Skip to content

Commit

Permalink
Fixed syntax errors.
Browse files Browse the repository at this point in the history
  • Loading branch information
@iamazadi
iamazadi committed Sep 13, 2024
1 parent e8945ad commit 74bca69
Showing 1 changed file with 35 additions and 37 deletions.
72 changes: 35 additions & 37 deletions docs/src/reactionwheelunicycle.md
View file
Original file line number Diff line number Diff line change
Expand Up @@ -163,101 +163,100 @@ Turning dynamics:
``m_{51} \ddot{\alpha} + m_{52} \ddot{\beta} + m_{53} \ddot{\gamma} + m_{54} \ddot{\delta} + m_{55} \ddot{\theta} + c_{51} \dot{\alpha}^2 + c_{52} \dot{\beta}^2 + c_{53} \dot{\gamma}^2 + c_{54} \dot{\alpha} \dot{\beta} + c_{55} \dot{\alpha} \dot{\gamma} + c_{56} \dot{\alpha} \dot{\delta} + c_{57} \dot{\alpha} \dot{\theta} + c_{58} \dot{\beta} \dot{\gamma} + c_{59} \dot{\beta} \dot{\delta} + c_{510} \dot{\gamma} \dot{\delta} + c_{511} \dot{\delta} \dot{\theta} = 0``

