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==Usage==
==Usage==
DIDO utilizes trademarked expressions and objects<ref name="R1" /> that facilitate a user to quickly formulate and solve [[optimal control]] problems.<ref name="hawkins-mit">A. M. Hawkins, ''Constrained Trajectory Optimization of a Soft Lunar Landing From a Parking Orbit,'' S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2005. http://dspace.mit.edu/handle/1721.1/32431
DIDO utilizes trademarked expressions and objects<ref name="R1" /> that facilitate a user to quickly formulate and solve [[optimal control]] problems.<ref name="hawkins-mit">A. M. Hawkins, ''Constrained Trajectory Optimization of a Soft Lunar Landing From a Parking Orbit,'' S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2005. http://dspace.mit.edu/handle/1721.1/32431
</ref><ref name="rea-mit" /><ref name="JR1" /><ref>{{cite journal|last=Infeld|first=S. I.|year=2005|title=Optimization of Mission Design for Constrained Libration Point Space Missions|url=http://www.stanford.edu/group/SOL/dissertations/samantha-thesis.pdf|format=[[PDF]]|publisher=Stanford University}}</ref> Rapidity in formulation is achieved through a set of DIDO expressions which are based on variables commonly used in optimal control theory.<ref name="R1" /> For example, the '''state''', '''control''' and '''time''' variables are formatted as:<ref name="R1" />
</ref><ref name="rea-mit" /><ref name="JR1" /><ref>{{cite journal|last=Infeld|first=S. I.|year=2005|title=Optimization of Mission Design for Constrained Libration Point Space Missions|url=http://www.stanford.edu/group/SOL/dissertations/samantha-thesis.pdf|publisher=Stanford University}}</ref> Rapidity in formulation is achieved through a set of DIDO expressions which are based on variables commonly used in optimal control theory.<ref name="R1" /> For example, the '''state''', '''control''' and '''time''' variables are formatted as:<ref name="R1" />
* primal.'''states''',
* primal.'''states''',
* primal.'''controls''', and
* primal.'''controls''', and
Line 19: Line 19:


==Theory==
==Theory==
DIDO implements a spectral algorithm<ref name="RF1" /><ref name="GFR1" /> based on [[pseudospectral optimal control]] theory founded by [[I. Michael Ross|Ross]] and his associates.<ref name="RK">{{cite journal | last1 = Ross | first1 = I. M. | last2 = Karpenko | first2 = M. | year = 2012 | title = A Review of Pseudospectral Optimal Control: From Theory to Flight | url = http://www.sciencedirect.com/science/article/pii/S1367578812000375 | journal = Annual Reviews in Control | volume = 36 | issue = | pages = 182–197 | doi=10.1016/j.arcontrol.2012.09.002}}</ref> The [[covector mapping principle]] of [[I. Michael Ross|Ross]] and [[Fariba Fahroo|Fahroo]] eliminates the curse of sensitivity<ref name="R1" /> associated in solving for the [[costate equations|costates]] in [[optimal control]] problems. [[DIDO (optimal control)|DIDO]] generates spectrally accurate solutions <ref name="GFR1">{{cite journal | last1 = Gong | first1 = Q. | authorlink2 = Fariba Fahroo | authorlink3 = I. Michael Ross | last2 = Fahroo | first2 = F. | last3 = Ross | first3 = I. M. | year = 2008 | title = A Spectral Algorithm for Pseudospectral Methods in Optimal Control | url = | journal = Journal of Guidance, Control and Dynamics | volume = 31 | issue = 3| pages = 460–471 | doi=10.2514/1.32908}}</ref> whose extremality can be verified using [[Pontryagin's Minimum Principle]]. Because no knowledge of pseudospectral methods is necessary to use it, DIDO is often used<ref name=":1" /><ref name="hawkins-mit" /><ref name="Gong+" /><ref name=":0" /> as a fundamental mathematical tool for solving [[optimal control]] problems. That is, a solution obtained from DIDO is treated as a candidate solution for the application of [[Pontryagin's minimum principle]] as a [[Necessary and sufficient conditions|necessary condition]] for optimality.
DIDO implements a spectral algorithm<ref name="RF1" /><ref name="GFR1" /> based on [[pseudospectral optimal control]] theory founded by [[I. Michael Ross|Ross]] and his associates.<ref name="RK">{{cite journal | last1 = Ross | first1 = I. M. | last2 = Karpenko | first2 = M. | year = 2012 | title = A Review of Pseudospectral Optimal Control: From Theory to Flight | url = http://www.sciencedirect.com/science/article/pii/S1367578812000375 | journal = Annual Reviews in Control | volume = 36 | issue = 2| pages = 182–197 | doi=10.1016/j.arcontrol.2012.09.002}}</ref> The [[covector mapping principle]] of [[I. Michael Ross|Ross]] and [[Fariba Fahroo|Fahroo]] eliminates the curse of sensitivity<ref name="R1" /> associated in solving for the [[costate equations|costates]] in [[optimal control]] problems. [[DIDO (optimal control)|DIDO]] generates spectrally accurate solutions <ref name="GFR1">{{cite journal | last1 = Gong | first1 = Q. | authorlink2 = Fariba Fahroo | authorlink3 = I. Michael Ross | last2 = Fahroo | first2 = F. | last3 = Ross | first3 = I. M. | year = 2008 | title = A Spectral Algorithm for Pseudospectral Methods in Optimal Control | url = | journal = Journal of Guidance, Control and Dynamics | volume = 31 | issue = 3| pages = 460–471 | doi=10.2514/1.32908}}</ref> whose extremality can be verified using [[Pontryagin's Minimum Principle]]. Because no knowledge of pseudospectral methods is necessary to use it, DIDO is often used<ref name=":1" /><ref name="hawkins-mit" /><ref name="Gong+" /><ref name=":0" /> as a fundamental mathematical tool for solving [[optimal control]] problems. That is, a solution obtained from DIDO is treated as a candidate solution for the application of [[Pontryagin's minimum principle]] as a [[Necessary and sufficient conditions|necessary condition]] for optimality.


