GPOPS-II

GPOPS-II (pronounced "GPOPS 2") is a general-purpose MATLAB software for solving continuous optimal control problems using hp-adaptive Gaussian quadrature collocation and sparse nonlinear programming. The acronym GPOPS stands for "General Purpose OPtimal Control Software", and the Roman numeral "II" refers to the fact that GPOPS-II is the second software of its type (that employs Gaussian quadrature integration).

GPOPS-II

Developer(s)	Michael Patterson [1] and Anil V. Rao[2]
Initial release	January 2013 (2013-01)
Stable release	2.0 / 1 September 2015 (2015-09-01)
Written in	MATLAB
Operating system	Mac OS X, Linux, Windows
Available in	English
Type	Numerical optimization software
License	Proprietary, Free-of-charge for K - 12 or classroom use. Licensing fees apply for all academic, not-for profit, and commercial use (outside of classroom use)
Website	gpops2.com

Problem Formulation

GPOPS-II[3] is designed to solve multiple-phase optimal control problems of the following mathematical form (where ${\displaystyle P}$ is the number of phases):

${\displaystyle \min J=\phi (\mathbf {e} ^{(1)},\ldots ,\mathbf {e} ^{(P)})}$

subject to the dynamic constraints

${\displaystyle {\dot {\mathbf {y} }}^{(p)}(t)=\mathbf {a} ^{(p)}(\mathbf {y} ^{(p)}(t),\mathbf {u} ^{(p)}(t),t,\mathbf {s} ),\quad (p=1,\ldots ,P),}$

the event constraints

${\displaystyle \mathbf {b} _{\min }\leq \mathbf {b} (\mathbf {e} ^{(1)},\ldots ,\mathbf {e} ^{(P)},\mathbf {s} )\leq \mathbf {b} _{\max },}$

the inequality path constraints

${\displaystyle \mathbf {c} _{\min }^{(p)}\leq \mathbf {c} (\mathbf {y} ^{(p)}(t),\mathbf {u} ^{(p)}(t),t,\mathbf {s} )\leq \mathbf {c} _{\max }^{(p)},\quad (p=1,\ldots ,P),}$

the static parameter constraints

${\displaystyle \mathbf {s} _{\min }\leq \mathbf {s} \leq \mathbf {s} _{\max },}$

and the integral constraints

${\displaystyle \mathbf {q} _{\min }^{(p)}\leq \mathbf {q} ^{(p)}\leq \mathbf {q} _{\max }^{(p)},\quad (p=1,\ldots ,P),}$

where

${\displaystyle \mathbf {e} ^{(p)}=\left[\mathbf {y} ^{(p)}(t_{0}^{(p)}),t_{0}^{(p)},\mathbf {y} ^{(p)}(t_{f}^{(p)}),t_{f}^{(p)},\mathbf {q} ^{(p)}\right],\quad (p=1,\ldots ,P),}$

and the integrals in each phase are defined as

${\displaystyle q_{i}^{(p)}=\int _{t_{0}^{(p)}}^{t_{f}^{(p)}}g_{i}^{(p)}(\mathbf {y} ^{(p)}(t),\mathbf {u} ^{(p)}(t),t,\mathbf {s} )dt,\quad (i=1,\ldots ,n_{q}^{(p)},\;p=1,\ldots ,P).}$

It is important to note that the event constraints can contain any functions that relate information at the start and/or terminus of any phase (including relationships that include both static parameters and integrals) and that the phases themselves need not be sequential. It is noted that the approach to linking phases is based on well-known formulations in the literature.[4]

Method Employed by GPOPS-II

GPOPS-II uses a class of methods referred to as ${\displaystyle hp}$-adaptive Gaussian quadrature collocation where the collocation points are the nodes of a Gauss quadrature (in this case, the Legendre-Gauss-Radau [LGR] points). The mesh consists of intervals into which the total time interval ${\displaystyle t^{(p)}\in [t_{0}^{(p)},t_{f}^{(p)}]}$ in each phase is divided, and LGR collocation is performed in each interval. Because the mesh can be adapted such that both the degree of the polynomial used to approximate the state ${\displaystyle \mathbf {y} ^{(p)}(t)}$ and the width of each mesh interval can be different from interval to interval, the method is referred to as an ${\displaystyle hp}$-adaptive method (where "${\displaystyle h}$" refers to the width of each mesh interval, while "${\displaystyle p}$" refers to the polynomial degree in each mesh interval). The LGR collocation method has been developed rigorously in Refs.,[5][6][7] while ${\displaystyle hp}$-adaptive mesh refinement methods based on the LGR collocation method can be found in Refs., .[8][9][10][11]

Development

The development of GPOPS-II began in 2007. The code development name for the software was OptimalPrime, but was changed to GPOPS-II in late 2012 in order to keep with the lineage of the original version of GPOPS [12] which implemented global collocation using the Gauss pseudospectral method. The development of GPOPS-II continues today, with improvements that include the open-source algorithmic differentiation package ADiGator [13] and continued development of ${\displaystyle hp}$-adaptive mesh refinement methods for optimal control.

Applications of GPOPS-II

GPOPS-II has been used extensively throughout the world both in academia and industry. Published academic research where GPOPS-II has been used includes Refs.,[14][15][16] where the software has been used in applications such as performance optimization of Formula One race cars, Ref.[17] where the software has been used for minimum-time optimization of low-thrust orbital transfers,[18] where the software has been used for human performance in cycling, Ref.[19] where the software has been used for soft lunar landing, and Ref.[20] where the software has been used to optimize the motion of a bipedal robot.

