June  2021, 11(2): 143-207. doi: 10.3934/naco.2020021

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

* Corresponding author: Stuart Rogers

Received  June 2019 Revised  January 2020 Published  June 2021 Early access  March 2020

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.

Citation: Vakhtang Putkaradze, Stuart Rogers. Numerical simulations of a rolling ball robot actuated by internal point masses. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 143-207. doi: 10.3934/naco.2020021
References:
[1]

Community portal for automatic differentiation, 2016, http://www.autodiff.org/.,

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E. Allgower and K. Georg, Continuation and path following, Acta Numerica, 2 (1993), 1-64.  doi: 10.1017/s0962492900002336.

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U. AscherJ. Christiansen and R. Russell, Algorithm 569: Colsys: Collocation software for boundary-value odes [d2], ACM Transactions on Mathematical Software (TOMS), 7 (1981), 223-229. 

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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.

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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.

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W. AuzingerG. KneislO. 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.

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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.

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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.

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D. Baraff, Physically based modeling: Rigid body simulation, SIGGRAPH Course Notes, ACM SIGGRAPH, 2 (2001), 1-2. 

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Z. Bashir-AliJ. 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.

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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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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. doi: 10.1137/1.9781611973525.

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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.  doi: 10.1016/j.cam.2005.01.016.

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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.

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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. 

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P. GillW. 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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Y. Hardy, K. Tan and W.-H. Steeb, Computer Algebra with SymbolicC++, World Scientific Publishing Company, 2008.

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J. Kierzenka and L. Shampine, A bvp solver that controls residual and error, JNAIAM J. Numer. Anal. Ind. Appl. Math, 3 (2008), 27-41. 

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G. KitzhoferO. KochG. PulvererC. 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. KitzhoferO. 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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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. 

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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.  doi: 10.1137/16M1068736.

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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. doi: 10.1145/2168773.2168774.

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Y. LeCunY. Bengio and G. Hinton, Deep learning, Nature, 521 (2015), 436-444. 

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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. doi: 10.1145/838250.838251.

[53]

J. MartinsP. 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.

[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.

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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, Constraint control of nonholonomic mechanical systems, Journal of Nonlinear Science, 28 (2018), 193-234.  doi: 10.1007/s00332-017-9406-1.

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V. Putkaradze and S. Rogers, On the dynamics of a rolling ball actuated by internal point masses, Meccanica, 53 (2018), 3839-3868.  doi: 10.1007/s11012-018-0904-5.

[61]

V. Putkaradze and S. Rogers, On the normal force and static friction acting on a rolling ball actuated by internal point masses, Regular and Chaotic Dynamics, 24 (2019), 145-170.  doi: 10.1134/S1560354719020023.

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L. Rall, Davidenko's Method for the Solution of Nonlinear Operator Equations, Technical report, University of Wisconsin, Madison, Mathematics Research Center, 1968.

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G. Rozenblat, On the choice of physically realizable parameters when studying the dynamics of spherical and ellipsoidal rigid bodies, Mechanics of Solids, 51 (2016), 415-423. 

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L. Shampine, J. Kierzenka and M. Reichelt, Solving boundary value problems for ordinary differential equations in matlab with bvp4c, Tutorial notes, 437–448.

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W. Squire and G. Trapp, Using complex variables to estimate derivatives of real functions, Siam Review, 40 (1998), 110-112.  doi: 10.1137/S003614459631241X.

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show all references

References:
[1]

Community portal for automatic differentiation, 2016, http://www.autodiff.org/.,

[2]

E. Allgower and K. Georg, Continuation and path following, Acta Numerica, 2 (1993), 1-64.  doi: 10.1017/s0962492900002336.

[3]

U. AscherJ. Christiansen and R. Russell, Algorithm 569: Colsys: Collocation software for boundary-value odes [d2], ACM Transactions on Mathematical Software (TOMS), 7 (1981), 223-229. 

[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. AuzingerG. KneislO. 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. 

[10]

Z. Bashir-AliJ. 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.

[13]

J. Boisvert, A Problem-Solving Environment for the Numerical Solution of Boundary Value Problems, PhD thesis, University of Saskatchewan, 2011.

[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.

[17] A. Bryson and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation and Control, CRC Press, 1975. 
[18]

J.-B. CaillauO. 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. CashG. 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.

[31]

J. Fike and J. Alonso, The development of hyper-dual numbers for exact second-derivative calculations, AIAA paper, 886 (2011), 124.

[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.

[34]

J. Frisvad, Building an orthonormal basis from a 3d unit vector without normalization, Journal of Graphics Tools, 16 (2012), 151-159. 

[35]

P. GillW. 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. GuptaA. 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.

[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. KitzhoferO. KochG. PulvererC. 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. KitzhoferO. 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. LeCunY. Bengio and G. Hinton, Deep learning, Nature, 521 (2015), 436-444. 

[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. MartinsP. 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.

