LocalPapa › Notes › Dev Log: A Map of the Gimbal Literature — Eight Starting Points, Five Methods, and Three Seams Nobody Owns

開發紀錄：把雲台的文獻攤成一張地圖——八個出發角度、五種方法、三個沒人管的接縫

2026-08-03
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這是 LocalPapa Notes 開發紀錄系列的第二十八篇，也是雲台系列的第十一篇。

前面十篇累積了大約六十篇引用。寫的時候是「需要什麼查什麼」，一篇一個主題；但全部攤開之後，浮出一個我在寫單篇時看不到的結構。

這篇不介紹新技術，它整理的是「這個領域怎麼分工」。 對缺乏經驗的人來說，這件事的實用價值可能比任何一篇單獨的論文都高——因為讀論文最容易踩的坑不是看不懂，是不知道這篇解的是哪一塊、以及它把什麼當成理所當然。

一、八個出發角度

文獻地圖：八個出發角度各自覆蓋設計鏈的哪一段 文獻地圖：八個出發角度各自覆蓋設計鏈的哪一段 架構選擇 結構／機構 致動器選型 傳動 感測 控制律 應用輸出 ⑥ 架構選擇 2 ⑤ 機構與結構 4 ④ 摩擦與非線性 5 ② 感測與量測架構 3 ⑧ 參數辨識 2 ① 動力學建模 4 ③ 擾動抑制控制 9 ⑦ 應用外延 4 （借用）航太機構標準 非雲台文獻 3 ↑ 雲台文獻沒有，只能借航太機構標準 相對論文量 橫條長度＝該群產出幫你決定的環節，粗細＝相對論文量。「致動器選型」那一欄唯一蓋到的是橘色那條——而它不是雲台文獻，是NASA-STD-5017B 與 ECSS 那類航太機構標準。雲台自己的八個群裡，建模群假設馬達理想、控制群假設力矩夠、結構群不管致動，沒有一群回答「要多大的馬達」。
圖 1：八個文獻群各自覆蓋設計鏈的哪一段。橫條長度是該群產出幫你決定的環節、粗細是相對論文量。注意「致動器選型」那一欄——唯一蓋到它的是橘色那條，而那不是雲台文獻。

同一個雲台，因為切入點不同，被當成常數的東西也不同。這是讀這批文獻最需要先看懂的一件事。

① 從「先寫出運動方程」出發。 Ekstrand 2001 用牛頓－尤拉法直接推兩軸耦合方程；Huang 2024 改用 Kane 法避開約束力；Abdo 2013 專攻動不平衡造成的交叉耦合。共同特徵是把馬達當理想力矩源、摩擦當常數。 這一群裡最有工程價值的是 Ekstrand 找出的慣量對稱條件——它說有一部分交叉耦合可以在機構階段設計掉，不必留給控制器。

② 從「你到底在量什麼」出發。 Kennedy & Kennedy 2003 的「直接 vs 間接 LOS 穩定」是整個領域的分水嶺概念；Hilkert 2008 與 Masten 2008 是兩篇並列的 tutorial。這一群不解方程，做的是分類學與量測學，價值在於它們定義了後面所有論文的詞彙。

③ 從「擾動打進來怎麼辦」出發。 這是主戰場，論文量比其他任何一群都多，下面第二節整節在講。

④ 從「低速為什麼會抖」出發。 Dahl 1968 把摩擦寫成微分方程，de Wit 1995 的 LuGre 六參數模型加進 Stribeck 效應成為事實標準，後面接一整批把 LuGre 塞進 ESO 或針對諧波減速做補償的論文。這群把「抖」這個現象接回一個具體的物理項。

⑤ 從「頻寬上限其實是機構給的」出發。 Mokbel 2012 用有限元模態分析驅動內框最佳化，減重 35%、一階扭轉模態提高 22%、整體動態性能提升 75%。這是唯一「往前端走」的一群——他們知道 f_res 決定速率環天花板，所以回頭改結構。

