1. Exact Numeric Meaning
1 arc-second is exactly 1/3600 of a degree or 4.848 µrad. On a workpiece at 100mm radius, this translates to an extremely tight linear error of 0.485 micrometers.
What Does 1 Arc Second Resolution Mean?
1 arc-second is 1/3600 of a degree. Convert it to workpiece displacement, then compare nominal resolution with measured CNC-axis accuracy.
Published: 2026-06-20Last Updated: 2026-09-28
1 Arc Second Converter
Interactive ToolConvert the angle to workpiece displacement. Optional checks flag entered values above it; they do not predict installed-axis accuracy.
Method limit: values are checked one at a time, not combined into an uncertainty budget. Verify installed positioning under stated conditions.
Enter a comparison angle and machine values, then check each value independently.
Executive Summary: The Reality of Arc-Second Engineering
Core Conclusions2. Resolution vs. Accuracy
A 1 arc-second encoder or command increment describes granularity; it does not establish axis positioning accuracy. Installation, mechanics, control, load, temperature, and measurement conditions also matter.
3. Reversal Error and Backlash
Geartrains can add reversal error and compliance. Direct drive removes the geartrain, while an output-side scale can observe table motion; either design still needs measurement under its intended load.
4. Thermal Sensitivity
For a steel length of 100 mm, a 1 °C uniform change at an illustrative 12 µm/m/°C coefficient gives 1.2 µm of length change. Its effect on rotary accuracy depends on machine geometry and temperature gradients.
5. Testing and Validation
Ask for test reports under stated conditions: ISO 230-2 covers positioning accuracy and repeatability; ISO 230-7 covers geometric error motion of rotation axes; ISO 230-3 covers thermal effects.
What to verify for a 1" requirement
Evidence to request:
- Bidirectional positioning results measured at the output table under the intended conditions.
- Encoder specification stating counts per revolution and how resolution is defined.
- Warm-up, load, compensation, and measurement conditions for the test report.
Limits to check:
- Motor-side feedback may not measure errors downstream of the motor encoder.
- Larger workpiece radii magnify the linear effect of the same angular deviation.
- Thermal gradients and load deflection depend on the machine and duty cycle.
Resolution and Positioning Accuracy
Compare a measured axis result with the process tolerance; nominal encoder resolution is only one input.
A finer increment does not guarantee a more accurate installed axis.
1. What Does 1 Arc Second Resolution Mean in CNC?
When machine tool builders or CNC catalogs list a rotary axis with "1 arc-second resolution", ask whether the value describes encoder counts, interpolation, or controller commands. Measuring-system resolution has a distinct metrology definition. The angle itself is:
Plane Angle Metric Identity:
- Degrees: 1 arc second (1") = 1 / 3,600 of a degree ≈ 0.0002778°
- Radians: 1" = π / 648,000 rad ≈ 4.848137 × 10^-6 rad (4.85 µrad)
- Linear Shift: At a radius of 100 mm, 1" is about 0.485 micrometers (µm) of tangential arc length.
Under the simple assumption of 2^23 equally spaced counts per revolution, a 23-bit encoder has a nominal increment of about 0.155 arc-second per count. That count spacing is not a prerequisite or proof of 1 arc-second positioning accuracy; installation, interpolation, noise, control, and axis testing still matter.
For example, HEIDENHAIN lists RCN 8001 variants with encoder system accuracy of ±1 or ±2 arc-seconds, depending on model. This is an encoder specification, not a test result for the assembled machine axis. Review the current model datasheet and test the installed axis under its intended conditions.
