Strapdown Mechanization
The strapdown mechanization covered here is restricted to what is presently implemented in state-estimation.
To Do: Decide what from below should survive when I write this page for real.
\[ \mathbf{g}_A=\begin{bmatrix}0\\0\\-g\end{bmatrix},\quad g=9.8>0 \]
\[ q_{AI}:\ I\to A \]
so to rotate gravity from \(A\) into IMU/body frame \(I\), use the inverse (conjugate):
\[ \mathbf{g}_I=R\!\left(q_{AI}^{*}\right)\,\mathbf{g}_A \]
Then convert specific force to total (kinematic) acceleration in body frame:
\[ \mathbf{a}_I=\mathbf{f}_I+\mathbf{g}_I \]
Using the doc’s quaternion action definition, this is equivalently:
\[ \mathbf{g}_I = \left(q_{AI}^{*}\otimes[0,\mathbf{g}_A]\otimes q_{AI}\right)_{\text{vector part}}, \quad \mathbf{a}_I=\mathbf{f}_I+\mathbf{g}_I. \]
This section covers the theory for the strapdown mechanization part of the State Estimation application.
Use
| term | definition |
|---|---|
| specific force | the raw input from the accelerometer, notated as \(f_B\) |
| kinematic (or translational) acceleration | \(a=\dot v\) |
Don’t use
linear acceleration, total acceleration, net acceleration, inertial acceleration
Specific force, \(f_B\) (or equivalently \(f_I\), since we have seen that the body frame, \(B\), and the IMU frame, \(I\), are collocated).
Specific force in the body frame: \(f_b\)
\(f_B = a_B - g_B\)
Kinematic acceleration in the body frame: \(a_b\)
\(a_B = f_B + g_B\)
Gravity in the body frame: \(g_b\)
- Start with the specific force reading from the IMU; since in the State Estimation application the IMU and body frame are collocated, we notate this as \(f_B\).
- Only one artifact of the gravitational force changes accelerometer readings: The normal contact force. Indeed, the proof mass can never “report” the direct influence the gravitational force has on it. This is because in the absence of a normal opposing force (i.e., in free fall), the accelerometer case and the proof mass respond exactly the same way to the gravitational force; that prevents the proof mass from shifting relative to its casing under free fall. And only the proof mass shifting relative to its casing causes the accelerometer to produce readings.
- A corollary to #2 is that any free fall experienced by both the accelerometer casing (which might as well be the car to which the casing is rigidly attached) and the proof mass is not reported by the accelerometer.
- Once #2 and #3 are clear, you can convince yourself that only if we add the acceleration due to gravity into the accelerometer reading can we actually know what acceleration, or force per unit mass, the car is experiencing. This is easiest to see with a [very] clean example. Assume a car is going down a hill. To make the math clean and the problem definite we’ll imagine the road is perfectly straight, the car is driving along a perfectly straight path along the road, and the car is perfectly “flat” at all times in the plane of the road… so no bumps, no passengers jostling around, etc. (never mind whether the driver is pushing on the pedal to speed up, braking, or neither it matters in general, but not for the gravity-related subset of the general case that we are ironing out). So how does this relate to #2 and #3. Well, the most important things related to gravity are the two key ideas in #2 and #3: How the normal force applies and how free fall applies. The normal force is responding to the component of gravity in the direction normal to the road. And that is baked into the accelerometer reading. So if we add gravitational acceleration decomposed into the directions normal to the road and along the road (in short, if we add gravitational acceleration in the IMU/body frame), then we have a term that cancels the normal force (good because the car isn’t going airborne in the direction normal to the road) and a term that contributes the free fall the car (and the accelerometer casing and the proof mass) would be experiencing along the sloped road (if there was no friction and, in general, no other forces than gravity in the direction along the road). Again good [because even though there will always be forces along the road other than gravity pushing on the car], we do need everything that represents an acceleration –force per unit mass– the car is experiencing in our expression for the car’s total acceleration that we integrate twice for position updates).