Motion Constraint Equation Research Articles

Fast monocular visual-inertial calibration and initialization with weak environmental control and motion control

Most existing visual-inertial systems require control of extrinsic environment and extrinsic motion to obtain calibration results and simplify the initialization model. The former limits the usage scenarios of the method, while the latter reduces the estimation accuracy of the state parameters of the navigation system. Therefore, a visual-inertial calibration and initialization method is necessary for weak environmental and motion control conditions. Firstly, to overcome the reliance on environmental control, we establish a motion constraint equation based on the relative consistency of visual-inertial sensing spatial motion and implement a visual-inertial online calibration method. Secondly, to eliminate the reliance on motion control, a dual graph optimization structure is be constructed. The initial optimization outcome serves as the foundation for achieving improved calibration results in the refined optimization. Thirdly, to improve initialization accuracy, a complete initial state model is considered to decouple and calculate the translation extrinsic parameters. These parameters are compensated into subsequent initialization, enhancing the robustness of the initialization process. The proposed method is validated on the EuRoC dataset, and experimental results demonstrate that it achieves more accurate calibration results at a faster rate than VINS-Mono in scenarios without calibration target and with limited vehicle motion. The navigation results of different scenarios demonstrate that our method can more effectively integrate visual-inertial data due to improved calibration and initialization estimates.

Endoscope FOV Autonomous Tracking Method for Robot-Assisted Surgery Considering Pose Control, Hand–Eye Coordination, and Image Definition

In laparoscopic surgeries, intelligent assisted endoscope-holding robots (AEHRs) with traditional servo control have limited performance due to the lack of endoscopic attitude control, hand-eye coordination, and image definition. This paper proposes an endoscope field-of-view (FOV) autonomous tracking method for robot-assisted surgery (RAS) considering pose (i.e., position and attitude) control, hand-eye coordination and image definition, it can realize that the AEHR system can safely, stably and quickly track the movement of surgical instruments. Firstly, the general framework of the intelligent tracking system with robot-assisted endoscope-holding is constructed according to the clinical needs and shortcomings of traditional research. Secondly, the kinematic model and remote center of motion (RCM) constraint equations of AEHRs are established. Then, surgical instrument tips are segmented and localized using the deep learning method (DLM), and further, a surgical FOV intelligent adjustment control model with multiple indexes is developed. Finally, an experimental system based on the hybrid of AEHRs and speech interaction is built. The simulation and experimental results demonstrate that the proposed method can achieve fast, accurate, and effective instrument tips tracking (ITT), reduce hand-eye incoordination, and improve the safety and stability of FOV intelligent adjustment.

A Fast Calibration of Laser Vision Robotic Welding Systems Using Automatic Path Planning

Calibration is a crucial issue in laser vision robotic welding systems (LVRWSs). Traditional calibration methods rely on complex manual teaching and fixed pattern of robot end-effector poses, which are time-consuming and will occasionally fall into local optimal solution. To address these issues, a fast calibration method based on automatic path planning is proposed in this article. By ensuring that the camera’s optical axis coincides with the center of the calibration board regardless of the robot end-effector’s attitude, images covering all feature points will be collected. Based on the camera optical axis constraint, we establish the robot end-effector motion constraint equation. Then, an automatic path planning method based on optical axis constraint equation, rotation constraint, and motion selection strategy is proposed, which improves the diversity of the calibration data. During data acquisition, a hand–eye parameters optimization process is performed simultaneously by minimizing the reprojection error of visual feature points. The actual calibration tests show that the proposed method can significantly improve the calibration efficiency and robustness. In addition, a variety of weld measurement experiments, such as S-type and C-type welds and filet weld, were conducted under arbitrary welding torch’s attitude to evaluate the accuracy and robustness of the method. The mean absolute error (MAE) and root mean square error (RMSE) of the position are within 0.2 mm.

Kinematic dexterity analysis of human-robot interaction of an upper limb rehabilitation robot.

