Accelerometer Tilt & Vector Calculator
Calculate three-axis accelerometer vector magnitude, normalized projection, roll, pitch, tilt from vertical, elevation from horizontal and gravity-consistency references from Ax, Ay and Az.
This calculator is for static accelerometer tilt reference. It does not calculate yaw, heading, gyroscope integration, Kalman filtering, vibration analysis or full IMU sensor fusion.
Engineering tool
Accelerometer Tilt & Vector Calculator
Calculate three-axis accelerometer magnitude, roll, pitch, tilt, vector normalization, gravity consistency and offset-corrected static tilt references.
Calculation mode
Parameter panel
Result console
- Roll
- 45 °
- Pitch
- -0 °
- Vector Magnitude
- 1g
- Gravity Reference Status
- Near 1g Reference
Magnitude is near the 1g reference, which may be consistent with gravity-dominated static orientation, but it does not prove the sensor is stationary. Roll range is typically -180° to +180°; pitch range is typically -90° to +90° with this convention.
Accelerometer tilt and vector formula audit
| Coordinate System | Right-hand XYZ coordinates entered directly by the user. |
|---|---|
| Axis Sign Convention | +X, +Y and +Z are positive sensor-axis readings; no automatic sign flip is applied. |
| Gravity Reference Convention | Examples assume +Z measures +1g when aligned with the adopted +Z gravity reference. |
| Magnitude Formula | |A| = sqrt(Ax² + Ay² + Az²). |
| Normalization Formula | nx=Ax/|A|, ny=Ay/|A|, nz=Az/|A|. |
| Roll Definition | Roll is rotation about the X-axis under the adopted static accelerometer convention. |
| Roll Formula | roll = atan2(Ay, Az). |
| Pitch Definition | Pitch is rotation about the Y-axis under the adopted static accelerometer convention. |
| Pitch Formula | pitch = atan2(-Ax, sqrt(Ay² + Az²)). |
| Tilt-from-Vertical Definition | Angle between acceleration vector and +Z. |
| Tilt Formula | theta = acos(Az/|A|), with numerical clamp to [-1, 1]. |
| Elevation Definition | Elevation relative to the XY plane, not a replacement for roll or pitch. |
| Elevation Formula | phi = atan2(Az, sqrt(Ax² + Ay²)). |
| Direction-Cosine Formulas | alpha=acos(Ax/|A|), beta=acos(Ay/|A|), gamma=acos(Az/|A|). |
| Angle Unit Convention | Internal radians; display can be degrees or radians. |
| Offset Correction Model | Ax,corr=Ax-Ox, Ay,corr=Ay-Oy, Az,corr=Az-Oz. |
| Gravity Consistency Model | |A| is compared with 1g using a user threshold; it is not static proof. |
| Near-Zero Magnitude Boundary | Tilt and normalization are undefined when vector magnitude is too small. |
| Dynamic Acceleration Boundary | Linear and centripetal acceleration can corrupt gravity-based tilt. |
| Free-Fall Boundary | Ideal free fall approaches 0g, so gravity-based tilt is undefined. |
| Yaw Boundary | Accelerometer-only data cannot determine yaw or absolute heading. |
| Euler-Angle Boundary | Roll can become sensitive near pitch close to ±90°. |
| SEN-009 Scope Boundary | Voltage-to-g conversion belongs to the Accelerometer Voltage & g Calculator. |
Formula reference
Accelerometer Tilt and Vector Formulas
The adopted convention uses user-entered right-hand XYZ accelerometer components and atan2-based static roll/pitch formulas.
