Constraint control of nonholonomic mechanical systems. (English) Zbl 1387.70027
Summary: We derive an optimal control formulation for a nonholonomic mechanical system using the nonholonomic constraint itself as the control. We focus on Suslov’s problem, which is defined as the motion of a rigid body with a vanishing projection of the body frame angular velocity on a given direction \(\xi\). We derive the optimal control formulation, first for an arbitrary group, and then in the classical realization of Suslov’s problem for the rotation group \(\mathrm{SO}(3)\). We show that it is possible to control the system using the constraint \(\xi (t)\) and demonstrate numerical examples in which the system tracks quite complex trajectories such as a spiral.
MSC:
| 70Q05 | Control of mechanical systems |
| 70E17 | Motion of a rigid body with a fixed point |
| 70B10 | Kinematics of a rigid body |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65P99 | Numerical problems in dynamical systems |
| 65-04 | Software, source code, etc. for problems pertaining to numerical analysis |
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 37J60 | Nonholonomic dynamical systems |
| 70F25 | Nonholonomic systems related to the dynamics of a system of particles |
| 93B05 | Controllability |
| 93B40 | Computational methods in systems theory (MSC2010) |
| 93C10 | Nonlinear systems in control theory |
| 93C15 | Control/observation systems governed by ordinary differential equations |
Software:
PMIRKDC; bvp4c; SNOPT; SOCS; TOMLAB; GPOPS; AUTO-07P; BvpSolve; Bocop; Ipopt; BNDSCO; COLSYS; TOMSYM; PROPT; ACADO; MIRKDC; COLNEW; AUTO; NAG; Chebfun; KNITRO; DIDO; bvpsuite; PSOPT; GitHub; ADiGator; Sbvp; ICLOCS; COCO; MISER3; OptimTrajReferences:
| [1] | Aceto, L., Mazzia, F., Trigiante, D.: The performances of the code TOM on the Holt problem. In: Antreich, K., Bulirsch, R., Gilg, A., Rentrop, P. (eds.) Modeling, Simulation, and Optimization of Integrated Circuits, pp. 349-360. Springer, Berlin (2003) · Zbl 1043.65086 |
| [2] | Agrachev, A., Sachkov, Y.: Control Theory from the Geometric Viewpoint. Springer, Berlin (2004) · Zbl 1062.93001 · doi:10.1007/978-3-662-06404-7 |
| [3] | Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods, vol. 45. SIAM, Philadelphia (2003) · Zbl 1036.65047 · doi:10.1137/1.9780898719154 |
| [4] | Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, 2nd edn. Spinger, Berlin (1997) · Zbl 0885.70001 |
| [5] | Ascher, U.M., Spiteri, R.J.: Collocation software for boundary value differential-algebraic equations. SIAM J. Sci. Comput. 15(4), 938-952 (1994) · Zbl 0804.65080 · doi:10.1137/0915056 |
| [6] | Ascher, U.M., Christiansen, J., Russell, R.D.: Algorithm 569: COLSYS: collocation software for boundary-value ODEs [D2]. ACM Trans. Math. Softw. (TOMS) 7(2), 223-229 (1981) · doi:10.1145/355945.355951 |
| [7] | Ascher, U.M., Mattheij, R.M.M., Russell, R.D.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, vol. 13. SIAM, Philadelphia (1994) · Zbl 0671.65063 |
| [8] | Auzinger, W., et al.: A collocation code for singular boundary value problems in ordinary differential equations. Numer. Algorithms 33(1-4), 27-39 (2003) · Zbl 1030.65089 · doi:10.1023/A:1025531130904 |
| [9] | Bader, G., Ascher, U.M.: A new basis implementation for a mixed order boundary value ODE solver. SIAM J. Sci. Stat. Comput. 8(4), 483-500 (1987) · Zbl 0633.65084 · doi:10.1137/0908047 |
