Three-dimensional Space Vector Research Articles

Collision Avoidance of UAV Formation Using Geometric Approach and Relative Velocity Vector in Three-dimensional Space

Collision Avoidance of UAV Formation Using Geometric Approach and Relative Velocity Vector in Three-dimensional Space

Four-Leg Inverter Based on Three-Dimensional Space Vector Modulation Method for Desired Output Frequency

Four-Leg Inverter Based on Three-Dimensional Space Vector Modulation Method for Desired Output Frequency

Octonions as Clifford-like algebras

The associative Cayley-Dickson algebras over the field of real numbers are also Clifford algebras. The alternative but nonassociative real Cayley-Dickson algebras, notably the octonions and split octonions, share with Clifford algebras an involutary anti-automorphism and a set of mutually anticommutative generators. On the basis of these similarities, we introduce Kingdon algebras: alternative Clifford-like algebras over vector spaces equipped with a symmetric bilinear form. Over three-dimensional vector spaces, our construction quantizes an alternative non-associative analogue of the exterior algebra. The octonions and split octonions, along with other real generalized Cayley-Dickson algebras in Albert's sense, arise as Kingdon algebras. Our construction gives natural characterizations of the octonion and split octonion algebras by a universality property endowing them with a selected superalgebra structure.

Three-Dimension Space Vector Modulation for Three-Phase Series-End Winding Voltage-Source Inverters

Series-end winding voltage-source inverters (SEW-VSIs) can effectively improve the dc-link voltage utilization and provide the capabilities to handle the zero-sequence loop. However, the existing modulation schemes for SEW-VSIs cannot fully utilize the output capability of SEW-VSIs while guaranteeing simple implementation. To this end, a three-dimensional space vector modulation (3D-SVM) scheme is proposed for driving three-phase SEW-VSIs in this paper. The modulation is conducted in the 3D-space made up of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α-β</i> plane and the zero-sequence-axis, and the reference voltage vector in the 3D-space is synthesized by three adjacent basic voltage vectors and the null voltage vector, which can fully utilize the output capability of three-phase SEW-VSI. Then, a convenient implementation method based on a look-up table is developed to find three basic voltage vectors adjacent to the reference voltage vector. Moreover, in the 3D-space, the feasible region and the linear modulation region of the proposed 3D-SVM are investigated. Accordingly, intuitive overmodulation between different subspaces and the switching sequence generation are developed. A series of simulation and experimental results are presented to validate the effectiveness of the proposed 3D-SVM for three-phase SEW-VSI drives.

Some geometric properties of the Padovan vectors in Euclidean 3-space

Padovan numbers were defined by Stewart (1996) in honor of the modern architect Richard Padovan (1935) and were first discovered in 1924 by Gerard Cordonnier. Padovan numbers are a special status of Tribonacci numbers with initial conditions and general terms. The ratio between Padovan numbers is one of the important algebraic numbers because it produces plastic numbers. Up to now, various studies have been conducted on Padovan numbers and Padovan polynomial sequences. In this study, Padovan vectors are defined for the first time by using the Padovan Binet-like formula and reduction relation. Then, geometric properties of Padovan vectors such as inner product, norm, and vector products are analyzed. In the last part of the study, Padovan vectors were calculated with Binet formulas in the Geogebra program. In addition, the first ten Padovan numbers and Padovan vectors were calculated using the Binet formulas and shown as points and vectors in three-dimensional space. According to the Padovan vectors found, the Padovan curve was drawn in space for the first time by using the curve fitting feature of the Geogebra program. Thus, with our study, a geometric approach to Padovan number sequences was brought for the first time.

Three-Dimensional Space Vector Modulation for Four-Leg Current-Source Inverters

Unbalanced loads are general in uninterruptible power supply (UPS), standalone power generation applications and the failure mode of three-phase inverters. Three-phase four-leg inverters are a well-known solution to handle neutral currents caused by unbalanced loads. In four-leg inverters, three-dimensional space vector modulation (3DSVM) is widely used. However, the 3DSVM of four-leg current source inverter (CSI) has not been systematically studied so far. To fill this gap, this paper proposes a 3DSVM for CSI. The definition of the current vector, the identification of the position of reference vector and the sequence of switches are introduced. Under unbalance loads, the symmetrical load voltage can be realized, which complies with IEEE Std. 1159-2019. The output current can comply with IEEE Std. 519-2014. The proposed 3DSVM has the potential to extend the advanced modulation strategies that have been implemented/ proposed on three-leg CSI.

