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To evaluate the effect of soil particle size on heavy metals removal by washing, two soil samples were collected around a lead-zinc mining area (SM) and lead-zinc smelter (SS). The total content of Cd, Pb and Zn in SM and SS were determined. And the effect of soil particle size on Cd removal by low molecular organic acids was studied. The results showed that Cd was the main pollutant and the total content of Cd in SS can reach to 24.8 mg Kg

To investigate the change in biochar's ability to directly immobilize Cd in soil, a successive wheat cultivation experiment was conducted. Three biochars with different Cd adsorption mechanisms were added to the soils, and a mesh bag was used to separate the soil particles ( 1 μm) from the biochar. The results showed that the ash contents and anionic contents (CO

Assembly interference matrix is one of foundation information model for assembly process planning like assembly sequence planning and assembly path planning, which supports the digital assembly simulation, intelligent assembly, digital twins based assembly etc. Assemble interference matrix represents each two parts are collision or not when they moving along specific directions. Traditional assembly interference matrix construction adopts geometric collision detections on approximate swept volume of part which composed by moving part step by step. This manner not only leads to heavy computational loading since small moving step but also easy to misjudging contact as collide as well as missing collide due to discrete spatial moving of part. To overcome the drawbacks, this work equivalent the collision detection of parts during assembly interference matrix establishing as visibility judgement, and develop an image-assisted calculation method to process the collision between part’s swept volume as occlusion of images via computer graphic rendering. This method is implemented on OpenGL to ensure its generality, and the implicit calculating is finished in computer graphic rendering process by arranging proper rendering conditions with adopting depth testing and stencil testing. Furthermore, the erosion operation of image process is employed to distinguish whether contact or collide. At last, an examples for gear box assembly validates the effectiveness of this method.

The flame-retardant finishing of cotton with pentaerythritol phosphate and small molecule organic amine, such as diethylenetriamine, triethylenetetramine, diethanolamine, or isopropanolamine, was studied. The mechanical and flame-retardant properties and thermal stability of cotton fabric before and after flame-retardant finishing were investigated. Results revealed that the longitudinal and latitudinal breaking strength increased. The limiting oxygen index increased from 18.7 % to 37.9–39.1 %. Vertical flame burning time and afterglow time decreased from 35.1 and 8.4 s, respectively, to 0 s. The damage length decreased from 30 cm to 7.1–8.4 cm. At 700 °C, the residual carbon increased from 17.69 % to 41.48–41.56 %. The experimental results show that the flame-retardant system has good flame retardant and carbonization effects on cotton and good foaming property for combustion. The experimental operation is simple, rapid, and has broad market prospects.

Traditional methods of coherent diffraction imaging using random masks result in an insufficient difference between the diffraction patterns, making it challenging to form a strong amplitude constraint, causing significant speckle noise in the measurement results. Hence, this study proposes an optimized mask design method combining random and Fresnel masks. Increasing the difference between diffraction intensity patterns enhances the amplitude constraint, suppresses the speckle noise effectively, and improves the phase recovery accuracy. The numerical distribution of the modulation masks is optimized by adjusting the combination ratio of the two mask modes. The simulation and physical experiments show that the reconstruction results of PSNR and SSIM using the proposed method are higher than those using random masks, and the speckle noises are effectively reduced.

Helical rake flank is an important structure of helical broach tools. Its perfect sharpening is crucial to maintain the working performance of the tool and prolong the tool life, which is obtained via proper wheel orientation and suitable grinding processes. In practical, wheels with diverse profiles are adopted to satisfy the sharpening for various rake flanks. Therefore, wheel orientation identification methods with strong robustness are expected to deal with general helical rake flank sharpening tasks. Besides, proper grinding processes should be devoted to helical rake flank sharpening to improve the quality of broach tool. This study developed a wheel orientation calculation method that combing the specific process for helical rake flank sharpening with involving discrete enveloping. Firstly, a general model for grinding wheel is established, and the swivel-tilt-based orientation and positioning parameters of wheel are given. Then, through approximating the grinding motion as a serial of discrete wheels, the machined profile on the broach tool axial-section is expressed as the external enveloping profile of the intersection curves between the wheels and the tool axial-section. After distinguishing the enveloping profile by points along radial direction, the derivation of the instant contact line was provided. Furthermore, the mathematic model of the specific process about helical broach rake flank sharpen is established to enhance the wheel orientation identification algorithm. At last, both experiments and simulations are implemented to verify the effectiveness of this method and to investigate the influences of the process constrains via solution space.