==Applications==
==Applications==
Line 35: Line 35:


==History==
==History==
The optimal control toolbox is named after [[Dido (Queen of Carthage)|Dido]], the legendary founder and first queen of [[Carthage]] who is famous in mathematics for her remarkable solution to a constrained [[optimal control]] [[Isoperimetric inequality|problem]] even before the invention of [[calculus]]. Invented by [[I. Michael Ross|Ross]], DIDO was first produced in 2001.<ref name="R1" /><ref name=":3" /><ref name="rea-mit">J. R. Rea, ''A Legendre Pseudospectral Method for Rapid Optimization of Launch Vehicle Trajectories,'' S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2001. http://dspace.mit.edu/handle/1721.1/8608</ref> The software is widely cited<ref name=":3">{{Cite journal|last=Rao|first=A. V.|year=2014|title=Trajectory Optimization: A Survey|url=|journal=Optimization and Optimal Control of Automotive Systems|volume=LNCIS 455|pages=3–21|via=}}</ref><ref name=":1">{{Cite journal|last=Conway|first=B. A.|year=2012|title=A Survey of Methods Available for the Numerical Optimization of Continuous Dynamical Systems|url=|journal=Journal of Optimization Theory and Applications|volume=152|pages=271–306|via=}}</ref><ref name=":0">D. Delahaye, S. Puechmorel, P. Tsiotras, and E. Feron, "Mathematical Models for Aircraft Trajectory Design : A Survey" Lecture notes in Electrical Engineering, 2014, Lecture Notes in Electrical Engineering, 290 (Part V), pp 205-247</ref><ref name=":2">S. E. Li, K. Deng, X. Zang, and Q. Zhang, "Pseudospectral Optimal Control of Constrained Nonlinear Systems," Ch 8, in Automotive Air Conditioning: Optimization, Control and Diagnosis, edited by Q. Zhang, S. E. Li and K. Deng, Springer 2016, pp. 145-166.</ref> and has many firsts to its credit:<ref name="NASA Fact Sheet">
The optimal control toolbox is named after [[Dido (Queen of Carthage)|Dido]], the legendary founder and first queen of [[Carthage]] who is famous in mathematics for her remarkable solution to a constrained [[optimal control]] [[Isoperimetric inequality|problem]] even before the invention of [[calculus]]. Invented by [[I. Michael Ross|Ross]], DIDO was first produced in 2001.<ref name="R1" /><ref name=":3" /><ref name="rea-mit">J. R. Rea, ''A Legendre Pseudospectral Method for Rapid Optimization of Launch Vehicle Trajectories,'' S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2001. http://dspace.mit.edu/handle/1721.1/8608</ref> The software is widely cited<ref name=":3">{{Cite journal|last=Rao|first=A. V.|year=2014|title=Trajectory Optimization: A Survey|url=|journal=Optimization and Optimal Control of Automotive Systems|volume=LNCIS 455|pages=3–21|via=|doi=10.1007/978-3-319-05371-4_1|series=Lecture Notes in Control and Information Sciences|isbn=978-3-319-05370-7}}</ref><ref name=":1">{{Cite journal|last=Conway|first=B. A.