Hilkert, J. M. (2004). A comparison of inertial line-of-sight stabilization techniques using mirrors.Proc. SPIE 5430, Acquisition, Tracking, and Pointing XVIII.
鏡面穩定的關鍵一篇。它指出反射定律在垂直於 LOS 的軸上帶來一個固有的 2:1 關係，而且——這是最重要的一句——單純把鏡面對慣性空間穩住並不會穩住 LOS，那個 2:1 反而讓鏡面穩定架構對基座運動特別敏感。它比較了幾種結合機構與慣性／相對運動感測器的作法，逐一討論各自的取捨。下面第五節整節在展開這句話。
Dynamics Modeling and Theoretical Study of the Two-Axis Four-Gimbal Coarse–Fine Composite UAV Electro-Optical Pod (2020).Applied Sciences, 10(6), 1923.
粗精複合軸的完整處理。用有限元分析與關鍵零件應力分析做結構、框架採 7075 鋁合金以達成全機 1 kg 以下的超輕量要求；依歐拉剛體動力學模型導出兩軸四框之間的傳動路徑與運動學耦合補償矩陣。致動採超音波馬達（USM）當粗級、音圈馬達（VCM）當精級。同組另一篇把該模型代進 DOB 抑制控制，報稱擾動抑制比 PID 好上 90%、比傳統 DOB 好 25%。
Ekstrand, B. (2001).IEEE TAES 37(3). 前面幾篇一直在引的那篇——它的慣量對稱條件說明部分交叉耦合可以在架構階段設計掉，這是「架構決策先於控制設計」最直接的學理依據。
Hilkert, J. M. (2004). A comparison of inertial line-of-sight stabilization techniques using mirrors. Proc. SPIE 5430, Acquisition, Tracking, and Pointing XVIII. DOI: 10.1117/12.541808
Dynamics Modeling and Theoretical Study of the Two-Axis Four-Gimbal Coarse–Fine Composite UAV Electro-Optical Pod (2020). Applied Sciences, 10(6), 1923. DOI: 10.3390/app10061923
Modeling and Stability Analysis of Coarse–Fine Composite Mechatronic System in UAV Multi-Gimbal Electro-Optical Pod (2020). Link
Ekstrand, B. (2001). Equations of Motion for a Two-Axes Gimbal System. IEEE Transactions on Aerospace and Electronic Systems, 37(3), 1083–1091. Link
Hilkert, J. M. (2008). Inertially Stabilized Platform Technology: Concepts and Principles. IEEE Control Systems Magazine, 28(1), 26–46.
Masten, M. K. (2008). Inertially Stabilized Platforms for Optical Imaging Systems. IEEE Control Systems Magazine, 28(1), 47–64.
Mokbel, H. F., Ying, L. Q., Roshdy, A. A., et al. (2012). Design Optimization of the Inner Gimbal for Dual Axis Inertially Stabilized Platform Using Finite Element Modal Analysis. International Journal of Modern Engineering Research.
This is the twenty-ninth post in the LocalPapa Notes dev-log series, and the twelfth on gimbals.
After the previous post laid the literature out as a map, one cell stood out: architecture choice has the fewest papers, and yet it is the first decision in the design flow.
The reason is not hard to see. "Two axes or three", "direct drive or geared" — the answers depend on mission requirements, which does not make a paper with general validity. So the topic drifts into vendor whitepapers and word of mouth.
But these can be computed. This post puts a number on each of the four decisions.
The literature
Academic work on architecture is thin, but not absent:
Hilkert, J. M. (2004). A comparison of inertial line-of-sight stabilization techniques using mirrors.Proc. SPIE 5430, Acquisition, Tracking, and Pointing XVIII.
The key paper on mirror stabilization. It notes that the law of reflection introduces an inherent 2:1 relationship in the axis perpendicular to the LOS, and — the most important sentence — that simply stabilizing the mirror in inertial space will not stabilize the LOS; the 2:1 in fact renders the mirror configuration particularly susceptible to base motions. It compares several techniques combining mechanisms with inertial and relative-motion sensors, discussing the trade-offs of each. Section six below unpacks that sentence.