Form error deflects the assembly surfaces and thus greatly affects the assembly accuracy of precision mechanical systems. Based on rigid motion, this paper proposes a prediction method for assembly surface contact considering form error to improve the simulation accuracy of virtual assembly. The method estimates the assembly equilibrium state by simulating the realistic behaviour of surfaces in assembly process: Firstly, one surface is fixed and the other is transferred along the assembly direction until the first contact occurs. Then, the other surface is rotated according the assembly force and the previous contact positions to successively reach the second and third contacts. The rotation operation is repeated until the assembly equilibrium conditions are finally met. Particularly, a space slicing algorithm is put forward to achieve faster calculation of the rotation information including rotation axis, rotation direction and rotation angle. This method can be applied in the field of assembly simulation and assembly verification. At the end of this paper, an assembly case of two rectangular plates is carried out to prove the feasibility and effectiveness of the method.

Precision contact probes have a huge demand in industrial manufacturing due to their high accuracy. However, the passive measurement mode of the existing probe easily causes wear of the probe ball and loss of accuracy during continuous contact scanning. In this paper, a PZT(piezoelectric actuators)-driven flexible probe is proposed with the dynamic time-domain function of the system response from low frequency to high drive frequency. The low-frequency measurement mode is established based on the linear response of the triangle wave. In order to avoid the trajectory nonlinearity of the stylus ball driven by PZT, the nonlinear mapping of the PZT hysteresis model is constructed by the BP neural network, and the linearly weighted particle swarm optimization (PSO) algorithm is used to realize the suppression of the response nonlinearity. Finally, the measurement experiment of the PZT-driven probe is conducted to reveal the measurement characteristics of low-frequency mode as well as measurement capability compared with commercial sensors.

Previous works have shown that only relying on measurement data is generally insufficient to identify root causes of the dimensional variation for the complex manufacturing system. It is also well known that for underdetermined multistage assembly processes (MAPs), the number of measurements is less than that of process errors so that the traditional methods are not available for the variation source identification. Therefore, there exists a substantial challenge for fault diagnosis of underdetermined MAPs, especially for the case with multiple fault patterns. To tackle this problem, a novel approach that integrates statistical analysis with domain knowledge is proposed in this paper. First, the variation propagation model is employed to reveal the inherent relationship between key control characteristics and key product characteristics, and then the corresponding variance model is constructed to interpret process faults by means of the abnormal variance of process errors. Considering that the probability of less process faults is far higher than that of more process faults in MAPs, the problem of fault diagnosis is further transformed into searching the sparse solution of abnormal variance changes for process faults. Afterwards, based on the non-negative property of the covariance matrix, an enhanced Bayesian hierarchical model is developed to favor sparse estimation of the variance for the underdetermined system in MAPs. Finally, experimental results of both the numeric simulation and practical case study demonstrate the proposed diagnosis methodology can effectively identify the process faults for different patterns with system noise.

The traditional finite element method has the advantages of strong versatility and high calculation accuracy. Thus it is widely applied in various numerical simulation problems. Although the traditional finite element method for solving the shaft-hole assembly problem can obtain better solution accuracy, the process of modeling and calculation take a long time and thus cannot meet the engineering application requirements when the equilibrium position after the assembly needs to be obtained quickly to guide the subsequent process. In this paper, an iterative method based on the spring contact model is proposed. Firstly, the actual form error of the shaft and hole is characterized by the Fourier series expansion method; Then, in the contact surface, we established a discrete spring contact model and obtained the equivalent stiffness of the coupling spring through the discrete solid model. The balance iterative calculation based on the form errors and contact deformation force can finally determine the equilibrium position after the assembly. Compared with the calculation results of the finite element model, the equilibrium position of the shaft-hole assembly can be quickly solved under the condition of sacrificing certain precision. From this aspect it can be estimated that this method has a good potential for engineering application.

