Represent as Modified Rodrigues Parameters (MRPs). On the other hand, by all accounts, it is not the actual derivative of the rotation in terms of the axis-angle parameters, but rather the derivative of an expression that has a corrective effect on the current estimate. University of Copenhagen, Copenhagen (1998), Dash, A.K., Chen, I.M., Yeo, S.H., Yang, G.: Workspace generation and planning singularity-free path for parallel manipulators. Further advantages of MRPs include the flexibility in constructing smooth quaternion curves with minimal distortion in more intuitive ways. is the identity mapping in the tangent space of R for any skew-symmetric matrix \(U_x\). The derivatives of the elements of a rotation matrix \(R=\left[ r_{ij}\right] \) in terms of MRPs are simple quadratic expressions of the quaternion components, \(\rho \) and \(\upsilon =\begin{bmatrix}\upsilon _1&\upsilon _2&\upsilon _3\end{bmatrix}^T\) and they are obtained via the chain rule, using Eqs. Even for a particular axes sequence, Euler angles are not unique since supplementary and/or negative angles can yield the same overall rotation[58]. Aided Geom. 5, the relationship between the Gibbs vector and the axis-angle vector given in Eq. It is therefore possible to devise a very simple retraction[1] \(R^{\prime }\), which maps the tangent space of R onto \(\mathcal{SO}(3)\) by applying a perturbing rotation on the rightFootnote 7 of R: It can be easily shown that \(R^{\prime }\) is a retraction, since the exponential map is smooth and \(R^{\prime }\left( \left[ 0\right] _{\times }\right) =Rexp\left( \left[ 0\right] _{\times }\right) =R\). Modified Rodrigues Parameters: An Efficient Representation of The proof is trivial for \(q=-1\). The common drawback of these approaches is that they are relatively complex to implement and not so flexible to configure under different circumstances. IEEE (2011), Hartley, R., Trumpf, J., Dai, Y., Li, H.: Rotation averaging. In other words, both the Jacobian computation as well as the update of the rotation matrix do not explicitly require the use of MRPs and both can be computed with a few primitive operations on previously stored numbers. More details on this representation peculiarity are given in Sect. Schaub et al. Tech. 26 for \(\upsilon ^{\prime }\), we have: Similarly, taking the stereographic projection formula in Eq. Interpolation of more than two key orientations is a far more challenging task, primarily because the constituent segments of the curve have to be pieced smoothly at the data points. the chord-length approximation method [67]). The idea is to observe the relationship between classical Rodrigues parameters (Gibbs vectors) and MRPs through axis-angle vectors as given in Eqs. IEEE Trans. Comput. 20 and 21: It should be stressed here that the components of \(\psi \) can assume infinite values. Figure9 illustrates plots of steps-to-convergence versus standard deviation of Gaussian noise. These two findings are very important for iterative optimization, because they allow both Jacobian computation and orientation updates to be carried out using exclusively quaternion components in simple additions and multiplications. JOSA A 5(7), 11271135 (1988), Hughes, J.F., van Dam, A., McGuire, M., Sklar, D.F., Foley, J.D., Feiner, S.K., Akeley, K.: Computer Graphics: Principles and Practice. Rep. 01-014, Dept. Springer, Berlin, Heidelberg (2013), Briales, J., Gonzalez-Jimenez, J.: Convex global 3D registration with Lagrangian duality. This block is intended for simulation (academic) purposes, where the time history of the attitude in term of Modified Rodrigues Parameters (MRPs) is provided. Pattern Anal. Based on the former, we introduce a novel approach for designing orientation splines by configuring their back-projections in 3D space. Vis. The method proposed in Sect. Two solutions are developed for the sensitivity matrix in the Kalman filter. Similarly to the BA experiment described above, all optimizations converged to the same poses for both parameterizations. Then, taking the stereographic projection formula in Eq. 2, followed by descriptions of common problems involving parameterized orientation and respective solutions in Sect. 10. 8, the components of the rotation matrix Jacobian tensor will in turn comprise simple polynomial expressions of the quaternion components (see AppendixA). Unscented Kalman filter for spacecraft attitude estimation using The execution times were generally lower for the parameterization based on MRPs, owing to the simpler calculations involved in the evaluation of the image projections and their derivatives. no. This local update is certain to lie within the aforementioned range, and the approach is also referred to as an incremental update in Sect. In: New Advances in Computer Graphics, pp. In modern mathematics, Euler-Rodrigues formula is used as an exponential map [25] that converts Lie algebra so (3) into Lie group SO (3), providing an algorithm for the exponential map without calculating the full matrix exponent [26], [27], [28] and for multi-body dynamics [29], [30], [31], [32]. However, our objective in this experiment was not to provide yet another solution, but rather to benchmark how MRPs compare against other parameterization schemes in the context of a basic, quadratic minimization problem in only the rotation parameters. Historically, this rotation-only formulation was originally introduced in astronautics as a satellite attitude estimation problem by Wahba[65]. 25 that \(1+\rho \left( u\right) =\frac{2}{1+\Vert \psi \left( u\right) \Vert ^2}\). 