Geometric Constraint Equations Research Articles

Motion Branch Transformation of 3(Rc)PU Parallel Mechanism with Reconfigurable Joint and Kinematic Performance Index

A novel 3(Rc)PU parallel mechanism based on the reconfigurable circular groove cross joint (Rc joint) is proposed. The motion branch transformation relationship and constraint performance evaluation method for the mechanism are studied. According to constrained screw analysis, the (Rc)PU limb provides a constraint couple to the moving platform in one motion mode and no constraint to the platform in another motion mode. According to the two configurations of the limb, the 3(Rc)PU mechanism has four different motion branches, namely 3T, 1R3T, 2R3T, and 3R3T. Based on the input selection principle, the input selection scheme of the 3(Rc)PU mechanism is determined, and it is concluded that, to achieve stable motion, at least two input drive pairs need to be applied on each limb. A unified kinematic model with four motion branches is established, and the geometric constraint equations of the 3(Rc)PU parallel mechanism are derived. After selecting the input actuated joints, the local minimization transmission index (LMTI) of the reconfigurable mechanism under different motion branches is established to evaluate the motion/force transmission performance of the reconfigurable mechanism under different motion branches. The conclusion shows that the LMTI values satisfy the corresponding constraint conditions, and the 3(Rc)PU parallel mechanism has good motion/force transmission performance.

Dynamic Analysis Strategy of Generalized Deployable Units via NCF-IGA with Force Constraint

Space deployable mechanisms typically employ modular designs, where each unit that can be independently deployed is defined as a generalized deployable unit (GDU), typically consisting of rigid components, flexible components and generalized kinematic pairs. Given the high-performance demands of spatial transmission mechanisms, generalized kinematic pairs frequently integrate flexible factors like torsion springs and flexible hinges, which control the motion according to the attitude and position of the component. In this investigation, a new modeling and solution strategy for rigid-flexible coupled dynamics is established by combining NCF and IGA. It can not only accurately describe the large deformation and large rotation of flexible components, but also consider the influence of flexible factors in the kinematic pairs. The rigid components are modeled by the Natural Coordinate Formula (NCF), and the flexible components are discretized in the inertial coordinate system using spline curves in Isogeometric Analysis (IGA). The intermediate reference coordinates were integrated into the NCF-IGA framework, overcoming the boundary constraints between B-spline and NCF elements, and the geometric constraint equation of the kinematic pair was established. Subsequently, this paper proposes the concept of force constraint in controllable deployable mechanisms. Based on the force constraint equation, the mechanical model of flexible joints is established. The dynamic equations of GDUs are derived considering both geometric and force constraints, and generalized-[Formula: see text] method is introduced to solve the equations. Finally, three numerical examples are provided to illustrate the applicability and superiority of the proposed method.

Geometric Design Methodology for Deployable Self-Locking Semicylindrical Structures

Due to their unique bistable characteristics, deployable self-locking structures are suitable for many engineering fields. Without changing the geometrical composition, such structures can be unfolded and locked solely by the elastic deformation of materials. However, their further applications are hampered by the lack of simple and systematic geometric design methodologies that consider arbitrary structural curvature profiles. This paper proposes such a methodology for double-layer semicylindrical grid structures to simplify their cumbersome geometric design. The proposed methodology takes joint sizes into account to ensure that the design results can be applied to actual projects without further adjustments. By introducing symmetry into the structural units (SUs) and selecting reasonable geometric parameters that describe the structural side elevation profile, a concise set of simultaneous nonlinear geometric constraint equations is established, the solution of which provides the geometric parameter values of the grid shape. On this basis, the remaining geometric parameter values, i.e., the geometric parameter values of the inner scissor-like elements (SLEs) of each SU, can be achieved from the equations derived from general SLEs. Design examples and the assembled physical grid structure indicate the feasibility and wide applicability of the proposed geometric design methodology.

