\documentclass{article}
\usepackage{ifthen}
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\usepackage{amsmath}

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\pagestyle{empty}
\begin{document}
$ q_{out} = q_{in} $
\pagebreak

$ f(q) = J * q + b $
\pagebreak

$ f(q) = C $
\pagebreak

\begin{eqnarray*}
\mathbf{f}(\mathbf{q}) &=&
\left(\begin{array}{c}
  ( x_{com} - x_{ref} ) \cdot u_z \\
  ( R^T (e \wedge u) ) \cdot u_z \\
  ( x_{com} - x_L ) \cdot (u)\\
  ( x_{com} - x_R ) \cdot (u)\\
\end{array}\right)
\end{eqnarray*}
\pagebreak

$\mathbf{x}_{com}$
\pagebreak

$\mathbf{x_L}$
\pagebreak

$\mathbf{x_R}$
\pagebreak

$\mathbf{x}_{ref}$
\pagebreak

$ u = x_R - x_L $
\pagebreak

$ e = x_{com} - (\frac{x_L + x_R}{2})$
\pagebreak

$ (o_i)_{i \in I } $
\pagebreak

$
(f_j)_{j \in J } $
\pagebreak

$o_i$
\pagebreak

$ f_j $
\pagebreak

$\textbf{n}_{f_j}$
\pagebreak

$d_{\parallel} = d(f_j, P (C_{o_i}, f_j))$
\pagebreak

$d_{\perp} =
\textbf{n}_{f_j}.\vec{C_{f_j}C_{o_i}}$
\pagebreak

$
f_j $
\pagebreak

$(x+m,y,z,0,0)$
\pagebreak

$(x+m,0,0,0,rz)$
\pagebreak

$(x+m,y,z,0,rz)$
\pagebreak

$(x+m,0,0,ry,rz)$
\pagebreak

$(x+m,y,z,ry,rz)$
\pagebreak

$(y,z,rx)$
\pagebreak

$(0,0,rx)$
\pagebreak

$(y+2jM,z+2iM,rx)$
\pagebreak

$(2jM,2iM,rx)$
\pagebreak

$M$
\pagebreak

$i$
\pagebreak

$j$
\pagebreak

\begin{equation}
   d(o_i,f_j)
   \end{equation}
\pagebreak

$ A + t*v, t \in
[0,1] $
\pagebreak

$ A + t*u $
\pagebreak

$ Q \in plane
\Leftrightarrow (P - Q) \cdot n = 0 $
\pagebreak

$ Pts_[1] - Pts_[0] $
\pagebreak

$ A + t*u
$
\pagebreak

$\epsilon$
\pagebreak

$f_{in}$
\pagebreak

$[0,L]$
\pagebreak

$[t_1,t_2]$
\pagebreak

$[0, |t_2-t_1|]$
\pagebreak

\begin{eqnarray*}
 f_{out}(t) = f_{in}(t_1+t) & \mbox{if} & t_1 \leq t_2 \\
 f_{out}(t) = f_{in}(t_1-t) & \mbox{if} & t_2 < t_1
 \end{eqnarray*}
\pagebreak

$E=(in,out,f)$
\pagebreak

$\mathcal{C}$
\pagebreak

$in\subset\{1,\cdots, \dim\mathcal{C}\}$
\pagebreak

$out\subset\{1,\cdots, \dim\mathcal{C}\}$
\pagebreak

$f$
\pagebreak

$\mathbf{R}^{|in|}$
\pagebreak

$\mathbf{R}^{|out|}$
\pagebreak

$in\cap out
= \emptyset$
\pagebreak

$\mathbf{p}\in\mathcal{C}$
\pagebreak

$\mathbf{q} =
E(\mathbf{p})$
\pagebreak

\begin{eqnarray}
    &\mathbf{q}_{\bar{out}} = \mathbf{p}_{\bar{out}}\\
    &\mathbf{q}_{out} = f (\mathbf{p}_{in}).
  \end{eqnarray}
\pagebreak

$rhs$
\pagebreak

\begin{equation}
\mathbf{q}_{out} = f (\mathbf{p}_{in}) + rhs.
\end{equation}
\pagebreak

$ g $
\pagebreak

$ g^{-1} $
\pagebreak

$E$
\pagebreak

\begin{equation}
g(\mathbf{q}_{out}) = f(\mathbf{p}_{in}) + rhs
\end{equation}
\pagebreak

$in$
\pagebreak

$out$
\pagebreak

$\mathbf{q}_{out} = g^{-1}
        \left(f(\mathbf{q}_{in}) + rhs\right)$
\pagebreak

$i\overline{n\cup ou}t$
\pagebreak

\begin{equation}
        \frac{\partial f}{\partial \mathbf{q}_{in}}
        (\mathbf{q}_{in}).
        \end{equation}
\pagebreak

$\mathbf{q} = E(\mathbf{p})$
\pagebreak

\begin{eqnarray}
\mathbf{q}_{\bar{out}} &=& \mathbf{p}_{\bar{out}} \\
\mathbf{q}_{out} &=& f (\mathbf{p}_{in}) + rhs
\end{eqnarray}
\pagebreak

\begin{equation}
 rhs = f (\mathbf{q}),
 \end{equation}
\pagebreak

\begin{equation}
 f(\mathbf{q}) = \mathbf{p}_{out} - f(\mathbf{q}_{in}),
 \end{equation}
\pagebreak

$\mathbf{q}$
\pagebreak

$(ic_{1}, \cdots, ic_{n_{ic}})$
\pagebreak

$(oc_{1}, \cdots, oc_{n_{oc}})$
\pagebreak

$(iv_{1}, \cdots, iv_{n_{iv}})$
\pagebreak

$(ov_{1}, \cdots, ov_{n_{ov}})$
\pagebreak

$n_{ic} + n_{oc}$
\pagebreak

$n_{iv} + n_{ov}$
\pagebreak

\begin{eqnarray*}
    \mathbf{q}_{out} - g \left(\mathbf{q}_{in}\right)
    \ \ &\mbox{with}&
    \mathbf{q}_{out} = \left(q_{oc_{1}} \cdots q_{oc_{n_{oc}}}\right)^T,
    \ \ \ \mathbf{q}_{in} = (q_{ic_{1}} \cdots q_{ic_{n_{ic}}})^T
    \end{eqnarray*}
\pagebreak

\begin{align*}
    \mathbf{q}_{out} &- g \left(\mathbf{q}_{in}\right) = rhs \\
    & \mbox {if and only if}\\
    \mathbf{q}_{out} &= g \left(\mathbf{q}_{in}\right) + rhs \\
    \end{align*}
\pagebreak

\begin{equation*}
    \mathbf{q} + \mathbf{v} = \mathbf{q}.\exp (\mathbf{v})
    \end{equation*}
\pagebreak

$\mathbf{v}$
\pagebreak

$\dot{q}_i$
\pagebreak

$\exists j$
\pagebreak

$1 \leq j \leq n_{iv}$
\pagebreak

$i=iv_{j}$
\pagebreak

$1\leq j \leq n_{ov}$
\pagebreak

$i=ov_{j}$
\pagebreak

\begin{equation*}
    J = \left(\begin{array}{cccccccccccc}
    \cdots & ov_1 & \cdots & iv_{1} & \cdots & ov_2 & \cdots & iv_2 & \cdots &
ov_{n_{ov}} & \cdots \\
           &  1   &        &        &        &  0   &        &      &        &
&        \\
           &  0   &        &        &        &  1   &        &      &        &
&        \\
           &       &        &  -\frac{\partial g}{q_1} & & &   & -\frac{\partial
g}{q_2} \\
      &&&&&\\
           & 0    &        &       &         &  0   &        &      &        & 1
    \end{array}\right)
    \end{equation*}
\pagebreak

\begin{equation*}
    J = \left(\begin{array}{cccccccccccc}
    ov_1 \ ov_2 \ ov_3 & iv_1 \cdots  iv_{n_{iv}} \\
    J_{log}(R_{g}^T R_{out}) & -J_{log}(R_{g}^T R_{out})R_{out}^T R_{g}
\frac{\partial g}{\partial q_{in}} \end{array}\right) \end{equation*}
\pagebreak

