Learning Rotations

Many problems in computer vision today are solved via deep learning. Tasks like pose estimation from images, pose estimation from point clouds or structure from motion can all be formulated as a regression on rotations. However, there is no unique way of parametrizing rotations mathematically: matrices, quaternions, axis-angle representation or Euler angles are all commonly used in the field. Some of them, however, present intrinsic limitations, including discontinuities, gimbal lock or antipodal symmetry. These limitations may make the learning of rotations via neural networks a challenging problem, potentially introducing large errors. Following recent literature, we propose three case studies: a sanity check, a pose estimation from 3D point clouds and an inverse kinematic problem. We do so by employing a full geometric algebra (GA) description of rotations. We compare the GA formulation with a 6D continuous representation previously presented in the literature in terms of regression error and reconstruction accuracy. We empirically demonstrate that parametrizing rotations as bivectors outperforms the 6D representation. The GA approach overcomes the continuity issue of representations as the 6D representation does, but it also needs fewer parameters to be learned and offers an enhanced robustness to noise. GA hence provides a broader framework for describing rotations in a simple and compact way that is suitable for regression tasks via deep learning, showing high regression accuracy and good generalizability in realistic high-noise scenarios.

A Sensor Fusion Based Nonholonomic Wheeled Mobile Robot for Tracking Control

In this paper, a detail design procedure of the real-time trajectory tracking for the nonholonomic wheeled mobile robot (NWMR) is proposed. A 9-axis micro electro-mechanical systems (MEMS) inertial measurement unit (IMU) sensor is used to measure the posture of the NWMR, the position information of NWMR and the hand-held device are acquired by global positioning system (GPS) and then transmit via radio frequency (RF) module. In addition, in order to avoid the gimbal lock produced by the posture computation from Euler angles, the quaternion is utilized to compute the posture of the NWMR. Furthermore, the Kalman filter is used to filter out the readout noise of the GPS and calculate the position of NWMR and then track the object. The simulation results show the posture error between the NWMR and the hand-held device can converge to zero after 3.928 seconds for the dynamic tracking. Lastly, the experimental results show the validation and feasibility of the proposed results.

An Image-Based Real-Time Georeferencing Scheme for a UAV Based on a New Angular Parametrization

Simultaneous localization and mapping (SLAM) of a monocular projective camera installed on an unmanned aerial vehicle (UAV) is a challenging task in photogrammetry, computer vision, and robotics. This paper presents a novel real-time monocular SLAM solution for UAV applications. It is based on two steps: consecutive construction of the UAV path, and adjacent strip connection. Consecutive construction rapidly estimates the UAV path by sequentially connecting incoming images to a network of connected images. A multilevel pyramid matching is proposed for this step that contains a sub-window matching using high-resolution images. The sub-window matching increases the frequency of tie points by propagating locations of matched sub-windows that leads to a list of high-frequency tie points while keeping the execution time relatively low. A sparse bundle block adjustment (BBA) is employed to optimize the initial path by considering nuisance parameters. System calibration parameters with respect to global navigation satellite system (GNSS) and inertial navigation system (INS) are optionally considered in the BBA model for direct georeferencing. Ground control points and checkpoints are optionally included in the model for georeferencing and quality control. Adjacent strip connection is enabled by an overlap analysis to further improve connectivity of local networks. A novel angular parametrization based on spherical rotation coordinate system is presented to address the gimbal lock singularity of BBA. Our results suggest that the proposed scheme is a precise real-time monocular SLAM solution for a UAV.

Alternative Representation of the Shoulder Orientation Based on the Tilt-and-Torsion Angles

The International Society of Biomechanics (ISB) has proposed standardized recommendations for recording human joint motion. The Euler angles—the orientation representation currently proposed by the ISB—have two drawbacks, namely, the issue of singularities (gimbal lock) and the difficulty to obtain clinical and interpretable orientation representation for compound movements. The orientation representation of the shoulder joint with the Euler angles is particularly challenging due to its broad range of motion. This paper proposes and evaluates an alternative orientation representation for shoulder movement based on the tilt-and-torsion representation, a method that aims at providing a more clinically interpretable solution for describing joint movements compared to the standard Euler angles. Three studies were performed to compare the different orientation representation methods. The first two studies consist in simulations of arm elevation in different planes. The third study is an experiment using inertial-measurement-units with one test subject performing shoulder elevation movements in different planes. The tilt-and-torsion representation is then compared with different Euler angle conventions. The results show that Euler angles are biased or clinically uninterpretable for compound movements. Conversely, tilt-and-torsion representation does not suffer from these limitations. Although not extensive, the experiments suggest that the tilt-and-torsion representation has the potential to better represent human movements and provide more clinically interpretable results than the Euler angles.