|year=2012|title=A Survey of Methods Available for the Numerical Optimization of Continuous Dynamical Systems|url=|journal=Journal of Optimization Theory and Applications|volume=152|issue=2|pages=271–306|via=|doi=10.1007/s10957-011-9918-z}}</ref><ref name=":0">D. Delahaye, S. Puechmorel, P. Tsiotras, and E. Feron, "Mathematical Models for Aircraft Trajectory Design : A Survey" Lecture notes in Electrical Engineering, 2014, Lecture Notes in Electrical Engineering, 290 (Part V), pp 205-247</ref><ref name=":2">S. E. Li, K. Deng, X. Zang, and Q. Zhang, "Pseudospectral Optimal Control of Constrained Nonlinear Systems," Ch 8, in Automotive Air Conditioning: Optimization, Control and Diagnosis, edited by Q. Zhang, S. E. Li and K. Deng, Springer 2016, pp. 145-166.</ref> and has many firsts to its credit:<ref name="NASA Fact Sheet">
National Aeronautics and Space Administration. "Fact Sheet: International Space Station Zero-Propellant Maneuver (ZPM) Demonstration." June 10, 2011. (Sept. 13, 2011) http://www.nasa.gov/mission_pages/station/research/experiments/ZPM.html</ref>
National Aeronautics and Space Administration. "Fact Sheet: International Space Station Zero-Propellant Maneuver (ZPM) Demonstration." June 10, 2011. (Sept. 13, 2011) http://www.nasa.gov/mission_pages/station/research/experiments/ZPM.html</ref>
<ref name="Kang2">W. Kang and N. Bedrossian, "Pseudospectral Optimal Control Theory Makes Debut Flight, Saves nasa $1m in Under Three Hours," SIAM News, 40, 2007.</ref>
<ref name="Kang2">W. Kang and N. Bedrossian, "Pseudospectral Optimal Control Theory Makes Debut Flight, Saves nasa $1m in Under Three Hours," SIAM News, 40, 2007.</ref>
Line 42: Line 42:
<ref name ="RF1" />
<ref name ="RF1" />
<ref name="JR1">{{cite journal | last1 = Josselyn | first1 = S. | last2 = Ross | first2 = I. M. | year = 2003 | title = A Rapid Verification Method for the Trajectory Optimization of Reentry Vehicles | url = | journal = Journal of Guidance, Control and Dynamics | volume = 26 | issue = 3| pages = 505–508 | doi=10.2514/2.5074}}</ref>
<ref name="JR1">{{cite journal | last1 = Josselyn | first1 = S. | last2 = Ross | first2 = I. M. | year = 2003 | title = A Rapid Verification Method for the Trajectory Optimization of Reentry Vehicles | url = | journal = Journal of Guidance, Control and Dynamics | volume = 26 | issue = 3| pages = 505–508 | doi=10.2514/2.5074}}</ref>
<ref name="FDN">{{cite journal | last1 = Fahroo | first1 = F. | last2 = Doman | first2 = D. B. | last3 = Ngo | first3 = A. D. | year = 2003 | title = Modeling Issues in Footprint Generation of Resuable Launch Vehicles | url = | journal = Proceedings of the IEEE Aerospace Conference | volume = 6 | issue = | pages = 2791–2799 | doi=10.1109/aero.2003.1235205}}</ref>
<ref name="FDN">{{cite journal | last1 = Fahroo | first1 = F. | last2 = Doman | first2 = D. B. | last3 = Ngo | first3 = A. D. | year = 2003 | title = Modeling Issues in Footprint Generation of Resuable Launch Vehicles | url = | journal = Proceedings of the IEEE Aerospace Conference | volume = 6 | issue = | pages = 2791–2799 | doi=10.1109/aero.2003.1235205| isbn = 978-0-7803-7651-9 }}</ref>