⑥ 從「架構要選哪一種」出發。 論文最少的一群，下一篇整篇在講。

⑦ 從「穩住之後要幹嘛」出發。 地理定位、視覺伺服、自由空間光通訊指向。第二十六篇已經走過這一塊。

⑧ 從「這些係數到底哪來」出發。 論文數量跟前面幾類差一個數量級，第二十七篇講的就是它。

圖上那個空欄

圖 1 有一欄只被橘色橫條蓋到：致動器選型。而橘色那條標了「非雲台文獻」——它是 NASA-STD-5017B 與 ECSS-E-ST-33-01C 那類航太機構標準。

換句話說：雲台領域自己的八個群裡，沒有一群回答「要多大的馬達」。 建模群假設馬達理想、控制群假設力矩夠、結構群不管致動。這不是我事後找出來的角度，是第二十二篇查文獻時撞到的——找不到雲台論文談力矩餘裕，只好去借航太機構的規範。

二、擾動抑制那一群：骨架相同，差在估計器

擾動抑制的五種方法：骨架相同，差在「怎麼估那個擾動」 擾動抑制的五種方法：骨架相同，差在「怎麼估那個擾動」 標稱控制器 受控體 Σ 指令 輸出 擾動 d 估計器 前饋抵消 −d̂ DOB 用標稱模型的逆推回擾動 需要標稱模型的逆 ADRC／ESO 把未建模項全打包成「總擾動」由 ESO 估 不需要模型的逆 滑模 SMC 不估，用切換律硬吃（代價：抖振） 需要擾動上界 強健 H∞ 不估，用頻域設計保證最壞情況 需要不確定性描述 學習型 從資料學補償量，不需模型 需要資料 DOB 與 ADRC 其實是同一個想法的兩種寫法——都是估擾動再前饋抵消，差別只在 ADRC 不需要標稱模型的逆。這批論文互相比較時常常沒把這件事講清楚。
圖 2：五種方法共用同一個骨架——估擾動、前饋抵消。虛線框住的估計器才是各家真正不同的那一塊。下半列出每一家怎麼估、以及各自需要什麼前提。

論文量最大的這一群，讀起來像五個互不相干的學派，但畫成方塊圖會發現它們的骨架是同一個。

  • DOB（擾動觀測器） —— Hilkert & Pautler 2011 的降階 DOB。用標稱模型的逆把擾動推回來。前提：你得有那個逆。
  • ADRC／ESO —— 把所有未建模項打包成「總擾動」，由擴張狀態觀測器估出來。前提：不需要模型的逆。
  • 滑模 SMC —— 不估，用切換律硬吃不確定性。前提：要知道擾動的上界。代價是抖振。
  • 強健 H∞ —— 不估，用頻域設計保證最壞情況。前提：要能描述不確定性集合。
  • 學習型 —— 從資料學補償量。前提：要有資料。
DOB 與 ADRC 其實是同一個想法的兩種寫法。 都是估擾動、再前饋抵消；差別只在 ADRC 不需要標稱模型的逆。這批論文互相比較時常常沒把這件事講清楚，讀起來會以為它們在解不同的問題。

看懂這個骨架之後，讀這一群論文可以只問三個問題：它怎麼估？它需要什麼前提？那個前提你有沒有？ 前提對不上，效能數字再漂亮都跟你無關。

三、三個結構性缺口

設計迴圈：想提高頻寬，最後會繞回摩擦 設計迴圈：想提高頻寬，最後會繞回摩擦 想提高 f_bw 得先提高 f_res 要減重 或加剛度 慣量降低 剛度也降 摩擦佔比 上升 低速更容易 stick-slip ↺ 而 stick-slip 又逼你降低 f_bw 這一段誰在做 想提高f_bw 第 ③ 群 得先提高f_res 第 ⑤ 群 要減重或加剛度 接縫：沒有一群以它為主題 慣量降低剛度也降 第 ① 群 摩擦佔比上升 第 ④ 群 低速更容易stick-slip 接縫：沒有一群以它為主題 這條迴圈沒有一篇論文完整走過。每一段都有人做，但接縫處（減重之後摩擦佔比變高、進而限制低速性能）落在所有人的邊界之外。
圖 3：設計迴圈。六段轉折每一段都有人在做，但其中兩段落在群與群的接縫上——右欄標成橘色的那兩段。