Spacecraft Homography Pose Estimation with Single-Stage Deep Convolutional Neural Network.

Spacecraft pose estimation using computer vision has garnered increasing attention in research areas such as automation system theory, control theory, sensors and instruments, robot technology, and automation software. Confronted with the extreme environment of space, existing spacecraft pose estimation methods are predominantly multi-stage networks with complex operations. In this study, we propose an approach for spacecraft homography pose estimation with a single-stage deep convolutional neural network for the first time. We formulated a homomorphic geometric constraint equation for spacecraft with planar features. Additionally, we employed a single-stage 2D keypoint regression network to obtain homography 2D keypoint coordinates for spacecraft. After decomposition to obtain the rough spacecraft pose based on the homography matrix constructed according to the geometric constraint equation, a loss function based on pixel errors was employed to refine the spacecraft pose. We conducted extensive experiments using widely used spacecraft pose estimation datasets and compared our method with state-of-the-art techniques in the field to demonstrate its effectiveness.

Conformal Geometric Algebra–Based Geometric Modeling Method for Forward Displacement Analysis of 6-4 Stewart Platforms

Abstract This paper presents a novel geometric modeling method for direct displacement analysis of 6-4 Stewart platforms based on conformal geometric algebra (CGA). First, a geometric constraint relationship of four lines and a plane intersecting at a point is published. Second, a new coordinate-invariant geometric constraint equation of 6-4 Stewart platforms is deduced by CGA operation. Third, five polynomial equations are established by CGA theory. Fourth, based on the above six equations, a 5 × 5 Sylvester’s matrix is formulated by using Sylvester’s Dialytic elimination method and Gröbner bases method under the graded reverse lexicographical order. Finally, the coordinates of four points on the moving platform are revealed. Besides, a numerical example is used to prove the validity of the proposed method. The novelty of this study is that a whole geometric modeling method by geometric constraint relationship of four lines and a plane intersecting at a point is put forward under the CGA framework, which has good intuition and offers a novel idea for solving the other complex mechanisms. At the same time, Sylvester’s matrix constructed by this method is the smallest one in the known literature for forward displacement analysis of 6-4 Stewart platforms.

Design and Modeling of Multi-vertebrae Continuum Robot for Minimally Invasive Surgery

The constant curvature assumption is normally utilized in kinematic modeling of continuum manipulators due to its simplicity. Nevertheless, in practical applications, factors such as frictional disturbances may cause the curvature of a continuum manipulator to deviate from being constant. In this article, we design a novel multi-vertebrae continuum manipulator that utilizes the uniform joint deformation curvature of a superelastic NITI alloy rod. We propose kinematic modeling approach for a multi-vertebrae continuum manipulator. To improve the unevenness of joint angles and increase motion accuracy, we inserted elastic beams into the multi-vertebrae continuum manipulator. The design takes into account frictional forces, and we established geometric constraint equations and force equilibrium equations for the hinged continuum manipulator. We employed an identification method to establish the physical equations of the deformation section. By solving these equations and analyzing the geometric relationships, we obtained the mapping relationship between the driving screw force and the end displacement of the continuum manipulator. To prove the effectiveness of the proposed method, we performed MATLAB simulations. This paper serves as a reference for the design and modeling of future minimally invasive surgical manipulators.

Structure optimization of the protection inductor for the high energy density pulse pump power supply

