a rigid body are proposed, in which unnormalized biquaternions of finite displacements, biquaternions of angular and linear velocities of the body and accelerations with nonzero dual scalar parts are used. The concept of solving the inverse problems of dynamics, the feedback control principle, and the approach based on the reduction of the equations of perturbed body motion to linear stationary differential forms of the selected structure invariant with respect to any selected programmed motion due to the corresponding choice of dual nonlinear feedbacks in the proposed biquaternion control laws is presented. Analytical solutions of biquaternion differential equations are designed to describe the dynamics of controlling the spatial body motion using the proposed biquaternion control laws. The properties and patterns of such control are analyzed.
Similar content being viewed by others
REFERENCES
W. Clifford, “Preliminary sketch of bi-quaternions,” Proc. London Math. Soc. 4 (64, 65), 381–395 (1873).
A. P. Kotel’nikov, Screw Theory and Some of Its Applications to Geometry and Mechanics (Kazan, 1895) [in Russian].
A. P. Kotel’nikov, “Screws and complex numbers,” Izv. Fiz.-Mat. O-va Kazan. Univ., Ser. 2, No. 6, 23–33 (1896).
Yu. N. Chelnokov, Quaternion and Biquaternion Models and Methods of Mechanics of Solid Body and Its Applications. Geometry and Kinematics of Motion (Fizmatlit, Moscow, 2006) [in Russian].
V. N. Branets and I. P. Shmyglevskii, Introduction to the Theory of Strapdown Inertial Navigation Systems (Nauka, Moscow, 1992) [in Russian].
V. V. Malanin and N. A. Strelkova, Optimal Control of Rigid Body Orientation and Screw Motion (Research Centre “Regular and chaotic dynamics”, Moscow-Izhevsk, 2004) [in Russian].
N. A. Strelkova, “Time optimal kinematic control of rigid body screw motion,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 4, 73–76 (1982).
Yu. N. Chelnokov, “On integration of kinematic equations of a rigid body’s screw-motion,” Appl. Math. Mech. 44 (1), 19–23 (1980).
Yu. N. Chelnokov, “One form of the equations of inertial navigation,” Mech. Solids 16 (5), 16–23 (1981).
D. Han, Q. Qing Wei, and Z. Li, “Kinematic control of free rigid bodies using dual quaternions,” Int. J. Autom. Comput. 05 (3), 319–324 (2008).
D. Han, Qing Wei, Z. Li, and Weimeng Sun, “Control of oriented mechanical systems: a method based on dual quaternion,” in Proc. 17th World Congress the International Federation of Automatic Control (Seoul, July 6–11, 2008), pp. 3836–3841.
D. Han, Qing Wei, and Z. Li, “A dual-quaternion method for control of spatial rigid body, networking, sensing and control,” in Proc. IEEE Int. Conf. on Networking, Sensing and Control (Sanya, Apr. 6–8, 2008), pp. 1–6.
E. Ozgur and Y. Mezouar, “Kinematic modeling and control of a robot arm using unit dual quaternions,” Rob. Auton. Syst. 77, 66–73 (2016).
Yu. N. Chelnokov, “Biquaternion solution of the kinematic control problem for the motion of a rigid body and its application to the solution of inverse problems of robot-manipulator kinematics,” Mech. Solids 48 (1), 31–47 (2013).
Yu. N. Chelnokov and E. I. Nelaeva, “Biquaternion solution of the kinematic problem on optimal nonlinear stabilization of arbitrary program movement of free rigid body,” Izv. Sarat. Univ. Nov. Ser., Ser.: Mat., Mekh., Inf. 16 (2), 198–206 (2016).
A. Perez and J. M. McCarthy, “Dual quaternion synthesis of constrained robotic systems,” J. Mech. Des. 126 (3), 425–435 (2004).
D. Han, Q. Wei, Z. Li, and W. Sun, “Control of oriented mechanical systems: a method based on dual quaternions,” in Proc. 17th IFAC World Congress (Seoul, 2008), pp. 3836–3841.
M. Schilling, “Universally manipulable body models – dual quaternion representations in layered and dynamic MMCs,” Auton. Rob. 30, 399–425 (2011).
F. Zhang and G. Duan, “Robust integrated translation and rotation finite-time maneuver of a rigid spacecraft based on dual quaternion,” in Proc. AIAA Guidance, Navigation, and Control Conf. (Portland, Aug. 8–11, 2011), Pap. No. AIAA 2011-6396.