$R_{out}$
\pagebreak

$(q_{oc1},q_{oc2},q_{oc3},q_{oc4})$
\pagebreak

$R_{g}$
\pagebreak

$J_{log}$
\pagebreak

$R$
\pagebreak

$\omega$
\pagebreak

\begin{equation*}
    R = \exp (\left[\omega\right]_{\times})
    \end{equation*}
\pagebreak

\begin{equation}
g \left((q_{ic_{1}} \cdots q_{ic_{n_{ic}}})^T\right) + rhs
\end{equation}
\pagebreak

\begin{eqnarray*}
J &=& \frac{\partial}{\partial\mathbf{q}_{in}}\left(g(\mathbf{q}_{in})
      + rhs\right).
\end{eqnarray*}
\pagebreak

$f(\mathbf{q}_{in})$
\pagebreak

\begin{equation}
       \mathbf{q}_{out} - g \left(\mathbf{q}_{in}\right),
       \end{equation}
\pagebreak

\begin{eqnarray*}
J &=& \frac{\partial}{\partial\mathbf{q}_{in}}
      \left(f(\mathbf{q}_{in}) + rhs\right).
\end{eqnarray*}
\pagebreak

\[
 \mathbf{q}_{out} = g(\mathbf{q}_{in})
\]
\pagebreak

$\mathbf{q}_{out}$
\pagebreak

${q}_{in}$
\pagebreak

$g$
\pagebreak

\[
 f(\mathbf{q}) = \mathbf{q}_{out} - g(\mathbf{q}_{in}).
 \]
\pagebreak

\[
 f(\mathbf{q}) = \log_{SE(3)}\left(g(\mathbf{q}_{in})^{-1}
                 \mathbf{q}_{out}\right)
 \]
\pagebreak

$g(\mathbf{q}_{in}$
\pagebreak

\begin{equation}
f \left(\mathbf{q}_{in}\right) + rhs_{expl}
\end{equation}
\pagebreak

$rhs_{expl}$
\pagebreak

\begin{equation}
rhs_{expl} = \log_{SE(3)}\left( F_{2/J_2} rhs_{impl} F_{2/J_2}^{-1}
             \right)
\end{equation}
\pagebreak

\begin{eqnarray*}
J &=& \frac{\partial}{\partial\mathbf{q}_{in}}\left(
      f(\mathbf{q}_{in}) + rhs\right).
\end{eqnarray*}
\pagebreak

\begin{eqnarray}
rhs_{expl} &=& \log_{SE(3)} \left(F_{2/J_2} F_{1/J_1}^{-1} J_1^{-1}
J_2\right)\\
rhs_{impl} &=& F_{1/J_1}^{-1} J_1^{-1}J_2 F_{2/J_2}
\end{eqnarray}
\pagebreak

\begin{equation}
rhs_{expl} = \log_{SE(3)}\left( F_{2/J_2}rhs_{impl}
F_{2/J_2}^{-1}\right)
\end{equation}
\pagebreak

\begin{equation}
rhs_{impl} = F_{2/J_2}^{-1} \exp_{SE(3)}(rhs_{expl}) F_{2/J_2}
\end{equation}
\pagebreak

$ f (q_0, ... , q_n) = f_{inner} (q_{left}) - f_{inner} (q_{right})
$
\pagebreak

$q_{left}$
\pagebreak

$q_{right}$
\pagebreak

\begin{equation}
f (q_1,\cdots,q_n) = g (q_{i},\cdots,q_{i+p-1})
\end{equation}
\pagebreak