Total Least-Squares Iterative Closest Point Algorithm Based on Lie Algebra

In geodetic surveying, input data from two coordinates are needed to compute rigid transformations. A common solution is a least-squares algorithm based on a Gauss–Markov model, called iterative closest point (ICP). However, the error in the ICP algorithm only exists in target coordinates, and the algorithm does not consider the source model’s error. A total least-squares (TLS) algorithm based on an errors-in-variables (EIV) model is proposed to solve this problem. Previous total least-squares ICP algorithms used a Euler angle parameterization method, which is easily affected by a gimbal lock problem. Lie algebra is more suitable than the Euler angle for interpolation during an iterative optimization process. In this paper, Lie algebra is used to parameterize the rotation matrix, and we re-derive the TLS algorithm based on a GHM (Gauss–Helmert model) using Lie algebra. We present two TLS-ICP models based on Lie algebra. Our method is more robust than previous TLS algorithms, and it suits all kinds of transformation matrices.

Research on Attitude Interpolation and Tracking Control Based on Improved Orientation Vector SLERP Method

SummaryIn order to make the end of the three-axis platform follow the control command and achieve stable control of the end attitude, an improved orientation vector spherical linear interpolation (SLERP) method is proposed for the requirements, which specifically handles the position of the gimbal lock, so that the platform can move smoothly around the gimbal lock position. A three-axis platform with a camera at the end is set up for the validity of the proposed algorithm. At first, an adaptive speed measurement method based on incremental encoder is introduced, which can automatically adapt to high and low speed, and estimate the ultra-low speed to realize the speed measurement of large dynamic range, and this is used for the motion control of the three-axis platform. Then, the SLERP method for the quaternion interpolation on the starting and ending attitudes represented in quaternion is introduced in detail, and it is continuously improved in response to its existing problems for the platform. Finally, an orientation vector SLERP method is proposed, which uses viscosity factor and rejection factor to adjust the algorithm near the platform’s gimbal lock position. A tracking experiment was designed using the red ball as the following target detected by the designed target tracking algorithm using the camera, which verified the effectiveness of the attitude tracking control based on the proposed improved orientation vector SLERP.

A Fusion Method for Combining Low-Cost IMU/Magnetometer Outputs for Use in Applications on Mobile Devices

This paper presents a fusion method for combining outputs acquired by low-cost inertial measurement units and electronic magnetic compasses. Specifically, measurements of inertial accelerometer and gyroscope sensors are combined with no-inertial magnetometer sensor measurements to provide the optimal three-dimensional (3D) orientation of the sensors’ axis systems in real time. The method combines Euler–Cardan angles and rotation matrix for attitude and heading representation estimation and deals with the “gimbal lock” problem. The mathematical formulation of the method is based on Kalman filter and takes into account the computational cost required for operation on mobile devices as well as the characteristics of the low-cost microelectromechanical sensors. The method was implemented, debugged, and evaluated in a desktop software utility by using a low-cost sensor system, and it was tested in an augmented reality application on an Android mobile device, while its efficiency was evaluated experimentally.

An Algorithm for Rigid-Body Angular Velocity and Attitude Estimation Based on Isotropic Accelerometer Strapdowns

A novel algorithm for the estimation of rigid-body angular velocity and attitude—the most challenging part of pose-and-twist estimation—based on isotropic accelerometer strapdowns, is proposed in this paper. Quaternions, which employ four parameters for attitude representation, provide a compact description without the drawbacks brought about by other representations, for example, the gimbal lock of Euler angles. Within the framework of quaternions for rigid-body angular velocity and attitude estimation, the proposed methodology automatically preserves the unit norm of the quaternion, thus improving the accuracy and efficiency of the estimation. By virtue of the inherent nature of isotropic accelerometer strapdowns, the centripetal acceleration is filtered out, leaving only its tangential counterpart, to be estimated and updated. Meanwhile, using the proposed integration algorithm, the angular velocity and the quaternion, which are dependent only on the tangential acceleration, are calculated and updated at appropriate sampled instants for high accuracy. This strategy, which brings about robustness, allows for relatively large time-step sizes, low memory demands, and low computational complexity. The proposed algorithm is tested by simulation examples of the angular velocity and attitude estimation of a free-rotating brick and the end-effector of an industrial robot. The simulation results showcase the algorithm with low errors, as estimated based on energy conservation, and high-order rate of convergence, as compared with other algorithms in the literature.