Develop Reference

Constrain rigidbody movement to spline at high speeds / sharp curves

I have a 2.5d platformer game. The character is using rigidbody movement on a spline (using the curvy splines asset) which curves into 3d space in all sorts of ways, while the camera stays fixed to the side so that you see the path and background turning, but maintain a 2d side scrolling perspective.
I'm essentially creating a look rotation based on the spline, then moving the player using that forward vector, and making sure to remove any velocity perpendicular to the path so that the player stays centered on the path even when curving. I'm removing the velocity on that vector instead of projecting all the velocity in the direction of the path so that the player can still jump and fall like normal.
void SetLookRotation()
{
// get nearest TF and point on spline
Vector3 p;
mTF = Spline.GetNearestPointTF(transform.localPosition, out p);
// Get forward and up vectors of point on spline
_localHorizontal = Spline.GetTangentFast(mTF);
_localVertical = Spline.GetOrientationUpFast(mTF);
// Set look rotation to path
transform.rotation = Quaternion.LookRotation(Vector3.Cross(_localHorizontal, _localVertical), _localVertical);
}
void Movement()
{
Vector3 m = transform.right * groundAcceleration * moveInput;
rb.AddForce(RemoveCrossVelocity(m));
rb.velocity = RemoveCrossVelocity(rb.velocity);
Vector3 localVelocity = transform.InverseTransformDirection(rb.velocity);
localVelocity.z = 0;
rb.velocity = transform.TransformDirection(localVelocity);
}
Vector3 RemoveCrossVelocity(Vector3 v)
{
// get magnitude going in the cross product / perpindicular of localHorizontal and localVertical vector
// (essentially the magnitude on "local Z" or to the sides of the player)
Vector3 crossVelocity = Vector3.Project(v, Vector3.Cross(transform.right, transform.up));
// and remove it from the vector
return v -= crossVelocity;
}
The first 2 functions are happening in FixedUpdate() in the order shown.
The problem is, when hitting sharp corners at high speeds, some inertia causes the player to deviate off the center of the path still just ever so slightly, and a lot of that momentum turns into upward momentum, launching the player upwards. Eventually the player can fall off the path completely (I do have a custom gravity acting towards the spline though). It works perfectly at lower speeds though, even when dealing with sharp corners. At least as far as I can tell.
I tried a bit of code from https://answers.unity.com/questions/205406/constraining-rigidbody-to-spline.html too but no luck.
Is there a way I could constrain the player rigidbody on a vector that is not one of the global x/y/z axes? I've tried a host of other solutions like setting the transform of the player towards at the center of the spline but I can't seem to get it without feeling very jerky. Using forces makes the player "rubber band" towards and past the center back and forth. Maybe there is something in my math wrong. In any case, I'm hoping someone could help me make sure that the player will always stay on the center of the spline but only on the vector to the sides of the player's face direction, so that it doesn't mess with jumping. Thank you very much in advance!