2. Resolution vs. Accuracy vs. Repeatability Comparison
In formal metrology (VIM definitions), these three parameters have distinct, non-interchangeable roles. Using them synonymously creates high engineering risk during procurement.
| Metric | Definition (VIM / ISO) | Typical CNC Target | Engineering Implication |
|---|---|---|---|
| Angular Resolution | For a measuring instrument, the smallest change that produces a perceptible indication. A CNC command increment is a related but distinct specification. | State counts/revolution and interpolation method | Granularity alone does not guarantee actual table position or accuracy under load. |
| Angular Positioning Accuracy | A system performance specification determined from measured positioning deviations under stated test conditions. | Use the supplier test result and conditions | Compare the measured axis result with the part tolerance; repeatability and error motion can also affect the finished part. |
| Angular Repeatability | The spread of repeated positions under specified measurement conditions. | Use the supplier test result and approach direction | Useful for consistency, but it does not by itself describe absolute positioning error. |
3. The Projection of Linear Error at Radius (Abbe Errors)
An angular positioning error does not manifest as a single number at the workpiece. The effective linear offset on the cutting tool increases in direct proportion to the radius. This geometric principle is critical for sizing high-precision indexing heads and 5-axis rotary tables.
Here is the linear tangent displacement for different angle offsets projected across typical workpiece radii (in micrometers µm):
| Workpiece Radius | 0.5" Target | 1.0" Target | 2.0" Target | 5.0" Target | 10.0" Target |
|---|---|---|---|---|---|
| 50 mm | 0.121 µm | 0.242 µm | 0.485 µm | 1.212 µm | 2.424 µm |
| 100 mm | 0.242 µm | 0.485 µm | 0.970 µm | 2.424 µm | 4.848 µm |
| 150 mm | 0.364 µm | 0.727 µm | 1.454 µm | 3.636 µm | 7.272 µm |
| 250 mm | 0.606 µm | 1.212 µm | 2.424 µm | 6.060 µm | 12.120 µm |
| 500 mm | 1.212 µm | 2.424 µm | 4.848 µm | 12.120 µm | 24.241 µm |
| 750 mm | 1.818 µm | 3.636 µm | 7.272 µm | 18.181 µm | 36.361 µm |
| 1000 mm | 2.424 µm | 4.848 µm | 9.696 µm | 24.241 µm | 48.481 µm |
At radii above 500 mm, a 1" angular deviation projects to more than about 2.4 µm of tangential displacement. Compare that value with the specific part tolerance; it does not create a universal requirement for sub-arcsecond tracking.
4. Thermal Expansion: Spindle Growth & Housing Drift
Thermal expansion is one possible source of position change. Its effect depends on material, geometry, constraints, heat flow, and measurement point. The example below uses an assumed coefficient of 12 µm/m/K to estimate uniform length change; it does not predict rotary-axis angular drift.
For machine-tool thermal tests, the ISO 230-3:2020 standard (Determination of thermal effects on machine tools) defines procedures including tests for environmental temperature variation, rotating-spindle heating, and rotary motion. Cooling and compensation choices depend on the motor and machine design; verify resulting drift across the operating temperature and duty cycle.
Illustrative length-change calculation
ΔL = αLΔT = (12 µm/m/K)(0.1 m)(1 K) = 1.2 µm
This is an assumed material coefficient and a uniform 100 mm length change. Converting that growth into an axis angle requires the actual structure, constraints, and temperature field.
5. Optical Encoder Runout & Centering Eccentricity
Many machine designs integrate an optical encoder scale on the output flange and assume catalog resolution is immediately achieved. However, the physical runout of the encoder ring installation introduces a dominant first-harmonic error.
If an optical ring encoder is mounted slightly off-center (eccentricity) relative to the spindle axis of rotation, it introduces a systematic angular measurement deviation. For a modular encoder ring without integral bearings (e.g., HEIDENHAIN ERA series or Renishaw REXM series), the peak sinusoidal angular error ($\Delta\theta"$) in arc-seconds is governed by the first-principles geometric relation:
Δθ" ≈ (e / R) × 206,265 arc-seconds
Where $e$ is the mounting eccentricity runout in micrometers and $R$ is the physical radius of the encoder scale ring in micrometers.
Applying this formula:
- On a 100mm diameter encoder ring ($R = 50,000$ µm), a radial center offset of $e = 5$ µm results in a peak sinusoidal angular error of: 5 / 50,000 × 206,265 ≈ 20.6 arc-seconds. This is 20 times the entire 1" precision budget.
- In this simplified model, keeping the uncorrected contribution below 1.0" at that radius would require a radial center offset below about 0.24 µm. Confirm whether the supplier reports center offset or total indicated runout; the error convention and any compensation affect the comparison.
