Engineering Guide
Motion, Distance, and Inertial Sensor Measurement Workflows
Reliable sensor results come from a declared physical model, consistent units, calibration, and an uncertainty budget. This guide connects acoustic time of flight and accelerometer signals to defensible distance, acceleration, and static-tilt measurements.
Measurement Workflow
| Step | Action | Engineering Check |
|---|---|---|
| Define the measurand | State whether the output is distance, one-axis acceleration, vector magnitude, or static tilt. | |
| Declare conventions | Document one-way versus round-trip time, axis signs, units, zero reference, and angle definition. | |
| Convert the raw signal | Apply timing, voltage sensitivity, offset, and ADC transfer functions before interpreting motion. | |
| Calibrate | Use traceable physical references across the intended range; retain residual error rather than only slope and offset. | |
| Budget uncertainty | Separate quantization, noise, temperature drift, alignment, target geometry, and model error. | |
| Filter with purpose | Set bandwidth from real motion dynamics; avoid hiding delay, ringing, or missed events behind excessive averaging. | |
| Validate boundaries | Test temperature, range endpoints, orientation, vibration, target material, supply, and mounting. | |
| Report honestly | Distinguish resolution, repeatability, accuracy, latency, and confidence in the final result. |
Measurement Models and Boundaries
| Workflow | Raw Input | Core Model | Dominant Boundaries |
|---|---|---|---|
| Ultrasonic pulse echo | Echo time | d = vt/2 | Temperature, target angle, ringing, multipath |
| Analog accelerometer | Voltage | a = (Vout - Vzero)/S | Offset, sensitivity, ADC, cross-axis error |
| Three-axis static tilt | Ax, Ay, Az | atan2 and vector geometry | Axis convention, dynamic acceleration, free fall |
| Calibration | Reference pairs | Linear or selected fit | Reference quality, nonlinearity, extrapolation |
| ADC digitization | Voltage code | Code width and transfer function | Noise and analog error remain separate |
Ultrasonic Time-of-Flight Distance
Formula reference
Pulse-echo ranging
Round-trip distance: d = vt/2One-way distance: d = vtRound-trip echo time: t = 2d/vAir approximation: v = 331.3 + 0.606T m/sRound-trip timer increment: delta d = v delta t/2Variable definitions
- t must be identified as one-way or round-trip time
- T is air temperature in degrees Celsius
- Humidity, pressure, airflow, target geometry, and transducer behavior are not fully modeled
Use the Ultrasonic Distance Calculator for echo timing, inverse timeout, temperature compensation, and timer increment. Verify sensor dead zone, beam angle, echo threshold, target reflectivity, multipath, and firmware latency with hardware.
Analog Accelerometer Voltage to Acceleration
Formula reference
Linear one-axis transfer function
Vout = Vzero + S x aa = (Vout - Vzero)/Sa(m/s squared) = a(g) x 9.80665g/code = ADC volts per code / |S x gain|Variable definitions
- Vzero is measured zero-g bias
- S must be converted to volts per g
- The first-order model excludes nonlinearity, cross-axis coupling, hysteresis, and dynamic bandwidth
The Accelerometer Voltage and g Calculator converts voltage, bias, and sensitivity. A two-point or multi-point calibration should be used when nominal bias and sensitivity do not meet the error budget.
Three-Axis Vector and Static Tilt
Formula reference
Adopted XYZ convention
|A| = sqrt(Ax squared + Ay squared + Az squared)roll = atan2(Ay, Az)pitch = atan2(-Ax, sqrt(Ay squared + Az squared))tilt from +Z = acos(Az/|A|)Variable definitions
- Axis labels and signs must match the physical mounting
- Equations assume acceleration is dominated by gravity for static tilt
- Yaw cannot be recovered from accelerometer-only data
- Near-zero magnitude makes normalization and tilt undefined
Use the Accelerometer Tilt and Vector Calculator only after axis conversion and offset correction. Dynamic attitude estimation requires gyroscope integration or sensor fusion and belongs outside this static model.