High energy density pulse pump power supply is an important part of a large laser fusion facility. When a short-circuit fault occurs, massive energy injection will cause the capacitor and switch to damage or even explode. Therefore, it is of great significance to set a series protection inductor. In this paper, the structure of the widely used solenoid inductor is optimized. The geometric mean distance principle is introduced, and Neumann’s formula is extended to calculate the inductance of the solenoid. The electromagnetic force on each turn is solved by the virtual displacement method. The analytical expression of the electromagnetic stress in the current-carrying solenoid is deduced by combining the force balance equation, constitutive equation and geometric constraint equation. The electromagnetic-structure coupling simulation verifies the accuracy of the expression. Based on the Taguchi method, the structure parameters of the solenoid are optimized. The results show that the solenoid’s wire radius and layer count are the two most important influencing factors. As the wire radius increases and the layer count increases, the maximum carrying energy per unit volume increases. The influence of insulation encapsulation on the stress distribution in the inductor is studied based on electromagnetic-structure coupling simulation. It can be known that insulation encapsulation can effectively reduce stress and restrain deformation. On this basis, a novel type of double-layer solenoid inductor is designed and processed. Several pulse discharge tests verify its large current-carrying capacity and high reliability. The above results are useful for structure design and application of the protection inductor.

CGA-based novel modeling method for solving the forward displacement analysis of 3-RPR planar parallel mechanism

This paper put forward a new geometric modeling method for forward displacement analysis of 3-RPR planar parallel mechanisms (PPMs) under the conformal geometric algebra (CGA) framework. A geometric constraint relationship of four lines intersecting at a point is first given. Then, a new coordinate-invariant geometric constraint equation of 3-RPR PPMs is elaborated via CGA operation. Next, an univariate polynomial equation without algebraic elimination is published. At last, the coordinates of three points on the mobile platform are figured out. The best part of this research is that a complete geometric modeling method by geometric constraint relationship of four lines intersecting at a point is formulated under the CGA framework, which has good intuition and offers a novel idea for solving the other complex mechanisms. Three numerical examples are utilized to confirm the effectiveness of the proposed method.

Design of clamping-pot-type planetary gear train transplanting mechanism for rice wide–narrow-row planting

To design a clamping-pot-type wide–narrow-row pot seedling transplanting (WPST) mechanism with desired spatial beak-shaped trajectory and working posture, a new design method of planetary gear train transplanting mechanism (PGTM) with non-circular gears based on several key spatial poses (position and posture) was proposed. The PGTM was simplified to a spatial open-loop chain with two-revolute (2R) joints. The geometric constraint equations containing only the structural parameters of the chain were then established on the basis of the three key spatial poses, and the homotopy algorithm was used to obtain all the required parameters of the mechanism. In accordance with the parameters obtained, the relative angular displacement relation between the planet carrier and the transplanting arm was optimized, the trajectory of the mechanism was replayed, and the total transmission ratio was determined. The degree of freedom of the spatial 2R mechanism was reduced by attaching to the unequal gear pair, and the transmission ratio was distributed in accordance with the gear type to realize the design of a non-circular gear pitch curve. Lastly, a clamping-pot-type PGTM for rice WPST driven by the combination of planar non-circular and non-conical gears was designed, and virtual simulation and prototype test were conducted. Results showed that the simulation and prototype test trajectories were consistent with the desired trajectory. Under the operating speeds of 50 r/min and 90 r/min, the success rates of seedling picking were 95.32% and 90.15%, respectively, which verified the feasibility of the theoretical method. This method could provide a reference for the design of a spatial PGTM with nonuniform transmission. Keywords: rice pot seedling, wide–narrow-row transplanting, spatial trajectory, planetary gear train, non-circular gear DOI: 10.25165/j.ijabe.20211402.5975 Citation: Wang L, Sun L, Huang H M, Yu Y X, Yu G H. Design of clamping-pot-type planetary gear train transplanting mechanism for rice wide–narrow-row planting. Int J Agric & Biol Eng, 2021; 14(2): 62–71.

Shape design of magnetic circuit of the magnetic lens based on medial axis transform

This paper proposes a method of designing a magnetic lens circuit based on axial transform theory. A unified mathematical description of the magnetic circuit profile is given within a medial axial transform in differential geometry. The objective constraint and geometric constraint equations of the magnetic circuit form a closed set of differential equations. Lastly, the optimized magnetic circuit is obtained using the solutions of the equations. The optical performance of the magnetic lens is significantly improved.