Integrated-bearing encoder designs reduce some mounting sensitivities. Dual-readhead arrangements can reduce first-harmonic eccentricity error when the encoder and signal processing are designed for it. Follow the manufacturer's installation procedure and verify the installed scale; neither option removes every axis error.
6. Gearbox Backlash and Reversal Error
Traditional gear reducers introduce lost motion that limits dynamic responsiveness. When a worm gear or planetary stage reverses direction, the gap between teeth prevents instant workpiece motion.
Gear reducers can add reversal error, compliance, heat, and wear. Direct drive removes the geartrain, while output-side feedback can measure downstream motion. Neither architecture guarantees a particular positioning result without an installed-axis test.
7. CNC Feedback Loop Architectures
Motor-side feedback does not directly measure errors downstream of the motor encoder. An output-side scale can include more of the transmission in the position loop, but it does not remove compliance, backlash, or thermal effects. Select and validate the feedback architecture against the required axis performance.
ISO 230-2 defines direct measurement methods for positioning accuracy and repeatability. Test points, cycles, approach directions, operating conditions, and reported parameters must follow the selected procedure and agreement between supplier and user. Compare measured results; a feedback architecture alone does not establish or guarantee sub-arcsecond performance.
8. Uncertainty Combination: When RSS Applies
Root-sum-square (RSS) is a method for combining standard uncertainties when the inputs and their relationships are known. It does not apply automatically to catalog limits, backlash ranges, nominal encoder increments, or estimated thermal shifts.
Under the uncorrelated-input case described in NIST Technical Note 1297, component standard uncertainties ($u_i$) combine as:
u_c = √(∑ u_i²)
To build a machine uncertainty budget, first define the measurand and test conditions, convert each measured or specified input to a standard uncertainty, and include covariance where relevant. This page’s screening calculator does not perform that analysis; validate the axis with measurements under the application’s load and temperature.
9. Drive Transmission Architectures for CNC Rotary Tables
Selecting the right physical transmission mechanism is the first step in aligning a spindle design with your target accuracy. The table below outlines how different configurations compare:
| Drive Type | Backlash Class | Encoder Setup | Thermal Behavior | Feasibility of 1" Target |
|---|---|---|---|---|
| Direct Drive DDR Motor | No gearbox; other axis errors remain | Choose feedback based on the axis and acceptance test | Motor heating depends on design and duty cycle | Potentially suitable; verify the installed axis under intended load and temperature. |
| Roller Gear Cam | Depends on preload, wear, and load | Compare input feedback with measured table position | Assess heat at the intended speed and duty cycle | Conditional on bidirectional positioning results and operating conditions. |
| Worm Gear / Dual-Pinion | Check reversal error and change with wear | Output feedback can observe downstream motion | Check lubrication and temperature effects | Do not infer from drive type; test both directions under the intended load. |
| Precision Planetary Gearbox | Varies by gearbox class, setup, and loading | Motor-side feedback may not observe transmission error | Check heating at the intended duty cycle | Not determined by the product label; compare measured axis results with the requirement. |
10. Supplier Validation Packet: Avoiding Catalog Pitfalls
During rotary table procurement, suppliers often publish best-case static repeatability claims and label them as "system accuracy." To verify physical capability, ensure your RFQ demands the following evidence package:
| Target Metric | Required Disclosures | Common Supplier Omissions | Risk If Omitted |
|---|---|---|---|
| Positioning Accuracy protocol | Bidirectional positioning accuracy (A) and repeatability (R) according to ISO 230-2. | Reporting only "nominal indexing accuracy" or one-directional repeatable snapshots. | A one-direction result can omit direction-dependent positioning error; request bidirectional results. |
| Encoder alignment eccentricity | Optical ring alignment runout tolerance, mounting axial deviation, and calibration log. | Assuming the ring has its catalog accuracy (e.g. ±1") regardless of installation runout. | A 5 µm radial center offset at a 50 mm ring radius corresponds to about ±20.6" before compensation; 5 µm TIR would imply about half that offset for pure eccentricity. |
| Spindle radial/axial error motion | Spindle axial error motion, tilt error motion, and synchronous radial runout per ISO 230-7. | Providing only static runout dial-indicator readings. | Spindle tilt under rotational dynamics translates to spatial abbe errors on workpiece height. |
11. Illustrative Scenarios: Calculated, Not Field Cases
These examples show how angular requirements translate to tangential displacement. They are calculations from the stated assumptions, not measured customer outcomes.
