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In this study, we obtained generalized Cayley formula, Rodrigues equation and Euler parameters of an orthogonal matrix in 3-dimensional generalized space E 3 (α , β). It is shown that unit generalized quaternion, which is defined by the generalized Euler parameters, corresponds to a rotation in E 3 (α , β) space.We found that the rotation in matrix equation forms using matrix form of the generalized quaternion product. Besides, in E 3 (α , β) space, we obtained the rotations determined by the unit quaternions and unit split quaternions, which are special cases of generalized quaternions for α , β ∈ ℝ , in 3-dimensional Eulidean space E 3 , in 3-dimensional Lorentzian space E 1 3 , respectively. [ABSTRACT FROM AUTHOR]

This study is devoted to investigating the robust adaptive finite-time attitude tracking control problem for a rigid body subject to unknown uncertainties. Considering about the nonlinear attitude dynamics for a rigid body, a homogeneous sliding variable is designed by employing the signum-function technique. For the presented homogeneous sliding variable, a real sliding manifold, on which the singularity problem is avoided, could be achieved. Subsequently, a novel signum-function-based MIMO adaptive homogeneous finite-time control (SMAHFTC) algorithm is proposed. Finite-time convergence of the tracking errors to a region around the origin is rigorously proved through the Lyapunov approach. The control gains will not be overestimated, which implies that minimal control gains will be obtained by the adaptation law. Moreover, the controls' signals are continuous for the SMAHFTC law. Then, by means of the proposed design method, an attitude tracking controller is designed for a rigid body described by the modified Rodrigues parameters' (MRPs) representation. A comparison simulation is carried out to show the effectiveness and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents a low-cost, efficient, and portable in vivo method for identifying axes of rotation of the proximal interphalangeal and distal interphalangeal joints in an index finger. The approach is associated with the screw displacement representation of rigid body motion. Using the matrix exponential method, a detailed derivation of general spatial displacement of a rigid body in the form of screw displacement including the Rodrigues' formulae for rotation is presented. Then, based on a gyroscope sensor, a test framework for determining axes of rotation of finger joints is established, and experiments on finding the directions of joint axes of the PIP and DIP joints are conducted. The results obtained highly agree with those presented in literature through traditional but complex methods. [ABSTRACT FROM AUTHOR]

Low-cost GPS/INS integration system is the ideal combination of navigation and positioning. However, the sensitivity of low-cost INS is not good enough for the initial alignment in statics base before navigation. Considering this problem, this paper presents an arbitrary misalignment angle error propagation model which does not rely on small misalignment angles assumption and two simplified versions. These models are presented in the ECEF frame approach and are suitable to implement the in-motion alignment with GPS aided. Another three error models based on quaternion, Rodrigues parameters and modified Rodrigues parameters were also proposed. Three experiments were designed to verify the accuracy and computational efficiency of these error models. The experiments' results showed that the small misalignment angle model was applicable to the small misalignment angle for higher computational efficiency but without improvement of accuracy. And the large misalignment angle is the best error model for the initial alignment of the arbitrary misalignment angle. [ABSTRACT FROM AUTHOR]

This research is devoted to present spacecraft attitude control via a finite-horizon nonlinear optimal control technique. The spacecraft kinematics are represented using the modified Rodrigues parameters, which possess singularities for eigenaxis rotations greater than 180 degree. The proposed technique is based on State Dependent Riccati Equation (SDRE). In this technique, the differential Riccati equation is converted to a linear Lyapunov differential equation. This technique can be applied for finite-horizon nonlinear regulation and tracking problems. The proposed technique is effective for a wide range of operating points. Simulation results are given to illustrate the effectiveness of the finite horizon technique. [ABSTRACT FROM AUTHOR]

This work complements the body of studies on the use of generalized Rodrigues parameters (GRPs) for describing the orientation of a rigid body. A simple decision logic is designed for handling the GRPs so that singular configurations are avoided, while enforcing the one-to-one mapping between the set of orientation parameters and the unique rotation matrix. [ABSTRACT FROM AUTHOR]

In this paper, we propose a finite-time sliding mode trajectory tracking control methodology for the vertical takeoff and landing unmanned aerial vehicle (VTOL UAV). Firstly, a system error model of trajectory tracking task is established based on Rodrigues parameters by considering both external and internal uncertainties. According to the cascade property, the system model is divided into translational and rotational subsystems, and a hierarchical control structure is hence proposed. Then, a finite-time generalized nonlinear disturbance observer (NDOB) is proposed, based on which the finite-time convergence result of equivalent disturbance estimation can be acquired. Finally, by introducing a tan-type compensator into the traditional terminal sliding mode control (SMC), the finite-time convergence result of the closed-loop control system is acquired based on Lyapunov stability analysis. Simulation results show the effectiveness of the proposed methodology. [ABSTRACT FROM AUTHOR]

Based on the representation of rigid body displacements as adjoint matrices, the article introduces the adjoint trigonometric representation of displacements (ATRD) as a further generalization of the trigonometric representation of rotations. In comparison to the dual Rodrigues–Euler–Gauß–Gelman equation, recently reported for affine screw displacements with arbitrary, fixed pitches, the ATRD is built upon a product of a unit line and a dual angle, instead of upon a product of a unit screw and a real angle. Due to this conceptual difference, the ATRD requires four independent parameters of a unit line instead of five when parametrizing a displacement along a unit screw. As a consequence for computational kinematics, the ATRD permits transferring the analytic solution to the inverse kinematics problem (IKP) of 3‐DOF, general, spherical 3R‐chains into a closed‐form solution to the IKP of 6‐DOF, general, affine 3C‐chains. [ABSTRACT FROM AUTHOR]

The fusion of motion data is key in the fields of robotic and automated driving. Most existing approaches are filter-based or pose-graph-based. By using filter-based approaches, parameters should be set very carefully and the motion data can usually only be fused in a time forward direction. Pose-graph-based approaches can fuse data in time forward and backward directions. However, pre-integration is needed by applying measurements from inertial measurement units. Additionally, both approaches only provide discrete fusion results. In this work, we address this problem and present a uniform B-spline-based continuous fusion approach, which can fuse motion measurements from an inertial measurement unit and pose data from other localization systems robustly, accurately and efficiently. In our continuous fusion approach, an axis-angle is applied as our rotation representation method and uniform B-spline as the back-end optimization base. Evaluation results performed on the real world data show that our approach provides accurate, robust and continuous fusion results, which again supports our continuous fusion concept. [ABSTRACT FROM AUTHOR]