The assembly accuracy of the extendible support structure is of importance for the imaging capability of synthetic aperture radar antennas. In general, due to manufacturing imperfections and installation variations, its assembly accuracy will be inevitably degraded. Therefore, controlling the assembly deviation is highly concerned in practice. To meet the accuracy requirement and make “the control” more efficient, this study proposes a novel method to quantitatively conduct dimensional adjustment of links for extendible support structures of synthetic aperture radar antennas. Since the extendible support structure is generally over-constrained in the deployed configuration, the relationship between the assembly deviation and the variation sources is first derived by means of equivalent transformation. Based on the mathematical expression of assembly deviation, an inequality constrained sparse optimization model for quantitatively resizing links is formulated. Then, an efficient algorithm integrating the alternating direction method of multipliers and binary search is developed to solve the above optimization model, thereby acquiring the optimal combination of link adjustment. Finally, numerical case studies are carried out to demonstrate the effectiveness of the proposed method in Matlab, which show that it can not only achieve satisfactory performance in prediction but also significantly improve the assembly efficiency.

In this work, a novel amino-modified chitosan/tailings composite (CS-PEI-nGT) was successfully synthesized from gold tailings particle treated by ball milling (nGT), chitosan (CS) and polyethyleneimine (PEI) as raw materials, for Lead (Pb(Ⅱ)) and Cadmium (Cd(Ⅱ)) removal from aqueous solutions. The CS-PEI-nGT was characterized by using FTIR, XRD, SEM, BET, TGA and XPS techniques. The results showed that CS-PEI-nGT had maximum adsorption capacity of 192.78 mg·g

Purpose This paper aims to develop a theoretical method to analyze the rotation accuracy of rotating machinery with multi-support structures. The method effectively considers the geometric errors and assembly deformation of parts. Design/methodology/approach A method composed of matrix and FEA methods is proposed to do the analysis. The deviation propagation analysis results and external loads are set as boundary conditions of the model which is built with Timoshenko beam elements to calculate the spatial pose of the rotor. The calculation is performed repeatedly as the rotation angle increased to get the rotation trajectories of concerned nodes, and further evaluation is done to get the rotation accuracy. Additionally, to get more reliable results, the bearing motion errors and stiffness are analyzed by a static model considering manufacturing errors of parts. Findings The feasibility of the proposed method is verified through a case study of a high-precision spindle. The method reasonably predicts the rotation accuracy of the spindle. Originality/value For rotating machinery with multi-support structures, the paper proposes a modeling method to predict the rotation accuracy, simultaneously considering geometric errors and assembly deformation of parts. This would improve the accuracy of tolerance analysis.

Contact scanning probes have higher accuracy than non-contact probes in industrial applications. But changes of the measured surface and adjustment of velocity and acceleration may cause random measurement deviations, which are mainly determined by dynamic characteristics of the probe system. To realize the prediction of measurement deviations related to scanning parameters, this paper proposes a simpler detection method of dynamic performance. In this article, the dynamic response characteristics of the probe to diverse external stimuli are studied theoretically and experimentally on the basis of the original probe design. Under different scanning variables including the amplitude, spacing and shape of the surface, velocity and acceleration, the integration of the second-order vibration model and external excitation forms the theoretical basis for quantitative analysis of the system response, and numerical simulation and three sets of experiments are designed to effectively verify the deviation distribution of the system response caused by different scanning parameters in the time and frequency domains.

To investigate the change in biochar’s ability to directly immobilize Cd in soil ，a successive wheat cultivation with experiment was conducted. Three biochar with different Cd adsorption mechanisms were added into soils and a mesh bag was used to separate the soil particles (> 1 μm) from biochar. The results showed that the ash contents and anionic contents (CO 3 2- and PO 4 3- ) of the biochar decreased with the cultivation time, while the oxygen-containing functional groups content and CEC of the biochar increased. Resultly, the Cd concentration on biochar decreased, highly decreased by 68.9% for WBC300, while unstable Cd species (acid soluble and reducible fraction of Cd) on biochar increased with successive cultivation, increasing from 3% to 17% for WBC300 in FS. Correspondingly, the ability of biochar to inhibit Cd accumulation in wheat decreased. The results of this study illustrated that the ability of biochar to directly immobilize Cd in soil is not permanent, it gradually decreases with aging in soil. The adsorption mechanism of Cd on biochar changed from precipitation to complexation and ion exchange processes could be the main reason.

