I don't understand how the pneumatic control precession actuation on the Brennan Monorail's gyros further balances the vehicle

I linked a YouTube video from Primal Space, The Self Balancing Monorail, discussing how the Brennan Monorail works.

@6:40 timestamp the narrator explains how the pneumatic pressure "[pushes] the gear rack and [forces] the gyros to precess in the opposite direction until the train returned to equilibrium." The video @5:58 initially shows the train being tipped over counterclockwise (CCW) from the front, indicating a CCW torque acting on the gyros. This would force the left (starboard) gyro to precess clockwise (CW) when viewed from the top, as the video accurately depicts. (For simplicity let's just focus on the left gyro for now.) This is because the left gyro's angular momentum is pointing to the right.

However, I fail to grasp how precessing the left gyro back to equilibrium (in the CCW direction), via the pneumatic piston & gear rack, would tilt the train back to its self balancing, upright position. In order for the left gyro to precess in this manner, wouldn't there have to be a torque acting on the gyro in the CW direction (when viewed from the front)?

If so, aren't the upper & lower bearings of the outer gimbal--housing the left gyro--going to experience a leftward & rightward force from the gyro's CCW torque reaction, respectively, causing the monorail to tip over even more CCW? If someone can help clarify any misconceptions, or pinpoint any incorrect assumptions that I'm making, I would greatly appreciate it. Perhaps I'm making the correct logical reasoning, and the claim from the narrator in the video (which I bolded above) is a bit misleading.

Example 22.4 Gyroscope on Rotating Platform from Physics LibreTexts might help some readers visualize the forces that act on gyroscopic mounts when a spinning disc is constrained on a rotating platform. The way I like to think how gyroscopic precession works is by fundamental definition that torque causes a change in angular momentum over change in time. When a torque vector is applied perpendicular to a gyro's angular momentum vector, then it causes its angular momentum to change direction (not magnitude). The gyro will hence undergo a change in spin orientation (aka precession). This precession angular velocity vector is orthogonal to the torque and angular momentum vectors.

On a side note I've seen another useful application of gyros that are used to stabilize the rocking motion of boats. For example, the Seakeeper is a marine gyro stabilizer that implements active control per hydraulic actuators to regulate the precession rate. Here are a couple YouTube videos about this:

Here is a document, How Gyros Create Stabilizing Torque, outlining the operating principle of VEEM marine gyros, which stabilize a vessel's rolling motion. It seems that the main purpose of the control system is to manage the "naturally occurring precession motion to ensure that it is maximized within the operating limits of the device. The operating limits are set during product design, and include precession range of movement, and maximum allowed precession rate" (Page 6, Step 4). It doesn't mention here that the control torque further stabilizes the vessel.

Note that there is a difference between the Brennan monorail and a boat/vessel when it comes to the vehicles leaning to one side: Without stabilization, the former has a tendency to tip over due to rolling torque induced by wheel reaction forces (unstable equilibrium), whereas the latter has a tendency to stay upright due to the self-balancing rolling torque caused by buoyancy forces from still water (stable equilibrium). Does this mean that if the monorail were to operate for an extended period of time, it will eventually tip over if there was an overall disturbance that would force it to roll to one side, due to the gyros reaching the operating limit of their precession range?

To make the history of this question clear: you had previously posted this question on physics.stackexchange

In my opinion your question squarely belongs on physics.stackexchange, but for unfathomable reasons your question was closed there.

While it is customary for physics authors to describe onset of gyroscopic precession in terms of angular momentum vector representation, using the angular momentum vector concept is a very opaque approach; you are pretty much guaranteed to be wrongfooting yourself.

So: by all means, do not use the concept of angular momentum vector for the context of gyroscopic precession; it introduces unnecessary complexity, it prevents understanding.

Video footage of marine gyro stabilization is helpful to understanding, because it shows a system in actual operation.

I will use the following three terms:
Rolling: defined relative to the gyro wheel: it is the spin of the gyro wheel
Pitching: defined relative to the gyro wheel: a tilt of the gyro wheel, perpendicular to the roll axis.
Swiveling: defined relative to the structure of the gyro stabilization assembly as it sits bolted to the hull of the ship

The standard gyro stabilization setup uses two gyro wheels so as to minimize side effects. The two gyro wheels are counter-rotating. The powerful actuators enforce a swiveling motion of the gyro wheels. The two gyro wheels are being counter-swiveled. Since the two wheels are counter-spining their pitching response is in a co-pitching direction.

The suspension of the gyro stabilization assembly is such that the gyro wheels do not have freedom to pitch relative to the enclosure that they are suspended in, and that enclosure is bolted to the hull of the ship. So: when the actuators enforce a swiveling motion of the gyro wheels the ship as a whole is forced to follow the pitching response.