Dynamics Modeling and Theoretical Study of the Two-Axis Four-Gimbal Coarse–Fine Composite UAV Electro-Optical Pod (2020).Applied Sciences, 10(6), 1923.
A full treatment of the coarse-fine architecture. Structure designed with finite element analysis and stress study of key components, frames in 7075 aluminium alloy to meet an under-1 kg ultralight requirement; the transmission path and kinematic coupling compensation matrix between the two-axis four-gimbal structures derived from a Euler rigid body dynamics model. Actuation uses an ultrasonic motor (USM) as the coarse stage and a voice coil motor (VCM) as the fine stage. A companion paper substitutes that model into DOB suppression control and reports disturbance rejection up to 90% better than PID and 25% better than a conventional DOB.
Ekstrand, B. (2001).IEEE TAES 37(3). The paper this series keeps returning to — its inertia symmetry condition shows that part of the cross-coupling can be designed out at the architecture stage, which is the most direct theoretical basis for putting architecture decisions before control design.
These papers model and control their respective architectures. None answers "which one should I choose" — that would require comparing architectures under one set of specs, and a paper usually covers only one. The comparisons below are my own, using the constants this series has carried throughout.
Scope of citation in this post
The method and contribution descriptions above come from abstracts and public paper pages. This environment returns 403 for most publishers, so I did not verify full texts and no unverified experimental values are quoted. The backlash magnitudes (harmonic 1–3 arcmin, planetary 3–10 arcmin) are the commonly published ranges for those transmission types, not any one vendor's data; the conversions and comparisons are my own.
1. Four decisions, and the order matters
Figure 1: The order of the four architecture decisions, with what sets each one and what each one fixes.
How many axes? — Set by the mission: how large a nadir keep-out is acceptable, and whether image derotation is needed.
Direct drive or geared? — Set by the stabilization spec: once backlash exceeds θ_max, no controller recovers it.
A coarse-fine second stage? — Set by the bandwidth requirement: only needed where one stage cannot reach.
Move the frame or a mirror? — Set by inertia and volume, but it costs you the 2:1.
The reason the order cannot be swapped is practical: axis count sets the frame layout, the frame layout sets inertia, and only then can transmission and actuation be discussed. Picking a motor first and then deciding the axis count means redoing all of posts 23 and 24.
2. Axis count: the third axis does not fix gimbal lock
Figure 2: What each axis count actually solves. The middle column is coloured because it is the most misunderstood.
This is the misconception this post most wants to correct.
A roll axis about the optical axis does not change where the line of sight points. It rotates the image, not the LOS direction. Therefore:
It does not fix gimbal lock. The nadir keep-out computed in post 24 exists on a three-axis gimbal exactly as it does on a two-axis one, because the keep-out comes from the azimuth axis's 1/cos(el) divergence and the third axis takes no part in that mapping.
What it does fix is image rotation. When the airframe rolls, the picture rolls with it; the third axis rolls it back. That has real value for observation and target interpretation — but it is a different requirement.
Actually removing gimbal lock requires another axis outboard — the two-axis four-gimbal layout. The extra outer frame takes over as the inner frame approaches the singularity, keeping the inner frame well away from cos(el) → 0.
So: if your problem is "it cannot keep up near nadir", a third axis will not help. That judgement is worth money, because a third axis costs a whole motor, frame, harness and power budget.
3. Direct or geared: backlash is a hard wall
Figure 3: The backlash range of three transmission types, on the same ruler as this series' θ_max. The green dashed line is θ_max.
Get the units straight first. 1 arcmin = π/180/60 rad = 0.2909 mrad.
Direct drive, backlash = 0 (rotor coupled straight to the load axis, torque through the magnetic field)
The stabilization spec this series has used throughout is θ_max = 0.1 mrad. Compare:
A harmonic drive's backlash alone is 2.9 to 8.7 times θ_max; a planetary gearbox's is 8.7 to 29.1 times.
What makes the comparison decisive is that backlash is not a disturbance feedback can suppress — it is a mechanical dead zone. At the instant of reversal the motor turns and the load does not, and the feedback has no idea what happened. No control law compensates something it cannot sense.
