# Optimization-based iterative learning for precise quadrocopter trajectory tracking

- 3.1k Downloads
- 63 Citations

## Abstract

Current control systems regulate the behavior of dynamic systems by reacting to noise and unexpected disturbances as they occur. To improve the performance of such control systems, experience from iterative executions can be used to anticipate recurring disturbances and proactively compensate for them. This paper presents an algorithm that exploits data from previous repetitions in order to learn to precisely follow a predefined trajectory. We adapt the feed-forward input signal to the system with the goal of achieving high tracking performance—even under the presence of model errors and other recurring disturbances. The approach is based on a dynamics model that captures the essential features of the system and that explicitly takes system input and state constraints into account. We combine traditional optimal filtering methods with state-of-the-art optimization techniques in order to obtain an effective and computationally efficient learning strategy that updates the feed-forward input signal according to a customizable learning objective. It is possible to define a termination condition that stops an execution early if the deviation from the nominal trajectory exceeds a given bound. This allows for a safe learning that gradually extends the time horizon of the trajectory. We developed a framework for generating arbitrary flight trajectories and for applying the algorithm to highly maneuverable autonomous quadrotor vehicles in the ETH Flying Machine Arena testbed. Experimental results are discussed for selected trajectories and different learning algorithm parameters.

## Keywords

Motion planning and control Aerial robotics Quadrocopter control Trajectory tracking Learning Input and state constraints Kalman filtering Convex optimization## Notes

### Acknowledgements

This research was funded in part by the Swiss National Science Foundation (SNSF). The authors thank the anonymous reviewers for their thoughtful comments.

## Supplementary material

(WMV 43.5 MB)