Illustrative scenario 1: Wafer dicing stage
Premise: Assume a process specification calls for a 0.5 arc-second angular tolerance at a 150 mm work radius.
Process: Compare candidate architectures by output-side positioning results, approach direction, load, temperature, and encoder specification. Do not infer workpiece performance from encoder bit depth alone.
Calculated implication: The angular allowance corresponds to about 0.364 µm of small-angle tangential displacement at 150 mm. This is a geometry conversion, not a measured machine result.
Illustrative scenario 2: Turbine blade rotary axis
Premise: Assume a 2 arc-second positioning requirement at a 500 mm work radius for a loaded 5-axis rotary axis.
Process: Request bidirectional axis measurements and repeatability under representative load. Compare the results with thermal and cutting-force effects at the tool point.
Calculated implication: Two arc-seconds correspond to about 4.848 µm of small-angle tangential displacement at 500 mm. Actual part error also depends on geometry, stiffness, and process conditions.
Illustrative scenario 3: Large gear inspection table
Premise: Assume a 1 arc-second angular requirement at a 1,000 mm work radius.
Process: Measure rotation-axis error motion, table positioning, workpiece offset, temperature, and load. Use correction only for error components shown to be repeatable within the test conditions.
Calculated implication: One arc-second corresponds to about 4.848 µm of small-angle tangential displacement at 1,000 mm. No installed accuracy or load performance is implied by this calculation.
Frequently Asked Questions
Common technical inquiries regarding angular resolution, CNC integration, and calibration.
Standards & Metrology
How is 1 arc second converted to linear error on a CNC workpiece?
To convert angular arc-seconds to linear deviation, use the formula: Linear Error = Spindle Radius × (Angle in Arc-Seconds / 206265). For example: at a 100 mm workpiece radius, 1" equals 0.485 µm. At a 500 mm radius, 1" scales linearly to 2.424 µm. This demonstrates why larger parts require much tighter angular control to maintain equivalent micron tolerances.
What is the main difference between resolution and accuracy on a rotary table?
In the VIM, measuring-instrument resolution is the smallest change in the measured quantity that produces a perceptible indication. A CNC command increment and an encoder count increment are separate specifications. Positioning accuracy describes measured axis performance under stated test conditions; a fine increment does not establish that accuracy.
What ISO standards govern CNC rotary axis accuracy verification?
ISO 230-2 specifies tests for positioning accuracy and repeatability of individual numerically controlled axes, including rotary axes. ISO 230-7 covers geometric accuracy and speed-induced shifts of rotation axes. ISO 230-3 covers thermal effects. These are test methods, not product certifications; request the relevant report and its conditions.
What is Abbe error in CNC rotary axis setups?
Abbe error occurs when the point of measurement (or cutting) is spatially offset from the axis of rotation or feedback scale. A tiny angular tilt error of 1" at the spindle bearing translates to a linear displacement that increases with distance. The further away the cutting tool is from the spindle bearings, the larger the linear positioning error becomes.
How do ISO 230-2 and ISO 230-3:2020 differ when verifying a 5-axis CNC rotary table?
ISO 230-2 provides methods for measuring positioning accuracy and repeatability of a CNC axis. ISO 230-3:2020 addresses thermal effects, while ISO 230-7:2015 addresses geometric accuracy and speed-induced shifts of rotation axes. Choose tests that match the acceptance question and record the load, environment, compensation state, and measurement uncertainty.
Feedback & Sensors
What does 1 arc second resolution mean in a CNC controller?
It usually refers to a stated encoder or command increment of 1 arc-second (1/3600° or about 0.0002778°). Ask whether the value describes encoder counts, interpolation, or controller commands. None of those alone establishes the installed axis positioning accuracy; that must be measured under stated conditions.
How many bits of encoder resolution are needed for 1 arc second control?