The definition of rehabilitation training trajectory is of great significance during rehabilitation training, and the dexterity of human-robot interaction motion provides a basis for selecting the trajectory of interaction motion. Aimed at the kinematic dexterity of human-robot interaction, a velocity manipulability ellipsoid intersection volume (VMEIV) index is proposed for analysis, and the dexterity distribution cloud map is obtained with the human-robot cooperation space. Firstly, the motion constraint equation of human-robot interaction is established, and the Jacobian matrix is obtained based on the speed of connecting rod. Then, the Monte Carlo method and the cell body segmentation method are used to obtain the collaborative space of human-robot interaction, and the VMEIV of human-robot interaction is solved in the cooperation space. Finally, taking the upper limb rehabilitation robot as the research object, the dexterity analysis of human-robot interaction is carried out by using the index of the approximate volume of the VMEIV. The results of the simulation and experiment have a certain consistency, which indicates that the VMEIV index is effective as an index of human-robot interaction kinematic dexterity. The VMEIV index can measure the kinematic dexterity of human-robot interaction, and provide a reference for the training trajectory selection of rehabilitation robot.

Motion in central gravitational field with Schwarzschild metric as a non-holonomic system with non-linear constraint: Geometrical setting

Mechanical systems with non-holonomic constraints are often studied in technical applications. In such situations linear or affine non-holonomic constraints are usually relevant. On the other hand, there exist mechanical systems which can be naturally interpreted as non-holonomic ones with non-linear constraint. Such systems occur typically in the theory of relativity. Within physical theories they are standardly described by physical methods, not as systems subjected to non-holonomic constraints. Nevertheless, every non-holonomic system can be treated by a universal purely mathematical approach — geometrical theory on fibred manifolds with Chetaev type constraint submanifolds. For describing all aspects of a system behaviour, including all symmetries and conservation laws, it is sufficient to have the corresponding unconstrained Lagrangian and the constraint. No additional physical assumptions are needed. We demonstrate this for a typical physical system with a non-linear non-holonomic constraint — a relativistic particle moving in central field of a non-rotating star. Within the geometrical theory we obtain constraint equations of motion and their solution. Especially, we obtain equations or constraint symmetries, their solution for coordinate transformations, and corresponding conservation laws from general relations obtained on the base of this universal geometrical theory.

Features of solving the inverse dynamic method equations for the synthesis of stable walking robots controlled motion

The problem of walking robots controlled motion synthesis by the inverse dynamic method is considered. The inverse dynamic method equations are represented by the methods of multibody system dynamics as free bodies motion equations and constraint equations. The variety of constraint equations group are introduced to specify the robot gait, to implement the robot stability conditions and to coordinate specified robot links movement. The key feature of the inverse dynamic method equations in this formulation is the presence of the second derivatives of the system coordinates in the constraint equations expressing the stability conditions that ensure the maintenance of the vertical position by the robot. The determined solution of such equations in general case is impossible due to the uncertainty of the initial conditions for the Lagrange multipliers. An approximate method for solving the inverse dynamic without taking into account the inertial components in the constraint equations that determine the stability of the robot is considered. Constraint equations that determine the coordinate movement of individual robot links and required for unique problem solving based on approximate equations are presented. The implementation of program motion synthesis methods in the control system of the humanoid robot AR-600 is presented. The comparison of theoretical and experimental parameters of controlled motion is performed. It has been established that with the achieved high accuracy of the robot links tracking drives control with an error of several percent, the indicators of the robot's absolute movements, in particular, the angles of roll, yaw and pitch, differ from the programmed by 30-40%. It’s shown that proposed method allows to synthesize robot control in quasistatic mode for different movement types such as moving forward, sideways, walking on stairs, inclinations etc.

Robust Optical Flow Estimation Using Tchebichef Moment Invariant Feature

In this paper, orthogonal moment invariant-based features are used to compute 2D optical flow from sequence of images. The gray level of pixel is described in terms of its local neighborhood using Tchebichef moment invariants rather than its individual intensity value. The description is then normalized in order to make it insensitive to intensity fluctuations due to noise or other perturbations such as varying illumination conditions. The principle of conservation of moment invariance is used to derive overdetermined system of 2D motion constraint equations in local neighborhood of each pixel. The velocity field is then estimated using the least square method. Experimental results are performed on sequential video and thermal image frames under varying environmental conditions. The run time, robustness against noisy and varying illumination conditions and rotation invariance of proposed method are compared with already existing moment and non-moment-based techniques.