References

1. http://www.anilvrao.com/People.html
2. Website of Anil V. Rao
3. Patterson, M. A.; Rao, A. V. (2014). "GPOPS-II: A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using hp-Adaptive Gaussian Quadrature Collocation Methods and Sparse Nonlinear Programming". ACM Transactions on Mathematical Software. 41 (1): 1:1–1:37. doi:10.1145/2558904.
4. Betts, John T. (2010). Practical Methods for Optimal Control and Estimation Using Nonlinear Programming. Philadelphia: SIAM Press. doi:10.1137/1.9780898718577. ISBN 9780898718577.
5. Garg, D.; Patterson, M. A.; Hager, W. W.; Rao, A. V.; Benson, D. A.; Huntington, G. T. (2010). "A Unified Framework for the Numerical Solution of Optimal Control Problems Using Pseudospectral Methods". Automatica. 46 (11): 1843–1851. doi:10.1016/j.automatica.2010.06.048.
6. Garg, D.; Hager, W. W.; Rao, A. V.; et al. (2011). "Pseudospectral Methods for Solving Infinite-Horizon Optimal Control Problems". Automatica. 47 (4): 829–837. doi:10.1016/j.automatica.2011.01.085.
7. Garg, D.; Patterson, M. A.; Darby, C. L.; Francolin, C.; Huntington, G. T.; Hager, W. W.; Rao, A. V.; et al. (2011). "Direct Trajectory Optimization and Costate Estimation of Finite-Horizon and Infinite-Horizon Optimal Control Problems Using a Radau Pseudospectral Method". Computational Optimization and Applications. 49 (2): 335–358. CiteSeerX 10.1.1.663.4215. doi:10.1007/s10589-009-9291-0.
8. Darby, C. L.; Hager, W. W.; Rao, A. V.; et al. (2011). "An hp-Adaptive Pseudospectral Method for Solving Optimal Control Problems". Optimal Control Applications and Methods. 32 (4): 476–502. doi:10.1002/oca.957.
9. Darby, C. L.; Hager, W. W.; Rao, A. V.; et al. (2011). "Direct Trajectory Optimization Using a Variable Low-Order Adaptive Pseudospectral Method". Journal of Spacecraft and Rockets. 48 (3): 433–445. Bibcode:2011JSpRo..48..433D. CiteSeerX 10.1.1.367.7092. doi:10.2514/1.52136.
10. Patterson, M. A.; Hager, W. W.; Rao, A. V. (2011). "A ph Mesh Refinement Method for Optimal Control". Optimal Control Applications and Methods. 36 (4): 398–421. doi:10.1002/oca.2114.
11. Liu, F.; Hager, W. W.; Rao, A. V. (2015). "Adaptive Mesh Refinement for Optimal Control Using Nonsmoothness Detection and Mesh Size Reduction". Journal of the Franklin Institute - Engineering and Applied Mathematics. 352 (10): 4081–4106. doi:10.1016/j.jfranklin.2015.05.028.
12. Rao, A. V.; Benson, D. A.; Darby, C. L.; Patterson, M. A.; Francolin, C.; Sanders, I.; Huntington, G. T. (2010). "GPOPS: A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using the Gauss Pseudospectral Method". ACM Transactions on Mathematical Software. 37 (2): 22:1–22:39. doi:10.1145/1731022.1731032.
13. Weinstein, M. J.; Rao, A. V. "ADiGator: A MATLAB Toolbox for Algorithmic Differentiation Using Source Transformation via Operator Overloading". ADiGator.
14. Perantoni, G.; Limebeer, D. J. N. (2015). "Optimal Control of a Formula One Car on a Three-Dimensional Track—Part 1: Track Modeling and Identification". Journal of Dynamic Systems, Measurement, and Control. 137 (2): 021010. doi:10.1115/1.4028253.
15. Limebeer, D. J. N.; Perantoni, G. (2015). "Optimal Control of a Formula One Car on a Three-Dimensional Track—Part 2: Optimal Control". Journal of Dynamic Systems, Measurement, and Control. 137 (5): 051019. doi:10.1115/1.4029466.
16. Limebeer, D. J. N.; Perantoni, G.; Rao, A. V. (2014). "Optimal Control of Formula One Car Energy Recovery Systems". International Journal of Control. 87 (10): 2065–2080. Bibcode:2014IJC....87.2065L. doi:10.1080/00207179.2014.900705.
17. Graham, K. F.; Rao, A. V. (2015). "Minimum-Time Trajectory Optimization of Many Revolution Low-Thrust Earth-Orbit Transfers". Journal of Spacecraft and Rockets. 52 (3): 711–727. doi:10.2514/1.a33187. S2CID 43633680.
18. Dahmen, T.; Saupeand, D. (2014). "Optimal pacing strategy for a race of two competing cyclists". Journal of Science and Cycling. 3 (2).
19. Moon, Y; Kwon, S (2014). "Lunar Soft Landing with Minimum-Mass Propulsion System Using H2O2/Kerosene Bipropellant Rocket System". Acta Astronautica. 99 (May–June): 153–157. Bibcode:2014AcAau..99..153M. doi:10.1016/j.actaastro.2014.02.003.
20. Haberland, M.; McClelland, H.; Kim, S.; Hong, D. (2006). "The Effect of Mass Distribution on Bipedal Robot Efficiency". International Journal of Robotics Research. 25 (11): 1087–1098. doi:10.1177/0278364906072449.

External links

• GPOPS-II home page
• GPOPS-II Journal Article Appearing in the ACM Transactions on Mathematical Software
• Website of Anil V. Rao