For potential future visitors, I have figured this out. There are a few components (and a lot more if you're trying to do full spline based physics, but just to start with movement...)
First we must orient our character, so that our local coordinate system can be referenced with transform.right etc. Luckily this package provides these functions which return useful vectors. I'm sure there is math beyond me to do this otherwise if you are building your own spline system.
void SetLookRotation()
{
// get nearest TF and point on spline
Vector3 p;
playerTF = currentSpline.GetNearestPointTF(transform.localPosition, out p);
// Get forward and up vectors of point on spline
_localHorizontal = currentSpline.GetTangentFast(playerTF);
_localVertical = currentSpline.GetOrientationUpFast(playerTF);
// Set look rotation to path
transform.rotation = Quaternion.LookRotation(Vector3.Cross(_localHorizontal, _localVertical), _localVertical);
}
Here I am setting a velocity directly but if you're using forces it's the same principle.
if (Mathf.Abs(localVelocityAs_X) <= maxDashSpeed * Mathf.Abs(moveInput))
{
Vector3 m = transform.right * maxDashSpeed * moveInput;
rb.velocity = RemoveCrossVelocity(m);
}
localVelocityAs_X is defined as (ran in fixedUpdate/ physics step):
float currLocalVelocityX = (playerTF - prevPositionX) / Time.deltaTime;
localVelocityAs_X = Mathf.Lerp(localVelocityAs_X, currLocalVelocityX, 0.5f);
prevPositionX = playerTF;
Where playerTF is your position on a spline (in this case, using the curvy spline package from the unity asset store. Those spline positions return very small floats so in my case I multiplied playerTF by around 10,000 to make it a more easily readable metric). This is essentially just manually calculating velocity of the player each frame by comparing current position on the spline to last frame's.
RemoveCrossVelocity is the same as above. Comment explanations should suffice.
Vector3 RemoveCrossVelocity(Vector3 v)
{
// get magnitude going in the cross product / perpendicular of local horizontal and local vertical vectors
// (essentially the magnitude on "local Z" of the player)
Vector3 crossVelocity = Vector3.Project(v, Vector3.Cross(transform.right, transform.up));
// and remove it from the vector
return v -= crossVelocity;
}
Finally the solution to the drift. My crude fix was essentially to just adjust the player to the center of the spline every frame. Horizontally, there is no change because it grabs the closest spline point which is calculated by this package to be sort of a float clamped between the start and end of the spline. Vertically, we are being set to the distance the player is from the spline in the local up direction - a fancy way of saying we're not moving vertically at all. The reason this must be done is to avoid the spline vertical position overwriting the players, and we obviously can't set this vector back to playerPos.y in our local coordinate space, so we must resort to using a direction vector * the distance from our everchanging floor. This isn't absolutely ideal at the end of the day, but it works, and there isn't any extra jitter from it (interpolate on your player's rigidbody and some camera dampening helps). All in all these together combine to make a player able to accelerate quickly around sharp corners of a spline with physics and intertia will never cause the player to fly off or drift from the center. Take that, rocket physics!
void ResetPlayerToSpline()
{
Vector3 P; //closest spline point to player
float pTf = currentSpline.GetNearestPointTF(transform.position, out P);
playerHeight = Vector3.Distance(transform.position, P);
transform.position = P + (transform.up * Vector3.Distance(transform.position, P));
}
Ultimately for those possibly looking to do some kind of implementation in the future, the biggest thing you'll run into is a lack of cardinal direction, global oriented axis-based functions and properties normally provided by a game engine. For a primer, here are a few I would use (not including gravity, which is simply opposite your up vector times whatever magnitude):
This one allows you to create a vector using x and y like normal (and z in theory) and run this function to convert it when you actually use the vector in a local space. That way, you don't have to try and think in directions without names. You can still think of things in terms of x and y:
Vector3 ConvertWorldToLocalVector(Vector3 v)
{
Vector3 c;
c = transform.right * v.x + transform.up * v.y;
return c;
}
This is basically the same as what is happening in RemoveCrossVelocity(), but it's important to reiterate this is how you set velocity in a direction to 0. The second part shows how to get velocity in a certain vector.
void Velocity_ZeroY()
{
rb.velocity -= GetLocalVerticalVelocity();
}
public Vector3 GetLocalVerticalVelocity()
{
return Vector3.Project(rb.velocity, _localVertical);
}
Getting height, since you cannot just compare y positions:
height = Vector3.Distance(transform.position, P);
I think that's all the good stuff I can think of. I noticed a severe lack of resources for created spline based physics movement in games, and I'm guessing now it's based on the fact that this was quite an undertaking. It has since been brought to my attention that the game "Pandemonium"(1996) is a curvy 3d spline based sidescrolling platformer - just like mine! The main difference seems to be that it isn't at all based on physics, and I'm not sure from what I can tell if it has pitch changes and gravity to compliment. Hope this helps someone someday, and thank you to those who contributed to the discussion.