A reduced-order approach to the adaptive fuzzy sliding mode control of the constrained manipulator

A reduced-order approach to the adaptive fuzzy sliding mode control of the constrained manipulator is proposed. Based on the Udwadia–Kalaba motion constraint equation, the dynamic equation of the constrained manipulator with both ideal and non-ideal constraints is obtained. Considering the uncertainty of the terminal non-ideal constrained force and the chattering phenomenon of sliding mode control, the adaptive fuzzy and the sliding mode control method are combined to control the constrained manipulator. Because the system is constrained, the model order reduction method is innovatively used in the control algorithm. The stability of the system is proved by Lyapunov theorem. For demonstrating the effectiveness of the control algorithm, the 2-degree-of-freedom manipulator is taken as the research object. Finally, the high-precision control of the manipulator is achieved and the chattering phenomenon caused by the sliding mode control is weakened.

Kinematics Simulation Analysis of Packaging Robot with Joint Clearance

Considering the influence of joint clearance on the motion error, repeated positioning accuracy and overall position of the machine, this paper presents simulation analysis of a packaging robot — 2 degrees of freedom(DOF) planar parallel robot based on the characteristics of high precision and fast speed of packaging equipment. The motion constraint equation of the mechanism is established, and the analysis and simulation of the motion error are carried out in the case of turning the revolute clearance. The simulation results show that the size of the joint clearance will affect the movement accuracy and packaging efficiency of the packaging robot. The analysis provides a reference point of view for the packaging equipment design and selection criteria and has a great significance on the packaging industry automation.

Pin-slot clearance joints in multibody systems

The pin-slot joint is one of the most commonly used form of kinematic pairs, which with clearance will have a significant effect on the motion and dynamic response of multibody systems. In this study, the multibody system with a pin-slot clearance joint is modeled and the kinematic and dynamic effects are studied. Firstly, the generalized coordinate systems were established and the motion constraint equations of the pin-slot clearance joint were derived. Then, the theories of kinematics and dynamics were used to deduce equations of motion and dynamics. Finally, using slider-crank mechanism as the application example, the simulation study of the multibody system with a pin-slot clearance joint was carried out. The results show that there is an impact phenomenon in the moving process of the end slider, and with the increase of the slot angle, the impact phenomenon is earlier and the impact strength is weaker, however, with the increase of the slot value, the time when the impact phenomenon appearing doesn't change while the impact strength increasing. Furthermore, the maximum displacement of the end slider reduces with the increase of the slot value while the maximum value of driving torque doesn't change and the fluctuation of driving torque increased.

Machining accuracy improvement of non-orthogonal five-axis machine tools by a new iterative compensation methodology based on the relative motion constraint equation

Abstract This paper proposes a new iterative compensation methodology of geometric errors to improve the machining accuracy of a non-orthogonal five-axis machine tool (NOFAMT). Firstly, based on homogeneous transform matrix (HTM) and multi-body system (MBS) theory, the relative motion constraint equations (TRMCEs) of the tool tip position and tool orientation vector related to a NOFAMT with a nutating rotary B axis are established. Then, by utilizing TRMCEs, the mapping relationships between tool path and the numerical control (NC) command without and with considering the geometric errors are constructed respectively. In order to truly reproduce tool motion trajectory of the machine tool driven by the given NC command, the mapping relationship between the NC command and tool cutting trajectory is also established. Meanwhile, procedures of iterative compensation are described by using the aforementioned mapping relationships without the traditional inverse calculation, and the actual NC code is generated in self-developed compensation software. It is not difficult to find that the new approach takes the difference between tool path and tool cutting trajectory as the control objective and can directly obtain the actual NC code controlling the machine tool to achieve the desired machining accuracy. Finally, a cutting test is carried out on the DMU60P NOFAMT. Experimental results show the developed iterative compensation methodology is precise and effective for NOFAMTs. Therefore, compared with the existing methods, the new method is more direct and accurate. And its basic idea can be applied to other type of machine tools.

