A geodesic feedback law to decouple the full and reduced attitude. (English) Zbl 1378.93104
Summary: This paper presents a novel approach to the problem of almost global attitude stabilization. The reduced attitude is steered along a geodesic path on the \(n-1\)-sphere. Meanwhile, the full attitude is stabilized on \(\mathsf{SO}(n)\). This action, essentially two maneuvers in sequel, is fused into one smooth motion. Our algorithm is useful in applications where stabilization of the reduced attitude takes precedence over stabilization of the full attitude. A two parameter feedback gain affords further trade-offs between the full and reduced attitude convergence speed. The closed loop kinematics on \(\mathsf{SO}(3)\) are solved for the states as functions of time and the initial conditions, providing precise knowledge of the transient dynamics. The exact solutions also help us to characterize the asymptotic behavior of the system such as establishing the region of attraction by straightforward evaluation of limits. The geometric flavor of these ideas is illustrated by a numerical example.
Software:
mftoolboxReferences:
| [2] | Chaturvedi, N. A.; Sanyal, A. K.; McClamroch, N. H., Rigid-body attitude control: Using rotation matrices for continuous singularity-free control laws, IEEE Control Syst. Mag., 31, 3, 30-51 (2011) · Zbl 1395.70027 |
| [3] | Bhat, S. P.; Bernstein, D. S., A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon, Systems Control Lett., 39, 1, 63-70 (2000) · Zbl 0986.93063 |
| [5] | Lee, T., Exponential stability of an attitude tracking control system on SO(3) for large-angle rotational maneuvers, Systems Control Lett., 61, 1, 231-237 (2012) · Zbl 1256.93088 |
| [6] | Mayhew, C. G.; Sanfelice, R. G.; Teel, A. R., Quaternion-based hybrid control for robust global attitude tracking, IEEE Trans. Automat. Control, 56, 11, 2555-2566 (2011) · Zbl 1368.93552 |
| [9] | Chaumette, F.; Hutchinson, S., Visual servo control. Part I: Basic approaches, IEEE Robot. Autom. Mag., 13, 4, 82-90 (2006) |
| [10] | Chaumette, F.; Hutchinson, S., Visual servo control. Part II: Advanced approaches, IEEE Robot. Autom. Mag., 14, 1, 109-118 (2007) |
| [11] | Hurtado, J. E.; Sinclair, A. J., Hamel coefficients for the rotational motion of an \(n\)-dimensional rigid body, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460, 2052, 3613-3630 (2004) · Zbl 1071.70003 |
| [12] | Maithripala, D. H.S.; Berg, J. M.; Dayawansa, W. P., Almost-global tracking of simple mechanical systems on a general class of Lie groups, IEEE Trans. Automat. Control, 51, 2, 216-225 (2006) · Zbl 1366.93406 |
| [14] | Lageman, C.; Trumpf, J.; Mahony, R., Gradient-Like observers for invariant dynamics on a Lie Group, IEEE Trans. Automat. Control, 55, 2, 367-377 (2010) · Zbl 1368.93050 |
| [17] | Thakur, S., Framework for Visualizing and Exploring High-Dimensional Geometry (Ph.D. thesis) (2008), Indiana University |
| [18] | Dwyer III, T., Exact nonlinear control of large angle rotational maneuvers, IEEE Trans. Automat. Control, 29, 9, 769-774 (1984) · Zbl 0542.93032 |
| [19] | Spindler, K., Optimal control on lie groups with applications to attitude control, Math. Control Signals Systems, 11, 3, 197-219 (1998) · Zbl 1081.49503 |
| [20] | Mahony, R.; Hamel, T.; Pflimlin, J.-M., Nonlinear complementary filters on the special orthogonal group, IEEE Trans. Automat. Control, 53, 5, 1203-1218 (2008) · Zbl 1367.93652 |
| [21] | Elipe, A.; Lanchares, V., Exact solution of a triaxial gyrostat with one rotor, Celestial Mech. Dynam. Astronom., 101, 49-68 (2008) · Zbl 1342.70014 |
| [22] | Doroshin, A. V., Exact solutions for angular motion of coaxial bodies and attitude dynamics of gyrostat-satellites, Int. J. Non-Linear Mech., 50, 68-74 (2012) |
| [23] | Ayoubi, M. A.; Longuski, J. M., Asymptotic theory for thrusting, spinning-up spacecraft maneuvers, Acta Astronaut., 64, 7, 810-831 (2009) |
| [25] | Markdal, J.; Hu, X., Exact solutions to a class of closed-loop systems on SO(n), Automatica, 63, 138-147 (2016) · Zbl 1329.93110 |
| [26] | Rudin, W., Real and Complex Analysis (1987), McGraw-Hill · Zbl 0925.00005 |
| [27] | Siciliano, B.; Khatib, O., Springer Handbook of Robotics (2008), Springer · Zbl 1171.93300 |
| [28] | Brockett, R. W., Lie theory and control systems defined on spheres, SIAM J. Appl. Math., 25, 2, 213-225 (1973) · Zbl 0272.93003 |
| [29] | Culver, W. J., On the existence and uniqueness of the real logarithm of a matrix, Proc. Amer. Math. Soc., 1146-1151 (1966) · Zbl 0143.26402 |
| [30] | Higham, N. J., Functions of Matrices: Theory and Computation (2008), SIAM · Zbl 1167.15001 |
| [31] | Aulbach, B., Continuous and Discrete Dynamics Near Manifolds of Equilibria (1984), Springer · Zbl 0535.34002 |
| [32] | Lageman, Christians, Convergence of Gradient-like Dynamical Systems and Optimization Algorithms (Ph.D. thesis) (2007), University of Würzburg |
| [34] | Khalil, H. K., Nonlinear Systems (2002), Prentice Hall · Zbl 0626.34052 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