Unity rotate object depending on other objects angle

I have two Gameobjects. The basic Question:
"When I spin circle 1, I want to spin circle 2 in the same way manipulated by factor x"
How do I sync the rotation around each of their local axis of circle 2 with the interactable rotation of circle one and scale that rotation by factor x? Setting the transform.right equal doesnt work, there are still to many degrees of freedom.
(Local Axis, because I want one or both gameobjects to be also tilted, but unrelated to one another.)
Trying math with rotational matrices didnt really work out based on the fact of them being evaluated every frame and thereby spinning Object 2 for eternity.
Thanks so much!

Assuming you rotate only around one single axis (as shown in the images) you can get the rotation difference in degrees of circle2 using e.g. Quaternion.Angle every frame
private Quaternion lastCircle2Rot;
//...
float rotDelta = Quaternion.Angle(lastCircle2Rot, circle2.transform.rotation);
lastCircle2Rot = circle2.transform.rotation;
than turn the circle1 accordignly using e.g. Transform.RotateAround
public float multiplier;
// e.g. rotate around local x = right
Vector3 YourAxis = circle1.transform.right;
circle1.transform.RotateAround(Vector3.zero, YourAxis, rotDelta * multiplier);
if using e.g. the right or another "clean" vector you could also simply use
circle1.transform.Rotate(Vector3.right, rotDelta * multiplier);
as it is done in local space by default.

Unity C# Firing a projectile at a target rotating around moving axis

I'm developing a 3D spacegame where the camera is in a constant 2D (top down) state. I am able to fire a projectile of speed (s) at a target moving at a given velocity and hit it every time. Great! Okay so what if that target has an angular velocity around a parent? I noticed that if the target has a parent object that is rotating, my projection isn't correct since it doesn't account for the angular velocity.
My initial code was built around the assumption that:
Position_target + Velocity_target * t = Position_shooter + Velocity_shooter * t + Bulletspeed * t
I assume that the shooter is stationary (or potentially moving) and needs to fire a bullet with a constant magnitude.
I simplify the above to this
Delta_Position = Position_target - Position_shooter
Delta_Velocity = Velocity_target - Velocity_shooter
Delta_Position + Delta_Velocity * t = BulletSpeed * t
Squaring both sides I come to a quadratic equation where I can solve for t given determinant outcomes or zeros. This works perfect. I return a t value and then project the target's position and current velocity out to that t, and then I have turret scripts that rotate at a given angular velocity towards that point. If the turret says its looking at that point within 1% on all axis, it fires the bullet at speed(s) and its a 100% hit if the target doesn't alter its course or velocity.
I started adding components on my ships / asteroids that were a child of the parent object, like a turret attached to a ship where the turret itself is a target. If the ship is rotating around an axis (for example Y axis) and the turret is not at x=0 and z=0 my projection no longer works. I thought that using r * sin ( theta + omega * t) as the angular velocity component for the X position and r * cos ( theta + omega * t) for the Z position could work. Theta is the current rotation (with respect to world coordinates) and the omega is the eulerAngle rotation around the y axis.
I've quickly realized this only works with rotating around the y axis, and I can't put the sin into a quadratic equation because I can't extract the t from it so I can't really project this appropriately. I tried using hyperbolics but it was the same situation. I can create an arbitrary t, let's say t=2, and calculate where the object will be in 2 seconds. But I am struggling to find a way to implement the bullet speed projection with this.
Position_targetparent + Velocity_targetparent * t + [ANGULAR VELOCITY COMPONENT] = Position_shooter + Velocity_shooter * t + Bulletspeed * t
Delta_Position_X + Delta_Velocity_X * t + S * t = r * sin (theta + Omegay * t)
Delta_Position_Z + Delta_Velocity_Z * t + S * t = r * cos (theta + Omegay * t)
From here I have been spinning my wheels endlessly trying to figure out a workable solution for this. I am using the eulerAngle.y for the omega which works well. Ultimately I just need that instantaneous point in space that I should fire at which is a product of the speed of the bullet and the distance of the projection, and then my turrets aiming scripts will take care of the rest.
I have been looking at a spherical coordinate system based around the parents position (the center of the rotation)
Vector3 deltaPosition = target.transform.position - target.transform.root.position;
r = deltaPosition .magnitude;
float theta = Mathf.Acos(deltaPosition.z / r);
float phi = Mathf.Atan2(deltaPosition.y,deltaPosition.x);
float xPos = r * Mathf.Sin(theta) * Mathf.Cos(phi)
float yPos = r * Mathf.Sin(theta) * Mathf.Sin(phi)
float zPos = r * Mathf.Cos(theta)
Vector3 currentRotation = transform.root.gameObject.transform.rotation.eulerAngles * Mathf.Deg2Rad;
Vector3 angularVelocity = transform.root.gameObject.GetComponent<Rigidbody>().angularVelocity;
I can calculate the position of the object given these angles ... but I am struggling to turn this into something I can use with the omega * t (angular velocity) approach.
I am wondering if there is a more elegant approach to this problem, or if someone can point me in the right direction of a formula to help me think this through? I am not the best with Quaternions and EulerAngles but I am learning them slowly. Maybe there's something clever I can do with those?

