Source of the problem is the actual meaning of the angular velocity of the IMU Gyroscope measurements and thereby aid in understanding the process of pre-integration IMU integration process Rotation Matrix (i.e., [1] in the formula (30) in the first equation).
To solve this problem, reference [2] State Estimation for Robotics of 6.2.4 Rotational Kinematicsand 6.4.4 Inertial Measurement Unit.
1. The angular velocity measured value Meaning
In for 6.2.4After basic impression, reading 6.4.4.
In 6.4.4the FIG. Figure 6.14There are three coordinate systems - inertia (world) coordinate system \ (\ underrightarrow {\ mathcal _i} {} F. \) , Carrier coordinate system \ (\ underrightarrow {\ mathcal _V} {} F. \) , IMU coordinate system \ (\ underrightarrow {\ mathcal _s} {} F. \) .
IMU coordinate system and the coordinate system do not coincide with the carrier, so the IMU measurements can not be directly used in the vehicle coordinate system, you need to be converted.
Equation (6.149) shows the measured values Gyroscope \ (\ mathbf {\ omega} \) and the desired angular velocity of the carrier \ (\ mathbf {\ omega} ^ {vi} _v \) relationship:
\[\mathbf{\omega} = \mathbf{C}_{sv}\mathbf{\omega}^{vi}_v，\]
Why official (6.3)
\[\underrightarrow{\mathcal{F}_1}^T = \underrightarrow{\mathcal{F}_2}^T \mathbf{C}_{21}\]
Get
\[\mathbf{\omega} = \underrightarrow{\mathcal{F}_s}\underrightarrow{\mathcal{F}_v}^T\mathbf{\omega}^{vi}_v，\]
Why official (6.40)
\[\underrightarrow{\omega}_{21} = \underrightarrow{\mathcal{F}_2}^T \mathbf{\omega}^{21}_2\]
Get
\[\begin{aligned} \mathbf{\omega} &= \underrightarrow{\mathcal{F}_s}\underrightarrow{\mathbf{\omega}}_{vi} \\ \underrightarrow{\mathcal{F}_s}^T \mathbf{\omega} &= \underrightarrow{\mathbf{\omega}}_{vi} \end{aligned}，\]
Because IMU and is formed between the rigid carrier, relative motion does not exist, from the space vector angular velocity (note, not the coordinates of the vector in a vector base) perspective, the angular velocity IMU same carrier. which is
\ [\ Under rightarrow {\ mathbf {\ omega}} _ {say} = \ under rightarrow {\ mathbf {\ omega}} _ {we}. \]
Therefore, the angular velocity measurements that
\[\mathbf{\omega} = \mathbf{\omega}^{si}_s。\]
Note that \ (\ underrightarrow {\ omega} _ {21} \) meaning, in 6.2.4a word quoted below
The angular velocity of frame 2 with respect to frame 1 is denoted by \(\underrightarrow{\omega}_{21}\).
Therefore, \ (\ underrightarrow {\ mathbf { \ omega}} _ {si} \) is understood to IMU coordinate system \ (\ underrightarrow {\ mathcal { F} _s} \) relative to an inertial coordinate system \ (\ underrightarrow {\ mathcal { F} _i} \) an angular velocity vector, \ (\ mathbf {\ Omega} ^ {Si} _s \) symbol is understood to IMU coordinate system \ (\ underrightarrow {\ mathcal { F} _s} \) relative to an inertial coordinate system \ (\ underrightarrow {\ mathcal {F } _i} \) in the coordinate system of the IMU angular velocity vector \ (\ underrightarrow {\ mathcal { F} _s} \) coordinates of.
2. From the angular velocity integration
Reference 6.2.4careful to understand the formula (6.36). It is appreciated that from the perspective of an angular velocity vector, and the vector from the coordinate angle is not understood.
The here \ (\ underrightarrow {\ mathcal { F} _1} \) corresponding to an inertial coordinate system \ (\ underrightarrow {\ mathcal _i} {} F. \) , \ (\ Underrightarrow {\ mathcal _2} {} F. \ ) corresponding to the IMU coordinate system \ (\ underrightarrow {\ mathcal _s} {} F. \) .
An angular velocity in the space can be represented as a vector, in accordance with right-hand helix diet, this vector indicating the direction of the rotational direction, the length of the angular velocity of this vector indicate scalars.
At a time \ (\ {\ mathcal {F } _1}, \ {\ {F} _2 mathcal} \ underrightarrow underrightarrow) exists between the angular velocity \ (\ underrightarrow {\ Omega 21 is {}} _ \) , considered \ (\ underrightarrow {\ mathcal {F } _2} \) for each axis (i.e. the vector of the substrate where the coordinate system) \ (\ underrightarrow {the 2_1-th}, \ underrightarrow {2_2}, \ underrightarrow {2_3} \) versus time the rate of change (of course, in (\ underrightarrow {\ mathcal {F } _1} \) \ a result of observation under the ({F _2 \ mathcal {} } \) \ \ underrightarrow the result is 0, so-called relative movement). The following results were obtained (see FIG appreciated):
\[\begin{aligned} \underrightarrow{2^{\bullet}_1} = \underrightarrow{\omega}_{21} \times \underrightarrow{2_1} \\ \underrightarrow{2^{\bullet}_2} = \underrightarrow{\omega}_{21} \times \underrightarrow{2_2} \\ \underrightarrow{2^{\bullet}_3} = \underrightarrow{\omega}_{21} \times \underrightarrow{2_3} \end{aligned}\]
Considering now after a sufficiently short time \ (\ Delta t \) , the coordinate system \ (\ underrightarrow {\ mathcal { F} _2} \) after \ (\ Delta t \) time to move to the \ (\ underrightarrow {\ mathcal { _2 ^ {}} F. \ Prime} \) .