Under the simple assumption of 2^N equally spaced counts per revolution, 20 bits gives about 1.24 arc-seconds per count and 23 bits about 0.155 arc-second per count. This nominal count spacing does not account for interpolation, noise, cyclic error, installation, or control-loop behavior, so there is no universal bit-depth threshold for a given positioning accuracy.
Why does mounting eccentricity degrade optical ring encoder accuracy?
A radial center offset can create a once-per-revolution angular error. For a small offset e and ring radius R, its first-order peak is approximately e/R radians. State whether a specification gives center offset or total indicated runout; for pure eccentricity, TIR is approximately twice the offset.
How does a dual-readhead optical ring encoder mathematically cancel out centering eccentricity?
Two readheads placed opposite one another can reduce the first-harmonic error from eccentricity when their signals and installation are properly matched. The amount of cancellation is system-dependent; follow the encoder maker’s installation and signal-processing guidance, then verify the installed result.
Mechanical & Drive Systems
Can I achieve 1 arc second accuracy with a worm-gear rotary table?
Possibly, but the answer depends on the complete axis and its test conditions. Measure bidirectional positioning performance at the table, under the intended load and temperature, and assess wear and compensation over time. A gearbox label or nominal backlash value alone cannot determine the result.
Why are direct-drive DDR motors preferred for arc-second CNC machines?
A direct-drive motor removes the geartrain and its associated gear backlash. Bearing error motion, structural compliance, thermal drift, encoder installation, and servo behavior remain. It can suit demanding rotary axes, but the installed performance still needs measurement.
Does dual-loop control solve all mechanical backlash issues?
It does not remove mechanical clearance. An output-side scale can measure table position across reversals, allowing the controller to respond to downstream motion, but achievable accuracy and stability still depend on stiffness, tuning, load, and the tested approach directions.
What are the dynamic limitations of direct-drive (DDR) motors under heavy, variable milling loads?
A DDR motor removes the gear transmission and its gear backlash, but the axis still has bearing, structural, thermal, encoder, and servo errors. Without a gearbox, the motor must supply the required torque directly; evaluate force deflection and dynamic response for the intended cut instead of assuming a universal drive choice.
Thermal & Calibration
How does thermal drift affect arc-second rotary spindle housings?
Thermal growth depends on the material, geometry, temperature field, and constraints. For illustration, a 100 mm steel length with an assumed coefficient of 12 µm/m/°C changes by about 1.2 µm for a uniform 1 °C change. Converting that length change to an axis angle requires the actual machine geometry; it is not a universal spindle-drift value.
Can software compensation tables replace high-quality mechanical components?
Compensation can correct characterized, repeatable errors within its modeled conditions. It cannot be assumed to remove changing errors from load, temperature gradients, friction, or setup changes. Validate the compensated axis over the intended operating envelope.
What are the key parameters for an effective active liquid cooling loop for a DDR rotary axis?
There is no universal coolant temperature or flow rate for every rotary axis. Size cooling from the motor and machine thermal design, monitor relevant component temperatures, and use the maker’s limits. Test the resulting drift over warm-up and representative duty cycles.
Methodology Sources & Standards References
Definitions and test-method notes link to the standards bodies and manufacturers. Geometry examples are calculated from the stated inputs; product specifications vary by model. Reviewed: 2026-09-28.
- ISO 230-2:2014 - positioning accuracy and repeatability tests for numerically controlled axes, including rotary axes.
- ISO 230-3:2020 - test methods for machine-tool thermal effects, including environmental variation and spindle heating.
- ISO 230-7:2015 - geometric accuracy and speed-induced shifts of rotation axes; thermal effects are addressed in ISO 230-3.
- JCGM 200:2012, VIM 4.14 - definition and notes for measuring-instrument resolution.
- HEIDENHAIN angle encoders with integral bearing - model-specific system-accuracy specifications for the RCN family.
- Renishaw REXM ultra-high-accuracy encoder data sheet - manufacturer context for dual-readhead compensation, resolution, repeatability, and installed accuracy.
- NIST TN 1297, Section 5 - combining standard uncertainties and accounting for covariances; RSS applies to uncertainty components, not arbitrary error limits.
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