3-D SVM for Three-Phase Open-End Winding Drives With Common DC Bus

Existing modulation strategies for three-phase open-end winding drives (OWDs) with common dc bus usually conduct the modulations for α-β subspace and zero-sequence loop (ZSL) separately, which cannot reflect the relationship between α-β subspace and ZSL and fully utilize the dc bus voltage. To this end, this paper proposes a three-dimensional space vector modulation (3-D SVM) strategy for three-phase OWDs with common dc bus, which can be conducted in both a-b-c and α-β- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> coordinates, leading to full utilization of dc bus voltage and convenient overmodulation adjustment. Voltage vector distributions of OWD in the 3-D spaces with a-b-c and α-β- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> coordinates are revealed at first. Subsequently, the three adjacent voltage vectors for reference voltage vector synthesis are selected based on the relationship between leg-voltage potentials, avoiding the complex tetrahedron identification in existing 3-D SVMs for other drive topologies. In addition, overmodulation in different coordinates is also investigated. Experimental results are presented to validate the feasibility of the proposed 3-D SVM for OWDs with common dc bus.

Geometric 3-space and multiplicative quaternions

In this paper, we introduce a new vector space, the three-dimensional geometric real vector space [Formula: see text], and a new number system, multiplicative quaternions [Formula: see text]. We give some basic algebraic properties of these concepts. We define spatial rotation in [Formula: see text] by using multiplicative quaternions. Finally, we give an example to illustrate our results.

Simulation of 3D-space vector modulation for neutral point clamped inverters

This paper gives an idea to simulation of three-dimensional space vector modulation for neutral point clamped multilevel inverter. Three dimensional-space vector modulation (3D-SVM) algorithm is progressed method of two dimensional-space vector modulation (2D-SVM) algorithm; it leads to reduce the complexity in reference vector identification and switching time calculation, also it includes the various advantages of 2D-SVM like minimized total harmonic distortion, reduced EMI issues. A simple system for the assortment of switching state vectors to track the reference voltage vectors without using any redundant switching vectors. This proposed method tracks the reference vector by identifyinglsubcubes and prisms by using mathematicallconditions. Here the cost of the proposedltechnique is independentlof voltagellevels oflinverter. This paper realizes the accomplishment of 3D-SVM using a neutral point clamped inverter. The simulation results of the proposed method are verified using MATLAB/Simulink.

Open-Circuit Fault-Tolerant Vector Control for Five-Phase SPMSM Based on Five-Phase Six-Leg Inverter

Five-phase permanent-magnet synchronous motor (PMSM) has the ability of fault-tolerant operation, which is beneficial for high reliability occasions. A fault-tolerant current vector control strategy based on five-phase six-leg inverter is proposed for open-circuit fault in this article. It deals with single-, double-, and triple-phase fault conditions. First, the overall control scheme is established based on the mathematical model of five-phase surface-mounted PMSM. To release more control degrees of freedom, the constraint for the sum of fault-tolerant phase currents is removed, and the control variates method is used to calculate the optimal fault-tolerant currents. Furthermore, to achieve the current tracking, the corresponding reduced-order current transformation matrices are derived, and the corresponding main power topology, the five-phase six-leg voltage source inverter (VSI), is established. The unbalanced operation of the topology is investigated, the equivalence to the full-bridge VSI is proved, and the voltage utilization and voltage unbalance of five phase half-bridge, full-bridge, and six-leg inverter are compared. Then, the corresponding carrier-based PWM control strategy based on zero-sequence injection has been proposed, which is equivalent with the three-dimensional space vector PWM algorithm and is computationally efficient. The experimental results confirm the effectiveness of the proposed fault-tolerant control strategy.

Novel Common Mode Voltage Elimination Methods in Three-Phase Four-Wire Grid-Connected Inverters

This paper presents two current control methods to attenuate the common mode (CM) conducted emissions in three-phase four-wire four-pole inverters connected to an AC distribution bus and operating in grid-following mode. Delta modulation and model predictive control both achieve elimination of the common mode voltage at the output of the three-phase four-pole inverter and meet the requirements of the military standard 461G without a full-size CM choke. The differential mode performance is comparable to that achieved with the benchmark three-dimensional space vector modulation, which does not cancel the CM voltage. A physics-based model of the system was developed to simulate the conducted emissions and DM performance of the two proposed methods. As the CM voltage cancellation methods result in increased switching frequency, wide bandgap devices were used to build a laboratory prototype, on which the simulated results were successfully validated.

