Object Localization by Construction of an Asymmetric Isobody of the Magnetic Gradient Tensor Contraction Using Two Identical Permanent Magnets

This article deals with predictive maintenance of infrastructure objects, focusing on the inspection of bridges. To achieve this goal, unmanned aerial vehicles (UAVs) place measuring units for data collection at different points of the bridge. This work deals with the question of how these measuring units can be found and removed by them later. For this purpose, a magnetic localization method based on the scalar triangulation and ranging (STAR) method is presented. In contrast to the STAR method, which only detects the center of a magnetic target, a modified version called scalar ranging with improved orientation detection (SRIOD) is implemented here, where the orientation of the magnetic target can also be detected. This is essential information if the UAV is to grab the measuring unit with a robotic arm. In this regard, two identical permanent magnets (PMs) are integrated into the measuring unit, whose gradient tensor contraction of the magnetic flux density distribution has a directed asymmetric surface when considering the contours of constant values. This creates a magnetic invisible reference plane that can be detected with a magnetic gradiometer. The method is verified both in simulation and experimentally in a laboratory setup.


I. INTRODUCTION
R OAD and rail infrastructure buildings are subject to inevitable aging processes, the progress of which must be monitored at regular intervals to minimize the risk to human life as much as possible. Late detection of hidden age-related damage can have fatal consequences, as demonstrated by the tragic accident in Genoa, Italy, in 2018, in which 43 people died when the busy Polcevera Bridge collapsed. One of the challenges of inspection is the sheer size of the infrastructure. For example, the aforementioned Polcevera Bridge is nearly 1200 m long, so inspection can take weeks, resulting in high maintenance costs. In addition, the inspection is very dangerous for the specialists, as they often have to work at great heights and parts of the infrastructure are very inaccessible. The ASAP 1 project was launched to address these challenges. It aims to develop autonomous robotic systems that can reach the most remote locations to perform the important inspection tasks more cost-effectively. For example, one important task is to place various sensors over the object to collect the data. Combined with numerical infrastructure models, the collected data will allow accurate prediction and assessment of the remaining load-bearing capacity and lifetime of the infrastructure. Deploying robots on infrastructure inspection is a rapidly growing research topic. To perform inspection tasks, robots are equipped with measurement units (MUs) consisting of various sensors connected to data loggers. For  [1], corrosion inspection of reinforced concrete is proposed using ground-penetrating radar in a wall-crawling robot to generate data. In [2], a method for inspection of bridge concrete based on passive infrared thermography is proposed. For this purpose, an unmanned aerial vehicle (UAV) was equipped with an infrared camera. Other UAVs have ultrasonic sensor systems to detect damage in concrete [3]. Laser-based, so-called LiDAR systems are also used, for example, in the inspection of wind turbine blades [4]. In [5], stereo cameras are mounted on a UAV to check the structural integrity of bridges. Other situations require the assembly of objects directly to the infrastructure. For example, Cacace et al. [6] present strategies for bird diverter and electrical spacer installation on power lines by UAVs. In this case, the UAV position information can be obtained from the magnetic field generated by the power lines or by using a camera. As mentioned above, object mounting is also a part of the project ASAP which requires the assembly of the MUs directly on the bridge, which then records and collects vibrations over a longer period of time, for example. For this purpose, intelligent UAVs are being developed that can cooperatively perform such assembly [7]. While Ivanovic et al. [7] addressed the question of how such an assembly can be performed by UAVs on bridges, this article presents a concept of how a UAV can find and remove the MU again, which is an unresolved issue. In this context, the coarse localization of the MU can already be considered solved, since various navigation systems fusing LiDAR, GPS, and ultrawide band (UWB) are available. These are usually combined with sensor fusion and simultaneous localization and mapping algorithms (SLAM). An overview of this concept can be found in [8]. However, these systems, which are within an affordable range, achieve an accuracy of about 10 cm, as shown by the example of UWB [9]. If one imagines that the dimensions of the MU are of this order of magnitude, it becomes clear that the level of accuracy required for a robotic arm to correctly grip the MU without damaging it is too low. Also, the fingers of the gripper could be damaged if they hit the wall. Therefore, a solution for fine localization is required, which is the subject of this article.

A. State of the Art
Since the MU records data over a relatively long period of time at remote locations, it must have a self-sufficient power supply-typically a lithium-ion battery-and also consume extremely little energy. Therefore, the variant in which the MU emits a signal to be detected and located by the UAV generally represents a disadvantage. A better variant, pursued in this work, is to use a small, strong, and lightweight permanent magnet (PM) that can be easily integrated into the MU. It passively provides a constant magnetic signal to the aerial robot, which can be equipped with several magnetometers that sample this signal and guide the end-effector toward the target. The concept is shown in Fig. 1. There are numerous scientific publications that have developed various methods for the precise location of magnetic targets. All are based on the fact that targets at a distance equal to three times their longest dimension can be mathematically modeled as magnetic dipoles. The same principle applies when the target is a war mine or a PM. In [10], for example, the motion of a capsule endoscope inside the human body containing a small PM is tracked using a cubic array of three-axis magnetoresistive sensors placed outside the human body. In [11], a cubic sensor array with symmetric sensor positions is also used to cancel the geomagnetic field and localize a medical microbot in vivo. The microbot contains two perpendicular PMs as a magnetic source. Its position and orientation are determined by solving the inverse dipole equation using the nonlinear least squares algorithm. Other research uses the Levenberg-Marquardt or the Powell algorithm to solve the inverse dipole equation [12], [13]. All of these methods have in common that they directly measure the magnetic flux density of the magnetic source and thus also the superimposed geomagnetic field, which leads to localization errors. This difficulty is solved by a calibration method that is often combined with a symmetric cubic array of sensors surrounding the magnetic target. However, such an arrangement is not feasible for all applications, such as in the underlying case. To minimize the influence of the Earth's field, several approaches have been developed that measure the magnetic gradient tensor components of the magnetic target instead of its magnetic flux density distribution. This takes advantage of the fact that the gradient of the Earth's field is much smaller than that of a typical magnetic target [14], [15], [16]. This method is also limited in its application because the magnetic gradient tensor depends directly on the attitude of the measurement system relative to the magnetic target. An improved approach was developed in [17] and [18]. Here, the rotationally invariant contraction of the magnetic gradient tensor is used to locate and navigate to the magnetic target. The approach is referred to as the Scalar Triangulation And Ranging (STAR) method and serves as the starting point for the approach proposed in this article.

B. Article Contributions and Organization
As the primary contribution of the article, we propose a new variant based on the STAR method that uses two PMs in the MU and provides additional information about the attitude of the gripper. The second contribution is the development of a novel magnetic gradiometer structure with a relatively small size. It is optimized for the task of uniquely localizing magnetic targets at close range. Due to the good signal-tonoise ratio (SNR), very low-cost magnetic field sensors are used for the first time, which significantly reduces the overall cost.
The article is organized as follows. Section II introduces the traditional STAR method and shows which of its characteristics are disadvantageous for the planned application in ASAP. Based on this, a new method Scalar Ranging with Improved Orientation Detection (SRIOD) using the same scalar ranging concept is derived in Section III, which guarantees reliable and efficient localization. The method is verified by simulations. In Section IV, the method is verified again using an experimental setup with a robotic arm, for which a new gradiometer measurement system was specially developed.

II. STAR METHOD
The magnetic flux density distribution of a magnetic target in a vacuum modeled as a dipole can be described by the following equation: with the magnetic field constant µ 0 , the magnetic moment ⃗ M of the magnetic target, and ⃗ r = (x, y, z) T describing an arbitrary point in space relative to its origin. The magnetic gradient tensor is Contours of equal C T value of 1.629 mT/m at a distance of 0.25 m from a magnetic target with a magnetic moment of 15 Am 2 . Isobody calculated with (2) and (3). where the Jacobian matrix elements G i j ≡ ∂ B i /∂ j are represented by The indices i and j take the values of the Cartesian coordinates x, y, or z, where i indicates the Cartesian vector component and j is the direction in which the component is being derived. The symbol δ i j is the Kronecker delta.
From this, the rotationally invariant gradient tensor contraction in an arbitrary point ⃗ r can be calculated via The contours of equal C T value-from now on called isobody-form surfaces that are approximately spheres, as depicted in Fig. 2. In [17], a magnetic gradiometer system is now proposed, consisting of five three-axis magnetometers arranged as shown in Fig. 3. The setup allows the computation of four discrete C T values that can be used to control the attitude of the gradiometer relative to the magnetic target. For example, with the sensors connected by solid lines an upper gradient tensor contraction C TU can be calculated based on (3) where L i j is the Euclidean distance between the corresponding sensors i and j [18]. In the same way, a downward value C TD can be obtained with sensors connected by dashed lines. The C T value of the sensor group being more close to the target will be larger [17], as it is a strictly monotonously decreasing function of distance. This property allows controlling the gradiometer's pitch angle G , which is zero when both C T values are equal. In the same way, the yaw angle G can be controlled with the other two sensor groups corresponding to the left and right contractions C TL and C TR . In summary, the heading line S in Fig. 3 points exactly to the center of the magnetic target when both the pitch and yaw angle are zero, i.e., if the opposite C T values are equal [18] C There is also another approach instead of (5) to ensure the correct heading of the gradiometer. It is based on the magnetic moment of the target and will be briefly explained. In the first step, according to [18], it is possible to determine the distance between the gradiometer and the magnetic target. If considered as a perfect sphere, the isobody in Fig. 2 can be described by the formula which can be used to triangulate the target position with at least two different C T values without knowing the magnitude | ⃗ M| of the magnetic moment, since it is canceled out during triangulation. The algorithm is explained in [19] and is not described again here. In the second step, the target position can be used as input to (2) to calculate the magnetic moment ⃗ M by inversion. Now, if the heading line S in Fig. 3 lies in the yz-plane of the local frame L T of the target and shows toward its center, the magnetic moment results in This is the condition for correct alignment. Compared to using (5), this method ensures not only that the gradiometer points to the target's center but also that it is parallel to the target's longitudinal axis. However, neither this information nor (5) are adequate for the intended ASAP robot arm to properly grip the MU, as will become clear below. Typically, the MU including the PM inside is mounted on the bridge wall so that the x-axis of the PM (see Fig. 3) is perpendicular to the surface of the earth. Further, the MU has dedicated bulges intended for the arm to attach to, as when a spaceship docks with the space station in the orbit. Hence, the relative attitude of the MU must be known, which is not possible to determine using (5) or the magnetic moment (7). This becomes clear if one imagines that the gradiometer is oriented parallel to the x-axis and rotates around the MU. Due to the symmetry of the spheroidal isobody and the magnetic dipole field, (5) and  (7) would always be satisfied for all rotation angles. Thus, there is a possibility that the arm could damage the MU or be damaged itself if it unexpectedly hits the wall of the bridge. A solution-based solely on a magnetic field is presented in Section III.

III. ADAPTED HEADING METHOD (SRIOD)
In order to be able to additionally determine the orientation of the MU on a bridge wall based on magnetic gradient tensor contractions, we have developed the SRIOD method. It combines two PMs as magnetic field sources with a new type of compact gradiometer structure. The entire system represents a new technique that can recognize the orientation of the MU for the first time and at the same time uses inexpensive magnetometers. In the following, the technique is explained in detail.
We start with the vertical heading process. After completion, the gradiometer and the MU are parallel in the vertical direction as shown in Fig. 1, i.e., C TU = C TD . It can be solved simply by fusing the vertical C T values with the data from the inertial MU (IMU) of the UAV. A traditional gradiometer as shown in Fig. 3 and only one PM inside the MU are sufficient for this.
However, the horizontal heading cannot be solved in this way. This is were a second identical PM and the new gradiometer structure come into play. First, we look at the PMs. The superposition of the magnetic fields of the two PMs using the dipole model (1) has now introduced a directional asymmetry into the gradient tensor contraction. Using (3), its corresponding isobody looks like a pill as depicted in Fig. 4. It contains additional information that can be used for horizontal alignment. Before starting with the horizontal heading, the PMs must be symmetrically embedded inside the MU. Then the MU has to be mounted at the wall so that the PMs are horizontally arranged next to each other and so that their longitudinal axis is perpendicular to the surface of the earth as depicted in Fig. 1. Further their magnetic moment must point in the same direction, i.e., the PMs must repel each other. Second, we look at the gradiometer. We turn to Fig. 1 once again to note that the gradiometer is aligned in the vertical direction, and has yet to align in the horizontal direction so that C TL = C TR . The situation is shown in Fig. 5 from the top view, with C TL and C TR shown as discrete measuring points on the gradiometer. Apparently, it can happen that both measuring points lie on the same isoline and thus the condition C TL = C TR , although the PMs and the gradiometer are not parallel. However, imagine two additional measurementsĈ TL andĈ TR taken symmetrically around the center of the gradiometer as depicted in Fig. 5. They never can be equal if the other measurements are, i.e.,Ĉ TL ̸ =Ĉ TL if C TL = C TR and vice versa. This is because the isolines morfed from circles to ellipsoids. Only if the heading line S looks toward the center and if the gradiometer is parallel to the PMs the condition applies, making (8) the new condition for the horizontal heading. Note, that with a traditional gradiometerĈ TL and C TR cannot be measured. Only our novel gradiometer structure shown in Fig. 6 makes it possible. Fig. 6(a) shows the sensor pairs involved to calculate C TL and C TR , where Fig. 6(b) shows the sensor combination formed to deriveĈ TL andĈ TR . Analogous to this, the vertical values C TU , C TD ,Ĉ TU , and C TD can be formed. However, as discussed above for the vertical heading only C TU and C TD are used. In the following condition (8)

A. Test Case A: Gradiometer Motion Along y-Axis
The first test scenario is shown in Fig. 7(a), where the local frames L T and L G are parallel, i.e., the gradiometer is parallel to the PMs and is moved in y-direction over the PMs. Its distance to the PMs is set to z 0 = 0.15 m. It is a realistic value considering that in the space between the target and the gradiometer in a real system there will be  mounted a robotic gripping hand. For each y, the values C TL , C TR ,Ĉ TL , andĈ TR are calculated as described above and are shown in Fig. 8. Evidently theĈ T curves in Fig. 8(b) show two intersection points at approximately ±0.2 m and one at exactly y = 0 m. However, the C T curves in Fig. 8(a) intersect only at y = 0 m, which confirms the correctness of the hypothesis, that the condition (8) is only fulfilled if the PMs and the gradiometer are parallel and the heading line S shows toward the target center. Not shown here are the curves of the vertical values C TU and C TD , which are equal. The reason is that (4) gives the same result because the sensors of the corresponding sensor groups encounter the same absolute magnetic field components due to the symmetry of the arrangement. Note, when approaching from the left, the right sensor group is closer to the target and thus C TR is larger than C TL . On further motion beyond the center point, the situation reverses. In the case of theĈ T values the whole situation is reversed in the range between the outer intersection points. The found results are as they should be and can be used to develop a control algorithm that enables a robotic arm to align the gradiometer correctly to a magnetic target. However, this is beyond the scope of this article as the focus here is the magnetic field-based measurement technique. Additional tests showed that the gradiometer must maintain a minimum distance from the target in all positions to ensure unambiguous detection. For example, Fig. 9 shows the result of the gradient tensor contractions at a distance of only z 0 = 0.07 m. Several intersections of the two curves are clearly visible, which is undesirable. Here in this configuration the spacing should be at least 1.5 times the spacing of the PMs. To verify if this is in general true, additional experiments with other magnets of different sizes are needed. This is also beyond the scope of this article.

B. Test Case B: Gradiometer Rotation Around x-Axis
In this test case, the gradiometer is rotated by the angle β as shown in Fig. 7(b) schematically, around the x-axis of the magnetic target. Thereby the heading line S always shows toward the target center and the gradiometer center to target center distance is always z 0 = 0.15 m again. The results are shown in Fig. 10. Both the C T curves in 10(a) and theĈ T curves in 10(b) intersect only at β = 0, which confirms the correctness of the hypothesis, that (8) is only fulfilled if the PMs and the gradiometer are parallel and the heading line S shows toward the target center. Not shown here are the curves of the vertical values C TU and C TD , which are equal because of the aforementioned symmetry of the arrangement. This confirms the hypothesis of an asymmetric isobody and the powerful concept of using two identical PMs.
This section has shown that the concept of an asymmetric isobody can be theoretically applied to navigate to a magnetic target and eliminate the orientation angles of the robot arm relative to the target. The information found in the gradient tensor contractions could be combined to implement such homing algorithms. However, this is beyond the scope of this article, as the goal here is to prove the simulation results of the novel localization method under real-world conditions. This will be presented in Section IV.

IV. EXPERIMENTS AND RESULTS
The test cases from Section III were repeated under real conditions with an experimental setup to verify the simulation  results found. For this purpose, a prototype magnetic gradiometer with magnetoresistive sensors was specially developed based on the considerations and results of the previous sections. In Section II, the prototype and the magnetic target used are described in technical detail. The position and orientation of the gradiometer with respect to the target were precisely adjusted during the measurements by a robotic arm, whose characteristics are also discussed. The section concludes with the results. It should be noted that the goal here is not to build a completely finished robotic arm capable of automatically grasping objects. That would be too big a step, as it is first necessary to ensure that the SNR of the PMs and sensor is sufficiently high for the underlying application. However, this is not so easy to estimate in advance, since this study is the first to use low-cost magnetometers that have significantly higher noise. The justification for this is that, unlike other approaches, the gradiometer is used here very close to the PM and therefore receives a strong signal. If the combination works, it would also provide a very low-cost engineering approach that far outperforms the expensive UWB and GPS systems.

A. Hardware Setup
The magnetic target is shown in Fig. 11(a) and consists of two PMs, the exact alloy of which is unknown. Their magnetic moment is also unknown. They are stacked from magnetic disks 2 cm wide and 5 mm thick, forming a cylinder 3 cm long. They produce a strong signal that makes good use of the sensor measurement range. The PMs are arranged in the plane at a distance of 10 cm. For this purpose, a plastic template was made with a 3-D printer, the cavities of which correspond to the dimensions of the PMs. When placing them, care was taken to ensure that they repel each other, as this ensures that their magnetic moment is in the same direction.
The gradiometer in Fig. 11(b) was implemented as a 25 × 25 cm printed circuit board (PCB) designed by us. Eight small LIS3MDL breakout boards from Pololu are soldered on the top side. They measure the magnetic flux density in all three spatial directions. They are arranged as described in Fig. 6. The maximum adjustable sampling rate and the measurement range are 1 kHz and ±1.6 mT. The sampled values are coded as 16 bit words and are output via a Serial Peripheral Interface (SPI). The sensors are connected via two SPI buses to a Teensy 2 4.1 development board which is soldered to the bottom side of the PCB. In this way, the two inner copper layers of the four-layer PCB form a shield that prevents the high-frequency electromagnetic fields of the Teensy from interfering with the sensors during measurement. The board offers a variety of interfaces, of which only the SPI buses are used here for communication with the LIS3MDL sensors and the SD card function for storing the measured values during the experiments.
The experiments were performed using the 7 DOF Franka Emika Panda robotic manipulator (FRANKA) in Fig. 12 with the PCB mounted on it. It is controlled via the Robot Operating System (ROS) interface. FRANKA has a reach of 855 mm with a maximum payload of 3 kg and a repeatability of 0.1 mm.
To enable robotic manipulation, an assembly interface between the FRANKA robotic arm and the PCB has been developed. During the development phase, the following requirements were considered: (i) the interface must be made of non-ferromagnetic materials to avoid unwanted magnetic material effects and (ii) the magnetic gradiometer must have a sufficient distance to the FRANKA mounting flange to avoid magnetic effects due to the robotic arm. The assembly interface consists of six parts which are shown in Fig. 13. They are the adapter, the base plate, and the four connectors. All parts were made of polylactic acid using the fused deposition modeling 3-D printing process, as it is a versatile and fast manufacturing process. To avoid magnetic influences on the gradiometer, the interface is assembled and fixed with brass fasteners.

B. Calibration and Motion Planning
Prior to conducting an experimental validation, a calibration process yielding the position and the orientation of the magnetic target in the robot base frame L 0 is conducted as shown in Fig. 12. The transformation T T 0 between the L 0 and the magnetic target's frame L T is estimated by positioning the gradiometer mounted on the robot flange (L f ) centrally above  L T where both frames are aligned. Four custom 3-D-printed spacers are used to accurately position the gradiometer 3 cm above L T . Once the robot is manually moved to this calibration position, T T 0 is extracted using the known robot kinematics T f 0 and the transformation T G f between the robot flange and the gradiometer, while taking into account the offset of 3 cm. The transformation T G f is generated using known dimensions of the assembly interface between the robot and the gradiometer. After estimating the transformation T T 0 , the gradiometer is positioned 15 cm above the magnetic target with the same orientation as the target. This initial position was the same for all three test cases. The positioning of FRANKA during the experiments was controlled by a laptop that processes user commands and sends them directly to the arm via an ethernet bus. The control interface and the motion planning algorithm were implemented in C++ in combination with MoveIt!, an open-source motion planning framework.

C. Results
The test cases described in Section III were repeated in the experimental environment. In test case A, the gradiometer moved along the y-axis of the target frame in incremental steps of 5 cm at a height of 15 cm, starting from −25 cm and up to 25 cm. In test case B, the gradiometer rotated around the x-axis of the target frame starting from −50 • up to 50 • in increments of 10 • . Finally, in test case C, the gradiometer At each discrete position, the gradiometer made measurements of 5 s length and stored the acquired magnetic flux density values from all sensors as a binary file on the SD card. The files were then imported and processed in MATLAB. Two measurements were made in each position, first without magnets in the plastic template and then with them. The first measurement was used to eliminate the magnetic offset of the sensors. This is necessary because the offsets in the range of −100 and 100 µT are relatively large and thus distort the calculation of the gradient tensor contractions. This calibration method also eliminates the geomagnetic field, but this does not call into question the approach presented in this article because the gradiometer would almost completely eliminate it anyway. The reason is that the gradient field of the magnetic target is much stronger than that of the Earth's field with about 0.02 nT m −1 [17]. Of course, this calibration method is not applicable in real operations, since the UAV is constantly moving. Here the offsets have to be determined once before in a Gaussian chamber, which cannot be penetrated by the Earth's magnetic field. For cost reasons, however, this was not done in this work. The gradient tensor contractions were then calculated from the second measurement. For this purpose, the average of the 5000 samples per sensor was first taken. In the last step, the C T andĈ T values were then plotted and are shown in the following figures.
In general, the resulting curves have a similar course as shown in the simulation results in Section III. However, deviations from the expected results can be observed. The test case A in Fig. 14 shows in (a) an intersection of the two curves shifted by 9 mm, although C TL and C TR should be equal only at y = 0 m. TheĈ T values in Fig. 14(b) are shifted by less than 1 mm. The curves in Fig. 14(c) are also not identical, but at 100 µT m −1 C TU is only slightly larger than C TD at y = 0 m. In test case B in Fig. 15, a similar picture emerges. In a) the intersection is shifted by 2 • , while in Fig. 15(b) it is shifted less than 1 • . The value of C TU in Fig. 15(c) is again about 90 µT m −1 larger than C TD at β = 0 • . This asymmetry is the result of the following inaccuracies: the flange-gradiometer transformation T G f , the calibration procedure, and the FRANKA positioning accuracy. The transformation T G f is generated based on the dimensions of the 3-D-printed assembly interface, assuming that the coordinate systems of the gradiometer L G is shifted only along the z-axis with respect to the flange coordinate system L f , keeping the same orientation as L f . However, the accuracy of the generated transformation T G f depends on the precision of the 3-D printing process. Additional inaccuracy comes from estimating the position of the target in a manual calibration process, where the position and orientation of the target in the robot base frame T T 0 is estimated by manually positioning the gradiometer mounted on the robot flange directly above the target. The calibration procedure could be further improved by estimating the pose of the target using RGB-D camera and visual markers.
The used PMs also contribute to this asymmetry. Although they are the same size and material, experiments have shown that they produce different magnetic field density distributions. When comparing the measured field components, an offset of up to 10 µT can be observed. One possible reason for this is the cutting process used to produce the small magnetic disks that make up a PM. Cutting can alter the ferromagnetic properties of the material.
In summary, however, the results are satisfactory, because the precision is already sufficient and will improve, since there will be no calibration process in the localization performed later by the robot arm. It was only needed here in connection with the experiments. Our video 3 shows an example of how well the method works, in which we control the FRANKA arm with two PMs. The arm tries constantly to align the gradiometer parallel to the PMs. For this purpose, special control algorithms were implemented based on the results found in this chapter. However, these are not discussed here, as the focus is on the basics of the presented magnetic field-based location technique.

V. CONCLUSION
In this work, an extension of the STAR method called SRIOD was presented, which also allows the orientation of objects to be detected at close range. This gives robots an easy way of grasping objects correctly without having to buy expensive sensors. The method is based on low-cost magnetoresistive sensors that together form a novel compact gradiometer structure to find the object. For this purpose, the object is equipped with two identical PMs that generate a static magnetic field. Due to the arrangement, its magnetic gradient tensor contraction exhibits directional asymmetry, which can be detected by the gradiometer and used to correctly align its position. The claims were verified under real conditions using a robotic arm. Currently, we develop an algorithm to automatically guide the robotic arm to the object. One of the further steps will be the development of a mechatronic gripping hand, which shall then use the gradiometer to grip a real object.

APPENDIX
In this section we provide the technical data of the magnetometers and the PMs used in the prototype presented in this article (Fig. 16).
The magnetometers used are from Pololu Corporation in USA. They come as very compact breakout board with the dimension 22.9 × 10.2 × 2.22 mm and are based on the three-axis magnetic sensor LIS3MDL from STMicroelectronics. The chip offers a maximum adjustable sampling rate of 1 kHz and a measurement range of ±1.6 mT. According to the data sheet, in this configuration the sensor exhibits a peak-to-peak noise level of 3.5 µT, which limits the effective resolution to 8 bit.
The exact alloy of the used PMs is unknown. They are stacked from magnetic disks 2 cm wide and 5 mm thick, forming a cylinder 3 cm long. We identified their unknown magnetic moment by comparing the measurements with values predicted using the dipole equation (1). It is about 2.6 Am 2 .