把八個群疊起來看，缺的不是某個技術，是接縫。

缺口一：沒有人回答「馬達要多大」。 上面說過了，這是圖 1 那個空欄。

缺口二：每一類都把別類當常數。 這是最要命的一個，因為真正的設計是這幾件事互相拉扯：

想提高 f_bw → 得先提高 f_res → 要減重或加剛度 → 慣量降低但剛度往往也降 → 摩擦在總力矩裡的佔比上升 → 低速更容易 stick-slip → 而 stick-slip 又逼你把 f_bw 降回來。

這條迴圈繞回原點。每一段都有人做：第 ③ 群管 f_bw、第 ⑤ 群管 f_res 與減重、第 ① 群管慣量、第 ④ 群管摩擦與 stick-slip。但接縫沒人管——「減重之後摩擦佔比變高」跨在第 ⑤ 群與第 ④ 群之間，「stick-slip 反過來限制 f_bw」跨在第 ④ 群與第 ③ 群之間。這兩段是整條迴圈的閉合處，也正是沒有論文以它為主題的地方。

缺口三：絕大多數是單軸，或「雙軸但分開處理」。 真正把兩軸耦合、俯仰角相依、天底禁區一起算的，只有 Ekstrand 和少數幾篇。第二十四篇算出的那個結論——方位軸的慣量在 0° 最大、但最壞力矩不在 0°，因為 yaw gain 同步放大——就是分開處理會漏掉的東西。

四、這張地圖怎麼用

實務上有三個用法：

第一，讀論文之前先定位它在哪一格。 知道它屬於第幾群，就知道它預設了什麼。看到一篇「擾動抑制改善 90%」的論文，先問它假設力矩餘裕多少——如果你的機構根本推不動，那 90% 是在解另一台機器的問題。

第二，遇到問題時反過來查該問哪一群。 畫面在慢速追蹤時一頓一頓 → 第 ④ 群（摩擦與 Stribeck）。位置環怎麼調都慢 → 先看第 ⑤ 群（共振壓住速率環），不是第 ③ 群。定位不準 → 第 ⑦ 群，而且第二十六篇算過，多半不是雲台的錯。

第三，接縫要自己走。 三個缺口都在接縫上，代表你不會找到一篇論文替你算完。這也是本系列第 22 到 27 篇實際在做的事——不是發明新方法，是把既有的方法接起來，然後把接縫處的數字算出來。

本文的引用範圍

這篇的分類與方法描述，來自各篇的摘要與公開說明頁。本環境對多數出版商（IEEE／MDPI／Springer／ScienceDirect）回 403，我沒有逐篇核對全文。引用的數值（35%／22%／75%、1–3 arcmin 等）都是論文摘要或說明頁自報的，我沒有獨立驗證。

「論文量」是本系列實際盤點到的相對數量，不是對整個領域的普查——它反映的是我在寫前面十篇時查到什麼，會低估我沒碰到的次領域。圖 1 的橫條粗細只該當成量級參考，不該當成統計。

至於「哪一群把什麼當常數」，那是我讀完之後的歸納，不是論文自己的宣稱。這是本篇最主觀的部分，也是最值得你自己驗證的部分。

想一起把這件事做完整

如果你手上有這個領域的論文清單，我最想知道的是這三件事：

  • 有沒有論文以致動器選型為主題（不是順帶提到，是主題）
  • 有沒有論文完整走過第三節那條迴圈
  • 三軸與四框的架構取捨，除了廠商白皮書之外有沒有學術處理

有的話請告訴我，我會修正這張地圖。

參考文獻

建模

  • Ekstrand, B. (2001). Equations of Motion for a Two-Axes Gimbal System. IEEE Transactions on Aerospace and Electronic Systems, 37(3), 1083–1091. Link
  • Huang, Q., et al. (2024). Modeling and Control of a Two-Axis Stabilized Gimbal Based on Kane Method. Sensors, 24(11), 3615.
  • Abdo, M., Vali, A. R., Toloei, A. R., & Arvan, M. R. (2013). Research on the Cross-Coupling of a Two Axes Gimbal System with Dynamic Unbalance. International Journal of Advanced Robotic Systems.

