In the paper "State Estimation for Legged Robots - Consistent Fusion of Leg Kinematics and IMU" (link above), the authors describe the application of an extended kalman filter to estimate states of a quadruped robot, where the equations used in the prediction phase of the kalman filter were equations 30 to 37:

[Link for paper][1]

[![Prediction model equations][2]][2]

Where (r,v,q,$p_1$..$p_n$,$b_f$,$b_w$) represent the states: position of the main body, velocity, quaternion of rotation from the intertial coordinate frame to the body coordinate frame, the absolute positions of the N foot contact points, accelerometer bias and gyroscope bias respectively. in addition, $C_k$, g, $f_k$ and $w_k$ represent: the rotation matrix corresponding to quaternion orientation, the gravity vector, the measured quantities of the accelerometer and gyroscope respectively.

The discrete linearized error dynamics matrix $F_k$ is represented by equation 44:
[![Discrete linearized error dynamic matrix][3]][3]

In addition, the authors introduce some auxiliary equations such as mappings and series, such as equation 13 which is a mapping of a rotation error vector in a quaternion, there is also the equation 20 where it is used to map the angular velocity in a matrix 4x4, and there are also equations 28 and 29 that are introduced to aid in the process of linearization and discretization of the models.

[![Mapping rotation vector in a quaternion (Equation 13)][4]][4]

[![Mapping of angular velocity in matrix 4x4 (Equation 20)][5]][5]

[![Exponencial Series (equation 28 and 29)][6]][6]

My first doubt is, what does this symbol below, used in the matrix Fk?

[![Unknown symbol][7]][7]

My second doubt is, the symbol $w^x$ represents a skew-symmetric matrix of angular velocity, the resulting matrix of mapping 20 is a skew-symetric matrix, I would like to know if the matrix $w^x$ used in equations 28 and 29 is the matrix resulting from mapping the angular velocity in a 4x4 matrix (equation 20)?

My third doubt is, how to apply equation 28-29? as it is a series of powers ranging from 0 to infinity. Do I need to truncate it?

Thank you, please help me clear this doubt.


  [1]: http://www.roboticsproceedings.org/rss08/p03.pdf
  [2]: https://i.sstatic.net/s1lKI.jpg
  [3]: https://i.sstatic.net/LYJq1.jpg
  [4]: https://i.sstatic.net/yqKTX.jpg
  [5]: https://i.sstatic.net/EYe0D.jpg
  [6]: https://i.sstatic.net/KC3tl.jpg
  [7]: https://i.sstatic.net/ltl5w.jpg