From b5cf5b55025f0622826afbb52c6db0282a58ee97 Mon Sep 17 00:00:00 2001 From: Stephanie Tsuei Date: Thu, 24 Oct 2019 09:29:04 -0700 Subject: [PATCH] fix derivation in section 5.1 --- doc/doc.tex | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/doc/doc.tex b/doc/doc.tex index 8e9b3c3c..f890f452 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -800,15 +800,15 @@ \subsection{Error state form of $X_s$} \begin{aligned} X_s &= g_{sb}(t_r) g_{bc} X_c \quad\text{(let } g_{sb}(t_r)=[R_r|T_r] \text{)}\\ &= R_r(R_{bc} X_c + T_{bc}) + T_r \\ -&= R_r(R_{bc} (\bar X_c + \tilde X_c) -+ T_{bc}) + T_r \quad\text{formally of } X_c = \bar X_c + \tilde X_c\\ -&= (\bar R_r + \asym{\tilde \omega_r})(\bar R_{bc} + \asym{\tilde \omega_{bc}})(\bar X_c + \tilde X_c) + (\bar R_r + \asym{\tilde \omega_r})(\bar T_{bc} + \tilde T_{bc}) + (\bar T_r + \tilde T_r) \\ +%&= R_r(R_{bc} (\bar X_c + \tilde X_c) +%+ T_{bc}) + T_r \quad\text{formally of } X_c = \bar X_c + \tilde X_c\\ +&= (\bar R_r + \bar R_r \asym{\tilde \omega_r})(\bar R_{bc} + \bar R_{bc}\asym{\tilde \omega_{bc}})(\bar X_c + \tilde X_c) + (\bar R_r + \bar R_r \asym{\tilde \omega_r})(\bar T_{bc} + \tilde T_{bc}) + (\bar T_r + \tilde T_r) \\ &= \bar R_r \bar R_{bc} \bar X_c + \bar R_r \bar R_{bc} \tilde X_c + -\bar R_r \asym{\tilde\omega_{bc}} \bar X_c + -\asym{\tilde \omega_r} \bar R_{bc} \bar X_c \quad\text{(1st term, drop higher-order terms)}\\ -&+ \bar R_r \bar T_{bc} + \bar R_r \tilde T_{bc} + \asym{\tilde\omega_r} \bar T_{bc} \quad\text{(2nd term, drop higher-order terms)}\\ -&+ \bar T_r + \tilde T_r. +\bar R_r \bar R_{bc} \asym{\tilde\omega_{bc}} \bar X_c + +\bar R_r \asym{\tilde \omega_r} \bar R_{bc} \bar X_c \quad\text{(1st term, drop higher-order terms)}\\ +&\quad + \bar R_r \bar T_{bc} + \bar R_r \tilde T_{bc} + \bar R_r \asym{\tilde\omega_r} \bar T_{bc} \quad\text{(2nd term, drop higher-order terms)}\\ +&\quad + \bar T_r + \tilde T_r. \end{aligned} \end{equation} @@ -816,7 +816,7 @@ \subsection{Error state form of $X_s$} \begin{equation} \begin{cases} \bar X_s &= \bar R_r \bar R_{bc} \bar X_c + \bar R_r \bar T_{bc} + \bar T_r\\ -\tilde X_s &= \bar R_r \bar R_{bc} \tilde X_c - \bar R_r \asym{\bar X_c} \tilde \omega_{bc} - \asym{\bar R_{bc}\bar X_c + \bar T_{bc}} \tilde \omega_r + \bar R_r \tilde T_{bc} + \tilde T_r. +\tilde X_s &= \bar R_r \bar R_{bc} \tilde X_c - \bar R_r \bar R_{bc} \asym{\bar X_c} \tilde \omega_{bc} - \bar R_r \asym{\bar R_{bc} \bar X_c + \bar T_{bc}} \tilde \omega_r + \bar R_r \tilde T_{bc} + \tilde T_r. \end{cases} \end{equation}