[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, Constraint control of nonholonomic mechanical systems, Journal of Nonlinear Science, 28 (2018), 193-234.  doi: 10.1007/s00332-017-9406-1.

[60]

V. Putkaradze and S. Rogers, On the dynamics of a rolling ball actuated by internal point masses, Meccanica, 53 (2018), 3839-3868.  doi: 10.1007/s11012-018-0904-5.

[61]

V. Putkaradze and S. Rogers, On the normal force and static friction acting on a rolling ball actuated by internal point masses, Regular and Chaotic Dynamics, 24 (2019), 145-170.  doi: 10.1134/S1560354719020023.

[62]

L. Rall, Davidenko's Method for the Solution of Nonlinear Operator Equations, Technical report, University of Wisconsin, Madison, Mathematics Research Center, 1968.

[63]

G. Rozenblat, On the choice of physically realizable parameters when studying the dynamics of spherical and ellipsoidal rigid bodies, Mechanics of Solids, 51 (2016), 415-423. 

[64]

L. Shampine, J. Kierzenka and M. Reichelt, Solving boundary value problems for ordinary differential equations in matlab with bvp4c, Tutorial notes, 437–448.

[65]

W. Squire and G. Trapp, Using complex variables to estimate derivatives of real functions, Siam Review, 40 (1998), 110-112.  doi: 10.1137/S003614459631241X.