The conclusion is hard: for 0.1 mrad-class stabilization, direct drive is very nearly the only option. Conversely, if your spec is 1 mrad rather than 0.1, a harmonic drive is back on the table. Fix the spec first, then choose the transmission — not the other way round.
Note also an internal contradiction in harmonic drives: the flex spline that makes them nearly backlash-free is, by design, a part that deforms. Under load it winds up — near-zero backlash and high torsional stiffness are mutually exclusive in a harmonic drive — and torsional stiffness feeds straight into f_res, which is post 25's ceiling on the rate loop.
4. Gearing does not let you escape the nadir keep-out
Figure 4: Keep-out half-angle against effective Kt, with four gear ratios marked. They all land on the same curve — that is the point.
This is the section I found most satisfying to work out.
Intuition says: add a gear ratio N, multiply torque by N, and surely you can use a low-Kt motor and shrink the keep-out?
No. Work the gearing through:
`` effective Kt at the load = N · Kt_motor ω_max at the load = (0.7 · V_bus / Kt_motor) / N = 0.7 · V_bus / (N · Kt_motor) ``
The ratio multiplies torque and divides speed by exactly the same factor, and the two cancel. Substituted into the keep-out formula, N disappears entirely — only the effective Kt survives.
With Kt_motor = 0.05, a 24 V bus and ω_LOS = 90°/s:
All four land on the same curve. Gearing does not let you escape the keep-out; it just moves where you pay: you buy torque capacity and pay with a larger keep-out, plus backlash on top.
That is why high-precision gimbals are nearly always direct drive — not because direct drive is better, but because the other road charges twice.
5. Coarse-fine: why the fine stage needs less torque
When one mechanical stage cannot reach the required bandwidth, add a second. The intuitive worry is that a fine stage with ten times the bandwidth faces α = (2πf)² · θ_max, a square law — surely the torque demand explodes?
The arithmetic says otherwise. Using post 24's inner-frame inertia J = 0.0036 at f_bw = 12 Hz as the coarse stage, and a fine stage with inertia three orders smaller (a voice-coil-driven small mirror versus a whole inner frame) and bandwidth an order higher:
Coarse — J = 3.6×10⁻³, f = 12 Hz → α = 0.568 rad/s² → T = 2.05×10⁻³ N·m
Fine — J = 3.6×10⁻⁶, f = 120 Hz → α = 56.85 rad/s² → T = 2.05×10⁻⁴ N·m
α is 100× larger, but inertia is 1000× smaller, and the net result is that the fine stage needs only 10% of the coarse stage's torque.
That is the physical reason coarse-fine works: the square term is frightening, but inertia is a linear term and you can make it very small. The Applied Sciences paper above uses an ultrasonic motor for the coarse stage and a voice coil motor for the fine stage — exactly this division: coarse handles travel, fine handles bandwidth.
The costs: another actuator, more optics, another layer of kinematic coupling to compensate (which is what that paper's compensation matrix is for), plus volume and mass. If one stage suffices, do not add a second.
6. Frame or mirror: the 2:1 and a trap
Figure 5: Two consequences of the law of reflection. Left is the 2:1 amplification; right is the trap — hold the mirror inertially still and 100% of the base motion reaches the line of sight.
From the law of reflection (2D, math convention): outgoing direction angle = 2α + 180° − ψ, where α is the mirror normal's angle and ψ the incoming direction. Differentiating in each variable:
`` ΔLOS = 2 · θ_mirror − φ_base ``
The benefit is the first term. Rotate the mirror by θ and the LOS moves 2θ — the same LOS rate needs only half the mechanical rate. Add that a moving mirror has far less inertia than a whole moving frame, and that is the real advantage of mirror stabilization.
The trap is the second term. Suppose you "stabilize the mirror in inertial space", i.e. θ_mirror = 0. Then:
`` ΔLOS = 2 · 0 − φ_base = −φ_base ``
100% of the base motion reaches the LOS; nothing has been stabilized at all. This is precisely what Hilkert 2004 means by "simply stabilizing the mirror in inertial space will not stabilize the LOS".
