Motion‐Compensation Control of Supernumerary Robotic Arms Subject to Human‐Induced Disturbances

Supernumerary robotic limbs (SRLs) are a novel type of wearable robot used as the third limb to assist the wearer to perform operations that are difficult or impossible for human hands. Although SRLs can compensate for and enhance human physiological abilities, the unpredictable disturbances caused by human movements significantly affect the coordinating control between the robot and wearer. In this study, a general modeling of supernumerary robotic arms (SRAs) on an omnidirectional floating base is presented. Using position and orientation feedback from sensors at the base and tip, three control methods based on different sensor feedback are proposed to improve tracking accuracy. Experiments on point and trajectory tracking are conducted on the SRAs. In the results (point and circular trajectory‐tracking errors with the manipulator as floating base: 1.18 ± 0.56 mm [mean ± standard deviation(SD) error] and 1.42 ± 0.43 mm, point trajectory‐tracking errors with the human shoulder as floating base: 1.37 ± 0.58 mm, and the performance in perforation positioning operation experiment), it is demonstrated that the proposed controllers enable the SRAs to achieve high‐precision tracking and good adaptability to different user movements and frequencies. Also, in the results, future studies on dynamic high‐precision manipulation of SRLs are motivated.


Introduction
Supernumerary robotic limbs (SRLs) are a new type of wearable robots used as the third arm to work with the human.Unlike exoskeletons and prosthetics, [1,2] SRL compensates and augments human abilities by providing the wearer with additional limbs.This advantage allows the wearer to break through the limitations of human physiological ability, such as expanding space to the human body, rather than replacing missing limbs or enhancing existing limbs.
Different types of SRL have been developed in recent years including the supernumerary robotic arms (SRAs) [3][4][5][6][7] mainly used to manipulate the target object when hands are occupied or the target objects are outside the range of human workplace, the supernumerary robotic legs [8][9][10][11][12][13] mainly used to assist human walking with load carriage or balance and stabilize the wearer, and the supernumerary robotic fingers [14][15][16][17][18][19] mainly used to enhance and compensate for the ability of hands.][33] When the wearer performs tasks with the assistance of SRL, human movements lead to changes in the position and attitude of the robot base, which change the dynamics of the SRL, thus greatly increasing the difficulty of auxiliary operations.However, most studies on control have focused on the direct manipulation of SRL by capturing the intention of the wearer, and far too little attention has been paid to the effects of human-machine coupling on SRL control.Parietti et al. presented a Kalman filter approach to estimate the SRL state despite the involuntary motion of the wearer.Facing the fine operation scene of the aircraft, they reduced human disturbance by proposing a bracing technique. [24]To compensate for the human-induced disturbances, Wu et al. proposed a data-driven latent-space impedance control method of supernumerary robotic fingers such that the robot secured the bottle and at the same time allowed natural human Supernumerary robotic limbs (SRLs) are a novel type of wearable robot used as the third limb to assist the wearer to perform operations that are difficult or impossible for human hands.Although SRLs can compensate for and enhance human physiological abilities, the unpredictable disturbances caused by human movements significantly affect the coordinating control between the robot and wearer.In this study, a general modeling of supernumerary robotic arms (SRAs) on an omnidirectional floating base is presented.Using position and orientation feedback from sensors at the base and tip, three control methods based on different sensor feedback are proposed to improve tracking accuracy.Experiments on point and trajectory tracking are conducted on the SRAs.In the results (point and circular trajectory-tracking errors with the manipulator as floating base: 1.18 AE 0.56 mm [mean AE standard deviation(SD) error] and 1.42 AE 0.43 mm, point