感測與量測架構

  • Kennedy, P. J., & Kennedy, R. L. (2003). Direct versus Indirect Line of Sight (LOS) Stabilization. IEEE Transactions on Control Systems Technology, 11(1), 3–15.
  • 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.

擾動抑制控制

  • Hilkert, J. M., & Pautler, D. (2011). Reduced-Order Disturbance Observer Applied to Inertially Stabilized Line-of-Sight Control. Proc. SPIE 8052.
  • Abdo, M. M., Vali, A. R., Toloei, A. R., & Arvan, M. R. (2015). Improving two axes gimbal seeker performance using cascade control approach. Proc. IMechE Part G, 229(1), 38–55. DOI: 10.1177/0954410014525130
  • Stabilization of two-axis line-of-sight system using active disturbance rejection control (2025). Multibody System Dynamics.
  • Fast terminal sliding mode control based on SDRE observer for two-axis gimbal with external disturbances (2022). Journal of the Brazilian Society of Mechanical Sciences and Engineering.
  • Learning-Based Control Compensation for Multi-Axis Gimbal Systems. arXiv:2112.02561

摩擦與非線性

  • Dahl, P. R. (1968). A Solid Friction Model. The Aerospace Corporation, TOR-0158(3107-18)-1.
  • de Wit, C. C., Olsson, H., Åström, K. J., & Lischinsky, P. (1995). A New Model for Control of Systems with Friction. IEEE Transactions on Automatic Control, 40(3), 419–425.
  • Two-Axis Optoelectronic Stabilized Platform Based on Active Disturbance Rejection Controller with LuGre Friction Model (2023). Electronics, 12(5), 1261.
  • Sightline Jitter Minimization and Shaping Using Nonlinear Friction Compensation (2007). International Journal of Optomechatronics, 1(3), 259–283.

機構與結構

  • 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.
  • Passive Isolator Design and Vibration Damping of EO/IR Gimbal Used in UAVs (2023). International Journal of Aviation Science and Technology.
  • Dynamic simulation and disturbance torque analyzing of motional cable harness based on Kirchhoff rod model (2012). Chinese Journal of Mechanical Engineering, 25(2), 346–354.
  • Jia, R., Nandikolla, V. K., Haggart, G., Volk, C., & Tazartes, D. (2017). System Performance of an Inertially Stabilized Gimbal Platform with Friction, Resonance, and Vibration Effects. Journal of Nonlinear Dynamics, 2017, 6594861.

架構選擇

  • 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

應用外延

  • Gautam, D., Watson, C., Lucieer, A., & Malenovský, Z. (2018). Error Budget for Geolocation of Spectroradiometer Point Observations from an Unmanned Aircraft System. Sensors, 18(10), 3465. DOI: 10.3390/s18103465
  • Dhruv, & Kaushal, H. (2025). A Review of Pointing Modules and Gimbal Systems for Free-Space Optical Communication in Non-Terrestrial Platforms. Photonics, 12(10), 1001.

參數辨識

  • Kim, S. (2019). Moment of Inertia and Friction Torque Coefficient Identification in a Servo Drive System. IEEE Transactions on Industrial Electronics, 66(1), 60–70. DOI: 10.1109/TIE.2018.2826456
  • System identification and mechanical resonance frequency suppression for servo control used in single gimbal control moment gyroscope (2022). PLOS ONE, 17(8), e0267450. DOI: 10.1371/journal.pone.0267450
  • Liang, M., & Zhou, D. (2022). A Nonlinear Friction Identification Method Combining Separable Least Squares Approach and Kinematic Orthogonal Property. International Journal of Precision Engineering and Manufacturing, 23, 139–152. DOI: 10.1007/s12541-021-00611-0

借來的：致動器選型

  • NASA-STD-5017B (2022). Design and Development Requirements for Mechanisms. NASA Technical Standard.
  • ECSS-E-ST-33-01C Rev.2 (2019). Space engineering — Mechanisms.
  • Nalbandian, R., Blais, M., & Horth, R. (2014). A Recommended New Approach on Motorization Ratio Calculations of Stepper Motors. 42nd Aerospace Mechanisms Symposium.

