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Dual-Manipulator Testing Technique
Published in Chunguang Xu, Robotic Nondestructive Testing Technology, 2022
In algebra, matrix exponential is a matrix function, similar to a general exponential function. The exponent of a matrix determines the mapping relation between the Lie algebra of the matrix and the region of its Lie group. On the other hand, matrix logarithm is the inverse function of matrix exponential [5]. Not all matrices have a logarithm. Those matrices with logarithm may have more than one logarithm. The study of logarithmic matrices has been accompanied by the development of the theory of Lie group, because a matrix with logarithm must be in a Lie group, and its logarithm must be the corresponding element of Lie algebra.
Riemannian Classification for SSVEP-Based BCI
Published in Chang S. Nam, Anton Nijholt, Fabien Lotte, Brain–Computer Interfaces Handbook, 2018
Sylvain Chevallier, Emmanuel K. Kalunga, Quentin Barthélemy, Florian Yger
Exp (⋅) and Log (⋅) are the matrix exponential and matrix logarithm, respectively. These two mappings are illustrated in Figure 19.3. The computation of these operators is straightforward for SPD matrices of ℳC. They are obtained from their eigenvalue decomposition:
Rotations in Three Dimensions
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
The matrix logarithm is defined as the inverse of the exponential X=log(expX).
Logarithmic control, trajectory tracking, and formation for nonholonomic vehicles on Lie group SE(2)
Published in International Journal of Control, 2019
Adopting the notation from Bullo and Murray (1995), we develop the exponential coordinates of SE(2) and its time derivative using the logarithm map. Since SE(2) is a matrix Lie group, the exponential map coincides with the matrix exponential as The inverse map is also given by the matrix logarithm as provided trace(g) ≠ −1 where q = (qx, qy)T = A−1(θ)p, , and . We refer to as the exponential coordinates of g which is parameterised by X = (θ, qx, qy)T. Note that even though and are both the Lie algebra elements corresponding to g but they represent velocity and configuration, respectively.
Consensus-based formation control for nonholonomic vehicles with parallel desired formations
Published in International Journal of Control, 2021
According to the property of the matrix logarithm in Lemma 2.2, there holds the equality In addition, Substitute (31) and (32) into (29), and it is obtained that Therefore, the relative system can be expressed as From (34), if is stabilised to the identity, the stabilisation of comes true. More significantly, in (34), the left side involves convex combination that contains , while the right side merely involves convex combination which does not contains . Denote , then where is the relative configuration of with respect to . Thus, the stabilisation of has been converted to that of , where the latter one does not have algebraic loop due to the nonexistence of in convex combination .
Continuous-time Laguerre-based subspace identification utilising nuclear norm minimisation
Published in International Journal of Systems Science, 2021
Miao Yu, Ge Guo, Jianchang Liu
It is worth mentioning that, compared with the identification methods of dealing with time-derivative in Li et al. (2003), Zhang et al. (2012) and Noël et al. (2014), the advantages of using the Laguerre filters can be mainly summarised with two aspects: (i) It avoids the complicated calculation of matrix logarithm due to that it deals with the identification problem via the bilinear relationship in Equation (6); (ii) It circumvents the non-differentiable problems of the continuous-time systems with both process and measurement noises due to that the input–output signal in Equation (1) can be dealt with by the Laguerre filters in Equation (6).