| [10] | Bashir-Ali, Z., Cash, J.R., Silva, H.H.M.: Lobatto deferred correction for stiff two-point boundary value problems. Comput. Math. Appl. 36(10), 59-69 (1998) · Zbl 0933.65086 · doi:10.1016/S0898-1221(98)80009-6 |
| [11] | Becerra, V.M.: Solving complex optimal control problems at no cost with PSOPT. In: 2010 IEEE International Symposium on Computer-Aided Control System Design, pp. 1391-1396. IEEE (2010) · Zbl 1154.65063 |
| [12] | Betts, J.T.: Survey of numerical methods for trajectory optimization. J. Guid. Control Dyn. 21(2), 193-207 (1998) · Zbl 1158.49303 · doi:10.2514/2.4231 |
| [13] | Betts, J.T.: Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, vol. 19. SIAM, Philadelphia (2010) · Zbl 1189.49001 · doi:10.1137/1.9780898718577 |
| [14] | Betts, J.T.: Sparse optimization suite (SOS). In: Applied Mathematical Analysis, LLC. (Based on the Algorithms Published in Betts, JT, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming. SIAM Press, Philadelphia, PA. (2010).) (2013) · Zbl 1293.65104 |
| [15] | Biegler, L.T.: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes, vol. 10. SIAM, Philadelphia (2010) · Zbl 1207.90004 · doi:10.1137/1.9780898719383 |
| [16] | Birkisson, Á., Driscoll, T.A.: Automatic Linearity Detection (2013a). http://eprints.maths.ox.ac.uk/1672/ · Zbl 1365.65192 |
| [17] | Birkisson, Á.: Numerical solution of nonlinear boundary value problems for ordinary differential equations in the continuous framework. Ph.D. thesis, University of Oxford (2013b) · Zbl 0844.65064 |
| [18] | Birkisson, Á., Driscoll, T.A.: Automatic Fréchet differentiation for the numerical solution of boundary-value problems. ACM Trans. Math. Softw. (TOMS) 38(4), 26 (2012) · Zbl 1365.65192 · doi:10.1145/2331130.2331134 |
| [19] | Bloch, A.M.: Nonholonomic Mechanics and Control, vol. 24. Springer, New York (2003) · Zbl 1045.70001 |
| [20] | Bloch, A.M., et al.: Nonholonomic mechanical systems with symmetry. Arch. Ration. Mech. Anal. 136, 21-99 (1996) · Zbl 0886.70014 · doi:10.1007/BF02199365 |
| [21] | Bloch, A.A., et al.: A geometric approach to the optimal control of nonholonomic mechanical systems. In: Bettiol, P., Cannarsa, P., Colombo, G., Motta, M., Rampazzo, F. (eds.) Analysis and Geometry in Control Theory and its Applications, pp. 35-64. Springer INdAM Series, Basel (2015a) · Zbl 1329.49032 |
| [22] | Bloch, A.M., Krupka, D., Zenkov, D.V.: The Helmholtz conditions and the method of controlled Lagrangians. In: Zenkov, D.V. (ed.) The Inverse Problem of the Calculus of Variations: Local and Global Theory, pp. 1-31. Atlantic Press, Minneapolis (2015b) · Zbl 1337.49062 |
| [23] | Boisvert, J.J., Muir, P.H., Spiteri, R.J.: py_bvp: a universal Python interface for BVP codes. In: Proceedings of the 2010 Spring Simulation Multiconference, p. 95. Society for Computer Simulation International (2010) · Zbl 1248.49025 |
| [24] | Boisvert, J.J., Muir, P.H., Spiteri, R.J.: A Runge-Kutta BVODE solver with global error and defect control. ACM Trans. Math. Softw. (TOMS) 39(2), 11 (2013) · Zbl 1295.65141 · doi:10.1145/2427023.2427028 |
| [25] | Bonnans, F., et al.: BOCOP: User Guide (2014). http://bocop.org/ · Zbl 1108.65082 |
| [26] | Bonnans, F., et al.: BocopHJB 1.0. 1-User Guide, PhD thesis. INRIA (2015) |
| [27] | Bonnard, B., Caillau, J.-B., Trélat, E.: Computation of conjugate times in smooth optimal control: the COTCOT algorithm. In: Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference 2005 (CDC-ECC’05), Séville, Spain. IEEE (2005) · Zbl 1082.70014 |