$n$
\pagebreak

$[i,i+p-1]$
\pagebreak

$[1,n]$
\pagebreak

$\mathbf{R}^p$
\pagebreak

\begin{equation*}
f (\mathbf{q}) = \left(\begin{array}{c}
\mathbf{translation}\left(T_{1/J_1}^T T_1^T T_2 T_{2/J_2}\right)\\
\log \left((R_1 R_{1/J_1})^T R_2 R_{2/J_2}\right) \end{array}\right)
\end{equation*}
\pagebreak

\begin{equation*}
\left(\begin{array}{c}
(R_1 R_{1/J_1})^T \left(\left[R_2 t_{2/J_2} + t_2 - t_1\right]_{\times}
R_1 J_{1\,\omega} - \left[R_2 t_{2/J_2}\right]_{\times} R_2 J_{2\,\omega} +
R_2 J_{2\,\mathbf{v}} - R_1 J_{1\,\mathbf{v}}\right) \\
J_{log}\left((R_1 R_{1/J_1})^T R_2 R_{2/J_2}\right)(R_1 R_{1/J_1})^T
(R_2 J_{2\,\omega} - R_1 J_{1\,\omega})
\end{array}\right)
\end{equation*}
\pagebreak

$T_2(\mathbf{q})$
\pagebreak

$T_1(\mathbf{q})^{-1} T_2(\mathbf{q})$
\pagebreak

$h$
\pagebreak

$\mathbf{L}$
\pagebreak

$(n_q,n_v)$
\pagebreak

$\mathbf{c}$
\pagebreak

$n_v$
\pagebreak

$\mathbf{Equality}$
\pagebreak

$\mathbf{EqualToZero}$
\pagebreak

$\mathbf{Inferior}$
\pagebreak

$\mathbf{Superior}$
\pagebreak

$n_q$
\pagebreak

$e$
\pagebreak

\[
\Delta = h (\mathbf{q}) - rhs \in \mathbf{R}^{n_v},
\]
\pagebreak

$i\in\{0,\cdots,n_v-1\}$
\pagebreak

$c_i$
\pagebreak

$e_i = \max (0,\Delta_i)$
\pagebreak

$e_i = \min (0,\Delta_i)$
\pagebreak

$e_i = \Delta_i$
\pagebreak

$\lambda$
\pagebreak

\[
rhs = \exp\left(\begin{array}{c}\lambda_1 \\ 0 \\ 0 \\ \lambda_2 \\
\vdots \end{array}\right)
 \ \ \ \ \mathbf{c} = \left(\begin{array}{c}\mathbf{Equality} \\
\mathbf{EqualToZero} \\ \mathbf{Inferior} \\ \mathbf{Equality} \\
\vdots \end{array}\right)
\]
\pagebreak

$\mathbf{R}$
\pagebreak

\begin{equation}
h (\mathbf{q}) = q_{out} - value
\end{equation}
\pagebreak

$value$
\pagebreak

$\mathbf{R}^{nv}$
\pagebreak

$nv$
\pagebreak

\begin{equation}
q_{out} = value + rhs
\end{equation}
\pagebreak

$f_1$
\pagebreak

\[
f(\mathbf{q}) = \log\det(J_1 J_1^T)
\]
\pagebreak

$J_1$
\pagebreak

\[
f(\mathbf{q}) = \max \left(-\log\det(J_1 J_1^T), 0\right)
\]
\pagebreak

$\det(J_1 J_1^T) \geq 1$
\pagebreak

$\det(J_1 J_1^T) < 1$
\pagebreak

\begin{eqnarray*}
\mathbf{f}(\mathbf{q}) &=& R^T \left(\mathbf{x} - \mathbf{t}\right)
- \mathbf{x}^{*}\\
\mathbf{\dot{f}} &=& R^T \left(
J_{com} + [\mathbf{x}-\mathbf{t}]_{\times}J_{joint}^{\omega}
- J_{joint}^{\mathbf{v}}\right)\mathbf{\dot{q}}
\end{eqnarray*}
\pagebreak

$
    \left(\begin{array}{cc} R & \mathbf{t} \\ 0 & 1\end{array}\right)
    $
\pagebreak

