On-Site Implementation of External Wrench Measurement via Non-Linear Optimization in Six-Axis Force–Torque Sensor Calibration and Crosstalk Compensation

Abstract

This study introduces a novel calibration method for accurate external wrench measurement using a six-axis FT (force–torque) sensor. We propose a sensor model and calibration method for FT sensors that enable precise separation of the force and torque components without the need for additional devices or sensors by estimating essential parameters: bias, crosstalk, CoM (center of mass), and inclination. By directly utilizing manufacturer-provided data, our approach eliminates the complexities of traditional calibration processes while achieving higher accuracy in force–torque measurements. This method simplifies the calibration workflow and enhances the practicality of FT sensor applications. A mobile manipulator installed with an FT sensor and a gripper is used to demonstrate calibration effectiveness across varying postures and incline conditions, with non-linear optimization based on the gradient descent method applied to minimize sensor-data errors. The tilt of the base is implemented by placing a step under the wheels of the mobile base to simulate roll or pitch scenarios. A digital level was used to measure the angle and verify that our predicted results were accurate. The proposed method addresses typical calibration challenges, including the effects of the end tool and base incline, which are not commonly covered in existing methods. The results show that, on a non-inclined base, crosstalk and CoM calibration reduces the MSE (mean squared error) by 55.8%, 56.2%, and 14.5% for the external force with respect to data without any calibration conducted. On an inclined base, our full calibration process reduces the MSE by a maximum of 98.6% for external mass measurement with respect to no calibration method applied. These findings highlight the importance of incline calibration for achieving accurate external force estimations, especially in mobile manipulator applications where the environment frequently changes.

1. Introduction

When performing tasks that involve interaction with the surrounding environment, it is essential to accurately understand the relationship between the manipulator and the object being interacted with or to assess the interaction based on force and torque feedback. Knowing the forces acting between the environment and the robot is crucial for generating the desired motions in response. In the field of robotics, researchers often utilize joint torque sensors or measure the current flowing through joint modules to estimate and control the forces acting on the robot’s end-effector or links. Control strategies that leverage such force and torque feedback include admittance and impedance control [1,2]. However, these methods require an accurate analysis of the manipulator’s model, including the inertia matrix, Coriolis forces, and friction, and they face the challenge of being heavily dependent on these parameters [2,3]. In particular, errors in friction compensation can lead to performance deviations of up to 25%. Moreover, these methods are complex to implement and require sensors on every link.

In general, when a robot is used to perform tasks, the task environment needs to be simplified. Therefore, using an FT sensor to measure forces at the robot’s end-effector simplifies the calculation of the required movements for the end-effector. Although this approach requires adding an FT sensor to the robot, it has the advantage of providing force measurements at the end-effector, even in robots without joint torque sensors.

To obtain the forces acting on the end-effector, an FT sensor is usually installed at the robot’s end-effector, and the end tool is attached after the FT sensor. The external force should be isolated from the FT sensor data. External force refers to the forces and torques acting on the end-effector itself, excluding any effects from the tool attached beyond the sensor. FT sensor calibration is necessary because the tool attached to the end (typically a gripper) has its own weight and inertia, causing the sensor to measure force and torque even when no external forces or torques are applied. For accurate calibration, it is essential to estimate the wrench caused by the tool and separate the external wrench from the sensor’s characteristic crosstalk through appropriate compensation.

Calibration using FT sensors is still an emerging research area. This is because the users typically use the data provided by the sensor as is. Generally, during the sensor manufacturing process, internal calibration is performed to obtain six-axis force and torque data, and the resulting calibration matrix is provided to users [4]. Some studies have proposed methods to collect the sensor’s raw data and determine a calibration matrix using artificial neural networks [5,6,7,8]. The calibration matrix obtained through this process enables users to retrieve force and torque data from the sensor. While this provides calibration at the sensor level, additional steps are required for users to obtain accurate data when an end tool is attached or the base is inclined. Research has been conducted to determine parameters such as bias, CoM, and gravity by sampling in various postures and modeling the sensor system [9,10,11,12]. Methods have been proposed to extract physical information using sensor data from specified postures and the robot’s kinematics, but these approaches have not considered base incline. More recently, studies have aimed to calibrate for an inclined base [13]. However, these methods have the drawback of requiring data collection with at least three end tools of known mass and CoM, making it unsuitable for applications like mobile manipulators, where the base’s incline changes frequently. Also, current researchers utilize the raw data of the sensor to make their own sensor model [14,15]. In the case of a mobile manipulator, if calibration is time-consuming, the preparation time can increase exponentially each time a new work location is added. Indeed, manufacturers of FT sensors typically calibrate the internal parameters of the sensor, enabling users to obtain six-axis force and torque measurements. However, our goal is not to recalibrate these internal parameters using raw data but rather to accurately decompose the forces and torques in real-world scenarios where the sensor is used, such as when an end tool is attached to the sensor or when the base is inclined.

Current research typically focuses on determining the parameters of a manipulator fixed in a specific environment and compensating for the effects on the FT sensor using the calculated parameters. However, these approaches have limitations: when the robot’s installation location changes, causing changes in leveling, excessive resistance, or degradation, the known joint position data may no longer be accurate, requiring the user to repeat the calibration process. As a result, these methods are not suitable for mobile manipulators or robots with small payloads that frequently encounter changing environments.