上一篇：參數辨識：把 J、B、T_c 從機器上量回來。下一篇：四個架構決策。要算馬達？雲台馬達選型計算機。想追這個系列？把 LocalPapa Notes 加進書籤。

Dev Log: A Map of the Gimbal Literature — Eight Starting Points, Five Methods, and Three Seams Nobody Owns

2026-08-03
dev-logengineeringgimballiterature-reviewresearch-methodbuilding-in-public

This is the twenty-eighth post in the LocalPapa Notes dev-log series, and the eleventh on gimbals.

The previous ten accumulated roughly sixty citations. They were gathered the way anyone gathers them — look up what you need for the post you are writing. But laid out all together, a structure surfaces that no single post could show.

This post introduces no new technique. What it organizes is how the field divides the labour. For someone without hands-on experience that may be worth more than any individual paper, because the trap in reading this literature is not comprehension — it is not knowing which piece a paper solves, and what it takes for granted.

1. Eight starting points

Literature map: which part of the design chain each angle covers Literature map: which part of the design chain each angle covers Architecture Structure Motor sizing Transmission Sensing Control law Application 6 Architecture choice 2 5 Mechanism and structure 4 4 Friction and nonlinearity 5 2 Sensing architecture 3 8 Parameter identification 2 1 Dynamic modeling 4 3 Disturbance rejection 9 7 Downstream application 4 (borrowed) Aerospace mechanism standards not gimbal literature 3 ↑ Absent from gimbal literature — borrowed from aerospace standards relative paper count Bar length is the stage a cluster's output helps you decide; thickness is relative paper count. The only bar over "motorsizing" is the orange one — and that is not gimbal literature, it is NASA-STD-5017B and ECSS-class aerospacemechanism standards. Of the eight gimbal clusters, the modeling papers assume an ideal motor, the control papers assumesufficient torque, and the structural papers ignore actuation. None of them answers "how big a motor".
Figure 1: Which part of the design chain each cluster covers. Bar length is the stage a cluster's output helps you decide; thickness is relative paper count. Note the "motor sizing" column — the only bar over it is the orange one, and that is not gimbal literature.

The same gimbal, approached from a different starting point, holds different things constant. That is the first thing to understand about this body of work.

1. Starting from "write down the equations of motion." Ekstrand 2001 derives the two-axis coupled equations by Newton–Euler; Huang 2024 switches to Kane's method to avoid constraint forces; Abdo 2013 targets cross-coupling from dynamic unbalance. All of them treat the motor as an ideal torque source and friction as a constant. The most engineering-useful result in this cluster is Ekstrand's inertia symmetry condition — it says part of the cross-coupling can be designed out at the mechanism stage rather than left for the controller.

2. Starting from "what are you actually measuring." Kennedy & Kennedy 2003's direct-versus-indirect LOS stabilization is the field's watershed concept; Hilkert 2008 and Masten 2008 are a paired tutorial. This cluster solves no equations; it does taxonomy and metrology — and its value is that it defined the vocabulary every later paper uses.

3. Starting from "what do I do about the disturbance." The main battlefield, with more papers than any other cluster. Section two is entirely about it.

4. Starting from "why does it judder at low speed." Dahl 1968 writes friction as a differential equation; de Wit 1995's six-parameter LuGre model adds the Stribeck effect and becomes the de facto standard; a whole run of papers then folds LuGre into an ESO or compensates a harmonic drive specifically. This cluster connects the symptom back to a concrete physical term.

5. Starting from "the bandwidth ceiling is set by the mechanism." Mokbel 2012 drives inner-gimbal optimization with finite element modal analysis: 35% mass reduction, first torsional mode up 22%, overall dynamic performance up 75%. This is the only cluster that walks upstream — they know f_res caps the rate loop, so they go back and change the structure.