A Novel Geometric Modeling and Calculation Method for Forward Displacement Analysis of 6-3 Stewart Platforms

A novel geometric modeling and calculation method for forward displacement analysis of the 6-3 Stewart platforms is proposed by using the conformal geometric algebra (CGA) framework. Firstly, two formulas between 2-blade and 1-blade are formulated. Secondly, the expressions for two spherical joints of the moving platform are given via CGA operation. Thirdly, a coordinate-invariant geometric constraint equation is deduced. Fourthly, a 16-degree univariate polynomial equation without algebraic elimination by using the Euler angle substitution is presented. Fifthly, the coordinates of three spherical joints on the moving platform are calculated without judging the radical symbols. Finally, two numerical examples are used to verify the method. The highlight of this paper is that a new geometric modeling and calculation method without algebraic elimination is obtained by using the determinant form of the CGA inner product algorithm, which provides a new idea to solve a more complex spatial parallel mechanism in the future.

A novel family of umbrella-shaped deployable mechanisms constructed by multi-layer and multi-loop spatial linkage units

This paper aims to construct a novel family of umbrella-shaped deployable mechanisms based on new multi-layer and multi-loop spatial deployable linkage units. Firstly, a general method is proposed for analyzing the degree-of-freedom (DOF) of the deployable units by identifying independent geometrical constraint equations of them. Then, multi-layer and multi-loop deployable units are synthesized systematically based on structural modules and DOF constraint conditions. Finally, three mobile assembly modes of the new deployable units are established, and many novel umbrella-shaped mechanisms are obtained by assembling the deployable units. This paper greatly extends our previous work on the deployable mechanism design based on two-layer and two-loop deployable units in breadth and in depth, and presents new design methods as well as many novel umbrella-shaped deployable mechanisms with potential applications in space exploration and civil engineering.

Decoupling Algorithm and Maximum Operation Frequency of a Novel Parallel Type Six-Axis Accelerometer

This paper presents a novel non-gyro six-axis accelerometer based on a derivative of Stewart parallel mechanism (SPM). To reduce the impact of interference noise, a set of geometric constraint equations are presented to compensate the errors of redundant outputs. A decoupling algorithm is introduced for easy modelling of this multiple-axis vibration problem. An auxiliary angular velocity is adopted so that the system dynamic equations can be directly solved by traditional explicit recursive formulas (ERFs) for real-time applications. Both an eigenvalue analytical algorithm and a matrix iteration approach are used to compute the fundamental frequency (FF) of the accelerometer for comparison. At the end, a four-step method is summarized in order to obtain an appropriate operation frequency range of this sensor. The analysis results reveal that the maximum operation frequency (MOF) of a parallel type six-axis accelerometer is always proportional to its FF in a ratio between 1/35~1/32. The predicted numerical results of the established decoupling algorithm are verified within the identified operation frequency bandwidth by comparing with virtual and actual experiment data, showing a maximum relative error of 0.47% and 8.87%, respectively.

Construction of a dynamic model for the interaction between the versatile tracks and a vehicle

Take hypothesis of wheel-rail rigid contact, this work presents a new method to characterize the vehicle-track interaction in the three-dimensional (3-D) space. The vehicle model is considered as a multi-rigid-body system with one car body, two bogie frames and four wheelsets that are connected by two-stage suspension systems. The track model is constructed by a versatile approach, where the finite elemental size and type of track components can be arbitrarily chosen and combined. The key matrices in depicting vehicle-track interaction are established with satisfaction of wheel-rail displacement complementarity by geometrical constraint equations and the wheel-rail force equilibrium by energy variational principle. Besides the functionality of depicting wheel-rail separations has been solved and compiled in the program. This model possesses the superiority of the previous models in computational stability. Besides the vehicle-track interaction at a 3-D space, which considers the wheel-rail contact geometries/creepages and infinite length calculation, are further formulated in the vehicle-track coupling matrices. Comparisons with general solutions built by well-known wheel-rail contact theories are conducted to show the engineering practicality of this model; moreover the differences of modelling results and the advancements of this model are also illustrated.

