Numerical simulations of a rolling ball robot actuated by internal point masses

Vakhtang Putkaradze; Stuart Rogers

doi:10.3934/naco.2020021

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

Numerical simulations of a rolling ball robot actuated by internal point masses

Vakhtang Putkaradze ^1, and Stuart Rogers ^2,,

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.

Keywords: Optimal control, rolling ball robots, trajectory tracking, obstacle avoidance, predictor-corrector continuation.

Mathematics Subject Classification: Primary: 37J60, 70E18, 70E60, 49J15; Secondary: 49K15, 34A34, 65L10.

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

References:

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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	J. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Vol. 19, SIAM, 2010. doi: 10.1137/1.9780898718577. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	H. Dankowicz and F. Schilder, Recipes for Continuation, SIAM, 2013. doi: 10.1137/1.9781611972573. Google Scholar
[27]	D. Davidenko, On a new method of numerical solution of systems of nonlinear equations, Dokl. Akad. Nauk SSSR, 88 (1953), 601-602. Google Scholar
[28]	D. Davidenko, The approximate solution of sets of nonlinear equations, Ukr. Mat. Zh, 5 (1953), 196-206. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[37]	B. Graf, Quaternions and dynamics, arXiv preprint arXiv: 0811.2889. Google Scholar
[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. Google Scholar
[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. Google Scholar
[41]	D. Hull, Optimal Control Theory for Applications, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4757-4180-3. Google Scholar
[42]	L. Kantorovich, On newton's method for functional equations, Dokl. Akad. Nauk SSSR, 59 (1948), 1237-1240. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[55]	U. Naumann, The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation, Vol. 24, SIAM, 2012. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[62]	L. Rall, Davidenko's Method for the Solution of Nonlinear Operator Equations, Technical report, University of Wisconsin, Madison, Mathematics Research Center, 1968. Google Scholar
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[64]	L. Shampine, J. Kierzenka and M. Reichelt, Solving boundary value problems for ordinary differential equations in matlab with bvp4c, Tutorial notes, 437–448. Google Scholar
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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

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
$ \dot {\boldsymbol{\theta}}_b $	$ \begin{bmatrix} 0 0 0 0 \end{bmatrix}^\mathsf{T} $
$ z_b $	$ 1 $
$ \dot z_b $	$ 0 $

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
$ \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(\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 $

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} $	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) $

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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) $

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) $

American Institute of Mathematical Sciences

Numerical simulations of a rolling ball robot actuated by internal point masses

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