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A modified Nelder-Mead barrier method for constrained optimization
Numerical simulations of a rolling ball robot actuated by internal point masses
1. | Department of Mathematical and Statistical Sciences, University of Alberta, CAB 632, Edmonton, AB T6G 2G1, Canada, and, ATCO SpaceLab, 5302 Forand ST SW, Calgary, AB T3E 8B4, Canada |
2. | Institute for Mathematics and its Applications, University of Minnesota, Twin Cities, 207 Church ST SE, 306 Lind Hall, Minneapolis, MN 55455, USA |
The controlled motion of a rolling ball actuated by internal point masses that move along arbitrarily-shaped rails fixed within the ball is considered. The controlled equations of motion are solved numerically using a predictor-corrector continuation method, starting from an initial solution obtained via a direct method, to realize trajectory tracking and obstacle avoidance maneuvers.
References:
[1] |
Community portal for automatic differentiation, , 2016, http://www.autodiff.org/., Google Scholar |
[2] |
E. Allgower and K. Georg,
Continuation and path following, Acta Numerica, 2 (1993), 1-64.
doi: 10.1017/s0962492900002336. |
[3] |
U. Ascher, J. Christiansen and R. Russell, Algorithm 569: Colsys: Collocation software for boundary-value odes [d2], ACM Transactions on Mathematical Software (TOMS), 7 (1981), 223-229. Google Scholar |
[4] |
U. Ascher, R. Mattheij and R. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Vol. 13, SIAM, 1994.
doi: 10.1137/1.9781611971231. |
[5] |
U. Ascher and R. Spiteri,
Collocation software for boundary value differential-algebraic equations, SIAM Journal on Scientific Computing, 15 (1994), 938-952.
doi: 10.1137/0915056. |
[6] |
W. Auzinger, G. Kneisl, O. Koch and E. Weinmüller,
A collocation code for singular boundary value problems in ordinary differential equations, Numerical Algorithms, 33 (2003), 27-39.
doi: 10.1023/A:1025531130904. |
[7] |
G. Bader and U. Ascher,
A new basis implementation for a mixed order boundary value ode solver, SIAM Journal on Scientific and Statistical Computing, 8 (1987), 483-500.
doi: 10.1137/0908047. |
[8] |
G. Bader and P. Kunkel,
Continuation and collocation for parameter-dependent boundary value problems, SIAM Journal on Scientific and Statistical Computing, 10 (1989), 72-88.
doi: 10.1137/0910007. |
[9] |
D. Baraff, Physically based modeling: Rigid body simulation, SIGGRAPH Course Notes, ACM SIGGRAPH, 2 (2001), 1-2. Google Scholar |
[10] |
Z. Bashir-Ali, J. Cash and H. Silva,
Lobatto deferred correction for stiff two-point boundary value problems, Computers & Mathematics with Applications, 36 (1998), 59-69.
doi: 10.1016/S0898-1221(98)80009-6. |
[11] |
J. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Vol. 19, SIAM, 2010.
doi: 10.1137/1.9780898718577. |
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Á. Birkisson, Numerical Solution of Nonlinear Boundary Value Problems for Ordinary Differential Equations in the Continuous Framework, PhD thesis, University of Oxford, 2013. Google Scholar |
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J. Boisvert, A Problem-Solving Environment for the Numerical Solution of Boundary Value Problems, PhD thesis, University of Saskatchewan, 2011. Google Scholar |
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J. Boisvert, P. Muir and R. Spiteri, A runge-kutta bvode solver with global error and defect control, , ACM Transactions on Mathematical Software (TOMS), 39 (2013), 11.
doi: 10.1145/2427023.2427028. |
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J. Boyd, Solving Transcendental Equations: The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation Series, and Oracles, Vol. 139, SIAM, 2014.
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![]() |
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Differential continuation for regular optimal control problems, Optimization Methods and Software, 27 (2012), 177-196.
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J. Cash, D. Hollevoet, F. Mazzia and A. Nagy, Algorithm 927: The matlab code bvptwp. m for the numerical solution of two point boundary value problems, , ACM Transactions on Mathematical Software (TOMS), 39 (2013), 15.
doi: 10.1145/2427023.2427032. |
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J. Cash and F. Mazzia,
A new mesh selection algorithm, based on conditioning, for two-point boundary value codes, Journal of Computational and Applied Mathematics, 184 (2005), 362-381.