6. Starting from "which architecture." The smallest cluster, and the subject of the next post.

7. Starting from "what happens after it is stable." Geolocation, visual servoing, free-space optical pointing. Post 26 covered this ground.

8. Starting from "where do these coefficients come from." An order of magnitude fewer papers than the clusters above; post 27 is about it.

The empty column

One column in figure 1 is covered only by the orange bar: motor sizing. And that bar is labelled "not gimbal literature" — it is NASA-STD-5017B and ECSS-E-ST-33-01C, aerospace mechanism standards.

Put plainly: none of the eight gimbal clusters answers "how big a motor." The modeling papers assume an ideal motor, the control papers assume sufficient torque, the structural papers ignore actuation. This is not an angle found in hindsight — it is what post 22 ran into while searching. No gimbal paper discusses torque margin, so the only option was to borrow from aerospace mechanism standards.

2. The disturbance-rejection cluster: same skeleton, different estimator

Five disturbance-rejection families: same skeleton, different estimator Five disturbance-rejection families: same skeleton, different estimator Nominal controller Plant Σ Command Output Disturbance d Estimator Feedforward cancel −d̂ DOB Invert the nominal model to recover d needs the nominal inverse ADRC / ESO Lump everything unmodeled into a "total disturbance", estimate with an ESO no inverse needed Sliding mode No estimate — a switching law absorbs it (cost: chatter) needs a disturbance bound Robust H∞ No estimate — frequency-domain design guarantees the worst case needs an uncertainty description Learning-based Learn the compensation from data, no model needed needs data DOB and ADRC are two spellings of one idea — estimate the disturbance, then cancel it forward. The only difference isthat ADRC does not need the nominal inverse. Papers comparing these families often leave that unsaid.
Figure 2: Five families sharing one skeleton — estimate the disturbance, cancel it forward. The dashed box is the estimator, the only part that genuinely differs. Below, how each family estimates and what each one requires.

The largest cluster reads like five unrelated schools, but drawn as a block diagram they share one skeleton.

  • DOB (disturbance observer) — Hilkert & Pautler 2011's reduced-order DOB. Recover the disturbance by inverting the nominal model. Requires: that inverse.
  • ADRC / ESO — Lump everything unmodeled into a "total disturbance" and estimate it with an extended state observer. Requires: no inverse.
  • Sliding mode — No estimate; a switching law absorbs the uncertainty. Requires: a bound on it. Costs: chatter.
  • Robust H∞ — No estimate; frequency-domain design guarantees the worst case. Requires: a description of the uncertainty set.
  • Learning-based — Learn the compensation from data. Requires: data.
DOB and ADRC are two spellings of one idea. Both estimate the disturbance and cancel it forward; the only difference is that ADRC needs no nominal inverse. Papers comparing these families often leave that unsaid, which makes them read as though they solve different problems.

Once the skeleton is visible, reading this cluster reduces to three questions: how does it estimate, what does it require, and do you have that? If the requirement does not match your setup, the performance figures are about someone else's machine.

3. Three structural gaps

The design loop: chasing bandwidth leads back to friction The design loop: chasing bandwidth leads back to friction Want higher f_bw Need higher f_res Cut mass or add stiffness Inertia drops, so does stiffness Friction share rises Stick-slip gets more likely ↺ and stick-slip forces f_bw back down Who works on this step Want higherf_bw cluster 3 Need higherf_res cluster 5 Cut massor add stiffness seam: no cluster owns it Inertia drops,so does stiffness cluster 1 Friction sharerises cluster 4 Stick-slip getsmore likely seam: no cluster owns it No single paper walks this whole loop. Every segment has someone working on it, but the seams — mass reduction raisingthe friction share, which then caps low-speed performance — fall outside everyone's boundary.
Figure 3: The design loop. Every one of the six steps has someone working on it, but two of them fall on the seams between clusters — the two marked orange in the right-hand column.