J. Wang and Z. Sun, “6DOF robust adaptive terminal sliding mode control for spacecraft formation flying,” Acta Astron. 73, 76–87 (2012).
J. Wang, H. Liang, Z. Sun, S. Zhang, and M. Liu, “Finite-time control for spacecraft formation with dual-number-based description,” J. Guid., Control, Dyn. 35 (3), 950–962 (2012).
J. Wang and C. Yu, “Unit dual quaternion-based feedback linearization tracking problem for attitude and position dynamics,” Syst. Control Lett. 62 (3), 225–233 (2013).
N. Filipe and P. Tsiotras, “Rigid body motion tracking without linear and angular velocity feedback using dual quaternions,” in Proc. IEEE European Control Conf. (Zurich, 2013), pp. 329–334.
U. Lee, “State-constrained rotational and translational motion control with applications to monolithic and distributed spacecraft,” A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Program Authorized to Offer Degree: Aeronautics and Astronautics (Univ. of Washington, 2014).
N. Filipe, M. Kontitsis, and P. Tsiotras, “Extended Kalman filter for spacecraft pose estimation using dual quaternions,” J. Guid., Control, Dyn. 38 (4), 1625–1641 (2015).
N. Filipe and P. Tsiotras, “Adaptive position and attitude-tracking controller for satellite proximity operations using dual quaternions,” J. Guid., Control, Dyn. 38 (4), 566–577 (2015).
U. Lee and M. Mesbahi, “Optimal power descent guidance with 6-DoF line of sight constraints via unit dual quaternions,” in Proc. AIAA Guidance, Navigation, and Control Conf. (Kissimmee, FL, 2015).
U. Lee and M. Mesbahi, “Optimal powered descent guidance with 6-DoF line of sight constraints via unit dual quaternions,” American Institute of Aeronautics and Astronautics, pp. 1–21.
N. Filipe and P. Tsiotras, “Adaptive position and attitude-tracking controller for satellite proximity operations using dual quaternions,” J. Guid., Control, Dyn. 38 (4), 566–577 (2015).
H. Gui and G. Vukovich, “Cite as dual-quaternion-based adaptive motion tracking of spacecraft with reduced control effort,” Nonlinear Dyn. 83 (4), 566–577 (2016).
S. A. Akhramovich, V. V. Malyshev, and A. V. Starkov, “Mathematical model of drone motion in the biquaternion form,” Polet 4, 9–20 (2018).
S. A. Akhramovich and V. V. Malyshev, “Biquaternions application in the aircraft control problems,” in System Analysis, Control and Navigation. Proceedings (Moscow State Aviation Institute, Moscow, 2018), pp. 117–120 [in Russian].
S. A. Akhramovich and A. V. Barinov, “The system for controlling drone’s motion with predicting model in the biquaternion form,” in System Analysis, Control and Navigation. Proceedings (Moscow State Aviation Institute, Moscow, 2018), pp. 120–122 [in Russian].
C. Garcia, D. M. Prett, and M. Morari, “Model predictive control: theory and practice,” Automatica, No. 3, 335–348 (1989).
Yu. N. Chelnokov, “Spacecraft attitude control using quaternions,” Cosmic Res. 32 (3), 244–253 (1994).
Yu. N. Chelnokov, “Quaternion research laws kinematic managements of orientation of a rigid body on angular speed,” J. Comput. Syst. Sci. Int. 40 (4), 655–661 (1995).
Yu. N. Chelnokov, “Construction of attitude controls of a rigid body using quaternions and reference forms of equations of transients. I,” Mech. Solids 37 (1), 1–12 (2002).
Yu. N. Chelnokov, “Construction of attitude controls of a rigid body using quaternions and reference forms of equationsof transients. II,” Mech. Solids 37 (2), 3–16 (2002).
Yu. N. Chelnokov, Quaternion Models and Methods of Dynamics, Navigation, and Control of Motion (Fizmatlit, Moscow, 2011) [in Russian].
Funding
This work was partially supported by the Russian Foundation for Basic Research, project no. 19-01-00205.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by A. Ivanov
About this article
Cite this article
Chelnokov, Y.N. Synthesis of Control of Spatial Motion of a Rigid Body Using Dual Quaternions. Mech. Solids 55, 977–998 (2020). https://doi.org/10.3103/S0025654420070080
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0025654420070080