Dev Log: How to Build an Optical LOS Measurement Rig — Verifying What the Motor Sizing Calculator Assumes

這是 LocalPapa Notes 開發紀錄系列的第十八篇。上一次做的是雲台馬達選型計算機：輸入氣動環境與穩定頻寬目標，反推每軸馬達所需的扭力與電流。工具上線後，我們回頭看了一下它的輸入欄位，發現一個沒有明講的假設——穩定頻寬 f_bw 跟 最大角偏移目標 θ_max，這兩個數字使用者要打哪裡來？

計算機只負責「已知穩定度目標，反推馬達規格」這半段。另一半——「你的雲台實際做出來之後，LOS 穩定度到底是多少、有沒有達標」——是一個量測問題，而且是光學量測問題，不是隨便拿手機錄影目測就能回答的。這篇整理我們讀過的文獻，把「LOS 光學量測系統要怎麼架」這件事說清楚。

什麼是 LOS，為什麼「算出來」還不夠

LOS（Line of Sight，視線）是感測器光軸實際指向的方向。理想狀況下，雲台把 LOS 死死釘在目標上，不管載具怎麼晃、馬達軸承怎麼有摩擦、線材怎麼拉扯，光軸都不該偏移。但現實中它會抖——這個抖動的角度量，業界統稱 LOS jitter。

NASA 藍格利研究中心的 Cornelius Dennehy 在一份系統性回顧報告裡把這個問題講得很直白：LOS jitter 的來源五花八門——反作用輪不平衡、冷卻器振動、機構共振、太陽能板展開後的殘留振盪——而且大多數來源沒辦法只靠分析模型精確預測，因為摩擦、線材阻力、共振頻率這些參數，實務上都得靠實測才拿得準（Dennehy, A Survey of the Spacecraft Line-of-Sight Jitter Problem, NASA NTRS 20200002690）。同一份調查後續擴充成的工程實務指南也強調同一件事：LOS 預算要在專案早期就排入實測驗證，不能只靠模擬（NASA/TM-20210017871, Spacecraft Line-of-Sight Jitter Management and Mitigation, 2021）。

我們的計算機用的公式——氣動阻力扭力、慣性扭力、摩擦餘量、安全係數——全部來自 J. M. Hilkert 那篇被廣泛引用的教科書級回顧文章（Hilkert, Inertially Stabilized Platform Technology: Concepts and Principles, IEEE Control Systems Magazine, vol. 28, no. 1, pp. 26–46, 2008）。但同一篇文章也講得很清楚：這些公式給的是「需求端」的扭力與頻寬估計，馬達選型只是控制迴路的一半；真正決定 LOS 能不能守住目標，還要看穩定迴路的閉迴路擾動抑制能力，而那必須用實測的頻率響應去驗證（Hilkert & Pautler, Reduced-Order Disturbance Observer Applied to Inertially Stabilized Line-of-Sight Control, Proc. SPIE 8052, 2011）。換句話說：計算機給的是「你大概需要多少扭力」，量測系統回答的是「你做出來的東西真的達標了嗎」——兩者缺一不可。

三種主流光學量測架構

讀了一輪文獻，LOS 量測的做法大致收斂成三種，精度、成本、架設難度依序遞減。

1. 自準直儀＋反射鏡（實驗室級）

這是航太與軍規光電系統最常見的做法。在雲台的（假）酬載上固定一面反射鏡，自準直儀（autocollimator）打出一束準直光，鏡子把光反射回自準直儀的偵測器；鏡面法線相對自準直儀光軸的偏轉角，直接就是 LOS 偏移角，解析度可以做到角秒（arcsec）等級。一篇針對電光炮塔摩擦誘發抖動的研究論文具體描述了這套架構：把校正過的商用自準直儀（文中用的是 UDT Model 1020）對準裝在雲台上的假酬載反射鏡，離軸角度就是視線偏移的直接量測值（Sightline Jitter Minimization and Shaping Using Nonlinear Friction Compensation, International Journal of Optomechatronics, 1(3), 259–283, 2007）。

