• Document: Indirect Kalman Filter for 3D Attitude Estimation
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Indirect Kalman Filter for 3D Attitude Estimation A Tutorial for Quaternion Algebra Nikolas Trawny and Stergios I. Roumeliotis Department of Computer Science & Engineering University of Minnesota Multiple Autonomous Robotic Systems Laboratory Technical Report Number 2005-002, Rev. 57 March 2005 Dept. of Computer Science & Engineering University of Minnesota 4-192 EE/CS Building 200 Union St. S.E. Minneapolis, MN 55455 Tel: (612) 625-2217 Fax: (612) 625-0572 URL: http://www.cs.umn.edu/˜trawny Indirect Kalman Filter for 3D Attitude Estimation A Tutorial for Quaternion Algebra Nikolas Trawny and Stergios I. Roumeliotis Department of Computer Science & Engineering University of Minnesota Multiple Autonomous Robotic Systems Laboratory, TR-2005-002, Rev. 57 March 2005 1 Elements of Quaternion Algebra 1.1 Quaternion Definitions The quaternion is generally defined as q̄ = q4 + q1 i + q2 j + q3 k (1) where i, j, and k are hyperimaginary numbers satisfying i2 = −1 , j2 = −1 , k2 = −1 , −ij = ji = k , − jk = kj = i , − ki = ik = j (2) Note that this does not correspond to the Hamilton notation. It rather is a convention resulting in multiplications of quaternions in “natural order” (see also section 1.4 and [1, p. 473]). This is in accordance with the JPL Proposed Standard Conventions [2]. The quantity q4 is the real or scalar part of the quaternion, and q1 i + q2 j + q3 k is the imaginary or vector part. The quaternion can therefore also be written in a four-dimensional column matrix q̄, given by   q  T q̄ = = q1 q2 q3 q4 (3) q4 If the quantities q and q4 fulfill   kx sin(θ/2) q = ky sin(θ/2) = k̂ sin(θ/2), q4 = cos(θ/2) (4) kz sin(θ/2) the elements q1 , . . . , q4 are called “quaternion of rotation” or “Euler symmetric parameters” [1]. In this notation, k̂ describes the unit vector along the axis and θ the angle of rotation. The quaternion of rotation is a unit quaternion, satisfying q p |q̄| = q̄ T q̄ = |q|2 + q42 = 1 (5) Henceforth, we will use the term “quaternion” to refer to a quaternion of rotation. The quaternion q̄ and the quaternion −q̄ describe a rotation to the same final coordinate system position, i. e. the angle–axis representation is not unique [1, p. 463]. The only difference is the direction of rotation to get to the target configuration, with the quaternion with positive scalar element q4 describing the shortest rotation [2]. 2 1.2 Quaternion Multiplication The quaternion multiplication is defined as q̄ ⊗ p̄ = (q4 + q1 i + q2 j + q3 k) (p4 + p1 i + p2 j + p3 k) = q4 p4 − q1 p1 − q2 p2 − q3 p3 + (q4 p1 + q1 p4 − q2 p3 + q3 p2 ) i + (q4 p2 + q2 p4 − q3 p1 + q1 p3 ) j + (q4 p3 + q3 p4 − q1 p2 + q2 p1 ) k   q4 p1 + q3 p2 − q2 p3 + q1 p4  −q3 p1 + q4 p2 + q1 p3 + q2 p4  =   q2 p1 − q1 p2 + q4 p3 + q3 p4   −q1 p1 − q2 p2 − q3 p3 + q4 p4 where we have used the relations defined in Eq. (2). The quaternion multiplication can alternatively be written in matrix form. For this, we first introduce the matr

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