## References

- Ahn, H.-S., Moore, K. L., & Chen, Y. (2007).
*Communications and control engineering*.*Iterative learning control: robustness and monotonic convergence for interval systems*(1st ed.). Berlin: Springer. zbMATHGoogle Scholar - Al-Hiddabi, S. (2009). Quadrotor control using feedback linearization with dynamic extension. In
*Proceedings of the international symposium on mechatronics and its applications (ISMA)*(pp. 1–3). CrossRefGoogle Scholar - Alexis, K., Nikolakopoulos, G., & Tzes, A. (2010). Constrained-control of a quadrotor helicopter for trajectory tracking under wind-gust disturbances. In
*Proceedings of the IEEE Mediterranean electrotechnical conference (MELECON)*(pp. 1411–1416). CrossRefGoogle Scholar - Amann, N., Owens, D. H., & Rogers, E. (1996). Iterative learning control using optimal feedback and feedforward actions.
*International Journal of Control*,*65*(2), 277–293. MathSciNetzbMATHCrossRefGoogle Scholar - Anderson, B. D. O., & Moore, J. B. (2005).
*Dover books on engineering*.*Optimal filtering*. New York: Dover. zbMATHGoogle Scholar - Arimoto, S., Kawamura, S., & Miyazaki, F. (1984). Bettering operation of robots by learning.
*Journal of Robotic Systems*,*1*(2), 123–140. CrossRefGoogle Scholar - Bamieh, B., Pearson, J. B., Francis, B. A., & Tannenbaum, A. (1991). A lifting technique for linear periodic systems with applications to sampled-data control.
*Systems & Control Letters*,*17*(2), 79–88. MathSciNetzbMATHCrossRefGoogle Scholar - Barton, K., Mishra, S., & Xargay, E. (2011). Robust iterative learning control: L1 adaptive feedback control in an ILC framework. In
*Proceedings of the American control conference (ACC)*(pp. 3663–3668). Google Scholar - Bauer, P., Kulcsar, B., & Bokor, J. (2009). Discrete time minimax tracking control with state and disturbance estimation ii: Time-varying reference and disturbance signals. In
*Proceedings of the Mediterranean conference on control and automation (MED)*(pp. 486–491). Google Scholar - Boost—Basic Linear Algebra Library (2012). Last accessed at 08 Feb 2012. [Online]. Available: http://www.boost.org/doc/libs/1_46_1/libs/numeric/ublas/doc/index.htm.
- Bouabdallah, S., & Siegwart, R. (2005). Backstepping and sliding-mode techniques applied to an indoor micro quadrotor. In
*Proceedings of the IEEE international conference on robotics and automation (ICRA)*(pp. 2247–2252). CrossRefGoogle Scholar - Bouktir, Y., Haddad, M., & Chettibi, T. (2008). Trajectory planning for a quadrotor helicopter. In
*Proceedings of the Mediterranean conference on control and automation*(pp. 1258–1263). CrossRefGoogle Scholar - Boyd, S., & Vandenberghe, L. (2004).
*Convex optimization*. Cambridge: Cambridge University Press. zbMATHGoogle Scholar - Bristow, D. A., Tharayil, M., & Alleyne, A. G. (2006). A survey of iterative learning control.
*IEEE Control Systems Magazine*,*26*(3), 96–114. CrossRefGoogle Scholar - Castillo, C. L., Moreno, W., & Valavanis, K. P. (2007). Unmanned helicopter waypoint trajectory tracking using model predictive control. In
*Proceedings of the Mediterranean conference on control automation*(pp. 1–8). CrossRefGoogle Scholar - Chen, Y., & Wen, C. (1999).
*Lecture notes in control and information sciences*.*Iterative learning control: convergence, robustness and applications*(1st ed.). Berlin: Springer. zbMATHGoogle Scholar - Chin, I., Qin, S. J., Lee, K. S., & Cho, M. (2004). A two-stage iterative learning control technique combined with real-time feedback for independent disturbance rejection.
*Automatica*,*40*(11), 1913–1922. MathSciNetzbMATHCrossRefGoogle Scholar - Cho, M., Lee, Y., Joo, S., & Lee, K. S. (2005). Semi-empirical model-based multivariable iterative learning control of an RTP system.
*IEEE Transactions on Semiconductor Manufacturing*,*18*(3), 430–439. CrossRefGoogle Scholar - Chui, C. K., & Chen, G. (1998).
*Springer series in information sciences*.*Kalman filtering: with real-time applications*. Berlin: Springer. zbMATHGoogle Scholar - Cowling, I., Yakimenko, O., Whidborne, J., & Cooke, A. (2007). A prototype of an autonomous controller for a quadrotor UAV. In
