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I'm building an orbit propagator that gives the position of two satellites at time T. The position are expressed in ECI and after that are being transformed in LVLH frame (as mentioned ECI to LVLH conversion). If during the integration the satellites will collide, I want to return at time dt and to activate the thrusters that will give to the chaser satellite a velocity dv of 5m/s. As one can see in the below image, we have 2 satellites that are going to collide at a certain time. In order to avoid this, satellite 2 will active its thruster that will reduce its velocity so that a collision won't happen. The velocity vector should be in the opposite direction to velocity sat 2. Problems: How to decompose this velocity vector on x,y,z axis so that the resultant to be 5 m/s and to decelerate the satellite. As I see, I need to take account for it when I integrate the acceleration.

MATLAB functions:

Function for the first sat

function yprim=sat1(t,y)
miu=398600.4418*10^9;
magn=(y(1)^2+y(2)^2+y(3)^2)^(3/2);
yprim=zeros(6,1);
yprim(1,1)=y(4);
yprim(2,1)=y(5);
yprim(3,1)=y(6);
yprim(4,1)=double(-miu*y(1)/magn); %acceleration on x
yprim(5,1)=double(-miu*y(2)/magn); %acceleration on y
yprim(6,1)=double(-miu*y(3)/magn); %acceleration on z
end

Function for the 2nd sat

function yprim=sat2(t,y1)
miu=398600.4418*10^9;
magn=(y1(1)^2+y1(2)^2+y1(3)^2)^(3/2);
yprim=zeros(6,1);
yprim(1,1)=y1(4);
yprim(2,1)=y1(5);
yprim(3,1)=y1(6);
yprim(4,1)=double(-miu*y1(1)/magn);
yprim(5,1)=double(-miu*y1(2)/magn);
yprim(6,1)=double(-miu*y1(3)/magn);
end

LVLH conversion

er = r1 /norm(r1);% r1-position of the first sat
eh = cross(r1, v1);%v1-velocity of the first sat
eh = eh / norm(eh);
et = cross(eh, er);

dri = r2-r1;
dvi = v2-v1;

dr(1) = dot(dri, er);
dr(2) = dot(dri, et);
dr(3) = dot(dri, eh);
dv(1) = dot(dvi, er);
dv(2) = dot(dvi, et);
dv(3) = dot(dvi, eh);

dist=norm(dr); % distance between the 2 sat in lvlh

enter image description here

Alexandru Lapusneanu
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    I believe your collision avoidance strategy is flawed, and will cause the chaser to get closer to the target. Use a radial burn instead. Also, it might be worth naming the programming language. –  Jan 27 '19 at 09:54
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    Language looks like Matlab. Also please clarify what you mean by"opposing velocity vector". You've formulated the derivatives for both satellites. If you intend to integrated, then at each step you have both positions and velocities. All you need should be there.You might want to make this question more concise. Also, why using LVLH frame? Have ou taken a look at Clohessy-Wiltshire equations ? – Mefitico Jan 27 '19 at 21:01
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    Hi Alexandru, your question has two votes to close for "unclear what you're asking Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. I have to say that it's a little hard to understand exactly what your question is. "fix my program" questions rarely get answered here, but I don't think that's what you are asking either. – uhoh Jan 28 '19 at 01:53
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    Your questions are usually quite clear! Can you try to adjust your wording here, and focus on stating the problem more clearly? Maybe your program is not actually necessary here, I think you are asking how to think about a strategy, not how to code it. Thanks! – uhoh Jan 28 '19 at 01:54
  • @Mefitico The LVLH frame is being used as a request of the contractor. I've looked over the Clohessy-Wiltshire equations , the problem is that both of the satellites are in HEO and as I've seen there ,one most consider one of the satellites as having a circular orbit. Unfortunatelly, I need a high precission when integrating so I can't really rely on those equations. – Alexandru Lapusneanu Jan 30 '19 at 12:30
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    Why not use linearized relative equations for elliptic orbits and pose the problem as an optimization one? Also, there is a technique called rotating hyperplane for collision avoidance problems of this type, although a bit rude. – Julio Jan 31 '19 at 16:54

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