From 3de18d80ed360bbaa584a7b7f56994c77169b27b Mon Sep 17 00:00:00 2001 From: tylee Date: Mon, 16 Jul 2018 13:56:56 -0400 Subject: [PATCH 1/2] tylee edits --- tex/appendix/inertial_eoms_derivation.tex | 4 ++-- tex/appendix/polyhedron_algorithms.tex | 4 ++-- tex/appendix/relative_eoms_derivation.tex | 6 +++--- tex/appendix/se3_control_appendix.tex | 2 +- tex/conclusion.tex | 4 +++- tex/glossaries_abbreviations.tex | 10 +++++----- tex/reachability_orbit_transfer.tex | 8 ++++---- tex/se3_control.tex | 12 ++++++------ tex/shape_reconstruction.tex | 9 +++++---- tex/system_model.tex | 10 +++++----- 10 files changed, 36 insertions(+), 33 deletions(-) diff --git a/tex/appendix/inertial_eoms_derivation.tex b/tex/appendix/inertial_eoms_derivation.tex index f7d7025..4c60d65 100644 --- a/tex/appendix/inertial_eoms_derivation.tex +++ b/tex/appendix/inertial_eoms_derivation.tex @@ -9,7 +9,7 @@ \chapter{Derivation of Inertial Equations of Motion}\label{proof:inertial_dumbbe where \( T(q, \dot{q} \) is the kinetic energy of the system while \( U(q) \) is the potential energy of the system. The derivation is simplified to the case of two spherical masses, but can be extended to any arbitrary configuration of masses. -The kinetic and potential energy of the dumbbell model is dupblicated from~\cref{eq:dumbbell_kinetic_and_potential_energy} +The kinetic and potential energy of the dumbbell model is duplicated from~\cref{eq:dumbbell_kinetic_and_potential_energy} \begin{align*} T &= \frac{1}{2} m \norm{\dot{\vb{x}}}^2 + \frac{1}{2} \tr{\vh{\Omega} J_d \vh{\Omega}^T} , \\ V( \vecbf{x}, R ) &= - m_1 U \parenth{R_A^T \parenth{\vecbf{x} + R \vecbf{\rho}_1}} - m_2 U \parenth{R_A^T \parenth{\vecbf{x} + R \vecbf{\rho}_2}} . @@ -126,7 +126,7 @@ \section{Hamilton's Principle}\label{sec:inertial_hamiltons_principle} \end{align*} where \( u_f \in \R^3 \) is an applied force that the system can generate, e.g.\ thrusters. -The attitude dynamics are found by applyging~\cref{eq:trPQ} which gives +The attitude dynamics are found by applying~\cref{eq:trPQ} which gives \begin{align} -\parenth{J \dot{\iangvel}}^\wedge - \parenth{ \hat{\iangvel} J \iangvel}^\wedge + \sum_{i=1}^{2} \parenth{m_i \apos_i \deriv{U}{\apos_i}^T \aatt^T \iatt }^T - \parenth{m_i \apos_i \deriv{U}{\apos_i}^T \aatt^T \iatt } = 0. \end{align} diff --git a/tex/appendix/polyhedron_algorithms.tex b/tex/appendix/polyhedron_algorithms.tex index ddd6fb7..dfc2171 100644 --- a/tex/appendix/polyhedron_algorithms.tex +++ b/tex/appendix/polyhedron_algorithms.tex @@ -6,7 +6,7 @@ \chapter{Volume of a Polyhedron}\label{sec:polyhedron_volume} The full derivation can be found in~\textcite{newson1899,orourke1998}. A polyhedron can be decomposed into a collection of tetrahedron by combining the vertices of each face with that of the origin. Without loss of generality, assume that each face of the polyhedron is a triangle, if not there are readily available algorithms to triangulate a surface mesh~\cite{berg2008}. -Assume that the vertices of a face are defined by \( \vc{v}_0, \vc{v}_2, \vc{v}_3 \) and numbered in a counterclock wise fashion such that the normal to the face is outward facing. +Assume that the vertices of a face are defined by \( \vc{v}_0, \vc{v}_2, \vc{v}_3 \) and numbered in a counterclockwise fashion such that the normal to the face is outward facing. A tetrahedron is created by combining the vertices of the face with any arbitrary point, such as the origin of the polyhedron reference frame. Then the volume of the tetrahedron is given by~\cref{eq:volume_tetrahedron}. \begin{align}\label{eq:volume_tetrahedron} @@ -18,7 +18,7 @@ \chapter{Volume of a