A stability observer for human-robot and environment-robot interaction with variable admittance control

When the robot interacts with the environment or people, if the stiffness of the environment or people suddenly increases, the robot is prone to instability. Traditional solutions, such as the stability observer based on frequency, are easily affected by high-frequency signal noise or filter phase error, resulting in misdiagnosis. In addition, the adaptive algorithm keeps the contact force stable in the environment-robot interaction by identifying the environmental stiffness. Still, the change in the environmental stiffness is too significant, which may lead to the failure of the adaptive algorithm. Therefore, this paper proposes an improved observer stabilization method, using the ratio of the standard deviation of force to the maximum allowable force to eliminate the influence of high-frequency noise and reduce misdiagnosis. In addition, the designed stability observer can monitor the interaction between the robot and the environment in real-time and ensure the stable operation of the adaptive algorithm by updating the initial environment stiffness. Finally, some comparative experiments are carried out. The results show that the proposed method has good accuracy and robustness in human-robot and environment-robot interaction.

and the environment or humans is also becoming more critical in robots, such as rehabilitation, rescue, medical surgery, and multi-robot cooperation [1][2][3]. In these tasks, the stability and safety of the interaction between the robot and the environment or human body must be ensured to avoid unexpected injuries.
In human-robot interaction, various techniques have been proposed to detect and prevent the instability of the interaction with the robot under admittance control guaranteeing the safe and stable interaction between the human and robot. In the time domain, the maximum and minimum values of the velocity signal were scanned through wide and narrow windows [4]. Then, vibration indicators were calculated from these extremes to indicate the degree of instability in the interaction process. The active vibration observer (AVO) method based on an artificial neural network could detect and reduce mechanical vibration more accurately than the variance and standard deviation observer in the time-frequency domain [5]. A heuristic algorithm was designed to detect vibration instability based on the interactive control model of the guide [6,7]. Still, the method strongly depended on the velocity and acceleration of the robot boundary and attendant parameters, leading to a time-varying threshold value for the calculation. Based on this, a statistical method was used to calculate the stability threshold [8], which improved the original method's sensitivity to the robot's physical characteristics. A haptic stability observer (HSO) was proposed by analyzing the position signals of stable human motion and dangerous robot activity in the frequency domain [9]. But compared with the position signal, the force signal had a higher sensitivity to high-frequency vibration and was preferred for detecting the instability state. However, the force signal was also sensitive to noises and may produce false positive detection, leading to an improved recursive algorithm proposed to solve this problem [10]. The discrete Fourier transform for frequency analysis had a compromise between sampling time and frequency resolution, and the choice of window size affects the fast response of the observer. Thus, a human-robot collaboration observer (HRCO) based on the infinite impulse response (IIR) Butterworth filter was proposed to overcome the limitations of HSO in terms of computational speed and resolution [11].
In environment-robot interaction, adaptive and robust intelligent interactive control methods [12][13][14][15][16] may encounter difficulties when applied in an uncertain environment because they need prior environment knowledge or time to adjust the specific parameter. At the same time, contact stability was assumed in the adaptation process, and these methods may be unstable [17][18][19][20][21] in case of a sudden increase in environmental stiffness and adaptation delay. Also, for some scenarios, the initial parameter settings significantly impact the control ability, especially in the adaptive methods. The initial stiffness setting will affect the convergence rate in the final contact. According to the hypothesis of [22], since the robot did not know the information before it contacted the environment, the stability of the interactive system in the uncertain environment can be guaranteed if the robot controller restricts an apparent environmental impedance to a level that can ensure the contact stability only by its ability. In other words, the contact performance (such as steady-state contact force) could also be enhanced by online identification of environmental information after establishing regular contact with its stability ability. Then, through the use of the reference trajectory correction or impedance, adapt to enhance as steady contact performance of the contact force [23,24].