Lie symmetries and their inverse problems of nonholonomic Hamilton systems with fractional derivatives

This letter focuses on studying Lie symmetries and their inverse problems of the fractional nonholonomic Hamilton systems. Based on the invariance of the fractional motion equations, constraint equations and virtual displacement restrictive conditions of the systems under the infinitesimal transformation with respect to the time and generalized coordinates, the Lie symmetries and conserved quantities of the fractional nonholonomic Hamilton system are discussed and the corresponding definitions, determining equations, limiting equations, additional restricting equations and Lie theorems are given. The letter also systematically studies inverse theorems of Lie symmetries of the fractional nonholonomic Hamilton systems. Finally, an example is discussed to illustrate theses results.

A concurrent multi-scale technique in modeling heterogeneous FCC nano-crystalline structures

In this paper, a multi-scale molecular dynamics–finite element coupling is presented to study the mechanical behavior of heterogeneous nano-crystalline structures. The stiffness and mass matrices of the continuum sub-domain are generated by applying a linear transformation on the matrices obtained via the atomic structure underlying the FE mesh. A Lagrange multiplier method is employed to the transition zone imposing velocity resemblance of the coupling regions. The constraint equations of motion are solved by the multi-time-step decomposition thus giving the opportunity to ascribe different time steps to each individual zone. The molecular dynamics is performed by employing the Sutton–RafiiTabar many body potential (Raffi-Tabar and Sutton, 1991) for FCC metallic alloys, and its integrity is attained by calculating the effective lattice parameter of different random alloys by minimizing the general form of Sutton–RafiiTabar interatomic potential energy. The authenticity and accuracy of the renovated concurrent scheme is remarkably acquired by comparing some numerical results of the proposed multi-scale model to those of implemented and validated molecular dynamics.

A twenty-segment kinematics and kinetics model for analysing golf swing mechanics

The golf swing is a complex, multi-planar, three-dimensional (3D) motion sequence performed at very high speeds. These properties make biomechanical analysis of the golf swing difficult. Hence, the aim of this study was to develop a computer model of the golf swing capable of calculating a diverse range of 3D kinematics and kinetics values based on motion analysis data collected in the laboratory. Five golfers performed six swings in the field of view of eight Falcon High Speed Resolution cameras (240 Hz), which captured the movements of 56 markers placed on the golfers and their clubs, resulting in marker trajectories that were processed into linear xyz-coordinates using the Eva Motion Analysis system. To perform the kinematics and kinetics calculations, a 20-segment rigid body model of the human body was designed in the Mechanical Systems Pack, connecting the segments by a selection of linear and spherical constraints, resulting in a system of segments with 58 degrees of freedom, with the constraint equations of motion calculated by the Newton–Lagrangian iteration method. The model allowed for the derivation of segmental sequencing, separation angles, segmental planes of motion, segmental velocity contributions, joint torques and muscle powers. The preliminary data suggest that such an integrated kinematics and kinetics analysis is necessary to understand the mechanical complexity of golf swing. Even with the small sample size analysed in this study, some interesting trends were found, such as certain violations of the classical proximal-to-distal sequencing scheme, differing swing plane and club head trajectories in the backswing and downswing phases, minimal hip angular velocity contribution to the ball at impact, concentric and eccentric muscle powers in the downswing phase, and increased lumbar loading factors from the mid-downswing phase to ball impact.

Dynamic modeling and simulation of multi-body systems using the Udwadia-Kalaba theory

Laboratory experiments were conducted for falling U-chain, but explicit analytic form of the general equations of motion was not presented. Several modeling methods were developed for fish robots, however they just focused on the whole fish’s locomotion which does little favor to understand the detailed swimming behavior of fish. Udwadia-Kalaba theory is used to model these two multi-body systems and obtain explicit analytic equations of motion. For falling U-chain, the mass matrix is non-singular. Second-order constraints are used to get the constraint force and equations of motion and the numerical simulation is conducted. Simulation results show that the chain tip falls faster than the freely falling body. For fish robot, two-joint Carangiform fish robot is focused on. Quasi-steady wing theory is used to approximately calculate fluid lift force acting on the caudal fin. Based on the obtained explicit analytic equations of motion (the mass matrix is singular), propulsive characteristics of each part of the fish robot are obtained. Through these two cases of U chain and fish robot, how to use Udwadia-Kalaba equation to obtain the dynamical model is shown and the modeling methodology for multi-body systems is presented. It is also shown that Udwadia-Kalaba theory is applicable to systems whether or not their mass matrices are singular. In the whole process of applying Udwadia-Kalaba equation, Lagrangian multipliers and quasi-coordinates are not used. Udwadia-Kalaba theory is creatively applied to dynamical modeling of falling U-chain and fish robot problems and explicit analytic equations of motion are obtained.

