From d00b977581fec05ad1410a7e564d28e9de285b68 Mon Sep 17 00:00:00 2001 From: Connor Date: Tue, 15 Mar 2022 20:34:31 -0600 Subject: [PATCH] Unless Bosanac has a last minute change, the paper is done! --- LaTeX/approach.tex | 20 +++++--------- LaTeX/fig/EMJS_thrust_components.png | Bin 40543 -> 44417 bytes LaTeX/fig/EMS_thrust_components.png | Bin 38940 -> 43045 bytes LaTeX/results.tex | 2 +- LaTeX/trajectory_design.tex | 38 +++++++++++---------------- LaTeX/trajectory_optimization.tex | 12 +++------ 6 files changed, 26 insertions(+), 46 deletions(-) diff --git a/LaTeX/approach.tex b/LaTeX/approach.tex index 178ac86..f29b0b0 100644 --- a/LaTeX/approach.tex +++ b/LaTeX/approach.tex @@ -264,7 +264,7 @@ \item The $v_{\infty,in}$ vector representing excess velocity at the planetary flyby (or completion of mission) at the end of the phase \item The time of flight for the phase - \item The unit-thrust profile in a sun-fixed frame represented by a + \item The unit-thrust profile in a sun-centered frame represented by a series of vectors with each element ranging from 0 to 1. \end{itemize} \end{itemize} @@ -397,18 +397,12 @@ non-powered flyby. From these two velocity vectors the turning angle, and thus the periapsis of the flyby, - can then be calculated by Equation~\ref{turning_angle_eq} and the following equation: - - \begin{equation} - r_p = \frac{\mu}{\vec{v}_{\infty,in} \cdot \vec{v}_{\infty,out}} \cdot \left( - \frac{1}{\sin(\delta/2)} - 1 \right) - \end{equation} - - If this radius of periapse is then found to be less than the minimum safe radius - (currently set to the radius of the planet plus 100 kilometers), then the process is - repeated with new random flyby velocities until a valid seed flyby is found. These - checks are also performed each time a mission is perturbed or generated by the NLP - solver. + can then be calculated by Equation~\ref{turning_angle_eq} and + Equation~\ref{periapsis_eq}. If this radius of periapse is then found to be less than + the minimum safe radius (currently set to the radius of the planet plus 100 kilometers), + then the process is repeated with new random flyby velocities until a valid seed flyby + is found. These checks are also performed each time a mission is perturbed or generated + by the NLP solver. The final requirement then, is the thrust controls, which are actually quite simple. Since the thrust is defined as a 3-vector of values between -1 and 1 representing some diff --git a/LaTeX/fig/EMJS_thrust_components.png b/LaTeX/fig/EMJS_thrust_components.png index 0fd0a94c7d11b80551f3447873bd1812d1fec495..b4868f1f19d45fd6dfd30b501f4f9db529fdd683 100644 GIT binary patch literal 44417 zcmeFZby$_%);_xEZlsY^y1PXhX=xCoyFognLj(jwT4_YOrMpBxy1P@lIdl2Gdw+Xh z=XcKezU!Rd`D-sPT+3%Y&wQRa=9u@m$2|r{s3^%`ppu|MAP@{WSxGer1P=Vb#3Cbt zBRR>caS#ZuzL&bTtD2EJrK7Whxs|OMrK_i-8Ks$rl{p0BF_W93<4oJ&EcsB5$^vsz zM)uBp=IzJQ)m~b*zKsTRmN-R%B1Sxk;IIayuY>oQ5270_myun1crOd<*KMyNS$G$( zvS&7n78ieZ-`w~;ap<_&shw23xg0$yc#$z+J2;em<8jxtygK4`FK>wW^~Bf2$3MGk zb7YbNt_csJxc8*zT6;uDCE&Rwi@mUuAR&=xZ~-gi4p_tre6xxN;#78!