# Apply forces such that a satellite reaches a given point in space

I'd like to find out a strategy for solving the following simplified problem.

• A small spherical craft at $$t=0$$ has a position and velocity vectors $$\mathbf{x_0}, \mathbf{v_0}$$ in a zero gravity environment.

• It has one outward-pointing thruster that can vector in any direction by rotating it's spherical shape using internal attitude control.

• Each second the craft can emit one pulse resulting in a standard, small $$\Delta v$$ via one outward-pointing thruster that can vector in any direction by the craft rotating it's spherical shape between pulses using internal attitude control.

• The craft must be moving with a speed (absolute value, not directional velocity) less than $$\mathbf{v_m}$$ at the position $$\mathbf{x_1}$$.

• ideally a least-time solution is sought, but a procedure that yields many solutions is still helpful because they can be compared and the optimal solution selected.

How would I go about trying to solve this problem? Are there existing solutions or methodologies?

• Are you making any considerations to fuel usage or mass or do you just assume that each pulse just changes the velocity? – Dragongeek Dec 12 '18 at 20:40
• I'm sure I read a question like this recently – JCRM Dec 13 '18 at 15:20
• what is it's rotational rate and acceleration - can we assume it can rotate to any position in the time between thrusts? Can we assume the thrusts are sufficiently short duration that they can be treated as impulses, otherwise can the acceleration be treated as constant during the thrust. Is $v_m$ known to be larger than $\delta v$ – JCRM Dec 13 '18 at 15:36
• I'm voting to close this question as off-topic because It isn't really about space exploration -- It's a geometry question. – JCRM Dec 13 '18 at 15:46
• The simplest way of doing this is to zero the velocity (isosceles triangle with current velocity as the base and $\Delta v$ as the sides) then point at the target and thrust. if $v_m$ is smaller than $\Delta v$ then another isosceles triangle is needed, and you need to adjust your aim by the distance travelled during the first thrust. – JCRM Dec 13 '18 at 15:55