In the above methods to ensure interaction stability, it is not easy to use the stability observer for robot-human interaction simultaneously in robot-environment interaction. Because in the case of constant contact with high stiffness, the observer receives high-frequency noise, which may lead to the inaccurate observation of the observed value. In addition, the adaptive method for environment interaction is also not applicable to human-robot interaction because the position, interaction force, and human impedance change in real-time.
When the operator increases the arm stiffness causing instability, the frequency of the interaction force signal rises, and the output of the observer increases to achieve the observation effect. However, the accuracy of the stability judgment may be affected by the advance and delay of the Butterworth high and low pass filters and the signal noise. In the environment-robot interaction, the convergence speed of the adaptive algorithm is subject to the parameter of the initial environment stiffness. When the difference between the initial environmental stiffness and the actual environmental stiffness is significant, the convergence speed of the adaptive algorithm is slow. Moreover, when the environmental stiffness of external interaction increases suddenly, it will lead to the failure of the adaptive algorithm. Therefore, this paper introduces a stability observer and a corresponding variable admittance control algorithm to monitor the current interaction in real-time. It can ensure the stability of the interaction and update the initial environmental stiffness in time to ensure the convergence speed of the algorithm. It also improves the robustness of the algorithm under the sudden change in environmental stiffness.
The rest of the paper is organized as follows. Section 2 introduces the limitations of the HRCO, and on this basis, this paper proposes an improved stability observer I os . Section 3 introduces the characteristics of admittance control in human-robot and environment-robot interaction. The corresponding variable admittance control algorithms based on these characteristics on the stability observer I os are designed to ensure stable interaction in both cases. Section 4 applies the proposed stability observer I os and controller at a six degree-of-freedom(DOF) robot arm to verify the effectiveness of the proposed method. Finally, Section 5 discusses the results of the study.

Stability observer design
The HRCO uses a second-order Butterworth filter to characterize the instability degree according to the ratio of the norm of the magnitude with the high-frequency signal and the lowfrequency signal [11]. But the advance of the high-pass filter (HPF) and the delay of the low-pass filter (LPF) may affect the judgment error, especially when the sampling period is short. In this paper, improvements are made to eliminate this influence. Firstly, HPF and LPF are constructed according to the difference equation of IIR filter: where u is the input signal, y is the output signal. P and Q are the filter order of the feedforward filter and feedback filter. a and b are the coefficients of the feedback filter and feedforward filter, respectively. To ensure that the designed observer can be applied in human-robot interaction and environment-robot interaction, we set the cut-off frequency of the second-order Butterworth high and low pass filters as 5 Hz, and the sampling period is 5 ms. Table 1 shows the detailed parameters of configuring these two filters. The observer is designed according to the amplitude response characteristics of HPF and LPF are as follows: where I o is a dimensionless value between 0 and 1. F n h and F n l are the Euclidean norm of n-DOF interactive force signals after HPF and LPF,respectively. To prevent abrupt changes in the output value I o , set the value I o to zero when F n l is less than 0.01N. Figure 1(a) shows the frequency variation of the simulated interaction force signal between 0 and 10 Hz, and the force signal size is set to 5 N and 10 N. The gray curve I o in Fig. 1(b) can be obtained by observing the analog signal according to Eq. (2). It can be seen that the curve is very rough, so a differentiator is needed for smoothing: Where the value of η is the smoothing coefficient, and its value is 0.02. After going through the differentiator, it can be seen that the red curve in Fig. 1(b) is the final HRCO output. The HRCO curve is smoother than the I o curve. Because of the advance and delay of the filter phase, the value at 20 s will be amplified and exceed the stability threshold. In addition, there is a more pronounced peak in the output value of HRCO at the beginning (at 0 s) and at the end (at 40 s), which may lead to misjudgments by the stability observer.
To solve the problem of output value mutation caused by filter phase advance and delay. This paper proposes an improved observer: where I std is a value that reflects the change in the amplitude of the force. η is the smoothing coefficient, which can be seen from the Fig. 2, the larger the value of η, the higher the accuracy of the observation, but the variation of the output value is rough. It does not facilitate the design of the control algorithm. The smaller the value of η, the smoother and more stable the observed output value, but the speed and accuracy The maximum value of the force signal, F max , normalizes the I std between 0 and 1. p is the window size for calculating the standard deviation. Its value is equal to 0.1/T s , which is used to calculate the change of I std within 0.1 s of the sampling period T s . At the beginning of the interaction, due to the advance of the HPF and the delay of the LPF, the value of HRCO may be too large to cause misjudgment. In contrast, the value of the introduced I std is relatively tiny. The final output value is reduced after multiplying with I o to curtail the output value and trim the misjudgment effect. Similarly, since the value of I o is frequency-dependent, high-frequency noise may lead to a rather significant output value of I o , resulting in false positives. And the introduction of I std can eliminate the frequency noise. The noise is relatively small compared with F max , so the value of I std will be minimal to compensate for the misjudgment problem caused by noises. During the 5 s at the beginning of Fig. 1(b), I os rises slowly relative to HRCO due to the effect of the maximum value setting, where 5 N is only half of the maximum value of 10 N, thus lowering the I os output value, but this does not affect the stable observation afterward. As shown in Fig. 1(c), the effect of I os output does not significantly change after we narrow the input force. And the prominent value of HRCO at the beginning and the stop is reduced, which indicates that the force size also has some effect on HRCO. Figure 1(d) shows the force signal within 1 Hz, but it is noisy. The force signal with the noise has a significantly affects the output value of HRCO, but the output value of I os is more robust to noise, and its output value is almost close to zero. When the robot interacts with a human, the frequency component of human upper limb movement is mainly within 5 Hz, while the frequency component of voluntary movement is below 2 Hz [25]. In environment-robot interaction, the frequency component of steady-state contact force control is relatively low, but when high-frequency vibration occurs, the frequency component is above 1 Hz [26]. Moreover, as shown in Fig. 3, the proposed observer I os , like HRCO, can also distinguish the frequency from 1 to 5 Hz, which is linear with the input frequency. Due to the introduction of I std , the output value of the stability observer I os has obvious sinusoidal variation, but it does not affect the stability criterion. Since the drag frequency of human-robot interaction is Fig. 1 The changes of observer output value under the same magnitude and frequency signals mainly within 2 Hz, the two observers' stability thresholds are set as ε H RC O = 0.16 and ε I os = 0.08. In environmental interaction, the frequency of force control is lower than the drag frequency of human-robot interaction, so the output value corresponding to 1 Hz is set as the stability threshold of environmental interaction. Namely, ε H RC O = 0.04 and ε I os = 0.01.

Interactive control analysis
The input of the admittance controller is the force/torque exerted by human or environment. The output is the desired position and pose of the robot in Cartesian space. The robot moves by obtaining external force information through a sixdimensional force/torque sensor installed on the end flange of the robot. To simplify the model, we analyze it as a one-DOF perspective. The admittance control block diagram of human-robot interaction is shown in Fig. 4. In the humanrobot interaction, the operator's hand keeps contacting with the robot's end-effector to ensure the robot can follow the operator to any position. In this case, the admittance equation is: where m and d are the virtual inertia and damping, respectively. According to Eq. (6) and (7), the accelerationẍ can where x r and x c represent the reference position and command position respectively. m, d, and k are admittance parameters: virtual inertia, damping, and stiffness. f ext is the external force and f d is the desired force in the interaction system. f s is the interaction force between the environment and the robot measured by the sensor. f e is the interaction force between the environment and the robot. x is the actual position of the robot. k e and x e are the environment's stiffness and position, respectively. It can be seen that in admittance control, whether the robot interacts with the human or the environment, it is necessary to acquire force signals to calculate the desired position of the robot in the next step. The only difference between the two is that interaction with the environment requires real-time force feedback to correct the reference position to achieve the desired force. To ensure the stability of the contact force, the stiffness k in the admittance parameter can not be ignored. On the contrary, in the human-robot interaction scenario, the primary consideration is that the robot can follow the operator to any position, so force feedback is not needed, and the stiffness k in the admittance parameter is set to zero.
