How can I implement gravity?

Question

How can I implement gravity? Not for a particular language, just pseudocode...

Answer 1

As others have noted in the comments, the basic Euler integration method described in tenpn's answer suffers from a few problems:

• Even for simple motion, like ballistic jumping under constant gravity, it introduces a systematic error.

• The error depends on the timestep, meaning that changing the timestep changes object trajectories in a systematic way that may be noticed by players if the game uses a variable timestep. Even for games with a fixed physics timestep, changing the timestep during development can noticeably affect the game physics such as the distance that an object launched with a given force will fly, potentially breaking previously designed levels.

• It doesn't conserve energy, even if the underlying physics should. In particular, objects that should oscillate steadily (e.g. pendulums, springs, orbiting planets, etc.) may steadily accumulate energy until the whole system blows apart.

Fortunately, it's not hard to replace Euler integration with something that is almost as simple, yet has none of these problems — specifically, a second-order symplectic integrator such as leapfrog integration or the closely related velocity Verlet method. In particular, where basic Euler integration updates the velocity and position as:

acceleration = force(time, position) / mass;
time += timestep;
position += timestep * velocity;
velocity += timestep * acceleration;

the velocity Verlet method does it like this:

acceleration = force(time, position) / mass;
time += timestep;
position += timestep * (velocity + timestep * acceleration / 2);
newAcceleration = force(time, position) / mass;
velocity += timestep * (acceleration + newAcceleration) / 2;

If you have multiple interacting objects, you should update all their positions before recalculating the forces and updating the velocities. The new acceleration(s) can then be saved and used to update the position(s) on the next timestep, reducing the number of calls to force() down to one (per object) per timestep, just like with the Euler method.

Also, if the acceleration is normally constant (like gravity during ballistic jumping), we can simplify the above to just:

time += timestep;
position += timestep * (velocity + timestep * acceleration / 2);
velocity += timestep * acceleration;

where the extra term in bold is the only change compared to basic Euler integration.

Compared to Euler integration, the velocity Verlet and leapfrog methods have several nice properties:

• For constant acceleration, they give exact results (up to floating point roundoff errors, anyway), meaning that ballistic jump trajectories stay the same even if the timestep is changed.

• They are second order integrators, meaning that, even with varying acceleration, the average integration error is only proportional to the square of the timestep. This can allow for larger timesteps without compromising accuracy.

• They are symplectic, meaning that they conserve energy if the underlying physics do (at least as long as the timestep is constant). In particular, this means that you won't get things like planets spontaneously flying out of their orbits, or objects attached to each other with springs gradually wobbling more and more until the whole thing blows up.

Yet the velocity Verlet / leapfrog method are nearly as simple and fast as basic Euler integration, and certainly much simpler than alternatives like fourth-order Runge-Kutta integration (which, while generally a very nice integrator, lacks the symplectic property and requires four evaluations of the force() function per time step). Thus, I would strongly recommend them for anyone writing any sort of game physics code, even if it's as simple as jumping from one platform to another.

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Edit: While the formal derivation of the velocity Verlet method is only valid when the forces are independent of the velocity, in practice you can use it just fine even with velocity-dependent forces such as fluid drag. For best results, you should use the initial acceleration value to estimate the new velocity for the second call to force(), like this:

acceleration = force(time, position, velocity) / mass;
time += timestep;
position += timestep * (velocity + timestep * acceleration / 2);
velocity += timestep * acceleration;
newAcceleration = force(time, position, velocity) / mass;
velocity += timestep * (newAcceleration - acceleration) / 2;

I'm not sure if this particular variant of the velocity Verlet method has a specific name, but I've tested it and it seems to work very well. It's not quite as accurate as fouth-order Runge-Kutta (as one would expect from a second-order method), but it's much better than Euler or naïve velocity Verlet without the intermediate velocity estimate, and it still retains the symplectic property of normal velocity Verlet for conservative, non-velocity-dependent forces.

Edit 2: A very similar algorithm is described e.g. by Groot & Warren (J. Chem. Phys. 1997), although, reading between the lines, it seems that they sacrificed some accuracy for extra speed by saving the newAcceleration value computed using the estimated velocity and reusing it as the acceleration for the next timestep. They also introduce a parameter 0 ≤ λ ≤ 1 which is multiplied with acceleration in the initial velocity estimate; for some reason, they recommend λ = 0.5, even though all my tests suggest that λ = 1 (which is effectively what I use above) works as well or better, with or without the acceleration reuse. Maybe it's got something to do with the fact that their forces include a stochastic Brownian motion component.

Answer 2

Every update loop of your game, do this:

if (collidingBelow())
    gravity = 0;
else gravity = [insert gravity value here];

velocity.y += gravity;

For instance, in a platformer, once you jump gravity would be enabled (collidingBelow tells you whether or not there is ground right below you) and once you hit the ground it would be disabled.

Besides this, to implement jumps, then do this:

if (pressingJumpButton() && collidingBelow())
    velocity.y = [insert jump speed here]; // the jump speed should be negative

And pretty obviously, in the update loop you also have to update your position:

position += velocity;

Answer 3

A proper frame-rate independent* newtonian physics integration:

Vector forces = 0.0f;

// gravity
forces += down * m_gravityConstant; // 9.8m/s/s on earth

// left/right movement
forces += right * m_movementConstant * controlInput; // where input is scaled -1..1

// add other forces in for taste - usual suspects include air resistence
// proportional to the square of velocity, against the direction of movement. 
// this has the effect of capping max speed.

Vector acceleration = forces / m_massConstant; 
m_velocity += acceleration * timeStep;
m_position += velocity * timeStep;

Tweak gravityConstant, movementConstant and massConstant until it feels right. It is an intuitive thing and can take a while to get feeling great.

It's easy to extend the forces vector to add new gameplay - for instance add a force away from any nearby explosion, or towards black holes.

*edit: these results will be wrong over time, but may be "good enough" for your fidelity or aptitude. See this link http://lol.zoy.org/blog/2011/12/14/understanding-motion-in-games for more info.

Answer 4

If you want to implement gravity on a slightly bigger scale, you can use this kind of calculation each loop:

for each object in the scene
  for each other_object in the scene not equal to object
    if object.mass * other_object.mass / object.distanceSquaredBetweenCenterOfMasses(other_object) < epsilon
      abort the calculation for this pair
    if object.mass is much, much bigger than other_object.mass
      abort the calculation for this pair
    force = gravitational_constant
            * object.mass * other_object.mass
            / object.distanceSquaredBetweenCenterOfMasses(other_object)
    object.addForceAtCenterOfMass(force * object.normalizedDirectionalVectorTo(other_object))
  end for loop
end for loop

For even bigger (galactic) scales, gravity alone won't suffice to create "real" motion though. The interaction of star systems is to a significant and very visible extent dictated by Navier-Stokes equations for fluid dynamics, and you'll have to keep the finite speed of light - and thus, gravity - in mind too.

Answer 5

Pecant's answser ignored frame time, and that makes your physics behavior differently from time to time.

If you are going to make a very simple game, you can make your own little physics engine -- assign mass and all kinds of physics parameters for every moving object, and do collision detection, then update their position and velocity every frame. In order to accelerate this progress, you need to simplify the collision mesh, reduce calls of collision detection, etc. In most cases, that's a pain.

It's better to use physics engine like physix, ODE and bullet. Any of them will be stable and efficient enough for you.

http://www.nvidia.com/object/physx_new.html

http://bulletphysics.org/wordpress/