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

Vakhtang Putkaradze; Stuart Rogers

doi:10.3934/naco.2020021

Article Contents

2021, Volume 11, Issue 2: 143-207. Doi: 10.3934/naco.2020021

This issue Previous Article Next Article Generalized Nash equilibrium problem based on malfatti's problem

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
^* Corresponding author: Stuart Rogers

Received: June 2019

Revised: January 2020

Early access: March 2020

Published: June 2021

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Abstract

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:

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

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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

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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 $

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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 $

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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

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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

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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

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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

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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 $

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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

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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

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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

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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)

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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

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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

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Figure 15. Predictor-corrector continuation

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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 $

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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 $

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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 $

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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} $

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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} $

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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 $

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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 $

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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) $

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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) $

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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) $

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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) $

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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) $

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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) $

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