Calibration, ADC Resolution, and Uncertainty
The Sensor Calibration Calculator supports slope, intercept, and residual checks; the Sensor ADC Resolution Calculator supports code-width planning. They are shared tools because calibration and digitization support many sensor types. Neither replaces a complete uncertainty budget.
Worked Examples
One-millisecond echo
At 20 degrees C, v = 343.42 m/s. A 1 ms round-trip echo gives d = 343.42 x 0.001 / 2 = 0.17171 m.
Temperature compensation
A 10 ms echo gives 1.6565 m at 0 degrees C and 1.7474 m at 30 degrees C, showing why fixed sound speed creates systematic error.
Timer resolution
At 20 degrees C, a 1 us round-trip timer tick represents about 0.17171 mm, but this is not complete measurement accuracy.
Voltage to g
With Vzero = 1.65 V, sensitivity = 330 mV/g, and Vout = 1.98 V, acceleration is +1.00 g.
Offset error
If a stationary zero-g axis reads 1.67 V instead of 1.65 V at 330 mV/g, indicated offset is about +0.0606 g.
ADC granularity
A 12-bit 3.3 V ADC has about 0.806 mV/code. At 330 mV/g and unity gain, that is about 0.00244 g/code before noise and error.
Static level orientation
Ax = 0, Ay = 0, Az = 1 g gives magnitude 1 g, roll 0 degrees, pitch 0 degrees, and tilt from +Z of 0 degrees.
Forty-five-degree roll
Ax = 0, Ay = 0.7071 g, Az = 0.7071 g gives roll = atan2(Ay, Az) = 45 degrees.
Free-fall boundary
When vector magnitude approaches zero, normalization and gravity-based tilt become undefined rather than zero.
Dynamic-motion boundary
A 1 g magnitude can still include linear acceleration; magnitude near 1 g does not prove the device is stationary.
Common Mistakes
Related Calculators
Related Resources
Support reference
FAQ
Why is ultrasonic echo distance divided by two?
The measured pulse travels to the target and back. Dividing total acoustic path length by two gives one-way target distance.
How does temperature affect ultrasonic distance?
Sound travels faster in warmer air. A temperature-compensated speed model reduces the systematic distance error caused by using one fixed speed.
Is ultrasonic timer resolution the same as accuracy?
No. Timer quantization is only one contributor; ringing, threshold timing, temperature, target geometry, multipath, and electronics latency also matter.
How do I convert accelerometer voltage to g?
Subtract the zero-g bias from output voltage and divide by sensitivity in volts per g: a = (Vout - Vzero)/S.
Why must accelerometer sensitivity units be explicit?
A value in mV/g differs from V/g by 1000. Converting units before division prevents a three-order-of-magnitude error.
How do I calculate accelerometer vector magnitude?
Use sqrt(Ax squared + Ay squared + Az squared), with all three components expressed in the same acceleration unit.
Why use atan2 for roll and pitch?
atan2 preserves quadrant information and remains defined when the denominator of a simple ratio approaches zero.
Can an accelerometer determine yaw?
No. Rotation around the gravity vector does not change its projection, so yaw needs another reference such as a magnetometer, vision, or integrated gyro information.
Why does static tilt fail during motion?
Linear, centripetal, and vibration acceleration add to gravity. The measured vector then no longer represents gravity alone.
What happens to tilt in free fall?
The measured proper-acceleration magnitude approaches zero, so vector normalization and gravity-based orientation are undefined or unreliable.
Does a higher-resolution ADC make a sensor more accurate?
Not automatically. More codes can improve quantization, but sensor noise, offset, drift, reference error, front-end error, and calibration still limit accuracy.
When should I recalibrate a sensor?
Recalibrate after mounting changes, significant temperature or aging exposure, mechanical shock, or whenever verification against references exceeds the allowed error budget.
Summary
Start from the physical measurand, declare timing and coordinate conventions, convert raw signals with consistent units, calibrate against references, and keep uncertainty sources separate. Calculators make the deterministic conversions repeatable; validation establishes whether the complete sensor system meets the required accuracy and dynamics.