In the “good old days” the only people worried about rendezvous were those designed space missions with crews. ISS established the need for robotic rendezvous and docking on a regular basis.

Orbit dynamics can be complex. If you are looking at rendezvous with any other spacecraft in a very different orbit you can start with Lambert’s Time-of-Flight algorithm. Given initial velocity and position vectors, and desired final position and velocity vectors and time of flight, it will give you the initial impulse velocity change needed to rendezvous. There are numerous formulation as it is complex math problem.

If you happen to be close to your target you can formulate your orbits as a relative orbit problem with Hill’s equations, shown below in state space form.

is the orbit rate of the target spacecraft. , and are the position of the “chase” spacecraft in the Hill’s frame. is the control acceleration. You want to reduce the positions and velocities to zero. This can be done with a Proportional Derivative (PD) Controller, or with a Linear Quadratic (LQ) Controller. If your chase and target spacecraft have GPS it is relatively easy to find this state vector in the above equation. A PD controller will ignore the coupling in the above equations while the LQ will accommodate the coupling.

It is interesting to look at the gain matrices for the two cases and the corresponding eigenvalues. We tweaked the PD to make its position gains close to that for the LQ. The PD is designed for a damping ratio of 1. The eigenvalues are identical. The cross-axis gains are small, but non-zero.

Gain Matrix LQ
0.0032 -0.0001 -0.0000 0.0796 0.0000 -0.0000
0.0001 0.0032 -0.0000 0.0000 0.0796 -0.0000
0.0000 -0.0000 0.0032 0.0000 -0.0000 0.0796

LQ eigenvalues
-0.0398 + 0.0409i
-0.0398 – 0.0409i
-0.0398 + 0.0386i
-0.0398 – 0.0386i
-0.0398 + 0.0397i
-0.0398 – 0.0397i

Gain Matrix PD
0.0036 0 0 0.1200 0 0
0 0.0036 0 0 0.1200 0
0 0 0.0036 0 0 0.1200

PD eigenvalues
-0.0398 + 0.0409i
-0.0398 – 0.0409i
-0.0398 + 0.0386i
-0.0398 – 0.0386i
-0.0398 + 0.0397i
-0.0398 – 0.0397i

The simulation results for the LQ are:

And for the PD are:

The results are very close. The PD has no overshoot, as expected. The LQ is slightly faster but has some overshoot. Both get the chase spacecraft to the target in a few minutes, assuming, of course, that you have the acceleration capability shown in the plots.

Both are linear controllers. You can approximate a linear controller with thrusters by using pulse width modulation. An issue will be the minimum impulse bit of the thrusters, that will lead to a minimum velocity and position error that can be achieved.

This script is included in the Spacecraft Control Toolbox 2020.1 coming soon!

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