Polyhedron}\label{sec:polyhedron_volume} -1 & 0 & 0 & 1 \end{vmatrix} \end{align} -The total volume of the polyhdron can be computed by summing the contributions of each face as shown in~\cref{eq:volume_polyhedron}. +The total volume of the polyhedron can be computed by summing the contributions of each face as shown in~\cref{eq:volume_polyhedron}. \begin{align}\label{eq:volume_polyhedron} V_T = \frac{1}{6} \sum_{f} V_f \end{align} diff --git a/tex/appendix/relative_eoms_derivation.tex b/tex/appendix/relative_eoms_derivation.tex index 33a993a..05b45f1 100644 --- a/tex/appendix/relative_eoms_derivation.tex +++ b/tex/appendix/relative_eoms_derivation.tex @@ -20,7 +20,7 @@ \section{Variations of the Relative Variables}\label{sec:relative_variations} In~\cref{proof:inertial_dumbbell_eoms} the variations of the inertial configuration variables were derived. Following a similar process the variations of the reduced variables must be derived such that they also satisfy the variations of the inertial variables. These variations must be defined precisely to ensure that the configuration space is preserved properly. -These variations will be critical for the following step of applying the Euler-Lagrage equations and deriving the equations of motion. +These variations will be critical for the following step of applying the Euler-Lagrange equations and deriving the equations of motion. The variations of the reduced variables are obtained from the variations of the original variables given in~\cref{eq:inertial_pos_variation,eq:inertial_vel_variation,eq:inertial_att_variation,eq:inertial_ang_vel_variation} and the definition of the reduced variables given in~\cref{eq:relative_position,eq:relative_velocity,eq:relative_attitude,eq:relative_angular_velocity}. \paragraph{Relative Position}\label{sec:relative_position_variation} @@ -46,7 +46,7 @@ \section{Variations of the Relative Variables}\label{sec:relative_variations} \begin{align}\label{eq:relative_attitude_variation_inverse} \gls{sym:iattvar} &= \gls{sym:iatt}[^T] \gls{sym:aatt} \gls{sym:rattvar} \gls{sym:aatt}[^T] \gls{sym:iatt}. \end{align} -Using this, the varation of the relative attitude is derived by substituting~\cref{eq:relative_attitude_variation_inverse} into~\cref{eq:relative_attitude_variation_1} as +Using this, the variation of the relative attitude is derived by substituting~\cref{eq:relative_attitude_variation_inverse} into~\cref{eq:relative_attitude_variation_1} as \begin{align}\label{eq:relative_attitude_matrix_variation} \delta \gls{sym:ratt} = \gls{sym:aatt}[^T] \gls{sym:iatt} \gls{sym:iattvar} = \gls{sym:rattvar} \gls{sym:aatt}[^T] \gls{sym:iatt} = \gls{sym:rattvar} \gls{sym:ratt} \end{align} @@ -59,7 +59,7 @@ \section{Variations of the Relative Variables}\label{sec:relative_variations} &= \delta \gls{sym:ratt} \hat{\gls{sym:iangvel}} \gls{sym:ratt}[^T] + \gls{sym:ratt} \delta \hat{\gls{sym:iangvel}} \gls{sym:ratt}[^T] + \gls{sym:ratt} \hat{\gls{sym:iangvel}} \delta \gls{sym:ratt}[^T], \end{align} where we used~\cref{eq:hatRxR} to arrive at the final form. -The variation \( \delta \gls{sym:iangvel} \) is defined in~\cref{eq:inertial_ang_vel_variation} and subsitituded into~\cref{eq:relative_angular_velocity_variation_1} to give +The variation \( \delta \gls{sym:iangvel} \) is defined in~\cref{eq:inertial_ang_vel_variation} and substituted into~\cref{eq:relative_angular_velocity_variation_1} to give \begin{align}\label{eq:relative_angular_velocity_variation_2} \parenth{\delta \gls{sym:rangvel}}^\wedge &= \delta \gls{sym:ratt} \hat{\gls{sym:iangvel}} \gls{sym:ratt}[^T] + \gls{sym:ratt} \delta \hat{\gls{sym:iangvel}} \gls{sym:ratt}[^T] + \gls{sym:ratt}\hat{\gls{sym:iangvel}}\delta \gls{sym:ratt}[^T], \nonumber \\ &= \gls{sym:rattvar} \gls{sym:ratt} \hat{\gls{sym:iangvel}} \gls{sym:ratt}[^T] + \gls{sym:ratt} \parenth{ - \gls{sym:iattvar} \hat{\gls{sym:iangvel}} + \hat{\gls{sym:iangvel}}\gls{sym:iattvar} + \dot{\gls{sym:iattvar}} } \gls{sym:ratt}[^T] + \gls{sym:ratt} \hat{\gls{sym:iangvel}} \gls{sym:ratt}[^T] \gls{sym:rattvar}[^T], \nonumber \\ diff --git a/tex/appendix/se3_control_appendix.tex b/tex/appendix/se3_control_appendix.tex index c00d0c2..4846004 100644 --- a/tex/appendix/se3_control_appendix.tex +++ b/tex/appendix/se3_control_appendix.tex @@ -116,7 +116,7 @@ \subsection{Proof of~\Cref{item:prop_A_critical_points}}\label{proof:prop_A_crit \end{enumerate} These conditions can be determined by noticing that~\cref{item:condition_1} is the trivial solution. For~\cref{item:condition_2}, consider the situation that \( q_{12} \neq 0 \) then from the fact that \( g_i \) are distinct yields the fact that \( \parenth{1 - \frac{g_1^2}{g_2^2} } \neq 0 \) and \( \parenth{1 - \frac{g_1^2}{g_3^2} } \neq 0 \) which means that \( q_{13}^2 \neq 0 \) since the terms must add to zero. -Similar arguments can be made for the other relations in~\cref{eq:era_critical_point_conditions} which means that the solutions must satisify either~\cref{item:condition_1} or~\cref{item:condition_2}. +Similar arguments can be made for the other relations in~\cref{eq:era_critical_point_conditions} which means that the solutions must satisfy either~\cref{item:condition_1} or~\cref{item:condition_2}. Now consider~\cref{item:condition_2} and assume that \( g_2 > g_1 \) and \( q_{12}, q_{13}, q_{23} \neq 0 \). From~\cref{eq:era_condition_1} this implies that \( \parenth{1 - \frac{g_1^2}{g_2^2} } q_{12}^2 > 0\) since \( q_{12}^2 > 0\). diff --git a/tex/conclusion.tex b/tex/conclusion.tex index 2270d20..c69e844 100644 --- a/tex/conclusion.tex +++ b/tex/conclusion.tex @@ -1,6 +1,8 @@ % !TEX root = ../dissertation.tex -\chapter{Conclusion}\label{sec:conclusion} +\chapter{Conclusions}\label{sec:conclusion} + +\section{Summary of Contributions} This dissertation combined concepts from the fields of astrodynamics, geometric mechanics and control, and computational geometry. The results provide a wide ranging solution for the operation of a spacecraft around a small body. diff --git a/tex/glossaries_abbreviations.tex b/tex/glossaries_abbreviations.tex index 0066437..e2a1e97 100644 --- a/tex/glossaries_abbreviations.tex +++ b/tex/glossaries_abbreviations.tex @@ -5,7 +5,7 @@ \newglossaryentry{filo}{ name={FILO}, type=\glsxtrabbrvtype, - description={first in last out}, + description={First In Last Out}, nonumberlist=true, first={first in last out (FILO)} } @@ -13,7 +13,7 @@ \newglossaryentry{tof}{ name={TOF}, type=\glsxtrabbrvtype, - description={time of flight}, + description={Time of Flight}, nonumberlist=true, first={time of flight (TOF)} } @@ -121,7 +121,7 @@ \newglossaryentry{uav}{ name={UAV}, type=\glsxtrabbrvtype, - description={unmanned aerial vehicle}, + description={Unmanned Aerial Vehicle}, first={\glsentrydesc{uav} (\glsentrytext{uav})}, plural={UAVs}, descriptionplural={unmanned aerial vehicles}, @@ -132,7 +132,7 @@ \newglossaryentry{rwa}{ name={RWA}, type=\glsxtrabbrvtype, - description={reaction wheel assembly}, + description={Reaction Wheel Assembly}, first={\glsentrydesc{rwa} (\glsentrytext{rwa})}, plural={RWAs}, descriptionplural={reaction wheel assemblies}, @@ -143,7 +143,7 @@ \newglossaryentry{cmg}{ name={CMG}, type=\glsxtrabbrvtype, - description={control moment gyroscope}, + description={Control Moment Gyroscope}, first={\glsentrydesc{cmg} (\glsentrytext{cmg})}, plural={CMGs}, descriptionplural={control