| [28] | Bonnard, B., Caillau, J.-B., Trélat, E.: Second order optimality conditions in the smooth case and applications in optimal control. ESAIM Control Optim. Calc. Var. 13(2), 207-236 (2007) · Zbl 1123.49014 · doi:10.1051/cocv:2007012 |
| [29] | Borisov, A.V., Kilin, A.A., Mamaev, I.S.: Hamiltonicity and integrability of the Suslov problem. Regul. Chaotic Dyn. 16(1-2), 104-116 (2011) · Zbl 1277.70008 · doi:10.1134/S1560354711010035 |
| [30] | Brauer, G.L., Cornick, D.E., Stevenson, R.: Capabilities and Applications of the Program to Optimize Simulated Trajectories (POST). Program Summary Document (1977) · Zbl 0727.65070 |
| [31] | Brockett, R.W.: System theory on group manifolds and coset spaces. SIAM J. Control 10(2), 265-284 (1972) · Zbl 0238.93001 · doi:10.1137/0310021 |
| [32] | Brugnano, L., Trigiante, D.: A new mesh selection strategy for ODEs. Appl. Numer. Math. 24(1), 1-21 (1997) · Zbl 0880.65066 · doi:10.1016/S0168-9274(97)00007-X |
| [33] | Bryson, A.E.: Applied Optimal Control: Optimization, Estimation and Control. CRC Press, Boca Raton (1975) |
| [34] | Bryson, A.E.: Dynamic Optimization, vol. 1. Prentice Hall, Englewood Cliffs (1999) |
| [35] | Bullo, F., Lewis, A.D.: Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems. Texts in Applied Mathematics. Springer, New York (2005) · Zbl 1066.70002 · doi:10.1007/978-1-4899-7276-7 |
| [36] | Byrd, R.H., Nocedal, J., Waltz, R.A.: KNITRO: an integrated package for nonlinear optimization. In: Di Pillo, G., Roma, M. (eds.) Large-Scale Nonlinear Optimization, pp. 35-59. Springer, Berlin (2006) · Zbl 1108.90004 |
| [37] | Caillau, J.-B., Cots, O., Gergaud, J.: Differential continuation for regular optimal control problems. Optim. Methods Softw. 27(2), 177-196 (2012) · Zbl 1248.49025 · doi:10.1080/10556788.2011.593625 |
| [38] | Cash, J.R., Wright, M.H.: A deferred correction method for nonlinear two-point boundary value problems: implementation and numerical evaluation. SIAM J. Sci. Stat. Comput. 12(4), 971-989 (1991) · Zbl 0727.65070 · doi:10.1137/0912052 |
| [39] | Cash, J.R., Mazzia, F.: A new mesh selection algorithm, based on conditioning, for two-point boundary value codes. J. Comput. Appl. Math. 184(2), 362-381 (2005) · Zbl 1076.65065 · doi:10.1016/j.cam.2005.01.016 |
| [40] | Cash, J.R., Mazzia, F.: Hybrid mesh selection algorithms based on conditioning for two-point boundary value problems. JNAIAM J. Numer. Anal. Ind. Appl. Math. 1(1), 81-90 (2006) · Zbl 1108.65082 |
| [41] | Cash, J.R., Moore, G., Wright, R.W.: An automatic continuation strategy for the solution of singularly perturbed nonlinear boundary value problems. ACM Trans. Math. Softw. (TOMS) 27(2), 245-266 (2001) · Zbl 1070.65554 · doi:10.1145/383738.383742 |
| [42] | Cash, J.R., et al.: The role of conditioning in mesh selection algorithms for first order systems of linear two point boundary value problems. J. Comput. Appl. Math. 185(2), 212-224 (2006) · Zbl 1077.65084 · doi:10.1016/j.cam.2005.03.007 |
| [43] | Cash, J.R., et al.: Algorithm 927: the MATLAB code bvptwp. m for the numerical solution of two point boundary value problems. ACM Trans. Math. Softw. (TOMS) 39(2), 15 (2013) · Zbl 1295.65142 · doi:10.1145/2427023.2427032 |