38, we make use of Lemma1, starting from the standard formula for the arc length of \(q\left( t\right) \): It can be easily inferred from Eq. Particularly in the case of animation, approximately constant speed in spherical curves is desirable because it can be warped into any desirable acceleration profile (e.g., trapezoidal moves) [66]. Springer (2013). Substituting the expression of \(r\left( t\right) \) from Eq. https://doi.org/10.1007/s10851-017-0765-x, DOI: https://doi.org/10.1007/s10851-017-0765-x. A visualization of stereographic projection in 3D. Previous work has used the MRP duality to avoid singular attitude descriptions but . A rotation matrix consists of nine elements but has only three DoFs due to the six independent constraints imposed by orthonormality. 3. In the last dataset, the parameterization employing MRPs required roughly four times more iterations but converged to a better minimum, which corresponded to over 60% lower average reprojection error compared to that obtained with quaternions. 4 is obtained by taking the trace of \(R+R^T\). This can be demonstrated with the aid of exponential notation: where \(R=\exp {\left( \left[ \omega \right] _{\times }\right) }\). The sba[37] package was used to optimize those datasets using its default, quaternion-based local rotation parameterization described in Sect. 4). Since it is minimal and does not require any additional constraints, the axis-angle representation is very often employed in vision and robotics problems. Recently, Terzakis et al. This means that there is no need to move through parameter spaces in iterative optimization, which is also an important benefit from a numerical and algorithmic standpoint. The resulting spherical curves have certain differential attributes which could be useful in manipulating their properties in the more familiar space \({\mathbb {R}}^3\). www.ics.forth.gr/~lourakis/sba, Lourakis, M., Zabulis, X.: Model-based pose estimation for rigid objects. Consequently, the 3 descent directions on the manifold are \(RG_1\), \(RG_2\), \(RG_3\), and they are obtained by differentiating \(R^{\prime }\) at the origin. 113118. For example, a point can be rotated using standard matrix-vector multiplication, two rotations can be composed via matrix multiplication, whereas a rotation can be inversed via matrix transposition. The horizontal asymptote \(y=-1\) is shown with a red dashed line (Color figure online). The ray \(r\left( t \right) =\mathrm {S}+t\left( \psi ^T\varphi -\mathrm {S}\right) \) intersects the unit sphere at q. IEEE (2007), Levenberg, K.: A method for the solution of certain non-linear problems in least squares. 20: It should be noted that the last formula is not valid for \(\psi = \left( 0, 0, 0\right) \), in which case the shadow quaternion coincides with the center of projection (i.e., the chosen South Pole) and the rotation has a single representation at the origin of the hyperplane. Abstract: In this article, attitude maneuver control without unwinding phenomenon is investigated for rigid spacecraft via modified Rodrigues parameters. In: Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, pp. Modified Rodrigues parameters is a formalism for the representation of orientation based on stereographic projection, originally introduced in the field of aerospace engineering by Wiener [68] in 1962. 724729 (1988), Rodrigues, O.: Des lois gomtriques qui rgissent les dplacements dun systme solide dans lespace, et de la variation des cordonnes provenant de ces dplacements considrs indpendamment des causes qui peuvent les produire. In our primary contribution, we show that the derivatives of a unit quaternion in terms of its MRPs are simple polynomial expressions of its scalar and vector part. Starting with the datasets employed for bundle adjustment in Sect. To study the descent behavior of MRPs against alternative parameterizations across multiple levels of noise, the optimization was carried out for 100 incremental standard deviation levels of noise from 0 to 2.5 using the LevenbergMarquardt algorithm [34, 42]. [27] introduced Modified Rodrigues Parameters, a projection of the unit quaternion sphere S 3 to R 3 used in attitude control [28], to a range of common computer. 33, No. Google Scholar, Feng-yun, L., Tian-sheng, L.: Development of a robot system for complex surfaces polishing based on CL data. Considering that a 3D rotation matrix has nine elements but only three degrees of freedom (DoF), suitable (and preferably minimal) parameterizations of rotation are thus necessary in order to intrinsically incorporate orthonormality constraints on rotations during the optimization. The latter suggests that the application scope of this approach is limited only to problems involving iterative optimization. Tech. At the very core of several key problems in computer graphics, vision and robotics lies the problem of estimating orientation. 10(34), 211229 (1993), Drummond, T., Cipolla, R.: Application of Lie algebras to visual servoing. ISBN: 0521540518, Book 2(2), 164168 (1944), Lourakis, M.: Sparse non-linear least squares optimization for geometric vision. Vis. scipy.spatial.transform.Rotation SciPy v1.11.2 Manual The reader is referred to [16, 29, 44, 61, 64] for more detailed introductions on quaternions and their properties. A brief overview of orientation representations with respect to various applications is given in Sect. In: International Conference on Pattern Recognition (ICPR), pp. 32 yields the second-order Cayley transform for MRPs given in Eq. J. Flight Vehicle Attitude Determination Using the Modified Rodrigues 8 that the same rotation matrix corresponds to quaternions q and \(-q\). For instance, Carlone et al. Standard CatmullRom interpolation. Canadian Information Processing Society, (1995), Kim, M.J., Kim, M.S., Shin, S.Y. The rotation is described by four Euler parameters due to Leonhard Euler.The Rodrigues formula (named after Olinde Rodrigues), a method of calculating the position of a rotated point, is used in . 3.2. PDF Modi ed Rodrigues Parameters: An E cient Representation of - OpenReview However, it has been largely overlooked in the computer graphics and vision communities as a practical means of parameterizing orientation.
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