* First general-purpose object-oriented optimal control software
* First general-purpose object-oriented optimal control software
Line 71: Line 71:
==Further reading==
==Further reading==


* {{cite journal | last = Ross | first = I. Michael |author2=Fahroo, Fariba | title = Legendre Pseudospectral Approximations of Optimal Control Problems| year = 2003 | publisher = Springer Verlag| url = http://www.elissarglobal.com/wp-content/uploads/2012/03/Legendre-Pseudospectral-Approximations-of-Optimal-Control-Problems.pdf| format = [[PDF]] }}
* {{cite journal | last = Ross | first = I. Michael |author2=Fahroo, Fariba | title = Legendre Pseudospectral Approximations of Optimal Control Problems| year = 2003 | publisher = Springer Verlag| url = http://www.elissarglobal.com/wp-content/uploads/2012/03/Legendre-Pseudospectral-Approximations-of-Optimal-Control-Problems.pdf}}
* {{cite journal | last = Bollino| first = K. |author2=Lewis, L. R. |author3=Sekhavat, P. |author4=Ross, I. M. | title = Pseudospectral Optimal Control: A Clear Road for Autonomous Intelligent Path Planning | year = 2007 | publisher = AIAA| url = http://www.elissarglobal.com/wp-content/uploads/2012/04/Pseudospectral-Optimal-Control-A-Clear-Road-for-Autonomous-Intelligent-Path-Planning.pdf| format = [[PDF]] }}
* {{cite journal | last = Bollino| first = K. |author2=Lewis, L. R. |author3=Sekhavat, P. |author4=Ross, I. M. | title = Pseudospectral Optimal Control: A Clear Road for Autonomous Intelligent Path Planning | year = 2007 | publisher = AIAA| url = http://www.elissarglobal.com/wp-content/uploads/2012/04/Pseudospectral-Optimal-Control-A-Clear-Road-for-Autonomous-Intelligent-Path-Planning.pdf}}
* {{cite journal | last = Kang| first = W. |author2=Ross, I. M. |author3=Gong, Q. | title = Pseudospectral Optimal Control and Its Convergence Theorems | year = 2007 | publisher = Springer Berlin Heidelberg| url = http://www.springerlink.com/content/v1874282v4541275/}}
* {{cite journal | last = Kang| first = W. |author2=Ross, I. M. |author3=Gong, Q. | title = Pseudospectral Optimal Control and Its Convergence Theorems | year = 2007 | publisher = Springer Berlin Heidelberg| url = http://www.springerlink.com/content/v1874282v4541275/}}
* {{cite journal | last = Ross| first = I. M. | title = A Primer on Pontryagin's Principle in Optimal Control | year = 2009 | publisher = Collegiate Publishers |ISBN=978-0-9843571-0-9}}
* {{cite book | last = Ross| first = I. M. | title = A Primer on Pontryagin's Principle in Optimal Control | year = 2009 | publisher = Collegiate Publishers |isbn=978-0-9843571-0-9}}


==External links==
==External links==

Revision as of 00:58, 16 March 2019

DIDO (/ˈdaɪdoʊ/ DY-doh) is a software product for solving general-purpose optimal control problems.[1][2][3][4] It is widely used in academia,[5][6][7] industry,[2][8] and NASA.[9][10][11] Hailed as a breakthrough software,[12][13] DIDO is based on the pseudospectral optimal control theory of Ross and Fahroo.[14]

Usage

DIDO utilizes trademarked expressions and objects[1] that facilitate a user to quickly formulate and solve optimal control problems.[7][15][16][17] Rapidity in formulation is achieved through a set of DIDO expressions which are based on variables commonly used in optimal control theory.[1] For example, the state, control and time variables are formatted as:[1]

  • primal.states,
  • primal.controls, and
  • primal.time

The entire problem is codified using the key words, cost, dynamics, events and path:[1]

  • problem.cost
  • problem.dynamics
  • problem.events, and
  • problem.path

A user runs DIDO using the one-line command:

[cost, primal, dual] = dido(problem, algorithm),

where the object defined by algorithm allows a user to choose various options. In addition to the cost value and the primal solution, DIDO automatically outputs all the dual variables that are necessary to verify and validate a computational solution.[1] The output dual is computed by an application of the covector mapping principle.