``
\begin{align}
\frac{\mathrm{d} x\left( t \right)}{\mathrm{d}t} &= r_{w} \cos\left( \delta\left( t \right) \right) \frac{\mathrm{d} \theta\left( t \right)}{\mathrm{d}t} \\
\frac{\mathrm{d} y\left( t \right)}{\mathrm{d}t} &= r_{w} \sin\left( \delta\left( t \right) \right) \frac{\mathrm{d} \theta\left( t \right)}{\mathrm{d}t} \\
\frac{\mathrm{d} z\left( t \right)}{\mathrm{d}t} &= 0 \\
I_{w} &= \left[
\frac{\mathrm{d} x\left( t \right)}{\mathrm{d}t} = r_{w} \cos\left( \delta\left( t \right) \right) \frac{\mathrm{d} \theta\left( t \right)}{\mathrm{d}t} \\
\frac{\mathrm{d} y\left( t \right)}{\mathrm{d}t} = r_{w} \sin\left( \delta\left( t \right) \right) \frac{\mathrm{d} \theta\left( t \right)}{\mathrm{d}t} \\
\frac{\mathrm{d} z\left( t \right)}{\mathrm{d}t} = 0 \\
I_{w} = \left[
\begin{array}{cccc}
I_{w1} & 0 & 0 & 0 \\
0 & I_{w2} & 0 & 0 \\
0 & 0 & I_{w3} & 0 \\
0 & 0 & 0 & 0 \\
\end{array}
\right] \\
I_{c} &= \left[
I_{c} = \left[
\begin{array}{cccc}
I_{c1} & 0 & 0 & 0 \\
0 & I_{c2} & 0 & 0 \\
0 & 0 & I_{c3} & 0 \\
0 & 0 & 0 & 0 \\
\end{array}
\right] \\
I_{r} &= \left[
I_{r} = \left[
\begin{array}{cccc}
I_{r1} & 0 & 0 & 0 \\
0 & I_{r2} & 0 & 0 \\
0 & 0 & I_{r3} & 0 \\
0 & 0 & 0 & 0 \\
\end{array}
\right] \\
\mathrm{w2}_{cp\_T}\left( t \right) &= \left[
\mathrm{w2}_{cp\_T}\left( t \right) = \left[
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \cos\left( \alpha\left( t \right) \right) & - \sin\left( \alpha\left( t \right) \right) & - r_{w} \sin\left( \alpha\left( t \right) \right) \\
0 & \sin\left( \alpha\left( t \right) \right) & \cos\left( \alpha\left( t \right) \right) & r_{w} \cos\left( \alpha\left( t \right) \right) \\
0 & 0 & 0 & 1 \\
\end{array}
\right] \\
\mathrm{cp}_{g\_T}\left( t \right) &= \left[
\mathrm{cp}_{g\_T}\left( t \right) = \left[
\begin{array}{cccc}
\cos\left( \delta\left( t \right) \right) & - \sin\left( \delta\left( t \right) \right) & 0 & x\left( t \right) \\
\sin\left( \delta\left( t \right) \right) & \cos\left( \delta\left( t \right) \right) & 0 & y\left( t \right) \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right] \\
\mathrm{w2}_{g\_T}\left( t \right) &= \mathrm{cp}_{g\_T}\left( t \right) \mathrm{w2}_{cp\_T}\left( t \right) \\
w2_{P\_w} &= \left[
\mathrm{w2}_{g\_T}\left( t \right) = \mathrm{cp}_{g\_T}\left( t \right) \mathrm{w2}_{cp\_T}\left( t \right) \\
w2_{P\_w} = \left[
\begin{array}{c}
0 \\
0 \\
0 \\
1 \\
\end{array}
\right] \\
g_{P\_w}\left( t \right) &= \mathrm{w2}_{g\_T}\left( t \right) w2_{P\_w} \\
c_{w2\_T}\left( t \right) &= \left[
g_{P\_w}\left( t \right) = \mathrm{w2}_{g\_T}\left( t \right) w2_{P\_w} \\
c_{w2\_T}\left( t \right) = \left[
\begin{array}{cccc}
\cos\left( \beta\left( t \right) \right) & 0 & \sin\left( \beta\left( t \right) \right) & l_{c} \sin\left( \beta\left( t \right) \right) \\
0 & 1 & 0 & 0 \\
- \sin\left( \beta\left( t \right) \right) & 0 & \cos\left( \beta\left( t \right) \right) & l_{c} \cos\left( \beta\left( t \right) \right) \\
0 & 0 & 0 & 1 \\
\end{array}
\right] \\
c_{g\_T}\left( t \right) &= \mathrm{w2}_{g\_T}\left( t \right) c_{w2\_T}\left( t \right) \\
c_{P\_c} &= \left[
c_{g\_T}\left( t \right) = \mathrm{w2}_{g\_T}\left( t \right) c_{w2\_T}\left( t \right) \\
c_{P\_c} = \left[
\begin{array}{c}
0 \\
0 \\
0 \\
1 \\
\end{array}
\right] \\
g_{P\_c}\left( t \right) &= c_{g\_T}\left( t \right) c_{P\_c} \\
r_{c\_T}\left( t \right) &= \left[
g_{P\_c}\left( t \right) = c_{g\_T}\left( t \right) c_{P\_c} \\
r_{c\_T}\left( t \right) = \left[
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \cos\left( \gamma\left( t \right) \right) & - \sin\left( \gamma\left( t \right) \right) & 0 \\
0 & \sin\left( \gamma\left( t \right) \right) & \cos\left( \gamma\left( t \right) \right) & l_{cr} \\
0 & 0 & 0 & 1 \\
\end{array}
\right] \\
r_{g\_T}\left( t \right) &= c_{g\_T}\left( t \right) r_{c\_T}\left( t \right) \\
r_{P\_r} &= \left[
r_{g\_T}\left( t \right) = c_{g\_T}\left( t \right) r_{c\_T}\left( t \right) \\
r_{P\_r} = \left[
\begin{array}{c}
0 \\
0 \\
0 \\
1 \\
\end{array}
\right] \\
g_{P\_r}\left( t \right) &= r_{g\_T}\left( t \right) r_{P\_r} \\
r_{w2\_T}\left( t \right) &= \mathrm{inv}\left( \mathrm{w2}_{g\_T}\left( t \right) \right) r_{g\_T}\left( t \right) \\
V_{w}\left( t \right) &= \frac{\mathrm{d} g_{P\_w}\left( t \right)}{\mathrm{d}t} \\
V_{c}\left( t \right) &= \frac{\mathrm{d} g_{P\_c}\left( t \right)}{\mathrm{d}t} \\
V_{r}\left( t \right) &= \frac{\mathrm{d} g_{P\_r}\left( t \right)}{\mathrm{d}t} \\
\Omega_{w}\left( t \right) &= \mathrm{broadcast}\left( +, \left[