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Figure 1.  A ball of radius $ r $ and mass $ m_0 $ rolls without slipping on a horizontal surface in the presence of a uniform gravitational field of magnitude $ g $. The ball's geometric center, center of mass, and contact point with the horizontal surface are denoted by GC, $ m_0 $, and CP, respectively. The spatial frame has origin located at height $ r $ above the horizontal surface and orthonormal axes $ \mathbf{e}_1 $, $ \mathbf{e}_2 $, and $ \mathbf{e}_3 $. The body frame has origin located at the ball's center of mass (denoted by $ m_0 $) and orthonormal axes $ \mathbf{E}_1 $, $ \mathbf{E}_2 $, and $ \mathbf{E}_3 $. The ball's motion is actuated by $ n $ point masses, each of mass $ m_i $, $ 1 \le i \le n $, and each moving along its own rail fixed inside the ball. The $ i^\mathrm{th} $ rail is depicted here by the dashed hoop. The trajectory of the $ i^\mathrm{th} $ rail, with respect to the body frame translated to the GC, is denoted by $ {\boldsymbol{\zeta}}_i $ and is parameterized by $ \theta_i $. All vectors inside the ball are expressed with respect to the body frame, while all vectors outside the ball are expressed with respect to the spatial frame
Figure 2.  A disk of radius $ r $ and mass $ m_0 $ rolls without slipping in the $ \mathbf{e}_1 $-$ \mathbf{e}_3 $ plane. $ \mathbf{e}_2 $ and $ \mathbf{E}_2 $ are directed into the page and are omitted from the figure. The disk's center of mass is denoted by $ m_0 $. The disk's motion is actuated by $ n $ point masses, each of mass $ m_i $, $ 1 \le i \le n $, and each moving along its own rail fixed inside the disk. The point mass depicted here by $ m_i $ moves along a circular hoop in the disk that is not centered on the disk's geometric center (GC). The disk's orientation is determined by $ \phi $, the angle measured counterclockwise from $ \mathbf{e}_1 $ to $ \mathbf{E}_1 $
Figure 3.  The disk of radius $ r = 1 $ actuated by $ 4 $ control masses, $ m_1 $, $ m_2 $, $ m_3 $, and $ m_4 $, each on its own circular control rail. The control rail radii are $ r_1 = .9 $, $ r_2 = .6\overline{3} $, $ r_3 = .3\overline{6} $, and $ r_4 = .1 $. The location of the disk's CM is denoted by $ m_0 $
Figure 4.  Numerical solutions of the rolling disk optimal control problem (19) using $ 4 $ control masses for $ \gamma_1 = \gamma_2 = \gamma_3 = \gamma_4 = .1 $ and for fixed initial and final times. The direct method results for $ \alpha = .1 $ are shown in the left column, while the predictor-corrector continuation indirect method results for $ \alpha \approx 272 $ are shown in the right column. The direct method solution tracks the desired GC path crudely, whereas the indirect method solution tracks the desired GC path very accurately at the expense of larger magnitude controls
Figure 5.  Numerical solutions of the rolling disk optimal control problem (19) using $ 4 $ control masses for $ \gamma_1 = \gamma_2 = \gamma_3 = \gamma_4 = .1 $ and for fixed initial and final times. The direct method results for $ \alpha = .1 $ are shown in the left column, while the predictor-corrector continuation indirect method results for $ \alpha \approx 272 $ are shown in the right column. The disk does not detach from the surface since the magnitude of the normal force is always positive. The disk rolls without slipping if $ \mu_\mathrm{s} \ge .07799 $ for the direct method solution and if $ \mu_\mathrm{s} \ge .3502 $ for the indirect method solution. That is, the indirect method solution requires a much larger coefficient of static friction
Figure 6.  Evolution of various parameters and variables during the predictor-corrector continuation indirect method, which starts from the direct method solution, used to solve the rolling disk optimal control problem (19). $ \mu $ decreases monotonically, while $ \alpha $ and $ J $ increase monotonically
Figure 7.  Evolution of various parameters and variables during an extended run of the predictor-corrector continuation indirect method, which starts from the direct method solution, used to solve the rolling disk optimal control problem (19). Note the turning points at solutions 7, 10, and 18. The minimum of the GC tracking error occurs at solution 7
Figure 8.  The ball of radius $ r = 1 $ actuated by $ 3 $ control masses, $ m_1 $, $ m_2 $, and $ m_3 $, each on its own circular control rail. The control rail radii are $ r_1 = .95 $, $ r_2 = .9 $, and $ r_3 = .85 $. The location of the ball's CM is denoted by $ m_0 $