*Proceedings of the European control conference (ECC)*(pp. 1–8). Google Scholar - Curve Fitting Toolbox Splines and MATLAB Splines (2012). Last accessed 08 Feb 2012. [Online]. Available: http://www.mathworks.com/help/toolbox/curvefit/f2-10452.html.
- Diao, C., Xian, B., Yin, Q., Zeng, W., Li, H., & Yang, Y. (2011). A nonlinear adaptive control approach for quadrotor UAVs. In
*Proceedings of the Asian control conference (ASCC)*(pp. 223–228). Google Scholar - Dierks, T., & Jagannathan, S. (2010). Output feedback control of a quadrotor UAV using neural networks.
*IEEE Transactions on Neural Networks*,*21*(1), 50–66. CrossRefGoogle Scholar - Gurdan, D., Stumpf, J., Achtelik, M., Doth, K.-M., Hirzinger, G., & Rus, D. (2007). Energy-efficient autonomous four-rotor flying robot controlled at 1 kHz. In
*Proceedings of the IEEE international conference on robotics and automation (ICRA)*(pp. 361–366). CrossRefGoogle Scholar - Hoffmann, G., Waslander, S., & Tomlin, C. (2008). Quadrotor helicopter trajectory tracking control. In
*Proceedings of the AIAA guidance, navigation and control conference and exhibit*, Honolulu, Hawaii. Google Scholar - How, J. P., Bethke, B., Frank, A., Dale, D., & Vian, J. (2008). Real-time indoor autonomous vehicle test environment.
*IEEE Control Systems Magazine*,*28*(2), 51–64. MathSciNetCrossRefGoogle Scholar - Huang, H., Hoffmann, G., Waslander, S., & Tomlin, C. (2009). Aerodynamics and control of autonomous quadrotor helicopters in aggressive maneuvering. In
*Proceedings of the IEEE international conference on robotics and automation (ICRA)*(pp. 3277–3282). Google Scholar - IBM ILOG CPLEX Optimizer (2012). Last accessed 08 Feb 2012. [Online]. Available: http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/.
- Lee, K. S., Lee, J., Chin, I., Choi, J., & Lee, J. H. (2001). Control of wafer temperature uniformity in rapid thermal processing using an optimal iterative learning control technique.
*Industrial & Engineering Chemistry Research*,*40*(7), 1661–1672. CrossRefGoogle Scholar - Lee, D., Burg, T., Dawson, D., Shu, D., Xian, B., & Tatlicioglu, E. (2009). Robust tracking control of an underactuated quadrotor aerial-robot based on a parametric uncertain model. In
*Proceedings of the IEEE international conference on systems, man and cybernetics (SMC)*(pp. 3187–3192). Google Scholar - Lin, S.-H., & Lian, F.-L. (2005). Study of feasible trajectory generation algorithms for control of planar mobile robots. In
*Proceedings of the IEEE international conference on robotics and biomimetics (ROBIO)*(pp. 121–126). Google Scholar - Longman, R. W. (2000). Iterative learning control and repetitive control for engineering practice.
*International Journal of Control*,*73*(10), 930–954. MathSciNetzbMATHCrossRefGoogle Scholar - Lupashin, S., & D’Andrea, R. (2011). Adaptive open-loop aerobatic maneuvers for quadrocopters. In
*Proceedings of the IFAC World congress*(Vol. 18(1), pp. 2600–2606). Google Scholar - Lupashin, S., Schoellig, A.P., Sherback, M., & D’Andrea, R. (2010). A simple learning strategy for high-speed quadrocopter multi-flips. In
*Proceedings of the IEEE international conference on robotics and automation (ICRA)*(pp. 1642–1648). Google Scholar - Madani, T., & Benallegue, A. (2006). Backstepping control for a quadrotor helicopter. In
*Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (IROS)*(pp. 3255–3260). Google Scholar - Mellinger, D., Michael, N., & Kumar, V. (2010). Trajectory generation and control for precise aggressive maneuvers with quadrotors. In
*Proceedings of the international symposium on experimental robotics*. Google Scholar - Michael, N., Mellinger, D., Lindsey, Q., & Kumar, V. (2010). The GRASP multiple micro UAV testbed.
*IEEE Robotics & Automation Magazine*,*17*(3), 56–65. CrossRefGoogle Scholar - Mokhtari, A., & Benallegue, A. (2004). Dynamic feedback controller of Euler angles and wind parameters estimation for a quadrotor unmanned aerial vehicle. In
*Proceedings of the IEEE international conference on robotics and automation (ICRA)*(Vol. 3, pp. 2359–2366). Google Scholar - Moore, K. L. (1998). Multi-loop control approach to designing iterative learning controllers. In