Actually holding the LOS requires:
`` θ_mirror = φ_base / 2 ``
The mirror must move at half the base rate, not stay still. That is counter-intuitive enough to be worth writing on a wall — in a mirror architecture, "stabilization" means holding the LOS still, not the mirror, and the two are entirely different control objectives.
The practical implication: a mirror architecture must measure base motion (or measure the LOS itself). Putting one gyro on the mirror and driving its reading to zero produces the worst possible outcome.
7. Putting the four together
For this series' 140 mm two-axis pod, the four decisions land like this:
How many axes? Two. The mission is ground observation, and the nadir keep-out at Kt = 0.10 is only 0.54° (post 24) — acceptable. No image derotation is needed, so the third axis's mass cannot be justified.
Direct or geared? Direct. The θ_max = 0.1 mrad spec rules out gearing outright — a harmonic drive's backlash starts at 2.9× it.
Coarse-fine? No. f_bw = 12 Hz is not demanding for one stage, provided f_res ≥ 60 Hz (post 25's separation rule). Coarse-fine is for higher bandwidth requirements.
Frame or mirror? Frame. The ball pod's inner-frame inertia is already small (the 12 mm moment arm in post 23 is exactly what the ball shape buys), so there is no reason to take on the 2:1 complexity and base sensitivity.
All four land on the simpler side — and each is backed by a specific number. That is what an architecture decision should look like: not chosen from experience, but able to name the number behind every choice.
The limits of this analysis
The backlash figures are generic type ranges, not measurements of any product. High-grade harmonic drives can beat 1 arcmin, and some preloaded planetary designs beat 3 arcmin. Recompute from the datasheet of a real part number.
The keep-out section assumes torque is sufficient. It computes the speed constraint only. Real selection needs both constraints at once — the "squeezed from both sides" in post 24.
The coarse-fine numbers are illustrative, not a design. Three orders of inertia and one order of bandwidth are ratios I chose to show the mechanism, namely that the square term is not the problem. A real fine stage is sized backwards from optical travel and actuator specs.
The mirror section is 2D with a single mirror. Real mirror architectures often use two mirrors, or a mirror plus a frame, with more complex coupling. The 2:1 itself does not change, but the full compensation law depends on the layout.
No thermal, environmental or life considerations. Temperature changes preload and clearance, which affects both backlash and friction. That is another topic.
How to work through this with me
To help judge an architecture, I need:
The stabilization spec θ_max, and where it came from (image quality? geolocation? optical comms alignment?)
The acceptable nadir keep-out half-angle (usually nobody thinks of this first, but it drives axis count and Kt directly)
Whether image derotation is needed
The required rejection bandwidth, and where the mechanism's first resonance sits
Volume and mass limits
Bus voltage and available current
If something's missing, say it's missing — I won't guess a value and fill it in for you.
References
Hilkert, J. M. (2004). A comparison of inertial line-of-sight stabilization techniques using mirrors. Proc. SPIE 5430, Acquisition, Tracking, and Pointing XVIII. DOI: 10.1117/12.541808
Dynamics Modeling and Theoretical Study of the Two-Axis Four-Gimbal Coarse–Fine Composite UAV Electro-Optical Pod (2020). Applied Sciences, 10(6), 1923. DOI: 10.3390/app10061923
Modeling and Stability Analysis of Coarse–Fine Composite Mechatronic System in UAV Multi-Gimbal Electro-Optical Pod (2020). Link
Ekstrand, B. (2001). Equations of Motion for a Two-Axes Gimbal System. IEEE Transactions on Aerospace and Electronic Systems, 37(3), 1083–1091. Link
Hilkert, J. M. (2008). Inertially Stabilized Platform Technology: Concepts and Principles. IEEE Control Systems Magazine, 28(1), 26–46.
Masten, M. K. (2008). Inertially Stabilized Platforms for Optical Imaging Systems. IEEE Control Systems Magazine, 28(1), 47–64.
Mokbel, H. F., Ying, L. Q., Roshdy, A. A., et al. (2012). Design Optimization of the Inner Gimbal for Dual Axis Inertially Stabilized Platform Using Finite Element Modal Analysis. International Journal of Modern Engineering Research.