Although the math is likely still tough, I suspect you can simplify the math substantially by having the "target" calculate its future position in local space. And then having it call that location to its parent, have that calculate it in local space, and so on up the hierarchy until you reach world space. Once you have its future position in world space you can aim your turret at that target.
For example an orbiting ship should be able to calculate its future orbit easily. This is an equation for an ellipse. Which can then send that local position to its parent (planet) which is presumably also orbiting and calculate that position relative to itself. The planet will then send this local position to its own parent (Star) and so on. Until you get to world space.
You can further simplify this math by making the bullet's travel time constant (flexible speed), so you can simplify figuring out the future position at a specific time. Depending on the scale of your game, the actual difference in speed might not be that different.
Another idea: Instead of doing all the calculations from brute force, you could "simulate" the target object forward in time. Make sure all the code that affects is position can be run separate from your actual update loop. Simply advance the clock way ahead, and see its future position without actually moving it. Then go back to the present and fire the gun at its future position.

I suggest to solve this problem approximately.
If you can describe the position of your target by a function over time, f(t), then you can approximate it using an divide and conquer strategy like this:
Algorithm (pseudo code):
Let f(t:float):Vector3 be a function that calculates the position of the target at time t
Let g(p:Vector3):float be a function that calculates how long the bullet would need to reach p
float begin = 0 // Lower bound of bullet travel time to hit the target
float end = g(target.position) // Upper bound
// Find an upper bound so that the bullet can hit the target between begin and end time
while g(f(end)) > end:
begin = end
end = end * 2 // Exponential growth for fast convergence
// Add break condition in case the target can't be hit (faster than bullet)
end
// Narrow down the possible aim target, doubling the precision in every step
for i = 1...[precision]:
float center = begin + (end - begin) / 2
float travelTime = g(f(center))
if travelTime > center: // Bullet can't reach target
begin = center
else // Bullet overtook target
end = center
end
end
float finalTravelTime = begin + (end - begin) / 2
Vector3 aimPosition = f(finalTravelTime) // You should aim here...
You need to experiment with the value for [precision]. It should be as small as possible, but large enough for the bullet to always hit the target.
You can also use another break condition, like restricting the absolute error (distance of the bullet to the target at the finalTravelTime).
In case that the target can travel faster than the bullet, you need to add a break condition on the upper bounds loop, otherwise it can become an infinite loop.
Why this is useful:
Instead of calculating a complex equality function to determine the time of impact, you can approximate it with a rather simple position function and this method.
This algorithm is independent of the actual position function, thus works with various enemy movements, as long as the future position can be calculated.
Downsides:
This function calculates f(t) many times, this can be CPU intensive for a complex f(t).
Also it is only an approximation, where the precision of the result gets worse the further the travel time is.
Note:
I wrote this algorithm from the top of my head.
I don't guarantee the correctness of the pseudo code, but the algorithm should work.