g_{P\_r}\left( t \right) = r_{g\_T}\left( t \right) r_{P\_r} \\
r_{w2\_T}\left( t \right) = \mathrm{inv}\left( \mathrm{w2}_{g\_T}\left( t \right) \right) r_{g\_T}\left( t \right) \\
V_{w}\left( t \right) = \frac{\mathrm{d} g_{P\_w}\left( t \right)}{\mathrm{d}t} \\
V_{c}\left( t \right) = \frac{\mathrm{d} g_{P\_c}\left( t \right)}{\mathrm{d}t} \\
V_{r}\left( t \right) = \frac{\mathrm{d} g_{P\_r}\left( t \right)}{\mathrm{d}t} \\
\Omega_{w}\left( t \right) = \mathrm{broadcast}\left( +, \left[
\begin{array}{c}
_{derivative}\left( \alpha\left( t \right), t, 1 \right) \\
_{derivative}\left( \theta\left( t \right), t, 1 \right) \\
Expand All @@ -272,7 +271,7 @@ _{derivative}\left( \delta\left( t \right), t, 1 \right) \\
0 \\
\end{array}
\right] \right) \\
\Omega_{c}\left( t \right) &= \mathrm{broadcast}\left( +, \mathrm{broadcast}\left( +, \left[
\Omega_{c}\left( t \right) = \mathrm{broadcast}\left( +, \mathrm{broadcast}\left( +, \left[
\begin{array}{c}
0 \\
_{derivative}\left( \beta\left( t \right), t, 1 \right) \\
Expand All @@ -294,7 +293,7 @@ _{derivative}\left( \delta\left( t \right), t, 1 \right) \\
0 \\
\end{array}
\right] \right) \\
\Omega_{r}\left( t \right) &= \mathrm{broadcast}\left( +, \mathrm{broadcast}\left( +, \mathrm{broadcast}\left( +, \left[
\Omega_{r}\left( t \right) = \mathrm{broadcast}\left( +, \mathrm{broadcast}\left( +, \mathrm{broadcast}\left( +, \left[
\begin{array}{c}
_{derivative}\left( \gamma\left( t \right), t, 1 \right) \\
0 \\
Expand Down Expand Up @@ -323,14 +322,13 @@ _{derivative}\left( \delta\left( t \right), t, 1 \right) \\
0 \\
\end{array}
\right] \right) \\
T_{w}\left( t \right) &= \mathrm{adjoint}\left( V_{w}\left( t \right) \right) \mathrm{broadcast}\left( *, V_{w}\left( t \right), \mathrm{Ref}\left( 0.5 m_{w} \right) \right)_{1} + \mathrm{adjoint}\left( \Omega_{w}\left( t \right) \right) \mathrm{broadcast}\left( *, I_{w} \Omega_{w}\left( t \right), 0.5 \right)_{1} \\
P_{w}\left( t \right) &= g g_{P\_w}\left( t \right)_{3} m_{w} \\
T_{c}\left( t \right) &= \mathrm{adjoint}\left( V_{c}\left( t \right) \right) \mathrm{broadcast}\left( *, V_{c}\left( t \right), \mathrm{Ref}\left( 0.5 m_{c} \right) \right)_{1} + \mathrm{adjoint}\left( \Omega_{c}\left( t \right) \right) \mathrm{broadcast}\left( *, I_{c} \Omega_{c}\left( t \right), 0.5 \right)_{1} \\
P_{c}\left( t \right) &= g g_{P\_c}\left( t \right)_{3} m_{c} \\
T_{r}\left( t \right) &= \mathrm{adjoint}\left( V_{r}\left( t \right) \right) \mathrm{broadcast}\left( *, V_{r}\left( t \right), \mathrm{Ref}\left( 0.5 m_{r} \right) \right)_{1} + \mathrm{adjoint}\left( \Omega_{r}\left( t \right) \right) \mathrm{broadcast}\left( *, I_{r} \Omega_{r}\left( t \right), 0.5 \right)_{1} \\
P_{r}\left( t \right) &= g g_{P\_r}\left( t \right)_{3} m_{r} \\
T_{total}\left( t \right) &= T_{r}\left( t \right) + T_{c}\left( t \right) + T_{w}\left( t \right) \\
P_{total}\left( t \right) &= P_{w}\left( t \right) + P_{c}\left( t \right) + P_{r}\left( t \right) \\
L\left( t \right) &= T_{total}\left( t \right) - P_{total}\left( t \right)
\end{align}
T_{w}\left( t \right) = \mathrm{adjoint}\left( V_{w}\left( t \right) \right) \mathrm{broadcast}\left( *, V_{w}\left( t \right), \mathrm{Ref}\left( 0.5 m_{w} \right) \right)_{1} + \mathrm{adjoint}\left( \Omega_{w}\left( t \right) \right) \mathrm{broadcast}\left( *, I_{w} \Omega_{w}\left( t \right), 0.5 \right)_{1} \\
P_{w}\left( t \right) = g g_{P\_w}\left( t \right)_{3} m_{w} \\
T_{c}\left( t \right) = \mathrm{adjoint}\left( V_{c}\left( t \right) \right) \mathrm{broadcast}\left( *, V_{c}\left( t \right), \mathrm{Ref}\left( 0.5 m_{c} \right) \right)_{1} + \mathrm{adjoint}\left( \Omega_{c}\left( t \right) \right) \mathrm{broadcast}\left( *, I_{c} \Omega_{c}\left( t \right), 0.5 \right)_{1} \\
P_{c}\left( t \right) = g g_{P\_c}\left( t \right)_{3} m_{c} \\
T_{r}\left( t \right) = \mathrm{adjoint}\left( V_{r}\left( t \right) \right) \mathrm{broadcast}\left( *, V_{r}\left( t \right), \mathrm{Ref}\left( 0.5 m_{r} \right) \right)_{1} + \mathrm{adjoint}\left( \Omega_{r}\left( t \right) \right) \mathrm{broadcast}\left( *, I_{r} \Omega_{r}\left( t \right), 0.5 \right)_{1} \\
P_{r}\left( t \right) = g g_{P\_r}\left( t \right)_{3} m_{r} \\
T_{total}\left( t \right) = T_{r}\left( t \right) + T_{c}\left( t \right) + T_{w}\left( t \right) \\
P_{total}\left( t \right) = P_{w}\left( t \right) + P_{c}\left( t \right) + P_{r}\left( t \right) \\
L\left( t \right) = T_{total}\left( t \right) - P_{total}\left( t \right)
``

0 comments on commit 74bca69

Please sign in to comment.