trajectory-tracking errors with the human shoulder as floating base: 1.37 AE 0.58 mm, and the performance in perforation positioning operation experiment), it is demonstrated that the proposed controllers enable the SRAs to achieve high-precision tracking and good adaptability to different user movements and frequencies.Also, in the results, future studies on dynamic high-precision manipulation of SRLs are motivated.movement to be carried out during the one-handed unscrewing of the bottle cap. [25]In many human-robot interaction tasks, accurate tracking of target points or trajectories guarantees effective completion of interactive tasks.However, the previous studies focus on reducing the disturbance and controlling the impedance under the disturbance, and do not consider the motion-compensation control problem of SRL under the influence of human-robot coupling.
[36][37] Andaluz et al. proposed a robust adaptive controller based on the dynamics of the mobile manipulator, which solves the problems of point stabilization and trajectory tracking. [34]Seraji et al. presented a simple online approach for mobile manipulator motion control by introducing nonholonomic base constraints. [35]The proposed approach is equally applicable to nonholonomic mobile robots such as rover-mounted manipulators and holonomic mobile robots.Fu et al. proposed a unified policy for whole-body control of a legged manipulator using reinforcement learning to control the arm end-effector to desired poses. [36]Caccavale et al. [37] presented a pipeline to combine learning-based locomotion policy with model-based manipulation for legged mobile manipulators and the method achieved precise control of the end-effector of the manipulator.Although the aforementioned methods can realize the trajectory-tracking control of floating-base manipulators, there are also clear differences between these robotic systems and SRL.First, the robotic arms and the mobile base behave as "common body, common brain" mentioned earlier.
During target tracking, the position and orientation of the base can be precisely controlled by the controller to improve the manipulation performance.The SRL and wearer behave as "common body, different brains" and the position and orientation of the base cannot be precisely controlled by the human.The wearer and the SRL are two relatively independent systems with their respective operational objectives and the wearer introduces disturbances to the SRL while performing its own objectives, which affects the accurate execution of the SRL tasks.Second, owing to the complexity of human motion, the uncertainty of base disturbances (such as disturbance frequency, amplitude, etc.) becomes more complex.The disturbance form is the superposition of the conscious movement with a large amplitude and unconscious shaking with a smaller amplitude and higher frequency of the wearer.Therefore, it is challenging to develop a general motion-compensation control method for SRL.
In this study, we explored manipulation-space motioncompensation control of SRAs via different feedback configurations to improve tracking accuracy.The control methods must stabilize the end-effector position to ensure accurate task execution under different working conditions.As shown in Figure 1, many auxiliary scenarios require precise path following to reach the target position, the control problem of concern is that the robot end-effector must meet the desired tracking trajectory to complete designated tasks despite human disturbances.The control algorithms were experimentally validated in our previously proposed SRAs prototype. [38]Our contributions can be summarized as follows: Three promising approaches based on basesensor, end-sensor, and dual-sensor feedback are proposed to address the challenge of coordinating control between humans and SRLs.The control algorithms are of great significance in improving the performance of SRLs and ensuring their widespread use.Experimental results validate the effectiveness of the proposed methods.
This article is organized as follows.Section 2 mainly introduces the SRA and human system models.Section 3 describes the exploration of the motion-compensation control algorithm.Section 4 introduces the experimental validation.Finally, conclusions and outlooks are drawn in Section 5.