Theory

DIDO implements a spectral algorithm[14][18] based on pseudospectral optimal control theory founded by Ross and his associates.[2] The covector mapping principle of Ross and Fahroo eliminates the curse of sensitivity[1] associated in solving for the costates in optimal control problems. DIDO generates spectrally accurate solutions [18] whose extremality can be verified using Pontryagin's Minimum Principle. Because no knowledge of pseudospectral methods is necessary to use it, DIDO is often used[6][7][8][19] as a fundamental mathematical tool for solving optimal control problems. That is, a solution obtained from DIDO is treated as a candidate solution for the application of Pontryagin's minimum principle as a necessary condition for optimality.

Applications

DIDO is used world wide in academia, industry and government laboratories.[8] Thanks to NASA, DIDO was flight-proven in 2006.[2] On November 5, 2006, NASA used DIDO to maneuver the International Space Station to perform the zero-propellant maneuver.

Since this flight demonstration, DIDO was used for the International Space Station and other NASA spacecraft.[11] It is also used in other industries.[1][8][19][20]

MATLAB optimal control toolbox

DIDO is also available as a MATLAB "toolbox" product.[21] It does not require the MATLAB Optimization Toolbox or any other third-party software like SNOPT or IPOPT or other nonlinear programming solvers.

The MATLAB/DIDO toolbox does not require a "guess" to run the algorithm. This and other distinguishing features have made DIDO a popular tool to solve optimal control problems.[3][6][13]

The MATLAB optimal control toolbox has been used to solve problems in aerospace,[10] robotics and search theory.

History

The optimal control toolbox is named after Dido, the legendary founder and first queen of Carthage who is famous in mathematics for her remarkable solution to a constrained optimal control problem even before the invention of calculus. Invented by Ross, DIDO was first produced in 2001.[1][5][15] The software is widely cited[5][6][19][20] and has many firsts to its credit:[9] [10] [11] [12] [14] [16] [22]

  • First general-purpose object-oriented optimal control software
  • First general-purpose pseudospectral optimal control software
  • First flight-proven general-purpose optimal control software
  • First embedded general-purpose optimal control solver
  • First guess-free general-purpose optimal control solver

Versions

Several different versions of DIDO are available from Elissar Global.[23]