Figure 9.  Numerical solutions of the rolling ball optimal control problem (37) for sigmoid obstacle avoidance using $ 3 $ control masses for $ \gamma_1 = \gamma_2 = \gamma_3 = 10 $ and for fixed initial and final times. The obstacle centers are located at $ {\boldsymbol{v}}_1 = \begin{bmatrix} v_{1,1} & v_{1,2} \end{bmatrix}^\mathsf{T} = \begin{bmatrix} .2 & .2 \end{bmatrix}^\mathsf{T} $ and $ {\boldsymbol{v}}_2 = \begin{bmatrix} v_{2,1} & v_{2,2} \end{bmatrix}^\mathsf{T} = \begin{bmatrix} .8 & .8 \end{bmatrix}^\mathsf{T} $ and the obstacle radii are $ \rho_1 = \rho_2 = .282 $. The direct method results for obstacle heights at $ h_1 = h_2 = 0 $ are shown in the left column, while the predictor-corrector continuation indirect method results for obstacle heights at $ h_1 = h_2 \approx 9.93 $ are shown in the right column
Figure 10.  Numerical solutions of the rolling ball optimal control problem (37) for sigmoid obstacle avoidance using $ 3 $ control masses for $ \gamma_1 = \gamma_2 = \gamma_3 = 10 $ and for fixed initial and final times. The direct method results for obstacle heights at $ h_1 = h_2 = 0 $ are shown in the left column, while the predictor-corrector continuation indirect method results for obstacle heights at $ h_1 = h_2 \approx 9.93 $ are shown in the right column. The ball does not detach from the surface since the magnitude of the normal force is always positive. The ball rolls without slipping if $ \mu_\mathrm{s} \ge .1055 $ for the direct method solution and if $ \mu_\mathrm{s} \ge .0988 $ for the indirect method solution
Figure 11.  Evolution of various parameters and variables during the predictor-corrector continuation indirect method, which starts from the direct method solution, used to solve the rolling ball optimal control problem (37) for sigmoid obstacle avoidance. Note the turning points at solutions 3 and 4
Figure 12.  Numerical solutions of the rolling ball optimal control problem (37) for $ {\rm ReLU} $ obstacle avoidance using $ 3 $ control masses. First a direct method (left column), followed by predictor-corrector continuation in the obstacle heights $ h_1 = h_2 $ (middle column), and finally predictor-corrector continuation in the control coefficients $ \gamma_1 = \gamma_2 = \gamma_3 $ (right column)
Figure 13.  A direct method (left column) is followed by two rounds of a predictor-corrector continuation indirect method to realize a $ {\rm ReLU} $ obstacle avoidance maneuver for the rolling ball. The first round (middle column) of predictor-corrector continuation increases the obstacle heights $ h_1 = h_2 $, and the second round (right column) of predictor-corrector continuation decreases the control coefficients $ \gamma_1 = \gamma_2 = \gamma_3 $. The ball does not detach from the surface since the magnitude of the normal force is always positive. The ball rolls without slipping if $ \mu_\mathrm{s} \ge .1055 $ for the direct method solution, if $ \mu_\mathrm{s} \ge .09942 $ for the first indirect method solution, and if $ \mu_\mathrm{s} \ge .09917 $ for the second indirect method solution
Figure 14.  Predictor-corrector continuation in the obstacle heights $ h_1 = h_2 $ (left column) is followed by predictor-corrector continuation in the control coefficients $ \gamma_1 = \gamma_2 = \gamma_3 $ (right column) to realize a $ {\rm ReLU} $ obstacle avoidance maneuver for the rolling ball
Figure 15.  Predictor-corrector continuation
Table 1.  Initial condition parameter values for the rolling disk. Refer to (22) and (23)
Parameter Value
$ {\boldsymbol{\theta}}_a $ $ \begin{bmatrix} - \frac{\pi}{2} - \frac{\pi}{2} - \frac{\pi}{2} - \frac{\pi}{2} \end{bmatrix}^\mathsf{T} $
$ \dot {\boldsymbol{\theta}}_a $ $ \begin{bmatrix} 0 0 0 0 \end{bmatrix}^\mathsf{T} $
$ \phi_a $ $ 0 $
$ z_a $ $ 0 $
$ \dot z_a $ $ 0 $
Parameter Value
$ {\boldsymbol{\theta}}_a $ $ \begin{bmatrix} - \frac{\pi}{2} - \frac{\pi}{2} - \frac{\pi}{2} - \frac{\pi}{2} \end{bmatrix}^\mathsf{T} $
$ \dot {\boldsymbol{\theta}}_a $ $ \begin{bmatrix} 0 0 0 0 \end{bmatrix}^\mathsf{T} $
$ \phi_a $ $ 0 $
$ z_a $ $ 0 $
$ \dot z_a $ $ 0 $
Table 2.  Final condition parameter values for the rolling disk. Refer to (23)
Parameter Value
$ \dot {\boldsymbol{\theta}}_b $ $ \begin{bmatrix} 0 0 0 0 \end{bmatrix}^\mathsf{T} $
$ z_b $ $ 1 $
$ \dot z_b $ $ 0 $
Parameter Value
$ \dot {\boldsymbol{\theta}}_b $ $ \begin{bmatrix} 0 0 0 0 \end{bmatrix}^\mathsf{T} $
$ z_b $ $ 1 $
$ \dot z_b $ $ 0 $
Table 3.  Integrand cost function coefficient values for the rolling disk when predictor-corrector continuation is performed in $ \alpha $. Refer to (27)
Parameter Value
$ \alpha(\mu) $ $ .1+\frac{.95-\mu}{.95-.00001}\left(5000-.1\right) $
$ \gamma_1=\gamma_2=\gamma_3=\gamma_4 $ $ .1 $
Parameter Value
$ \alpha(\mu) $ $ .1+\frac{.95-\mu}{.95-.00001}\left(5000-.1\right) $
$ \gamma_1=\gamma_2=\gamma_3=\gamma_4 $ $ .1 $
Table 4.  Initial condition parameter values for the rolling ball. Refer to (43)
Parameter Value
$ {\boldsymbol{\theta}}_a $ $ \begin{bmatrix} 0 2.0369 .7044 \end{bmatrix}^\mathsf{T} $
$ \dot {\boldsymbol{\theta}}_a $ $ \begin{bmatrix} 0 0 0 \end{bmatrix}^\mathsf{T} $
$ \mathfrak{q}_a $ $ \begin{bmatrix} 1 0 0 0 \end{bmatrix}^\mathsf{T} $
