-
Notifications
You must be signed in to change notification settings - Fork 3
/
ElemPB.m
251 lines (204 loc) · 13.1 KB
/
ElemPB.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
function [ dE, PBJ ] = ElemPB( dWdE,d2WdE2, E, ElemType, mu, JacobianOn)
%ElemPB Evaluates Poisson Brackets of the specified Orbital Elements (functions
%of the canonical Delaunay elements) with respect to a second quantity using
% its partials provided. The second quantity is typically a generating function,
% in that case the output is the variations of the elements due to that
% generating function.
%
% Author: Bharat Mahajan (https://github.com/princemahajan)
if strcmp(ElemType,'classical') == true
% Elements
a = E(1);
e = E(2);
i = E(3);
% h = E(4);
% g = E(5);
% f = E(6);
% partials of the generating function
dWda = dWdE(1);
dWde = dWdE(2);
dWdi = dWdE(3);
dWdh = dWdE(4);
dWdg = dWdE(5);
dWdl = dWdE(6);
% intermediate quantities
eta = sqrt(1-e^2);
L = sqrt(mu*a);
G = L*eta;
H = G*cos(i);
dadL = 2*L/mu;
dedL = eta^2/(e*L);
dedG = -eta^2/(e*G);
didG = H/(G^2*sin(i));
didH = -1/(G*sin(i));
% Element Variations
% SMA
da = -dadL*dWdl;
% Ecc
de = -dedL*dWdl - dedG*dWdg;
% Inc
di = -didG*dWdg - didH*dWdh;
% RAAN
dh = didH*dWdi;
%AOP
dg = didG*dWdi + dedG*dWde;
% MA
dl = dWda*dadL + dWde*dedL;
dE = [da, de, di dh, dg, dl]';
elseif strcmp(ElemType,'equinoctial') == true
% Elements
a = E(1);
e = E(2);
i = E(3);
h = E(4);
g = E(5);
% f = E(6);
% partials of the generating function
dWda = dWdE(1);
dWde = dWdE(2);
dWdi = dWdE(3);
didGWgdidHWh = dWdE(4);
dWdletadWdg = dWdE(5);
dWdl = dWdE(6);
% intermediate quantities
eta = sqrt(1-e^2);
L = sqrt(mu*a);
G = L*eta;
dadL = 2*L/mu;
% dfdl = (1 + e*cos(f))^2/eta^3;
% Element Variations
% SMA corrections
DelSMA = -dadL*dWdl;
% Mean Argument of Latitude
DelMAOL = dadL*dWda - eta/L*dWde*sqrt((1-eta)/(1+eta)) - dWdi*tan(i/2)/G;
% p1 and p2
Delp1 = sin(h)/(G*(1+cos(i)))*dWdi - cos(h)/(1+cos(i))*didGWgdidHWh;
Delp2 = -cos(h)/(G*(1+cos(i)))*dWdi - sin(h)/(1+cos(i))*didGWgdidHWh;
% q1,q2
Delq1 = -sin(g+h)*(-eta/L*dWde-e*dWdi*tan(i/2)/G) - cos(g+h)*eta^2/L*dWdletadWdg;
Delq2 = cos(g+h)*(-eta/L*dWde-e*dWdi*tan(i/2)/G) - sin(g+h)*eta^2/L*dWdletadWdg;
dE = [DelSMA, DelMAOL, Delp1, Delp2, Delq1, Delq2]';
if JacobianOn == true
% equinoctial and classical element relations
si = sin(i);
ci = cos(i);
ci2 = cos(i/2);
ti2 = tan(i/2);
sh = sin(h);
ch = cos(h);
cgh = cos(g+h);
sgh = sin(g+h);
dedq1 = cgh;
dedq2 = sgh;
didp1 = ch*(1+ci);
didp2 = sh*(1+ci);
dgdp1 = sh/ti2;
dgdp2 = -ch/ti2;
dgdq1 = -sgh/e;
dgdq2 = cgh/e;
dhdp1 = -sh/ti2;
dhdp2 = ch/ti2;
dldq1 = sgh/e;
dldq2 = -cgh/e;
% dfde = (2+e*cos(f)).*sin(f)/eta^2;
% d2fdfdl = 2*(1+e*cos(f))/eta^3*(-e*sin(f));
% d2fdl2 = dfdl*d2fdfdl;
% d2fdlde = -3/eta^4*(-e/eta)*(1+e*cos(f))^2 + 2*(1+e*cos(f))/eta^3*(cos(f)-e*sin(f)*dfde);
% Del SMA partials w.r.t. equinoctial elements
DelSMAa = -1/L*dWdl - dadL*d2WdE2(6,1);
DelSMAML = -dadL*d2WdE2(6,6);
DelSMAp1 = -dadL*(d2WdE2(6,3)*didp1 + d2WdE2(6,5)*dgdp1 + d2WdE2(6,4)*dhdp1);
DelSMAp2 = -dadL*(d2WdE2(6,3)*didp2 + d2WdE2(6,5)*dgdp2 + d2WdE2(6,4)*dhdp2);
DelSMAq1 = -dadL*((d2WdE2(6,2))*dedq1 + d2WdE2(6,5)*dgdq1 + (d2WdE2(6,6))*dldq1);
DelSMAq2 = -dadL*((d2WdE2(6,2))*dedq2 + d2WdE2(6,5)*dgdq2 + (d2WdE2(6,6))*dldq2);
% Del MOL partials w.r.t. equinoctial elements
DelMAOLa = (1/L*dWda+dadL*d2WdE2(1,1)) - eta*sqrt((1-eta)/(1+eta))*(-mu/(2*L^3)*dWde + 1/L*d2WdE2(2,1)) ...
