Question

So I have a ship, that has thrusters at the bottom and that can only use these to move forward. It can also rotate itself around its center. Its thrusters gives it acceleration, so it doesn’t move at a constant velocity. What I want to do is to tell it “move to point B”.

I have come up with a solution but it doesn’t work very well and it doesn’t rotate smoothly, it moves jerkily and it doesn’t end up exactly where it should be, so I have to have a big margin of error.

Is this a normal problem, and if so is there a “standard” way of doing it? Is this an easy problem? I want to make it look like the ship is steering itself to that point, using the constraints (thrusters, rotation) the player has. This excludes just lerping it from point A to B. Or does it?

I’d love some help in solving this problem. Positions are stored in vectors, and it’s a 2D problem. Just for reference I’m including my solution, which basically is accelerating the ship until and rotating it to point to the point. I think my implementation of this idea is the problem:

Vector diff = vector_sub(to_point, pos);
float angle = vector_getangle(diff);
float current_angle = vector_getangle(dir);
float angle_diff = rightrange(angle) - rightrange(current_angle);
float len = vector_getlength(diff);

// "Margin of error"
float margin = 15.0;

// Adjust direction, only if we're not stopping the next thing we do (len <= margin)
if ( len > margin && fabs(angle_diff) > 2.0 )
{
    dir = vector_setangle(dir, current_angle + (angle_diff)*delta*(MY_PI) - MY_PI/2);
}
else if ( len > margin )
{
    dir = vector_normalize(diff);
}

// accelerate ship (if needed)
acc.x = acc.y = speed;
acc = vector_setangle(acc, vector_getangle(dir));

if ( len <= margin )
{
    // Player is within margin of error
}

Answer 1

If you are not looking for a very general solution that works online, then there is a simple solution. What I mean by online is continuously re-calculating the actions along the complete trajectory.

Assuming the ship is at rest at start, simply rotate it towards your target point (while still at rest). Now, your ship can reach the target by accelerating for t seconds, rotating back while in motion (for 0.5 seconds as per your constraint), and decelerating for another t seconds. If the distance between current point and destination is d, then the equation you need to solve is:

d = 0.5*a*t^2 + 0.5*a*t + 0.5*a*t^2

The first term is distance traveled while accelerating. The second term is distance traveled while rotating (v*t_rot, v=a*t, t_rot=0.5). The final term is the distance traveled while decelerating. Solve the above for t, and you have your trajectory.

If the ship is moving at start, I would first stop it (just rotate in opposite direction of its speed vector, and decelerate until at rest). Now we know how to reach destination.

The problem with offline trajectory calculation is that it is not very accurate. There is a good chance that you will end up in the vicinity of the target, but not exactly on top of it.

---

Let’s make the problem a little more interesting: the ship cannot rotate without acceleration. Let’s call this acceleration vector a_r, a vector that is at a certain angle against the ship’s direction (somewhat like having a thruster at an angle at the back). Your task now is to rotate the ship and accelerate in such a direction that the speed component perpendicular to the vector connecting the current position to the target is canceled out. Instead of trying to calculate the vectors offline, I would go with an online approach with this.

The easiest thing to do would be to add the following algorithm calculated at every time interval:

1. Calculate the vector pointing from ship to destination.
2. Split your current speed vector into two components: towards the destination, and perpendicular to it.
3. If perpendicular speed is zero, skip 4
4. Start rotating towards the negative of the perpendicular vector’s direction. If already looking away from it (not exact opposite, but just looking away), also fire main thruster.

This will oscillate a bit, I suspect it will also stabilize after a while. I must admit, I don’t know how I would make it stop at destination.

---

And the final approach is to model the ship’s dynamics, and try to linearize it. It will be a non-linear system, so the second step will be necessary. Then convert the model to a discrete time system. And finally apply a control rule to make it reach target point. For this, you can change your state-space from position and speed to error in position and (maybe) error in speed, and finally add a regulation control (a control loop that takes the current state, and generates an input such that the state variables will approach zero).

This last one is fairly difficult in the maths compartment, and you’d probably need to study control engineering a bit to do it. However, you’ll get much better results than the above simplistic algorithm – which admittedly might not even work. In addition, you can now apply various optimization rules to it: minimize time to reach target, minimize fuel consumption, minimize distance traveled, etc.