Using filamentous fungi to capture unicellular microalgae is an effective way for microalgae recovery in water treatment. Here, fungi Aspergillus flavus ZJ-1 and microalgae Chlorella vulgaris WZ-1 isolated from a copper tailings pond were used to study the capture effect of ZJ-1 on WZ-1. The highest capture efficiency (97.85%) was obtained within 6 h under the optimized conditions of 30 °C, 150 rpm, fungi-algae biomass ratio of 2.24:1, and initial pH of 9.24 in microalgae medium. The formed fungi-algae pellets (FAPs) were further used to remove Cu(II) from aqueous solution. Results showed that the FAPs formed at different capture times all adsorbed Cu(II) well, and the PAFs formed within 2 h (PAFs

The aim of current study was to develop a new material for the fast and efficient removal of hexavalent molybdenum (Mo(VI)) from contaminated water. In this work, a novel adsorbent was synthesized through the polypyrrole intercalation modification of bentonite (PPy-BT) via in-situ chemical polymerization method for effectively removal of Mo(VI) from aqueous solution. The surface morphology and chemical composition of PPy-BT composites were investigated by X-ray diffraction, energy-dispersive X-ray spectroscopy, Fourier transform infrared spectrometer, scanning electron microscopy techniques and X-ray photoelectron spectroscopy. PPy and BT could well resist the aggregation of each other, and therefore resulted in a loose-packed structure and good exposure of active sites. Using materials for the adsorption of Mo(VI) revealed has a maximum adsorption capacity of 100.17 mg/g at 25 °C and pH 4.0 by the Langmuir model. The adsorption kinetics and isotherm data are found to be well elucidated through pseudo-second-order and Langmuir models. Moreover, high regeneration ability (89.3%) of PPy-BT was noted for five consecutive adsorption-desorption cycles. These findings highlight the potential of PPy-BT for practical water treatment applications. The intercalation material of PPy-BT could provide a new strategy to develop cost-effective clay-based nanomaterials for wastewater treatment.

The non-contact R-test is used for the dynamic measurement of five-axis machine tools. However, due to spherical curvature, calculating the spherical center displacement (SCD) using the non-contact R-test is challenging and inefficient. Thus, in this paper, an efficient calculation method based on multiple Gaussian process regression (GPR) is proposed to calculate the SCDs of the continuous trajectory by using the non-contact R-test. In the proposed method, first, the transformation matrix is constructed to roughly calculate the SCD. Second, the measurement error caused by the spherical curvature is solved and compensated by using the GPR model. Next, the threshold is set to remove some of the set points and reconstruct GPR to improve the model accuracy. Finally, the transformation matrix and the GPR model are outputted to calculate the SCDs of the continuous trajectory. We constructed a micro S trajectory to verify the proposed method. Experimental results showed that compared with the neural network method, the proposed method achieves higher model accuracy with shorter model construction time and improves the calculation efficiency by 99%.

In non-cooperative target frequency scanning interferometry, the return optical power is low, the quality of the interferometric signal is poor, and the signal-to-noise ratio (SNR) is low. Moreover, the power change accompanying the use of the frequency scanning laser modulates the interferometric signal’s amplitude and shifts the amplitude centre. Traditional signal enhancement techniques, such as filtering, can only solve some of the problems affecting the measurement accuracy, and the full-factor processing of such signals is difficult. This paper proposes a non-cooperative target frequency scanning interferometry signal enhancement method based on a generative adversarial network. By learning the sample dataset, the SNR of the signal can be improved within a certain range, and the signal waveform can be corrected simultaneously. The simulation results reveal that the SNR of the non-cooperative target signal is improved and the signal waveform is satisfactorily corrected. Finally, the effectiveness of the enhancement method was experimentally confirmed.

Rotary-laser automatic theodolite (R-LAT) system is a distributed large-scale metrology system, which provides parallel measurement in scalable measurement room without obvious precision losing. Each of R-LAT emits two nonparallel laser planes to scan the measurement space via evenly rotation, while the photoelectric sensors receive these laser planes signals and performs the coordinate calculation based on triangulation. The accurate geometric parameters of the two laser planes plays a crucial role in maintaining the measurement precision of R-LAT system. Practically, the geometry of the two laser plane, which is termed as intrinsic parameters, is unknown after assembled. Therefore, how to figure out the accurate intrinsic parameters of each R-LAT is a fundamental question for the application of R-LAT system. This paper proposed an easily operated intrinsic parameter calibration method for R-LAT system with adopting coordinate measurement machine. The mathematic model of laser planes and the observing equation group of R-LAT are established. Then, the intrinsic calibration is formulated as a nonlinear least square problem that minimize the sum of deviations of target points and laser planes, and the ascertain of its initial guess is introduced. At last, experience is performed to verify the effectiveness of this method, and simulations are carried out to investigate the influence of the target point configuration on the accuracy of intrinsic parameters.

Instead of the common analytical model for involute gears machining, a general mathematical model for two-parameter generating machining is developed in this study with adopting discrete enveloping to increase its robustness and generality for machined profile calculation and instant contact analysis. The geometric parameters and motion relationships are integrated according to the cross-axis gear meshing at first, and the enveloping theory for two-parameter generating process is analyzed briefly with involving middle gear rack. By transforming the multiple cutting edges relative to the gear blank according to the generating motions, a numerical algorithm is proposed to distinguish the external enveloping profile on the shaft section, then the derivation of instant contact points via the implied time sequences of profile points is introduced. Finally, the coordinate transformations for the spatial trajectory of instant contact points identifying and the engaged cutting edge segments distinguishing are investigated. At last, a non- involute profile skiving is performed to verify the generality and the robustness of proposed method, and several typically generating machining operations for cylindrical involute gear are simulated numerically to proof the accuracy of the proposed model by finding the deviations from the standard involute curves and the geometries of the instant contact points.

Input uncertainties inevitably result in the output error for parallel mechanisms, which will lead to significant influence on good work performance. To control the output error within the specification boundary, this article proposes a novel approach of input tolerance design based on the level set method for the driving joint. The implementation of the proposed method can be divided into two subtasks. First, using the level set method, the exact input error boundary is determined by means of evolving an initial input error interface with a defined normal speed field, thereby transforming the problem of exploring the exact boundary into solving a partial differential equation with initial value. On this basis, according to the equal and scaled principles, the tolerance width of each input is evaluated in an intuitionally geometrical manner, that is, searching the maximum geometry corresponding to the principle inside the exact input error boundary. Finally, two planar parallel mechanisms with different degrees of freedom are introduced as numerical examples to demonstrate the execution and effectiveness of the proposed method.

Accuracy is of great importance for deployable mechanisms because it greatly determines the performance of the antenna. The main purpose of this paper is to propose a method to analyze the angular errors of the multi-closed-loop deployable mechanism of the planar synthetic-aperture radar antenna considering link deviations and joint clearances. First, a two-link unit is explored to calculate the position errors of the joints due to the manufacturing imperfection only in the assembly process. On this basis, conducting a geometrical analysis, the formulations of the studied angular errors of two planar antenna panels after the introduction of clearances are established. Utilizing the rotatability law of the single-loop linkage, the worst-case scenario associated with a chaotic clearance-induced uncertainty is reduced to a purely geometrical problem. Accordingly, the maximum angular errors caused by these two error sources are obtained in an intuitive manner. Finally, the proposed method is demonstrated by a numerical example of the deployable mechanism and verified by experimental measurements.

Because of the existence of link errors, assembly deviations, and redundant constraints, link adjustments for deployable structures must be implemented during the assembly process to ensure accurate performance and deployable reliability. Thus, this study proposes an optimization method to conduct quantitative link adjustments for spatial multi-loop deployable structures considering redundant constraints. First, the equations of the deformation compatibility are derived based on static analysis. Second, by formulating the objective functions of surface accuracy and strain energy, a multiobjective discrete optimization model of link adjustment, subjected to the equations of the deformation compatibility and adjustable scopes of the support links, is established accordingly. Third, the optimization model is solved using the successive Taguchi approach, which uses the grey relational analysis coupled with principal component analysis as the multicriteria decision-making model, leading to the optimal adjustment for each link. Finally, a numerical example is presented to verify the validity of the proposed method.

Strict requirements for concentricity of the multistage high pressure rotor of an aero-engine are employed to guarantee performances such as vibration. Tedious and time-wasting trial assembly by adjusting the installation angles of stages is needed to meet the requirements due to the lack of effective analysis methods. Furthermore, there is no quick way to find out where the problem is and how to repair the parts when the installation-angle-adjusting method fails. This article focuses on a solution to optimize the installation angle of each stage and to make repair decisions in the assembly process. The run-out data are processed by least square method to get the spatial positions and attitudes of flanges and a deviation propagation analysis model is built by virtue of homogeneous coordinate transformation theory to predict the accumulative errors of each stage. The eccentricities of stages are evaluated with reference to the common axis and the installation angles of stages are optimized by minimizing the sum of eccentricities. Sensitivities of eccentricity, eccentric angle and parallelism of each stage are analyzed and repair decisions for parts are made to meet more strict requirements. An example of a three-stage subassembly is presented to demonstrate the solution.

Antimony (Sb) is a highly toxic heavy metal, and its amount in soil is increasing due to anthropogenic activities. Excessive Sb intake could ultimately threat human health. Recently, biochar (BC) has been accepted for remediation of Sb-contaminated soil. Understanding the interaction between BC and Sb and the effect of BC-induced changes in soil properties on immobilization/mobilization of Sb will help, therefore, to elucidate the mechanism of BC in immobilization/mobilization of Sb in contaminated soils. Wheat straw-derived BC (SBC) and fruit (apple) tree-derived BC (FBC) were obtained at the pyrolysis temperatures of 500 °C. Sb-contaminated soil was incubated with/without 0.5, 5, or 10 wt% of SBC and FBC for 130 days. Change of soil properties induced by BC was explored during the incubation. Dynamic change of Sb fraction and speciation were assessed by the sequential chemical extraction and citric acid extraction, respectively. The X-ray photoelectron spectroscopy (XPS) was employed to check elemental change of soil with 10% SBC at different incubation times. The correlation analysis and principal component analysis (PCA) were used to analyze relationship between Sb immobilization/mobilization and change of soil properties induced by BC. The obvious change of soil properties can be observed when the soil was treated with 10% BC instead of 5 and 0.5% BC. During the first 20 days with SBC incubation and 50 days with FBC incubation, Sb mobilization increased may be because of the electrostatic repulsion of functional groups in BC; OM and functional group of BC govern the reduction reactions, anionic competition, electrostatic repulsion, and biological reduction in soil induced by BC. By contrast, after 20 days with SBC incubation and 50 days with FBC incubation, the mobilization of Sb decreased, which may be attributed to the formation of complexes between Sb and OM of BC, secondary mineral precipitation, and organic complex between Sb and humus acid in soil induced by BC. It is noteworthy that the application of BC has a potential mobilizing risk for Sb and the final effect depends on BC characteristic and the change of soil properties induced by BC. The possible risks induced by BC should be considered before applying the BC to Sb-contaminated soil.

Cadmium (Cd) and lead (Pb) are toxic heavy metals that cause severe soil pollution and pose health risks to humans. It is urgent to develop feasible strategies for Pb and Cd remediation. In this study, a bacteria consortium (Enterobacter asburiae G3, Enterobacter tabaci I12 and Klebsiella variicola J2 in a 1:3:3 proportion) with optimal Cd, Pb adsorption ability was constructed and immobilized on biochar (BC)/activated carbon (AC) via physisorption and sodium alginate encapsulation. The effects of mixed bacteria-loaded BC/AC on Cd and Pb remediation were investigated. The results indicated that their application reduced the DTPA-extractable Cd, Pb in soil by 22.05%-55.84% and 31.64%-48.13%, respectively. The residual Pb, Cd were increased while the exchangeable fractions were decreased. Soil urease, catalase and phosphatase activities were enhanced and soil bacterial community was improved, indicating a soil quality improvement. Consequently, the biomass of pakchoi plants was significantly increased. Cd and Pb in the shoots of pakchoi plants were decreased by 28.68%-51.01% and 24.18%-52.87%, respectively. Collectively, the bacteria-loaded BC/AC showed superior performance than free bacteria, BC and AC alone. Our study may provide a better understanding of the development of green and sustainable materials for remediation of heavy metal by the combination of BC/AC and functional bacteria.

Time-dependent system kinematic reliability of robotic manipulators, referring to the probability of the end-effector’s pose error falling into the specified safe boundary over the whole motion input, is of significant importance for its work performance. However, investigations regarding this issue are quite limited. Therefore, this work conducts time-dependent system kinematic reliability analysis defined with respect to the pose error for robotic manipulators based on the first-passage method. Central to the proposed method is to calculate the outcrossing rate. Given that the errors in robotic manipulators are very small, the closed-form solution to the covariance of the joint distribution of the pose error and its derivative is first derived by means of the Lie group theory. Then, by decomposing the outcrossing event of the pose error, calculating the outcrossing rate is transformed into a problem of determining the first-order moment of a truncated multivariate Gaussian. Then, based on the independent assumption that the outcrossing events occur independently, the analytical formula of the outcrossing rate is deduced for the stochastic kinematic process of robotic manipulators via taking advantage of the moment generating function of the multivariate Gaussian, accordingly leading to achievement of the time-dependent system kinematic reliability. Finally, a six-degrees-of-freedom (6-DOF) robotic manipulator is used to demonstrate the effectiveness of the proposed method by comparison with the Monte Carlo simulation and finite-difference-based outcrossing rate method.