moment gyroscopes}, diff --git a/tex/reachability_orbit_transfer.tex b/tex/reachability_orbit_transfer.tex index 901210c..bcaf875 100644 --- a/tex/reachability_orbit_transfer.tex +++ b/tex/reachability_orbit_transfer.tex @@ -380,8 +380,8 @@ \section{Numerical Examples of Transfers using Reachability Sets}\label{sec:simu The second example is a transfer from geostationary orbit of the Earth to a period orbit of \( L_1 \). This examples demonstrates the ability to extend the reachability process to multiple iterations, to allow for a much larger and more general transfer. Utilizing both examples it is possible to depart from the vicinity of the Earth to a Moon orbit via a series of reachable sets defined on \Poincare sections. -Finally, the results are extended from the planar problem to three dimensions with a simulation about asteroid 4769 Castlia. -This numerical example demonstrates the extension of the method to the more complicated dynamic enviornment around an asteroid. +Finally, the results are extended from the planar problem to three dimensions with a simulation about asteroid 4769 Castalia. +This numerical example demonstrates the extension of the method to the more complicated dynamic environment around an asteroid. \subsection{Three Body Examples} @@ -481,7 +481,7 @@ \subsection{Three Body Examples} Increases in \(t_f\) are less critical and have minimal impact on the distribution of the reachability set on the \Poincare section.~\label{fig:varying_tf_reachability_sets}} \end{figure} -\Cref{fig:varying_tf_reachability_sets} shows the reachabilty set for twelve combinations of \(t_f\) and \( u_m\) listed in~\cref{tab:varying_tf}. +\Cref{fig:varying_tf_reachability_sets} shows the reachability set for twelve combinations of \(t_f\) and \( u_m\) listed in~\cref{tab:varying_tf}. With a small maximum control bound, \( u_m = \num{0.05} \) shown using square markers, the reachable set is not dramatically changed from that of the no control solution. The reachable set is denoted using square markers on the left most portion of \cref{fig:varying_tf_poincare}. Variations of \( t_f\) are indicated using different colors and also demonstrate that this parameters has a smaller impact than changes in \( u_m \). @@ -844,7 +844,7 @@ \subsection{4769 Castalia Example}\label{sec:castalia_transfer} This amount of thrust is typical of many current ion or hall effect thrusters~\cite{goebel2008,choueiri2009}. The objective is to transfer the spacecraft between two periodic equatorial orbits about Castalia. -The initial and target orbits are periodic solutions about Castlia computed using the method introduced by Reference~\cite{scheeres2003}. +The initial and target orbits are periodic solutions about Castalia computed using the method introduced by Reference~\cite{scheeres2003}. The initial conditions for both orbits are defined in the body-fixed frame as \begin{align}\label{sec:initial_transfer} \vc{x}_i &= diff --git a/tex/se3_control.tex b/tex/se3_control.tex index f2e891f..4ad0ea2 100644 --- a/tex/se3_control.tex +++ b/tex/se3_control.tex @@ -14,17 +14,17 @@ \chapter{Geometric Control for Spacecraft}\label{sec:se3_control} Furthermore, the coupled geometric controller explicitly considers the attitude coupling of the body in contrast to many of the previous approaches. This chapter considers the controlled motion of a rigid dumbbell spacecraft about a small body. -We assume that the desired trajectory, \( R_d(t), \vc{x}_d(t) \), are defined as the rotation matrix of the spacecraft body frame with respect to the inertial frame and the relative position of the spacecraft center of