|A| = sqrt(Ax² + Ay² + Az²)roll = atan2(Ay, Az)pitch = atan2(-Ax, sqrt(Ay² + Az²))theta = acos(Az / |A|)phi = atan2(Az, sqrt(Ax² + Ay²))nx = Ax/|A|, ny = Ay/|A|, nz = Az/|A|alpha = acos(Ax/|A|), beta = acos(Ay/|A|), gamma = acos(Az/|A|)Variable definitions
- Ax, Ay, Az
- acceleration components along sensor axes
- |A|
- acceleration vector magnitude
- roll
- rotation reference about X
- pitch
- rotation reference about Y
- theta
- tilt from +Z vertical
- phi
- elevation from the XY plane
Coordinate Convention and Formula Audit
| Coordinate System | Right-hand XYZ coordinates entered directly by the user. |
|---|---|
| Axis Sign Convention | +X, +Y and +Z are positive sensor-axis readings; no automatic sign flip. |
| Gravity Reference Convention | Known examples assume +Z gives +1g in the adopted reference orientation. |
| Magnitude Formula | |A| = sqrt(Ax² + Ay² + Az²). |
| Normalization Formula | nx=Ax/|A|, ny=Ay/|A|, nz=Az/|A|. |
| Roll Definition | Roll is the adopted static rotation reference about X. |
| Roll Formula | roll = atan2(Ay, Az). |
| Pitch Definition | Pitch is the adopted static rotation reference about Y. |
| Pitch Formula | pitch = atan2(-Ax, sqrt(Ay² + Az²)). |
| Tilt-from-Vertical Definition | Angle between A and +Z. |
| Tilt Formula | theta = acos(Az/|A|). |
| Elevation Definition | Angle above or below the XY plane. |
| Elevation Formula | phi = atan2(Az, sqrt(Ax² + Ay²)). |
| Direction Cosines | alpha=acos(Ax/|A|), beta=acos(Ay/|A|), gamma=acos(Az/|A|). |
| Angle Unit Convention | Internal radians; display as degrees or radians. |
| Offset Correction Model | Ax,corr=Ax-Ox, Ay,corr=Ay-Oy, Az,corr=Az-Oz. |
| Gravity Consistency Model | |A| compared with 1g using a user threshold. |
| Near-Zero Magnitude Boundary | Tilt and normalization are undefined near 0g. |
| Dynamic Acceleration Boundary | Motion can corrupt gravity-based tilt. |
| Free-Fall Boundary | Free fall magnitude approaches 0g and tilt is undefined. |
| Yaw Boundary | Yaw is not computed from accelerometer-only data. |
| SEN-009 Scope Boundary | Voltage-to-g conversion remains in the accelerometer voltage calculator. |
Worked Examples
+Z orientation
Known: Ax=0, Ay=0, Az=1g
Magnitude=1g, roll=0°, pitch=0°, tilt from +Z=0°.
+Y orientation
Known: Ax=0, Ay=1g, Az=0
Roll=+90°, pitch=0°, tilt from +Z=90°.
-Y orientation
Known: Ax=0, Ay=-1g, Az=0
Roll=-90°.
+X orientation
Known: Ax=1g, Ay=0, Az=0
Pitch=-90° under the adopted convention.
-X orientation
Known: Ax=-1g, Ay=0, Az=0
Pitch=+90°.
-Z orientation
Known: Ax=0, Ay=0, Az=-1g
Tilt from +Z=180°; roll policy gives 180° for atan2(0,-1).
45° roll
Known: Ax=0, Ay=sqrt(0.5), Az=sqrt(0.5)
Roll=45°, pitch=0°.
45° pitch
Known: Ax=-sqrt(0.5), Ay=0, Az=sqrt(0.5)
Pitch=45°.
3-4-5 vector
Known: Ax=3, Ay=4, Az=0
Magnitude=5, normalized vector=(0.6,0.8,0).
SI scaling
Known: Vector multiplied by 9.80665
Roll and pitch are unchanged after m/s² to g conversion.
Combined orientation
Known: Ax=0.1, Ay=0.2, Az≈0.974679
Magnitude≈1; roll and pitch follow atan2 formulas.
Above 1g
Known: |A|=1.1g
Deviation=+0.1g or +10%.
Below 1g
Known: |A|=0.9g
Deviation=-10%.
Zero magnitude
Known: Ax=0, Ay=0, Az=0
Tilt and normalization are undefined.
Offset correction
Known: Raw Ax=0.02g, X offset=0.02g
Corrected Ax=0g.
Scale invariance
Known: Vector multiplied by 100
Angles are unchanged before display rounding.
Engineering Notes
| Accelerometer Tilt | Static tilt estimation assumes the measured acceleration vector is dominated by gravity. |
|---|---|
| Acceleration Vector | A three-axis accelerometer returns components along X, Y and Z. |
| Vector Magnitude | Magnitude is sqrt(Ax²+Ay²+Az²), commonly near 1g in gravity-dominated static cases. |
| Roll | Roll depends on the adopted coordinate convention and is calculated here with atan2(Ay, Az). |
| Pitch | Pitch uses atan2(-Ax, sqrt(Ay²+Az²)) under the adopted convention. |
| Tilt | Tilt from +Z is a direct vector angle, not a full attitude solution. |
| Gravity Vector | The calculator uses user-entered acceleration signs and does not flip gravity convention automatically. |
| Static Orientation | Near-1g magnitude is only a reference; it is not proof of no motion. |
| Dynamic Acceleration | Vehicle motion, shaking, vibration and centripetal acceleration can corrupt tilt estimates. |
| Free Fall | In free fall, accelerometer magnitude approaches zero and gravity-based tilt is not defined. |
| Normalization | Normalization removes scale but not dynamic acceleration or calibration error. |
| Sensor Offset | Simple offset subtraction helps with bias but is not a full calibration matrix. |
| Scale Error | X/Y/Z sensitivity mismatch can create orientation error. |
| Cross-Axis Sensitivity | Cross-axis coupling and non-orthogonal axes are not modeled. |
| Euler Angles | Roll/pitch interpretation becomes sensitive near extreme pitch orientations. |
Common Mistakes
- Using atan instead of atan2.
