Implementation and use of the Levenberg-Marquard algorithm in the problems of calibration of robotic manipulators

The well-known problem of calibration of an arbitrary robotic manipulator, which is formulated in the most general form, is considered. To solve the direct problem of kinematics, an alternative to the Denavit-Hartenberg method, a universal analytical description of the kinematic scheme, taking into account possible errors in the manufacture and assembly of robot parts, is proposed. At the same time, a universal description of the errors in the orientation of the axes of the articulated joints of the links is proposed. On the basis of such a description, the direct and inverse problem of kinematics of robots as spatial mechanisms can be solved, taking into account the distortions of dimensions, the position of the axes of the joints and the positions of the zeros of the angles of their rotation. The problem of calibration of manipulators is formulated as a problem of the least squares method. Analytical formulas of the objective function of the least squares method for solving the problem are obtained. Expressions for the gradient vector and the Hessian of the objective function for the direct algorithm, Newton-Gauss and Levenberg-Marquardt algorithms are obtained by analytical differentiation using a special computer algebra system KiDyM. The procedures in the C ++ language for calculating the elements of the gradient and hessian are automatically generated. On the example of a projected angular 6-degree robot-manipulator, the results of modeling the solution to the problem of its calibration, that is, determination of 36 unknown angular and linear errors, are presented. A comparison is made of the solution of the calibration problem for simulated 64 and 729 experiments, in which the generalized coordinates - the angles in the joints took the values ±90° and -90°, 0, +90°.

Design and implementation of the movement gauge for rail clearance based on combination measurement method

Purpose As the infrastructure of the railway, the rail could sink or deform to different degrees due to the impact of train operation or the geological changing force for years, which will lead to the possibility that the facilities on both sides of the rail invade the rail clearance and bring hidden dangers to the safe operation of the train. The purpose of this paper is to design the gauge to measure the clearance parameters of rail. Design/methodology/approach Aiming at the problem, the gauge for clearance measurement was designed based on a combination measurement method in this paper. It consists of the measurement box and the rail measurement vehicle, which integrates a laser displacement sensor, inclination sensor, gauge sensor and mileage sensor. The measurement box was placed outside the rail vehicle. Through the design of a hardware circuit and software system, the movement measurement of the clearance parameters was realized. Findings In this paper, the measurement equations of horizontal distance and vertical height were established, the optimal solutions of the structural parameters in the equations were obtained by Levenberg–Marquardt method, then the parameter calibration problem was also solved. Originality/value The gauge has high precision; its measurement uncertainty reaches 1.27 mm. The gauge has manual and automatic working modes, which are convenient to operate and have practical popularization value.

Calibration and Validation of a Cone Crusher Model with Industrial Data

This paper reports the calibration and validation of a cone crusher model using industrial data. Usually, there are three calibration parameters in the condensed breakage function; by contrast, in this work, every entry of the lower triangular breakage function matrix is considered a calibration parameter. The calibration problem is cast as an optimization problem based on the least squares method. The results show that the calibrated model is able to fit the validation datasets closely, as seen from the low values of the objective function. Another significant advantage of the proposed approach is that the model can be calibrated on data that are usually available from industrial operation; no additional laboratory tests are required. Calibration and validation tests on datasets collected from two different mines show that the calibrated model is a strong candidate for use in various dynamic simulation applications, such as control system design, equipment sizing, operator training, and optimization of crushing circuits.

fSDE: efficient evolutionary optimisation for many-objective aero-engine calibration

Engine calibration aims at simultaneously adjusting a set of parameters to ensure the performance of an engine under various working conditions using an engine simulator. Due to the large number of engine parameters to be calibrated, the performance measurements to be considered, and the working conditions to be tested, the calibration process is very time-consuming and relies on the human knowledge. In this paper, we consider non-convex constrained search space and model a real aero-engine calibration problem as a many-objective optimisation problem. A fast many-objective evolutionary optimisation algorithm with shift-based density estimation, called fSDE, is designed to search for parameters with an acceptable performance accuracy and improve the calibration efficiency. Our approach is compared to several state-of-the-art many- and multi-objective optimisation algorithms on the well-known many-objective optimisation benchmark test suite and a real aero-engine calibration problem, and achieves superior performance. To further validate our approach, the studied aero-engine calibration is also modelled as a single-objective optimisation problem and optimised by some classic and state-of-the-art evolutionary algorithms, compared to which fSDE not only provides more diverse solutions but also finds solutions of high-quality faster.

Multi-Objective Surrogate-Assisted Stochastic Optimization for Engine Calibration

A surrogate assisted optimization approach is an attractive way to reduce the total computational budget for obtaining optimal solutions. This makes it special for its application to practical optimization problems requiring a large number of expensive evaluations. Unfortunately, all practical applications are affected by measurement noises, and not much work has been done to address the issue of handling stochastic problems with multiple objectives and constraints. This work tries to bridge the gap by demonstrating three different frameworks for performing surrogate assisted optimization on multiobjective constrained problems with stochastic measurements. To make the algorithms applicable to real-world problems, heteroscedastic (non-uniform) noise is considered for all frameworks. The proposed algorithms are first validated on several multiobjective numerical problems (unconstrained and constrained) to verify their effectiveness, and then applied to the diesel engine calibration problem, which is expensive to perform and has measurement noises. A GT-SUITE model is used to perform the engine calibration study. Three control parameters, namely variable geometry turbocharger vane position, exhaust-gas-recirculating valve position, and the start of injection, are calibrated to obtain the trade-off between engine fuel efficiency performance (brake specific fuel consumption) and NOx emissions within the constrained design space. The results show that all three proposed extensions can handle the problems well with different measurement noise levels at a reduced evaluation budget. For the engine calibration problem, a good approximation of the optimal region is observed with more than 80\% reduction in evaluation budget for all the proposed methodologies.

Self-Calibration for the General Cable-Driven Serial Manipulator with Multi-Segment Cables

This paper focuses on the kinematic calibration problem for the general cable-driven serial manipulator (CDSM) with multi-segment cables to improve its motion control accuracy. Firstly, to fully describe the calibration parameters of cables, links, joint positions, and the transmission system, this paper proposes a new cable routing description method named cable-routing configuration struct (CRCS), which provides a complete set of parameters to be calibrated for the proposed self-calibration algorithm. Then, a self-calibration algorithm for CDSM with motor incremental encoders is proposed, which can calibrate the robot at one time only using sufficient measured motor and joint positions. Its premise, the initial cable length, needs to be calibrated. Finally, the parameters of a three-DOF (degree of freedom) six-cable CDSM were described using the CRCS description method, and a comparative experiment was carried out on the same motion controller using the parameters before and after calibration. The experiment results of trajectory tracking error showed that the calibration parameters obtained by the proposed calibration algorithm can significantly improve the motion control accuracy of the three-DOF six-cable CDSM. This verified the correctness and effectiveness of the proposed calibration algorithm.

Image Distortion and Rectification Calibration Algorithms and Validation Technique for a Stereo Camera

This paper focuses on the calibration problem using stereo camera images. Currently, advanced vehicle systems such as smart cars and mobile robots require accurate and reliable vision in order to detect obstacles and special marks around. Such modern vehicles can be equipped with sensors and cameras together or separately. In this study, we propose new methodologies of stereo camera calibration based on the correction of distortion and image rectification. Once the calibration is complete, the validation of the corrections is presented followed by an evaluation of the calibration process. Usually, the validation section is not jointly considered with the calibration in other studies. However, the mass production of cameras widely uses the validation techniques in calibrations owned by manufacturing businesses. Here, we aim to present a single process for the calibration and validation of stereo cameras. The experiment results showed the disparity maps in comparison with another study and proved that the proposed calibration methods can be efficient.

Calibrating Range Measurements of Lidars Using Fixed Landmarks in Unknown Positions

We consider the problem of calibrating range measurements of a Light Detection and Ranging (lidar) sensor that is dealing with the sensor nonlinearity and heteroskedastic, range-dependent, measurement error. We solved the calibration problem without using additional hardware, but rather exploiting assumptions on the environment surrounding the sensor during the calibration procedure. More specifically we consider the assumption of calibrating the sensor by placing it in an environment so that its measurements lie in a 2D plane that is parallel to the ground. Then, its measurements come from fixed objects that develop orthogonally w.r.t. the ground, so that they may be considered as fixed points in an inertial reference frame. Moreover, we consider the intuition that moving the distance sensor within this environment implies that its measurements should be such that the relative distances and angles among the fixed points above remain the same. We thus exploit this intuition to cast the sensor calibration problem as making its measurements comply with this assumption that “fixed features shall have fixed relative distances and angles”. The resulting calibration procedure does thus not need to use additional (typically expensive) equipment, nor deploy special hardware. As for the proposed estimation strategies, from a mathematical perspective we consider models that lead to analytically solvable equations, so to enable deployment in embedded systems. Besides proposing the estimators we moreover analyze their statistical performance both in simulation and with field tests. We report the dependency of the MSE performance of the calibration procedure as a function of the sensor noise levels, and observe that in field tests the approach can lead to a tenfold improvement in the accuracy of the raw measurements.