- tan(i/2)/eta*(d2WdE2(3,1)/L - dWdi*mu/(2*L^3));
DelMAOLML = dadL*d2WdE2(1,6) - eta/L*d2WdE2(2,6)*sqrt((1-eta)/(1+eta)) - d2WdE2(3,6)*tan(i/2)/G;
DelMAOLp1 = dadL*(d2WdE2(1,3)*didp1 + d2WdE2(1,5)*dgdp1 + d2WdE2(1,4)*dhdp1) ...
- eta/L*sqrt((1-eta)/(1+eta))*(d2WdE2(2,3)*didp1 + d2WdE2(2,5)*dgdp1 + d2WdE2(2,4)*dhdp1) ...
- 1/G*((d2WdE2(3,5)*dgdp1 + d2WdE2(3,4)*dhdp1 + d2WdE2(3,3)*didp1)*tan(i/2) + dWdi*(sec(i/2)^2/2)*didp1);
DelMAOLp2 = dadL*(d2WdE2(1,3)*didp2 + d2WdE2(1,5)*dgdp2 + d2WdE2(1,4)*dhdp2) ...
- eta/L*sqrt((1-eta)/(1+eta))*(d2WdE2(2,3)*didp2 + d2WdE2(2,5)*dgdp2 + d2WdE2(2,4)*dhdp2) ...
- 1/G*((d2WdE2(3,5)*dgdp2 + d2WdE2(3,4)*dhdp2 + d2WdE2(3,3)*didp2)*tan(i/2) + dWdi*(sec(i/2)^2/2)*didp2);
DelMAOLq1 = dadL*(d2WdE2(1,2)*dedq1 + d2WdE2(1,5)*dgdq1 + d2WdE2(1,6)*dldq1)...
- 1/L*((eta*sqrt((1-eta)/(1+eta))*d2WdE2(2,2) + ((2*eta-1)/eta-eta/(1+eta))*dWde)*dedq1 ...
+ eta*sqrt((1-eta)/(1+eta))*(d2WdE2(2,5)*dgdq1 + d2WdE2(2,6)*dldq1)) ...
- 1/L*((-1/eta^2)*(-e/eta)*dedq1*tan(i/2)*dWdi + 1/eta*tan(i/2)*d2WdE2(3,2)*dedq1 ...
+ 1/eta*tan(i/2)*(d2WdE2(3,5)*dgdq1 + d2WdE2(3,6)*dldq1));
DelMAOLq2 = dadL*(d2WdE2(1,2)*dedq2 + d2WdE2(1,5)*dgdq2 + d2WdE2(1,6)*dldq2)...
- 1/L*((eta*sqrt((1-eta)/(1+eta))*d2WdE2(2,2) + ((2*eta-1)/eta-eta/(1+eta))*dWde)*dedq2 ...
+ eta*sqrt((1-eta)/(1+eta))*(d2WdE2(2,5)*dgdq2 + d2WdE2(2,6)*dldq2)) ...
- 1/L*((-1/eta^2)*(-e/eta)*dedq2*tan(i/2)*dWdi + 1/eta*tan(i/2)*d2WdE2(3,2)*dedq2 ...