The main reasons for instability in human-robot interaction tasks are very low admittance and increased arm stiffness during the dragging process. When the operator interacts with the unstable robot, the operator will subconsciously contract the arm muscles to control the robot. In this case, the instability worsens because the arm muscles' contraction increases the stiffness in contact with the robot. The main reason for instability in the interaction task between the robot and the environment is the sudden increase of external impedance. The stability is restored by increasing the admittance value after the instability is observed to ensure the safety of the interaction task.  Figure 6 shows the amplitude-frequency response of the system with varying admittance parameters. In human-robot interaction, the operator's effort is minimized by setting the smallest possible admittance parameter. m = 5 kg, d = 50 Ns/m is the minimum admittance parameter at the stability boundary of the human-robot interaction system, and its damping ratio is ζ min = 0.1581. Its resonance peak M max = −64 dB. The minimum amplitude in the low-frequency band is M min = −73 dB. The ratio of amplitude changes from minimum to maximum R m = 1.14. With the same stiffness, increasing the inertia may cause the amplitude ratio R to exceed R m , and the system will tend to be unstable. On the contrary, expanding inertia and damping or only damping can make the amplitude ratio R less than R m . In the process of a parameter change, the system can ensure stable operation if the amplitude ratio is not greater than R m . From another point of view, when the damping ratio ζ is less than ζ min , the system is prone to instability. Therefore, the damping ratio should not be set smaller than ζ min either.

Variable admittance control method in human-robot interaction
In human-robot interaction tasks, with the operator's arm stiffness increasing, the high-frequency vibration of the robot can be caused. At this time, the value of I os increases rapidly. According to this characteristic, it can establish a linear relationship between admittance parameters and observer. Among the methods in Fig. 6, increasing damping is the most intuitive method to restore stability. However, it will aggravate the frustration of the operator in the task process, and only increasing inertia may lead to excessive acceleration and make the robot out of control. Therefore, this paper adopts the method in Fig. 6(a) to keep the ratio of damping and inertia constant. The high-frequency vibration was eliminated by increasing the damping and inertia simultaneously. Human-robot interaction focuses on interaction flexibility, which requires that the admittance parameter be kept at a low value. During the operator's task process, the stiffness k in the admittance parameter is set to zero to ensure that the robot follows the operator's intention to move. Therefore, in human-robot interaction, the variable admittance control method based on the stable observer I os is as follows: where m 0 and d 0 are the initial values of inertia m and damping d, which are the lowest admittance parameters to ensure stable interaction with the robot. ε is the stability threshold, which is the stable output value of the stability observer I os under the input signal of 2 Hz in Fig. 3. It uses to judge whether the current interaction is stable or not. α is the weight coefficient for adjusting admittance parameters [11].