優點是精度最高、頻寬最寬（自準直儀偵測器可以取樣到千赫茲等級），缺點是需要專用儀器、光路要精密對準，架設門檻高，比較適合有實驗室環境的專案。

2. 雷射＋位置感測探測器（PSD／象限探測器）

概念類似自準直儀，但用一顆準直雷射打向雲台酬載上的靶點，靶點後方或反射路徑上放一顆位置感測探測器（PSD）或四象限光電二極體——這類元件的原理是量測光點打在感測面上的重心位置，換算成角度偏移（RP Photonics Encyclopedia, Position-Sensitive Detectors）。這套架構在早期的追瞄系統文獻裡也很常見，象限探測器訊號常拿來當低頻追蹤迴路的誤差輸入，同時兼作 LOS jitter 的量測手段。

比自準直儀親民一些（雷射二極體＋PSD 模組的成本遠低於精密自準直儀），但仍需要穩定的光路架設與電子訊號處理鏈，且量測範圍受限於 PSD 的線性動態範圍。

3. 目標板＋望遠攝影機（親民型）

前兩種都是「光學儀器級」做法。如果雲台的規模比較小（例如我們計算機鎖定的 BaseCam 級雲台，酬載通常幾公斤到幾十公斤），一個更容易上手的替代方案，是把攝影機本身當量測工具：架設一塊有已知圖案的目標板在固定距離外，讓雲台鏡頭對準它，用影像處理演算法逐幀追蹤目標圖案在畫面中的像素位移，再依鏡頭焦距與目標距離換算成角度抖動。小型無人機雲台的相關研究就是走這條路，用飛行測試搭配影像資料分析雲台實際的姿態穩定表現（相關綜述可見 Dhruv & Kaushal, A Review of Pointing Modules and Gimbal Systems for Free-Space Optical Communication in Non-Terrestrial Platforms, Photonics, 12(10), 1001, 2025，文中對指向精度、頻寬、慣性等設計權衡有系統性整理）。

精度不如前兩種（受限於攝影機解析度與影像追蹤演算法的子像素精度），但架設成本最低——只要有一顆望遠鏡頭、一塊列印的靶板、和一支能做子像素邊角/斑點追蹤的程式，中小型專案就能自己動手做。

怎麼注入擾動、量出頻寬

光有靜態偵測還不夠——LOS 穩定系統的重點是抗擾動能力，也就是載具晃動時穩定迴路能不能把 LOS 誤差壓下去、多快壓下去。標準做法是把雲台裝在擾動台（rate table / shaker table）上，用已知的角速度或加速度輪廓去激振雲台基座，同步記錄前述任一種光學方式量到的 LOS 誤差，再用系統識別的方式算出擾動抑制的閉迴路頻率響應——這正是計算機裡「穩定頻寬 f_bw」這個輸入欄位理論上該從哪裡量出來的地方（Hilkert 2008；Hilkert & Pautler 2011）。簡化版做法則是手持晃動＋IMU 記錄擾動輸入，雖然不如擾動台精確，但足以抓出量級級的頻寬估計，適合驗證性質的測試。

資料怎麼分析：RMS 與功率頻譜密度

量到一段時間的 LOS 誤差時間序列之後，兩種分析方式最常見：

RMS 抖動量：在特定時間窗（例如幾秒到幾十秒）內計算誤差的均方根值，直接對應計算機裡的 θ_max（最大角偏移目標）。詹姆斯韋伯太空望遠鏡的指向穩定度規格就是這樣定義的——在固定目標情境下，以 15 秒區間內導星星位置的 RMS 誤差、相對一萬秒觀測期間的平均位置來評估，並用雜訊等效角（Noise Equivalent Angle, NEA）描述感測器本身的量測精度極限（JWST 使用者文件，JWST Fine Guide Stability）。
功率頻譜密度（PSD）分析：把時域誤差轉到頻域，找出抖動能量集中在哪些頻率——這能幫你分辨「是馬達伺服頻寬不夠」還是「某個結構共振被踢到」。JWST 在軌測試階段就是靠 PSD 分析，抓出了一個 0.3 Hz 的減振器振動模態，以及一個 0.045 Hz 的燃料晃動訊號，兩者都跟反作用輪或冷卻器無關——如果沒有做頻域分析，光看 RMS 數字完全無從得知抖動根因（STScI, JWST Line-of-Sight Jitter Measurement during Commissioning, JWST-STScI-008271, 2022）。

即使是遠比太空望遠鏡小的雲台，這套「RMS 看達標與否、PSD 看根因」的分析邏輯完全可以照搬：先確認 RMS 抖動有沒有壓在 θ_max 目標以下，壓不下去再翻 PSD 找是哪個頻段在扯後腿——是穩定迴路頻寬不夠（該調控制器或加大馬達），還是某段結構共振被打到（該加強機構剛性或做減振）。