Stacking the eight clusters, what is missing is not a technique. It is the seams.

Gap one: nobody answers "how big a motor." As above — the empty column in figure 1.

Gap two: every cluster holds the others constant. This is the damaging one, because real design is these things pulling against each other:

Want a higher f_bw → need a higher f_res → cut mass or add stiffness → inertia drops but so, usually, does stiffness → friction's share of total torque rises → stick-slip becomes more likely at low speed → and stick-slip forces f_bw back down.

The loop closes on itself. Every segment has an owner: cluster 3 owns f_bw, cluster 5 owns f_res and mass, cluster 1 owns inertia, cluster 4 owns friction and stick-slip. The seams have none. "Mass reduction raises the friction share" sits between clusters 5 and 4; "stick-slip caps f_bw" sits between clusters 4 and 3. Those two are where the loop closes, and they are exactly where no paper takes the subject.

Gap three: most work is single-axis, or "two-axis but treated separately." Only Ekstrand and a handful of others compute the two-axis coupling, the elevation dependence and the nadir keep-out together. The result in post 24 — that azimuth inertia peaks at 0° but the worst-case torque does not, because the yaw gain amplifies in step — is precisely what separate treatment misses.

4. How to use the map

Three practical uses:

First, locate a paper before reading it. Knowing which cluster it belongs to tells you what it assumes. Faced with a paper claiming 90% better disturbance rejection, ask what torque margin it assumed — if your mechanism cannot physically move the load, that 90% is solving a different machine's problem.

Second, work backwards from a symptom to a cluster. Image judders while tracking a slow target → cluster 4 (friction and Stribeck). Position loop stays sluggish no matter the tuning → look at cluster 5 first (resonance caps the rate loop), not cluster 3. Geolocation is off → cluster 7, and post 26 showed it is usually not the gimbal's fault.

Third, the seams are yours to walk. All three gaps are on seams, which means no paper will finish the calculation for you. That is what posts 22 through 27 of this series actually did — not invent new methods, but join existing ones together and work out the numbers where they meet.

Scope of citation in this post

The classification and method descriptions come from abstracts and public paper pages. This environment returns 403 for most publishers (IEEE, MDPI, Springer, ScienceDirect), so I did not verify full texts. The quoted figures (35%/22%/75%, 1–3 arcmin, and so on) are self-reported in abstracts or paper pages; I did not verify them independently.

"Paper count" is the relative count this series actually accumulated, not a survey of the field — it reflects what I found while writing the previous ten posts and will undercount subfields I never touched. The bar thickness in figure 1 should be read as an order of magnitude, not a statistic.

As for "what each cluster holds constant": that is my inference after reading, not a claim the papers make about themselves. It is the most subjective part of this post, and the part most worth checking yourself.

How to work through this with me

If you have a reading list in this field, these are the three things I most want to know:

  • Is there a paper with actuator sizing as its subject (not a passing mention — the subject)
  • Is there a paper that walks the whole loop in section three
  • Beyond vendor whitepapers, is there academic treatment of the three-axis versus four-gimbal trade

If so, tell me and I will correct the map.

References

Modeling

  • Ekstrand, B. (2001). Equations of Motion for a Two-Axes Gimbal System. IEEE Transactions on Aerospace and Electronic Systems, 37(3), 1083–1091. Link
  • Huang, Q., et al. (2024). Modeling and Control of a Two-Axis Stabilized Gimbal Based on Kane Method. Sensors, 24(11), 3615.
  • Abdo, M., Vali, A. R., Toloei, A. R., & Arvan, M. R. (2013). Research on the Cross-Coupling of a Two Axes Gimbal System with Dynamic Unbalance. International Journal of Advanced Robotic Systems.

Sensing architecture

  • Kennedy, P. J., & Kennedy, R. L. (2003). Direct versus Indirect Line of Sight (LOS) Stabilization. IEEE Transactions on Control Systems Technology, 11(1), 3–15.
  • 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.