Variable admittance control methods in environment-robot interaction
In environment-robot interaction, the stiffness k of the admittance parameter is not zero, high-frequency vibration can also occur when the contact environmental stiffness suddenly increases. Therefore, the stability observer I os is needed to observe the vibration caused by a sudden increase in environmental stiffness. In the control of steady-state contact forces with the environment, this paper uses the most intuitive and effective method of Fig. 6(c) to quickly restore contact stability. Hence, in the environment-robot interaction, when I os ≥ ε the variable admittance control method based on the stable observer I os is as follows: To ensure stable contact, the admittance parameters are kept constant after stabilization. Because it is a steady-state contact force control, it is also necessary to ensure that the contact force reaches the desired steady-state contact force after the contact has been stabilized. In the traditional adaptive control method [27], the required reference position x r is calculated by online estimation of environmental stiffnesŝ k e and positionx e to achieve the desired steady-state contact force f d . However, the convergence speed of this method is affected by the setting of initial environmental stiffnessk e (0). Figure 7(a) showed the convergence of contact force f under different initial environmental stiffnessk e (0). The desired contact force f d = 5 N and the actual environmental stiffness are 3800 N/m. It can be seen from Fig. 7(b) that the smaller the difference between the initial environmental stiffness and the actual environmental stiffness is, the faster the interactive contact force f converges to the desired contact force f d . Otherwise, the slower it is. Therefore, the initial environmental stiffness can be updated using the stability observer when I os < ε: where γ 1 and γ 2 are positive scalar constants [27], which can be used to improve the convergence rate of stiffness identification. However, the larger the value is, the more unstable the contact will be.k e (0) is the set initial stiffness.f and f are theoretical contact force and actual contact force, respectively, where the actual contact force f can be obtained from the force sensor, andx e (0) is the initial environmental position.

Experiments
The experimental equipment of this study is a lightweight 6-DOF collaborative robot AUBO-i5, which is equipped with an ATI Gamma six-dimensional force/torque sensor at its end, as shown in Fig. 8. A real-time Linux PC is utilized

Human-robot interaction experiment
In order to verify the effectiveness of the proposed stability observer and variable admittance controller, the point-topoint drag experiment is carried out, and two experimental scenarios is designed. One is to verify the observation effect of the stability observer without using variable admittance  Table 2.
In the first experiment, the parameters of the admittance controller are fixed, as shown in Fig. 9, respectively m = 5 kg and d = 50 Ns/m, which are the minimum stability parameters of the robot. In the stability observer I os , its maximum force F max = 10 N. The sampling period of the robot is 200 Hz, and the sliding window p size is 20. Figure 9(a) shows the position information of the robot end-effector in the X-axis direction. The operator starts dragging from the initial position to position A and turns again at position B. Then, the operator suddenly increases the arm stiffness at position C for about 3 s and returns to position A. Figure 9(b) depicts the interaction force information collected by the sensor during the dragging process. In this case, 0 to 1.5 seconds is the fixed phase. From 1.5 to 5 s is a stable towing phase, during which the operator turns twice. At 5 s, the operator suddenly increased the arm stiffness, resulting in high-frequency vibration of the robot. At 8 s, the operator relaxes his arm to resume stable interaction. Figure 9(c) shows the output values of the two stability observers during the interaction. The solid blue line and dotted line are the output value, and stability threshold of I os , respectively. The solid red line and dotted line are the output  value and stability threshold of HRCO, respectively. When the operator suddenly increases the stiffness of the arm and causes the robot to vibrate at high frequency, the output values of both observers are higher than the corresponding stability threshold, thus achieving the observation effect. However, when dragging starts and stops, the HRCO output value also exceeds its stability threshold. In contrast, the output value of I os is close to zero at all these moments and will respond only when unstable vibration occurs.
