scholarly journals WRITING OUT OF FORMULAS FOR CALCULATING FORCES IN THE JOINTS OF MANIPULATORS IN STATICS

2021 ◽  
Vol 21 (3) ◽  
pp. 47-58
Author(s):  
S.G. Pudovkina ◽  
◽  
A.I. Telegin
Keyword(s):  
Coordinate System ◽  
Degrees Of Freedom ◽  
Industrial Robots ◽  
Coordinate Systems ◽  
Systematic Analysis ◽  
Stationary Coordinate System ◽  
Reaction Forces ◽  
Vector Mechanics ◽  
Assembly Operations ◽  
Analytical Verification

The problem of bulkiness of mathematical models of manipulative systems of industrial robots is solved. Here we consider formulas for calculating static reactions in joints and formulas for active forces that balance the forces of gravity acting on the manipulator's bodies in its stationary state. The manipulator can be in such a state when it is before capturing the object of manipulation and releasing it, or when it is performing some assembly operations, or it is during spot welding and in slow (quasi-static) arc-welding and painting processes. Aim. The aim is to derive general recur-rence and finite formulas for calculating the reaction forces in joints and their projections to the ax-es of the coordinate system rigidly connected with the selected body. Express the formulas of force projections in terms of guiding cosines and justify their optimality in terms of the minimum of arithmetic operations. Derive general inverse recurrence formulas for writing out the guide cosines of the axes associated with the moving bodies of the coordinate system with respect to the stationary coordinate system. Research methods. The methods of research relate to vector mechanics and sys-tems analysis, and the algorithmization of calculations by reducing them to the use of recurrent formulas. Results. A systematic analysis of general formulas, in which all possible regular expres-sions are highlighted which are corresponding unambiguously to the kinematic parameters of ma-nipulators, is performed. These regular expressions are used in software for analytical modeling of manipulator, in particular, for the analytical solution of problems of statics of a manipulator. The method of analytical verification of the prescribed formulas is described. The tasks of writing out optimal formulas for calculating the projections of static reaction forces in joints have been solved. And the tasks of writing out optimal formulas for calculating active forces in progressive joints of universal manipulators with six degrees of freedom, operating in Cartesian, cylindrical, spherical and angular coordinate systems, have been solved also. Analytical verification of the derived equations of stat-ics is performed. Examples of the reuse of the derived formulas for manipulators with the same kin-ematic schemes of their subsystems. Conclusion. Expressions of the equations of statics of manipu-lators through the guide cosines of the axes of the associated coordinate systems of their bodies al-low us to write these equations through the known parameters of body orientation. The recurrent formulas for calculating directional cosines allows to use recursive functions in their software im-plementation, i.e. to increase the computational efficiency of the software.

scholarly journals A coordinate-system-independent method for comparing joint rotational mobilities

2020 ◽  
Vol 223 (18) ◽  
pp. jeb227108
Author(s):  
Armita R. Manafzadeh ◽  
Stephen M. Gatesy
Keyword(s):  
Coordinate System ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Rotational Mobility ◽  
Coordinate Systems ◽  
Map Projection ◽  
Fold Reduction ◽  
Euler Space ◽  
Rotational Degrees Of Freedom ◽  
Alternative Spaces

ABSTRACTThree-dimensional studies of range of motion currently plot joint poses in a ‘Euler space’ whose axes are angles measured in the joint's three rotational degrees of freedom. Researchers then compute the volume of a pose cloud to measure rotational mobility. However, pairs of poses that are equally different from one another in orientation are not always plotted equally far apart in Euler space. This distortion causes a single joint's mobility to change when measured based on different joint coordinate systems and precludes fair comparison among joints. Here, we present two alternative spaces inspired by a 16th century map projection – cosine-corrected and sine-corrected Euler spaces – that allow coordinate-system-independent comparison of joint rotational mobility. When tested with data from a bird hip joint, cosine-corrected Euler space demonstrated a 10-fold reduction in variation among mobilities measured from three joint coordinate systems. This new quantitative framework enables previously intractable, comparative studies of articular function.

scholarly journals Development a model of robot movement with five degrees of freedom for a warehouse

2021 ◽  
Vol 5 (1(61)) ◽  
pp. 12-17
Author(s):  
Volodymyr Shvets ◽  
Viktor Tkachov
Keyword(s):  
Coordinate System ◽  
Degrees Of Freedom ◽  
Kinematic Model ◽  
Emergent Behavior ◽  
Position Vector ◽  
Coordinate Systems ◽  
Robot Calibration ◽  
Initial State ◽  
The Matrix ◽  
Robot Position

The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy. The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy.

Reference Coordinate System Requirements for Geophysics

1975 ◽  
Vol 26 ◽  
pp. 87-92
Author(s):  
P. L. Bender
Keyword(s):  
Coordinate System ◽  
Polar Motion ◽  
Main Part ◽  
Measurement Techniques ◽  
Reference Points ◽  
Angular Motion ◽  
Coordinate Systems ◽  
Axis Of Rotation ◽  
The Earth ◽  
Fixed System

AbstractFive important geodynamical quantities which are closely linked are: 1) motions of points on the Earth’s surface; 2)polar motion; 3) changes in UT1-UTC; 4) nutation; and 5) motion of the geocenter. For each of these we expect to achieve measurements in the near future which have an accuracy of 1 to 3 cm or 0.3 to 1 milliarcsec.From a metrological point of view, one can say simply: “Measure each quantity against whichever coordinate system you can make the most accurate measurements with respect to”. I believe that this statement should serve as a guiding principle for the recommendations of the colloquium. However, it also is important that the coordinate systems help to provide a clear separation between the different phenomena of interest, and correspond closely to the conceptual definitions in terms of which geophysicists think about the phenomena.In any discussion of angular motion in space, both a “body-fixed” system and a “space-fixed” system are used. Some relevant types of coordinate systems, reference directions, or reference points which have been considered are: 1) celestial systems based on optical star catalogs, distant galaxies, radio source catalogs, or the Moon and inner planets; 2) the Earth’s axis of rotation, which defines a line through the Earth as well as a celestial reference direction; 3) the geocenter; and 4) “quasi-Earth-fixed” coordinate systems.When a geophysicists discusses UT1 and polar motion, he usually is thinking of the angular motion of the main part of the mantle with respect to an inertial frame and to the direction of the spin axis. Since the velocities of relative motion in most of the mantle are expectd to be extremely small, even if “substantial” deep convection is occurring, the conceptual “quasi-Earth-fixed” reference frame seems well defined. Methods for realizing a close approximation to this frame fortunately exist. Hopefully, this colloquium will recommend procedures for establishing and maintaining such a system for use in geodynamics. Motion of points on the Earth’s surface and of the geocenter can be measured against such a system with the full accuracy of the new techniques.The situation with respect to celestial reference frames is different. The various measurement techniques give changes in the orientation of the Earth, relative to different systems, so that we would like to know the relative motions of the systems in order to compare the results. However, there does not appear to be a need for defining any new system. Subjective figures of merit for the various system dependon both the accuracy with which measurements can be made against them and the degree to which they can be related to inertial systems.The main coordinate system requirement related to the 5 geodynamic quantities discussed in this talk is thus for the establishment and maintenance of a “quasi-Earth-fixed” coordinate system which closely approximates the motion of the main part of the mantle. Changes in the orientation of this system with respect to the various celestial systems can be determined by both the new and the conventional techniques, provided that some knowledge of changes in the local vertical is available. Changes in the axis of rotation and in the geocenter with respect to this system also can be obtained, as well as measurements of nutation.

2.2. Working Group 2: Conceptual Definitions of Reference Coordinate Systems for Earth Dynamics

1975 ◽  
Vol 26 ◽  
pp. 21-26
Keyword(s):  
Coordinate System ◽  
Working Group ◽  
General Character ◽  
Coordinate Systems ◽  
Reference Coordinate System ◽  
Group 2 ◽  
Definition Of ◽  
Earth Dynamics

An ideal definition of a reference coordinate system should meet the following general requirements:1. It should be as conceptually simple as possible, so its philosophy is well understood by the users.2. It should imply as few physical assumptions as possible. Wherever they are necessary, such assumptions should be of a very general character and, in particular, they should not be dependent upon astronomical and geophysical detailed theories.3. It should suggest a materialization that is dynamically stable and is accessible to observations with the required accuracy.

Technique for relation global reference system and local realization of global reference system by continuously operated reference stations

2020 ◽  
Vol 962 (8) ◽  
pp. 24-37
Author(s):  
V.E. Tereshchenko
Keyword(s):  
Coordinate System ◽  
Russian Federation ◽  
Reference System ◽  
Main Part ◽  
Precise Point Positioning ◽  
Global Coordinate System ◽  
Coordinate Systems ◽  
Novosibirsk Region ◽  
The Russian Federation

The article suggests a technique for relation global kinematic reference system and local static realization of global reference system by regional continuously operated reference stations (CORS) network. On the example of regional CORS network located in the Novosibirsk Region (CORS NSO) the relation parameters of the global reference system WGS-84 and its local static realization by CORS NSO network at the epoch of fixing stations coordinates in catalog are calculated. With the realization of this technique, the main parameters to be determined are the speed of displacement one system center relativly to another and the speeds of rotation the coordinate axes of one system relatively to another, since the time evolution of most stations in the Russian Federation is not currently provided. The article shows the scale factor for relation determination of coordinate systems is not always necessary to consider. The technique described in the article also allows detecting the errors in determining the coordinates of CORS network in global coordinate system and compensate for them. A systematic error of determining and fixing the CORS NSO coordinates in global coordinate system was detected. It is noted that the main part of the error falls on the altitude component and reaches 12 cm. The proposed technique creates conditions for practical use of the advanced method Precise Point Positioning (PPP) in some regions of the Russian Federation. Also the technique will ensure consistent PPP method results with the results of the most commonly used in the Russian Federation other post-processing methods of high-precision positioning.

Analysis and verification of harmonic interaction between DDWTs based on αβ stationary coordinate system

2020 ◽  
Vol 14 (10) ◽  
pp. 1893-1901
Author(s):  
Pingping Han ◽  
Longjian Wang ◽  
Sheng Dou ◽  
Lei Wang ◽  
Rui Bi ◽  
...  
Keyword(s):  
Coordinate System ◽  
Stationary Coordinate System ◽  
Harmonic Interaction

scholarly journals Research of Calibration Method for Industrial Robot Based on Error Model of Position

2021 ◽  
Vol 11 (3) ◽  
pp. 1287
Author(s):  
Tianyan Chen ◽  
Jinsong Lin ◽  
Deyu Wu ◽  
Haibin Wu
Keyword(s):  
Coordinate System ◽  
Industrial Robot ◽  
Calibration Method ◽  
Error Model ◽  
Position Error ◽  
Industrial Robots ◽  
Positioning Error ◽  
Robot Position ◽  
General Measurement ◽  
Parameter Error

Based on the current situation of high precision and comparatively low APA (absolute positioning accuracy) in industrial robots, a calibration method to enhance the APA of industrial robots is proposed. In view of the "hidden" characteristics of the RBCS (robot base coordinate system) and the FCS (flange coordinate system) in the measurement process, a comparatively general measurement and calibration method of the RBCS and the FCS is proposed, and the source of the robot terminal position error is classified into three aspects: positioning error of industrial RBCS, kinematics parameter error of manipulator, and positioning error of industrial robot end FCS. The robot position error model is established, and the relation equation of the robot end position error and the industrial robot model parameter error is deduced. By solving the equation, the parameter error identification and the supplementary results are obtained, and the method of compensating the error by using the robot joint angle is realized. The Leica laser tracker is used to verify the calibration method on ABB IRB120 industrial robot. The experimental results show that the calibration method can effectively enhance the APA of the robot.

scholarly journals The Celestial Reference System in Relativistic Framework

1990 ◽  
Vol 141 ◽  
pp. 99-110
Author(s):  
Han Chun-Hao ◽  
Huang Tian-Yi ◽  
Xu Bang-Xin
Keyword(s):  
Coordinate System ◽  
Reference Frame ◽  
Reference System ◽  
Light Deflection ◽  
Coordinate Systems ◽  
Definition Of ◽  
Celestial Sphere ◽  
Celestial Reference System ◽  
Relativistic Framework

The concept of reference system, reference frame, coordinate system and celestial sphere in a relativistic framework are given. The problems on the choice of celestial coordinate systems and the definition of the light deflection are discussed. Our suggestions are listed in Sec. 5.

Quaternion Algorithm for Initial Alignment of Strapdown INS Using the A. N. Tikhonov Regularization Method

2021 ◽  
Vol 22 (4) ◽  
pp. 217-224
Author(s):  
Yu. N. Chelnokov ◽  
A. V. Molodenkov
Keyword(s):  
Angular Velocity ◽  
Coordinate System ◽  
Regularization Method ◽  
Numerical Experiments ◽  
Tikhonov Regularization Method ◽  
Coordinate Systems ◽  
Initial Alignment ◽  
Unknown Variable ◽  
Absolute Angular Velocity ◽  
Acceleration Vector

For the functioning of algorithms of inertial orientation and navigation of strapdown inertial navigation system (SINS), it is necessary to conduct a mathematical initial alignment of SINS immediately before the operation of these algorithms. An efficient method of initial alignment (not calibration!) of SINS is the method of vector matching. Its essence is to determine the relative orientation of the instrument trihedron Y (related to the unit of SINS sensors) and the reference trihedron X according to the results of measuring the projections of at least two non-collinear vectors of the axes on both trihedrons. We address the estimation of the initial orientation of the object using the method of gyrocompassing, which is a form of vector matching method. This initial alignment method is based upon using the projections of the apparent acceleration vector a and the absolute angular velocity vector ω of the object in the coordinate systems X and Y. It is assumed that the three single-axis accelerometers and the three gyroscopes (generally speaking, the three absolute angular velocity sensors of any type), which measure the projections of the vectors a and ω, are installed along the axes of the instrument coordinate system Y. If the projections of the same vectors on the axes of the base coordinate system X are known, then it is possible to estimate the mutual orientation of X and Y trihedrons. We are solving the problem of the initial alignment of SINS for the case of a fixed base, when the accelerometers measure the projection gi (i = 1, 2, 3) of the gravity acceleration vector g, and the gyroscopes measure the projections u i of the vector u of angular velocity of Earth’s rotation on the body-fixed axes. The projections of the same vectors on the axes of the normal geographic coordinate system X are also estimated using the known formulas. The correlation between the projections of the vectors u and g in X and Y coordinate system is given by known quaternion relations. In these relations the unknown variable is the orientation quaternion of the object in the X coordinate system. By separating the scalar and vector parts in the equations, we obtain an overdetermined system of linear algebraic equations (SLAE), where the unknown variable is the finite rotation vector θ, which aligns the X and Y coordinate systems (it is assumed that there is no half-turn of the X coordinate system with respect to the Y coordinate system). Thus, the mathematical formulation of the problem of SINS initial alignment by means of gyrocompassing is to find the unknown vector θ from the derived overdetermined SLAE. When finding the vector θ directly from the SLAE (algorithm 1) and data containing measurement errors, the components of the vector q are also determined with errors (especially the component of the vector θ, which is responsible for the course ψ of an object). Depending on the pre-defined in the course of numerical experiments values of heading ψ, roll ϑ, pitch γ angles of an object and errors of the input data (measurements of gyroscopes and accelerometers), the errors of estimating the heading angle Δψ of an object may in many cases differ from the errors of estimating the roll Δϑ and pitch Δγ angles by two-three (typically) or more orders. Therefore, in order to smooth out these effects, we have used the A. N. Tikhonov regularization method (algorithm 2), which consists of multiplying the left and right sides of the SLAE by the transposed matrix of coefficients for that SLAE, and adding the system regularization parameter to the elements of the main diagonal of the coefficient matrix for the newly derived SLAE (if necessary, depending on the value of the determinant of this matrix). Analysis of the results of the numerical experiments on the initial alignment shows that the errors of estimating the object’s orientation angles Δψ, Δϑ, Δγ using algorithm 2 are more comparable (more consistent) regarding their order.

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