You are the pilot of a spaceship at the origin P_start = (0,0,0) at time t = 0. Your mission is to reach a space station located at G = (x_g, y_g, z_g ) in the minimum possible time T.
1.Spaceship Dynamics:
The spaceship can move in any direction in 3D space with a maximum constant speed V_max. Let S(t) be the position of the ship at time t, The constraint on its velocity is:
2.Moving Obstacles ( Asteroids ):
There are N spherical asteroids in the field. Each asteroid i is defined by its initial position.
P_i0 = (x_i, y_i, z_i ), a constant velocity vector V_i = ( v*x_i, v*y_i, v*z_i ), and a radius R_i.
The position of the center of asteroid i at any time t >= 0 is:
3.Safety Constraint:
To avoid dustruction, the spaceship must never enter the volume of any asteroid. At all times t ∈ [0,T], the distance between the ship and every asteroid center must satisfy:
4.Goal:
Find the minimum time T such that S(T) = G. If the goal is unreachable due to obstacles, return Inf.
Input:
- goal: A 1x3 vector [x_g, y_g, z_g]
- Vmax: A scalar representing your maximum speed.
- asteroids: An Nx7 matrix. Each row represents [x_0, y_0, z_0, vx, vy, vz, R].
Output:
- min_T: A scalar representing the shortest time to reach the goal
P/s: Sorry for not being clean in presentation, hope you guys sympathize with me :((
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