\[\begin{aligned} \underrightarrow{2^{\prime}_1} &= \underrightarrow{2^{\bullet}_1} \Delta t + \underrightarrow{2_1} \\ &= (\underrightarrow{\omega}_{21} \Delta t) \times \underrightarrow{2_1} + \underrightarrow{2_1} \\ \underrightarrow{2^{\prime}_2} &= \underrightarrow{2^{\bullet}_2} \Delta t + \underrightarrow{2_2} \\ &= (\underrightarrow{\omega}_{21} \Delta t) \times \underrightarrow{2_2} + \underrightarrow{2_2} \\ \underrightarrow{2^{\prime}_3} &= \underrightarrow{2^{\bullet}_3} \Delta t + \underrightarrow{2_3} \\ &= (\underrightarrow{\omega}_{21} \Delta t) \times \underrightarrow{2_3} + \underrightarrow{2_3} \end{aligned}\]
Merger, writing
\[\underrightarrow{\mathcal{F}^T_2}^{\prime} = (\underrightarrow{\omega}_{21} \Delta t) \times \underrightarrow{\mathcal{F}^T_2} + \underrightarrow{\mathcal{F}^T_2} \]
among them
\[\begin{aligned} \underrightarrow{\omega}_{21} &= \underrightarrow{\mathcal{F}^T_2} \mathbf{\omega}^{21}_2 \\ \underrightarrow{\omega}_{21} &= \underrightarrow{\mathcal{F}^T_1} \mathbf{\omega}^{21}_1 \end{aligned}\]
Consider \ (\ underrightarrow {\ mathcal { F} _1}, \ underrightarrow {\ mathcal {F} _2} \) rotation matrix between \ (\ mathbf {} _ {C} 21 is \) , the elapsed time \ ( \ Delta t \) after the change into the \ (\ mathbf {C} {_} ^ {21 is \ Prime} \) .
There
\[\begin{aligned} \mathbf{C}_{21}^T &= \underrightarrow{\mathcal{F}_1}\underrightarrow{\mathcal{F}^T_2} \\ {\mathbf{C}_{21}^{\prime}}^T &= \underrightarrow{\mathcal{F}_1}\underrightarrow{\mathcal{F}^T_2}^{\prime} \end{aligned}\]
For \ ({\ mathbf {C} _ {21} ^ {\ prime}} ^ T \) is difficult to calculate. Given a set of orthogonal base unit in this space, for simplicity will be set in the set-yl \ (\ underrightarrow {\ mathcal { F} _1} \) of the respective arbor, have a length of 1. These vectors are used to coordinate them in the units of the set of orthogonal group represented, to consider each coordinate dimension vector:
- \(\underrightarrow{\mathcal{F}_1}, \underrightarrow{\mathcal{F}_2}\)，\(9 \times 1\)；
- \ (\ underrightarrow {\ mathcal T_l} ^ {} F., \ underrightarrow {\ mathcal T_2} ^ {} F. \) , \ (. 3 \. 3 Times \) , noted here conflict with customary transpose symbol system ;
- \(\underrightarrow{\omega}_{21}\)，\(3 \times 1\)，并且 \({\underrightarrow{\omega}_{21}}_{3 \times 1} = \underrightarrow{\mathcal{F}^T_2}_{3 \times 3} {\mathbf{\omega}^{21}_2}_{3 \times 1}\) 。
Use properties \ ((\ mathbf {C} \ mathbf {V}) ^ {\ Wedge} = \ mathbf {C} \ mathbf {V} ^ {\ Wedge} \ mathbf {C} ^ T \) , available for the following derivation:
\[\begin{aligned} {\mathbf{C}_{21}^{\prime}}^T &= \underrightarrow{\mathcal{F}_1}((\underrightarrow{\omega}_{21} \Delta t) \times \underrightarrow{\mathcal{F}^T_2} + \underrightarrow{\mathcal{F}^T_2}) \\ &= \underrightarrow{\mathcal{F}_1}(\underrightarrow{\omega}_{21} \Delta t) \times \underrightarrow{\mathcal{F}^T_2} + \underrightarrow{\mathcal{F}_1}\underrightarrow{\mathcal{F}^T_2} \\ &= \underrightarrow{\mathcal{F}_1}(\underrightarrow{\mathcal{F}^T_2} \mathbf{\omega}^{21}_2 \Delta t)^{\wedge} \underrightarrow{\mathcal{F}^T_2} + \mathbf{C}_{21}^T \\ &= \underrightarrow{\mathcal{F}_1} \underrightarrow{\mathcal{F}^T_2} ( \mathbf{\omega}^{21}_2 \Delta t)^{\wedge} \underrightarrow{\mathcal{F}^T_2}^T \underrightarrow{\mathcal{F}^T_2} + \mathbf{C}_{21}^T \\ &= \mathbf{C}_{21}^T (\mathbf{\omega}^{21}_2 \Delta t)^{\wedge} \mathbf{I} + \mathbf{C}_{21}^T \\ &= \mathbf{C}_{21}^T (\mathbf{I} + (\mathbf{\omega}^{21}_2 \Delta t)^{\wedge}) \\ &= \mathbf{C}_{21}^T \text{Exp}(\mathbf{\omega}^{21}_2 \Delta t) \end{aligned}\]
Since then, get IMU attitude integral formula.
references
[2] Barfoot, Timothy D. State Estimation for Robotics. Cambridge University Press, 2017.