- Using roll and pitch formulas without defining the coordinate system.
- Changing Ax/Ay/Az sign conventions between examples and UI.
- Trying to calculate yaw from accelerometer-only data.
- Treating |A|≈1g as proof that the sensor is stationary.
- Assuming normalization removes dynamic acceleration.
- Reporting reliable tilt in free fall.
- Dividing by magnitude when magnitude is zero.
- Letting floating-point acos arguments leak NaN.
- Mixing g and m/s² without conversion.
- Subtracting offset with the wrong sign.
- Duplicating SEN-009 voltage-to-g conversion in this page.
- Claiming accurate dynamic attitude from accelerometer-only static equations.
Related Calculators
Accelerometer Voltage & g Calculator
AvailableConvert analog accelerometer voltage and mV/g sensitivity into signed axis acceleration.
Open calculatorSensor Calibration Calculator
AvailableFit calibration curves and residuals before using corrected sensor readings.
Open calculatorSensor ADC Resolution Calculator
AvailableEstimate ADC code resolution for sensor measurement channels.
Open calculatorUltrasonic Distance Calculator
AvailableCompare another sensor measurement workflow with explicit boundary conditions.
Open calculatorAnalog Sensor Linear Scaling Calculator
AvailableScale generic analog sensor endpoints into engineering units.
Open calculatorSupport reference
FAQ
How do I calculate accelerometer vector magnitude?
Use |A| = sqrt(Ax² + Ay² + Az²), with all acceleration components in the same unit.
How do I calculate roll from Ax, Ay and Az?
With the adopted convention, roll is atan2(Ay, Az). atan2 preserves quadrant information and handles Az near zero.
How do I calculate pitch from accelerometer data?
With the adopted convention, pitch is atan2(-Ax, sqrt(Ay² + Az²)). This gives a typical pitch range of about -90° to +90°.
Why should I use atan2 instead of atan?
atan(Ay/Az) loses quadrant information and fails when the denominator is zero. atan2 uses both arguments and returns the correct quadrant.
How do I calculate tilt from the vertical?
Tilt from +Z is acos(Az/|A|), guarded against near-zero vector magnitude and with numerical clamp for floating-point safety.
How do I calculate tilt from the horizontal?
Elevation from the XY plane is atan2(Az, sqrt(Ax² + Ay²)). It is not the same as roll or pitch.
Why does a stationary accelerometer measure about 1 g?
An accelerometer measures proper acceleration or specific force. At rest, gravity support creates about 1 g along the appropriate axis projection.
Can an accelerometer calculate yaw?
No. Accelerometer-only data cannot determine yaw because rotation around the gravity vector does not change gravity projection.
Why is accelerometer-only tilt inaccurate during motion?
Linear acceleration, vibration and centripetal acceleration add to gravity projection, so the measured vector may not represent gravity alone.
What happens to tilt calculations in free fall?
In ideal free fall the accelerometer magnitude approaches 0 g, so gravity-based roll, pitch and tilt become undefined or unreliable.
Does normalizing the acceleration vector remove motion effects?
No. Normalization only scales the vector to unit length. It does not remove dynamic acceleration or sensor error.
How do accelerometer offsets affect tilt?
Additive offsets shift Ax, Ay and Az before angle calculation, which can create tilt error, especially near small angles.
What is the difference between roll, pitch and tilt angle?
Roll and pitch are convention-dependent Euler-style references. Tilt from vertical is simply the angle between the vector and +Z.
Why do coordinate-system conventions matter?
Changing axis labels or signs changes roll and pitch sign. This calculator uses a fixed convention and does not auto-detect sensor mounting.
How can I tell whether the acceleration vector is gravity dominated?
A magnitude near 1 g is a useful reference, but it does not prove the device is stationary. Motion can still produce a near-1g magnitude.
Why are gyroscopes combined with accelerometers?
Gyroscopes help track dynamic rotation, while accelerometers provide gravity reference. Fusion is commonly used for dynamic attitude estimation.
What is the difference between SEN-009 and this calculator?
SEN-009 converts analog accelerometer voltage to g on one axis. SEN-010 starts from Ax, Ay and Az acceleration components and calculates vector and static tilt references.