_2r?;4I9ovM8&q@5ddhE0P_iSH69x*1T@$GY54pr9)*;S6tO*3+__2k-maF##Z zIGoh-rDdy8ww~oJh+Lf+v}w-urTMuK6FY5USbvAPzr4T7ZA9e!gm0E8OH;N=Y)pHB zg%_N<)fy7fldrF@a|j6zFT{&-dW#;*W8Gn95p~fe0IMUXVzDhMDb^eCqK}PW+XOjz zPQA3SKDW-hP&p$Bqg2A&lsbeAwchGW{5K-JONR2mFB#7x8FJI)zZT|ZNW2$VPkTYY zaE>d_{UlLezA!h@Kq8uFT>nLJ_4tAhTCBNseWIqGO%s=e#oEvLbTlstzhE30PMR3b z9m{OxC9YKkt|87{+e)^U_CPUB1GDjQ%|&~-Typmr&!kuGvv-hcydM#C=|261`RVt` z)qKVNJ`Yk8bE!Q3T)Aii784%T2{BY@TeJ)ov_Dd}dOcK8AyryGZQVG$%#;Kw2VXww ztL)85W66IP*H)LV+~d3U^O7#Rf9`?FA+OWH^WZAptCELx%SjE@Jy(up4N*~fi@N)ae?TUA#H6C`9j1daaw06?qQ|m{(5iG8NbDb=YG2qgheeJDDTV6fkYvl<)IQ;MXMZ0@C1%-dF?Z^*~*)%~l#v7I*8O~x}G z{OF6XOMYpYkMo>RN#AB;4!<+@Es|%jJ`5R*tP!=0TL#|e$nt6ZHNhkmOlMAWg_jA6 z!cjI2{M1vk%w00ITvNt1a-WlG-BLu4-;3f5+6~NjkLg@7WlWjLO~9<-sh_L3Cvj_B z`eWSlgfcQI+tU)j3n}p1Vuxzt)&}MJQXj!4bl}7ptO|R_H#J&x(T~M5d~RyIPxL>Ka%iY2{9a!uq!FV+uauRZ4#Iy%oSxO&T(VV>9+Dqvx4w_!i1=*V z69J7`e&{2IdR)wFA*{B)8gU?>`JIyu$AGN0g--zinzw>^zY**3_<)D8;+rXPj zSkraF_8P7K?UVjb*^+4O;)s+bYpLvbGMcs_#l!U1yI@IB3uN;vx zi!2^Laxb+}BEnia{B`@r0#g`Cm>dqVJ3W*v&~~EFfD7-fJ)IxCG&TAzeXJ{e>MKnu zsy-=4G(_!WemtwrBQ<^ed0Y!+_D)U_v%X=>pVTZ(0SlYNOZMea%eu zr6GG2LFE4Xjhfxt^&WIj!C2Ukr)aJ4eEqyu6BDFHfjd)D)`%6_0^9;WK~r>ewrRKRrC4V$~oiP zc}oPA%&U9lMs*fcdG0LK!o2!L`k;TmYR!7j;+Zd{No}XY_h8fWibK=}%QZDM^ypM#! z?}(LVQ`xzY86Bog?Moom$BWNw2wu5|C;D*P^e$2)q9?<)E9#MMcjh48h0;}YGS1Kl zP7xq6TA-Bi$#PR)RMnjJ<!>1KtcXQK|^6dVU3f|0nR zaT9gV-8G%$zs+^Uad6f$)?}PD_W9Ti8}n>gZ86|dCBY*G^NLOI>ja`>>QI3%OxG_Z z>DQRY2HadZrA!&(dp*O43hZ1T$Wwz_klkJAx7&H7D$pE%5zrJ`GXAVy_bB_+6|wX* zu$WThgxcz-ii%~7$QE0P%TMHo`k+EV)U62k=3mcP=qu_1#Hr}YP#(Y6W)(th_xC!_ z%Ra(PP@D8n`DD96q7puPo`g9x{>#bKzL!chiq`BH;f_k92k$-oIEVep)o^Ov&`kO% z!;CsD8Ultb;nZ?EgZedmq}U93cM>}CW6M|ER|^q-cj1QjAH@ietebNQ$8S>%PuNiM zR$7(qBGJ&%wA0ug6IfSS!amZn)3CiBa>e)XTMV}UV265*@iGSMOuE3UwNBS<-3}n zTTjjL?TBlIkRyG?AC(Ibu{z2UCULPMCoOwguw(fV&%Nh}eHx^f!)wj=>Y z(Zbi05H$lNE-GDAv~Q7UQAAZlGO~T#tg(zos3IwG+MN+FbkEP|P!8wbr}Si%@%$iB z#J)Mu7{xu|O{xFcEK7Y}+9=t(_7mys+2XD&3^l>}6Rf>WYeWLLrN}9t{EMj9=t$xS z5MNa6+9)Pg&OjPxnp-)&@|yAFft>XkZ9Lb5VU+lgXkxnzFR5GdS46w9Fe%2v$fTE+!s&V_I zHTO8Mn5m7C-}IOJcQi_&3~$?Kyc~m$V@tjXNEgzzl~u2XN39h6KjG`((`Rds7O{nM+XE+ZkCQ1aD^IkP#gzE>dG_KI#_e z8aPe`lo#+alWD>zerZJ_zX+@@w=Z+ABsHkX3ZpD6POa$}iKJ$WK<&W|dh|)t91$j) zjNOfq=p73KQrnQOyO$67!3~MVO`}7zKK7gb9WB-bhq0_k%)pO=46F&Mk1c*jM)@1R z(Ei1+FVy9ydJ7E!?K@4PQ~!~ChQQO4xQPv(PQc-2^Oi$tP$h1mfw z98~7|=sf|73Yk@rfH$mFfzd00xfUEp6x@nKF^}LAPiCVj7BHUfeL5f>BI7OPRpXVj

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- \sqrt{-\psi}}{(-\psi)^{3/2}} \quad &\text{if} \, \psi < -10^{-6} \\ 1/6 \quad &\text{if} \, 10^{-6} > \psi > -10^{-6} \end{cases} @@ -578,7 +577,7 @@ flight: \begin{equation} - \chi = sqrt{\frac{y}{c_2}} + \chi = \sqrt{\frac{y}{c_2}} \end{equation} \begin{equation} @@ -691,7 +690,7 @@ $v_{eq}$, such that the thrust equation becomes: \begin{align} - v_{eq} &= v_e - \frac{\Delta p A_e}{\dot{m}} \\ + v_{eq} &= v_e + \frac{\Delta p A_e}{\dot{m}} \\ F &= \dot{m} v_{eq} \label{isp_1} \end{align} @@ -717,7 +716,7 @@ \subsection{Sims-Flanagan Transcription} - this thesis chose to use a model well suited for modeling low-thrust paths: the + This thesis chose to use a model well suited for modeling low-thrust paths: the Sims-Flanagan transcription (SFT)\cite{sims1999preliminary}. The SFT allows for flexibility in the trade-off between fidelity and performance, which makes