Rigid Body Simulation builds off of the dynamics of Particles.
Each particle has an associated position and velocity ¹ . The entire system has a state The dynamics follow that of Newton (see the ODE below) With initial conditions and .

From Particles to Rigid Bodies

Theory

Rigid Bodies are characterized with both a translation and a rotation .
Rigid Bodies are not scaled. Therefore, we define a fixed space called the Body Space. and transform the body space to world space.
Assume: The Center of Mass is at the origin.
- can be interpreted the location of the center of mass at time (see below). This can be proven mathematically.
- can be interpreted as the position of the axes in body space at time .
Every point on the rigid space (in Body Space) is transformed as follows:
Linear velocity is defined as
If we have a rotation matrix, then we can define the angular velocity as follows

Where

And we apply this column wise so that
To calculate mass, we conceptually assume that the rigid body is composed of particles. Let denote the location of the particle in rigid body space. The dynamics of this particle are the same as that of the rigid body

The total mass is simply the sum of all individual particle masses
- The velocity of a particle can be calculated using its dynamics ²
- The center of mass of the system is defined as The body coordinate system is defined so that
- One important relation is as follows Intuitively this says that the first moment of mass is .

To introduce dynamics, we assume that at a geometric location of interest on the rigid body, there is a particle. Let be the total force from external forces on particle . Also let be the external torque.

The torque is defined as
The total force and total torque is defined as ³
- The torque tells us the distribution of forces on the body.

The momentum of the system is defined as the total momentum of our particles
- Thus, the linear momentum and the force are related as follows
The angular momentum of the body is defined as

Where is the inertia tensor. The Torque and the Angular Momentum are related as follows:
As a sanity check, we have that linear momentum is not dependent on the body’s rotation and angular momentum is not dependent on the body’s translation.
The Inertia Tensor is defined as follows. Let be the displacement of the -th particle from the center of mass, then is defined using the following
- Another way to write this is as follows. Denote $𝟙$ as the Identity Matrix. Then $𝟙 𝟙 𝟙 𝟙 𝟙$ Note that is an orthogonal matrix so that $𝟙$ . We also treat as a scalar.
- Now define the mass of the body as $𝟙$ The Inertial Tensor is then

The state of the rigid body is now defined as

We assume that and are constants.

We also compute

To calculate the derivative
The rotation transformation can be represented either as a matrix or as a Quaternion.
The Quaternion Approach is better since it is less susceptible to drift when numerically computed.
- We represent the quaternion as .
- Each quaternion is assumed to be a unit quaternion.
- Rotation of radians about a unit axis is then represented as
- The product of the quaternions indicate composite rotation.
- Angular velocity is defined using
- The Rotation Matrix is then derived from the quaternion when we need it.