mass with respect to the asteorid and defined in the inertial frame, respectively. +We assume that the desired trajectory, \( R_d(t), \vc{x}_d(t) \), are defined as the rotation matrix of the spacecraft body frame with respect to the inertial frame and the relative position of the spacecraft center of mass with respect to the asteroid and defined in the inertial frame, respectively. In contrast to controllers derived for quadrotor \gls{uav}, the attitude and translational motion are only lightly coupled in the spacecraft scenario. From the inertial equations of motion given in~\cref{eq:inertial_position_dynamics,eq:inertial_velocity_dynamics,eq:inertial_attitude_dynamics,eq:inertial_angvel_dynamics}, the coupling between the translational and rotational dynamics is due solely to the gravitational moment on the spacecraft. Furthermore, we assume that we have a fully actuated spacecraft such that we can apply a torque about all three rotational axes and a force in all three directions. This is in contrast to \glspl{uav} where the system is underactuated and in order to produce certain translational forces the system must first rotate. -This is a relatively standard assumption in the astrodynamics community as most spacecraft contain seperate systems for attitude control, e.g.\ \glspl{rwa}, \glspl{cmg}, or cold gas thrusters, and translational control, e.g.\ cold-gas thrusters, electric propulsion, or large chemical rockets~\cite{hughes2004,wertz1978}. +This is a relatively standard assumption in the astrodynamics community as most spacecraft contain separate systems for attitude control, e.g.\ \glspl{rwa}, \glspl{cmg}, or cold gas thrusters, and translational control, e.g.\ cold-gas thrusters, electric propulsion, or large chemical rockets~\cite{hughes2004,wertz1978}. However, there are also many examples of underactuated spacecraft, such as those with damaged components~\cite{petersen2015a} or cubesats with limited cost and/or size budgets. In these situations, there are a variety of methods to handle the underactuation, ranging from optimal control techniques, exploiting external forces, or utilizing multiple control loops. This chapter presents two controllers for the attitude motion and one controller for the translational motion. -These controllers are computed independently and used to manuever the spacecraft around the small body. +These controllers are computed independently and used to maneuver the spacecraft around the small body. The controllers presented in this section and the low thrust orbital transfer scheme of~\cref{sec:lowthrust_transfers} provide a complete solution for the controlled motion of a spacecraft around a small body. The reachability based transfers of~\cref{sec:lowthrust_transfers} are ideal for arrival, departure, or other large scale orbital transfers around a small body. The controllers presented in this chapter are best suited for precise trajectory tracking, such as the shape reconstruction and landing guidance presented in~\cref{sec:shape_reconstruction}. @@ -54,7 +54,7 @@ \subsection{Problem Formulation}\label{sec:control_problem_formulation} In this section, we consider the attitude dynamics of a rigid body, and more specifically those of a dumbbell spacecraft around a small body. We consider the inertial attitude dynamics given by~\cref{eq:inertial_attitude_dynamics,eq:inertial_angvel_dynamics} and repeated here as \begin{align*} - \dot{\iatt} &= \iatt \iangvel, \\ + \dot{\iatt} &= \iatt \hat\iangvel, \\ J \dot{\iangvel} + \hat{\iangvel} J \iangvel &= M_{ext} + u_m, \end{align*} where we combined the gravitational moment into a single term \( M_{ext} \in \R^3 \) which defines the total external moment on the spacecraft. @@ -90,7 +90,7 @@ \subsection{Problem Formulation}\label{sec:control_problem_formulation} \end{subequations} Then the following properties hold: \begin{enumerate} - \item \label{item:prop_A_psd} \( A \) is locally positive definite about \( R = R_d \) on \( \SO \). + \item \label{item:prop_A_psd} \( A \) is positive definite about \( R = R_d \) on \( \SO \). \item \label{item:prop_eRA} The variation of \( A \) with respect to a variation of \( \delta R = R \hat{\eta} \) for \( \eta \in \R^3 \) is given by \begin{align}\label{eq:dirDiff_A} \dirDiff{A}{R} &= \eta \cdot e_{R} , @@ -702,7 +702,7 @@ \section{Summary} In addition, a new adaptive attitude control for state inequality constraints is developed. This enables the accurate stabilization and tracking of a desired attitude while avoiding obstacles. -The control approaches presented in this chapter, when combined with the results of~\cref{sec:lowthrust_transfers}, provide a complete approach for the manuevering of a spacecraft around a small body. +The control approaches presented in this chapter, when combined with the results of~\cref{sec:lowthrust_transfers}, provide a complete approach for the maneuvering of a spacecraft around a small body. The low thrust propulsion scheme of~\cref{sec:lowthrust_transfers} is ideal for large magnitude orbital changes, while the results of this chapter provide precise trajectory tracking ideal for low altitude operations. These nonlinear controllers are ideally suited for low altitude operations. The results are utilized in the subsequent chapters to explore an asteroid and land on the surface. diff --git a/tex/shape_reconstruction.tex b/tex/shape_reconstruction.tex index 8b99e41..4f5234d 100644 --- a/tex/shape_reconstruction.tex +++ b/tex/shape_reconstruction.tex @@ -118,7 +118,8 @@ \section{Range Measurement Model} C = \vc{c}_1 - H \vc{c}_3 + W \vc{c}_2, \\ D = \vc{c}_1 - H \vc{c}_3 - W \vc{c}_2. \end{align*} -The view frustrum is visualized in~\cref{fig:view_frustrum} and allows for any vector in the field of view to befined as a linear combination of the extents of the far plane. +The view frustum is visualized in~\cref{fig:view_frustrum} and allows for any vector in the field of view to befined as a linear combination of the extents of the far plane. +% tylee: not sure what "befined" means? \section{Incremental Shape Reconstruction}\label{sec:radius_update} @@ -134,7 +135,7 @@ \section{Incremental Shape Reconstruction}\label{sec:radius_update} We assume that upon arrival at a target body, the spacecraft contains an initial estimate for the shape of the small body. This shape can be a coarse estimate computed from ground measurements or it can be a triaxial ellipsoid based on the semimajor axes of the asteroid, such as that shown in~\cref{fig:uniform_mesh} which represents the maximum axes for asteroid Castalia. Additionally, we assume that the shape estimate is closed and a triangular faceted surface mesh, emulating those used in practice to represent asteroids. -Furthermore, the number of vertices in the estimate can be scaled according to the desired final acccuracy or computational capabilities. +Furthermore, the number of vertices in the estimate can be scaled according to the desired final accuracy or computational capabilities. For example,~\cref{fig:uniform_mesh} demonstrates two surface mesh representation for a triaxial ellipsoid at varying levels of detail. A wide variety of algorithms are available to generate a near uniformly spaced mesh for an arbitrary surface~\cite{persson2004,boissonnat2005}. @@ -213,7 +214,7 @@ \section{Incremental