2 vectors difference to rotation

I've been hanging my head around this issue for some time now and I'm at the top of my head not figuring this out.
The issue:
Currently trying to do a fake "swing" moment. Where I have the player object becoming a child to an object you can latch onto during a keypush.
With this in mind, I need to get the correct rotation on the grappleobject, making the velocity of the player becoming the correct rotation on the grappleobject.
This is not meant to be 2D, but 3D. Which causes this headache.
http://img843.imageshack.us/img843/7434/rotations.jpg
I figured that if I could get the vector of position, and the vector of the direction where the player is going towards, as in velocity of the character motor, to get an unique Vector, and this vector info should be able to rotate the grappleobject correctly.
However, that's what I thought was easy, but I've been hacking at this for hours to get that vector difference to do the rotation as it was thought out to be.
tl:dr
2 Vectors, the difference of these two to one vector, this vector controls the rotation of grappleobject to "fake" a swing motion with the proper velocity.
Thank you in advance if there'll be an answer.
Added Info:
Current testings has been these.
///
Parent Orb is the grappleobjects which updates its transform depending on trigger events.
CurDirection is where the player is heading within the vector in world.
CurPos is where the player is at that point in time within the world.
CurDirection = (transform.position-ParentOrb.position);
CurDirection.Normalize();
motor.movement.velocity = (CurDirection);
CurPos = transform.position;
////
Also tried out to get the angle from the grappleobject to the player.
///
otherDirection = The direction of velocity in space.
OtherDirectionPre = Current position in space.
Vector3 targetDir = otherDirection;
Vector3 forward = otherDirectionPre;
angle = Vector3.Angle(targetDir, forward);
///
I assume these may not be helpful, but better to show where I have gotten so far.

I think you know the radius between the hit point and the center, let's call it r. With the velocity v of the player and the angular velocity ω (=omega) of the axis the player should be connected to you have the vector equation:
v = ω × r
Assuming all 3 vectors are perpendicular to each other you have ω = v / r.
EDIT: You get the radius r from the position of the center and the point of contact of your collision:
Vector3 r = collision.contacts[0].point - center;
float radius = r.magnitude;
float angularSpeed = rigidbody.veloctiy.magnitude / radius;

C# XNA - Calculating next point in a vector direction

lets say i have two points A & B in my 3D Space
now i want to start calculate points from A to B in direction of B and i want to continue calculation farther from B on the same line.
How t do it ?
what i am actually working on is bullets from plane.

If I understood your question correctly, you should first get the direction vector by calculating
dir = B - A
and then you can continue the travel by
C = B + dir
Otherwise, please clarify your question, like for example what you mean by "calculate points from A to B", because mathematically there is an infinite amount of points between A and B.
Edit:
If you want to trace a bullet path you have two options:
1) You implement it as hitscan weapon like done in many FPS; any bullets fired will immediately hit where they were aimed. This can be best achieved by doing a raytrace via Ray.Intersects and is probably the simplest and least computationally intensive way to do it. Of course, it's also not terribly realistic.
2) You make the bullet a physical entity in your world and move it during your Update() call via a "normal" combination of current position vector and movement/velocity vector, doing a manual collision detection with any hittable surfaces and objects. The advantage of this is that you can implement proper physics like travel time, drop, friction, etc., however it is also orders of magnitude more difficult to implement in a robust way. If you want to go this way, I suggest using either a pre-made physics API (like Bullet, but I'm not sure if there's a version for XNA) or at least doing extensive Google research on the subject, so you can avoid the many pitfalls of collision detection with fast-moving objects.

its a try to implement a 2D technique in 3D, i calculate the following once
position = start;
dx = destination.X - start.X;
dy = destination.Y - start.Y;
dz = destination.Z - start.Z;
distance = (float)Math.Sqrt( dx * dx + dy * dy + dz * dz );
scale = 2f / distance;
then i go on calculating
position.X += dx * scale;
position.Y += dy * scale;
position.Z += dz * scale;
but the result is still is not working of 3D Space, i am getting result for only 2 Dimension, the third axis is not being changed