Model-based Control

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

Camera

From the perspective of PID control and gimbal axis coupling, the camera should ideally have these properties:

• I_roll ≈ I_yaw
• I_pitch ≈ I_roll, I_yaw or 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, or a long "needle-like" camera with the "needle" along the pitch axis.

Gimbal

For the roll arm a condition I⁽²⁾_yaw ≈ I⁽²⁾_pitch, similar to the first condition for the camera, would ideally be fulfilled. This is however unrealistic in practical builds.

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 is much reduced and the PID controller can work better. Also vibrations in the roll arm limiting the controller performance are lesser.

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 ( a² + b² )

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 ∝ 2.52 + 5.52 = 36.5	1
roll	I ∝ 5.52 + 92 = 111.25	3.05
yaw	I ∝ 2.52 + 92 = 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 ∝ 2.52 + 4.52 = 26.5	1
roll	I ∝ 4.52 + 6.32 = 59.94	2.26
yaw	I ∝ 2.52 + 6.32 = 45.94	1.73

Camera Mobius

width	height	length
6 cm	2 cm	3 cm

axis	approximate moment of inertia	relative ratio
pitch	I ∝ 62 + 22 = 40	1
roll	I ∝ 22 + 32 = 13	0.325
yaw	I ∝ 62 + 32 = 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.