# Gimbal Model

The STorM32's model-based gimbal control can take into account, to a certain extend, the moments of inertia of the camera, and those of the roll and yaw arms. It also allows us to model the different KV values of the pitch, roll and yaw motors.

For the underlying theory see Camera Gimbals: A Robotics Approach.

# Parameters

The camera is described by the three principal moments of inertia

*I*,_{pitch}*I*,_{roll}*I*_{yaw}

The roll arm is modeled with these moments of inertia

*I*,^{(2)}_{roll}*I*^{(2)}_{yaw}

and the yaw arm with

*I*^{(3)}_{yaw}

In the GUI they are accessible as the T parameters. The model-based PID controller can also take into account different KV values of the pitch, roll and yaw motors via the K parameters.

# Ideal Camera/Gimbal Design

From the perspective of PID control and gimbal axis coupling, these suggestions can be deduced:

### Camera

The camera should ideally have these properties:

*I*≈_{roll}*I*_{yaw}*I*≈_{pitch}*I*,_{roll}*I*or_{yaw}*I*<<_{pitch}*I*,_{roll}*I*_{yaw}

The first condition is more important than the second, and should be approached if possible.

The second condition suggests a sphere or cube shaped camera, or a long "needle-like" camera with the "needle" along the pitch axis.

### Gimbal

For the roll arm a condition *I ^{(2)}_{yaw}* ≈

*I*, similar to the first condition for the camera, would ideally be fulfilled. This is however almost always unrealistic in practical builds.

^{(2)}_{pitch}Whenever possible, the roll arm should be designed as a U-shaped bracket and not as an L-shaped arm. This brings two benefits: First, the un-modelled non-linearity in the gimbal dynamics is much reduced and the PID controller can work better. Also vibrations in the roll arm limiting the controller performance tend to be lesser.

The yaw arm is a bit of a problem, as it essentially always will have to be of an L-shaped form, bringing in serious gimbal-axis couplings. The situation can be improved by mounting a weight on the roll axis opposite to the camera, e.g. a battery.

# Moments of Inertia

Estimates for the relative ratios of the moments of inertia of the camera can be obtained by approximating it by a cuboid.

For a homogeneous solid cuboid holds

*I* = 1/12 *M* ( *a*^{2} + *b*^{2} )

Links:

- https://en.wikipedia.org/wiki/Moment_of_inertia
- https://en.wikipedia.org/wiki/List_of_moments_of_inertia

### Camera Panasonic

width | height | length |
---|---|---|

2.5 cm | 5.5 cm | 9 cm |

axis | approximate moment of inertia | relative ratio |
---|---|---|

pitch | I ∝ 2.5^{2} + 5.5^{2} = 36.5 |
1 |

roll | I ∝ 5.5^{2} + 9^{2} = 111.25 |
3.05 |

yaw | I ∝ 2.5^{2} + 9^{2} = 87.25 |
2.39 |

### Camera GoPro Hero5

width | height | length |
---|---|---|

2.5 cm | 4.5 cm | 6.3 cm |

axis | approximate moment of inertia | relative ratio |
---|---|---|

pitch | I ∝ 2.5^{2} + 4.5^{2} = 26.5 |
1 |

roll | I ∝ 4.5^{2} + 6.3^{2} = 59.94 |
2.26 |

yaw | I ∝ 2.5^{2} + 6.3^{2} = 45.94 |
1.73 |

### Camera Mobius

width | height | length |
---|---|---|

6 cm | 2 cm | 3 cm |

axis | approximate moment of inertia | relative ratio |
---|---|---|

pitch | I ∝ 6^{2} + 2^{2} = 40 |
1 |

roll | I ∝ 2^{2} + 3^{2} = 13 |
0.325 |

yaw | I ∝ 6^{2} + 3^{2} = 45 |
1.125 |

As evidenced by these estimates, the shape of the Mobius camera is not ideal from a gimbal control perspective, and this camera is thus more difficult to stabilize than others.