For citation:
Chelnokov Y. N., Nelaeva E. I. Solving Kinematic Problem of Optimal Nonlinear Stabilization of Arbitrary Program Movement of Free Rigid Body. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 2, pp. 198-207. DOI: 10.18500/1816-9791-2016-16-2-198-207, EDN: WCNQLP
Solving Kinematic Problem of Optimal Nonlinear Stabilization of Arbitrary Program Movement of Free Rigid Body
The kinematic problem of nonlinear stabilization of arbitrary program motion of free rigid body is studied. Biquaternion kinematic equation of perturbed motion of a free rigid body is considered as a mathematical model of motion. Instant speed screw of body motion is considered as a control. There are two functionals that are to be minimized. Both of them characterize the integral quantity of energy costs of control and squared deviations of motion parameters of a free rigid body from their program values. Optimal control laws and differential equations of optimization problem are determined using the Pontryagin’s maximum principle. Analytical solution of this problem has been found. The control law obtained is used for numerical solution of the inverse kinematics of a Stanford robot arm. The analysis of the numerical solution is carried out.
- Branec V. N., Shmyglevskij I. P. Using Biquaternions in Problem of Rigid Body Position Control. Izv. AN SSSR. MTT, 1972, no. 4, pp. 24–31 (in Russian).
- Branec V. N., Shmyglevskij I. P. Kinematic Problem of Orientation in Rotating Coordinate Frame. Izv. AN SSSR. MTT, 1972, no. 6, pp. 36–43 (in Russian).
- Branec V. N., Shmyglevskij I. P. Primenenie kvaternionov v zadachah orientacii tverdogo tela [Using Biquaternions in Problem of Rigid Body Orientation]. Moscow, Nauka, 1973, 320 p. (in Russian).
- Plotnikov P. K., Sergeev A. N., Chelnokov Yu. N. Kinematic control problem for the orientation of a rigid body. Mech. Solids, 1991, vol. 37, no. 5, pp. 7–16.
- Pankov A. A., Chelnokov Yu. N. Investigation of quaternion laws of kinematic control of solid body orientation in angular velocity. Mech. Solids, 1995, vol. 33, no. 6, pp. 3–13.
- Branec V. N., Shmyglevskij I. P. Vvedenie v teoriju besplatformennyh inercial’nyh navigacionnyh sistem [Introduction to the Theory of Strapdown Inertial Navigation Systems]. Moscow, Nauka, 1992, 280 p. (in Russian).
- Molodenkov A. V. Kvaternionnoe reshenie zadachi optimal’nogo razvorota tverdogo tela so sfericheskim raspredeleniem mass [Quaternion Solution of the Problem of Optimal Rotation of a Rigid Body With a Spherical Mass Distribution]. Problemy mehaniki i upravlenija: Mezhvuz. sb. nauch. tr. Perm’, Perm Univ. Press, 1995, pp. 122–131 (in Russian).
- Birjukov V. G., Chelnokov Ju. N. Kinematicheskaja zadacha optimal’noj nelinejnoj stabilizacii uglovogo dvizhenija tverdogo tela [Kinematic Problem of Optimal Nonlinear Stabilization of Rigid Body Angular Motion]. Matematika. Mehanika [Mathematics. Mechanics]. Saratov, Saratov Univ. Press, 2002, iss. 4, pp. 172–174 (in Russian).
- Malanin V. V., Strelkova N. A. Optimal’noe upravlenie orientaciej i vintovym dvizheniem tverdogo tela [Optimal Control of Rigid Body Orientation and Screw Motion]. Moscow; Izhevsk, NIC “Reguljarnaja i haoticheskaja dinamika”, 2004, 204 p. (in Russian).
- Chelnokov Yu. N. Quaternion and Biquaternion Models and Methods of Mechanics of Solid Bodies and its Applications. Geometry and Kinematics of Motion. Moscow, Fizmatlit, 2006, 511 p. (in Russian).
- Strelkova N. A. Optimal’noe po bystrodejstviju kinematicheskoe upravlenie vintovym peremeshheniem tverdogo tela [Time Optimal Kinematic Control of Rigid Body Screw Motion]. Izv. AN SSSR. MTT, 1982, no. 4, pp. 73–76 (in Russian).
- Chelnokov Yu. N. On integration of kinematic equations of a rigid body’s screw-motion. Applied mathematics and mechanics, 1980, vol. 44, no. 1, pp. 19–23.
- Chelnokov Yu. N. 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, 2013, vol. 48, no. 1, pp. 31–46.
- Lomovceva E. I., Chelnokov Ju. N. Dual matrix and biquaternion methods of solving direct and inverse kinematics problems of manipulators for example Stanford robot arm. II. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 1, pp. 88–95 (in Russian).
- Nelaeva E. I., Chelnokov Ju. N. Solution to the Problems of Direct and Inverse Kinematics of the Robots-Manipulators Using Dual Matrices and Biquaternions on the Example of Stanford Robot Arm. Pt. 1. Mekhatronika, Avtomatizatsiya, Upravlenie, 2015, vol. 16, no. 6, pp. 373–380. DOI: https://doi.org/10.17587/mau.16.373-380 (in Russian).
- 1404 reads