$ {{\boldsymbol{\Omega}}}_a $ $ \begin{bmatrix} 0 0 0 \end{bmatrix}^\mathsf{T} $
$ {\boldsymbol{z}}_a $ $ \begin{bmatrix} 0 0 \end{bmatrix}^\mathsf{T} $
Parameter Value
$ {\boldsymbol{\theta}}_a $ $ \begin{bmatrix} 0 2.0369 .7044 \end{bmatrix}^\mathsf{T} $
$ \dot {\boldsymbol{\theta}}_a $ $ \begin{bmatrix} 0 0 0 \end{bmatrix}^\mathsf{T} $
$ \mathfrak{q}_a $ $ \begin{bmatrix} 1 0 0 0 \end{bmatrix}^\mathsf{T} $
$ {{\boldsymbol{\Omega}}}_a $ $ \begin{bmatrix} 0 0 0 \end{bmatrix}^\mathsf{T} $
$ {\boldsymbol{z}}_a $ $ \begin{bmatrix} 0 0 \end{bmatrix}^\mathsf{T} $
Table 5.  Final condition parameter values for the rolling ball. Refer to (44)
Parameter Value
$ \dot {\boldsymbol{\theta}}_b $ $ \begin{bmatrix} 0 0 0 \end{bmatrix}^\mathsf{T} $
$ {{\boldsymbol{\Omega}}}_b $ $ \begin{bmatrix} 0 0 0 \end{bmatrix}^\mathsf{T} $
$ {\boldsymbol{z}}_b $ $ \begin{bmatrix} 1 1 \end{bmatrix}^\mathsf{T} $
Parameter Value
$ \dot {\boldsymbol{\theta}}_b $ $ \begin{bmatrix} 0 0 0 \end{bmatrix}^\mathsf{T} $
$ {{\boldsymbol{\Omega}}}_b $ $ \begin{bmatrix} 0 0 0 \end{bmatrix}^\mathsf{T} $
$ {\boldsymbol{z}}_b $ $ \begin{bmatrix} 1 1 \end{bmatrix}^\mathsf{T} $
Table 6.  Integrand cost function coefficient values for the rolling ball when predictor-corrector continuation is performed in the obstacle heights. Refer to (48)
Parameter Value
$ \gamma_1=\gamma_2=\gamma_3 $ $ 10 $
$ h_1(\mu)=h_2(\mu) $ $ \frac{.95-\mu}{.95-.00001}\left(1000\right) $
$ {\boldsymbol{v}}_1 $ $ \begin{bmatrix} .2 .2 \end{bmatrix}^\mathsf{T} $
$ {\boldsymbol{v}}_2 $ $ \begin{bmatrix} .8 .8 \end{bmatrix}^\mathsf{T} $
$ \rho_1=\rho_2 $ $ .282 $
Parameter Value
$ \gamma_1=\gamma_2=\gamma_3 $ $ 10 $
$ h_1(\mu)=h_2(\mu) $ $ \frac{.95-\mu}{.95-.00001}\left(1000\right) $
$ {\boldsymbol{v}}_1 $ $ \begin{bmatrix} .2 .2 \end{bmatrix}^\mathsf{T} $
$ {\boldsymbol{v}}_2 $ $ \begin{bmatrix} .8 .8 \end{bmatrix}^\mathsf{T} $
$ \rho_1=\rho_2 $ $ .282 $
Table 7.  Integrand cost function coefficient values for the rolling ball when a second round of predictor-corrector continuation is performed in the control coefficients. Refer to (48)
Parameter Value
$ \gamma_1(\mu)=\gamma_2(\mu)=\gamma_3(\mu) $ $ 10+\frac{.95-\mu}{.95-.00001}\left(-1000-10\right) $
$ h_1=h_2 $ $ 7.846\mathrm{e}{8} $
$ {\boldsymbol{v}}_1 $ $ \begin{bmatrix} .2 .2 \end{bmatrix}^\mathsf{T} $
$ {\boldsymbol{v}}_2 $ $ \begin{bmatrix} .8 .8 \end{bmatrix}^\mathsf{T} $
$ \rho_1=\rho_2 $ $ .282 $
Parameter Value
$ \gamma_1(\mu)=\gamma_2(\mu)=\gamma_3(\mu) $ $ 10+\frac{.95-\mu}{.95-.00001}\left(-1000-10\right) $
$ h_1=h_2 $ $ 7.846\mathrm{e}{8} $
$ {\boldsymbol{v}}_1 $ $ \begin{bmatrix} .2 .2 \end{bmatrix}^\mathsf{T} $
$ {\boldsymbol{v}}_2 $ $ \begin{bmatrix} .8 .8 \end{bmatrix}^\mathsf{T} $
$ \rho_1=\rho_2 $ $ .282 $
Table 8.  Explanation of shorthand notation for zeroth and first derivatives of $ \hat{\mathbf{f}} $ and first and second derivatives of $ \hat{H} $ used in (86) and (87)
Shorthand $ \mathbf{\vert} $ Extended $ \mathbf{\vert} $ Normalized $ \mathbf{\vert} $ Un-Normalized
$ \mathbf{\vert} $ Shorthand $ \mathbf{\vert} $ $ \mathbf{\vert} $
$ \hat{\mathbf{f}} $ = $ \left. \hat{\mathbf{f}} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{\mathbf{f}} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{\mathbf{f}} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{\mathbf{f}}_{ {\boldsymbol{\lambda}}} $ = $ \left. \hat{\mathbf{f}}_{ {\boldsymbol{\lambda}}} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{\mathbf{f}}_{ {\boldsymbol{\lambda}}} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{\mathbf{f}}_{ {\boldsymbol{\lambda}}} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{\mathbf{f}}_{{{\boldsymbol {x} }}} $ = $ \left. \hat{\mathbf{f}}_{{{\boldsymbol {x} }}} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{\mathbf{f}}_{{{\boldsymbol {x} }}} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{\mathbf{f}}_{{{\boldsymbol {x} }}} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{\mathbf{f}}_{t} $ = $ \left. \hat{\mathbf{f}}_{t} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{\mathbf{f}}_{t} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{\mathbf{f}}_{t} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{\mathbf{f}}_{\mu} $ = $ \left. \hat{\mathbf{f}}_{\mu} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{\mathbf{f}}_{\mu} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{\mathbf{f}}_{\mu} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{H}_{{{\boldsymbol {x} }}}^\mathsf{T} $ = $ \left. \hat{H}_{{{\boldsymbol {x} }}}^\mathsf{T} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{H}_{{{\boldsymbol {x} }}}^\mathsf{T} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{H}_{{{\boldsymbol {x} }}}^\mathsf{T} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{H}_{{{\boldsymbol {x} }} {{\boldsymbol {x} }}} $ = $ \left. \hat{H}_{{{\boldsymbol {x} }} {{\boldsymbol {x} }}} \right|_{\left( s,\tilde {\boldsymbol{z}}(s) ,\mu\right)} $ = $ \hat{H}_{{{\boldsymbol {x} }} {{\boldsymbol {x} }}} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{H}_{{{\boldsymbol {x} }} {{\boldsymbol {x} }}} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{H}_{{{\boldsymbol {x} }} t} $ = $ \left. \hat{H}_{{{\boldsymbol {x} }} t} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{H}_{{{\boldsymbol {x} }} t} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{H}_{{{\boldsymbol {x} }} t} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{H}_{{{\boldsymbol {x} }} \mu} $ = $ \left. \hat{H}_{{{\boldsymbol {x} }} \mu} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{H}_{{{\boldsymbol {x} }} \mu} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{H}_{{{\boldsymbol {x} }} \mu} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
Shorthand $ \mathbf{\vert} $ Extended $ \mathbf{\vert} $ Normalized $ \mathbf{\vert} $ Un-Normalized
$ \mathbf{\vert} $ Shorthand $ \mathbf{\vert} $ $ \mathbf{\vert} $
$ \hat{\mathbf{f}} $ = $ \left. \hat{\mathbf{f}} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{\mathbf{f}} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{\mathbf{f}} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{\mathbf{f}}_{ {\boldsymbol{\lambda}}} $ = $ \left. \hat{\mathbf{f}}_{ {\boldsymbol{\lambda}}} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{\mathbf{f}}_{ {\boldsymbol{\lambda}}} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{\mathbf{f}}_{ {\boldsymbol{\lambda}}} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{\mathbf{f}}_{{{\boldsymbol {x} }}} $ = $ \left. \hat{\mathbf{f}}_{{{\boldsymbol {x} }}} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{\mathbf{f}}_{{{\boldsymbol {x} }}} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{\mathbf{f}}_{{{\boldsymbol {x} }}} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{\mathbf{f}}_{t} $ = $ \left. \hat{\mathbf{f}}_{t} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{\mathbf{f}}_{t} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{\mathbf{f}}_{t} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{\mathbf{f}}_{\mu} $ = $ \left. \hat{\mathbf{f}}_{\mu} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{\mathbf{f}}_{\mu} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{\mathbf{f}}_{\mu} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{H}_{{{\boldsymbol {x} }}}^\mathsf{T} $ = $ \left. \hat{H}_{{{\boldsymbol {x} }}}^\mathsf{T} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{H}_{{{\boldsymbol {x} }}}^\mathsf{T} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{H}_{{{\boldsymbol {x} }}}^\mathsf{T} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{H}_{{{\boldsymbol {x} }} {{\boldsymbol {x} }}} $ = $ \left. \hat{H}_{{{\boldsymbol {x} }} {{\boldsymbol {x} }}} \right|_{\left( s,\tilde {\boldsymbol{z}}(s) ,\mu\right)} $ = $ \hat{H}_{{{\boldsymbol {x} }} {{\boldsymbol {x} }}} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{H}_{{{\boldsymbol {x} }} {{\boldsymbol {x} }}} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{H}_{{{\boldsymbol {x} }} t} $ = $ \left. \hat{H}_{{{\boldsymbol {x} }} t} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{H}_{{{\boldsymbol {x} }} t} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{H}_{{{\boldsymbol {x} }} t} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
$ \hat{H}_{{{\boldsymbol {x} }} \mu} $ = $ \left. \hat{H}_{{{\boldsymbol {x} }} \mu} \right|_{\left( s,\tilde {\boldsymbol{z}}(s),\mu \right)} $ = $ \hat{H}_{{{\boldsymbol {x} }} \mu} \left(t(s),\tilde {{\boldsymbol {x} }}(s),\tilde {\boldsymbol{\lambda}}(s),\mu\right) $ = $ \hat{H}_{{{\boldsymbol {x} }} \mu} \left(t(s),{{\boldsymbol {x} }}(t(s)), {\boldsymbol{\lambda}}(t(s)),\mu\right) $
Table 9.  Explanation of shorthand notation for $ \hat{\mathbf{f}} $ and first derivatives of $ \hat{H} $ evaluated at $ a $ used in (113), (114), and (115). Note that $ \left. \hat{H}_{ {\boldsymbol{\lambda}}} \right|_a = \hat{H}_{ {\boldsymbol{\lambda}}} \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu \right) = \left. \hat{\mathbf{f}}^\mathsf{T} \right|_a $
Shorthand $ \mathbf{\vert} $ Meaning $ \mathbf{\vert} $ Simplification
$ \left. \hat{H}_{{{\boldsymbol {x} }}} \right|_a $ = $ \hat{H}_{{{\boldsymbol {x} }}} \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu \right) $ = $ H_{{{\boldsymbol {x} }}} \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a), {\boldsymbol{\pi}}\left(a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu\right),\mu \right) $
$ \left. \hat{\mathbf{f}}^\mathsf{T} \right|_a $ = $ \hat{\mathbf{f}}^\mathsf{T} \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu \right) $ = $ \mathbf{f}^\mathsf{T} \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a), {\boldsymbol{\pi}}\left(a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu\right),\mu \right) $