Eddy Current Sensor System for Blade Tip Clearance Measurement Based on a Speed Adjustment Model

Blade tip clearance (BTC) measurement and active clearance control (ACC) are becoming crucial technologies in aero-engine health monitoring so as to improve the efficiency and reliability as well as to ensure timely maintenance. Eddy current sensor (ECS) offers an attractive option for BTC measurement due to its robustness, whereas current approaches have not considered two issues sufficiently. One is that BTC affects the response time of a measurement loop, the other is that ECS signal decays with increasing speed. This paper proposes a speed adjustment model (SAM) to deal with these issues in detail. SAM is trained using a nonlinear regression method from a dynamic training data set obtained by an experiment. The Levenberg–Marquardt (LM) algorithm is used to estimate SAM characteristic parameters. The quantitative relationship between the response time of ECS measurement loop and BTC, as well as the output signal and speed are obtained. A BTC measurement method (BTCMM) based on the SAM is proposed and a geometric constraint equation is constructed to assess the accuracy of BTC measurement. Experiment on a real-time BTC measurement during the running process for a micro turbojet engine is conducted to validate the BTCMM. It is desirable and significative to effectively improve BTC measurement accuracy and expand the range of applicable engine speed.

Polyp Shape Recovery from Single Endoscope Image using Medical Suture

Background:Polyp shapes play an important role in colorectal diagnosis. However, endoscopy images are usually composed of nonrigid objects such as a polyp. Hence, it is challenging for polyp shape recovery. It is demanded to establish a support system of the colorectal diagnosis system based on polyp shape.Introduction:Shape from Shading (SFS) is one valuable approach based on photoclinometry for polyp shape recovery. SFS and endoscope image are compatible on the first sight, but there are constraints for applying SFS to endoscope image. Those approaches need some parameters like a depth from the endoscope lens to the surface, and surface reflectance factor . Furthermore, those approaches assume the whole surface which has the same value of for the Lambertian surface.Methods:This paper contributes to mitigating constraint for applying SFS to the endoscope image based on a cue from the medical structure. An extracted medical suture is used to estimate parameters, and a method of polyp shape recovery method is proposed using both geometric and photometric constraint equations. Notably, the proposed method realizes polyp shape recovery from a single endoscope image.Results:From experiments it was confirmed that the approximate polyp model shape was recovered and the proposed method recovered absolute size and shape of polyp using medical suture information and obtained parameters from a single endoscope image.Conclusion:This paper proposed a polyp shape recovery method which mitigated the constraint for applying SFS to the endoscope image using the medical suture. Notably, the proposed method realized polyp shape recovery from a single endoscope image without generating uniform Lambertian reflectance.

EXPERIMENTAL CONFIRMATION OF EQUATIONS’ ADEQUACY OF GEOMETRIC CONNECTIONS IN THE MUNICIPAL MACHINE WITH THE BRUSH WORKING BODY PARALLELOGRAPHIC SUSPENSION

