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6.2. Sensor Calibration 105The errors due to accelerometer alignment and population mean estimation are independent.The errors in the minimum and maximum accelerometer measurements are therefore:δmin,δmax = δ ¯x 2 + δa 2 = 0.001 2 + 0.0013 2 ≈ 0.0016 ≡ 0.16%. (6.10)The scale error, δS i , for each axis is: ∂S i 2 ∂S i 2δS i =δmax i + δmin i∂max i ∂min i 2= 2 δmax(max − min) 2 2i + δmin 2 i1 = δmax24 i + δmin 2 i= 0.1%, (6.11)similarly, the offset error, δO i , is: ∂O i 2 ∂O i 2δO i =δmax i + δmin i∂max i ∂min i= 0.1%. (6.12)The effect of accelerometer scale and offset errors is to cause an error in the estimatedgravity vector. This orientation error can be defined as the error angle between the true gravityvector, g re f , and the measured vector, g meas :θ ε = cos −1 g re f ·g measg re f . (6.13)· g meas The error is also dependent on the orientation of the device. This dependence occurs as thegravity vector is measured by a combination of device axes.A Monte Carlo simulation was performed to generate the expected error distribution. Onethousand simulations were performed with the scale and offset errors normally distributed withσ = 0.001. For each simulation the orientation of the device was parameterised by two angles:φ ∈ [−45 ◦ ,−44 ◦ ,...,44 ◦ ,45 ◦ ] and θ ∈ [−45 ◦ ,−44 ◦ ,...,44 ◦ ,45 ◦ ]. The resulting estimatedCDF is shown in Figure 6.2. It can been seen that 90% of the resulting errors are lessthan 0.15 ◦ .6.2.1.2 Magnetometer CalibrationAs with the accelerometers, suitable scale and offset must be determined for the magnetometers.The calibration is complicated by the fact that the exact direction of the magnetic fieldvector is hard to estimate due to the inclination angle. Measuring the minimum and maximumvalues for each axis is not simple.