State-Space Transformation 

Obtain the governing equations of a spherical pendulum in Cartesian coordinates, put them into the affine state-space form, and convert them to spherical coordinates.

The governing equations from the Lagrangian.

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X \[ScriptCapitalL] = 1/2 m (x'[t]^2 + y'[t]^2 + (x[t] x'[t] + y[t] y'[t])^2/( r^2 - x[t]^2 - y[t]^2)) + m g Sqrt[r^2 - x[t]^2 - y[t]^2];

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X Short[eqns = Table[D[D[\[ScriptCapitalL], q'[t]], t] - D[\[ScriptCapitalL], q[t]] == Subscript[F, q][t], {q, {x, y}}]]

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The affine representation with states and inputs .

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X Short[eqns = Table[D[D[\[ScriptCapitalL], q'[t]], t] - D[\[ScriptCapitalL], q[t]] == Subscript[F, q][t], {q, {x, y}}]]; AffineStateSpaceModel[ eqns, {x[t], y[t], x'[t], y'[t]}, {Subscript[F, x][t], Subscript[F, y][t]}, {Sqrt[r^2 - x[t]^2 - y[t]^2]}, t]; assm = AffineStateSpaceModel[%, {x, y, Subscript[x, d], Subscript[y, d]}] // Simplify

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The transformation between spherical and Cartesian coordinates.

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X ctrans = Most@ CoordinateTransform[ "Spherical" -> "Cartesian", {r, \[Theta][t], \[Phi][t]}]

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The transformation of the coordinate derivatives.

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X ctransd = D[ctrans, t] /. {\[Theta]' -> Subscript[\[Theta], d], \[Phi]' -> Subscript[\[Phi], d]}

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The complete state transformation.

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X trans = Thread[{x[t], y[t], Subscript[x, d][t], Subscript[y, d][t]} -> Join[ctrans, ctransd]]

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The expressions in spherical coordinates are much simpler.

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X StateSpaceTransform[ assm, {trans, {\[Theta][t], Subscript[\[Theta], d][t], \[Phi][t], Subscript[\[Phi], d][t]}}] // PowerExpand

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