| [44] | Chen, Y.Q., Schwartz, A.L.: RIOTS95: a MATLAB toolbox for solving general optimal control problems and its applications to chemical processes (2002). http://www.schwartz-home.com/riots/ |
| [45] | Chow, Y.T., et al.: Algorithm for overcoming the curse of dimensionality for certain non-convex Hamilton-Jacobi equations, projections and differential games. Ann. Math. Sci. Appl. (2016, to appear) |
| [46] | Cizniar, M., et al.: Dynopt-dynamic optimisation code for MATLAB. In: Technical Computing Prague 2005 (2005) · Zbl 1432.65100 |
| [47] | Community Portal for Automatic Differentiation (2016). http://www.autodiff.org/. Visited 10/08/2016 · Zbl 1293.65104 |
| [48] | Cortés, J., Martínez, E.: Mechanical control systems on Lie algebroids. IMA J. Math. Control Inf. 21, 457-492 (2004) · Zbl 1106.93020 · doi:10.1093/imamci/21.4.457 |
| [49] | Dahlquist, G., Björck, A.: Numerical Methods. Series in Automatic Computation. Prentice-Hall, Englewood Cliffs (1974) · Zbl 1029.65002 |
| [50] | Dankowicz, H., Schilder, F.: Recipes for Continuation. SIAM, Philadelphia (2013) · Zbl 1277.65037 · doi:10.1137/1.9781611972573 |
| [51] | Darbon, J., Osher, S.: Algorithms for Overcoming the Curse of Dimensionality for Certain Hamilton-Jacobi Equations Arising in Control Theory and Elsewhere. arXiv preprint arXiv:1605.01799 (2016) · Zbl 1348.49026 |
| [52] | Doedel, E.J., et al.: AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations (2007) |
| [53] | Driscoll, T.A., Hale, N., Trefethen, L.N.: Chebfun Guide (2014). Pafnuty Publications. http://www.chebfun.org/docs/guide/ · Zbl 0238.93001 |
| [54] | Enright, W.H., Muir, P.H.: Runge-Kutta software with defect control for boundary value ODEs. SIAM J. Sci. Comput. 17(2), 479-497 (1996) · Zbl 0844.65064 · doi:10.1137/S1064827593251496 |
| [55] | Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence. ISBN: 9780821849743, 0821849743 (2010) · Zbl 1194.35001 |
| [56] | Falugi, P., Kerrigan, E., Van Wyk, E.: Imperial College london Optimal Control Software User Guide (ICLOCS). Department of Electrical and Electronic Engineering, Imperial College London, London (2010) |
| [57] | Gay-Balmaz, F., Putkaradze, V.: On noisy extensions of nonholonomic constraints. J. Nonlinear Sci. 26(6), 1571-1613 (2016). doi:10.1007/s00332-016-9313-x · Zbl 1378.70013 |
| [58] | Gerdts, M.: Optimal Control of ODEs and DAEs. Walter de Gruyter, Berlin (2012) · Zbl 1275.49001 · doi:10.1515/9783110249996 |
| [59] | Gill, P.E., Wong, E.: Users Guide for SNCTRL. (2015). https://ccom.ucsd.edu/research/abstract.php?id=102 · Zbl 1365.65192 |
| [60] | Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99-131 (2005) · Zbl 1210.90176 · doi:10.1137/S0036144504446096 |
| [61] | Goh, C.J., Teo, K.L.: MISER: a FORTRAN program for solving optimal control problems. Adv. Eng. Softw. (1978) 10(2), 90-99 (1988) · doi:10.1016/0141-1195(88)90005-8 |
| [62] | Hale, N., Moore, D.R.: A sixth-order extension to the MATLAB package bvp4c of J. Kierzenka and L. Shampine (2008) |
| [63] | Holm, D.D.: Geometric mechanics: rotating, translating, and rolling. In: Geometric Mechanics. Imperial College Press, London. ISBN: 9781848167773 (2011) · Zbl 1381.70001 |
| [64] | Holmström, K., Göran, A., Edvall, M.M.: Users Guide for Tomlab 7 (2010). Tomlab Optimization, Vallentuna and Västerås. http://tomopt.com/tomlab/download/manuals.php · Zbl 1218.49002 |
| [65] | Houska, B., Ferreau, H.J., Diehl, M.: ACADO toolkitAn open-source framework for automatic control and dynamic optimization. Optim. Control Appl. Methods 32(3), 298-312 (2011) · Zbl 1218.49002 · doi:10.1002/oca.939 |