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J. Cash and F. Mazzia,
Hybrid mesh selection algorithms based on conditioning for two-point boundary value problems, JNAIAM J. Numer. Anal. Indust. Appl. Math, 1 (2006), 81-90.
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J. Cash, G. Moore and R. Wright,
An automatic continuation strategy for the solution of singularly perturbed nonlinear boundary value problems, ACM Transactions on Mathematical Software (TOMS), 27 (2001), 245-266.
doi: 10.1006/jcph.1995.1212. |
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J. Cash and M. Wright,
A deferred correction method for nonlinear two-point boundary value problems: implementation and numerical evaluation, SIAM Journal on Scientific and Statistical Computing, 12 (1991), 971-989.
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F. Chernousko and A. Lyubushin,
Method of successive approximations for solution of optimal control problems, Optimal Control Applications and Methods, 3 (1982), 101-114.
doi: 10.1002/oca.4660030201. |
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G. Corliss, C. Faure, A. Griewank, L. Hascoet and U. Naumann, Automatic Differentiation Of Algorithms: From Simulation To Optimization, Vol. 1, Springer Science $ & $ Business Media, 2002. |
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H. Dankowicz and F. Schilder, Recipes for Continuation, SIAM, 2013.
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D. Davidenko,
On a new method of numerical solution of systems of nonlinear equations, Dokl. Akad. Nauk SSSR, 88 (1953), 601-602.
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D. Davidenko,
The approximate solution of sets of nonlinear equations, Ukr. Mat. Zh, 5 (1953), 196-206.
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J. Fike and J. Alonso, The development of hyper-dual numbers for exact second-derivative calculations, , AIAA paper, 886 (2011), 124. Google Scholar |
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J. Fike and J. Alonso, Automatic differentiation through the use of hyper-dual numbers for second derivatives, , in Recent Advances in Algorithmic Differentiation, Springer, (2012), 163–173.
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J. Frisvad, Building an orthonormal basis from a 3d unit vector without normalization, Journal of Graphics Tools, 16 (2012), 151-159. Google Scholar |
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P. Gill, W. Murray and M. Saunders,
SNOPT: An SQP algorithm for large-scale constrained optimization, SIAM Rev., 47 (2005), 99-131.
doi: 10.1137/S0036144504446096. |
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P. Gill, W. Murray, M. Saunders and E. Wong, User's Guide for SNOPT 7.6: Software for Large-Scale Nonlinear Programming, Center for Computational Mathematics Report CCoM 17-1, Department of Mathematics, University of California, San Diego, La Jolla, CA, 2017.
doi: 10.1137/S1052623499350013. |
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B. Graf, Quaternions and dynamics, , arXiv preprint arXiv: 0811.2889. |
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R. Gupta, A. Bloch and I. Kolmanovsky,
Combined homotopy and neighboring extremal optimal control, Optimal Control Applications and Methods, 38 (2017), 459-469.
doi: 10.1002/oca.2253. |
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Y. Hardy, K. Tan and W.-H. Steeb, Computer Algebra with SymbolicC++, World Scientific Publishing Company, 2008. Google Scholar |
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G. Kitzhofer, O. Koch, G. Pulverer, C. Simon and E. Weinmüller,
The new matlab code bvpsuite for the solution of singular implicit bvps, J. Numer. Anal. Indust. Appl. Math, 5 (2010), 113-134.
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G. Kitzhofer, O. Koch and E. Weinmüller,
Pathfollowing for essentially singular boundary value problems with application to the complex ginzburg-landau equation, BIT Numerical Mathematics, 49 (2009), 141-160.
doi: 10.1007/s10543-008-0208-6. |
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An algorithm for the method of successive approximations in optimal control problems, USSR Computational Mathematics and Mathematical Physics, 12 (1972), 15-38.
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V. Kungurtsev and J. Jäschke,
A predictor-corrector path-following algorithm for dual-degenerate parametric optimization problems, SIAM Journal on Optimization, 27 (2017), 538-564.
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G. Lantoine, R. Russell and T. Dargent, Using multicomplex variables for automatic computation of high-order derivatives, , ACM Transactions on Mathematical Software (TOMS), 38 (2012), 16.