+ 1/eta*tan(i/2)*(d2WdE2(3,5)*dgdq2 + d2WdE2(3,6)*dldq2));
% Del p1 and p2 partials w.r.t. equinoctial elements
Delp1a = sh/(1+ci)/eta*(-mu/(2*L^3)*dWdi + 1/L*d2WdE2(3,1)) - ch/(1+ci)*d2WdE2(7,1);
Delp2a = -ch/(1+ci)/eta*(-mu/(2*L^3)*dWdi + 1/L*d2WdE2(3,1)) - sh/(1+ci)*d2WdE2(7,1);
Delp1ML = sh/(G*(1+ci))*d2WdE2(3,6) - ch/(1+ci)*d2WdE2(7,6);
Delp2ML = -ch/(G*(1+ci))*d2WdE2(3,6) - sh/(1+ci)*d2WdE2(7,6);
Delp1p1 = 1/G*(sh*(si/(1+ci)^2*dWdi + 1/(1+ci)*d2WdE2(3,3))*didp1 + 1/(1+ci)*(sh*d2WdE2(3,5)*dgdp1 + (ch*dWdi + sh*d2WdE2(3,4))*dhdp1)) ...
-ch*(si/(1+ci)^2*didGWgdidHWh + 1/(1+ci)*d2WdE2(7,3))*didp1 - 1/(1+ci)*(ch*d2WdE2(7,5)*dgdp1+(-sh*didGWgdidHWh+ch*d2WdE2(7,4))*dhdp1);
Delp1p2 = 1/G*(sh*(si/(1+ci)^2*dWdi + 1/(1+ci)*d2WdE2(3,3))*didp2 + 1/(1+ci)*(sh*d2WdE2(3,5)*dgdp2 + (ch*dWdi + sh*d2WdE2(3,4))*dhdp2)) ...
-ch*(si/(1+ci)^2*didGWgdidHWh + 1/(1+ci)*d2WdE2(7,3))*didp2-1/(1+ci)*(ch*d2WdE2(7,5)*dgdp2+(-sh*didGWgdidHWh+ch*d2WdE2(7,4))*dhdp2);
Delp2p1 = 1/G*(-ch*(si/(1+ci)^2*dWdi + 1/(1+ci)*d2WdE2(3,3))*didp1 + 1/(1+ci)*(-ch*d2WdE2(3,5)*dgdp1 + (sh*dWdi - ch*d2WdE2(3,4))*dhdp1)) ...
-sh*(si/(1+ci)^2*didGWgdidHWh + 1/(1+ci)*d2WdE2(7,3))*didp1 - 1/(1+ci)*(sh*d2WdE2(7,5)*dgdp1+(ch*didGWgdidHWh+sh*d2WdE2(7,4))*dhdp1);
Delp2p2 = 1/G*(-ch*(si/(1+ci)^2*dWdi + 1/(1+ci)*d2WdE2(3,3))*didp2 + 1/(1+ci)*(-ch*d2WdE2(3,5)*dgdp2 + (sh*dWdi - ch*d2WdE2(3,4))*dhdp2)) ...
-sh*(si/(1+ci)^2*didGWgdidHWh + 1/(1+ci)*d2WdE2(7,3))*didp2 - 1/(1+ci)*(sh*d2WdE2(7,5)*dgdp2+(ch*didGWgdidHWh+sh*d2WdE2(7,4))*dhdp2);
Delp1q1 = 1/L*sh/(1+ci)*((-1/eta^2*(-e/eta)*dWdi + 1/eta*d2WdE2(3,2))*dedq1 + 1/eta*(d2WdE2(3,5)*dgdq1 + d2WdE2(3,6)*dldq1)) ...
- ch/(1+ci)*(d2WdE2(7,2)*dedq1 + d2WdE2(7,5)*dgdq1 + d2WdE2(7,6)*dldq1);
Delp1q2 = 1/L*sh/(1+ci)*((-1/eta^2*(-e/eta)*dWdi + 1/eta*d2WdE2(3,2))*dedq2 + 1/eta*(d2WdE2(3,5)*dgdq2 + d2WdE2(3,6)*dldq2)) ...
- ch/(1+ci)*(d2WdE2(7,2)*dedq2 + d2WdE2(7,5)*dgdq2 + d2WdE2(7,6)*dldq2);
Delp2q1 = -1/L*ch/(1+ci)*((-1/eta^2*(-e/eta)*dWdi + 1/eta*d2WdE2(3,2))*dedq1 + 1/eta*(d2WdE2(3,5)*dgdq1 + d2WdE2(3,6)*dldq1)) ...