| [66] | Hull, D.G.: Optimal Control Theory for Applications. Springer, Berlin (2013) |
| [67] | Isidori, A.: Nonlinear Control Systems. Springer, Berlin (2013) · Zbl 1293.93666 |
| [68] | Jansch, C., Well, K.H., Schnepper, K.: GESOP-Eine Software Umgebung Zur Simulation Und Optimierung. In: Proceedings des SFB (1994) · Zbl 1106.93020 |
| [69] | Jean, F.: Control of Nonholonomic Systems: From Sub-Riemannian Geometry to Motion Planning. Springer, Cham (2014) · Zbl 1309.93002 |
| [70] | Kelly, M.P.: OptimTraj User’s Guide, Version 1.5. (2016). https://github.com/MatthewPeterKelly/OptimTraj/tree/master/docs/UsersGuide · Zbl 0880.65066 |
| [71] | Kierzenka, J., Shampine, L.F.: A BVP solver that controls residual and error. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3(1-2), 27-41 (2008) · Zbl 1154.65063 |
| [72] | Kitzhofer, G., et al.: The new Matlab code bvpsuite for the solution of singular implicit BVPs. J. Numer. Anal. Ind. Appl. Math 5, 113-134 (2010) · Zbl 1432.65100 |
| [73] | Koon, W.-S., Marsden, J.E.: Optimal control for holonomic and nonholonomic mechanical system with symmetry and Lagrangian reduction. SIAM J. Control Optim. 35, 901-929 (1997) · Zbl 0880.70020 · doi:10.1137/S0363012995290367 |
| [74] | Kozlov, V.V.: On the integration theory of equations of nonholonomic mechanics. Regul. Chaotic Dyn. 7(2), 161-176 (2002) · Zbl 1006.37040 · doi:10.1070/RD2002v007n02ABEH000203 |
| [75] | Kunkel, P.: Differential-Algebraic Equations: Analysis and Numerical Solution. European Mathematical Society, Germany (2006) · Zbl 1095.34004 · doi:10.4171/017 |
| [76] | Lee, J.M.: Smooth Manifolds. Springer, Berlin (2003) · doi:10.1007/978-0-387-21752-9 |
| [77] | Lewis, A.D.: Simple mechanical control systems with constraints. IEEE Trans. Automat. Control 45, 14201436 (2000) · Zbl 0994.70023 |
| [78] | Lewis, A.D., Murray, R.M.: Variational principles for constrained systems: theory and experiment. Int. J. Non Linear Mech. 30(6), 793-815 (1995) · Zbl 0864.70008 · doi:10.1016/0020-7462(95)00024-0 |
| [79] | Lewis, A.D., Murray, R.M.: Controllability of simple mechanical control systems. SIAM J. Control Optim. 35, 766-790 (1997) · Zbl 0870.53013 · doi:10.1137/S0363012995287155 |
| [80] | Li, Z., Canny, J.F. (eds.): Nonholonomic Motion Planning. Springer, New York (1992) · Zbl 0875.00053 |
| [81] | Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, vol. 17. Springer, Berlin (2013) · Zbl 0933.70003 |
| [82] | Mazzia, F., Sgura, I.: Numerical approximation of nonlinear BVPs by means of BVMs. Appl. Numer. Math. 42(1), 337-352 (2002) · Zbl 0999.65076 · doi:10.1016/S0168-9274(01)00159-3 |
| [83] | Mazzia, F., Trigiante, D.: A hybrid mesh selection strategy based on conditioning for boundary value ODE problems. Numer. Algorithms 36(2), 169-187 (2004) · Zbl 1050.65072 · doi:10.1023/B:NUMA.0000033132.99233.c8 |
| [84] | Mazzia, F., Cash, J.R., Soetaert, K.: Solving boundary value problems in the open source software R: package bvpSolve. Opusc. Math. 34(2), 387-403 (2014) · Zbl 1293.65104 · doi:10.7494/OpMath.2014.34.2.387 |
| [85] | Nijmeijer, H., Van der Schaft, A.: Nonlinear Dynamical Control Systems. Springer, Berlin (2013) · Zbl 0701.93001 |
| [86] | Nikolayzik, T., Büskens, C., Gerdts, M.: Nonlinear large-scale optimization with WORHP. In: Proceedings of the 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, vol. 13, no. 15.09 (2010) |