Modeling of SRA System
This section introduces the kinematics and dynamics of the human-SRA system including the mutual coupling influence between the wearer, the SRA, and the environment.In this study, we refer to modeling as the process of mapping from the task space to configuration space (or joint space to actuator space).The detailed model of an SRA system with an omnidirectional floating base is derived using frames depicted in Figure 2: frame fOg is an arbitrarily placed inertial frame such that the z axis is opposite to the gravity vector; frame {User} is rigidly attached to the wearer at its center of inertia; frame {Base} is rigidly attached to the backplane such that the frame {Base} in frame {User} is given by a constant transformation ( U p B ∈ ℜ 3 ); and orientation ( U R B ∈ SOð3Þ.θ i and u i (i = 1 ≈ 6) represent SRA joint position and torque states, respectively.The external forces at the endeffector are represented by F envir and τ envir , respectively and the human-induced disturbance force and torque at the robot base are F user and τ user , respectively.F base and τ base represent the force and torque of the robot base acting on the wearer.The Jacobian matrix of the entire system and SRA are represented by O J t and B J s , respectively.

Kinematic Model
The configuration of the human is given by the position and orientation of frame {User} with respect to frame fOg, denoted by the vector O p U ∈ ℜ 3 and rotation matrix O R U ∈ SOð3Þ.We complete the human state with the linear and angular velocities of frame {User} with respect to frame fOg denoted by the vectors O v U ∈ ℜ 3 and O w U ∈ ℜ 3 , respectively.The specific expressions of O p U , O R U , O v U , and O w U are reported in Appendix A. We define the state of the human body using the variables To describe the SRA configuration, the actuated joint positions θ s ∈ ℜ 6 are chosen as generalized coordinates.In a feasible and singularity-free workspace, there is a one-to-one correspondence between θ s ∈ ℜ 6 and the pose of the end-effector with respect to frame {Base}, denoted by the vector B p E ∈ ℜ 3 and rotation matrix B R E ∈ SOð3Þ.For a 6-degree-of-freedom (DOF) universal configuration series manipulator, the mapping relationship between joint states (configuration and corresponding velocity) and end-effector states (pose and linear velocity) is well known.We define the state of the end-effector using the variable The forward kinematic relation, B FK E ∶ℜ 6 !SEð3Þ, maps actuated joint positions (θ s ) to the end-effector position and orientation ( B p E , B R E ) and the differential kinematic function maps actuated joint velocities ( θs ) to the end-effector velocities ( B v E , B w E ).The specific equality relationships between θ s , S end , θs , B v E , and B w E are reported in Appendix A. The output states of the whole system are given by the position, orientation, linear, and angular velocities of frame fEndg with respect to frame fOg denoted by the vector O p E ∈ ℜ 3 , the rotation matrix O R E ∈ SOð3Þ, the vectors O v E ∈ ℜ 3 , and O w E ∈ ℜ 3 , respectively.The input states of the whole system are denoted by q ¼ ðθ h , θ s Þ ∈ SEð3Þ Â ℜ 6 and q ¼ ð θh , θs Þ T ∈ ℜ 12 .Combining the pose and velocity of the human body, we derive the kinematic relations of the whole system where U P E ¼ U P B þ U R B ⋅ B P E and O J t represent the Jacobian matrix of the whole system relative to the frame fOg, B P E is obtained by solving Equation (4).

Dynamic Model
The modeling of the dynamics is completed by considering the interaction forces between SRA and humans, SRA and environment, as well as the motion states of the wearer, as follows where H ∈ ℜ 12Â12 is the inertia matrix; λ ∈ ℜ 12 is a vector of forces due to Coriolis, gravity, and friction forces; as well as the dynamic system changes caused by the floating base, q ∈ ℜ 12 and τ ∈ ℜ 12 , represent the joint angles; and torques of SRA and operator and are defined as where τ ext ∈ ℜ 12 is a vector of joint torque from external forces, typically calculated as where O J t is the Jacobian matrix of the whole system mapping the velocity of the end-effector to all joint velocities and F ext represents the force of the end operation object and the wearer on SRA.

SRA Motion-Compensation Control
In this section, we explore three motion-compensation control methods based on different sensor feedback configurations for operating conditions with different stability requirements to contribute to the long-term success of practical assistance applications.The three control methods based on base-sensor, endsensor, and dual-sensor feedback are proposed to achieve stable control.We introduce the control system hardware platform and control methods.

The System and Control Methods Overview
The hardware architecture ingredient of SRAs control system is illustrated in Figure 3.The single arm mounted on the shoulder K i is the transmission-ratio conversion factor between the joint and motor.The proportion integration differentiation (PID) feedback controller intends to precisely track the motor angle required to keep the end-effector position stable as calculated.
The dynamic feedforward compensation is used to promote the fast-tracking ability of the SRA system.

Stability-Control Method Based on Base-Sensor Feedback
A control method based on feedback from the camera located on the back was proposed.The expected stable trajectories for the end-effector position and orientation  Based on the kinematics model, we can obtain the controlled position and attitude in the frame fBaseg as follows where the expression of U p E ðtÞ is as follows According to the mapping relationship between the task space ( B p E ðtÞ, B R E ðtÞ) and the configuration space (θ s ∈ ℜ 6 ), we can obtain the configuration space control state as follows Through the transmission-ratio relationship mentioned in the previously proposed SRA prototype, [38] we get the motor control angle θ m ðtÞ from the joint control angle θ s ðtÞ.Considering the changeable SRA working environment, there are various uncertain disturbances caused by the wearer and the environment.Therefore, the feedforward compensation and PID feedback control are combined to perform closed-loop control at the motor level.The control flow is shown in Figure 4, where the light blue area represents the implementation part.In this control, Switch 1 is closed and Switch 2 is open.The real-time position and attitude of the wearer detected by the tracking camera T265 located on the base are transformed by the transformation matrix T for feedback control.First, the main controller obtains the desired joint control position through transformation matrix T and the inverse kinematics part of the manipulator ( B IK E ).The joint command position is multiplied by the transmission-ratio conversion factor K i to obtain the motor-command position.Then, the desired motor-command position is sent to the position-control loop implemented by the PID algorithm.