병렬 메커니즘을 이용한 무인 비행체 조종기

It is difficult for a human operator to find roll, pitch, yaw (RPY) that indicates the desired direction of unmanned aerial vehicle (UAV) in a three-dimensional space. Herein, a controller for UAV was developed allowing the human operator controlling the direction of UAV without finding RPY information. The algorithm implemented in the controller automatically calculated RPY information of UAV from the normal vector of the end effector. The developed controller was designed using a parallel mechanism. The joint angles of the controller were measured using potentiometers to estimate the normal vector of the end effector. Five subjects participated in an experiment to control a vector in three-dimensional space to follow a randomly generated target vector using the developed controller and the thumb sticks. The performance of the two controllers was evaluated by two methods: measuring the required time to reduce the error between the controlled vector and the target vector to be less than 0.1 cm and calculating a normalized error between the controlled vector and the target vector after manipulating the controlled vector for 10 seconds. When using the developed controller, the difference in control ability between subjects was reduced, and both required time and normalized error were generally reduced.

Study of degradation of fuel cell stack based on the collected high-dimensional data and clustering algorithms calculations

Accurate perception of the performance degradation of fuel cell is very important to detect its health state. However, inconsistent operating conditions of fuel cell vehicles in the test result in errors in the data. In order to obtain a more credible degradation rate, this study proposes a novel method to classify the experimental data collected under different working conditions into similar operating conditions by using dimensionality reduction and clustering algorithms. Firstly, the experimental data collected from fuel cell vehicles belong to high-dimensional data. Then projecting high-dimensional data into three-dimensional feature vector space via principal component analysis (PCA). The dimension-reduced three-dimensional feature vectors are input into the clustering algorithm, such as K-means and density-based noise application spatial clustering(DBSCAN). According to the clustering results, the fuel cell voltage data with similar operating conditions can be classified. Finally, the selected voltage data can be used to precisely represent the true performance degradation of an on-board fuel cell stack. The results show that the voltage using the K-means algorithm declines the fastest, followed by the DBSCAN algorithm, finally the original data, which indicates that the performance of the fuel cell actually declines faste. Early intervention can prolong its life to the greatest extent.

Octonions and the two strictly projective tight 5-designs

In addition to the vertices of the regular hexagon and icosahedron, there are precisely two strictly projective tight 5-designs: one constructed from the short vectors of the Leech lattice and the other corresponding to a generalized hexagon structure in the octonion projective plane. This paper describes a new connection between these two strictly projective tight 5-designs -- a common construction using octonions. Certain octonion involutionary matrices act on a three-dimensional octonion vector space to produce the first 5-design and these same matrices act on the octonion projective plane to produce the second 5-design. This result uses the octonion construction of the Leech lattice due to Robert Wilson and provides a new link between the generalized hexagon Gh(2,8) and the Leech lattice.

A Spatial Biarc Method for Inverse Kinematics and Configuration Planning of Concentric Cable-Driven Manipulators

Superior dexterity and extreme flexibility are typical advantages for concentric cable-driven manipulators working in confined spaces. However, its inverse kinematics and configuration planning are very complicated. In this article, we propose a spatial biarc method for the above problem. The distinguishing feature of this method is that input parameters are two positions and two direction vectors in three-dimensional (3-D) space, and the output is a reasonable spatial biarc for controlling a concentric cable-driven manipulator in 3-D space. This method has the following three advantages. First, the positions and direction vectors of the base and inner distal tip are considered simultaneously. In addition, the length and ratio of the overlapped section and separated section can be adjusted by changing the length of the direction vectors. Furthermore, by judging the angular value of the direction vectors, one can predetermine whether the spatial configuration of the entire arm is C- or S-shaped. The proposed method realizes the parameterization of a concentric cable-driven manipulator, which makes it convenient to intuitively control the manipulator to achieve interference-free motion trajectory planning in confined spaces. Finally, trajectory tracking inspections are simulated and experimentally executed. It can be seen from results that the proposed spatial biarc method can provide reasonable solutions for concentric cable-driven manipulators. The method is especially favorable in terms of 3-D-pose-determination problem and trajectory-planning problem. It can also be applied to other manipulators with similar configurations. Without loss of generality, when the given points and direction vectors are coplanar, the proposed spatial biarc method can be transformed to a planar biarc method.

3D object detection for autonomous driving: Methods, models, sensors, data, and challenges

Detection of the surrounding objects of a vehicle is the most crucial step in autonomous driving. Failure to identify those objects correctly in a timely manner can cause irreparable damage, impacting our safety and society. Several studies have been introduced to identify these objects in the two-dimensional (2D) and three-dimensional (3D) vector space. The 2D object detection method has achieved remarkable success; however, in the last few years, detecting objects in 3D have received more remarkable adoption. 3D object recognition has several advantages over 2D detection methods, as more accurate information about the environment is obtained for better detection. For example, the depth of the images is not considered in the 2D detection, which reduces the detection accuracy. Despite considerable efforts in 3D object detection, it has not yet reached the stage of maturity. Therefore, in this paper, we aim at providing a comprehensive overview of the state-of-the-art 3D object detection methods, with a focus on 1) identifying advantages and limitations, 2) revelling a novel categorization of the literature, 3) outlying the various training procedures, 4) highlighting the research gap in the existing methods and 5) building a road map for future directions.

