The estimation of the six-degree-of-freedom position and orientation of an end effector is of high interest in industrial robotics. High precision and data rates are important requirements when choosing an adequate measurement system. In this work, a six-degree-of-freedom pose estimation setup based on laser multilateration is described together with the measurement principle and self-calibration strategies used in this setup. In an experimental setup, data rates of 200
The estimation of the six-degree-of-freedom (6-DoF) pose (position plus orientation) of an object is a common task in different scientific areas such as robotics, automation, navigation and augmented reality. Especially in industrial robotics, high precision of the determined pose is of interest. Depending on the use case, this problem can be solved by using different types of input data, including those based on inertial sensors, interferometric length measurements and image processing.
In parallel robotics, the 6-DoF pose estimation of the end effector is a common direct kinematic problem. Different solutions using additional angular or linear sensors to solve this problem are summarized in
The different solutions proposed by the aforementioned publications have several disadvantages. Setups which include CCD imaging are limited in their data rate due to their limited frame rates and the need for extensive image processing. For online pose correction, this is a major drawback. Furthermore, systems combining different measurement techniques for pose and orientation may reach different levels of accuracy for the different values. For high-precision assembly tasks, higher absolute accuracy of the pose estimation is required. In addition, traceability to the International System of Units (SI) is complex depending on the measurement techniques used.
In the following sections, we describe a 6-DoF pose estimation setup with high accuracy and data rates which is based on multilateration using a set of tracking laser interferometers. The position and orientation of the observed target are calculated based on interferometric length measurements, allowing data rates of up to 1
Laser multilateration is a common concept for determining the unknown position of a retroreflector target by measuring a set of distances between the target and known base stations. It is commonly used in different coordinate measuring systems
The residual functions in Eq. (
Due to the lack of information about the rotation angles, the orientation of an observed target cannot be calculated from one single 3D point in space. Hence, the observed target needs to be extended to include three or more retroreflectors in a non-collinear arrangement. The target assembly is observed by at least six tracking laser interferometers in a non-coplanar setup, while each retroreflector is observed by at least one of the interferometers. The principle of such a setup is depicted in Fig.
Measurement principle of 6-DoF multilateration. A set of tracking laser interferometers is placed in a non-coplanar arrangement and measures a multitude of distances to a target assembly of three retroreflectors.
For the calculation of the pose of the moving target, precise knowledge of the system parameters is required. These parameters include the position of the laser interferometer base stations, the unknown dead paths of the interferometers, and the distances between the different retroreflectors of the target assembly. To identify the system parameters, a self-calibration procedure can be used
The equation system defined in the previous section contains
Depending on the mover used to position the target, full 6-DoF movements may not be possible. Different self-calibration strategies can be used to calibrate the system parameters for movements with 6 DoF (e.g. industrial robots), 4 DoF (e.g. Cartesian manipulators with one rotational axis) or 3 DoF (e.g. Cartesian coordinate measuring machines, CMMs). To provide a set of reasonable starting values for the optimization problem, the existence of nominal positions and orientations of the manipulator (e.g. a tool centre point or reference point) is assumed. By using such nominal values of the manipulator, the identified system parameters can be orientated in the machine coordinate system of the manipulator.
Using a 6-DoF manipulator, self-calibration can be performed by positioning the target to
In case the mover used to position the target is not capable of performing a 6-DoF movement, self-calibration can be performed in two or more steps. If one rotational axis is available, two steps are sufficient.
In the first step, all interferometers are locked into one of the retroreflectors. This configuration results in
In the second step, the interferometers are split up to allow the three retroreflectors to be observed. To change the observed retroreflectors, the laser beams of the interferometers must be interrupted, which results in a change in the dead paths. Hence, only the position of the interferometers can be used from the previous calibration step, and
Depending on the stability of the measurement setup, the calibration of the interferometer positions may need to be repeated from time to time. This may be necessary if the positions of the interferometers change, e.g. due to thermal influences or vibration.
Self-calibration of a measurement setup that has only 3 DoF is possible if movements on the different axes result in a 3D translation of the retroreflectors. This may be the case in 3D translational movement, 2D translation and rotation along one axis in the plane of translation or rotations around three orthogonal axes. The following strategy refers to a setup with 3D translation.
Similarly to the self-calibration with 4-DoF movement, the first step is used to calibrate the interferometer position by observing one retroreflector. In two consecutive steps, the same calibration is performed by observing the remaining two retroreflectors. Assuming that each retroreflector is positioned on the reference point of the mover, three different positions for each interferometer are calculated. The distances between the different interferometer positions correspond to the distances between the retroreflectors. These first three steps allow the interferometer positions and the target distances to be identified.
In the following step, the interferometers are split up again to observe all three retroreflectors. Consequently,
The result of this serial calibration of the target distances depends on the stability of the target structure and the position feedback of the mover.
The measurement principle and self-calibration strategies described form the theoretical basis for a 6-DoF multilateration setup. Experimental proof of this concept is described in the following section.
Different experiments were performed to evaluate the measurement principle described. In a static measurement setup, self-calibration strategies for 3-DoF and 4-DoF movement were compared. Additionally, dynamic measurements with 3-DoF movement were performed to investigate the system behaviour at high data rates and the evaluation run times. Both experiments are designed to confirm the concept of the 6-DoF multilateration system. Further analysis of the system, e.g. the estimation of the measurement uncertainty from a detailed Monte Carlo simulation, is pending.
A precision CMM was used as a mover and position reference with an additional rotational axis in the probing head providing 4-DoF movements. Three cat's-eye retroreflectors were mounted on the probing head in a right-angled triangle arrangement whose legs were 140 and 150
Seven tracking laser interferometers were installed along the short sides of the CMM working space in different
A grid of
In a static measurement, self-calibration with 3-DoF and 4-DoF movement was performed according to the strategies described above.
The results for the retroreflector distances are summarized in Table
Resulting retroreflector distances from self-calibration with 3-DoF and 4-DoF movement.
The difference between the calibrated target distances was below 2.6
Continuous measurement was evaluated with a trigger frequency of 200
Due to the high number of data points acquired at such frequencies, evaluation of a whole dataset, which results in equation systems of several thousand equations, was not reasonable. When considering the use case of the online pose correction of an industrial robot, only the actual position is of interest. Hence, the continuous evaluation took place by solving the equation system in a serial manner for each data point using the pose information of the previous data point as a starting value for the next data point. This required the system parameters to be calibrated beforehand. If all system parameters were known, one data point consisted of seven measured lengths and six unknowns for the target pose.
Evaluation of the continuous dataset took place using
Due to the delicate surfaces of the retroreflectors, it was not possible to calibrate the position of the retroreflectors in relation to the CMM reference point. In order to compare the results of the multilateration setup with the reference values of the CMM, alignment of the datasets was required. To align the data points from CMM and multilateration data, cross-correlation was calculated for
Figure
Relative coordinate deviation between multilateration and CMM reference for each retroreflector. The primary
Assuming a maximum deviation of 5
To explain the deviation of 20
In this work, a multilateration setup to evaluate the full 6-DoF pose of an observed target was described. The setup used seven tracking laser interferometers to calculate the position and orientation of the target at high data rates. In a proof of concept, measurement points were recorded at a frequency of 200
To identify unknown system parameters, different self-calibration strategies for movements with 3, 4 and 6 DoF were developed. Experimental evaluation of the self-calibration using 3- and 4-DoF movements showed good agreement. In a dynamic experiment, deviations between the multilateration evaluation and a reference CMM of 20
In further experiments, the capability of tracking full 6-DoF movements will be analysed. For such an experiment, establishing a precise reference with a wide measuring range for the orientation of the target will be a major challenge.
The underlying measurement data used in this work can be requested from the authors if required.
JN developed the 6-DoF multilateration evaluation and self-calibration strategies, evaluated the experiments, and wrote the article, JN and MF devised and performed the experiment, and NH and DH provided support for the experiments and the evaluation and reviewed the article.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Sensors and Measurement Science International SMSI 2020”. It is a result of the Sensor and Measurement Science International, Nuremberg, Germany, 22–25 June 2020.
This open-access publication was funded by the Physikalisch-Technische Bundesanstalt.
This paper was edited by Thomas Fröhlich and reviewed by two anonymous referees.