A Local Algorithm for the Computation of Image Velocity via Constructive Interference of Global Fourier Components

A novel Fourier-based technique for local motion detection from image sequences is proposed. In this method, the instantaneous velocities of local image points are inferred directly from the global 3D Fourier components of the image sequence. This is done by selecting those velocities for which the superposition of the corresponding Fourier gratings leads to constructive interference at the image point. Hence, image velocities can be assigned locally even though position is computed from the phases and amplitudes of global Fourier components (spanning the whole image sequence) that have been filtered based on the motion-constraint equation, reducing certain aperture effects typically arising from windowing in other methods. Regularization is introduced for sequences having smooth flow fields. Aperture effects and their effect on optic-flow regularization are investigated in this context. The algorithm is tested on both synthetic and real image sequences and the results are compared to those of other local methods. Finally, we show that other motion features, i.e. motion direction, can be computed using the same algorithmic framework without requiring an intermediate representation of local velocity, which is an important characteristic of the proposed method.

A Discussion of Low-Order Numerical Integration Formulas for Rigid and Flexible Multibody Dynamics

The premise of this work is that the presence of high stiffness and/or frictional contact/impact phenomena limits the effective use of high order integration formulas when numerically investigating the time evolution of real-life mechanical systems. Producing a numerical solution relies most often on low-order integration formulas of which the paper investigates three alternatives: Newmark, HHT, and order 2 BDFs. Using these methods, a first set of three algorithms is obtained as the outcome of a direct index-3 discretization approach that considers the equations of motion of a multibody system along with the position kinematic constraints. The second batch of three algorithms draws on the HHT and BDF integration formulas and considers, in addition to the equations of motion, both the position and velocity kinematic constraint equations. Numerical experiments are carried out to compare the algorithms in terms of several metrics: (a) order of convergence, (b) energy preservation, (c) velocity kinematic constraint drift, and (d) efficiency. The numerical experiments draw on a set of three mechanical systems: a rigid slider-crank, a slider-crank with a flexible body, and a seven body mechanism. The algorithms investigated show good performance in relation to the asymptotic behavior of the integration error and, with one exception, result in comparable CPU simulation times with a small premium being paid for enforcing the velocity kinematic constraints.

A Computational Method for Planar Kinematic Analysis of Beam Pumping Units

A generalized computational method for planar kinematic analysis of pumping units is presented in this study. In this method, a local coordinate system is assigned to each body with respect to a fixed global coordinate system. The position of each point in a body is determined by specifying the global translational coordinates of the local coordinate system origin and its rotational angle relative to the global coordinate system. Constraint equations of motion are developed using the vector of coordinates of the connected bodies. These equations are solved to yield the position, velocity, and acceleration of the individual linkages at each instance of time. Both rotational and translational types of joints are considered in the analysis. The translational joint analysis is not discussed in this paper as they are not applicable for beam pumping units. This method can be used as an effective tool for pumping unit design and optimization. An example is provided to show the application of this method.

Research on Software Error Compensation of Ultra-Precision Lathe

Based on the theory of kinematics for multi-body system, the relative motion constraint equation is deduced according to the structure layout and the error distribution of the ultra-precision lathe Nanoform200. By solving the constraint equation, the corrective NC code is derived that can compensate the geometric errors. The error compensation software is developed aimed at the ultra precision manufacturing of optics parts. The cutting experiments show that the method and model in this paper can improve the accuracy about 50% for the ultra-precision turning of optical parts.

Constraint algebra and equations of motion in the Bohmian interpretation of quantum gravity

It is shown that introducing quantum effects using de Broglie–Bohm theory in the canonical formulation of gravity changes the constraint algebra. The new algebra is a clear projection of general coordinate transformations to the spatial and temporal diffeomorphisms. The quantum Einstein equations are manifestly covariant under the above diffeomorphisms.