OpenServo

Mathematical Model

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

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:

$J \dot{\omega} = \tau - F_l - F_f(\omega,\tau-F_l)$

where $\omega$ is the angular velocity of the motor. The friction moment $F_f$ is a function of the angular velocity and the sum of all other moments. It consists 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 $u_A, i_A, R_A, L_A$ are the voltage, current, resistance, inductance of the armature, respectively, and $u_q$ 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 $\Phi$ . The dependency can be expressed linear as followed:

$u_q = \underbrace{C_1 \Phi}_{K_{\tau}} \omega$

where $C_1$ is a motor constant. Assuming the magnetic field to be constant, it can be expressed together with $C_1$ in a compound parameter $K_{\tau}$ . The generated torque can be derived over a simple power balance:

$P_{Electrical}=P_{Mechanical}$

with

$\begin{array}{lll}P_{Electrical}&=&u_q i_A\\P_{Mechanical}&=&\omega \tau\end{array}$

which yields in a expression for the torque:

$\tau=\frac{ u_q i_A}{\omega}=\frac{\omega K_{\tau} i_A}{\omega}=K_{\tau} i_A$

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

$i_a R_A + \frac{d}{dt} i_A L_A=u_A-u_q$

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.

$\begin{array}{lll}\dot{\theta}&=&\omega\\ \dot{\omega}&=&\frac{K_{\tau}}{J}i_A-\frac{1}{J}F_l-\frac{1}{J}F_f(\omega,\frac{K_{\tau}}{J}i_A-\frac{1}{J}F_l)\\ \dot{i_a}&=&-\frac{R_A}{L_A}i_a-\frac{K_{\tau}}{L_A}\omega+\frac{1}{L_A}u_A\end{array}$

where the new state $\theta$ is the position of the servo. Introducing the state vector

$x=\left[\begin{array}{c} \theta\\ \omega\\ i_a \end{array}\right]$

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

$\dot{x} = \underbrace{\left[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & \frac{K_{\tau}}{J} \\ 0 & -\frac{K_{\tau}}{L_A} & -\frac{R_A}{L_A} \end{array}\right]}_{A} x + \underbrace{\left[\begin{array}{c} 0\\0\\\frac{1}{L_A} \end{array}\right]}_{B} u_A + \left[\begin{array}{c} 0\\\frac{1}{J}F_l-\frac{1}{J}F_f(\omega,\frac{K_{\tau}}{J}i_A-\frac{1}{J}F_l)\\0 \end{array}\right]$