- sh/(1+ci)*(d2WdE2(7,2)*dedq1 + d2WdE2(7,5)*dgdq1 + d2WdE2(7,6)*dldq1);
Delp2q2 = -1/L*ch/(1+ci)*((-1/eta^2*(-e/eta)*dWdi + 1/eta*d2WdE2(3,2))*dedq2 + 1/eta*(d2WdE2(3,5)*dgdq2 + d2WdE2(3,6)*dldq2)) ...
- sh/(1+ci)*(d2WdE2(7,2)*dedq2 + d2WdE2(7,5)*dgdq2 + d2WdE2(7,6)*dldq2);
% Del q1,q2 Partials w.r.t equinoctial elements
Delq1a = -sgh*((-mu/(2*L^3)*(-eta*dWde-e*dWdi*ti2/eta) + 1/L*(-eta*d2WdE2(2,1)-e*d2WdE2(3,1)*ti2/eta))) ...
-cgh*eta^2*(-mu/(2*L^3)*dWdletadWdg + 1/L*d2WdE2(8,1));
Delq2a = cgh*((-mu/(2*L^3)*(-eta*dWde-e*dWdi*ti2/eta) + 1/L*(-eta*d2WdE2(2,1)-e*d2WdE2(3,1)*ti2/eta))) ...
-sgh*eta^2*(-mu/(2*L^3)*dWdletadWdg + 1/L*d2WdE2(8,1));
Delq1ML = -sgh*(-eta/L*d2WdE2(2,6)-e*d2WdE2(3,6)*tan(i/2)/G) - cgh*eta^2/L*d2WdE2(8,6);
Delq2ML = cgh*(-eta/L*d2WdE2(2,6)-e*d2WdE2(3,6)*tan(i/2)/G) - sgh*eta^2/L*d2WdE2(8,6);
Delq1p1 = -sgh*(-eta/L*d2WdE2(2,3)-e*d2WdE2(3,3)*ti2/G-e*dWdi/(2*ci2^2*G))*didp1 - cgh*(dgdp1+dhdp1)*(-eta/L*dWde-e*dWdi*ti2/G) ...
-sgh*(-eta/L*(d2WdE2(2,5)*dgdp1+d2WdE2(2,4)*dhdp1)-e*ti2/G*(d2WdE2(3,5)*dgdp1+d2WdE2(3,4)*dhdp1)) ...
-eta^2/L*(cgh*d2WdE2(8,3)*didp1 -sgh*(dgdp1+dhdp1)*dWdletadWdg + cgh*(d2WdE2(8,5)*dgdp1+d2WdE2(8,4)*dhdp1));
Delq2p1 = cgh*(-eta/L*d2WdE2(2,3)-e*d2WdE2(3,3)*ti2/G-e*dWdi/(2*ci2^2*G))*didp1 - sgh*(dgdp1+dhdp1)*(-eta/L*dWde-e*dWdi*ti2/G) ...
+cgh*(-eta/L*(d2WdE2(2,5)*dgdp1+d2WdE2(2,4)*dhdp1)-e*ti2/G*(d2WdE2(3,5)*dgdp1+d2WdE2(3,4)*dhdp1)) ...
-eta^2/L*(sgh*d2WdE2(8,3)*didp1 + cgh*(dgdp1+dhdp1)*dWdletadWdg + sgh*(d2WdE2(8,5)*dgdp1+d2WdE2(8,4)*dhdp1));
Delq1p2 = -sgh*(-eta/L*d2WdE2(2,3)-e*d2WdE2(3,3)*ti2/G-e*dWdi/(2*ci2^2*G))*didp2 - cgh*(dgdp2+dhdp2)*(-eta/L*dWde-e*dWdi*ti2/G) ...
-sgh*(-eta/L*(d2WdE2(2,5)*dgdp2+d2WdE2(2,4)*dhdp2)-e*ti2/G*(d2WdE2(3,5)*dgdp2+d2WdE2(3,4)*dhdp2)) ...
-eta^2/L*(cgh*d2WdE2(8,3)*didp2 -sgh*(dgdp2+dhdp2)*dWdletadWdg + cgh*(d2WdE2(8,5)*dgdp2+d2WdE2(8,4)*dhdp2));
Delq2p2 = cgh*(-eta/L*d2WdE2(2,3)-e*d2WdE2(3,3)*ti2/G-e*dWdi/(2*ci2^2*G))*didp2 - sgh*(dgdp2+dhdp2)*(-eta/L*dWde-e*dWdi*ti2/G) ...
+cgh*(-eta/L*(d2WdE2(2,5)*dgdp2+d2WdE2(2,4)*dhdp2)-e*ti2/G*(d2WdE2(3,5)*dgdp2+d2WdE2(3,4)*dhdp2)) ...