See also

References

  1. ^ a b c d e f g h i Ross, I. M. A Primer on Pontryagin's Principle in Optimal Control, Second Edition, Collegiate Publishers, San Francisco, 2015.
  2. ^ a b c d Ross, I. M.; Karpenko, M. (2012). "A Review of Pseudospectral Optimal Control: From Theory to Flight". Annual Reviews in Control. 36 (2): 182–197. doi:10.1016/j.arcontrol.2012.09.002.
  3. ^ a b Eren, H., "Optimal Control and the Software," Measurements, Instrumentation, and Sensors Handbook, Second Edition, CRC Press, 2014, pp.92-1-16.
  4. ^ Ross, I. M.; D'Souza, C. N. (2005). "A Hybrid Optimal Control Framework for Mission Planning". Journal of Guidance, Control and Dynamics. 28 (4): 686–697. doi:10.2514/1.8285.
  5. ^ a b c Rao, A. V. (2014). "Trajectory Optimization: A Survey". Optimization and Optimal Control of Automotive Systems. Lecture Notes in Control and Information Sciences. LNCIS 455: 3–21. doi:10.1007/978-3-319-05371-4_1. ISBN 978-3-319-05370-7.
  6. ^ a b c d Conway, B. A. (2012). "A Survey of Methods Available for the Numerical Optimization of Continuous Dynamical Systems". Journal of Optimization Theory and Applications. 152 (2): 271–306. doi:10.1007/s10957-011-9918-z.
  7. ^ a b c A. M. Hawkins, Constrained Trajectory Optimization of a Soft Lunar Landing From a Parking Orbit, S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2005. http://dspace.mit.edu/handle/1721.1/32431
  8. ^ a b c d Q. Gong, W. Kang, N. Bedrossian, F. Fahroo, P. Sekhavat and K. Bollino, Pseudospectral Optimal Control for Military and Industrial Applications, 46th IEEE Conference on Decision and Control, New Orleans, LA, pp. 4128-4142, Dec. 2007.
  9. ^ a b National Aeronautics and Space Administration. "Fact Sheet: International Space Station Zero-Propellant Maneuver (ZPM) Demonstration." June 10, 2011. (Sept. 13, 2011) http://www.nasa.gov/mission_pages/station/research/experiments/ZPM.html
  10. ^ a b c W. Kang and N. Bedrossian, "Pseudospectral Optimal Control Theory Makes Debut Flight, Saves nasa $1m in Under Three Hours," SIAM News, 40, 2007.
  11. ^ a b c L. Keesey, "TRACE Spacecraft's New Slewing Procedure." NASA's Goddard Space Flight Center. National Aeronautics and Space Administration. Dec. 20, 2010. (Sept. 11, 2011) http://www.nasa.gov/mission_pages/sunearth/news/trace-slew.html.
  12. ^ a b B. Honegger, "NPS Professor's Software Breakthrough Allows Zero-Propellant Maneuvers in Space." Navy.mil. United States Navy. April 20, 2007. (Sept. 11, 2011) http://www.elissarglobal.com/wp-content/uploads/2011/07/Navy_News.pdf.
  13. ^ a b Kallrath, Josef (2004). Modeling Languages in Mathematical Optimization. Dordrecht, The Netherlands: Kluwer Academic Publishers. pp. 379–403.
  14. ^ a b c Ross, I. M.; Fahroo, F. (2004). "Pseudospectral Knotting Methods for Solving Optimal Control Problems". Journal of Guidance, Control and Dynamics. 27 (3): 397–405. doi:10.2514/1.3426.
  15. ^ a b J. R. Rea, A Legendre Pseudospectral Method for Rapid Optimization of Launch Vehicle Trajectories, S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2001. http://dspace.mit.edu/handle/1721.1/8608
  16. ^ a b Josselyn, S.; Ross, I. M. (2003). "A Rapid Verification Method for the Trajectory Optimization of Reentry Vehicles". Journal of Guidance, Control and Dynamics. 26 (3): 505–508. doi:10.2514/2.5074.
  17. ^ Infeld, S. I. (2005). "Optimization of Mission Design for Constrained Libration Point Space Missions" (PDF). Stanford University. {{cite journal}}: Cite journal requires |journal= (help)
  18. ^ a b Gong, Q.; Fahroo, F.; Ross, I. M. (2008). "A Spectral Algorithm for Pseudospectral Methods in Optimal Control". Journal of Guidance, Control and Dynamics. 31 (3): 460–471. doi:10.2514/1.32908.
  19. ^ a b c D. Delahaye, S. Puechmorel, P. Tsiotras, and E. Feron, "Mathematical Models for Aircraft Trajectory Design : A Survey" Lecture notes in Electrical Engineering, 2014, Lecture Notes in Electrical Engineering, 290 (Part V), pp 205-247
  20. ^ a b S. E. Li, K. Deng, X. Zang, and Q. Zhang, "Pseudospectral Optimal Control of Constrained Nonlinear Systems," Ch 8, in Automotive Air Conditioning: Optimization, Control and Diagnosis, edited by Q. Zhang, S. E. Li and K. Deng, Springer 2016, pp. 145-166.
  21. ^ "DIDO: Optimal control software". Promotional web page. Mathworks.
  22. ^ Fahroo, F.; Doman, D. B.; Ngo, A. D. (2003). "Modeling Issues in Footprint Generation of Resuable Launch Vehicles". Proceedings of the IEEE Aerospace Conference. 6: 2791–2799. doi:10.1109/aero.2003.1235205. ISBN 978-0-7803-7651-9.
  23. ^ "Elissar Global". web site. distributes the software.

Further reading

External links

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