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Y. LeCun, Y. Bengio and G. Hinton, Deep learning, Nature, 521 (2015), 436-444. Google Scholar |
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A. Lyubushin,
Modifications of the method of successive approximations for solving optimal control problems, USSR Computational Mathematics and Mathematical Physics, 22 (1982), 29-34.
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J. Martins, P. Sturdza and J. Alonso, The connection between the complex-step derivative approximation and algorithmic differentiation, , AIAA Paper, 921 (2001), 2001.
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J. Martins, P. Sturdza and J. Alonso,
The complex-step derivative approximation, ACM Transactions on Mathematical Software (TOMS), 29 (2003), 245-262.
doi: 10.1145/838250.838251. |
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P. Muir,
Optimal discrete and continuous mono-implicit runge–kutta schemes for bvodes, Advances in Computational Mathematics, 10 (1999), 135-167.
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U. Naumann, The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation, Vol. 24, SIAM, 2012. |
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M. Neuenhofen, Review of theory and implementation of hyper-dual numbers for first and second order automatic differentiation, , arXiv preprint arXiv: 1801.03614. Google Scholar |
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M. Patterson and A. Rao, 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 (TOMS), 41 (2014), 1.
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show all references
References:
[1] |
Community portal for automatic differentiation, , 2016, http://www.autodiff.org/., Google Scholar |
[2] |
E. Allgower and K. Georg,
Continuation and path following, Acta Numerica, 2 (1993), 1-64.
doi: 10.1017/s0962492900002336. |
[3] |
U. Ascher, J. Christiansen and R. Russell, Algorithm 569: Colsys: Collocation software for boundary-value odes [d2], ACM Transactions on Mathematical Software (TOMS), 7 (1981), 223-229. Google Scholar |
[4] |
U. Ascher, R. Mattheij and R. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Vol. 13, SIAM, 1994.
doi: 10.1137/1.9781611971231. |
[5] |
U. Ascher and R. Spiteri,
Collocation software for boundary value differential-algebraic equations, SIAM Journal on Scientific Computing, 15 (1994), 938-952.
doi: 10.1137/0915056. |
[6] |
W. Auzinger, G. Kneisl, O. Koch and E. Weinmüller,
A collocation code for singular boundary value problems in ordinary differential equations, Numerical Algorithms, 33 (2003), 27-39.
doi: 10.1023/A:1025531130904. |
[7] |
G. Bader and U. Ascher,
A new basis implementation for a mixed order boundary value ode solver, SIAM Journal on Scientific and Statistical Computing, 8 (1987), 483-500.
doi: 10.1137/0908047. |
[8] |
G. Bader and P. Kunkel,
Continuation and collocation for parameter-dependent boundary value problems, SIAM Journal on Scientific and Statistical Computing, 10 (1989), 72-88.
doi: 10.1137/0910007. |
[9] |
D. Baraff, Physically based modeling: Rigid body simulation, SIGGRAPH Course Notes, ACM SIGGRAPH, 2 (2001), 1-2. Google Scholar |
[10] |
Z. Bashir-Ali, J. Cash and H. Silva,
Lobatto deferred correction for stiff two-point boundary value problems, Computers & Mathematics with Applications, 36 (1998), 59-69.
doi: 10.1016/S0898-1221(98)80009-6. |
[11] |
J. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Vol. 19, SIAM, 2010.
doi: 10.1137/1.9780898718577. |
[12] |
Á. Birkisson, Numerical Solution of Nonlinear Boundary Value Problems for Ordinary Differential Equations in the Continuous Framework, PhD thesis, University of Oxford, 2013. Google Scholar |
[13] |
J. Boisvert, A Problem-Solving Environment for the Numerical Solution of Boundary Value Problems, PhD thesis, University of Saskatchewan, 2011. Google Scholar |
[14] |
J. Boisvert, P. Muir and R. Spiteri, A runge-kutta bvode solver with global error and defect control, , ACM Transactions on Mathematical Software (TOMS), 39 (2013), 11.
doi: 10.1145/2427023.2427028. |
[15] |
J. Boyd, Solving Transcendental Equations: The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation Series, and Oracles, Vol. 139, SIAM, 2014.
doi: 10.1137/1.9781611973525. |
[16] |
A. Bryson, Dynamic Optimization, Vol. 1, Prentice Hall, 1999. Google Scholar |
[17] |
A. Bryson and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation and Control, CRC Press, 1975.
![]() |
[18] |
J.-B. Caillau, O. Cots and J. Gergaud,
Differential continuation for regular optimal control problems, Optimization Methods and Software, 27 (2012), 177-196.
