trick/trick_sims/SIM_singlerigidbody/README.md

Single Rigid Body

SIM_singlerigidbody is a simulation of the motion of a sphere that is a single rigid body with six degrees of freedom.

The motion of the sphere is found by using Euler's equations of motion, and can be controlled by changing several initial input values such as force, position, angle, angular velocity, velocity, and position of the force. The recorded data of the motion can then be viewed and analyzed.

Dynamics model

Equations of Motion

To determine the sphere's motion, we will use Euler's equations of motion for a single rigid body. Euler's equations describe the combined translational and rotational dynamics of a rigid body in three-dimensional space.

Where,

• is a 6x1 column vector.
• F is the applied force.
• τ is the torque.

• is a 6x6 matrix.
• m is the mass.
• I _3x3 is the identity matrix.
• I is the inertia.
• C _m is the center of mass.

• is a 6x1 column vector.
• a _o is the acceleration.

• is a 6x1 column vector.
• ω ^A is the angular velocity.

Force

The force acting on the sphere is calculated by the cross product of the rotation matrix and initial force.

Torque

Torque is found by the cross product of the rotation matrix and the cross product of the position of force and initial force.

Mass

The default value of the mass of the sphere is:

Radius

The default value of the radius is:

Center of mass

The center of mass of a sphere is equal to its radius.

Inertia

Inertia of a sphere is calculated by:

Acceleration and angular acceleration

To determine the acceleration and angular acceleration, dLU_Choleski must be used. dLU_Choleski solves linear sets of equations in the form of [A]x = b and returns the vector x, which is the acceleration. It decomposes the symmetric, positive definite [A] matrix into a lower triangular form, where we can then substitute and our equation becomes:

The term 'a' is a 6x1 vector. Acceleration is equal to the first three values of vector a and angular acceleration is equal to the last three values of vector a.

State Integration

The state is defined by position, velocity, angle, and angular velocity. They are calculated by numerically integrating velocity, acceleration, angular velocity, and angular acceleration, respectively.

Position

The position of the sphere is determined by integrating the velocity over time.

Velocity

The velocity is determined by integrating the acceleration of the sphere over time.

Angle

The angle is determined by integrating the angular velocity over time.

Angular velocity

The angular velocity is determined by integrating the angular acceleration over time.

Inputs

Variable	Type	Units
dyn.body.POS_INIT	double[3]	m
dyn.body.ANGLE_INIT	double[3]	rad
dyn.body.OMEGA_INIT	double[3]	rad*s^-1
dyn.body.FORCE_INIT	double[3]	N
dyn.body.TORQUE_INIT	double[3]	N*m
dyn.body.VEL_INIT	double[3]	m/s
dyn.body.POS_FORCE	double[3]	m

Outputs

Variable	Type	Units
dyn.body.pos	double[3]	m
dyn.body.vel	double[3]	m/s
dyn.body.acc	double[3]	m/s^2
dyn.FORCE_INIT	double[3]	N
dyn.force	double[3]	N
dyn.body.angle_force	double[3]	rad
dyn.body.omegaDot	double[3]	rad/^s2
dyn.body.omega	double[3]	rad*s^-1
dyn.body.rotate	double[3]