$ \left. \hat{H}_t \right|_a $ = $ \hat{H}_t \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu \right) $ = $ H_t \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a), {\boldsymbol{\pi}}\left(a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu\right),\mu \right) $
$ \left. \hat{H}_\mu \right|_a $ = $ \hat{H}_\mu \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu \right) $ = $ H_\mu \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a), {\boldsymbol{\pi}}\left(a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu\right),\mu \right) $
Shorthand $ \mathbf{\vert} $ Meaning $ \mathbf{\vert} $ Simplification
$ \left. \hat{H}_{{{\boldsymbol {x} }}} \right|_a $ = $ \hat{H}_{{{\boldsymbol {x} }}} \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu \right) $ = $ H_{{{\boldsymbol {x} }}} \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a), {\boldsymbol{\pi}}\left(a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu\right),\mu \right) $
$ \left. \hat{\mathbf{f}}^\mathsf{T} \right|_a $ = $ \hat{\mathbf{f}}^\mathsf{T} \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu \right) $ = $ \mathbf{f}^\mathsf{T} \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a), {\boldsymbol{\pi}}\left(a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu\right),\mu \right) $
$ \left. \hat{H}_t \right|_a $ = $ \hat{H}_t \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu \right) $ = $ H_t \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a), {\boldsymbol{\pi}}\left(a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu\right),\mu \right) $
$ \left. \hat{H}_\mu \right|_a $ = $ \hat{H}_\mu \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu \right) $ = $ H_\mu \left( a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a), {\boldsymbol{\pi}}\left(a,{{\boldsymbol {x} }}(a), {\boldsymbol{\lambda}}(a),\mu\right),\mu \right) $
Table 10.  Explanation of shorthand notation for first derivatives of $ {\boldsymbol{\sigma}} $ used in (113), (114), and (115)
Shorthand $ \mathbf{\vert} $ Meaning
$ {\boldsymbol{\sigma}}_{{{\boldsymbol {x} }}(a)} $ = $ {\boldsymbol{\sigma}}_{{{\boldsymbol {x} }}(a)} \left( a,{{\boldsymbol {x} }}(a),\mu \right) $
$ {\boldsymbol{\sigma}}_{a} $ = $ {\boldsymbol{\sigma}}_{a} \left( a,{{\boldsymbol {x} }}(a),\mu \right) $
$ {\boldsymbol{\sigma}}_{\mu} $ = $ {\boldsymbol{\sigma}}_{\mu} \left( a,{{\boldsymbol {x} }}(a),\mu \right) $
Shorthand $ \mathbf{\vert} $ Meaning
$ {\boldsymbol{\sigma}}_{{{\boldsymbol {x} }}(a)} $ = $ {\boldsymbol{\sigma}}_{{{\boldsymbol {x} }}(a)} \left( a,{{\boldsymbol {x} }}(a),\mu \right) $
$ {\boldsymbol{\sigma}}_{a} $ = $ {\boldsymbol{\sigma}}_{a} \left( a,{{\boldsymbol {x} }}(a),\mu \right) $
$ {\boldsymbol{\sigma}}_{\mu} $ = $ {\boldsymbol{\sigma}}_{\mu} \left( a,{{\boldsymbol {x} }}(a),\mu \right) $
Table 11.  Explanation of shorthand notation for $ \hat{\mathbf{f}} $ and first derivatives of $ \hat{H} $ evaluated at $ b $ used in (113), (114), and (115). Note that $ \left. \hat{H}_{ {\boldsymbol{\lambda}}} \right|_b = \hat{H}_{ {\boldsymbol{\lambda}}} \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu \right) = \left. \hat{\mathbf{f}}^\mathsf{T} \right|_b $
Shorthand $ \mathbf{\vert} $ Meaning $ \mathbf{\vert} $ Simplification
$ \left. \hat{H}_{{{\boldsymbol {x} }}} \right|_b $ = $ \hat{H}_{{{\boldsymbol {x} }}} \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu \right) $ = $ H_{{{\boldsymbol {x} }}} \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b), {\boldsymbol{\pi}}\left(b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu\right),\mu \right) $
$ \left. \hat{\mathbf{f}}^\mathsf{T} \right|_b $ = $ \hat{\mathbf{f}}^\mathsf{T} \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu \right) $ = $ \mathbf{f}^\mathsf{T} \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b), {\boldsymbol{\pi}}\left(b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu\right),\mu \right) $
$ \left. \hat{H}_t \right|_b $ = $ \hat{H}_t \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu \right) $ = $ H_t \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b), {\boldsymbol{\pi}}\left(b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu\right),\mu \right) $
$ \left. \hat{H}_\mu \right|_b $ = $ \hat{H}_\mu \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu \right) $ = $ H_\mu \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b), {\boldsymbol{\pi}}\left(b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu\right),\mu \right) $
Shorthand $ \mathbf{\vert} $ Meaning $ \mathbf{\vert} $ Simplification
$ \left. \hat{H}_{{{\boldsymbol {x} }}} \right|_b $ = $ \hat{H}_{{{\boldsymbol {x} }}} \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu \right) $ = $ H_{{{\boldsymbol {x} }}} \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b), {\boldsymbol{\pi}}\left(b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu\right),\mu \right) $
