Rigid Body Attitude Estimation- An Overview and Comparative Stud.pdf

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One of the receivers can be placed at the origin of the body. This receiver is denoted as receiver 1 with position p 1 and the positions of other receivers are denoted by p i + 1 . Let x i : = p i + 1 - p 1 , (4.106) that is expressed and known in B . By taking the x i vectors as the columns of the matrix X : = [ x 1 x 2 ... x r - 1 ] R 3 × ( r - 1) , a linear combination of the body vectors can be expressed as y i : = r - 1 X i = 1 b i j x i Y X = XB X , (4.107) where B X R ( r - 1) × ( r - 1) is invertible by construction and Y X : = [ y 1 y 2 ... y r - 1 ] R 3 × ( r - 1) . The vector Y X can be transformed to the inertial frame coordinated by ¯ Y X : = R Y X and to the observer frame by ˆ Y X : = ˆ R Y X . The nonlinear observer with bias correction is given by ˙ ˆ R = ˆ RS ( ˆ ω ) , ˆ ω = ˆ R T ¯ Y X ˆ Y T X ˆ R ( ω y - ˆ ω b ) - k ω σ, ˙ ˆ ω b = k b σ, σ = ˆ R T n X i = 1 ( ¯ Y X e i ) × ( ˆ Y X e i ) , (4.108) with e i being the unit vector where e j = 1 , for j = i . The observer inputs ¯ Y X = - [ f p ( ρ 2 ) - f p ( ρ 1 ) , ... , f p ( ρ r ) - f p ( ρ 1 )] B X and ˆ Y X = ˆ R XB X can be calculated using the vectors ρ j : = [ ρ 1 j ... ρ m j ] , j = 1 , ..., r , obtained from (4.105) with a constant range bias assumption and known coordinates of pseudo-satellites installed at ground level. f p ( ρ j ) , j = 1 , ..., r are functions of the sensor measurements and observer estimates that includes matrices described by the pseudoranges measurements and satellite’s position. The definition and derivation of this function can be found in the original work.

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C hapter 4. D ynamic A ttitude F iltering and E stimation 82 Exponential stability of the origin of the error system for the proposed observer for both biased and unbiased velocity measurements was shown. The work was also extended to the design of a position and linear-velocity observer of the following form ˙ ˆ p = ˆ v - k p ˜ p , ˙ ˆ v = ˆ Rb a + ge 3 - k v ˜ v , (4.109) where b a is the accelerometer reading in the body frame. The same stability analysis was applied to the cascaded system for biased and unbiased velocity measurements. It should be noted that the complete observer is in fact on SE (3) since the translational motion dynamics of the moving vehicle were considered in the observer design. How- ever, this observer is discussed in this section since the structure of the attitude observer of (4.108) is on the Special Orthogonal group SO (3) and even without the position and veloc- ity observer, which does not have any e ff ects on the main attitude observer, the estimated attitude is obtained from a compatible observer on the rotation group. Moreover, the basic idea of designing such an attitude observer is based on the use of multiple GPS receivers installed onboard the flying vehicle, which results in a technique that relies only on GPS data for its estimation. The method, thus, has a fundamental di ff erence in structure with most other SE (3) based observers that require position data obtained from a set of on-board cameras or image-based position / velocity estimations. The work is partially an extension to the authors’ previous work in [Vasconcelos et al., 2008a], where an attitude observer on SO (3) with biased angular velocity readings was designed.
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