Introduction. The authors arise the question of increasing the efficiency of the municipal machine. Moreover, the influence of the clamping force of the brush body on the surface and on the brush deterioration and, as a consequence, on the quality of the roadway cleaning is highlighted in the article.Methods and materials. The results of the changing process of the brushing tool vertical coordinate in the municipal machine were presented, depending on the displacement of the hydraulic cylinder rod. The analysis of the existing types of the brushing working body suspension was carried out. As a result of such analysis, the authors established the most frequent usage of the three-point and parallelogram suspensions, the schemes of which were also presented in the article.Results. The equations of geometrical connections of the municipal machine elements, which could be used in mathematical model, are presented. In addition, the kinematic diagram of the municipal machine with the brush working body suspension is drawn up. Therefore, the presented model takes into account such parameters as the length of the levers’ arms, the linear movements of the hinges, the rod stroke, the levers’ angles and the distances between the hinges. The methods and the course of the experiment for determining the displacement dependence of the brush body from the displacement of the hydraulic cylinder rod are described. The authors also present the photographs of the experimental research fragments.Discussion and conclusions. On the basis of the obtained data, the authors construct the experimental and theoretical dependences’ graph of the brush body displacement and the actuator hydraulic cylinder rod displacement. The analysis of such graphs confirms the adequacy of the geometric constraint equations and also proves that these equations could be used in the mathematical model of the position controlling process of the brush operating body and also for determination the optimum value of the brush working element clamping force to the surface.

보행자 추적용 이동로봇의 주행을 위한 최적 자세 및 경로결정

One of the goals in the field of mobile robotics is the appropriate action of security service robots responding to pedestrians behaving suspiciously in security areas. In this paper, we propose an algorithm for planning an optimal position and trajectory to block a pedestrian by a mobile robot in real-time, and describe a security robot system and ongoing research results in order to forbid pedestrian from entering sensitive areas like security facilities, nuclear power plants, and industrial facilities. The proposed robot system usually surveils a security area with equipped active camera sensors. The direction and rotational angular velocity of the pedestrian are estimated using the Kalman filter, a state estimator. It is demonstrated that the pedestrian is tracked by using a 2-DOF active camera mounted on the mobile robot and then their entrance is blocked by a mobile robot. The pedestrian is assumed to be a point-object and projected onto a view plane to form a geometrical constraint equation that provides position data of the human based on the kinematics of the moving mobile robot. While moving, the robot continues these processes for adapting to situational changes. After arriving in the path of the pedestrian’s moving direction, the robot advises him not to proceed and to change his course. The simulation and experimental results of estimating and tracking of the pedestrian moving in the wrong direction with the mobile robot are presented.

Polyp Shape Recovery Based on Blood Vessel Structure Analysis

Obtaining polyp size and shape is important for precise diagnosis. The 3-D shape reconstruction technology of computer vision is tried to be used in the medical diagnosis for this reason. Some approaches based on Shape from Shading are proposed for polyp shape recovery from endoscope image. These approaches need some parameters such as surface reflectance coefficient parameter C. It is important to obtain those parameters for accurate polyp shape recovery. This paper proposed a method to extract corresponding blood vessel regions from two endoscope images based on blood vessel structure analysis. Parameters for polyp shape recovery are calculated from geometric constraint equations of the pair of extracted regions.

Design and motion analysis of load-carrying and transmit robot (LCT-Robot) for small openings

This paper proposes a Load-carrying and transmit robot (LCT-Robot), which is based on a novel modular joint, for the purpose of telemaintenance of the Tokamak First wall (FW). The LCT-Robot, which is installed on a Multipurpose transport cask (MTC) and plays the role of transmitting and load-carrying, has the adaptability to several of maintenance tasks by carrying different types of micro-robots. The transmission mechanism of LCT-Robot modular joint and structural “lightweight” design is comprehensively considered through establishing the geometric parameter constraint equations related to Tokamak FW. It can steadily provide the ability of long-distance transmitting and performs maintenance tasks when the weight of the micro-robot is changed. Without any gravity compensation, the LCT-robot equipped with ordinary drive motor can bear a wide range of load. It not only satisfies the higher load, but also enhances the efficiency of the drive motor. For avoiding frequent acceleration and deceleration of the LCT-robot, the entire process through small openings from the placement state (0-state) to the flattened state (3-state) is analyzed and the effective translational distance of MTC is confirmed. As a result, experiment verifies the effectiveness of the LCT-robot, which has a length of 2.95 meters and can smoothly pass through small openings (Width: 200 mm, Height: 250 mm).