$\mathbf{x}$
\pagebreak

$\mathbf{x}^{*}$
\pagebreak

$g_1 (\mathbf{q}) = 0, g_2 (\mathbf{q}) = 0, \cdots$
\pagebreak

\[
\textbf{q}_{res} = \left(I_n -
J^{+}J(\textbf{q}_{from})\right) (\textbf{v})
\]
\pagebreak

\[
\textbf{q}_{res} = \textbf{q}_{from} + \left(I_n -
J^{+}J(\textbf{q}_{from})\right) (\textbf{q}_{to} -
                                  \textbf{q}_{from})
\]
\pagebreak

$f (\mathbf{q}) = 0$
\pagebreak

$C^1$
\pagebreak

$\mathcal{L}$
\pagebreak

$\mathbf{q}_0$
\pagebreak

$\mathbf{q}_i$
\pagebreak

\begin{eqnarray*}
\mathbf{q}_{i+1} = \mathbf{q}_i -
   \alpha_i \frac{\partial f}{\partial \mathbf{q}}(\mathbf{q}_i)^{+}
   f (\mathbf{q}_i)
\end{eqnarray*}
\pagebreak

$\frac{\partial f}{\partial \mathbf{q}}(\mathbf{q}_i)^{+}$
\pagebreak

$\alpha_i$
\pagebreak

$\|f(\mathbf{q})\|$
\pagebreak

$f(\mathbf{q}) = 0$
\pagebreak

$f(\mathbf{q}) = rhs$
\pagebreak

$f(\mathbf{q}) > 0$
\pagebreak

$f(\mathbf{q}) < 0$
\pagebreak

$\alpha_i = 1$
\pagebreak

$ \alpha_{i+1} = \alpha - K \times (\alpha_{max} - \alpha_i) $
\pagebreak

$K$
\pagebreak

$\alpha_{max}$
\pagebreak

$ \alpha_{i} = C - K \times \text{tanh}(a
\frac{\|f(\mathbf{q}_i)\|}{\epsilon^2} + b) $
\pagebreak

$\|f(\mathbf{q}_i)\|<\epsilon$
\pagebreak

$\mathbf{q}_2 = f (\mathbf{q}_1)$
\pagebreak

$\mathbf{q}_2 - f (\mathbf{q}_1) = 0$
\pagebreak

$\mathbf{q}_1$
\pagebreak

$\mathbf{q}_2$
\pagebreak

$rhs = f(\mathbf{q})$
\pagebreak

$\mathbf{r}$
\pagebreak

$R=\exp [\mathbf{r}]_{\times}$
\pagebreak

$R=\exp \left[\mathbf{r}\right]_{\times}\in SO(3)$
\pagebreak

$SO(3)$
\pagebreak

$\mathbf{R}^3$
\pagebreak

\begin{equation*}
 \dot{\mathbf{r}} = J_{log}(R)\ \omega\,\,\,\mbox{with}\,\,\,
 \dot {R} = \left[\omega\right]_{\times} R
 \end{equation*}
\pagebreak

$\dot{R} = \left[\omega\right]_{\times}R$
\pagebreak

$\dot{R} = R\left[\omega\right]_{\times}$
\pagebreak

$\|r\|$
\pagebreak

$J_{log} (R)$
\pagebreak

\begin{align*}
 J_{log} (R) &=&
\frac{\|\mathbf{r}\|\sin\|\mathbf{r}\|}{2(1-\cos\|\mathbf{r}\|)} I_3 - \frac
{1}{2}\left[\mathbf{r}\right]_{\times} + (\frac{1}{\|\mathbf{r}\|^2} -
\frac{\sin\|\mathbf{r}\|}{2\|\mathbf{r}\|(1-\cos\|\mathbf{r}\|)})
\mathbf{r}\mathbf{r}^T\\
  &=& I_3 +\frac{1}{2}\left[\mathbf{r}\right]_{\times} +
\left(\frac{2(1-\cos\|\mathbf{r}\|) -
\|\mathbf{r}\|\sin\|\mathbf{r}\|}{2\|\mathbf{r}\|^2(1-\cos\|\mathbf{r}\|)}\right)\left[\mathbf{r}\right]_{\times}^2
 \end{align*}
\pagebreak

$(\mathbf{v},\mathbf{r})$
\pagebreak

\end{document}
