Terms Of Unit Quaternions Research Articles

Efficient Relative Pose Estimation for Cameras and Generalized Cameras in Case of Known Relative Rotation Angle

We propose two minimal solvers to the problem of relative pose estimation for a camera with known relative rotation angle. In practice, such angle can be derived from the readings of a 3D gyroscope. Different from other relative pose formulations fusing a camera and a gyroscope, the use of relative rotation angle does not require extrinsic calibration between the two sensors. The first proposed solver is formulated for a calibrated regular camera and requires four-point correspondences from two views. The second solver extends the problem to a generalized camera and requires five-point correspondences. We represent the rotation part of the motion in terms of unit quaternions in order to construct polynomial equations encoding the epipolar constraints. The Grobner basis technique is then used to efficiently derive the polynomial solutions. Our first solver for regular cameras significantly improves the existing state-of-the-art solution. The second solver for generalized cameras is novel. The presented minimal solvers can be used in a hypothesize-and-test architecture such as RANSAC for reliable pose estimation. Experiments on synthetic and real datasets confirm that our algorithms are numerically stable, fast, and robust.

Robust precision attitude tracking of an uncertain rigid spacecraft based on regulation theory

SummaryThis paper explores regulation theory for the design of robust precision attitude tracking of an uncertain rigid spacecraft with external disturbances. Focusing on the attitude system in terms of unit quaternions with an unknown inertia matrix and unmodeled input disturbances, we first introduce specific nonlinear logic such that the resultant error‐quaternion system has an input‐to‐state stability property. Then, we establish an attitude deviation system with an output feedback normal form that has a strict vector relative degree of unity. This enables us to achieve robust output regulation based on an internal model. In particular, we can construct a high‐gain stabilizer for the relevant augmented system. As a major consequence, our study achieves not only precision attitude tracking with exponential convergence but also the input‐to‐state stability disturbance attenuation for the closed‐loop system. Finally, we show extensive simulation and experimental results to illustrate the approach.

A novel approach to Lie group structured configuration spaces of rigid bodies

AbstractIn rigid body mechanics, a body's configuration consisting of orientation and position have to be described, as well as angular velocity and velocity. There are several different ways to do this, leading to different configuration spaces. Popular choices include SO(3) × ℝ3, SE(3) or unit dual quaternions. All these configuration spaces possess a Lie group structure. We present the novel approach 𝕊 ⋉ ℝ3, which is a configuration space similar to SE(3), but describes the orientation in terms of unit quaternions instead of rotation matrices. Furthermore we show its relation to and advantages over other configuration spaces. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Isogeometric collocation methods for Cosserat rods and rod structures

We present a novel method for the mechanical simulation of slender, elastic, spatial rods and rod structures subject to large deformation and rotation. We develop an isogeometric collocation method for the geometrically exact, nonlinear Cosserat rod theory. The rod centerlines are represented as spatial NURBS curves and cross-section orientations are parameterized in terms of unit quaternions as 4-dimensional NURBS curves. Within the isogeometric framework, the strong forms of the equilibrium equations of forces and moments of the discretized Cosserat model are collocated, leading to an efficient method for higher-order discretizations. For rod structures consisting of multiple, connected rods we introduce a formulation with rigid, quasi-G1-coupling. It is based on the strong enforcement of continuity of displacement and change of cross-section orientation at interfaces. We also develop a mixed isogeometric formulation, which is based on an independent discretization of internal forces and moments and alleviates shear locking for thin rods. The novel rod simulation methods are verified by numerical convergence studies. Further computational examples include realistic applications with large deformations and rotations, as well as a large-scale rod structure with several hundreds of coupled rods and complex buckling behavior.

Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations

A novel ODE time stepping scheme for solving rotational kinematics in terms of unit quaternions is presented in the paper. This scheme inherently respects the unit-length condition without including it explicitly as a constraint equation, as it is common practice. In the standard algorithms, the unit-length condition is included as an additional equation leading to kinematical equations in the form of a system of differential-algebraic equations (DAEs). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form a system of ordinary differential equations (ODEs) on the Lie algebra $\mathit{so}(3)$ of the rotation group $\mathit{SO}(3)$ . This rotation vector defines an incremental rotation (and thus the associated incremental unit quaternion), and the rotation update is determined by the exponential mapping on the quaternion group. Since the kinematic ODE on $\mathit{so}(3)$ can be solved by using any standard (possibly higher-order) ODE integration scheme, the proposed method yields a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions, avoiding integration of DAE equations. Besides being ‘more elegant’—in the opinion of the authors—this integration procedure also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation. As presented in the paper, the numerical integration of three non-linear ODEs in terms of the rotation vector as canonical coordinates achieves a higher accuracy compared to integrating the four (linear in ODE part) standard-quaternion DAE system. In summary, this paper solves the long-standing problem of the necessity of imposing the unit-length constraint equation during integration of quaternions, i.e. the need to deal with DAE’s in the context of such kinematical model, which has been a major drawback of using quaternions, and a numerical scheme is presented that also allows for longer integration steps during kinematic reconstruction of large three-dimensional rotations.

Path Planning of Spatial Rigid Motion with Constrained Attitude

This paper presents a general quadratic optimization methodology for autonomous path planning of spatial rigid motion with constrained attitude. The motion to be planned has six degrees of freedom and is assumed under constant velocity in the body frame. The objective is to determine the motion orientation (or attitude), handled as control variables, along the planned paths. A procedure is discussed to transform the rotational constraints and attitude constraints as quadratic functions in terms of unit quaternions, and the path-planning problem is reformulated as a general, quadratically constrained, quadratic programming problem. A semidefinite relaxation method is then applied to obtain a bound on the global optimal value of the nonconvex, quadratically constrained, quadratic programming problem. Subsequently, an iterative rank minimization approach is proposed to find the optimal solution. Application examples of aircraft path-planning problems are presented using the proposed method and compared with those obtained from the other method.