Calculate gravity with inclometer

Question

How would one convert an inclinometers (Pitch, Yaw and Roll) into the gravitational pull expected on the system in [X,Y,Z]?

A system at rest in a certain Pitch, Yaw and Roll angle should be pulled to earth at a certain [X*g,Y*g,Z*g], lets say this is for simulation purposes. I want to make a function whoose input is Pitch, Yaw and Roll and the output is a Vector3(X,Y,Z) of the downard moment.

Meaning a object at rest with it's back downwards would output something like [0,-1,0] from the accelerometers and a [pitch,yaw,roll]->[0,-1,0], where [0,-1,0] minus [0,-1,0] resulting in [0,0,0]. or if we pull it left at the speed 1g we have a accelerometer showing [1,-1,0] making the new value [1,0,0].

With the system on its back [pitch,yaw,roll]->[0,-1,0] function is what i'm after

Vector3 OriToXYZ(float pitch, float yaw, float roll){
    Vector3 XYZ = Vector.Zero;
    //Simulate what the XYZ should be on a object in this orientation
    //oriented against gravity
    ...
    return XYZ;
}

Yes I know as the explanation below shows I'm not able to detect if the systems upside down or not based on the roll as roll only gives (-90 to 90) but that's a different problem).

This is how the orientation is laid out.

For extra information about why, what and how to use this information keep reading.

The plan is to use the incinometer as an alternative to the gyrometer for removing the gravity component to the accelerometer data, by simulating/calculating the expected value of gravity at orientation (Pitch,Yaw,Roll).

As the accelerometer(XYZ) is a combination of two components gravity(XYZ) and movement(XYZ), I'm assuming that gravity(XYZ)-calc_g(XYZ) = 0, allowing me to perform accelerometer(XYZ)- calc_g(XYZ) =movement(XYZ)

to show why i think this is possible. when i graph the values from the phone and move the phone sideways hard in a somewhat pendulum motion the lines that looks like sine/cosine motions are inclinometre and the other lines are XYZ accelerometer:

• red = (Pitch & accell-X)
• green = (Yaw & accell-Y)
• blue = (Roll & accell-Z)

Acceleration value is multiplied by 90 as it ranges from (-2 to 2) so that it in the drawing ranges from -180 to 180, Pitch yaw and roll ranges as seen in the instructable above. The middle of the image is Y = 0, left side is X=0 (X=time)

Solved Solution by Romasz

VectorX = Cos(Pitch)*Sin(Roll);
VectorY = -Sin(Pitch);
VectorZ = -Cos(Pitch)*Cos(Roll);

Result

*The graphs are not from the same measurement.

Answer 1

(Edited (completely) after comments)

If you want to calculate a gravity components in the direction of inclination then you will need only Pitch and Roll (in WP convention) - the rotation about Z (Yaw) doesn't have influance on accelerometers. The formula should look like this:

VectorX = Cos(Pitch)*Sin(Roll);
VectorY = -Sin(Pitch);
VectorZ = -Cos(Pitch)*Cos(Roll);

(Similar formula you can find for example here or here)

There can be some problems with accuracy from many reasons:

• inclination is single precision, acceleration double
• it may take a while for inclinometers to stabilize
• different timing while performing readouts from Inclinometer and Accelerometer (lookout becasue those sensors have different minimum report interval)
• accelerometers have different accuracy dependant on their position

Also watch out because accelerometers can be overloaded (their range is +-2g) - for example if you snap the phone.

---

To test it I have writted a simple App (which you can download here) - comparing Values indicated by Accelerometers and those calculated via inclination. Because values of accelerometers are relative to gravity, its strightforward:
In XAML - few TextBlocks:

<Grid x:Name="LayoutRoot" Background="Transparent" Margin="20">
        <Grid.RowDefinitions>
            <RowDefinition Height="1*"/>
            <RowDefinition Height="1*"/>
            <RowDefinition Height="1*"/>
        </Grid.RowDefinitions>
        <StackPanel Grid.Row="0" VerticalAlignment="Center" Orientation="Horizontal">
            <TextBlock Text="Incliation:" FontSize="16"/>
            <TextBlock Name="incXTB" Margin="10"/>
            <TextBlock Name="incYTB" Margin="10"/>
        </StackPanel>
        <StackPanel Grid.Row="1" VerticalAlignment="Center" Orientation="Horizontal">
            <TextBlock Text="Accelerometers:" FontSize="16"/>
            <TextBlock Name="accXTB" Margin="10"/>
            <TextBlock Name="accYTB" Margin="10"/>
            <TextBlock Name="accZTB" Margin="10"/>
        </StackPanel>
        <StackPanel Grid.Row="2" VerticalAlignment="Center" Orientation="Horizontal">
            <TextBlock Text="Through Inc:" FontSize="16"/>
            <TextBlock Name="accincXTB" Margin="10"/>
            <TextBlock Name="accincYTB" Margin="10"/>
            <TextBlock Name="accincZTB" Margin="10"/>
        </StackPanel>
</Grid>

In code behind:

public partial class MainPage : PhoneApplicationPage
{
    private Inclinometer myIncMeter = null;
    private float inclX = 0;
    private float inclY = 0;

    private Accelerometer myAccel = null;
    private double accX = 0;
    private double accY = 0;
    private double accZ = 0;

    public MainPage()
    {
        InitializeComponent();
        this.DataContext = this;
        myIncMeter = Inclinometer.GetDefault();
        myIncMeter.ReportInterval = myIncMeter.MinimumReportInterval;
        myAccel = Accelerometer.GetDefault();
        myAccel.ReportInterval = myIncMeter.MinimumReportInterval;

        CompositionTarget.Rendering += CompositionTarget_Rendering;
    }

    private void CompositionTarget_Rendering(object sender, EventArgs e)
    {
        InclinometerReading incRead = myIncMeter.GetCurrentReading();
        AccelerometerReading accRead = myAccel.GetCurrentReading();

        accX = accRead.AccelerationX;
        accY = accRead.AccelerationY;
        accZ = accRead.AccelerationZ;

        inclX = incRead.RollDegrees;
        inclY = incRead.PitchDegrees;

        incXTB.Text = "X: " + inclX.ToString("0.00");
        incYTB.Text = "Y: " + inclY.ToString("0.00");

        accXTB.Text = "X: " + accX.ToString("0.00");
        accYTB.Text = "Y: " + accY.ToString("0.00");
        accZTB.Text = "Z: " + accZ.ToString("0.00");

        accincXTB.Text = "X: " + ((Math.Cos(inclY * Math.PI / 180) * Math.Sin(inclX * Math.PI / 180))).ToString("0.00");
        accincYTB.Text = "Y: " + (-Math.Sin(inclY * Math.PI / 180)).ToString("0.00");
        accincZTB.Text = "Z: " + (-(Math.Cos(inclX * Math.PI / 180) * Math.Cos(inclY * Math.PI / 180))).ToString("0.00");
    }
}