1. The Description of Rotation
can project vectors from global to camera frame.
where and represent the vector in camera and global frame, respectively. There are two descriptions of (1) as following:
- The rotation matrix from global to camera frame.
- The rotation matrix representing global direction with respect to camera frame.
Translation works the same way.
2. Possion Equation
2.1 Rotation Matrix Version
A vector of angular velocity is the product of an angular and an axis .
The velocity of vector is defined by the first derivative with repect to time:
We define that axes of the body frame are . Usually the axes in body frame are , , . And the axes in the world frame is computed as following:
where is a 3x3 identity matrix. The equation (3) means columns are vectors representing body axes with respect to global frame.
We can decompose the rotation about into three axes. in (2) is with respect to global frame. Then the ordinary differential equation is
Sensors usually are mounted on the robot and measure the angular velocity in body frame. Thus, we need to express (4) with using equations and
The equation (5) is consistent with the formula (6.45) in State Estimation for Robotics,
where the coordinate is the body frame in the book and is the global frame.
Equations (5) and (6) are Poisson's equation.
2.2 Quaternion Version
The relationship between rotation vector and quaternion can be expressed as:
where the rotation vector is , is the angular and is the unit vector.
In the limit, as , the angle of the rotation will become very small. Quaternion can be simplified as:
The derivative of with respect to represents the angular velocity in the body frame.
is the angle between times and , thus it represent the local angular velocity.
We now deduce the quaternion derivative as
where is a pure quaternion and is the left-quaternion-product matrix, which can be writen by:
where is the imaginary unit. The mathematical deduction in the page can be found in section 4.5 of Quaternion kinematics for the error-state Kalman filter.
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