給中小型雲台建置者的最小可行方案

看完文獻，我們自己歸納出一套適合 BaseCam 級專案（不是衛星等級）、成本跟精度都比較平衡的組合：

量測端：優先選「目標板＋望遠攝影機」；預算允許的話，升級成雷射＋PSD 模組（精度高一個量級，元件成本仍在業餘可負擔範圍）。自準直儀通常只有在真的要衝角秒級精度時才划算。
擾動端：先用手持晃動＋IMU 記錄粗抓量級，確認方向對了之後，如果要精確量測穩定頻寬，再考慮租用或自製簡易搖振台。
分析端：把量到的時間序列同時算 RMS（拿去跟計算機的 θ_max 對照）與 PSD（找抖動根因），兩者都做、缺一不可。
回饋迴路：量出來的 RMS／頻寬數字如果沒達到計算機當初設定的目標，回頭調整計算機裡的安全係數 SF 或摩擦阻力比例，重新反推一輪馬達規格——量測跟計算彼此校正，而不是算完就結案。

參考文獻

Dennehy, C. J. (2020). A Survey of the Spacecraft Line-of-Sight Jitter Problem. NASA Langley Research Center, NASA NTRS 20200002690. PDF
NASA (2021). Spacecraft Line-of-Sight Jitter Management and Mitigation — Lessons Learned and Engineering Best Practices. NASA/TM-20210017871. PDF
Hilkert, J. M. (2008). Inertially Stabilized Platform Technology: Concepts and Principles. IEEE Control Systems Magazine, 28(1), 26–46.
Hilkert, J. M., & Pautler, D. (2011). Reduced-Order Disturbance Observer Applied to Inertially Stabilized Line-of-Sight Control. Proc. SPIE 8052, Acquisition, Tracking, Pointing, and Laser Systems Technologies XXV.
Sightline Jitter Minimization and Shaping Using Nonlinear Friction Compensation. (2007). International Journal of Optomechatronics, 1(3), 259–283. DOI: 10.1080/15599610701548837
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. DOI: 10.3390/photonics12101001
STScI (2022). JWST Line-of-Sight Jitter Measurement during Commissioning. JWST-STScI-008271. PDF
JWST User Documentation. JWST Fine Guide Stability. 連結
RP Photonics Encyclopedia. Position-Sensitive Detectors. 連結

想搭配文獻裡的公式自己反推馬達規格？雲台馬達選型計算機就是為了這件事做的。想追蹤系列後續？把 LocalPapa Notes 加入書籤吧。

This is the eighteenth post in the LocalPapa Notes dev-log series. Last time we shipped the Gimbal Motor Sizing Calculator: plug in aerodynamic environment and stabilization bandwidth targets, and it works backward to the torque and current each axis motor needs. After it went live, we looked back at its input fields and noticed an unstated assumption — where do the numbers for stabilization bandwidth f_bw and maximum angular deviation target θ_max actually come from?

The calculator only handles half the problem: "given a known stability target, derive the motor spec." The other half — "once you've actually built the gimbal, what is its real LOS stability, and does it hit the target?" — is a measurement problem, and specifically an optical measurement problem, not something you can eyeball from a phone video. This post pulls together what we read while figuring out how to actually set up a LOS optical measurement rig.

What LOS is, and why "calculating it" isn't enough

LOS (Line of Sight) is the direction the sensor's optical axis actually points. Ideally the gimbal pins the LOS to the target no matter how the carrier shakes, how the motor bearings introduce friction, or how cables tug on the mechanism — the optical axis shouldn't move. In practice it does, and the angular magnitude of that motion is what the field calls LOS jitter.

Cornelius Dennehy at NASA Langley Research Center put it plainly in a systematic survey: LOS jitter sources are diverse — reaction wheel imbalance, cryocooler vibration, structural resonance, residual oscillation after solar-array deployment — and most of them can't be predicted precisely from analysis alone, because parameters like friction, cable drag, and resonant frequency only become accurate through actual measurement in practice (Dennehy, A Survey of the Spacecraft Line-of-Sight Jitter Problem, NASA NTRS 20200002690). A follow-on engineering practice guide built from the same survey makes the same point even more directly: LOS budgets need measurement-based verification baked in from early in the project, not simulation alone (NASA/TM-20210017871, Spacecraft Line-of-Sight Jitter Management and Mitigation, 2021).