$ \left. \hat{\mathbf{f}}^\mathsf{T} \right|_b $ = $ \hat{\mathbf{f}}^\mathsf{T} \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu \right) $ = $ \mathbf{f}^\mathsf{T} \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b), {\boldsymbol{\pi}}\left(b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu\right),\mu \right) $
$ \left. \hat{H}_t \right|_b $ = $ \hat{H}_t \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu \right) $ = $ H_t \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b), {\boldsymbol{\pi}}\left(b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu\right),\mu \right) $
$ \left. \hat{H}_\mu \right|_b $ = $ \hat{H}_\mu \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu \right) $ = $ H_\mu \left( b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b), {\boldsymbol{\pi}}\left(b,{{\boldsymbol {x} }}(b), {\boldsymbol{\lambda}}(b),\mu\right),\mu \right) $
Table 12.  Explanation of shorthand notation for first derivatives of $ {\boldsymbol{\psi}} $ used in (113), (114), and (115)
Shorthand $ \mathbf{\vert} $ Meaning
$ {\boldsymbol{\psi}}_{{{\boldsymbol {x} }}(b)} $ = $ {\boldsymbol{\psi}}_{{{\boldsymbol {x} }}(b)} \left( b,{{\boldsymbol {x} }}(b),\mu \right) $
$ {\boldsymbol{\psi}}_{b} $ = $ {\boldsymbol{\psi}}_{b} \left( b,{{\boldsymbol {x} }}(b),\mu \right) $
$ {\boldsymbol{\psi}}_{\mu} $ = $ {\boldsymbol{\psi}}_{\mu} \left( b,{{\boldsymbol {x} }}(b),\mu \right) $
Shorthand $ \mathbf{\vert} $ Meaning
$ {\boldsymbol{\psi}}_{{{\boldsymbol {x} }}(b)} $ = $ {\boldsymbol{\psi}}_{{{\boldsymbol {x} }}(b)} \left( b,{{\boldsymbol {x} }}(b),\mu \right) $
$ {\boldsymbol{\psi}}_{b} $ = $ {\boldsymbol{\psi}}_{b} \left( b,{{\boldsymbol {x} }}(b),\mu \right) $
$ {\boldsymbol{\psi}}_{\mu} $ = $ {\boldsymbol{\psi}}_{\mu} \left( b,{{\boldsymbol {x} }}(b),\mu \right) $
Table 13.  Equality between Jacobians of two-point boundary condition functions in normalized and un-normalized coordinates
Normalized $ \mathbf{\vert} $ Un-Normalized
$ \tilde {\boldsymbol{\Upsilon}}_{\tilde {\boldsymbol{z}}(0)}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{ {\boldsymbol{z}}(a)}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{1,\tilde {\boldsymbol{z}}(0)}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{1, {\boldsymbol{z}}(a)}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{2,\tilde {\boldsymbol{z}}(0)}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{2, {\boldsymbol{z}}(a)}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{\tilde {\boldsymbol{z}}(1)}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{ {\boldsymbol{z}}(b)}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{1,\tilde {\boldsymbol{z}}(1)}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{1, {\boldsymbol{z}}(b)}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{2,\tilde {\boldsymbol{z}}(1)}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{2, {\boldsymbol{z}}(b)}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{\mu}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{\mu}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{1,\mu}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{1,\mu}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{2,\mu}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{2,\mu}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
Normalized $ \mathbf{\vert} $ Un-Normalized
$ \tilde {\boldsymbol{\Upsilon}}_{\tilde {\boldsymbol{z}}(0)}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{ {\boldsymbol{z}}(a)}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{1,\tilde {\boldsymbol{z}}(0)}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{1, {\boldsymbol{z}}(a)}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{2,\tilde {\boldsymbol{z}}(0)}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{2, {\boldsymbol{z}}(a)}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{\tilde {\boldsymbol{z}}(1)}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{ {\boldsymbol{z}}(b)}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{1,\tilde {\boldsymbol{z}}(1)}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{1, {\boldsymbol{z}}(b)}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{2,\tilde {\boldsymbol{z}}(1)}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{2, {\boldsymbol{z}}(b)}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{\mu}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{\mu}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{1,\mu}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{1,\mu}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
$ \tilde {\boldsymbol{\Upsilon}}_{2,\mu}\left(\tilde {\boldsymbol{z}}(0),\tilde {\boldsymbol{z}}(1),\mu\right) $ = $ {\boldsymbol{\Upsilon}}_{2,\mu}\left( {\boldsymbol{z}}(a), {\boldsymbol{z}}(b),\mu\right) $
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