In the second experiment, the parameters of the admittance controller are adjust based on the stability observer. This experiment compares HRCO and I os observers, as shown in Figs. 10 and 11. For the initial admittance parameters and adjustment coefficient, the two stability observers are the same, m = 5 kg, d = 50 Ns/m and α = 20, respectively, while the stability threshold, maximum force value and sliding window size of I os and HRCO are ε I os = 0.08, ε H RC O = 0.16, F max = 10 N and p = 20 respectively. Figures 10(a) and 11(a) show the position curves of the HRCO and I os based variable admittance control experiments. Their operation is consistent with the first experiment. Figure 10(b) shows the interaction force information based on the HRCO variable admittance control experiment. From 0 s to 8.6 s is the stable motion phase. The operator suddenly increases the arm stiffness at 8.6 s, resulting in high-frequency vibration in the interactive force, and stops dragging at 14 s after recovery of stability. Figure 11(b) shows the interaction force information based on the I os variable admittance control experiment. From 0 s to 6.6 s is the stable motion phase. The robot's high-frequency vibrates at 6.6 s and stops dragging at 10 s after recovering stability. Because it is manually dragged, the robot moves to the reference point time vary, but the trajectory and the position of the stiffness change are the same for both movements. As can be seen from the figure, in high-frequency interactions, the number of interaction force vibrations based on I os is smaller than that based on HRCO. Figures 10(c) and 11(c) show the variation of the output values of the two stabilization observers, respectively. It can be seen that the observed value of HRCO exceeds the stability threshold not only during vibration (at 8.6 s) but also during steering (at 3.5 s) and stopping (at 14 s). In contrast, the observed value of I os exceeded the stability threshold only during vibration (at 6.6 s). When the observed value exceeds the stability threshold, it means that the current interaction is unstable and needs to be restored to stability by changing the admittance parameters, as shown in Figs. 10(d) and 11(d). The change of admittance parameters caused by the misjudgment of HRCO will affect the smoothness and comfort of the operator's task. However, I os does not have a misjudgment problem. For I os , it is a normal movement during steering and stopping. The interaction force at this moment is small  relative to the maximum allowed force, which means that the value of I std is small.

Environment-robot interaction experiment
Two experiments are carried out to verify the application of the proposed stability observer in environment-robot interaction. The first experiment is a static constant force control experiment to verify that the convergence speed of the proposed method is not affected by the initial environment stiffness setting. The second is a moving constant force control experiment to verify that the proposed method stabilizes and maintains good contact performance for a short time during sudden changes in environmental stiffness. Both experiments are compared with HRCO and traditional adaptive algorithms.
Firstly, the experimental scene of the static constant force control is shown in Fig. 12. This device can indirectly change the stiffness of the interactive environment by changing the stiffness of springs. There is two environmental stiffness value of robot interaction in this experiment, 3500 N/m and 7500 N/m, respectively. Under the two environmental stiffness conditions, the initial admittance parameters of the adaptive method based on the stable observer I os and HRCO In order to ensure the stable operation of the traditional adaptive method under two kinds of stiffness conditions, the damping term in the initial admittance parameter is increased to d = 2000 Ns/m when the environmental stiffness is 7500 N/m. The initial environment stiffness of the traditional adaptive method is set ask e (0) =3500 N/m, and the positive scalar constant γ 1 and γ 2 are 100 and 1, respectively. Compared with human-robot interaction, the value of instability in constant force control is lower, so the stability threshold of the stable observer I os and HRCO is selected as the stable output value at 1 Hz in Fig. 3, which is ε I os = 0.01 and ε H RC O = 0.04, respectively. In variable admittance control, adjustment coefficient α = 40, maximum force F max = 10 N, and sliding window size p = 20. Figure 13 shows the static constant force control experiment when the environmental stiffness is 3500 N/m. The black dotted line is the traditional adaptive method, and the red dotted line and blue dotted line are the methods of the stable observer I os and HRCO, respectively. The experimental desired contact force is 10 N. Figure 13(a) can be seen in the traditional adaptive method of the initial stiffness being set to the actual conditions, and the three methods are convergence around 2 s. In the HRCO method, after convergence, the contact force will appear unstable and fluctuate; this is due to the high-frequency force signal noise. Figure 13(b) shows the output values of the two observers. After the force control at the beginning, the observer detects that the force signal changes at a high frequency. The output values exceed the set stability threshold, so the stability is quickly restored by increasing the damping as shown in Fig. 13(e). After the stability, the initial stiffness is updated, as shown in Fig. 13(c), and then the reference position is calculated, as shown in Fig. 13(d). It can be seen that the observer I os approach stabilizes after the first update of the initial stiffness, but HRCO does not. This is because in the process of maintaining force equilibrium received the influence of high-frequency force signal noise, the output value of HRCO will keep changing. It causes misdiagnosis so that the corresponding damping value keeps increasing to remain  Figure 14 shows the static constant force control experiment when the environmental stiffness is 7500 N/m. As shown in Fig. 14(a), when the initial environmental stiffness differs greatly from the actual environmental stiffness in the traditional adaptive method, the convergence rate of contact force in the traditional adaptive method slows down. However, in the method using the stability observer I os and HRCO, the convergence speed is not affected by the initial stiffness setting and still converges around 2 s. In summary, the convergence speed of the conventional adaptive algorithm is influenced by the initial environment stiffness setting. In contrast, the proposed method is not influenced by the initial environment stiffness setting and has better robustness compared with HRCO.