The formulas our calculator uses — aerodynamic drag torque, inertial torque, friction margin, safety factor — all trace back to J. M. Hilkert's widely-cited review, essentially the textbook reference in this field (Hilkert, Inertially Stabilized Platform Technology: Concepts and Principles, IEEE Control Systems Magazine, vol. 28, no. 1, pp. 26–46, 2008). But that same paper is equally clear that these formulas give you the demand-side torque and bandwidth estimate — motor sizing is only half of the stabilization loop. Whether LOS actually holds the target depends on the loop's closed-loop disturbance rejection, and that has to be verified with a measured frequency response (Hilkert & Pautler, Reduced-Order Disturbance Observer Applied to Inertially Stabilized Line-of-Sight Control, Proc. SPIE 8052, 2011). In other words: the calculator tells you roughly how much torque you need; the measurement rig tells you whether what you built actually meets the target — you need both.

Three common optical measurement architectures

After a round of reading, LOS measurement approaches roughly converge into three tiers, decreasing in precision, cost, and setup difficulty.

1. Autocollimator + mirror (lab-grade)

This is the most common approach in aerospace and mil-spec EO/IR systems. A mirror is mounted on the gimbal's (dummy) payload; an autocollimator projects a collimated beam that reflects off the mirror back into the autocollimator's detector. The deflection angle of the mirror's normal relative to the autocollimator's optical axis is a direct measurement of LOS deviation, resolvable down to arcsecond levels. A paper on friction-induced jitter in electro-optic turrets describes this setup concretely: a calibrated commercial autocollimator (a UDT Model 1020, in that paper) aimed at a mirror on a dummy payload mounted to the gimbal, with the off-axis angle serving as the direct sightline-deflection measurement (Sightline Jitter Minimization and Shaping Using Nonlinear Friction Compensation, International Journal of Optomechatronics, 1(3), 259–283, 2007).

The upside is the highest precision and widest bandwidth (autocollimator detectors can sample at kilohertz-class rates); the downside is dedicated instrumentation and precise optical alignment, which raises the setup bar — better suited to projects with lab access.

2. Laser + position-sensitive detector (PSD / quadrant photodiode)

Conceptually similar to an autocollimator, but a collimated laser is aimed at a target point on the gimbal payload, and a position-sensitive detector (PSD) or quadrant photodiode is placed in the return or transmission path. These devices work by measuring the centroid position of the light spot on the sensing surface and converting it to an angular offset (RP Photonics Encyclopedia, Position-Sensitive Detectors). This architecture shows up throughout earlier tracking-system literature too, where quad-cell signals often double as both the low-bandwidth tracking error input and the LOS jitter measurement.

More accessible than an autocollimator (a laser diode + PSD module costs far less than precision autocollimator instrumentation), but still requires a stable optical path and a signal-processing chain, and the measurable range is limited by the PSD's linear dynamic range.

3. Target board + telephoto camera (accessible)

The first two are "optical-instrument-grade" approaches. For smaller gimbals — like the BaseCam-class systems our calculator targets, typically carrying payloads from a few to a few tens of kilograms — a more approachable alternative is to use the camera itself as the measurement instrument: mount a target board with a known pattern at a fixed distance, point the gimbal's camera at it, and use image-processing to track the pattern's pixel displacement frame by frame, converting it to angular jitter via the lens focal length and target distance. Small-UAV gimbal research takes this route, combining flight tests with image-based analysis of actual attitude stability (see the survey by Dhruv & Kaushal, A Review of Pointing Modules and Gimbal Systems for Free-Space Optical Communication in Non-Terrestrial Platforms, Photonics, 12(10), 1001, 2025, which systematically covers the pointing-accuracy, bandwidth, and inertia trade-offs across gimbal designs).

Less precise than the first two (limited by camera resolution and the sub-pixel accuracy of the tracking algorithm), but the cheapest to set up — a telephoto lens, a printed target board, and software capable of sub-pixel corner/blob tracking is enough for a small-to-mid project to build in-house.

Injecting disturbance and measuring bandwidth

Static detection alone isn't enough — what actually matters for a LOS stabilization system is its disturbance rejection: can the stabilization loop suppress LOS error when the carrier shakes, and how fast? The standard approach is to mount the gimbal on a rate table / shaker table, excite the base with a known angular-rate or acceleration profile, simultaneously record the LOS error using one of the optical methods above, and use system identification to derive the loop's closed-loop frequency response — this is precisely where the calculator's "stabilization bandwidth f_bw" input should, in principle, come from in the first place (Hilkert 2008; Hilkert & Pautler 2011). A simplified version uses handheld shaking with IMU-logged disturbance input — less precise than a rate table, but good enough for an order-of-magnitude bandwidth estimate, which is fine for verification-grade testing.