doi: 10.1080/10556788.2011.593625. |
[19] |
J. Cash, D. Hollevoet, F. Mazzia and A. Nagy, Algorithm 927: The matlab code bvptwp. m for the numerical solution of two point boundary value problems, , ACM Transactions on Mathematical Software (TOMS), 39 (2013), 15.
doi: 10.1145/2427023.2427032. |
[20] |
J. Cash and F. Mazzia,
A new mesh selection algorithm, based on conditioning, for two-point boundary value codes, Journal of Computational and Applied Mathematics, 184 (2005), 362-381.
doi: 10.1016/j.cam.2005.01.016. |
[21] |
J. Cash and F. Mazzia,
Hybrid mesh selection algorithms based on conditioning for two-point boundary value problems, JNAIAM J. Numer. Anal. Indust. Appl. Math, 1 (2006), 81-90.
|
[22] |
J. Cash, G. Moore and R. Wright,
An automatic continuation strategy for the solution of singularly perturbed nonlinear boundary value problems, ACM Transactions on Mathematical Software (TOMS), 27 (2001), 245-266.
doi: 10.1006/jcph.1995.1212. |
[23] |
J. Cash and M. Wright,
A deferred correction method for nonlinear two-point boundary value problems: implementation and numerical evaluation, SIAM Journal on Scientific and Statistical Computing, 12 (1991), 971-989.
doi: 10.1137/0912052. |
[24] |
F. Chernousko and A. Lyubushin,
Method of successive approximations for solution of optimal control problems, Optimal Control Applications and Methods, 3 (1982), 101-114.
doi: 10.1002/oca.4660030201. |
[25] |
G. Corliss, C. Faure, A. Griewank, L. Hascoet and U. Naumann, Automatic Differentiation Of Algorithms: From Simulation To Optimization, Vol. 1, Springer Science $ & $ Business Media, 2002. |
[26] |
H. Dankowicz and F. Schilder, Recipes for Continuation, SIAM, 2013.
doi: 10.1137/1.9781611972573. |
[27] |
D. Davidenko,
On a new method of numerical solution of systems of nonlinear equations, Dokl. Akad. Nauk SSSR, 88 (1953), 601-602.
|
[28] |
D. Davidenko,
The approximate solution of sets of nonlinear equations, Ukr. Mat. Zh, 5 (1953), 196-206.
|
[29] |
P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Vol. 35, Springer Science $ & $ Business Media, 2011.
doi: 10.1007/978-3-642-23899-4. |
[30] |
E. Doedel, T. Fairgrieve, B. Sandstede, A. Champneys, Y. Kuznetsov and X. Wang, Auto-07p: Continuation and bifurcation software for ordinary differential equations., Google Scholar |
[31] |
J. Fike and J. Alonso, The development of hyper-dual numbers for exact second-derivative calculations, , AIAA paper, 886 (2011), 124. Google Scholar |
[32] |
J. Fike and J. Alonso, Automatic differentiation through the use of hyper-dual numbers for second derivatives, , in Recent Advances in Algorithmic Differentiation, Springer, (2012), 163–173.
doi: 10.1007/978-3-642-30023-3_15. |
[33] |
J. Fike, S. Jongsma, J. Alonso and E. Van Der Weide, Optimization with gradient and hessian information calculated using hyper-dual numbers, , AIAA paper, 3807 (2011), 2011. Google Scholar |
[34] |
J. Frisvad, Building an orthonormal basis from a 3d unit vector without normalization, Journal of Graphics Tools, 16 (2012), 151-159. Google Scholar |
[35] |
P. Gill, W. Murray and M. Saunders,
SNOPT: An SQP algorithm for large-scale constrained optimization, SIAM Rev., 47 (2005), 99-131.
doi: 10.1137/S0036144504446096. |
[36] |
P. Gill, W. Murray, M. Saunders and E. Wong, User's Guide for SNOPT 7.6: Software for Large-Scale Nonlinear Programming, Center for Computational Mathematics Report CCoM 17-1, Department of Mathematics, University of California, San Diego, La Jolla, CA, 2017.