The second experimental environment is shown in Fig. 15. The robot end effector is equipped with a roller that can ensure smooth movement, and the robot will move from a foam board with a stiffness of about 4000 N/m to a white brick with a stiffness of about 10000 N/m. Their heights are different. To ensure smooth movement, the surface of the foam board and the white brick is connected by a thin cushion of 200 mm in length. In this experiment, the initial stiffness of the adaptive method is set as the same as that of the foam board environment, that is,k e (0) = 4000 N/m. The initial admittance parameters of all methods are m = 20 kg, d = 200 Ns/m, and k = 7000 N/m. The initial parameters of I os and HRCO methods are damping adjustment coefficient α = 40, maximum force F max = 30 N, and sliding window size p = 10, respectively. Figure 16(a) shows the interaction force information for the three methods in the force tracking experiment. The traditional adaptive method of gray dotted line converges at about 2 s with a desired value of 30 N on the foam board with environmental stiffness of 4000 N/m. Then the robot starts to move along the positive direction of the X-axis at 5 s. When the robot moves to the white brick (at 18 s), the environmental stiffness changes abruptly to 10,000 N/m, the conventional adaptive algorithm fails, and the robot occurs high-frequency vibration. The methods based on the stability observer I os (blue dotted line) and HRCO(red dotted line) are not affected by the initial stiffness setting, and both can converge at about 2 s. In addition, when the environmental stiffness changes at 18 s, both can adjust the admittance parameters and update the initial environmental stiffness in response so that the robot can quickly recover stability.  Figure 16(b) shows the output values of the stability observer I os and HRCO during the environmental interaction. It can be seen that the output value of I os exceeds the stability threshold only at the beginning of the force control and the sudden change of the environmental stiffness (around 18 s). While the output value of HRCO receives the influence of high-frequency noise, the output value keeps Fig. 15 Force control experimental setup fluctuating around the stability threshold, which makes the initial environmental stiffness and reference position constantly updated, making it difficult to keep the contact force stable. It can be seen from the change curves of environmental stiffness and reference position in Figs. 16(c) and 16(d) that the traditional adaptive method fails to identify environmental stiffness after environmental stiffness mutation and the reference position jumps at high frequency, resulting in high-frequency vibration of the robot. Similarly, the HRCO method is affected by the high-frequency noise, and misjudgment constantly occurs, resulting in the constant rise of damping to 5000 Ns/m(the maximum damping value that the experimental robot can achieve in this paper) in Fig. 16(e).

Conclusions
In this study, a new stability observer I os is designed by introducing the ratio of the standard deviation of force to the maximum allowable force to improve the limit of HRCO and update the initial stiffness in time according to the observed value. On this basis, corresponding variable admittance controllers are designed for human-robot interaction and environment-robot interaction. Through the comparison of several experimental scenarios designed, it is found that compared with the HRCO method, the I os method proposed in this study can enable operators to perform tasks in human-robot interaction more efficiently and safely, while in the environment-robot scenario, I os can suppress high-frequency interference and significantly improve the robustness of interaction.
Future research will further expand and optimize the application scenarios of the stability observer. It can be applied to the interaction of coupled human, robot, and environment and can distinguish whether the current robot interaction is with the operator, the environment, or both. Regardless of the interaction scenario, it can ensure the system's stability and the operator's safety.