Stability-Control Method Based on End-Sensor Feedback
A control method based on the feedback of the end camera was proposed to compensate for the SRA end-position and attitude errors.To put it simply, we constantly adjust the target position of the end-effector through the feedback of the end error using the errors between the desired pose ( O p d E , O R d E ) and real pose ( O p r E , O R r E ) as a modification term of the input to the control system.The position feedback control law is expressed as follows where p d k ðtÞ is the desired stable position of the tracking point; p r k ðtÞ is the actual position of the end-effector of SRA; k represents the number of iterations; Γ, Ψ represent 3 Â 3 the constant matrix diagonal-learning-gain matrices, which are empirically set through experimental effects; and Γ, Ψ matrices determine the response speed and control accuracy of the system, respectively.First, the value of Γ is determined by the system response speed, and then Ψ is adjusted to achieve the desired accuracy effect.The specific results of Γ, Ψ are reported in Appendix B. For the attitude control, the feedback control law is expressed as follows where R δ k represent the iterative gain matrix is related to the rotation matrix of attitude error and the specific expressions of R δ k are reported in Appendix B, and R d k ðtÞ represents the expected orientation command output by the kth iteration.The specific control flow block is shown in Figure 4.The orange and light blue areas represent the implementation part; in this control method, Switch 1 is open and Switch 2 is closed.The position and attitude of the wearer are not directly feedback through the base-tracking camera but are obtained by the following inverse transformation equation O FK U ðqÞ where θ s represents joint states, S base represents the motion state of the base

Stability-Control Method Based on Dual-Sensors Feedback
On the basis of the control method based on the base-camera configuration proposed earlier, a control method based on the dual-camera configuration was proposed.We use the end feedback error to correct the desired command position and attitude to eliminate the error caused by the model.The specific expressions of the control law are as follows epðtÞ where epðtÞ and eoðtÞ represent the errors between the control command and the actual output on the position and attitude of the SRA end-effector, respectively.The actual pose ( O p r E ðtÞ, O R r E ðtÞ) is obtained from the feedback of the tracking camera T265.K p and K o represent the gain feedback matrices in position and attitude, respectively.They are set according to experience effects for tracking accuracy, and the results are reported in Appendix B. The control flow is shown in Figure 4.The pink area represents the realization of the end feedback correction, and the light blue area represents the implementation part; Switches 1 and 2 are both closed in this control method.The actual position and attitude of the wearer detected directly by the tracking camera T265 located on the base were passed to transformation matrix T for feedback control.In these experiments, the SRA was fixed at the end-effector of the manipulator and was commanded to track the fixed target position (0.693, À0.01, 0.277) in the geodetic coordinate system, and the target poses were all 0°.First, experiments were performed to compare the performance of the aforementioned three control methods.Considering the uncertainty of human motion, the 6D spatial motion as the base movement was selected, which was achieved by polynomial trajectory planning for the manipulator end-effector.The initial spatial position P 0 and attitude O 0 of manipulator end-effector were (0, 0, 0) and (0, 0, 0), and the target position P    Further experiments were performed in the 1D z direction using the SCM-DSF method to evaluate the robustness of the algorithm to the motion amplitude and velocity of the base.The end-effector of the manipulator was planned to move from the initial position (0, 0, 0) to points (0, 0, À0.10), (0, 0, À0.20), (0, 0, À0.30), (0, 0, À0.40), (0, 0, À0.50), and (0, 0, À0.60), respectively, at a constant speed of 0.1 m s À1 to investigate the influence of base motion amplitude on the control performance.The orientation angle of all the aforementioned target points was set to  (0, 0, 0).The error histogram of different motion amplitude is shown in Figure 8 and the accompanying table displays the interquartile range and mean of the error.Within the reasonable SRA motion range, the target point can be accurately tracked under different base motion amplitudes, indicating that the system is robust to large human body motion.The manipulator endeffector was set to move from the initial position (0, 0, 0) to the point (0, 0, À0.60) at speeds of 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6 m s À1 to evaluate the influence of the base motion velocity on the control performance.The average tracking error of the SRA end-effector increased with the increase in base motion velocity, and the mean Euclidean distance error was 5.34 mm when the base speed reached 0.6 m s À1 .The results demonstrate that the proposed controller has a good adaptability for the high velocity of the base.

Trajectory-Tracking Experiment
The performance of the three control methods was further investigated via trajectory-tracking tasks.The SRA end-effector was commanded to track target points on a circle with a radius of 0.075 m on the x = 0.693 m plane in the geodetic coordinate system, and the target poses of the SRA end-effector were all 0°.In this experiment, the movement of the base was the same as the point-tracking experiment shown in Figure 5b.2).