Exponential Set-Point Stabilization of Underactuated Vehicles Moving in Three-Dimensional Space

This paper investigates the stabilization of underactuated vehicles moving in a three-dimensional vector space. The vehicle's model is established on the matrix Lie group SE(3), which describes the configuration of rigid bodies globally and uniquely. We focus on the kinematic model of the underactuated vehicle, which features an underactuation form that has no sway and heave velocity. To compensate for the lack of these two velocities, we construct additional rotation matrices to generate a motion of rotation coupled with translation. Then, the state feedback is designed with the help of the logarithmic map, and we prove that the proposed control law can exponentially stabilize the underactuated vehicle to the identity group element with an almost global domain of attraction. Later, the presented control strategy is extended to set-point stabilization in the sense that the underactuated vehicle can be stabilized to an arbitrary desired configuration specified in advance. Finally, simulation examples are provided to verify the effectiveness of the stabilization controller.

ПРЕДСТАВЛЕНИЕ ПАРНОГО ПОТЕНЦИАЛА ВЗАИМОДЕЙСТВИЯ В ВИДЕ МНОГОМЕРНЫХ РАЦИОНАЛЬНЫХ РЯДОВ В ПЕРЕМЕННЫХ ЯКОБИ ДЛЯ ЗАДАЧ МНОГИХ ТЕЛ

Scalar power functions of the form x1 + + xN -v Î are in some cases found in physical problems and applications, especially in many-body problems with paired interactions. There are known decompositions for two vectors in three-dimensional space. In this paper, we consider analogous decompositions with any number of N arbitrary M-dimensional vectors in Euclidean space as a product of a multidimensional rational series with respect to spatial variables and hyperspheric functions on the unit sphere SM-1. Such an advantage of expansion arises in three-body problems when solving the Faddeev equation, where it is known that the main problem is the approximate choice of approximation of interaction potentials, in which the t-matrix scattering elements acquired a separable form.

Analysis and control of a Stewart platform as base motion compensators - Part I: Kinematics using moving frames

Kinematics and its control application are presented for a Stewart platform whose base plate is installed on a floor in a moving ship or a vehicle. With a manipulator or a sensitive equipment mounted on the top plate, a Stewart platform is utilized to mitigate the undesirable motion of its base plate by controlling actuated translational joints on six legs. To reveal closed loops, a directed graph is utilized to express the joint connections. Then, kinematics begins by attaching an orthonormal coordinate system to each body at its center of mass and to each joint to define moving coordinate frames. Using the moving frames, each body in the configuration space is represented by an inertial position vector of its center of mass in the three-dimensional vector space ℝ3, and a rotation matrix of the body-attached coordinate axes. The set of differentiable rotation matrices forms a Lie group: the special orthogonal group, SO(3). The connections of body-attached moving frames are mathematically expressed by using frame connection matrices, which belong to another Lie group: the special Euclidean group, SE(3). The employment of SO(3) and SE(3) facilitates effective matrix computations of velocities of body-attached coordinate frames. Loop closure constrains are expressed in matrix form and solved analytically for inverse kinematics. Finally, experimental results of an inverse kinematics control are presented for a scale model of a base-moving Stewart platform. Dynamics and a control application of inverse dynamics are presented in the part II-paper.

Asymmetrical four-wire cascaded h-bridge multi-level inverter based shunt active power filter supplied by a photovoltaic source

&lt;span&gt;This paper presents a novel shunt active power filter (SAPF). The power converter that is used in this SAPF is constructed from a four-leg asymmetric multi-level cascaded H-bridge (CHB) inverter that is fed from a photovoltaic source. A three-dimensional space vector modulation (3D-SVPWM) technique is adopted in this work. The multi-level inverter can generate 27-level output with harmonic content is almost zero. In addition to the capability to inject reactive power and mitigating the harmonics, the proposed SAPF has also, the ability to inject real power as it is fed from a PV source. Moreover, it has a fault-tolerant capability that makes the SAPF maintaining its operation under a loss of one leg of the multi-level inverter due to an open-circuit fault without any degradation in the performance. The proposed SAPF is designed and simulated in MATLAB SIMULINK using a single nonlinear load and the results have shown a significant reduction in total harmonics distortion (THD) of the source current under the normal operating condition and post a failure in one phase of the SAPF. Also, similar results are obtained when IEEE 15 bus network is used.&lt;/span&gt;