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

Is a point arbitrarily taken on l i and d i is its

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is a point arbitrarily taken on l i , and d i is its direction vector. The scalar s de- termines the length of vector in each direction. In the body frame, the line is represented by l i = { x B R 3 : x B = R T ξ i , I + d i , I s - p , s R } , (4.142) with p denoting the rigid body position expressed in the inertial frame. The camera is assumed to be placed in the rigid body such that its focal point coincides with the origin of B . With such assumption, each line l i is projected onto the image plane P I as an intersection of this plane with the camera focal point P l . The normal vector η i to P l is given by η i = R T [ d i × ( ξ i - p )] . (4.143) Since the camera focal plane never becomes perpendicular to the plane in which the line exists, i.e. , d i and ξ i - p ( t ) are not parallel, it can be assumed that the norm of η i never becomes zero. Therefore, the observations in the body frame can be taken as y i = μ i η i k η i k , (4.144) with η i ∈ {- 1 , 1 } being an unknown parameter for sign ambiguity. The existence of this sign parameter for the unknown line directions does not a ff ect the structure of the observer. The vectors y i correspond to the normalized normal vectors of some known lines in the inertial frame projected onto the camera plane. Assuming that at least three non-collinear inertial lines are available, by intuition a reconstruction of the orientation is possible, since each observer normal vector can be assigned as a basis to generate orthogonal bases in B . In this way, the attitude observer with point and line observations is given by ˙ ˆ R = ˆ R [ S ( ω y ) + σ ] , σ = k i N i = 1 y T i ( ˆ R T d i ) S ( y i × ˆ R T d i ) , (4.145)
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C hapter 4. D ynamic A ttitude F iltering and E stimation 98 where k i are positive scalars. In order to show the convergence of this observer, the follow- ing Frobenius norm Lyapunov function is considered V = 1 2 k X k 2 F = 1 2 k I - ˆ R T R k 2 F . (4.146) The local representation of the error rotation matrix ˜ R = ˆ R T R is derived as a function of k X k F using the Rodrigues attitude representation ˜ R = I + S ( ˆ k ) k X k 2 + O ( k X k 2 ) , (4.147) where ˆ k R 3 with k ˆ k k = 1. The term O ( x k ) contains higher order terms with x k as its lowest degree term. Local exponential convergence of the estimated attitude is shown by taking the time derivative of the Lyapunov function (4.146) ˙ V = - ( R T ˆ k ) T { N X i = 1 k i p i p T i k p i k 2 } ( R T ˆ k ) k X k 2 - O ( k X k 3 ) . (4.148) In order for the observer to be locally convergent, a condition of trivial observability is investigated. Letting p i ( t ) be the shortest vector from the position p ( t ) to the line l i , i.e. , p i ( t ) = p ( t ) - ξ i ( t ) is chosen such that d T i ( t )( p ( t ) - ξ ) = 0, and the information matrix be given by Q ( t ) = N X i = 1 p i ( t ) p T i ( t ) k p i ( t ) k 2 , (4.149) the system is called trivially observable if the following condition is met det Q ( t ) q , (4.150) for a non-negative q . Assuming that this observability condition holds, the Lyapunov func- tion derivative of (4.148) leads to - ˙ V ≥ q min { k i } ( R T ˆ k ) T ( R T ˆ k ) k X k 2 + O ( k X k 3 ) q min { k i }k ˆ k k 2 k X k 2 + O ( k X k 3 ) (4.151)
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