*Proceedings of the IEEE conference on decision and control (CDC)*(Vol. 1, pp. 666–671). Google Scholar - Nicol, C., Macnab, C. J. B., & Ramirez-Serrano, A. (2011). Robust adaptive control of a quadrotor helicopter.
*Mechatronics*,*21*(6), 927–938. CrossRefGoogle Scholar - Norrloef, M., & Gunnarsson, S. (2002). Time and frequency domain convergence properties in iterative learning control.
*International Journal of Control*,*75*(14), 1114–1126. MathSciNetzbMATHCrossRefGoogle Scholar - Phan, M. Q., & Longman, R. W. (1988). A mathematical theory of learning control for linear discrete multivariable systems. In
*Proceedings of the AIAA/AAS astrodynamics conference*(pp. 740–746). Google Scholar - Piazzi, A., & Guarino Lo Bianco, C. (2000). Quintic
*G*^{2}-splines for trajectory planning of autonomous vehicles. In*Proceedings of the IEEE intelligent vehicles symposium*(pp. 198–203). Google Scholar - Pounds, P., Mahony, R., & Corke, P. (2006). Modelling and control of a quad-rotor robot. In
*Proceedings of the Australasian conference on robotics and automation*. Google Scholar - Purwin, O., & D’Andrea, R. (2009). Performing aggressive maneuvers using iterative learning control. In
*Proceedings of the IEEE international conference on robotics and automation (ICRA)*(pp. 1731–1736). Google Scholar - Raffo, G., Ortega, M., & Rubio, F. (2008). Backstepping/nonlinear
*H*_{∞}control for path tracking of a quadrotor unmanned aerial vehicle. In*Proceedings of the American control conference (ACC)*(pp. 3356–3361). Google Scholar - Rice, J. K., & Verhaegen, M. (2010). A structured matrix approach to efficient calculation of LQG repetitive learning controllers in the lifted setting.
*International Journal of Control*,*83*(6), 1265–1276. MathSciNetzbMATHCrossRefGoogle Scholar - Ritz, R., Hehn, M., Lupashin, S., & D’Andrea, R. (2011). Quadrotor performance benchmarking using optimal control. In
*Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (IROS)*(pp. 5179–5186). Google Scholar - Schoellig, A. P., & D’Andrea, R. (2009). Optimization-based iterative learning control for trajectory tracking. In
*Proceedings of the European control conference (ECC)*(pp. 1505–1510). Google Scholar - The Mathworks Optimization Toolbox (2012). Last accessed 08 Feb 2012. [Online]. Available: http://www.mathworks.com/help/toolbox/optim/ug/fmincon.html.
- Tousain, R., van der Meche, E., & Bosgra, O. (2001). Design strategy for iterative learning control based on optimal control. In
*Proceedings of the IEEE conference on decision and control (CDC)*(Vol. 5, pp. 4463–4468). Google Scholar - Waslander, S. L., Hoffmann, G. M., Jang, J. S., & Tomlin, C. (2005). Multi-agent quadrotor testbed control design: integral sliding mode vs. reinforcement learning. In
*Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (IROS)*, August 2005 (pp. 3712–3717). Google Scholar - Zhang, C., Wang, N., & Chen, J. (2010). Trajectory generation for aircraft based on differential flatness and spline theory. In
*Proceedings of the international conference on information networking and automation (ICINA)*(Vol. 1, pp. V1-110–V1-114). CrossRefGoogle Scholar - Zhou, Q.-L., Zhang, Y., Qu, Y.-H., & Rabbath, C.-A. (2010a). Dead reckoning and Kalman filter design for trajectory tracking of a quadrotor UAV. In
*Proceedings of the IEEE/ASME international conference on mechatronics and embedded systems and applications (MESA)*(pp. 119–124). CrossRefGoogle Scholar - Zhou, Q.-L., Zhang, Y., Rabbath, C.-A., & Theilliol, D. (2010b). Design of feedback linearization control and reconfigurable control allocation with application to a quadrotor UAV. In
*Proceedings of the conference on fault-tolerant systems (SysTol)*(pp. 371–376). CrossRefGoogle Scholar - Zhu, B., & Huo, W. (2010). Trajectory linearization control for a quadrotor helicopter. In
*Proceedings of the IEEE international conference on control and automation (ICCA)*(pp. 34–39). Google Scholar - Zuo, Z. (2010). Trajectory tracking control design with command-filtered compensation for a quadrotor.
*IET Control Theory & Applications*,*4*(11), 2343–2355. MathSciNetCrossRefGoogle Scholar