In my answer on physics.stackexchange I had already linked to the explanation of why, when a spinning gyro wheel is subjected to swiveling motion the response of the gyro wheel is to pitch.

In the case of marine gyro stabilization I presume it is standard to have the gyro unit use data input from a separate Inertial Measurement Unit

According to the Primal Space video Brennnan had opted to use the gyro wheels themselves as tilt sensors. The engineering problem that Brennan had to solve is that he needed a slight overreaction from the stabilization unit. On its own (without the additional pneumatic system) the response of the gyro wheels would only slow down a listing of the train carriage, and not restore back to proper orientation.

The engineering problem was one of boosting the response to listing motion of the train carriage, so that the orientation of the train carriage would be restored, while still keeping that response gentle enough to avoid inducing oscillation.

Comparison: in aviation there is an overreaction danger named pilot induced oscillation

• I answered at the Physics stack exchange as well and I agree with you. Like I commented there, it is a grey area question, but if I had to pick I would call it physics, since it is a "why does this machine work" (which means "why does it behave this way") and not an engineering design question. Physics is a bit "close happy" IMO. Jan 14 at 2:11
• By overreaction from the stabilization unit it seems that the video has it backwards? To provide the overcorrection to force the monorail to lean the other way to balance itself, then I believe the gyros would have to swivel (via the actuators) at a faster rate than its natural precession. This positive feedback loop will provide the amplified counter torque to bring the monorail back to equilibrium after the initial disturbance. That means that the directional control valves should work in the opposite way. Jan 14 at 4:30
• I have only skimmed the video, so I can't comment on the video content. But yeah, without actuators (passive system) a developing lean of the train carriage will be slowed, but not reversed. A passive gyro system acts as a source of tons of extra inertia against developing a lean, way more inertia than the amount of gravitational mass that is added. The function of the actuators is to push the swivel of the gyro wheel ahead of the amount of swivel as compared to the passive configuration. In response to the actuated swiveling motion there will be a pitching torque. Jan 14 at 10:37
• @AdamYassine About the video: at 4:44 the narrator gets it right: "the key to all of this was taking control of the gyro's precession. By purposefully precessing them quicker than they would normally precess a stronger force would be put into righting the vehicle." However, at 6:43 the narration goes off the rails: "forcing the gyros to precess in the opposite direction until the train returned to equilibrium". At 6:19 the view is zoomed out. I think there is an error. Jan 14 at 11:41

This was my answer, which I believe is correct and would take comments / criticisms on:

In the version shown in the video, each gyro has angular momentum pointing inward. As you say,"the left gyro's angular momentum is pointing to the right." The system is arranged so that when the train begins to tip, the pneumatics precess the gyros so that the angular momentum of each one turns toward the angular momentum of the tip. Take a moment to think through that a bit. Note that this is what happens naturally, so the pneumatic controls are augmenting the motion of the gyros.

At the timestamp you give, the train tips with angular momentum pointing out the front of the train (when being precise, I always say which way angular velocity / momentum points, but this means that when viewed from the front the train tips counter clockwise as you say).

The gyros are precessed so that the angular momentum of each is also turned toward the front. This means that the outer faces of the gyros are turned toward the rear, as is shown in the video.

Forget the train entirely for a minute and just think through what happens when you turn over a gyroscopic object on a system with the freedom to rotate. The general demo is when you sit in an office chair with a spinning wheel and turn the wheel so that the angular momentum points up or down. In response, the office chair will rotate with with the opposite sense.

Similarly, if you actuate these gyros so that the initially cancelling transverse angular momenta rotate so that each has a component in the forward direction (which now add rather than cancel), then the train will roll with angular velocity pointing toward the rear (the opposite sense). This is basic angular momentum conservation.

The device shown in the video has sufficient gain so that the gyros turn more than the train rolled, so that they provide a restoring torque.

Please let me know if this is not totally clear. I would say that like almost all pop science and engineering videos, the language the narrator uses is very imprecise and leads to confusion. This one is no exception, and I agree with your opinion that the explanation that "the gyros are precessed back to equilibrium to stabilize the train" is wrong. The pneumatics, like I said augment the precession of the gyros away from their equilibrium points.

The reason for this is to augment the counter torque that rights the train. Just think of the train as a box, and like all boxes, it is subject to angular momentum conservation. When the gyros are precessed in some way, how does that change the angular momentum inside the box? The train must rotate the opposite way.

Okay, very good catch. The video description is wrong. In the video, the pneumatic system is described as having a restoring force on the rack. That isn't how it works. The pneumatic system is acting as an amplifier. It acts in the same direction as the perturbation and increases the procession rate. This improves the response, and the carriage recovers from the perturbation more quickly. The pneumatic system will need to provide a damping response also, or the delay in pressurizing and venting the cylinder will tend to produce a dynamic wobble.