-eta^2/L*(sgh*d2WdE2(8,3)*didp2 +cgh*(dgdp2+dhdp2)*dWdletadWdg + sgh*(d2WdE2(8,5)*dgdp2+d2WdE2(8,4)*dhdp2));
Delq1q1 = -1/L*(sgh*(e/eta*dWde-eta*d2WdE2(2,2)-(1/eta^3*dWdi+e/eta*d2WdE2(3,2))*ti2)*dedq1 ...
+(cgh*dgdq1*(-eta*dWde-e*dWdi*ti2/eta)+sgh*((-eta*d2WdE2(2,5)-e*d2WdE2(3,5)*ti2/eta)*dgdq1...
+(-eta*d2WdE2(2,6)-e*d2WdE2(3,6)*ti2/eta)*dldq1))) ...
- 1/L*((cgh*2*eta*(-e/eta)*dWdletadWdg + cgh*eta^2*d2WdE2(8,2))*dedq1 + eta^2*(-sgh*dWdletadWdg + cgh*d2WdE2(8,5))*dgdq1 ...
+ eta^2*cgh*d2WdE2(8,6)*dldq1);
Delq2q1 = -1/L*(-cgh*(e/eta*dWde-eta*d2WdE2(2,2)-(1/eta^3*dWdi+e/eta*d2WdE2(3,2))*ti2)*dedq1 ...
+(sgh*dgdq1*(-eta*dWde-e*dWdi*ti2/eta)-cgh*((-eta*d2WdE2(2,5)-e*d2WdE2(3,5)*ti2/eta)*dgdq1...
+(-eta*d2WdE2(2,6)-e*d2WdE2(3,6)*ti2/eta)*dldq1))) ...
- 1/L*((sgh*2*eta*(-e/eta)*dWdletadWdg + sgh*eta^2*d2WdE2(8,2))*dedq1 + eta^2*(cgh*dWdletadWdg + sgh*d2WdE2(8,5))*dgdq1 ...
+ eta^2*sgh*d2WdE2(8,6)*dldq1);
Delq1q2 = -1/L*(sgh*(e/eta*dWde-eta*d2WdE2(2,2)-(1/eta^3*dWdi+e/eta*d2WdE2(3,2))*ti2)*dedq2 ...
+(cgh*dgdq2*(-eta*dWde-e*dWdi*ti2/eta)+sgh*((-eta*d2WdE2(2,5)-e*d2WdE2(3,5)*ti2/eta)*dgdq2...
+(-eta*d2WdE2(2,6)-e*d2WdE2(3,6)*ti2/eta)*dldq2))) ...
- 1/L*((cgh*2*eta*(-e/eta)*dWdletadWdg + cgh*eta^2*d2WdE2(8,2))*dedq2 + eta^2*(-sgh*dWdletadWdg + cgh*d2WdE2(8,5))*dgdq2 ...
+ eta^2*cgh*d2WdE2(8,6)*dldq2);
Delq2q2 = -1/L*(-cgh*(e/eta*dWde-eta*d2WdE2(2,2)-(1/eta^3*dWdi+e/eta*d2WdE2(3,2))*ti2)*dedq2 ...
+(sgh*dgdq2*(-eta*dWde-e*dWdi*ti2/eta)-cgh*((-eta*d2WdE2(2,5)-e*d2WdE2(3,5)*ti2/eta)*dgdq2...
+(-eta*d2WdE2(2,6)-e*d2WdE2(3,6)*ti2/eta)*dldq2))) ...
- 1/L*((sgh*2*eta*(-e/eta)*dWdletadWdg + sgh*eta^2*d2WdE2(8,2))*dedq2 + eta^2*(cgh*dWdletadWdg + sgh*d2WdE2(8,5))*dgdq2 ...
+ eta^2*sgh*d2WdE2(8,6)*dldq2);
% Short-period Jacobian
PBJ = [DelSMAa, DelSMAML, DelSMAp1, DelSMAp2, DelSMAq1, DelSMAq2;
DelMAOLa, DelMAOLML, DelMAOLp1, DelMAOLp2, DelMAOLq1, DelMAOLq2;
Delp1a, Delp1ML, Delp1p1, Delp1p2, Delp1q1, Delp1q2;
Delp2a, Delp2ML, Delp2p1, Delp2p2, Delp2q1, Delp2q2;
Delq1a, Delq1ML, Delq1p1, Delq1p2, Delq1q1, Delq1q2;
Delq2a, Delq2ML, Delq2p1, Delq2p2, Delq2q1, Delq2q2];
else
PBJ = 0;
end
end