Shape Reconstruction}\label{sec:radius_update} \begin{align}\label{eq:region_of_interest} \Delta \sigma_{max} = \sqrt \frac{\Delta S}{r_b^2} \end{align} -where \( r_b \) defines the Brillouin sphere radius, or the radius of the circumscribing sphere of the asteroid. +where \( r_b \) defines the Bernoulli sphere radius, or the radius of the circumscribing sphere of the asteroid. Only vertices which satisfy \( \Delta \sigma_i \leq \Delta \sigma_{max} \) are considered in the Bayesian update shown in~\cref{eq:posterior_probability}. The approach presented in this section allows one to update the shape of small body given a single range measurement of the surface. @@ -280,7 +281,7 @@ \section{Optimal Guidance for Shape Reconstruction}\label{sec:explore_asteroid} where \( \theta : \bracket{0, \frac{\rpos \cdot \vc{v}_i}{\norm{\rpos}\norm{\vc{v}_i}}} \to \R^1\) parameterizes the desired trajectory. \Cref{eq:spherical_waypoint} simply describes a portion of a great circle trajectory between the current state, \( \rpos \), and the desired vertex \( \vc{v}_i \)~\cite{chen2016}. The altitude of the spacecraft, \( r_d \in \R \), can be chosen based on sensor characteristics of safety concerns. -For example, \( r_d \) can be chosen as the distance of the Biroullin sphere with an additional safety margin to mitigate any surface collision~\cite{scheeres2012a}. +For example, \( r_d \) can be chosen as the distance of the Bernoulli sphere with an additional safety margin to mitigate any surface collision~\cite{scheeres2012a}. The translational controller presented in~\cref{eq:translational_control} is used to determine the control input to follow \( x_d\). We assume that the tracking errors are small, such that \( e_x, e_v \) are negligible, therefore the control becomes \begin{align}\label{eq:tracking_control_cost} diff --git a/tex/system_model.tex b/tex/system_model.tex index 6d706b1..1b5261b 100644 --- a/tex/system_model.tex +++ b/tex/system_model.tex @@ -86,7 +86,7 @@ \subsection{Reference Frames}\label{ssec:dumbbell_eoms_reference_frames} As a result, the configuration space for the general motion of a rigid body is defined by at a minimum of three coordinates to represent translation and three coordinates to represent the orientation. This configuration space is the semi-direct product, \( \gls{sym:special_euclidean_group} = \R^3 \otimes \gls{sym:special_orthogonal_group} \), or the special euclidean group. The special euclidean group is the group of all possible rigid body motions and defines the six degrees of freedom possible by our system. -An element of the special euclidean group can be expressed using the homogenous representation as +An element of the special euclidean group can be expressed using the homogeneous representation as \begin{align*} \begin{bmatrix} R & \vc{x} \\ @@ -321,7 +321,7 @@ \subsection{Inertial Frame Equations of Motion}\label{sec:inertial_dumbbell_eoms \paragraph{Inertial Equations of Motion: Hamiltonian Form}\label{sec:inertial_hamiltonian_form} -Hamilton's equations allows for the representation of the second order equations derived using the Euler-Lagrange equation as a system of \( 2n \) first order equations called a hamiltonian system of equations or canonical equations~\cite{arnold1989}. +Hamilton's equations allows for the representation of the second order equations derived using the Euler-Lagrange equation as a system of \( 2n \) first order equations called a Hamiltonian system of equations or canonical equations~\cite{arnold1989}. Hamilton's equations are derivable directly from the Lagrangian through the use of the Legendre transformation which is a mapping \( \left( q, \dot{q},t\right) \rightarrow \left(q, p, t \right) \) where \( p_i\) is the generalized momenta, \begin{align}\label{eq:legendre_transform} p_i = \deriv{L}{\dot{q}}. @@ -532,7 +532,7 @@ \section{Gravitational Models around Small Bodies}\label{sec:gravitational_model \begin{align}\label{eq:volume_integral_potential} \vc{F}_g ( r) = \nabla U = G \int_{\mathcal{B}} \frac{1}{\norm{\vc{r} - \vc{\rho} }} dm (\vc{\rho}), \end{align} -where \( \vc{r} \) is the position of the field point, \( \vc{\rho} \) is the position of the differential mass element of the body and \( \nabla U \) is defined as the gradient of \( U \) with respect to the standard cartesian basis vectors +where \( \vc{r} \) is the position of the field point, \( \vc{\rho} \) is the position of the differential mass element of the body and \( \nabla U \) is defined as the gradient of \( U \) with respect to the standard Cartesian basis vectors \begin{align*} \nabla = \deriv{}{x} \vc{i} + \deriv{}{y} \vc{j} + \deriv{}{z} \vc{k}. \end{align*} @@ -559,7 +559,7 @@ \section{Gravitational Models around Small Bodies}\label{sec:gravitational_model \end{align} where \( \gls{sym:sigma} \) is the local density of the body. Laplace's equation can be solved using the separation of variables in terms of spherical coordinates~\cite{scheeres2012a}. -The transformation between cartesian, \( \vc{r} = \begin{bmatrix} x & y & z \end{bmatrix}\), and spherical coordinates is given by +The transformation between Cartesian, \( \vc{r} = \begin{bmatrix} x & y & z \end{bmatrix}\), and spherical coordinates is given by \begin{subequations} \begin{align*} r &= \sqrt{x^2 + y^2 + z^2}, \\ @@ -657,7 +657,7 @@ \subsection{Polyhedron Potential Model}\label{sec:polyhedron_potential} Furthermore, the accuracy of the model is solely dependent on the fidelity of the shape in representing the true body and the constant density assumption. For bodies with varying density, it is possible to augment the polyhedron potential model to account for the density variations~\cite{scheeres2000a}. This approach alleviates the divergence issue of the spherical harmonic model and is applicable for all points on or above the surface and is ideal for general operations around small bodies. -The polyhedron potential model has become the defacto standard for small body missions and has been used operationally. +The polyhedron potential model has become the de facto standard for small body missions and has been used operationally. In this section we summarize the polyhedron potential model based on the derivations of Werner. We additionally present some of the implementation details in our model and the adaptations for use with the \texttt{Surface\_mesh} data structure presented in~\cref{sec:polyhedron_data_structures}. From 246c053cb7228fd4937e906253e4d733e2f41c3d Mon Sep 17 00:00:00 2001 From: tylee Date: Mon, 16 Jul 2018 13:57:37 -0400 Subject: [PATCH 2/2] tylee edits --- tex/appendix/polyhedron_algorithms.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/tex/appendix/polyhedron_algorithms.tex b/tex/appendix/polyhedron_algorithms.tex index dfc2171..6c15a9e 100644 --- a/tex/appendix/polyhedron_algorithms.tex +++ b/tex/appendix/polyhedron_algorithms.tex @@ -16,9 +16,9 @@ \chapter{Volume of a Polyhedron}\label{sec:polyhedron_volume} x_1 & y_2 & z_2 & 1 \\ x_2 & y_3 & z_3 & 1 \\ -1 & 0 & 0 & 1 - \end{vmatrix} + \end{vmatrix}. \end{align} The total volume of the polyhedron can be computed by summing the contributions of each face as shown in~\cref{eq:volume_polyhedron}. \begin{align}\label{eq:volume_polyhedron} - V_T = \frac{1}{6} \sum_{f} V_f + V_T = \frac{1}{6} \sum_{f} V_f. \end{align}