To address these challenges, we propose a method for quickly estimating the parameters necessary for accurate external force estimation using FT sensor data collected from several specific postures. First, we propose a generalized sensor model consisting of parameters such as the CoM, crosstalk matrix, and inclination, ensuring that sensor compensation is complete once all parameters are identified. By using the force and torque data provided by the sensor, rather than relying on raw sensor data, our model enhances generality and applicability across various sensor types and manufacturers. Second, instead of directly utilizing the sensor’s raw data, we perform calibration using only the force and torque data provided by the sensor. This ensures that our proposed method can be applied generally, regardless of the sensor type or manufacturer. Third, our method enables on-site calibration without requiring additional sensors or equipment, even when precise information about the operating environment of the manipulator-mounted FT sensor (such as the CoM of the end tool or the inclination of the base) is unknown. This is particularly relevant in scenarios where the FT sensor is used with an end tool, as users often lack accurate knowledge of the end tool’s physical properties or the degree of ground inclination. Our approach allows these environmental factors to be determined through calibration and compensated for in the sensor data, enabling accurate force and torque measurements. Unlike other studies that require attaching an object with a precisely known CoM for calibration, our method does not rely on such prior information and offers a more flexible and practical solution. This approach enables rapid calibration to adapt to varying environments without the need to replace the end tool or perform additional tasks during the calibration process. Accurate external force estimation following calibration offers a range of applications. One such application is determining the mass of an object grasped by the end tool. Accurately assessing the mass of a grasped object enhances the safety of environmental interactions involving the object. In some cases, it enables the clear definition of safety boundaries, such as impact forces during object–environment contact, and provides the ability to estimate the force the grasped object exerts on the environment when contact is necessary.

The remainder of this paper is as follows. Section 2 describes the details of the proposed sensor calibration method. Section 3 shows the experiments and verifies the proposed method. Finally, Section 4 concludes this paper and suggests future work.

2. Method

In this section, we introduce the definitions used to explain our calibration method and describe the calibration process in detail. The goal of the calibration is to determine the parameters that allow for the removal of the effects caused by the end tool, enabling accurate estimation of the external forces.

2.1. Background

If the FT sensor data are directly used by the robot, discrepancies can arise between the actual wrench acting on the robot’s end effector and the sensor readings, potentially causing the system to misinterpret these discrepancies as external forces. For instance, the gravity resulting from the end effector’s own weight can lead to varying sensor readings depending on the inclination of the base, even when the robot is in the same posture. Additionally, if the CoM of the end tool is inaccurate, external torque may be detected even when no object is grasped. To address these issues, our proposed method estimates the parameters and enables consistent external force estimation.

To accurately estimate the external wrench, we need to remove all other influences from the wrench values measured by the sensor. In our approach, we aim to perform tasks using a mobile manipulator equipped with an FT sensor and gripper at the end-effector. If we can eliminate the effect of the gripper’s mass m, which influences the FT sensor due to gravity, we can achieve an accurate estimation of the external wrench. The wrench values obtained from the sensor include sensor bias. Bias refers to the offset between the ideal value of zero and the actual force and torque values measured by the FT sensor when no external forces are applied. Additionally, the sensor readings are affected by crosstalk. In the context of this FT sensor, crosstalk refers to errors caused by mutual interference between the changes in internal capacitance, which prevent the independent separation of forces and torques.

2.2. Notation and Frame Definition

The following notations and definitions are used throughout this paper:

• The vector $\mathbf{f}\in {\mathbb{R}}^3$ is the force vector composed of the x, y, and z axes.
• The vector $\mathit{\tau }\in {\mathbb{R}}^3$ is the torque vector composed of the x, y, and z axes.
• The vector $\mathbf{w}\in {\mathbb{R}}^6$ is the wrench vector composed of f and $\tau$.
• The vector $\mathbf{g}\in {\mathbb{R}}^3$ is the gravity vector composed of the x, y, and z axes.
• The vector $\mathbf{r}\in {\mathbb{R}}^3$ is the CoM vector of the end tool in sensor coordinates.
• The matrix $\mathbf{C}\in {\mathbb{R}}^{3\times 6}$ is the crosstalk matrix.
• The subscript ${\mathbf{A}}_i$ represents the coordinates on which the vector and matrix $\mathbf{A}$ are defined.
• The superscript ${\mathbf{A}}^i$ describes the type of vector and matrix $\mathbf{A}$.
• The operator ${[·]}^{+}$ is the pseudo-inverse matrix.
• The operator ${[·]}_{\times }$ is the skew-symmetric matrix.

The coordinates described in Figure 1 are used throughout this paper. Each rotation matrix between coordinates satisfies the following relationship: ${\mathbf{g}}_f={\mathbf{R}}_s{\mathbf{g}}_e={\mathbf{R}}_s{\mathbf{R}}_f{\mathbf{g}}_b={\mathbf{R}}_s{\mathbf{R}}_f{\mathbf{R}}_0{\mathbf{g}}_g$.

2.3. Calibration Model

The typical crosstalk value can be interpreted based on changes in the structure of the FT sensor due to deformation. These changes are estimated by analyzing the variation in strain gauges or capacitance and are then converted into force or torque [16]. For strain gauges, the variation in the sensors installed on the structure is calculated through an internal matrix and converted into a force and torque that the user can understand. On the other hand, in the case of capacitance-type sensors, raw data are generated by utilizing changes in the distance between electrodes on the installed substrate (parallel plates with normal displacement), the area changes due to shear force (parallel plates with shear displacement), and the angle and distance changes between electrodes orthogonal to the direction of the shear force (orthogonal plates with shear displacement). Similar to the strain gauge type, the data are processed through internal calculations to provide force and torque values to the user [17].

We model the wrench measured by the FT sensor as follows. Let the superscript $ft$ denote the value measured by the FT sensor, $abs$ the theoretical applied value, $bias$ the bias present in the sensor, and $est$ our estimated value. Then, the wrench read by the sensor is defined as shown in Equation (1).

$${\mathbf{w}}_f^{ft}={\mathbf{w}}_f^{abs}+{\mathbf{w}}_f^{bias}+\mathbf{C}\left[\begin{array}{c}{\mathit{\tau }}_f^{est}\\ {\mathbf{0}}_{3\times 1}\end{array}\right]$$

(1)

${\mathbf{w}}^{abs}$ represents the ideal force applied by the end tool attached to the sensor, where force is expressed as $m\mathbf{g}$ and torque as $\mathbf{r}\times m\mathbf{g}$. ${\mathbf{w}}^{bias}$ is the initial default value of each data point when the sensor is first activated, which varies with each boot. The C matrix is the crosstalk matrix, which maps the influence of torque on force. Our objective is to determine all the terms on the right side of Equation (1) to compensate for the wrench caused by the end tool attached to the sensor, thereby enabling the measurement of the pure external force.

Since the ${\mathbf{w}}^{abs}$ caused by the end tool is due to gravity, it is necessary to calculate the body gravity in the $\left\{f\right\}$ coordinate. If the base frame coincides with the global coordinate system $\left\{g\right\}$, this calculation can be simplified using the robot’s forward kinematics. However, if the base is inclined, it is essential to know the angle to accurately determine the direction of gravity which is described in frame $\left\{b\right\}$.

$${\mathbf{g}}_b={\mathbf{R}}_0{\mathbf{g}}_g=g\left[\begin{array}{c}sin\beta \\ -sin\alpha ·cos\beta \\ -cos\alpha ·cos\beta \end{array}\right]$$

(2)

Defining the orientation of the $\left\{b\right\}$ frame relative to the global $\left\{g\right\}$ frame by the roll, pitch, and yaw angles $\alpha$, $\beta$, and $\gamma$, respectively, the gravity acting in the $\left\{b\right\}$ frame can be formulated as in Equation (2). It is important to note that since the normal vectors of the $\left\{g\right\}$ and $\left\{b\right\}$ frames can be aligned in opposite directions through two rotations, as in a pan-tilt configuration, it is sufficient to determine only $\alpha$ and $\beta$ to obtain ${\mathbf{g}}_g$ without fully calculating ${\mathbf{R}}_0$.

To estimate the force ${\mathbf{f}}_f^{abs}$ physically exerted by the end tool on the sensor, ${\mathbf{f}}_f^{est}$ is defined as shown in Equation (3).

$${\mathbf{f}}_f^{est}=m{\mathbf{R}}_s{\mathbf{R}}_f{\mathbf{R}}_0{\mathbf{g}}_g=m{\mathbf{R}}_f{\mathbf{g}}_b$$

(3)

The body gravity of frame $\left\{b\right\}$ obtained from Equation (2) is transformed to the sensor frame $\left\{f\right\}$ using forward kinematics, based on data received from the controller. The transformation matrix between the end-effector $\left\{e\right\}$ frame and the force sensor f frame represents any rotational misalignment due to sensor assembly error; however, this effect is minimal relative to other factors and can be reasonably approximated as an identity matrix.

Since the torque arises from the CoM displacement of the end tool and its weight in the $\left\{f\right\}$ frame, it can be expressed as shown in Equation (4) using Equation (2).

$$\begin{array}{cc}\hfill {\mathit{\tau }}_f^{est}& ={\mathbf{r}}_f\times {\mathbf{f}}_f^{est}\hfill \\ \hfill & \hfill =\left[\begin{array}{c}\left(r_{31}{d}_y-r_{21}{d}_z\right)sin\beta -\left(r_{33}{d}_y-r_{23}{d}_z\right)cos\alpha cos\beta -\left(r_{32}{d}_y-r_{22}{d}_z\right)cos\beta sin\alpha \\ \left(r_{33}{d}_x-r_{13}{d}_z\right)cos\alpha cos\beta -\left(r_{31}{d}_x-r_{11}{d}_z\right)sin\beta +\left(r_{32}{d}_x-r_{12}{d}_z\right)cos\beta sin\alpha \\ \left(r_{21}{d}_x-r_{11}{d}_y\right)sin\beta -cos\alpha \left(r_{23}{d}_x-r_{13}{d}_y\right)cos\beta -\left(r_{22}{d}_x-r_{12}{d}_y\right)cos\beta sin\alpha \end{array}\right]\hfill \end{array}$$

(4)

The ${\mathbf{r}}_f$ vector represents the vector of CoM on the end tool in the $\left\{f\right\}$ coordinate system. Expanding the forward kinematics from the $\left\{b\right\}$ coordinate system to the $\left\{e\right\}$ coordinate system, if each element of ${\mathbf{R}}_f$ is defined as $r_{i,j}$, ${\mathit{\tau }}_f^{est}$ can be expressed in terms of the known variables $r_{11}$ through $r_{33}$ and the unknown variables $d_x$, $d_y$, $d_z$, $\alpha$, and $\beta$.

By using the sensor values obtained from the four postures shown in Figure 2, the bias values can be determined through Equation (5).

$$\begin{array}{cc}\hfill {\mathbf{w}}_1^{ft}& \hfill ={\left[\begin{array}{c}{\mathbf{f}}_f^{abs}\\ {\mathit{\tau }}_f^{abs}\end{array}\right]}_1+{\mathbf{w}}^{bias}+\mathbf{C}{\left[\begin{array}{c}{\mathit{\tau }}_f^{est}\\ {\mathbf{0}}_{3\times 1}\end{array}\right]}_1,\mathrm{where}{\mathbf{f}}_{f,1}^{abs}=\left[\begin{array}{c}+f_x\\ +f_y\\ +f_z\end{array}\right],{\mathit{\tau }}_{f,1}^{abs}=\left[\begin{array}{c}-d_z{f}_y+d_y{f}_z\\ +d_z{f}_x-d_x{f}_z\\ -d_y{f}_x+d_x{f}_y\end{array}\right]\hfill \\ \hfill {\mathbf{w}}_2^{ft}& \hfill ={\left[\begin{array}{c}{\mathbf{f}}_f^{abs}\\ {\mathit{\tau }}_f^{abs}\end{array}\right]}_2+{\mathbf{w}}^{bias}+\mathbf{C}{\left[\begin{array}{c}{\mathit{\tau }}_f^{est}\\ {\mathbf{0}}_{3\times 1}\end{array}\right]}_2,\mathrm{where}{\mathbf{f}}_{f,2}^{abs}=\left[\begin{array}{c}-f_x\\ -f_y\\ +f_z\end{array}\right],{\mathit{\tau }}_{f,2}^{abs}=\left[\begin{array}{c}+d_z{f}_y+d_y{f}_z\\ -d_z{f}_x-d_x{f}_z\\ +d_y{f}_x-d_x{f}_y\end{array}\right]\hfill \\ \hfill {\mathbf{w}}_3^{ft}& \hfill ={\left[\begin{array}{c}{\mathbf{f}}_f^{abs}\\ {\mathit{\tau }}_f^{abs}\end{array}\right]}_3+{\mathbf{w}}^{bias}+\mathbf{C}{\left[\begin{array}{c}{\mathit{\tau }}_f^{est}\\ {\mathbf{0}}_{3\times 1}\end{array}\right]}_3,\mathrm{where}{\mathbf{f}}_{f,3}^{abs}=\left[\begin{array}{c}-f_x\\ +f_y\\ -f_z\end{array}\right],{\mathit{\tau }}_{f,3}^{abs}=\left[\begin{array}{c}-d_z{f}_y-d_y{f}_z\\ -d_z{f}_x+d_x{f}_z\\ +d_y{f}_x+d_x{f}_y\end{array}\right]\hfill \\ \hfill {\mathbf{w}}_4^{ft}& \hfill ={\left[\begin{array}{c}{\mathbf{f}}_f^{abs}\\ {\mathit{\tau }}_f^{abs}\end{array}\right]}_4+{\mathbf{w}}^{bias}+\mathbf{C}{\left[\begin{array}{c}{\mathit{\tau }}_f^{est}\\ {\mathbf{0}}_{3\times 1}\end{array}\right]}_4,\mathrm{where}{\mathbf{f}}_{f,4}^{abs}=\left[\begin{array}{c}+f_x\\ -f_y\\ -f_z\end{array}\right],{\mathit{\tau }}_{f,4}^{abs}=\left[\begin{array}{c}+d_z{f}_y-d_z{f}_z\\ +d_z{f}_x+d_x{f}_z\\ -d_y{f}_x+d_x{f}_y\end{array}\right]\hfill \\ \hfill \sum _{i=1}^4{\mathbf{w}}_i^{ft}& =4{\mathbf{w}}^{bias}\hfill \end{array}$$

(5)

When the same force is applied in the $\left\{g\right\}$ frame, the forces and torques from the four postures in Figure 2 are composed of elements with identical magnitudes but opposite signs. Additionally, since the crosstalk effect we modeled is a linear combination of ${\mathit{\tau }}_f^{est}$, it is dependent on the sign of ${\mathit{\tau }}_f^{est}$. If two postures could be identified where each of the three axes has an opposite sign, it would be possible to cancel out all terms except ${\mathbf{w}}^{bias}$ using only those two postures. However, as such postures do not exist, we achieve the same effect through a combination of four postures. Consequently, by summing the sensor data from all four postures, we isolate the term composed solely of ${\mathbf{w}}^{bias}$.

The crosstalk matrix $\mathbf{C}$ is defined as shown in Equation (6). This matrix represents the effect that a linear combination of torque values exerts on the force term. As mentioned above, forces and torques along the same axis do not affect each other in a capacitance-type FT sensor. Therefore, ${\mathbf{C}}_{t2f}$, representing the crosstalk from torque to force, is designed as a diagonal matrix with zeros to prevent mutual influence along corresponding axes. ${\mathbf{C}}_{f2t}$, representing the crosstalk from force to torque, is set to zero in our sensor configuration, as there is no observed impact of force on torque. The reason why ${\mathbf{C}}_{f2t}$ can be designed in this way will be addressed later in conjunction with the sensor data.

$$\mathbf{C}=\left[\begin{array}{cc}{\mathbf{C}}_{t2f}& {\mathbf{C}}_{f2t}\end{array}\right]=\left[\begin{array}{cccccc}0& c_1& c_2& 0& 0& 0\\ c_3& 0& c_4& 0& 0& 0\\ c_5& c_6& 0& 0& 0& 0\end{array}\right]$$

(6)

2.3.1. Linear Method

The CoM of the end tool can be obtained using Equation (7). From the relation $\mathit{\tau }=\mathbf{r}\times m\mathbf{g}={\left[-m\mathbf{g}\right]}_{\times }\mathbf{r}$, we derive a linear equation in the form $\mathbf{r}={\left[-m\mathbf{g}\right]}_{\times }^{+}\mathit{\tau }$ which can be solved by the least squares method. By constructing a matrix from the predicted values obtained across n postures, we can determine the CoM that minimizes the torque error in each posture through the pseudo-inverse.

$$\left[\begin{array}{c}{d}_x\\ d_y\\ d_z\end{array}\right]={\displaystyle \frac{1}{m}}{\left[\begin{array}{c}{\left[-{\mathbf{g}}_{f,1}\right]}_{\times }\\ {\left[-{\mathbf{g}}_{f,2}\right]}_{\times }\\ ⋮\\ {\left[-{\mathbf{g}}_{f,n}\right]}_{\times }\end{array}\right]}^{+}\left[\begin{array}{c}{\mathit{\tau }}_{f,1}^{ft}\\ {\mathit{\tau }}_{f,2}^{ft}\\ ⋮\\ {\mathit{\tau }}_{f,n}^{ft}\end{array}\right]$$

(7)

The elements of the crosstalk matrix can be obtained using linear Equation (8). Since ${\mathbf{C}}_{t2f}{\mathit{\tau }}_f^{est}={\mathbf{f}}^{ft}-{\mathbf{f}}_f^{est}$, we can vectorize $\mathbf{C}$ and convert $\mathit{\tau }$ into matrix form by following the definition of Equation (8) to achieve a least squares method structure, allowing the elements to be solved through the pseudo-inverse.

$$\begin{array}{cc}\hfill & \hfill vec\left({\mathbf{C}}_{t2f}\right)={\left[\begin{array}{c}〈{\mathit{\tau }}_1^{ft}〉\\ 〈{\mathit{\tau }}_2^{ft}〉\\ ⋮\\ 〈{\mathit{\tau }}_n^{ft}〉\end{array}\right]}^{+}\left[\begin{array}{c}{\mathbf{f}}_1^{ft}-{\mathbf{f}}_{f,1}^{est}\\ {\mathbf{f}}_2^{ft}-{\mathbf{f}}_{f,2}^{est}\\ ⋮\\ {\mathbf{f}}_n^{ft}-{\mathbf{f}}_{f,n}^{est}\end{array}\right],\hfill \\ \hfill & \hfill \mathrm{where}vec\left({\mathbf{C}}_{t2f}\right)={\left[\begin{array}{ccccccccc}0& c_1& c_2& c_3& 0& c_4& c_5& c_6& 0\end{array}\right]}^{⊺}\hfill \\ \hfill & \hfill 〈{\mathit{\tau }}_i^{ft}〉=\left[\begin{array}{ccccccccc}0& {\tau }_y& {\tau }_z& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& {\tau }_x& 0& {\tau }_z& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& {\tau }_x& {\tau }_y& 0\end{array}\right]\hfill \end{array}$$

(8)

The tilt information of the base obtained from Equation (2) is non-linear, making it impossible to determine the parameters using a linear equation. Therefore, while compensation is possible with a linear equation on a flat surface, it is insufficient on an inclined surface. The cases where CoM and crosstalk are calibrated using Equations (7) and (8) with the LSM are labeled as LSM Calibration for the following experiments.

2.3.2. Non-Linear Method

The variables required to predict the sensor’s wrench include a total of 11 parameters: the CoM vector of the end tool, the incline angle of the base, and the elements of the crosstalk matrix. To determine these variables, we design an objective function as shown in Equation (10) and aim to minimize it using a non-linear optimization technique.

$$w^{err}={∥{\mathbf{w}}_{f,i}^{sensor}-\left({\left[\begin{array}{c}{\mathbf{f}}_f^{est}\\ {\mathit{\tau }}_f^{est}\end{array}\right]}_i+{\mathbf{w}}_f^{bias}+\mathbf{C}{\left[\begin{array}{c}{\mathit{\tau }}_f^{est}\\ {\mathbf{0}}_{3\times 1}\end{array}\right]}_i\right)∥}^2,$$

(9)

$$H\left(\mathbf{X}\right)=\sum _{i=1}^n{w}_i^{err},\mathrm{where}\mathbf{X}=\left[\begin{array}{ccccccccccc}{d}_x& d_y& d_z& \alpha & \beta & c_1& c_2& c_3& c_4& c_5& c_6\end{array}\right]$$

(10)

By using the predicted values obtained through Equations (3) and (4), along with the bias values from Equation (5), we minimize the error with respect to the sensor data. Defining $w^{err}$ as in Equation (9), we set the objective function H as the summation of the $w_i^{err}$ values across n specified postures, allowing us to find the variables through gradient descent [18]. In general, performing gradient descent on n data points for an m-dimensional vector over k iterations is known to have a time complexity of $O(n·m·k)$. In our case, the optimization process involves 60 data points and an 11-dimensional vector. Since the optimization was completed within fewer than 1000 iterations during the experiment, the computational cost was not a significant burden. The cases where CoM, crosstalk, and inclination are calibrated using Equation (10) are labeled as Full Calibration for the following experiments.

3. Experiments and Results

In this section, we apply our calibration method to a real robot and sensor and present the results. The mobile manipulator used is the Moby manufactured by Neuromeka (Seoul, Republic of Korea), as shown in Figure 1, which consists of a 7-DOF manipulator and a mobile 4WIS-4WID platform. The FT sensor used is the AFT200 manufactured by AIDIN ROBOTICS (Anyang, Republic of Korea), with detailed specifications provided in

Table 1. Specification of AFT200 FT sensor from ADIN ROBOTICS.

Index	Value	Unit
Type	capacitance	-
Nominal force range	200	N
Nominal torque range	15	Nm
Limit force ($f_{xyz}$)	300	N
Limit torque (${\tau }_{xyz}$)	25	Nm
Resolution ($f_{xyz}$)	0.15	N
Resolution (${\tau }_{xyz}$)	0.015	Nm
Maximum sample rate	1000	Hz

. The end tool used is the Robotiq’s 3-Finger Adaptive Gripper (Lévis, QC, Canada).

The data acquisition postures for calibration are shown in Figure 3. In our case, the CoM of the end tool is the furthest along the z-axis in the $\left\{f\right\}$ coordinate system, resulting in the greatest torque generation. Therefore, data were collected from the 10 postures that produced the highest torque. Depending on the shape of the end tool, users may choose different postures for calibration.

To assess the accuracy of the base incline estimation, we first adjust the inclination of the mobile platform as shown in Figure 4 and compare the results. The estimated base incline angles are then compared with the geometric analysis, and the results are presented in

Table 2. The result of the incline estimation. Since the kinematically computed angles do not indicate the axis of rotation, they are expressed using the screw axis. Also, kinematic angles are used as the approximation value to verify if the calibration result is reasonable.

Parameter	Kinematics	Calibration Result [°]
$\alpha$ [°]	-	1.38
$\beta$ [°]	-	4.88
screw angle [°]	4.8	5.07

.

To analyze the impact of each calibration parameter, we define the calibration levels for the following experiments as follows: cases where only bias calibration is performed are labeled as No Calibration; cases where CoM and crosstalk are calibrated using Equations (7) and (8) with the LSM are labeled as LSM Calibration; and cases where floor incline calibration using Equation (10) is additionally performed using Equation (3) are labeled as Full Calibration.

3.1. CoM Estimation

The CoM estimation based on the data obtained from Figure 3 yielded the results shown in

Table 3. The result of CoM estimation. The CoM specified in the datasheet is also an approximation and therefore not entirely accurate.

Parameter	Datasheet	Estimation
$d_x$ [mm]	−8	4.7
$d_y$ [mm]	0.0	−7.4
$d_z$ [mm]	105 1	104.4

¹ These data represent the values specified in the datasheet, with the addition of the height of the mount we designed.

. Considering the data specified in the Robotiq 3-Finger Adaptive Gripper datasheet and the length of the mount we designed, the CoM along the z-axis is approximately 105 mm, while the estimated result is 104.4 mm. The difference in the CoM results for the x- and y-axes is due to the fact that our calibration aimed to estimate the wrench acting on the sensor. Therefore, the calibration was performed to predict the CoM in a way that minimizes the effect of the end tool on the sensor.

3.2. Crosstalk Calibration

The reason for setting ${\mathbf{C}}_{f2t}$ to zero when designing the crosstalk matrix $\mathbf{C}$ is that torque is minimally affected by crosstalk. Figure 5 shows the results when the wrist angle of the end-effector is rotated 360 degrees, starting from a posture with the highest torque, as in the calibration data collection postures. In this case, the CoM is estimated without inclination, and no crosstalk calibration was performed. As a result, the ${\mathbf{f}}_{ext}$ exhibited significant errors due to the influence of crosstalk. In contrast, the ${\mathit{\tau }}_{ext}$ showed a maximum error of 0.1 Nm, indicating that the influence of crosstalk on torque is minimal. As shown in

Table 4. The table of the MSE (mean squared error) with respect to 0, the mean, and the variance of the external torque for each axis in the data collected by rotating only the wrist in the sampling pose, without performing crosstalk calibration. To assess the relationship with the ideal value of zero, the mean and variance were calculated based on the absolute values of the data.

	${\mathit{\tau }}_{\mathit{x}}^{\mathit{ext}}$	${\mathit{\tau }}_{\mathit{y}}^{\mathit{ext}}$	${\mathit{\tau }}_{\mathit{z}}^{\mathit{ext}}$
MSE	0.0075	0.0037	0.0004
Mean	0.059	0.062	0.015
Var	0.002	0.002	0.0001

, the mean and variance of the external torque are very small. Therefore, our approach to modeling the crosstalk matrix is reasonable.

To evaluate whether the crosstalk calibration can compensate for the effects of the end tool’s own weight, we conducted an experiment using the Robotiq 3F gripper, trying a pick-and-place scenario on a non-inclined base. The path of the scenario is described in Figure 6. The robot moves to predefined joint positions using its internal controller, which is based on H-infinity control [19]. When the CoM is not estimated, the data from the gripper’s datasheet are used directly. During the experiment, the gripper does not grasp any object, and no external forces are applied. Ideally, the external wrench ${\mathbf{w}}^{ext}={\mathbf{w}}^{ft}-{\mathbf{w}}^{comp}$ should be zero, where ${\mathbf{w}}^{comp}={\mathbf{w}}_f^{est}+{\mathbf{w}}_f^{bias}+\mathbf{C}{\left[{\mathit{\tau }}_f^{est⊺}{\mathbf{0}}_{1\times 3}\right]}^{⊺}$.

The first experiment is conducted on a non-inclined surface to investigate the impact of crosstalk. Figure 7a shows the external force and torque graphs during the pick-and-place operation in the No Calibration case, while Figure 7b shows the same data for LSM Calibration. As you can see in Figure 7, in postures where torque is generated due to the influence of the end tool, significant force errors are observed due to crosstalk. The analysis of the external force and torque data along each axis, as shown in Figure 7a, is summarized in

Table 5. The table of the MSE of force and torque with respect to 0 for each axis when calibration is performed using two methods on a non-inclined base.

		${\mathit{f}}_{\mathit{x}}^{\mathit{ext}}$	${\mathit{f}}_{\mathit{y}}^{\mathit{ext}}$	${\mathit{f}}_{\mathit{z}}^{\mathit{ext}}$	${\mathit{\tau }}_{\mathit{x}}^{\mathit{ext}}$	${\mathit{\tau }}_{\mathit{y}}^{\mathit{ext}}$	${\mathit{\tau }}_{\mathit{z}}^{\mathit{ext}}$
MSE	No Calib.	1.086	2.607	1.932	0.009	0.012	0.000
	LSM Calib.	0.480	1.142	1.652	0.009	0.012	0.000
	Improved (%)	55.8	56.2	14.5	-	-	-

.

When crosstalk calibration was applied, the MSE reduced 55.8%, 56.2%, and 14.5% with respect to No Calibration for external force. In contrast, the external torque showed almost identical results. This is because the estimated CoM closely matches the CoM specified in the datasheet, and no crosstalk compensation was applied for torque. Therefore, crosstalk compensation effectively reduces errors due to force.

3.3. Inclination Calibration

To investigate the impact of the inclined base, we conduct the second experiment with three calibration steps: No Calibration, LSM Calibration, and Full Calibration. The parameters and initial values for non-linear optimization are set as follows: the learning rate for CoM is $0.00005$, the learning rate for alpha and beta is $0.0001$, and the learning rate for crosstalk is $0.001$. For the ground inclination, since it is measured in radians and the base inclination is generally small, it is advantageous to optimize with a small learning rate. Similarly, for CoM, as its value is in meters and relatively small, a small learning rate is also appropriate. For initial values, the CoM can converge easily by directly using the values provided in the datasheet, as they are close to the optimal point. For ground inclination, even if the inclination is unknown, setting it to 0 allows convergence due to the small learning rate. For crosstalk, it is recommended to set the initial value to 0, as sensor manufacturers usually perform internal calibration. As will be shown in the experimental results discussed later, the inclination significantly impacts accurate force–torque decoupling. Therefore, setting the tolerance to $1\times {10}^{-6}$ to ensure sufficient convergence is advantageous for obtaining precise results.

The results of the experiment are shown in Figure 8. The base is inclined by approximately 5 degrees as we set up in Figure 4. Comparing Figure 8a,b, one can see that while the external force error in Figure 8b has decreased due to crosstalk compensation, the torque error remains due to the base’s incline, indicating that the estimated CoM and crosstalk are insufficient to reduce the torque error. In fact, the torque error is the same as in Figure 8a, where No Calibration is applied. Comparing Figure 8b,c, one can observe that both force and torque have converged close to zero in Figure 8c. This result demonstrates effective compensation for the errors in force and torque caused by the inclined base, and compared to Figure 8a, where No Calibration is applied, it shows that the influence of the end tool has been effectively compensated for. Additionally, as mentioned in the CoM results, our calibration method compensates using the data provided by the sensor, enabling effective calibration from the sensor’s perspective even if there are discrepancies with the datasheet.

The analyzed experimental data are presented in the table. Figure 8a,b show a trend similar to that in Figure 7. Although crosstalk compensation reduced the force error, the presence of the base incline prevented convergence close to zero. Comparing the data in Figure 8b,c, one can see that estimating the incline angle allowed the error to converge close to zero. The Full Calibration reduced MSE with respect to No Calibration by 85.8%, 5.4%, and 56.8% for external force and 27.3% and 65.3% for external torque as you can see in

Table 6. The table of the MSE of force and torque for each axis with respect to 0 when calibration is performed using three methods on an inclined base.

		${\mathit{f}}_{\mathit{x}}^{\mathit{ext}}$	${\mathit{f}}_{\mathit{y}}^{\mathit{ext}}$	${\mathit{f}}_{\mathit{z}}^{\mathit{ext}}$	${\mathit{\tau }}_{\mathit{x}}^{\mathit{ext}}$	${\mathit{\tau }}_{\mathit{y}}^{\mathit{ext}}$	${\mathit{\tau }}_{\mathit{z}}^{\mathit{ext}}$
MSE	No Calib.	3.113	0.650	4.413	0.011	0.060	0.000
	LSM Calib.	2.454	0.468	2.826	0.011	0.056	0.000
	Full Calib.	0.442	0.615	1.905	0.008	0.013	0.000
	Improved (%)	85.8	5.4	56.8	27.3	65.3	-

. Additionally, the final results in Table 5, obtained from experiments on a non-inclined base, are nearly identical to the results obtained on an inclined base. Therefore, our calibration method achieves the same external force estimation accuracy on an inclined base as on an even floor.

From Equations (3) and (4), it can be observed that the predicted force and torque are dependent on the ground inclination and the CoM. In the case of the crosstalk matrix used in Equation (8) with LSM or in Equation (9) in the non-linear form, it can be interpreted as a constant value multiplied by the error of the predicted force and torque. Therefore, in the absence of inclination, multiplying the torque by the crosstalk matrix can reduce the error. However, when inclination is present, the inclination and CoM are coupled, making it impossible to reduce the error using only the crosstalk matrix. Furthermore, as seen in Equation (3), when there is no inclination, the sine term becomes 0, eliminating many terms. However, when inclination is present, a sine value appears, increasing the non-linearity. Hence, the influence of inclination can be considered the most significant factor. This is notable in the experimental results shown in Figure 8, where the error remains large until the inclination is compensated for.

We conduct the same pick-and-place motion with the gripper holding an object with 2 kg of mass. Since the gripper is grasping the object, the FT sensor measures the wrench resulting from both the gripper’s weight and the object’s weight. Given that the calibration was performed using the previously described method to compensate for the gripper’s weight, the external wrench should reflect only the effect of the object’s weight. The results of estimating the object’s mass under three conditions are shown in Figure 9. To estimate the mass of an object grasped by the gripper, we calculated the l2-norm of external force. However, the actual wrench applied is not ideally separated into force and torque components. In the No Calibration results, during the period between 2 and 8 s, where both force and torque are applied together, there is a significant fluctuation in the magnitude of the external force. This is due to the lack of ideal separation between force and torque, which requires the crosstalk calibration we propose. With LSM Calibration, force and torque separation is achieved, resulting in a more stable external force magnitude over the same period. However, due to the inclined base, there remains an offset from the true mass of 2 kg. Finally, in the Full Calibration results, where the base incline is accounted for, the estimated mass converges closer to the actual 2 kg.

The analysis of mass estimation data from the three calibration stages in the pick-and-place scenario is shown in

Table 7. The table of mass estimation results [kg] for the three different calibration scenarios.

	MSE	Mean	MAE	Variance
No Calibration	0.180	1.730	0.302	0.107
LSM Calibration	0.109	2.308	0.308	0.014
Full Calibration	0.009	1.993	0.081	0.009
Improved (%)	95.0	97.7	73.2	92.6

. In the No Calibration results, the error is due to crosstalk from external torque, while in the LSM Calibration results, it is due to an offset from the inclined base. This is evident from the MSE result, which decreased 40% with respect to No Calibration. However, the remaining error reflects the failure to accurately separate force and torque due to the inclined base. With Full Calibration, compensating for the gravity axis error brought the estimated mass closer to 2 kg, and alignment of the gravity axis further reduced the MSE by an additional 91.7%.

The above results were obtained from dynamic data collected during the pick-and-place scenario. To ensure an accurate comparison, we also performed mass estimation under static conditions across various postures with different orientations, as shown in

Table 8. This table summarizes the mass estimation results [kg] across five different orientations for each of the three calibration scenarios.

	Pose 1 $\mathit{\alpha }:28.8°$ $\mathit{\beta }:165.7°$ $\mathit{\gamma }:27.7°$	Pose 2 $\mathit{\alpha }:53.5^{\circ }$ $\mathit{\beta }:-171.4^{\circ }$ $\mathit{\gamma }:99.7^{\circ }$	Pose 3 $\mathit{\alpha }:140.6^{\circ }$ $\mathit{\beta }:24.2^{\circ }$ $\mathit{\gamma }:109.2^{\circ }$	Pose 4 $\mathit{\alpha }:-43.9^{\circ }$ $\mathit{\beta }:-154.2^{\circ }$ $\mathit{\gamma }:46.1^{\circ }$	Pose 5 $\mathit{\alpha }:-66.1^{\circ }$ $\mathit{\beta }:-170.7^{\circ }$ $\mathit{\gamma }:179.{74}^{\circ }$	MSE	Mean	MAE	Variance
No Calib.	2.58	2.26	1.14	1.08	1.25	0.511	1.66	0.674	0.496
LSM Calib.	2.26	2.28	1.64	2.10	2.07	0.058	2.07	0.214	0.067
Full Calib.	2.22	2.28	1.70	1.97	2.08	0.044	2.05	0.182	0.053

. The overall trends are consistent with the previous experiment: crosstalk compensation enabled force and torque separation, while incline compensation allowed for more accurate estimations.

Additionally, we evaluate whether the external force converges to 0 when no object is grasped and to 2 kg when a 2 kg object is grasped by repeatedly performing the same pick-and-place scenario under various base inclination angles, as shown in

Table 9. This table summarizes the calibration results obtained using Robotiq and Barrett grippers under various base angles. It includes the MSE calculated relative to an external wrench of 0 when no object was grasped and the MSE calculated relative to a 2 kg reference when a 2 kg object was grasped.

			Robotiq				Barret
			Roll		Pitch		Roll		Pitch
			−7°	−15°	−8°	−17°	−7°	−15°	−8°	−17°
MSE	w/o mass	No.	0.217	0.639	0.217	0.749	0.044	0.137	0.049	0.153
		LSM.	0.136	0.535	0.145	0.625	0.036	0.124	0.038	0.129
		Full.	0.011	0.008	0.021	0.058	0.005	0.004	0.002	0.006
		Improved (%)	94.9	98.7	90.3	90.3	88.6	97.1	95.9	96.1
	w/mass	No.	0.090	0.060	0.131	0.484	0.063	0.031	0.072	0.170
		LSM.	0.069	0.040	0.099	0.382	0.031	0.038	0.039	0.135
		Full.	0.021	0.026	0.012	0.088	0.015	0.011	0.017	0.056
		Improved (%)	76.7	56.7	84.0	81.8	76.2	64.5	76.4	67.1

. The base inclinations are set to roll angles of $-7°$ and $-15°$ and pitch angles of $-8°$ and $-17°$. The grippers used are the Robotiq 3f and Barrett hand. Similar to previous experimental results, the No Calibration scenario exhibits significant errors. While the LSM Calibration reduces errors to some extent, substantial errors still remain. With Full Calibration, the errors are significantly reduced compared to both No Calibration and LSM Calibration, achieving a maximum reduction of 98.7%. Consistent with our experimental analysis, these results indicate that compensating for inclination plays the most significant role in accurately estimating a consistent mass.

4. Conclusions

In this paper, we introduce a calibration method for accurately obtaining external forces by modeling an FT sensor to determine essential parameters, including bias, crosstalk, CoM, and inclination. Although discrepancies can exist between the FT sensor’s measured data and actual physical quantities, our goal is to model the sensor to minimize the error between the FT sensor data and the predicted values to obtain accurate external forces. Tasks requiring interaction with the environment through a robot necessitate the attachment of an end tool. To isolate contact forces with the external environment, it is essential to compensate for the wrench due to the end tool’s weight, which we achieve through calibration. Notably, adding incline angle calibration, found to be the most critical factor, improves compensation performance. In Section 3, we evaluated whether calibration was appropriately applied in response to changes in the environment. Our calibration results showed that on a non-inclined base, crosstalk and CoM adjustments reduced the MSE by 55.8%, 56.2%, and 14.5% for external force with respect to No Calibration. On an inclined base, the MSE decreased by 85.8%, 5.4%, and 56.8% for external force and 27.3% and 65.3% for external torque with respect to No Calibration. These results indicate that working on an inclined base can significantly impair accurate external force estimation using FT sensors, underscoring the necessity of our calibration method, including incline compensation, to ensure consistent FT sensor performance. We expect that the proposed calibration method will be valuable in applications like mobile manipulators, where environments change frequently and interaction with the environment is essential.