You need to know two things to design the system:
- Where you want the systems's center of mass to be.
- How to calculate such a thing.
Having read the wikipedia and watch some video I'm guess the answer to (1) is "a short distance directly below the gimbal" (for both high and low mode, though that means slightly different places).
How to calculate the CoM for a set of point masses.
Establish a coordinate system. Say $z$ increases up and starts at the camera platform, and $x$ increase to the front and starts at the location of the gimbal. This makes $y$ increase to the left if we want the usual convention, and we'll start it at the gimbal as well.
Find the (three dimensional!) location and mass of each part of the system (remember so far I'm only talking about little dense masses) and call these $(x,y,z)_i$ and $m_i$ where $i$ is just an index that runs over all you bits. Write
$$ x_{com} = \frac{\sum_i x_i m_i}{\sum_i m_i} $$
and an identical equation for $y$ and $z$. And now we know the location of the the Center of Mass.
For continuous objects, we could turn those sums into integrals, but you probably don't want to do that. Especially as you don't know the mass distribution of the camera.
Instead I will suggest finding the center of mass for each object (and this is why the wsclater link calls the red dots center of mass, they are for each object), by the simple expedient of balancing or hanging them.
- If you hang an object from any point, it's CoM will lie below the point you hang it from.
- If you balance an object on a thin edge, the objects CoM will lie above the edge.
Combine several such measurements and you can find the three diminsional position of the CoM.
Now here's the trick. Once you know the CoM for an extended object, you can treat the whole thing as a point mass at that location (for the purposes of calculating the CoM of a system that includes the object).
OK. Making progress, but were, still not done.
You know where you want the CoM, and need to find where to put the masses. Which is backwards from what we did above.
You need to control three values (the $x$, $y$, and $z$ positions of the CoM), and you have three equations. Which means that you can have only three adjustable parameters in your system.
Side note: The value of $y$ is easy, if you position the camera so that it's CoM is centered. Then, because everything else is symmetric, it happens automatically. So I'm going to only worry about $x$ and $z$
What you do is fix all the position relationships in the system except for two. Maybe the height of the lower control mass and the horizontal position of the middle one. Those two you make variables. Now you write down the CoM equations above *with those two variables in there and the desired values for the location of the CoM. The resulting system is solvable. Do the algebra and you're done.
But leave some ability to fine tune things (say by putting the masses on threaded rods). This lets you only worry about getting close with the calculation. The pictured one appears to use a flexible support to provide the adjustability. That will work, but may require constant tinkering.
Here is a brief analysis of the problem which involves:
i) Momentum and the concept of impulse
ii) Analysis of a force into two components
iii) Moment of a force
iv) Moment of inertia and the parallel axis theorem
For the sake of simplicity let us assume we have a spherical spaceship of radius R in the outer space. Let there be a rocket at the surface of the spaceship pointing in a direction that does not pass through the centre of the sphere. Imagine now we are firing the rocket.
The impulse will generate a force, $F$, in the usual way and push the spaceship in the direction of the rocket. The force exerted on the sphere in that direction can be analysed into the tangent and the perpendicular to the surface of the sphere. If $\theta$ is the angle between the rocket direction and the normal to the sphere we have:
Tangent component: $F_T=F\sin(\theta)$
Normal component: $F_N=F\cos(\theta)$.
The normal component is parallel to the radius of the sphere and passes through the centre (CM) and has no moment with respect to the centre. This component will push the sphere in the normal direction.
The tangent component has a moment with respect to the centre
$M=FR\sin(\theta)$.
This component would rotate the sphere, should the axis of the sphere be pivoted, but it is not! However, due to the inertia of the mass of the sphere, it would be sufficient to give a pivotal leverage for the tangent force to rotate the sphere. The law of conservation of energy must be written, for a short time interval of application of the force, as it moves the spaceship by a displacement ${\bf x}$, in the form
${\bf F.x}=\frac {1}{2}mv^2+ {\frac {1}{2}I{\omega}^2}$
The first term on the RHS is the kinetic energy due to the linear motion, and the second is the kinetic energy due to the rotational motion.
The spaceship will rotate about the centre of mass of the spaceship, and the axis of rotation will be perpendicular to the plane made between the tangent force and the radius of the sphere. The axis will pass through the centre of the sphere because, according the parallel axis theorem, the moment of inertial is minimum when the axis of rotation passes through the centre of mass. Hence the spaceship will have the minimum energy, and this is the preferred energy state for the spaceship.
Best Answer
This link should help:
https://www.faa.gov/data_research/research/med_humanfacs/oamtechreports/1960s/media/AM62-14.pdf
For a natural sitting position with hands in the lap, the center of gravity is about (from what I can make out from the text - it is not clear) 8 3/8" and 9 1/8" from the horizontal and vertical reference points, which I believe are taken to be the horizontal and vertical surfaces of the seat. This would put the centre of mass somewhere around the position of your navel.
Including a wheelchair in the centre of mass would shift it downward and rearward probably by a small amount - maybe somewhere closer to your lap, but it all would depend on your mass and the mass of your wheelchair, etc.