Analyzing the data: RMS and power spectral density

Once you have a time series of LOS error over some duration, two analysis methods are standard:

RMS jitter: compute the root-mean-square error over a specific time window (seconds to tens of seconds), which maps directly onto the calculator's θ_max (maximum angular deviation target). The James Webb Space Telescope's pointing stability spec is defined exactly this way — for fixed targets, as the RMS error of guide-star position over a 15-second interval relative to the mean position over a 10,000-second observation, with a Noise Equivalent Angle (NEA) describing the sensor's own measurement precision floor (JWST User Documentation, JWST Fine Guide Stability).
Power spectral density (PSD) analysis: transform the time-domain error into the frequency domain to find where the jitter energy concentrates — this is what tells you whether "the servo bandwidth isn't wide enough" or "a specific structural resonance got excited." During JWST's in-orbit commissioning, PSD analysis is exactly what surfaced a 0.3 Hz vibration-damper mode and a 0.045 Hz fuel-slosh signature — neither related to the reaction wheels or cryocooler. Without frequency-domain analysis, the RMS number alone would have given no clue as to the root cause (STScI, JWST Line-of-Sight Jitter Measurement during Commissioning, JWST-STScI-008271, 2022).

Even for gimbals far smaller than a space telescope, this "RMS tells you pass/fail, PSD tells you why" analysis logic carries over directly: check whether RMS jitter is under the θ_max target first; if it isn't, dig into the PSD to find which frequency band is the culprit — insufficient stabilization loop bandwidth (tune the controller or size up the motor) versus an excited structural resonance (stiffen the mechanism or add damping).

A minimal practical rig for small-to-mid gimbal builders

After going through the literature, here's the combination we'd recommend for a BaseCam-class project (not a satellite-class one), balancing cost against precision:

Measurement side: default to "target board + telephoto camera"; upgrade to a laser + PSD module if budget allows (an order of magnitude more precise, still within hobbyist-affordable component cost). An autocollimator is usually only worth it if you genuinely need arcsecond-class precision.
Disturbance side: start with handheld shaking + IMU logging to get a rough order-of-magnitude estimate; once the direction looks right, consider renting or building a simple shaker table if you need a precise bandwidth measurement.
Analysis side: compute both RMS (compare against the calculator's θ_max) and PSD (find the jitter's root cause) from the recorded time series — do both, neither is optional.
Feedback loop: if the measured RMS/bandwidth doesn't meet the target you originally set in the calculator, go back and adjust the safety factor SF or friction margin, then re-derive the motor spec — measurement and calculation should correct each other, not end once you've run the numbers once.

References

Dennehy, C. J. (2020). A Survey of the Spacecraft Line-of-Sight Jitter Problem. NASA Langley Research Center, NASA NTRS 20200002690. PDF
NASA (2021). Spacecraft Line-of-Sight Jitter Management and Mitigation — Lessons Learned and Engineering Best Practices. NASA/TM-20210017871. PDF
Hilkert, J. M. (2008). Inertially Stabilized Platform Technology: Concepts and Principles. IEEE Control Systems Magazine, 28(1), 26–46.
Hilkert, J. M., & Pautler, D. (2011). Reduced-Order Disturbance Observer Applied to Inertially Stabilized Line-of-Sight Control. Proc. SPIE 8052, Acquisition, Tracking, Pointing, and Laser Systems Technologies XXV.
Sightline Jitter Minimization and Shaping Using Nonlinear Friction Compensation. (2007). International Journal of Optomechatronics, 1(3), 259–283. DOI: 10.1080/15599610701548837
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. DOI: 10.3390/photonics12101001
STScI (2022). JWST Line-of-Sight Jitter Measurement during Commissioning. JWST-STScI-008271. PDF
JWST User Documentation. JWST Fine Guide Stability. link
RP Photonics Encyclopedia. Position-Sensitive Detectors. link

Want to plug the formulas from this literature straight into motor sizing? That's exactly what the Gimbal Motor Sizing Calculator is for. Want to follow the rest of the series? Bookmark LocalPapa Notes.