In the integrated navigation system using extended Kalman filter (EKF), the state error conventionally uses linear approximation to tackle the commonly nonlinear problem. However, this error definition can diverge the filter in some adverse situations due to significant distortion of the linear approximation. By contrast, the nonlinear state error defined in the Lie group satisfies the autonomous equation, which thus has distinctively better convergence property. This work proposes a novel strapdown inertial navigation system (SINS) nonlinear state error defined in the Lie group and derives the SINS equations of the Lie group EKF (LG-EKF) for the MIMU/GNSS/magnetometer integrated navigation system. The corresponding measurement equations are also derived. A land vehicle field test has been conducted to evaluate the performance of EKF, ST-EKF (state transformation extended Kalman filter) and LG-EKF, which verifies LG-EKF's superior estimation accuracy of the heading angle as well as the other two horizontal angles (pitch and roll). The LG-EKF proposed in this paper is unlimited in the choice of sensors, which means it can be applied with both high-end and low-end inertial sensors.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/95715528C501DBBE12816626024D6123/S037346332100014Xa.pdf/div-class-title-lie-group-based-nonlinear-state-errors-for-mems-imu-gnss-magnetometer-integrated-navigation-div.pdf

Lie group based nonlinear state errors for MEMS-IMU/GNSS/magnetometer integrated navigation

Registration is highly demanded in many real-world scenarios such as robotics and automation. Registration is challenging partly due to the fact that the acquired data is usually noisy and has many outliers. In addition, in many practical applications, one point set (PS) usually only covers a partial region of the other PS. Thus, most existing registration algorithms cannot guarantee theoretical convergence. This article presents a novel, robust, and accurate three-dimensional (3D) rigid point set registration (PSR) method, which is achieved by generalizing the state-of-the-art (SOTA) Bayesian coherent point drift (BCPD) theory to the scenario that high-dimensional point sets (PSs) are aligned and the anisotropic positional noise is considered. The high-dimensional point sets typically consist of the positional vectors and normal vectors. On one hand, with the normal vectors, the proposed method is more robust to noise and outliers, and the point correspondences can be found more accurately. On the other hand, incorporating the registration into the BCPD framework will guarantee the algorithm’s theoretical convergence. Our contributions in this article are three folds. First, the problem of rigidly aligning two general PSs with normal vectors is incorporated into a variational Bayesian inference framework, which is solved by generalizing the BCPD approach while the anisotropic positional noise is considered. Second, the updated parameters during the algorithm’s iterations are given in closed-form or with iterative solutions. Third, extensive experiments have been done to validate the proposed approach and its significant improvements over the BCPD. Note to Practitioners—This paper was motivated by the problem of 3D rigid PSR for computer-assisted surgery (CAS), especially in orthopedic applications. The proposed algorithm is also suitable for other scenarios where the initial coarse registration is conducted. The traditional registration methods are susceptible to noise (especially anisotropic noise), outliers, and incomplete partial data. This paper generalizes the recently proposed BCPD method to the six-dimensional scenario where anisotropic positional noise is considered and normal vectors are incorporated. The proposed noise model is decomposed into three parts to be solved alternately: the membership probability of mixture distributions, the soft correspondence estimation, and the model parameters (i.e., the rotation matrix, translation vector, the covariance matrix with the anisotropic positional error, and the concentration parameter with the estimation of the normal vectors). Especially, the convergence is guaranteed at the theoretical level using the variational inference theory. The experimental results demonstrate the superiority of our algorithm on registration accuracy, convergence speed, and robustness to noise, outliers, and partial data. 

/pdf/anisotropic-generalized-bayesian-coherent-point-drift-for-1utxljsu.pdf

Anisotropic Generalized Bayesian Coherent Point Drift for Point Set Registration

The problem of calibrating an extrinsic parameter between a camera and an inertial measurement unit (IMU) using an industrial robotic manipulator has been studied. This generates a result of hand–eye/robot–world/camera–IMU calibration in a simultaneous fashion. The developed method is free of inertial integration over time and, thus, is robust to uncertain IMU biases. It is derived that the problem can be solved via a simultaneous optimization of hand–eye/robot–world/camera–IMU transformations. The resulted optimization is highly nonconvex on the special Euclidean group, and we give globally optimal solutions. Experiments verify that the proposed method is capable of estimating accurate calibration parameters. Comparative studies between representatives show the global optimality of the proposed method. The new simultaneous method is capable of conducting calibration of a robot/camera/IMU combination. The designed method guarantees the global optimality; thus, the accuracy is ensured. The developed globally optimal solutions will also be computationally efficient on modern industrial computers. Finally, we show that the proposed method can give accurate calibration results for a stereo/IMU sensor combination. 