it very useful for this sort of preliminary analysis. @@ -829,10 +828,3 @@ \caption{Graphic of an orbit-raising with a high efficiency cutoff} \label{high_efficiency_fig} \end{figure} - - All of this is, of course, also true for impulsive trajectories. However, since the - thrust presence for those trajectories are generally taken to be impulse functions, the - control laws can afford to be much less complicated for a given mission goal, by simply - thrusting only at the moment on the orbit when the transition will be most efficient. - For a low-thrust mission, however, the control law must be continuous rather than - discrete and therefore the control law inherently gains a lot of complexity. diff --git a/LaTeX/trajectory_optimization.tex b/LaTeX/trajectory_optimization.tex index 7d016ec..a3e403f 100644 --- a/LaTeX/trajectory_optimization.tex +++ b/LaTeX/trajectory_optimization.tex @@ -20,7 +20,7 @@ The other category is the direct methods. In a direct optimization problem, the cost function itself provides a value that an iterative numerical optimizer can measure - itself against. The optimal solution is then found by varying the inputs $x$ until the + itself against. The optimal solution is then found by varying the inputs $\vec{x}$ until the cost function is reduced to a minimum value, often determined by its derivative jacobian. A number of tools have been developed to optimize NLPs via this direct method in the general case. @@ -48,7 +48,7 @@ University. It uses a sparse sequential quadratic programming algorithm as its back-end optimization scheme. - Another common NLP optimization packages (and the one used in this implementation) + Another common NLP optimization package (and the one used in this implementation) is the Interior Point Optimizer or IPOPT\cite{wachter2006implementation}. It uses an Interior Point Linesearch Filter Method and was developed as an open-source project by the organization COIN-OR under the Eclipse Public License. @@ -74,11 +74,6 @@ step the initial guess, now labeled $x_{k+1}$ after the addition of the ``step'' vector and iterates this process until predefined termination conditions are met. - In this case, the IPOPT algorithm was used, not as an optimizer, but as a solver. For - reasons that will be explained in the algorithm description in Section~\ref{algorithm} it - was sufficient merely that the non-linear constraints were met, therefore optimization (in - the particular step in which IPOPT was used) was unnecessary. - \subsubsection{Shooting Schemes for Solving a Two-Point Boundary Value Problem} One straightforward approach to trajectory corrections is a single shooting @@ -154,8 +149,7 @@ very well to low-thrust arcs and, in fact, Sims-Flanagan Transcribed low-thrust arcs in particular, because there actually are control thrusts to be optimized at a variety of different points along the orbit. This is, however, not an exhaustive - description of ways that multiple shooting can be used to optimize a trajectory, - simply the most convenient for low-thrust arcs. + description of ways that multiple shooting can be used to optimize a trajectory. \section{Monotonic Basin Hopping Algorithms}