We can group an air vehicle’s 12 state variables into 4 buckets.
3D position: x,y,z
3D translational velocity: x_dot, y_dot, z_dot
3D orientation: ϕ,θ,ψ
3D rotational velocity: p,q,r
One reason is redundancy: if one sensor fails than another sensor is there for backup. But that's not the only reason! It's also important to consider how a sensor is inaccurate! Some sensors are noisy but unbiased. Some are biased but low-noise. Ideally we want our estimate of the vehicle's state to be both unbiased and low-noise and often that means using multiple sensors to measure the same quantity.
7. Gyro Measurement Model
Noise is zero mean gaussian noise
Bias might be temperature dependent and changes slowly but is constant for a long time.
Now we’d like to use this measurement to track the angle theta We can use this omega_tilda measurement to update our angle from timestamp to timestamp.
This is a discreet approximation for computing the integral so there really would be some epsilon e approximation error but we’re going to ignore that for now.
We can also assume bias has been removed with calibration, so we eliminate that below.
We also see that we need to have a theta_0 in the beginning for
10. Full 3D attitude update
Euler Forward Method
Derivative of the Euler angles = some matrix that depends on the vehicle attitude x the body rates.
Euler angles at time t is the last set of angles plus the derivative x Delta t
Assuming the vehicle is not moving, the accelerometers can measure attitude
Solving for World-Frame Attitude with Body-Frame Acceleration Measurements
We get the body frame measurements are equal to a rotation matrix x the gravity vector. When using the euler angles, the rotation matrix is expressed with roll angle(phi) and pitch(theta). When you solve these equations for theta and phi, you get the arctan and arcsine equations.
NB: error in graphic, should be-g
2. Solving for position and velocity from linear acceleration
We can only do this when we know the attitude because
if we know the attitude, we know the rotation matrix
if we know the rotation matrix, we can solve for acceleration with the formula below
3. Introduction to Sensors
7. Gyro Measurement Model
10. Full 3D attitude update
Euler Forward Method
13. Two Things Accelerometers Measure
1. Attitude (without yaw)
Solving for World-Frame Attitude with Body-Frame Acceleration Measurements
2. Solving for position and velocity from linear acceleration