Simultaneous Hand–Eye/Robot–World/Camera–IMU Calibration

Registration is highly demanded in many real-world scenarios such as robotics and automation. Registration is challenging partly due to the fact that the acquired data is usually noisy and has many outliers. In addition, in many practical applications, one point set (PS) usually only covers a partial region of the other PS. Thus, most existing registration algorithms cannot guarantee theoretical convergence. This article presents a novel, robust, and accurate three-dimensional (3D) rigid point set registration (PSR) method, which is achieved by generalizing the state-of-the-art (SOTA) Bayesian coherent point drift (BCPD) theory to the scenario that high-dimensional point sets (PSs) are aligned and the anisotropic positional noise is considered. The high-dimensional point sets typically consist of the positional vectors and normal vectors. On one hand, with the normal vectors, the proposed method is more robust to noise and outliers, and the point correspondences can be found more accurately. On the other hand, incorporating the registration into the BCPD framework will guarantee the algorithm’s theoretical convergence. Our contributions in this article are three folds. First, the problem of rigidly aligning two general PSs with normal vectors is incorporated into a variational Bayesian inference framework, which is solved by generalizing the BCPD approach while the anisotropic positional noise is considered. Second, the updated parameters during the algorithm’s iterations are given in closed-form or with iterative solutions. Third, extensive experiments have been done to validate the proposed approach and its significant improvements over the BCPD. Note to Practitioners—This paper was motivated by the problem of 3D rigid PSR for computer-assisted surgery (CAS), especially in orthopedic applications. The proposed algorithm is also suitable for other scenarios where the initial coarse registration is conducted. The traditional registration methods are susceptible to noise (especially anisotropic noise), outliers, and incomplete partial data. This paper generalizes the recently proposed BCPD method to the six-dimensional scenario where anisotropic positional noise is considered and normal vectors are incorporated. The proposed noise model is decomposed into three parts to be solved alternately: the membership probability of mixture distributions, the soft correspondence estimation, and the model parameters (i.e., the rotation matrix, translation vector, the covariance matrix with the anisotropic positional error, and the concentration parameter with the estimation of the normal vectors). Especially, the convergence is guaranteed at the theoretical level using the variational inference theory. The experimental results demonstrate the superiority of our algorithm on registration accuracy, convergence speed, and robustness to noise, outliers, and partial data.

The generalized rigid registration problem in high-dimensional Euclidean spaces is studied. The loss function is minimized with an equivalent error formulation by the Cayley formula. The closed-form linear least-square solution to such a problem is derived which generates the registration covariances, i.e., uncertainty information of rotation and translation, providing quite accurate probabilistic descriptions. Simulation results indicate the correctness of the proposed method and also present its efficiency on computation-time consumption, compared with previous algorithms using singular value decomposition (SVD) and linear matrix inequality (LMI). The proposed scheme is then applied to an interpolation problem on the special Euclidean group $SE(n)$ with covariance-preserving functionality. Finally, experiments on covariance-aided Lidar mapping show practical superiority in robotic navigation.

Generalized n-Dimensional Rigid Registration: Theory and Applications

In robotic applications, many pose problems involve solving the homogeneous transformation based on the special Euclidean group SE(n). However, due to the nonconvexity of SE(n), many of these solvers treat rotation and translation separately, and the computational efficiency is still unsatisfactory. A new technique called the SE(n)++ is proposed in this article that exploits a novel mapping from SE(n) to SO(n + 1). The mapping transforms the coupling between rotation and translation into a unified formulation on the Lie group and gives better analytical results and computational performances. Specifically, three major pose problems are considered in this article, that is, the point-cloud registration, the hand-eye calibration, and the SE(n) synchronization. Experimental validations have confirmed the effectiveness of the proposed SE(n)++ method in open datasets.

SE(n)++: An Efficient Solution to Multiple Pose Estimation Problems.

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SE(n)++: An Efficient Solution to Multiple Pose Estimation Problems.