Disturbance rejection

  • Hilkert, J. M., & Pautler, D. (2011). Reduced-Order Disturbance Observer Applied to Inertially Stabilized Line-of-Sight Control. Proc. SPIE 8052.
  • Abdo, M. M., Vali, A. R., Toloei, A. R., & Arvan, M. R. (2015). Improving two axes gimbal seeker performance using cascade control approach. Proc. IMechE Part G, 229(1), 38–55. DOI: 10.1177/0954410014525130
  • Stabilization of two-axis line-of-sight system using active disturbance rejection control (2025). Multibody System Dynamics.
  • Fast terminal sliding mode control based on SDRE observer for two-axis gimbal with external disturbances (2022). Journal of the Brazilian Society of Mechanical Sciences and Engineering.
  • Learning-Based Control Compensation for Multi-Axis Gimbal Systems. arXiv:2112.02561

Friction and nonlinearity

  • Dahl, P. R. (1968). A Solid Friction Model. The Aerospace Corporation, TOR-0158(3107-18)-1.
  • de Wit, C. C., Olsson, H., Åström, K. J., & Lischinsky, P. (1995). A New Model for Control of Systems with Friction. IEEE Transactions on Automatic Control, 40(3), 419–425.
  • Two-Axis Optoelectronic Stabilized Platform Based on Active Disturbance Rejection Controller with LuGre Friction Model (2023). Electronics, 12(5), 1261.
  • Sightline Jitter Minimization and Shaping Using Nonlinear Friction Compensation (2007). International Journal of Optomechatronics, 1(3), 259–283.

Mechanism and structure

  • 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.
  • Passive Isolator Design and Vibration Damping of EO/IR Gimbal Used in UAVs (2023). International Journal of Aviation Science and Technology.
  • Dynamic simulation and disturbance torque analyzing of motional cable harness based on Kirchhoff rod model (2012). Chinese Journal of Mechanical Engineering, 25(2), 346–354.
  • Jia, R., Nandikolla, V. K., Haggart, G., Volk, C., & Tazartes, D. (2017). System Performance of an Inertially Stabilized Gimbal Platform with Friction, Resonance, and Vibration Effects. Journal of Nonlinear Dynamics, 2017, 6594861.

Architecture

  • 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

Downstream applications

  • Gautam, D., Watson, C., Lucieer, A., & Malenovský, Z. (2018). Error Budget for Geolocation of Spectroradiometer Point Observations from an Unmanned Aircraft System. Sensors, 18(10), 3465. DOI: 10.3390/s18103465
  • Dhruv, & Kaushal, H. (2025). A Review of Pointing Modules and Gimbal Systems for Free-Space Optical Communication in Non-Terrestrial Platforms. Photonics, 12(10), 1001.

Parameter identification

  • Kim, S. (2019). Moment of Inertia and Friction Torque Coefficient Identification in a Servo Drive System. IEEE Transactions on Industrial Electronics, 66(1), 60–70. DOI: 10.1109/TIE.2018.2826456
  • System identification and mechanical resonance frequency suppression for servo control used in single gimbal control moment gyroscope (2022). PLOS ONE, 17(8), e0267450. DOI: 10.1371/journal.pone.0267450
  • Liang, M., & Zhou, D. (2022). A Nonlinear Friction Identification Method Combining Separable Least Squares Approach and Kinematic Orthogonal Property. International Journal of Precision Engineering and Manufacturing, 23, 139–152. DOI: 10.1007/s12541-021-00611-0

Borrowed: actuator sizing

  • NASA-STD-5017B (2022). Design and Development Requirements for Mechanisms. NASA Technical Standard.
  • ECSS-E-ST-33-01C Rev.2 (2019). Space engineering — Mechanisms.
  • Nalbandian, R., Blais, M., & Horth, R. (2014). A Recommended New Approach on Motorization Ratio Calculations of Stepper Motors. 42nd Aerospace Mechanisms Symposium.

Previous post: Identifying J, B and T_c From the Actual Machine. Next: Four Architecture Decisions. Want to size a motor? Gimbal Motor Sizing Calculator. Want to follow the series? Bookmark LocalPapa Notes.

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