| [87] | Oberle, H.J., Grimm, W.: BNDSCO: A Program for the Numerical Solution of Optimal Control Problems. Inst. für Angewandte Math. der Univ, Hamburg (2001) |
| [88] | Osborne, J., Zenkov, D.V.: Steering the Chaplygin sleigh by a moving mass. In: Proceedings of CDC 2005, vol. 44, pp. 1114-1118 |
| [89] | Patterson, M.A., Rao, A.V.: GPOPS-II: a MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming. ACM Trans. Math. Softw. (TOMS) 41(1), 1 (2014) · Zbl 1369.65201 · doi:10.1145/2558904 |
| [90] | Poincaré, H.: Sur une forme nouvelle des équations de la mécanique. CR Acad. Sci 132, 369-371 (1901) · JFM 32.0715.01 |
| [91] | Rao, A.V.: A survey of numerical methods for optimal control. Adv. Astronaut. Sci. 135(1), 497-528 (2009) |
| [92] | Rao, A.V., et al.: Algorithm 902: Gpops, a matlab software for solving multiple-phase optimal control problems using the gauss pseudospectral method. ACM Trans. Math. Softw. (TOMS) 37(2), 22 (2010) · Zbl 1364.65131 · doi:10.1145/1731022.1731032 |
| [93] | Rieck, M., et al.: FALCON.m User Guide. Institute of Flight System Dynamics, Technische Universität München, Munich (2016) |
| [94] | Ross, I.M.: Users Manual for DIDO: A MATLAB Application Package for Solving Optimal Control Problems. Elissar Global (2004). http://www.elissarglobal.com/academic/products/ |
| [95] | Rutquist, P.E., Edvall, M.M.: Propt-matlab Optimal Control Software, vol. 260. Tomlab Optimization Inc, Vallentuna and Västerås (2010) |
| [96] | Shampine, L.F., Kierzenka, J., Reichelt, M.W.: Solving boundary value problems for ordinary differential equations in MATLAB with bvp4c. In: Tutorial notes, pp. 437-448 (2000) · Zbl 1117.65114 |
| [97] | Shampine, L.F., Muir, P.H., Xu, H.: A user-friendly Fortran BVP solver1. JNAIAM 1(2), 201-217 (2006) · Zbl 1117.65114 |
| [98] | Suslov, G.K.: Theoretical Mechanics, vol. 3. Gostekhizdat, Moscow (1946) |
| [99] | The Numerical Algorithms Group (NAG). The NAG Library. www.nag.com |
| [100] | Vlases, W.G., et al.: Optimal trajectories by implicit simulation. In: Boeing Aerospace and Electronics, Technical Report WRDC-TR-90-3056, Wright-Patterson Air Force Base (1990) |
| [101] | von Stryk, O.: Users Guide for DIRCOL. Technische Universität Darmstadt, Darmstadt (2000) |
| [102] | Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25-57 (2006) · Zbl 1134.90542 · doi:10.1007/s10107-004-0559-y |
| [103] | Weinstein, M.J., Patterson, M.A., Rao, A.V.: Utilizing the algorithmic differentiation package ADiGator for solving optimal control problems using direct collocation. In: AIAA Guidance, Navigation, and Control Conference, p. 1085 (2015) |
| [104] | Weinstein, M.J., Rao, A.V.: Algorithm: ADiGator, a toolbox for the algorithmic differentiation of mathematical functions in MATLAB using source transformation via operator overloading. ACM Trans. Math. Softw. (2016, in revision). http://www.anilvrao.com/Publications/SubmittedJournalPublications/adigator-CALGO.pdf · Zbl 1484.65363 |
| [105] | Zenkov, D.V., et al.: Matching and stabilization of low dimensional nonholonomic systems. In: Proc CDC 2000, vol 39. pp. 1289-1295 (2000) · Zbl 1134.90542 |
| [106] | Zenkov, D.V., Bloch, A.M., Marsden, J.E.: Flat nonholonomic matching. In: Proceedings of ACC 2002, pp. 2812-2817 (2002) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
