Exact solutions to a class of feedback systems on \(\mathsf{SO}(n)\). (English) Zbl 1329.93110
Summary: This paper provides a novel approach to the problem of attitude tracking for a class of almost globally asymptotically stable feedback laws on \(\mathsf{SO}(n)\). The closed-loop systems are solved exactly for the rotation matrices as explicit functions of time, the initial conditions, and the gain parameters of the control laws. The exact solutions provide insight into the transient dynamics of the system and can be used to prove almost global attractiveness of the identity matrix. Applications of these results are found in model predictive control problems where detailed insight into the transient attitude dynamics is utilized to approximately complete a task of secondary importance. Knowledge of the future trajectory of the states can also be used as an alternative to the zero-order hold in systems where the attitude is sampled at discrete time instances.
MSC:
| 93D15 | Stabilization of systems by feedback |
| 93B25 | Algebraic methods |
| 93B40 | Computational methods in systems theory (MSC2010) |
| 93C57 | Sampled-data control/observation systems |
Keywords:
attitude control; global stability; Lie groups; nonlinear systems; predictive control; sampled-data systemsSoftware:
mftoolboxReferences:
| [1] | Åström, K. J.; Wittenmark, B., Computer controlled systems: theory and design (1997), Prentice Hall |
| [2] | Ayoubi, M. A.; Longuski, J. M., Asymptotic theory for thrusting, spinning-up spacecraft maneuvers, Acta Astronautica, 64, 7, 810-831 (2009) |
| [3] | Bhat, S. P.; Bernstein, D. S., A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon, Systems & Control Letters, 39, 1, 63-70 (2000) · Zbl 0986.93063 |
| [5] | Chaturvedi, N. A.; McClamroch, N. H.; Bernstein, D. S., Asymptotic smooth stabilization of the inverted 3-D pendulum, IEEE Transactions on Automatic Control, 54, 6, 1204-1215 (2009) · Zbl 1367.93492 |
| [6] | Chaturvedi, N. A.; Sanyal, A. K.; McClamroch, N. H., Rigid-body attitude control: Using rotation matrices for continuous singularity-free control laws, IEEE Control Systems Magazine, 31, 3, 30-51 (2011) · Zbl 1395.70027 |
| [7] | Chaumette, F.; Hutchinson, S., Visual servo control. Part I: basic approaches, IEEE Robotics & Automation Magazine, 13, 4, 82-90 (2006) |
| [8] | Chaumette, F.; Hutchinson, S., Visual servo control. Part II: advanced approaches, IEEE Robotics & Automation Magazine, 14, 1, 109-118 (2007) |
| [9] | Culver, W. J., On the existence and uniqueness of the real logarithm of a matrix, Proceedings of the American Mathematical Society, 1146-1151 (1966) · Zbl 0143.26402 |
| [10] | Doroshin, A. V., Exact solutions for angular motion of coaxial bodies and attitude dynamics of gyrostat-satellites, International Journal of Non-Linear Mechanics, 50, 68-74 (2012) |
| [11] | Dwyer, T., Exact nonlinear control of large angle rotational maneuvers, IEEE Transactions on Automatic Control, 29, 9, 769-774 (1984) · Zbl 0542.93032 |
| [12] | Elipe, A.; Lanchares, V., Exact solution of a triaxial gyrostat with one rotor, Celestial Mechanics and Dynamical Astronomy, 101, 49-68 (2008) · Zbl 1342.70014 |
| [13] | Filippov, A. F., Differential equations with discontinuous righthand sides: control systems (1988), Kluwer · Zbl 0664.34001 |
| [14] | Higham, N. J., Functions of matrices: theory and computation (2008), SIAM · Zbl 1167.15001 |
| [15] | Hinrichsen, D.; Pritchard, A. J., Mathematical systems theory I: modelling, state space analysis, stability and robustness (2005), Springer · Zbl 1074.93003 |
| [16] | Hu, G.; MacKunis, W.; Gans, N.; Dixon, W. E.; Chen, J.; Behal, A., Homography-based visual servo control with imperfect camera calibration, IEEE Transactions on Automatic Control, 54, 6, 1318-1324 (2009) · Zbl 1367.93309 |
| [17] | Hurtado, J. E.; Sinclair, A. J., Hamel coefficients for the rotational motion of an \(n\)-dimensional rigid body, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 460, 2052, 3613-3630 (2004) · Zbl 1071.70003 |
| [18] | Krstic, M.; Kokotovic, P. V.; Kanellakopoulos, I., Nonlinear and adaptive control design (1995), Wiley · Zbl 0763.93043 |
| [20] | Lee, T., Exponential stability of an attitude tracking control system on SO(3) for large-angle rotational maneuvers, Systems & Control Letters, 61, 1, 231-237 (2012) · Zbl 1256.93088 |
| [21] | Lee, T.; Leok, M.; McClamroch, N. H., Lagrangian mechanics and variational integrators on two-spheres, International Journal for Numerical Methods in Engineering, 79, 9, 1147-1174 (2009) · Zbl 1176.70004 |
| [23] | Liberzon, D., Switching in systems and control (2003), Springer · Zbl 1036.93001 |
| [24] | 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 Transactions on Automatic Control, 51, 2, 216-225 (2006) · Zbl 1366.93406 |
| [30] | Mayhew, C. G.; Sanfelice, R. G.; Teel, A. R., Quaternion-based hybrid control for robust global attitude tracking, IEEE Transactions on Automatic Control, 56, 11, 2555-2566 (2011) · Zbl 1368.93552 |
| [31] | Roza, A.; Maggiore, M., Position control for a class of vehicles in SE(3), (Proceedings of the 51st IEEE conference on decision and control (2012), IEEE), 5631-5636 |
| [33] | Sastry, S. S., Nonlinear systems: analysis, stability, and control (1999), Springer · Zbl 0924.93001 |
| [34] | Spindler, K., Optimal control on Lie groups with applications to attitude control, Mathematics of Control, Signals, and Systems, 11, 3, 197-219 (1998) · Zbl 1081.49503 |
| [35] | Thakur, S., Framework for visualizing and exploring high-dimensional geometry (2008), Indiana University, (Ph.D. thesis) |
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