Experiment with the Human Shoulder as the Floating Base
This experiment aimed to evaluate the adaptability of control algorithms to random uncertain human disturbances.First, the SRA end-effector was commanded to track the fixed target position (0.626, À0.187, 0.10) in the geodetic coordinate system using the SCM-DSF method, and the target poses were all 0°.
Figure 12a shows the experimental setup of point tracking.
After the stability-control algorithm was started, the wearer performed reciprocating movements in 6D directions.The human  disturbance was considered the SRA base movement in this experiment; it could be directly detected by the base-tracking camera T265 mounted on the backplane.The end-tracking camera T265 was used to detect the position and attitude of the SRA end-effector for feedback to achieve high-precision position tracking.The human disturbances are shown in Figure 12b, and Figure 12c presents the point-tracking error result during movement.The average error of the whole motion was 1.37 AE 0.58 mm (mean AE SD error), showing that the proposed controller is well adaptable to different user movements, and can achieve highprecision tracking of target points under human-induced uncertain disturbances.Then, the SRA was commanded to complete the tracking of the fixed straight trajectory under random disturbances by the wearer at different frequencies using SCM-DSF.
Figure 13a shows the experimental setup of the trajectory tracking.Figure 13b shows the 6D human disturbance curves at different frequencies during this experiment.The wearer produced relatively large disturbances in z directions, and the maximum disturbance reached 0.156, 0.154, and 0.188 m, respectively.under different frequencies are shown in Figure 14d-f.During tracking at low disturbance frequency, the root-mean-square error in Euclidean distance was 1.2 mm and the maximum attitude root-mean-square error in three directions was 0.009 rad in the α direction.At middle disturbance frequency, the rootmean-square error in Euclidean distance was 2 mm and the maximum attitude root-mean-square error in three directions was 0.015 rad in the α direction.Figure 14f depicts that the Euclidean distance root-mean-square error in space reached 3.1 mm and the maximum pose root-mean-square error in three directions reached 0.029 rad in the α direction for tracking under high disturbance frequency.The results demonstrate that the proposed controller has a good adaptability to different user frequencies.
The accompanying video illustrates that SRA can dynamically compensate for human-induced disturbances at different frequencies and realize the auxiliary perforation positioning operation.

Conclusions and Outlook
This study designed three control methods for improved tracking accuracy based on different sensor feedback.We performed point and trajectory-tracking experiments with the manipulator end-effector as the floating base to evaluate the effectiveness of the three control algorithms.The results demonstrated that SCM-ESF and SCM-DSF methods can achieve higher accuracy in position and attitude tracking under the same base disturbance.Experiments with the human shoulder as the floating base were conducted to demonstrate the adaptability of the control algorithm.The results indicated that the SCM-DSF method could handle random human disturbances at different user movements and frequencies while tracking.Overall, the proposed controllers based on the operator-SRA model are effective.
The control accuracy of SCM-BSF method depends on the accuracy of the modeling.The SCM-ESF method does not rely on modeling and can achieve high control accuracy through direct end-error feedback.However, high model errors may lead to poor stability of the control system.The SCM-DSF method uses the model for feedforward and can achieve high control accuracy by correcting model errors using end-error feedback.The three control methods can be weighed in terms of cost and control accuracy.In terms of cost, the SCM-BSF and SCM-ESF methods have fewer sensors than the SCM-DSF method, incurring lower costs.In terms of control accuracy, the SCM-ESF and SCM-DSF methods have higher accuracy than the SCM-BSF method, which can be traded off according to different target control accuracy requirements and the accuracy of the model of the SRA.For example, when low cost is desired, the SCM-BSF method can be considered for rigid SRL with an accurate model, and the SCM-ESF method can be considered for flexible SRL with an inaccurate model.At present, the wrist joints were driven by the servos, and the movement speed was limited.DC motors with highspeed capability can be considered later to improve the system response.In this study, the gain regulation coefficient of the controller was kept constant.In future, we will adaptively adjust the control gain coefficient according to disturbance amplitude and frequency to improve the robustness of the algorithm.We will add related prediction algorithms, such as long short-term memory, to predict human-induced uncertain disturbances in advance to further improve the control performance.The quaternion expression q k of the rotation matrix of attitude error is as follows 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : x ¼ eo k ð3, 2Þ À eo k ð2, 3Þ 4w y ¼ eo k ð1, 3Þ À eo k ð3, 1Þ 4w Then, the expression of R δ k is 8 > > > > > < > > > > > : where α represent the learning factor, and the parameter is set to 0.35 according to experience effects on pose tracking, θ ¼ cos À1 ðq 0 ⋅ q k Þ, and it represents the angle between the quaternions q 0 and q k .If the angle between two quaternions is greater than 90°, a quaternion is negated to ensure the shortest path when the rotation goes.If the angle between two quaternions is less than 2°, the expression of q δ k is adjusted as follows to avoid the denominator being 0.

