Understanding Euler Angles and Co-rotational basis

<<Notation`
Symbolize[(NotationTemplateTag[(Subscript[γ, i_])])]

ring3={Red,CapForm->None,Tube[{{-.1,0,0},{.1,0,0}},3],Line[{{{0,0,4},{0,0,3}},-{{0,0,4},{0,0,3}}}]};
ring2={Blue,CapForm->None,Tube[{{-.1,0,0},{.1,0,0}},2],Line[{{{0,3,0},{0,2,0}},-{{0,3,0},{0,2,0}}}]};
ring1={Green,CapForm->None,Tube[{{-.1,0,0},{.1,0,0}},1],Black,Arrow[{{-1,0,0},{1,0,0}}],Line[{{{0,0,1},{0,0,2}},-{{0,0,1},{0,0,2}}}]};

(Subscript[γ, 1]), (Subscript[γ, 2]), and (Subscript[γ, 3]) are the Euler-Angles, with respect to the body axis, a, b and c, e.g., (a, b, c)=(3,2,1).

gimbal[{(Subscript[γ, 1]):_,(Subscript[γ, 2]):_,(Subscript[γ, 3]):_},{a_,b_,c_}]:=
Graphics3D[{GeometricTransformation[ring3,EulerMatrix[{α,0,0},{a,b,c}]],GeometricTransformation[ring2,EulerMatrix[{α,β,0},{a,b,c}]],GeometricTransformation[ring1,m=EulerMatrix[{α,β,γ},{a,b,c}]]},
ViewPoint->{1.3, -2.4, 2.},PlotRange->3.5,PlotLabel->NumberForm[MatrixForm[{{α,β,γ}}],2]]

3-2-1 Euler angles

Manipulate[gimbal[{ψ,θ,ϕ},{3,2,1}],{{ψ,0},0,2Pi,Pi/32},{{θ,0},0,2Pi,Pi/32},{{ϕ,0},0,2Pi,Pi/32},SaveDefinitions->True]

3-2-3 Euler Angles

Manipulate[gimbal[{ψ,θ,ϕ},{3,2,3}],{{ψ,0},0,2Pi,Pi/32},{{θ,0},0,2Pi,Pi/32},{{ϕ,0},0,2Pi,Pi/32},SaveDefinitions->True]

Co-rotational basis and Yaw-Pitch-Roll

(Subscript[(UnderBar[(OverHat[E])]), i]) are the laboratory Cartesian basis vectors and (Subscript[(UnderBar[(OverHat[e])]), i]) are co-rotational Cartesian basis vectors

s1=;
With[{a=3,b=2,c=1},
Manipulate[
m=EulerMatrix[{ψ,θ,ϕ},{a,b,c}];
Show[{
Graphics3D[Arrow[{{0,0,0},{10,0,0}}]],Graphics3D[Text["(Subscript[(UnderBar[(OverHat[E])]), 1])",{11,0,0}]],
Graphics3D[Arrow[{{0,0,0},{0,10,0}}]],Graphics3D[Text["(Subscript[(UnderBar[(OverHat[E])]), 2])",{0,11,0}]],
Graphics3D[Arrow[{{0,0,0},{0,0,10}}]],Graphics3D[Text["(Subscript[(UnderBar[(OverHat[E])]), 3])",{0,0,11}]],
Map[TransformedRegion[#,AffineTransform[m]]&,{s1}],
Graphics3D[{Red,Arrow[{{0,0,0},m.{10,0,0}}]}],Graphics3D[{Red, Text["(Subscript[(UnderBar[(OverHat[e])]), 1])",m.{12,0,0}]}],
Graphics3D[{Blue,Arrow[{{0,0,0},m.{0,10,0}}]}],Graphics3D[{Blue, Text["(Subscript[(UnderBar[(OverHat[e])]), 2])",m.{0,12,0}]}],
Graphics3D[{Green,Arrow[{{0,0,0},m.{0,0,10}}]}],Graphics3D[{Darker[Green], Text["(Subscript[(UnderBar[(OverHat[e])]), 3])",m.{0,0,12}]}]
},
PlotRange->{{-10,10},{-10,10},{-10,10}}
],{ψ,0,π/2},{θ,0,π/2},{ϕ,0,π/2}
]]