Robot Arm Motion Research Articles

Effects of Human-Like Characteristics in Sampling-Based Motion Planning on the Legibility of Robot Arm Motions

Conveying the intended goal of a robot arm motion has been shown to increase the quality of human-robot collaboration drastically. To this end, optimization-based approaches have been proposed that optimize the legibility of a robot’s motion. However, they are limited in two ways. First, they are typically not validated in environments with obstacles and narrow passages that require collision-free motion planning. Second, they do not consider the influence of the anthropomorphization process that might be caused by a human-like motion or appearance of the arm. This leads to the question of to what extent the legibility of motions is influenced by these factors. In this work, we study the influence of our previously proposed human-likeness function on the legibility of robot arm motions in the context of sampling-based motion planning. We evaluate it against three other motions: a functional motion, a recorded expert motion, and a legible motion based on a heuristic for the observer’s prediction. For this, we conduct an extensive user study with 94 participants. In contrast to other works, we manipulate the robot’s appearance and the complexity of the environment. We thus provide insights into how the legibility of robot motions is influenced by human-like characteristics in motion, appearance and restricting workspace conditions. The complete stimulus material used in this work is provided at https://mytuc.org/zpvt .

Liquid Metal Elastomer-Based U-Shaped Electrode Flexible Strain Sensors with Enhanced Stability and Sensitivity.

Soft and stretchable strain sensors are crucial for applications in human-machine interfaces, flexible robotics, and electronic skin. Among these, capacitive strain sensors are widely used and studied; however, they face challenges due to material and structural constraints, such as low baseline capacitance and susceptibility to external interference, which result in low signal-to-noise ratios and poor stability. To address these issues, we propose a U-shaped electrode flexible strain sensor based on liquid metal elastomer (LME). By incorporating liquid metal into the elastomer, we create a high-dielectric constant LME dielectric layer. Additionally, we innovatively introduce an external U-shaped flexible textile electrode in conjunction with an internal planar flexible textile electrode to form the flexible strain sensor, effectively resolving the trade-off between stability and integration in capacitive sensors. We also investigated the dielectric and mechanical properties of the LME at varying liquid metal volume fraction (ϕLM). A comparative simulation was conducted between parallel plate structure and U-shaped package structure designs, leading to the fabrication of the U-shaped electrode flexible strain sensor based on LME. Furthermore, we demonstrated its potential applications in monitoring human motion, soft gripper electronic skin, and robotic arm movements.

Time-optimal trajectory planning for a six-degree-of-freedom manipulator: a method integrating RRT and chaotic PSO

This study proposes an innovative algorithm based on a hybrid optimization strategy, integrating the rapidly-exploring random tree (RRT) and an improved particle swarm optimization (PSO) method to address the time-optimal trajectory planning problem for six-degree-of-freedom robotic arms. The proposed approach emphasizes obstacle avoidance and motion smoothness. RRT is utilized within a dynamic three-dimensional environment to rapidly generate an initial collision-free path. Subsequently, improved PSO enhances global search performance by introducing a multi-source chaotic mapping-based population initialization strategy, dynamically adjusted inertia weights, and nonlinear learning factors. These enhancements effectively mitigate the limitations of traditional PSO methods, which are prone to premature convergence in complex optimization problems. Furthermore, the proposed 3-5-3 polynomial interpolation method significantly smooths the trajectory, reducing fluctuations in velocity and acceleration, and thereby improving the precision and energy efficiency of trajectory planning. Experimental results demonstrate that the proposed algorithm outperforms existing methods, such as the improved RRT and improved non-dominated sorting genetic algorithm, across multiple metrics. Notable achievements include a reduction of total motion time by approximately 21%, improved stability in robotic arm motion, and enhanced adaptability to dynamic environments. Particularly, the method achieves superior trajectory diversity and uniformity through joint optimization of multiple chaotic sources, overcoming the inherent limitations of single chaotic mappings. This research expands the application scenarios of RRT and PSO and provides a novel solution for intelligent control and real-time planning of high-degree-of-freedom robotic arms.

Design of an Active-Passive Composite Impedance Controller for a Soft Robotic Arm Under Contact Constraints.

The inherent passive impedance characteristics of soft robotic arms provide excellent environmental adaptability. When a soft robotic arm interacts with its surroundings, its passive impedance responds swiftly, preventing rigid collisions that could damage the arm and ensuring high safety. However, during the movement of the soft robotic arm, these passive impedance properties are uncontrollable, making it impossible to achieve precise impedance control in constrained environments by relying solely on passive mechanisms. Therefore, this paper integrated active impedance control with the passive impedance characteristics of soft robotic arms, proposing an active-passive composite impedance controller. Additionally, a position-based impedance controller was designed for comparative analysis. Finally, this article developed both control systems and conducted simulations and experiments, demonstrating that the composite active-passive impedance controller offers superior control performance and environmental adaptability.

Adaptive Step RRT*-Based Method for Path Planning of Tea-Picking Robotic Arm.

The Adaptive Step RRT* (AS-RRT*) path planning algorithm for tea-picking robotic arms was proposed as a solution to the autonomy, safety, and efficiency problems inherent to tea-picking robots in tea plantations. The algorithm employs an accumulator-based sampling point selection strategy to enhance the efficiency of path planning and the quality of the resulting path. It combines fast connectivity and pruning optimization methods to identify collision-free paths in a shorter time and to reduce the computational burden. It also incorporates a dynamic step length adjustment mechanism following collision detection, ensuring that the robot arm can avoid obstacles in real time. Furthermore, the generated paths were optimized through the introduction of redundant node removal and curve smoothing techniques. In the robotic arm motion planning experiments, the depth vision sensor was employed to obtain three-dimensional information within the tea plantation as the data source. The experimental results demonstrate that the AS-RRT* algorithm reduces the path length by 14.18% and the path planning time is less than 1 s, indicating that the proposed method enhances the efficiency of path planning and obstacle avoidance performance of the tea-picking robot arm.

Operator Experience With Robot-Assisted Laparoscopic Rectal Resection and the Impact of Using an Extra Arm on Surgical Outcomes.

Although robot-assisted surgery has been rapidly gaining popularity in recent years, few reports have examined the impact of individual robotic arm movements on surgical outcomes. This study investigated the relationship between extra arm (EA) usage and experience of surgeons in robot-assisted surgery. A total of 176 robot-assisted rectal resections were performed by three surgeons from 2018 to 2022. Each surgeon's first 20 cases categorized in the Early group and the 21st-40th cases in the Late group. Sixty cases in the Early group and 60 cases in the Late group were included to compare patient background, EA motion, and surgical outcomes. More anastomoses were performed closer to the anus in the Late group (p = 0.001). Console time was significantly shorter in the Late group (196 min vs. 153 min, p = 0.020). EA injuries, improper grasping, and intraoperative problems (intraoperative bleeding, misidentification of layers) were significantly less frequent in the Late group (p = 0.001). As experience with robot-assisted surgery increases, appropriate and safe use of an EA was achieved, and operative time and intraoperative problems were reduced.

How to arrange the robotic environment? Leveraging experience in both motion planning and environment optimization.

This study presents an experience-based hierarchical-structure optimization algorithm to address the robotic system environment design problem, which combines motion planning and environment arrangement problems together. The motion planning problem, which could be defined as a multiple-degree-of-freedom (m-DOF) problem, together with the environment arrangement problem, which could be defined as a free DOF problem, is a high-dimensional optimization problem. Therefore, the hierarchical structure was established, with the higher layer solving the environment arrangement problem and lower layer solving the problem of motion planning. Previously planned trajectories and past results for this design problem were first constructed as datasets; however, they cannot be seen as optimal. Therefore, this study proposed an experience-reuse manner, which selected the most "useful" experience from the datasets and reused it to query new problems, optimize the results in the datasets, and provide better environment arrangement with shorter path lengths within the same time. Therefore, a hierarchical structural caseGA-ERTC algorithm was proposed. In the higher layer, a novel approach employing the case-injected genetic algorithm (GA) was implemented to tackle optimization challenges in robot environment design, leveraging experiential insights. Performance indices in the arrangement of the robot system's environment were determined by the robotic arm's motion and path length calculated using an experience-driven random tree (ERT) algorithm. Moreover, the effectiveness of the proposed method is illustrated with the 12.59% decrease in path lengths by solving different settings of hard problems and 5.05% decrease in easy problems compared with other state-of-the-art methods in three small robots.

Design of intelligent human-machine collaborative robot-assisted craniotomy system

ObjectivesTo develop an intelligent human-machine collaborative control robot-assisted craniotomy system, and test its efficacy by experiments. MethodsThe system integrated a UR5 robotic arm (Universal Robots, Denmark), a host computer, a double six-degree-of-freedom force sensor(Nanjing Yuli Instrument Co., Ltd.), a medical drill(AESCULAP®, Germany), a Polaris Optical navigation system(NDI,Canada), with a self-designed navigation procedure and a visual graphical user interface(GUI). According to a preoperative CT and resection plan, the motion of robotic arm can be restricted in a precise and safe area. Through experiments of the 3D-printed skull models and animals (Bama mini pig), we tested the accuracy, efficiency and safety of the robot system. ResultsAfter successfully developed the robot-assisted craniotomy system, we tested the collaborative controlling fluency of robotic arm with the average response time less than 1 s, as well as feedback sensitivity of force sensor with an average result of 60 N and 50 N when drilling on skull models and mini pigs respectively. In addition, compared with “surgeon” group, “robot” group had less average positioning error (1.87 ± 0.66 mm VS 3.14 ± 0.73 mm, P < 0.001) and time spent (6.64 ± 1.15min VS 8.06 ± 1.10min, P = 0.001) in skull model experiments. Also, in mini pig experiments, “robot” group had less average positioning error (3.26 ± 0.51 mm VS 4.39 ± 0.75 mm, P = 0.008) and time spent (11.83 ± 0.92min VS 26.10 ± 1.62min, P < 0.001) compared with “surgeon” group. No matter in skull model experiments or in mini pig experiments, the durations of robot startup and navigation process were not different between the experimental group and control group (3.44 ± 0.98 VS 3.75 ± 1.00min, P = 0.39 [skull model experiments]; 6.42 ± 0.65 VS 7.10 ± 1.12min, P = 0.11 [mini pig experiments]). Because of limited samples, we compared the incidence of tissue injury between “robot” and “surgeon” group jointly (3.8 % [1/26] VS 19.2 % [5/26], P = 0.193). ConclusionSuccessfully developed, the human-machine collaborative robot-assisted craniotomy system achieved craniotomy procedure fluently providing a sensitive force feedback to surgeon and did better than manual work by surgeon in accuracy, efficiency and safety. Further experimental research needs to be performed to testify its applicability in neurosurgery in future.

Motion controller for multi-joint robotic arm with deep cascade gated Bayesian broad learning system

Intelligent controllers based on the broad learning system can simplify the process of model parameter adjustment, finding wide applications in the motion control of multi-joint robotic arms. However, motion controllers for multi-joint robotic arms based on broad learning system exhibit insufficient precision and overlook the impact of joint motion commonalities on controller design. Therefore, this paper proposes a novel motion control strategy for a multi-joint robotic arm based on a deep cascade feature-enhancement gated Bayesian broad learning system. Firstly, the motion controller of the deep cascade feature-enhancement Bayesian broad learning system is constructed to enhance the robotic arm motion control precision. Secondly, an incremental node generation module with an attention-gated mechanism is constructed to capture the unique motion characteristics of the target joints, which is further combined with model generalization to simplify the motion control process of the multi-joint robotic arm. Finally, controller convergence is enhanced by combining it with the Lyapunov theory to constrain the learning parameters. Simulations and physical experiments are designed to verify the feasibility and superiority of the proposed motion control strategy. The results demonstrated that the strategy improved the accuracy of robotic arm motion control, with the root mean square error in position tracking reduced to 0.0019 rad. This represents a 93.39% reduction in error compared to existing techniques.

A Multimodal Perception-Enabled Flexible Memristor with Combined Sensing-Storage-Memory Functions for Enhanced Artificial Injury Recognition.

With the continuous advancement of wearable technology and advanced medical monitoring, there is an increasing demand for electronic devices that can adapt to complex environments and have high perceptual sensitivity. Here, a novel artificial injury perception device based on an Ag/HfOx/ITO/PET flexible memristor is designed to address the limitations of current technologies in multimodal perception and environmental adaptability. The memristor exhibits excellent resistive switching (RS) performance and mechanical flexibility under different bending angles (BAs), temperatures, humid environment, and repetitive folding conditions. Further, the device demonstrates the multimodal perception and conversion capabilities toward voltage, mechanical, and thermal stimuli through current response tests under different conditions, enabling not only the simulation of artificial injury perception but also holds promise for monitoring and controlling the movement of robotic arms. Moreover, the logical operation capability of the memristor-based reconfigurable logic (MRL) gates is also demonstrated, proving the device has great potential applications with sensing, storage, and memory functions. Overall, this study not only provides a direction for the development of the next-generation flexible multimodal sensors, but also has significant implications for technological advancements in many fields such as robotic arms, electronic skin (e-skin), and medical monitoring.

Design of an Optimal Multivariable PID Controller for a Two-Link Robot Arm

Incorporating manipulator robots into various industries has revolutionized automation and manufacturing processes. Manipulator robots are versatile machines equipped with robotic arms and end-effectors, enabling them to handle complex tasks with precision and efficiency. This research aims to design an optimal Mutivariable PID (MPID) controller that ensures precise and accurate movements of two-link robot arm. Theoretical foundations of the MPID controller are discussed together with its relevance to robotic arm systems, and the challenges associated with its implementation. The research methodology involves mathematical modelling of the two-link robot arm dynamics, and the use of an optimization algorithm to determine the optimal MPID controller coefficients. Simulation results demonstrate performance enhancement and productivity of the two-link robot arm through the improved MPID controller.

Trajectory Planning of Robotic Arm Based on Particle Swarm Optimization Algorithm

Achieving vibration-free smooth motion of industrial robotic arms in a short period is an important research topic. Existing path planning algorithms often sacrifice smoothness in pursuit of efficient motion. A robotic trajectory planning particle swarm optimization algorithm (RTPPSO) is introduced for optimizing joint angles or paths of mechanical arm movements. The RTPPSO algorithm is enhanced through the introduction of adaptive weight strategies and random perturbation terms. Subsequently, the RTPPSO algorithm is utilized to plan selected parameters of an S-shaped velocity profile, iterating to obtain the optimal solution. Experimental results demonstrate that this velocity planning algorithm significantly improves the acceleration of the robotic arm, surpassing traditional trial-and-error velocity planning methods.

Low modulus hydrogel-like elastomer sensors with ultra-fast self-healing, underwater self-adhesion, high durability/stability and recyclability for bioelectronics

Flexible sensors simultaneously with property of hydrogel-like low modulus/room temperature ultra-fast self-healing and with elastomer-like durability /environmental stability and underwater adhesion for bioelectronics has not been reported. A low modulus hydrogel-like elastomer that achieves ultrafast self-healing through molecular chain entanglement at room temperature was prepared based on furfuryl alcohol-modified poly(sebacate glyceride) (PGS) prepolymer, furfuryl alcohol-modified poly(ionic liquid) and bismaleimide by Diels-Alder (DA) reaction. The conductive elastomer-based flexible sensors exhibit hydrogel-like properties of low modulus (6.41 kPa) and ultra-fast self-healing (98 % self-healing efficiency within 5 s). The elastomer also possesses rapid subzero and underwater self-healing properties within 5 s. Moreover, PGS-0.2DA-0.2PIL exhibits pressure sensitive adhesive properties and can be adhered/re-adhered in water. The flexible sensor shows elastomer-like high durability, high environmental stability, multiple recyclability and reusability, and it exhibits wide detection ranges, fast response time, low hysteresis, anti-freezing, anti-bacterial and good biocompatibility. The flexible sensors can accurately identify micro-expressions/eye rotation, monitor human movement/health, detect ECG/EMG signals and control robotic arm movements. In conclusion, a new strategy for design of hydrogel-like conductive elastomers via molecular structure design is proposed, and the elastomers-based flexible sensors with low modulus, rapid self-healing and durability/environmental stability show great promising for bioelectronic applications.

Comparative Analysis of Structure and Motion of Vision Logistics Robotic Arms

This paper combines visual recognition and robotic arm, optimizes the design of robotic arm structure, and researches the logistics robotic arm with six degrees of freedom and visual perception ability to enrich the logistics robotic use scene. Firstly, the visual recognition system is installed on the existing six-degree-of-freedom robotic arm, and the coordinate system is established by the light bar recognition method combined with the D-H parameter method. Secondly, a kinematic simulation model was established by using MATLAB, and the state of the joints was analyzed by using forward and inverse kinematic simulation and fifth degree polynomial interpolation. Finally, the actual logistics fixed-point transportation task is designed for trajectory planning. The results show that the robotic arm is able to be able to perform different grasping tasks in non-structural environments, around the visual analysis of the robotic arm as well as the optimization of the motion track, to enhance the efficiency of the comprehensive use of visual and tactile information, to improve the adaptability of the robotic hand grasping, and to be able to meet the precise control of the pick-up process and the use of the realization of the payload grasping and dexterous movement. The research in this paper provides a certain reference value for the propulsion robot arm to perform intelligent grasping tasks.

Adversarial generative learning and timed path optimization for real-time visual image prediction to guide robot arm movements

Adversarial generative learning and timed path optimization for real-time visual image prediction to guide robot arm movements

Reduced-port robotic pancreaticoduodenectomy with optimized surgical field deployment: early results of single-site plus-two ports method

BackgroundThe adoption of Robotic Pancreaticoduodenectomy (RPD) is increasing globally. Meanwhile, reduced-port RPD (RPRPD) remains uncommon, requiring robot-specific techniques not possible with laparoscopy. We introduce a unique RPRPD technique optimizing surgical field exposure.MethodsOur RPRPD utilizes a single-site plus-two ports technique, facilitated by a single-port platform through a 5-cm incision. The configuration of robotic arms (arm1, arm2, arm3, and arm4) were strategically designed for optimal procedural efficiency, with the arms2 and arm3, alongside the assistant trocar, mounted on the single-port platform, while the arms1 and arm4 were positioned laterally across the abdomen. Drainage was established via channels created at the arm1 and arm4 insertion sites. A “gooseneck traction” was principally employed with the robotic instrument to prop up the specimen rather than grasp, improving the surgical field’s visibility and access. Clinical outcomes of patients who underwent RPRPD performed between August 2020 and September 2023 by a single surgeon across two centers in Taiwan and Japan were reviewed.ResultsFifty patients underwent RPRPD using the single-site plus-two ports technique. The gooseneck traction technique enabled goodsurgical field deployment and allowed for unrestricted movement of robotic arms with no collisions with the assistant instruments. The median operative time was 351 min (250–488 min), including 271 min (219–422 min) of console time and three minutes (2–10 min) of docking time. The median estimated blood loss was 80 mL (1–872 mL). All RPRPD procedures were successfully performed without the need for conversion to open surgery. Postoperative major morbidity (i.e., Clavien-Dindo grade ≥ IIIa) was observed in 6 (12%) patients and median postoperative hospital stay was 13 days.ConclusionsThe single-site plus-two ports RPRPD with the gooseneck traction proves to be a safe, feasible option, facilitating surgical field visibility and robotic arm maneuverability.Graphical

Machine Learning-Based Feature Extraction and Classification of EMG Signals for Intuitive Prosthetic Control

Signals play a fundamental role in science, technology, and communication by conveying information through varying patterns, amplitudes, and frequencies. This paper introduces innovative methodologies for processing electromyographic (EMG) signals to develop artificial intelligence systems capable of decoding muscle activity for controlling arm movements. The study investigates advanced signal processing techniques and machine learning classification algorithms using the GRABMyo dataset, aiming to enhance prosthetic control systems and rehabilitation technologies. A comprehensive analysis was conducted on signal processing techniques, including signal filtering and discrete wavelet transform (DWT), alongside a composite feature set comprising Mean Absolute Value (MAV), Waveform Length (WL), Zero Crossing (ZC), Slope Sign Changes (SSC), Root Mean Square (RMS), Enhanced Waveform Length (EWL), and Enhanced Mean Absolute Value (EMAV). These features, refined through Linear Discriminant Analysis (LDA) for dimensionality reduction, were classified using Support Vector Machine (SVM) algorithms. Signal filtering and DWT improved signal quality, facilitating better feature extraction, while the diverse feature set enhanced classification accuracy. LDA further improved accuracy by isolating the most informative features, and the SVM achieved optimal performance in decoding complex EMG patterns. Machine learning models, including K-Nearest Neighbor (KNN), Naïve Bayes (NB), and the SVM, were evaluated, with the SVM outperforming the others. The significance of these results lies in their potential applications in prosthetic control systems and rehabilitation technologies. By accurately decoding muscle activity, the developed systems can facilitate more intuitive and responsive robotic arm movements, contributing to the advancement of innovative solutions for individuals requiring prosthetic devices or undergoing rehabilitation, hence improving the quality of life for users. This research marks a significant step forward in the integration of advanced signal processing and machine learning in the field of EMG analysis.

Logistic Regression Analysis of Factors Affecting Robot Arm Movement Testing

Currently, with the rapid growth of robots in the industrial and service sectors, the robotic arm product is in high demand, and manufacturers need to deliver it on time. The manufacturer has a new product called “robot arm”. The issue is a high failure rate at the test station called the robot arm movement test. The manufacturer focused on the test process in order to reduce any variance that may result in a failure rate, with the operator's performance being their primary interest. Since the first group was built and tested, totaling 233 units of the robot arm, the logistic regression was applied with three independent variables. There are operators, working shifts, and product models. The results indicate day or night shifts are not related to the test failing. The operator and product model are important factors in the test failing. The 1.5–meter–long model has a higher chance of passing the test than the 1–meter–long model by about 13.66 times and the 2–meter–long model by 25.25 times. Operator D is the best at performing the robotic arm test and has a better chance of passing than the other operators (2.07 times for Operator A, 6.53 times for Operator B, and 7.01 times for Operator C). The action is that the software test needs to be updated for the 1– and 2–meter–long models. Moreover, Operators B and C need to be retrained as a priority. Then the manufacturer needs to focus more on the assembly process for yield improvement.

A robot-assisted tracheal intubation system based on a soft actuator?

Tracheal intubation is the gold standard of airway protection and constitutes a pivotal life-saving technique frequently employed in emergency medical interventions. Hence, in this paper, a system is designed to execute tracheal intubation tasks automatically, offering a safer and more efficient solution, thereby alleviating the burden on physicians. The system comprises a tracheal tube with a bendable front end, a drive system, and a tip endoscope. The soft actuator provides two degrees of freedom for precise orientation. It is fabricated with varying-hardness silicone and reinforced with fibers and spiral steel wire for flexibility and safety. The hydraulic actuation system and tube feeding mechanism enable controlled bending and delivery. Object detection of key anatomical features guides the robotic arm and soft actuator. The control strategy involves visual servo control for coordinated robotic arm and soft actuator movements, ensuring accurate and safe tracheal intubation. The kinematics of the soft actuator were established using a constant curvature model, allowing simulation of its workspace. Through experiments, the actuator is capable of 90° bending as well as 20° deflection on the left and right sides. The maximum insertion force of the tube is 2N. Autonomous tracheal intubation experiments on a training manikin were successful in all 10 trials, with an average insertion time of 45.6s. Experimental validation on the manikin demonstrated that the robot tracheal intubation system based on a soft actuator was able to perform safe, stable, and automated tracheal intubation. In summary, this paper proposed a safe and automated robot-assisted tracheal intubation system based on a soft actuator, showing considerable potential for clinical applications.

Geometrically exact 3D arbitrarily curved rod theory for dynamic analysis: Application to predicting the motion of hard-magnetic soft robotic arm

Magnetorheological elastomers are active materials which can be actuated by the applied magnetic field. Hard magnetic soft (HMS) materials, a type of magnetorheological elastomers, show great potential in the fields of biomedical engineering and soft robotics, due to their short response time, remote operation, and shape programmability. To exploit its potential, a series of theoretical frameworks of HMS rods have been developed, but they are mainly limited to the static rod models or classical curved rod models that fail to consider the effect of the “initial curvature” on the distribution of the stress. In this work, we develop a curved rod theory to predict the 3D dynamic motion of the rod-like HMS robotics under large deformation. Based on the geometrically exact rod theory, we include the heterogeneous initial length of the longitudinal fiber caused by “initial curvature” into our model and obtain the reduced balance equations of the HMS robotics. As a result, the “tension-bending” and “shear-torsion” coupling effects of curved rods emerge in the present model. A numerical implementation of our model based on the classical Newmark algorithm is presented. To validate our model, three numerical examples, including the dynamic snap-through behavior of a bistable arch, are performed and compared with the simulation or experiment results reported in literatures, which show a good agreement. Finally, we experimentally study the 2D and 3D static and dynamic motion of a quarter arc HMS robotic arm under an applied magnetic field of 10 mT, and our model gives a satisfactory prediction, especially for static deformation.