# OpenServo

## Mathematical Model

The OpenServo inclusive gear can be modeled simplified as a mass with moment of inertia mounted to a DC-motor. The motor generates a torque depending on the applied voltage and the gear generates a converse torque due to the friction . Furthermore a mounted load is modeled by torque .

The transfair behavior of the whole servo system can be sperated in two submodels. A mechanical part and a electro-magnetic part.

### Mechanical Part

The rotation of the servo is described by the conservation of angular momentum. This means, the angular momentum of a rigid body only changes due to torques exerting to the center of mass, which yields to the equation of motion:

where is the angular velocity of the motor. The friction force is consisting of three components as shown in the following figure:

- Columb-Friction is the offset with the same sign as the velocity
- Viscose-Friction is the proportional component
- Stribeck-Effect is the increasing friction with decreasing velocity in the low-velocity region

### Electro-Magnetic Part

The schematic diagram of the electic motor can be modeled as:

where are the voltage, current, resistance, inductance of the armature, respectively, and is the inner source volatage generated by the back-emf. The inner source volatage is caused by the coil while moving through the magnetic field . The dependency can be expressed linear as followed:

where is a motor constant. Assuming the magnetic field to be constant, it can be expressed together with in a compound parameter . The generated torque can be derived over a simple power balance:

with

which yields in a expression for the torque:

The dependency between the current and the outer volatage can be derived with Kirchhoff's law and yields to

### Complete Model

Combining the differntial equation of 2nd order for the mechanical model and the differential equation of 1st order for the electro-magnetic model the complete model can be written in 3 differential equations of 1st order, the so called state-space-form.

where the new state is the position of the servo. Introducing the state vector

and seperating the nonlinear friction term and the unkown load torque, the equations can be written in matrix notation: