Homogeneous Coordinate Transformation Research Articles

Error-compensation model for simultaneous measurement of five degrees of freedom motion errors of a rotary axis

This paper introduces an error-compensation model for our measurement method to measure five motion errors of a rotary axis based on fibre laser collimation. The error-compensation model is established in a matrix form using the homogeneous coordinate transformation theory. The influences of the installation errors, error crosstalk, and manufacturing errors are analysed. The model is verified by both ZEMAX simulation and measurement experiments. The repeatability values of the radial and axial motion errors are significantly suppressed by more than 50% after compensation. The repeatability experiments of five degrees of freedom motion errors and the comparison experiments of two degrees of freedom motion errors of an indexing table were performed by our measuring device and a standard instrument. The results show that the repeatability values of the angular positioning error εz and tilt motion error around the Y axis εy are 1.2″ and 4.4″, and the comparison deviations of the two motion errors are 4.0″ and 4.4″, respectively. The repeatability values of the radial and axial motion errors, δy and δz, are 1.3 and 0.6 µm, respectively. The repeatability value of the tilt motion error around the X axis εx is 3.8″.

Kinematic models and simulations for trajectory planning in the cutting of Spatially-Arbitrary crosssections by a robotic roadheader

In this study, a boom-type roadheader’s mechanical composition was simplified into a kinematic chain composed of a series of linked translational or rotational joints, and a coordinate system was constructed for the spatial pose of the roadheaders. Homogeneous coordinate transformations and robot kinematics were used to construct a spatial pose matrix for the pose of a roadheader’s body and cutting head relative to the geodetic coordinate system. The compressive breaking forces of a cutting head during vertical and rotary cutting of coal walls was analyzed, and the trajectory planning for the cutting of arbitrary crosssections was described in detail. In addition, simulations of the limits of a roadheader in the cutting of crosssections were performed. Underground roadheader cutting trajectories were measured via the pose measurement sensors installed on the roadheader to verify the accuracy of the mathematical modeling, and it was determined that cuttings performed using manually-operated levers were highly random and had lower levels of crosssection forming quality. Therefore, trajectory planning and simulations for actual coal mine roadways as well as field tests on an S-shaped bottom-up trajectory plan for automated cutting were performed. The results of these tests indicated that the maximum absolute error on both sides of a coal roadway that has a 4500 × 4000 mm crosssection was 65 mm, which corresponds to a maximum relative control error of 1.4%. Hence, the control system developed in this work was observed to exhibit accuracy and quality that satisfy requirements for cross sections in a coalmine’s roadways, effective repeatability, and absence of cumulative deviations. Hence, the results of this research study are highly significant and are of value for practical applications.

Ground target geolocation based on digital elevation model for airborne wide-area reconnaissance system

To obtain the geographical position of the ground target accurately, a geolocation algorithm based on the digital elevation model (DEM) is developed for an airborne wide-area reconnaissance system. According to the platform position and attitude information measured by the airborne position and orientation system and the gimbal angles information from the encoder, the line-of-sight pointing vector in the Earth-centered Earth-fixed coordinate frame is solved by the homogeneous coordinate transformation. The target longitude and latitude can be solved with the elliptical Earth model and the global DEM. The influences of the systematic error and measurement error on ground target geolocation calculation accuracy are analyzed by the Monte Carlo method. The simulation results show that this algorithm can improve the geolocation accuracy of ground target in rough terrain area obviously. The geolocation accuracy of moving ground target can be improved by moving average filtering (MAF). The validity of the geolocation algorithm is verified by the flight test in which the plane flies at a geodetic height of 15,000 m and the outer gimbal angle is <47°. The geolocation root mean square error of the target trajectory is <45 and <7 m after MAF.

Inverse kinematics analysis of 6 – DOF Stewart platform based on homogeneous coordinate transformation

ABSTRACTAt present, the mainstream method is directly reduced the platform of the six series branch to six drive rods for analysis and modeling refers to inverse kinematic of the parallel six-degree-of-freedom (6-DOF) Stewart platform, and ignored the influence of the joints that excluded the driving leg on the Stewart platform, resulted in the mathematical model lack of accuracy, there is error in the actual platform. In this paper, the mathematical analysis method of inverse kinematic of the parallel six-degree-of-freedom Stewart platform based on homogeneous coordinate transformation is studied. The relationship of spatial coordinate system is worked up by using the Denavit-Hartenberg (DH) method on the upper and lower platforms and each branch of parallel 6-DOF Stewart platform. Considering the influence among the series branch joints, the transfer relationship is established by using the three-dimensional vector and the homogeneous coordinate transformation matrix between the adjacent joint coordinate systems. The length of the branch drive lever is treated as a variable, the vector equation of the platform system is deduced based on the geometric relation of the parallel 6–DOF Stewart platform. The application of this method in the kinematic inverse solution of parallel 6–DOF Stewart platform is given, verified the correctness of this method. Thinks to the data validation of the actual Stewart platform, it is proved that the proposed method can be applied to the kinematic inverse of parallel 6–DOF Stewart platform that composed of series branch.

An automatic generation method of the coordinate system for automatic assembly tolerance analysis

The automation of tolerance analysis is the ambitious goal of the researcher. In order to achieve this aim, the homogeneous coordinate transformation is a usual method for the position calculation of geometric features and parts in the assembly tolerance analysis method. The hierarchy definition and the automatic creation of the coordinate system are the key work for the automation of tolerance analysis. In this paper, the position calculation equations of the geometric feature in assembly model are described firstly, then the error propagation relation graph for geometric features in the part model is analyzed based on the datum–target mechanism of geometric tolerance. The assembly errors propagation graph of the part in the machine model is described based on the assembly relation and assembly precedence. The controlling points variation model (CPVM) of a geometric feature is used to generate and represent the position of the substitute geometry of the geometric feature, and the position parameters of substitute geometry are generated using Monte Carlo simulations. Based on the CPVM model and datum–target mechanism, the hierarchy of the coordinate system for a geometric feature is defined on the part level. Based on the assembly relations and the real part model, the assembly coordinate system between the locating part and the assembling part is established. According to these two errors propagation graphs, the homogeneous coordinate transformation matrix between the related coordinate systems is generated automatically. Finally, a case study is provided to illustrate the proposed method.

Design and investigation of a cycloid helical gear drive

In this paper, an original analysis of the geometric design of a parallel-axis cycloid helical gear drive and an investigation on its basic meshing characteristics are presented. This type of gear drive consists of a pinion with one tooth and a cycloid-helix gear in continuous line contact along the whole length of the tooth. Firstly, mathematical formulations of their tooth surfaces are derived based on the envelope theory and homogeneous coordinate transformation. Secondly, the undercutting analysis is applied to determine the design constraint to avoid the undercutting on the cycloid-helix gear tooth surface. Then, the meshing characteristics including the contact lines, surface of action, induced normal curvature are analyzed. Numerical example is presented to demonstrate the influence of design parameters on the contact stress distribution. Finally, the loaded tooth contact analysis is performed to simulate the tooth contact condition and to validate the stress distribution by the finite element method.

Using an Imaginary Planar Rack Cutter to Create a Spherical Gear Pair with Continue Involute Teeth

This paper presents an imaginary planar rack cutter having two parameters of motion to create a spherical gear pair with two degrees of freedom. The working region of the imaginary planar rack cutter is that of symmetrical straight lines. The relationship between the coordinate system of the imaginary planar rack cutter and that of the spherical gear pair having two parameters of motion is illustrated. The two-parameter family of imaginary planar rack cutter surfaces is obtained by using the homogeneous coordinate transformation matrix to transfer the coordinate system of the planar rack cutter to that of the spherical pinion or the spherical gear. Based on the two-parameter enveloping theory, the mathematical models of the spherical gear pair with two degrees of freedom were proposed. The developed mathematical model of the gear pair with two degrees of freedom can be used to analyze undercutting analysis of the imaginary planar rack cutter. Based on the undercutting equation, a minimum number of teeth of the spherical pinion are determined. A line of singular points on the spherical pinion’s teeth are displayed. Results show that a minimum number of teeth of the spherical pinion can be determined in order to avoid undercutting.

DESIGN AND ANALYSIS OF CONSTANT-BREADTH CAM MECHANISM WITH OSCILLATING FLAT-FACED FOLLOWER

A simple yet comprehensive method is presented for the design and analysis of a constant-breadth cam mechanism with an oscillating flat-faced follower. In the proposed approach, the kinematic characteristics of the cam mechanism are first derived. The cam profile is then designed using homogenous coordinate transformation and conjugate surface theory. Moreover, the sliding velocity at the cam-follower contact points is determined. Finally, the pressure angle of the constant-breadth cam mechanism and the principal curvatures of the cam are analyzed. The validity of the proposed design methodology is verified by means of motion simulations performed using CAD software.

A New Decoupling Measurement Method of Five-axis Machine Tools’ Geometric Errors Based on Cross Grid Encoder and DBB

In order to obtain all the geometric errors of five-axis CNC machine tool, a measuring strategy with cross grid encoder and double ball-bar (DBB) is proposed. According to the theory of multi-body systems, by using the homogeneous coordinate transformation method the model of geometric errors is established. The cross grid encoder is used to detect the translation geometric errors and the DBB is used to measure the rotation ones. Based on the kinematic chain and the properties of geometric errors, the translation geometric errors are measured before the rotation ones. For each axis, both PIGEs (position-independent errors) and PDGEs (position-dependent errors) are under consideration. Then a complete compensated model can be established.

Comprehensive error modelling and error compensation for complex optical free-form surface polishing platform

Four-axis polishing platform plays an important role in machining a complex optical free-form surface and the kinematic error generally determine the accuracy of the optical free-form surface fabricated by this platform. This paper analyses kinematics of the polishing platform based on the homogeneous coordinate transformation theory and the error of kinematics synthesis model is established. Based on minimum error assumption, transition matrix is obtained from machine tool coordinate system to the cutting tool coordinate system and changing rule of comprehensive error in different position and direction is gained in two-axis motion worktable. Laser interferometer is used to measure the geometric error of polishing platform, it is found that different feed rate and measured spacing had no significant effect on the positioning error of moving axis and positioning error rather than the straightness and angle error has more significant influence on the accuracy of machine tool. Compensation experiment was conducted based on integrated error model, the x-axis and z-axis positioning error was reduced by 87.37% and 90.32% after compensation and the compensation result is remarkable.

Frequency analysis of a three-translation vibration isolation device

To solve the problem of multi-dimensional vibration of some equipment effectively, a three-translation vibration isolation device based on a three-translation parallel mechanism is proposed. The kinematics model and the Jacobian matrix of the mechanism are obtained through the homogeneous coordinate transformation method. The dynamics model of the device is established by the Lagrange's equation. The comparison of three results indicates that the assumption of ignoring the mass of rods and sliding blocks is reasonable and the calculative process is correct. Frequency response analysis reflects frequency requirement for the device. The relationships between the maximal natural frequency and the parameters of the vibration isolation device are obtained, which can be used to design the device with special natural frequency, and analyse the frequency changes of some special equipment as well. The result provides guidance for the optimisation design and practical application of the vibration isolation device. [Received 21 July 2016; Accepted 16 November 2016]

Frequency Analysis of a Three-Translation Vibration Isolation Device

To solve the problem of multi-dimensional vibration of some equipment effectively, a three-translation vibration isolation device based on a three-translation parallel mechanism is proposed. The kinematics model and the Jacobian matrix of the mechanism are obtained through the homogeneous coordinate transformation method. The dynamics model of the device is established by the Lagrange's equation. The comparison of three results indicates that the assumption of ignoring the mass of rods and sliding blocks is reasonable and the calculative process is correct. Frequency response analysis reflects frequency requirement for the device. The relationships between the maximal natural frequency and the parameters of the vibration isolation device are obtained, which can be used to design the device with special natural frequency, and analyse the frequency changes of some special equipment as well. The result provides guidance for the optimisation design and practical application of the vibration isolation device. [Received 21 July 2016; Accepted 16 November 2016]

Comprehensive error modelling and error compensation for complex optical free-form surface polishing platform

Four-axis polishing platform plays an important role in machining a complex optical free-form surface and the kinematic error generally determine the accuracy of the optical free-form surface fabricated by this platform. This paper analyses kinematics of the polishing platform based on the homogeneous coordinate transformation theory and the error of kinematics synthesis model is established. Based on minimum error assumption, transition matrix is obtained from machine tool coordinate system to the cutting tool coordinate system and changing rule of comprehensive error in different position and direction is gained in two-axis motion worktable. Laser interferometer is used to measure the geometric error of polishing platform, it is found that different feed rate and measured spacing had no significant effect on the positioning error of moving axis and positioning error rather than the straightness and angle error has more significant influence on the accuracy of machine tool. Compensation experiment was conducted based on integrated error model, the x-axis and z-axis positioning error was reduced by 87.37% and 90.32% after compensation and the compensation result is remarkable.

Real-Time Multi-Target Localization from Unmanned Aerial Vehicles.

In order to improve the reconnaissance efficiency of unmanned aerial vehicle (UAV) electro-optical stabilized imaging systems, a real-time multi-target localization scheme based on an UAV electro-optical stabilized imaging system is proposed. First, a target location model is studied. Then, the geodetic coordinates of multi-targets are calculated using the homogeneous coordinate transformation. On the basis of this, two methods which can improve the accuracy of the multi-target localization are proposed: (1) the real-time zoom lens distortion correction method; (2) a recursive least squares (RLS) filtering method based on UAV dead reckoning. The multi-target localization error model is established using Monte Carlo theory. In an actual flight, the UAV flight altitude is 1140 m. The multi-target localization results are within the range of allowable error. After we use a lens distortion correction method in a single image, the circular error probability (CEP) of the multi-target localization is reduced by 7%, and 50 targets can be located at the same time. The RLS algorithm can adaptively estimate the location data based on multiple images. Compared with multi-target localization based on a single image, CEP of the multi-target localization using RLS is reduced by 25%. The proposed method can be implemented on a small circuit board to operate in real time. This research is expected to significantly benefit small UAVs which need multi-target geo-location functions.

Kinematic performance of a parabolic gear tooth with two parabolic coefficients

This paper presents design imaginary skew rack cutters having teeth of two parabolic coefficients to generate spur and helical gears. First, a mathematical model of the imaginary rack cutter having teeth of two parabolic coefficients is derived using geometric relations and coordinate transformation. After the relationship between the coordinate system of the imaginary rack cutter and that of the helical gear pair is set up, a family of imaginary rack cutter surfaces is obtained using the homogeneous coordinate transformation matrix to transfer the coordinate system of the rack cutters to that of the helical gear. Based on the gear theory, the equations of meshing between the rack cutter and the helical gear are determined. Substituting the equations of meshing into the family of imaginary rack cutter surfaces, a pinion and a gear are generated and their geometries are plotted using an in-house software package. Based on the assembly errors, tooth contact analysis is performed to determine the influences of kinematic assembly errors and the parabolic coefficients. Results show that maximum parabolic coefficient value of the imaginary skew rack cutter exists in order to avoid undercutting and it can be determined.

Stream of Variation Modeling and Analysis for Compliant Composite Part Assembly—Part I: Single-Station Processes

Composite structures are widely used due to their superior properties, such as low density, high strength, and high stiffness-to-weight ratio (Mallick, 1993, Fiber-Reinforced Composites: Materials, Manufacturing, and Design, Marcel Dekker, New York). However, the lack of methodologies for variation modeling and analysis of composite part assembly has imposed a significant constraint on developing dimensional control for composite assembly processes. This paper develops a modeling method to predict assembly deviation for compliant composite parts in a single-station assembly process. The approach is discussed in two steps: considering the part manufacturing error (PME) only and considering both the PME and the fixture position error (FPE). Finite element method (FEM) and homogenous coordinate transformation are used to reveal the impact of the PME and the FPE. The validity of the method is verified with two case studies on assembly deviation prediction of two composite laminated plates considering the PME only and both the PME and the FPE, respectively. The proposed method provides the basis for assembly deviation prediction in the multistation composite assembly.

A simple and quick control strategy for a class of first-order nonholonomic manipulator

This paper presents a simple and quick control strategy for a class of first-order nonholonomic manipulator: planar n-link manipulator with a passive first joint (no gravity). The control target is driving its endpoint from any initial position to any target position. First, we reduce the planar n-link manipulator to a planar three-link one by maintaining the states (angles and angular velocities) of n-3 active links in the initial value all the time. That is, we only adjust the states of the remaining two active links in the whole control process, and these two adjusted active links are chosen to guarantee that the target position is in the reachable region of the planar n-link manipulator by using the enumeration method. Then, we divide the whole control process into two stages for the reduced planar three-link manipulator. In each stage, the manipulator becomes a planar two-link one by maintaining the states of one adjusted active link to be the constant value. The state constraint existing between the passive first link and the adjusted active link is obtained by using the integral characteristics of a planar two-link manipulator. Meantime, the geometric constraint between the position of the endpoint and angles of all joints is obtained based on the homogeneous coordinate transformation method. According to the above two kinds of constraint, the target angles of the two adjusted active links are calculated by employing the particle swarm optimization algorithm. When the two adjusted active links are controlled to their target angles in turn, the control target of the planar n-link manipulator is completed. Finally, simulation results demonstrate that the proposed control strategy is valid and rapid.

Design and kinematics evaluation of a gear pair with asymmetric parabolic teeth

This paper presents design of imaginary spur and skew rack cutters having teeth of asymmetric parabolic profiles to generate spur and helical gear pairs, respectively. First, a mathematical model of the imaginary rack cutter having teeth of an asymmetric parabolic profile is derived using geometric relations and coordinate transformation. After the relationship between the coordinate system of the imaginary rack cutter and that of the gear pair is set up, a family of imaginary rack cutter surfaces is obtained using the homogeneous coordinate transformation matrix to transfer the coordinate system of the rack cutter to that of the gear pair. Substituting the equations of meshing into the family of imaginary rack cutter surfaces, a pinion and a gear are generated and their geometries are plotted using an in-house software package. Based on the derived mathematical model of the gear pair with assumed assembly errors, tooth contact analysis is performed to determine the influences of kinematic errors and the asymmetric tooth profile. A helical gear pair made by CNC machining is used to verify the influences of asymmetric parabolic tooth profiles on contact conditions and kinematic performance.

Note: Optimal choice of the reflector by phase analysis for heterodyne interferometric roll angle measurement.

The choice of an actual reflector instead of an ideal or perfect reflector has to be dealt with for heterodyne-interferometric roll angle measurement methods. The candidates of regular reflectors such as corner cube prism and right angle prism are analyzed by the homogeneous coordinate transformation method, which provides a theoretical basis for the determination of the actual reflector. Furthermore, the influences of the phase loss of the determined reflector are strictly argued due to non-perfect or natural characteristics itself. Several tactics are considered to solve this problem by phase analysis. By contrast, it is an optimal choice that mirrors-based right angle prism can improve evidently the phase loss by experiment.

A New Algorithm for the Identification of CNC Geometric Errors

This paper proposes a geometric error model for the three-axis machine tool using the homogeneous transformation matrix. An error-prediction model for the three-axis CNC machine tools is presented. This model takes the geometric errors into consideration, using the homogeneous coordinate transformation matrix(HTM) to analyze the geometric error model, which can change the tool's complex position and orientation changes in space into matrix calculations to make the model simpler and more intuitive. A simple method using the cubic polynomial functions to fit the geometric error components is proposed. Using this method, the high-order, non-linear spatial geometric problem can be transformed into algebraic equation. To large extent, the difficulty of calculation is reduced. The QC20 ball-bar is used to measure, analyze and separate all the geometric error components. A new kind of particle swarm algorithm based on the genetic algorithm is developed to solve the algebra equation.