doi: 10.1137/S1052623499350013. |
[37] |
B. Graf, Quaternions and dynamics, , arXiv preprint arXiv: 0811.2889. |
[38] |
R. Gupta, A. Bloch and I. Kolmanovsky,
Combined homotopy and neighboring extremal optimal control, Optimal Control Applications and Methods, 38 (2017), 459-469.
doi: 10.1002/oca.2253. |
[39] |
Y. Hardy, K. Tan and W.-H. Steeb, Computer Algebra with SymbolicC++, World Scientific Publishing Company, 2008. Google Scholar |
[40] |
D. Holm, Geometric Mechanics: Rotating, Translating, and Rolling, $2^{nd}$ edition, Geometric Mechanics, Imperial College Press, 2011.
doi: 10.1142/p802.![]() ![]() |
[41] |
D. Hull, Optimal Control Theory for Applications, Springer Science & Business Media, 2013.
doi: 10.1007/978-1-4757-4180-3. |
[42] |
L. Kantorovich,
On newton's method for functional equations, Dokl. Akad. Nauk SSSR, 59 (1948), 1237-1240.
|
[43] |
J. Kierzenka and L. Shampine,
A bvp solver that controls residual and error, JNAIAM J. Numer. Anal. Ind. Appl. Math, 3 (2008), 27-41.
|
[44] |
G. Kitzhofer, O. Koch, G. Pulverer, C. Simon and E. Weinmüller,
The new matlab code bvpsuite for the solution of singular implicit bvps, J. Numer. Anal. Indust. Appl. Math, 5 (2010), 113-134.
|
[45] |
G. Kitzhofer, O. Koch and E. Weinmüller,
Pathfollowing for essentially singular boundary value problems with application to the complex ginzburg-landau equation, BIT Numerical Mathematics, 49 (2009), 141-160.
doi: 10.1007/s10543-008-0208-6. |
[46] |
I. Krylov and F. Chernousko,
On a method of successive approximations for the solution of problems of optimal control, USSR Computational Mathematics and Mathematical Physics, 2 (1963), 1371-1382.
|
[47] |
I. Krylov and F. Chernousko,
An algorithm for the method of successive approximations in optimal control problems, USSR Computational Mathematics and Mathematical Physics, 12 (1972), 15-38.
|
[48] |
V. Kungurtsev and J. Jäschke,
A predictor-corrector path-following algorithm for dual-degenerate parametric optimization problems, SIAM Journal on Optimization, 27 (2017), 538-564.
doi: 10.1137/16M1068736. |
[49] |
G. Lantoine, R. Russell and T. Dargent, Using multicomplex variables for automatic computation of high-order derivatives, , ACM Transactions on Mathematical Software (TOMS), 38 (2012), 16.
doi: 10.1145/2168773.2168774. |
[50] |
Y. LeCun, Y. Bengio and G. Hinton, Deep learning, Nature, 521 (2015), 436-444. Google Scholar |
[51] |
A. Lyubushin,
Modifications of the method of successive approximations for solving optimal control problems, USSR Computational Mathematics and Mathematical Physics, 22 (1982), 29-34.
|
[52] |
J. Martins, P. Sturdza and J. Alonso, The connection between the complex-step derivative approximation and algorithmic differentiation, , AIAA Paper, 921 (2001), 2001.
doi: 10.1145/838250.838251. |
[53] |
J. Martins, P. Sturdza and J. Alonso,
The complex-step derivative approximation, ACM Transactions on Mathematical Software (TOMS), 29 (2003), 245-262.
doi: 10.1145/838250.838251. |
[54] |
P. Muir,
Optimal discrete and continuous mono-implicit runge–kutta schemes for bvodes, Advances in Computational Mathematics, 10 (1999), 135-167.
doi: 10.1023/A:1018926631734. |
[55] |
U. Naumann, The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation, Vol. 24, SIAM, 2012. |
[56] |
M. Neuenhofen, Review of theory and implementation of hyper-dual numbers for first and second order automatic differentiation, , arXiv preprint arXiv: 1801.03614. Google Scholar |
[57] |
M. Patterson and A. Rao, 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 (TOMS), 41 (2014), 1.
doi: 10.1145/2558904. |
[58] |
V. Putkaradze and S. Rogers, On the optimal control of a rolling ball robot actuated by internal point masses, Journal of Dynamic Systems, Measurement, and Control, 142 (2020), 051002, 22 pages.
doi: 10.1115/1.4046104. |
[59] |
V. Putkaradze and S. Rogers,
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