# Rigid body game physics

24 Oct 2019Inspired by Hubert Eichner's articles on game physics I decided to write about game physics as well. My motivation is to create a space flight simulator. The idea is to simulate a space ship with landing gear and docking adapter using multiple rigid bodies.

## The Newton-Euler equation

Each rigid body has a center with the 3D position *c*.
The orientation of the object can be described using a quaternion *q* which has four components.
The speed of the object is a 3D vector *v*.
Finally the 3D vector *ω* specifies the current rotational speed of the rigid body.

The time derivatives (indicated by a dot on top of the variable name) of the position and the orientation are (Sauer et al.):
The multiplication is a quaternion multiplication where the rotational speed *ω* is converted to a quaternion:

In a given time step *Δt* the position changes as follows

The orientation *q* changes as shown below
Note that *ω* again is converted to a quaternion.

Given the mass *m* and the sum of external forces *F* the speed changes like this
In other words this means

Finally given the rotated inertia matrix *I(t)* and the overall torque *τ* one can determine the change of rotational speed
Or written as a differential equation

These are the Newton-Euler equations.
The "x" is the vector cross product.
Note that even if the overall torque *τ* is zero, the rotation vector still can change over time if the inertial matrix *I* has different eigenvalues.

The rotated inertia matrix is obtained by converting the quaternion *q* to a rotation matrix *R* and using it to rotate *I₀*:

The three column vectors of *R* can be determined by performing quaternion rotations on the unit vectors.
Note that the vectors get converted to quaternion and back implicitely.

According to David Hammen, the Newton-Euler equation can be used unmodified in the world inertial frame.

## The Runge-Kutta Method

The different properties of the rigid body can be stacked in a state vector *y* as follows.
The equations above then can be brought into the following form
Using *h=Δt* the numerical integration for a time step can be performed using the Runge-Kutta method:

The Runge-Kutta algorithm can also be formulated using a function which instead of derivatives returns infitesimal changes

The Runge-Kutta formula then becomes

Using time-stepping with *F=0* and *τ=0* one can simulate an object tumbling in space.