1\documentclass{article} 2 3%\usepackage[pdftex]{graphicx} 4\usepackage[T1]{fontenc} 5\usepackage{ae,aecompl} 6\usepackage{graphicx} 7\usepackage{psfrag} 8\usepackage{bm, amsmath, amsfonts, amssymb} 9%\usepackage[dvips,breaklinks=true,colorlinks=false]{hyperref} 10\usepackage{html} 11\usepackage{comment} 12 13\input{../../manual/stdmacro} 14%poor man's bold symbol 15\newcommand{\T}[1]{\bm{#1}} 16\newcommand{\TT}[1]{\bm{#1}} 17\newcommand{\TTT}[1]{\bm{#1}} 18\newcommand{\TTTT}[1]{\mathbb{#1}} 19\newcommand{\equu}{\overset{\text{uu}}{=}} 20\newcommand{\dof}{DOF} 21\newcommand{\dofs}{DOFs} 22 23% Custom 24\topmargin 0.31cm % 0.00cm 25\oddsidemargin 0.00cm 26\evensidemargin 0.00cm 27\marginparsep 0.00cm 28\textwidth 15.92cm 29\textheight 21.00cm % 23.62cm 30 31 32\begin{document} 33 34\title{Module IMU} 35\author{Pierangelo Masarati \\ \texttt{masarati@aero.polimi.it}} 36\date{\today} 37\maketitle 38 39 40\section{Introduction} 41\begin{subequations} 42\begin{align} 43 \TT{R} 44 &= 45 \TT{R}_n \tilde{\TT{R}}_h 46 \\ 47 \T{x} 48 &= 49 \T{x}_n 50 + 51 \TT{R}_n \tilde{\T{f}} 52\end{align} 53\end{subequations} 54 55\begin{align} 56 \T{f} 57 &= 58 \TT{R}_n \tilde{\T{f}} 59\end{align} 60 61\begin{align} 62 \overline{(\cdot)} 63 &= 64 \TT{R}^T (\cdot) 65\end{align} 66 67\section{Angular Velocity Measurement} 68\begin{align} 69 \T{\omega}\times{} 70 &= 71 \dot{\TT{R}}\TT{R}^T 72 = 73 \T{\omega}_n\times{} 74\end{align} 75 76\begin{align} 77 \overline{\T{\omega}} 78 &= 79 \TT{R}^T \T{\omega}_n 80\end{align} 81 82\section{Linear Acceleration Measurement} 83\begin{align} 84 \dot{\T{x}} 85 &= 86 \dot{\T{x}}_n 87 + 88 \T{\omega}_n \times \T{f} 89\end{align} 90 91\begin{align} 92 \overline{\dot{\T{x}}} 93 &= 94 \TT{R}^T \dot{\T{x}}_n 95 + 96 \plbr{\TT{R}^T \T{\omega}_n} \times \overline{\T{f}} 97 = 98 \overline{\T{v}} 99\end{align} 100 101\begin{align} 102 \ddot{\T{x}} 103 &= 104 \ddot{\T{x}}_n 105 + 106 \dot{\T{\omega}} \times \T{f} 107 + 108 \T{\omega}_n \times \T{\omega}_n \times \T{f} 109\end{align} 110 111\begin{align} 112 \overline{\ddot{\T{x}}} 113 &= 114 \TT{R}^T \ddot{\T{x}}_n 115 + 116 \plbr{\TT{R}^T \dot{\T{\omega}}} \times \overline{\T{f}} 117 + 118 \plbr{\TT{R}^T \T{\omega}} \times \plbr{\TT{R}^T \T{\omega}} \times \overline{\T{f}} 119\end{align} 120 121\begin{align} 122 \overline{\T{f}} 123 &= 124 \TT{R}^T \T{f} 125 = 126 \tilde{\TT{R}}_h^T \tilde{\T{f}} 127\end{align} 128 129\section{IMU Constraint} 130Constraint equations 131\begin{subequations} 132\begin{align} 133 \TT{R}^T \T{\omega}_n - \overline{\T{\omega}} 134 &= 135 \T{0} 136 \\ 137 \TT{R}^T \dot{\T{x}}_n 138 + 139 \plbr{\TT{R}^T \T{\omega}_n} \times \overline{\T{f}} 140 - 141 \overline{\T{v}} 142 &= 143 \T{0} 144 \\ 145 \dot{\overline{\T{v}}} 146 - 147 \overline{\T{a}} 148 &= 149 \T{0} 150\end{align} 151\end{subequations} 152 153Linearization of constraint equations 154\begin{subequations} 155\begin{align} 156 \TT{R}^T \plbr{ 157 \delta\T{\omega}_n 158 + 159 \T{\omega}_n \times \T{\theta}_{n\delta} 160 } 161 &\equu 162 \TT{R}^T \delta\dot{\T{g}} 163 \\ 164 \TT{R}^T \plbr{ 165 \delta\dot{\T{x}}_n 166 + 167 \dot{\T{x}}_n \times \T{\theta}_{n\delta} 168 } 169 + 170 \plbr{ 171 \TT{R}^T \plbr{ 172 \delta\T{\omega}_n 173 + 174 \T{\omega}_n \times \T{\theta}_{n\delta} 175 } 176 } \times \overline{\T{f}} 177 - 178 \delta\overline{\T{v}} 179 &\equu 180 \TT{R}^T \plbr{ 181 \delta\dot{\T{x}}_n 182 + 183 \dot{\T{x}}_n \times \delta\T{g}_n 184 - 185 \T{f} \times \delta\dot{\T{g}}_n 186 } 187 - 188 \delta\overline{\T{v}} 189 \\ 190 \delta\dot{\overline{\T{v}}} 191 &\equu 192 \delta\dot{\overline{\T{v}}} 193\end{align} 194\end{subequations} 195 196Forces and moments 197\begin{subequations} 198\begin{align} 199 \T{f}_n 200 &= 201 \TT{R} \T{\lambda}_{\T{v}} 202 \\ 203 \T{m}_n 204 &= 205 \TT{R} \T{\lambda}_{\T{\omega}} 206 + 207 \T{f} \times \TT{R} \T{\lambda}_{\T{v}} 208\end{align} 209\end{subequations} 210 211Linearization of forces and moments 212\begin{subequations} 213\begin{align} 214 \delta\T{f}_n 215 &= 216 \TT{R} \delta \T{\lambda}_{\T{v}} 217 - \plbr{\TT{R} \T{\lambda}_{\T{v}}} \times \T{\theta}_{n\delta} 218 \\ 219 \delta \T{m}_n 220 &= 221 \TT{R} \delta \T{\lambda}_{\T{\omega}} 222 + 223 \T{f} \times \TT{R} \delta \T{\lambda}_{\T{v}} 224 - 225 \plbr{ 226 \TT{R} \T{\lambda}_{\T{\omega}} 227 + 228 \T{f} \times \TT{R} \T{\lambda}_{\T{v}} 229 } \times \T{\theta}_{n\delta} 230\end{align} 231\end{subequations} 232 233$\T{\lambda}_{\T{v}}$, $\T{\lambda}_{\T{\omega}}$: Lagrange's multipliers (algebraic); 234$\overline{\T{v}}$: local velocity (differential). 235 236\begin{align} 237 \sqbr{\matr{cc|cc|c}{ 238 \TT{0} & -c\plbr{\TT{R} \T{\lambda}_{\T{v}}}\times{} 239 & \TT{R} & \TT{0} & \TT{0} 240 \\ 241 \TT{0} & -c\plbr{\TT{R} \T{\lambda}_{\T{\omega}} + \T{f}\times\TT{R} \T{\lambda}_{\T{v}}}\times{} 242 & \T{f}\times\TT{R} & \TT{R} & \TT{0} 243 \\ 244 \hline 245 \TT{R}^T & \TT{R}^T \plbr{\T{f}\times{}^T - c \dot{\T{x}}_n\times{}^T} 246 & \TT{0} & \TT{0} & -c \TT{I} 247 \\ 248 \TT{0} & \TT{R}^T & \TT{0} & \TT{0} & \TT{0} 249 \\ 250 \hline 251 \TT{0} & \TT{0} & \TT{0} & \TT{0} & \TT{I} 252 }} \cubr{\cvvect{ 253 \delta\dot{\T{x}}_n 254 \\ 255 \delta\dot{\T{g}}_n 256 \\ 257 \delta \T{\lambda}_{\T{v}} 258 \\ 259 \delta \T{\lambda}_{\T{\omega}} 260 \\ 261 \delta\overline{\T{v}} 262 }} 263\end{align} 264 265 266\end{document} 267