Figure 1 .
Figure 1.Path following in the case of human-induced uncertain disturbances of the wearer.

Figure 2 .
Figure 2. The general model of the human-SRAs system.System model including world(fOg), user base ({User}), SRAs base ({Base}), end effector ({End}) frames and the states of operator, SRAs, and the environment.

Figure 3 .
Figure 3.The hardware architecture ingredient of SRAs control system.The single arm has 6 DOFs, of which three rotational DOFs of the shoulder and elbow joints are designed as 4, 4, 2 transmission ratios and the 3-DOF wrist joint is directly driven.The main controller completes the real-time control of the motor through the CAN bus and the 485 bus, and obtains the data of the tracking camera T265 at a frequency of 200 Hz through the USB device.

(
O p U ðtÞ, O R U ðtÞ) are obtained through the T265 tracking camera.

Figure 4 .
Figure 4.The control flow block diagram of stability control method based on different camera configurations.The light green area is the whole SRAs hardware system and the light gray area represents the algorithm implementation part of the whole control system.The switches 1 and 2 are selectively closed according to the three control methods.The base and end tracking cameras T265 can directly feedback on the position and orientation of where they are located for control.

Figure 5 .Figure 6 .
Figure 5. a) Experimental configuration and b) the disturbance of the base.

Figure 7 .
Figure 7.The error histogram of different control methods for point tracking.

Figure 8 .
Figure 8.The error histogram of different motion amplitude at a constant speed of 0.1 m s À1 .

Figure 9 .Figure 10 .
Figure 9.The error histogram of different motion velocity at a constant motion amplitude of 0.6 m.

Figure 11 .
Figure 11.The error histogram of different control methods for trajectory tracking.

Figure 9
presents the error histogram of different motion velocities.The interquartile range and mean of the error are shown in the accompanying table.
Figure10a-cillustrate the tracking performance using SCM-BSF, SCM-ESF, and SCM-DSF.Figure10ddepicts the tracking accuracy of the SCM-BSF method and Figure10edepicts the tracking accuracy of the SCM-ESF and SCM-DSF methods.Table1summarizes the specific error results of circle trajectory under the three controllers and the error histogram of different control methods is shown in Figure11.The error result of SCM-BSF was 6.17 AE 2.16 mm (mean AE SD error) and these errors may have been caused by the error of the theoretical model itself.The average errors of SCM-ESF and SCM-DSF were reduced to 2.79 AE 0.48 and 1.42 AE 0.43 mm, respectively.These results demonstrate that SCM-ESF and SCM-DSF can achieve path following with higher accuracy in the case of overcoming the same base disturbances.Finally, compared with Figure10b,c, SCM-DSF improved the stability of motion along the circular trajectory and achieved smoother tracking compared to SCM-ESF (Table

Figure 12 .
Figure 12. a) Experimental setup of point tracking.b) Disturbances of human at different movement.c) Performance of tracking on a point at different movement.

Figure 13 .Figure 14 .
Figure 13.a) Experimental setup of trajectory tracking.b) The 6D disturbances of human at different frequencies.
expression rotation matrix of attitude error iseo k ðtÞ ¼ ½R r k ðtÞ À1 ⋅ R d k ðtÞ (A7)where R d k ðtÞ is the desired stable orientation of the tracking point, and R r k ðtÞ is the actual orientation of the end-effector of SRAs.
SEð3Þ in frame fEndg, and E P U ¼E P B þ E R B ⋅ B P U and U p B ∈ ℜ3 and U R B ∈ SOð3Þ are fixed constant transformations.The actual position and attitude ( O p E ∈ ℜ 3 , O R E ∈ SOð3Þ) detected by the tracking camera T265 located on the SRA end-effector are passed into end feedback control law and inverse transformation equation for control.

Table 1 .
The results of point-tracking experiment for three stability control methods.Stability control methods Mean error [mm] Standard deviation (SD) [mm] Maximum error [mm] Q1 [mm] Median [mm] Q3 [mm]

Table 2 .
The results of trajectory-tracking experiment for three stability control methods.