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Hindawi Journal of Robotics Volume 2021, Article ID 8142161, 9 pages https://doi.org/10.1155/2021/8142161 Research Article Variable Impedance Control for Bipedal Robot Standing Balance Based on Artificial Muscle Activation Model Kaiyang Yin , Yaxu Xue , Yadong Yu , and Shuangxi Xie School of Electrical and Mechanical Engineering, Pingdingshan University, Pingdingshan 467000, China Correspondence should be addressed to Yaxu Xue; kaiyangyin@163.com Received 13 April 2021; Accepted 23 June 2021; Published 2 July 2021 Academic Editor: Gordon R. Pennock Copyright © 2021 Kaiyang Yin et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. *e bipedal robot should be able to maintain standing balance even in the presence of disturbing forces. *e control schemes of bipedal robot are conventionally developed based on system models or ﬁxed torque-ankle states, which often lack robustness. In this paper, a variable impedance control based on artiﬁcial muscle activation is investigated for bipedal robotic standing balance to address this limitation. *e robustness was improved by applying the artiﬁcial muscle activation model to adjust the impedance parameters. In particular, an ankle variable impedance model was used to obtain the antidisturbance torque which combined with the ankle dynamic torque to estimate the desired ankle torque for robotic standing balance. *e simulation and prototype experimentation results demonstrate that the control method improves the robustness of bipedal robotic standing balance control. complete the bipedal robotic standing balance control [5, 6], 1. Introduction and this approach relies on the robotic dynamic model Nowadays, a vast variety of bipedal robots have been created which is diﬃcult to improve the robustness of the standing to help humans and they have been applied to a myriad of balance control. With the development of intelligent control social applications, such as military training, medical ser- algorithms which are increasingly being used to solve the vices, industrial manufacturing, and other ﬁelds [1]. In these robotic standing balance control problems [7], the intelli- applications, the working environments are usually un- gent control algorithms rely on a large amount of test data. known, which have the risk to interfere with the standing In addition, robots have some hardware limits. For example, balance of the bipedal robots. How to prevent robots from the input saturation and actuator dead zones aﬀect the falling, that is, standing balance control, is a fundamental control robustness, and the intelligent algorithms and control problem for bipedal robots. *e ankle plays an adaptive methods provide eﬀective solutions [8, 9]. important role in bipedal robotic standing balance control *e bipedal robots have a complex structure, with the [2], which raises concerns for robotic ankle control. characteristics of system nonlinearity and structural vari- Researchers have done a lot of works on bipedal robotic ability, which brings about great challenges to motion standing balance control. Vukobratovic et al. [3] referred to control [10]. *rough a long period of evolution, human the principle of mechanical arm balance control and pro- beings have the ability to adapt and swiftly respond to posed the Zero Moment Point (ZMP) control method which environmental changes. *ese abilities of human beings adjusts the joint torque according to the trajectory of real- have provided the best guidance to advance the design of time ZMP. However, the ZMP control method has some robotic controllers. *e authors in [11] attempted to build limitations: the ZMP was calculated by sensor information the virtual neuromuscular model for robotic control. *is feedback which lags behind the actual attitude change, and approach can generate human-like diverse and robust lo- the delay will cause the controller ring [4]. According to the comotion behaviors. However, the major components of this law of conservation of momentum, some scholars simul- control strategy are twofold: the virtual muscle model and taneously adjust the angular and linear momentum to the muscle activation model. However, the virtual model 2 Journal of Robotics involves many control model parameter sets, such as virtual the impedance model which is based on the robotic ankle muscular parameters, which limit the general applicability of and its change rate output the disturbance torque (τ ). From this approach [12]. this, the ankle desired torque (τ ) for bipedal robotic To make robotic joints present a “gloppy” or “springy” standing balance control is aggregated output of τ and τ . r e compliant control behavior similar to human joints, the concept of impedance control in the ﬁeld of robots has been 2.2. Dynamic Torque Estimation. *e dynamic model de- proposed by Hogan [13]. Impedance control is extensively scribes the relationship between the motion of the bipedal employed in robotic control and its robustness and feasi- robot and the dynamic torque (τ ) of the robotic joint. In bility have been acknowledged by many research studies order to study the bipedal robotic standing balance ankle [14, 15]. However, a ﬁxed impedance model may not suﬃce strategy, the complex actions such as arm swing, curved in many applications, and variable impedance is necessary to body, and step can be ignored. Without loss of generality, the achieve optimal performance of the system [16]; for ex- bipedal robot was simpliﬁed as an inverted pendulum model ample, human beings have the ability to adjust their joint with swinging around the ankle, in this paper, and all the impedance through muscle contraction. robotic weight is concentrated on the center of mass (CoM). In this paper, the robotic ankle is streamlined into an *e robotic ankle in the initial state is marked as the co- impedance model, and an artiﬁcial muscle activation model ordinate origin, and the horizontal and vertical directions is built to adjust the impedance parameters. *en, the are marked as the x-axis and y-axis, respectively. *e bipedal variable impedance control based on artiﬁcial muscle acti- robotic inverted pendulum model is illustrated in Figure 2. vation for bipedal robot balance was proposed. Speciﬁcally, Based on the established x-y coordinate system, the the ankle antidisturbance torque is obtained by constructing diﬀerential equation of the robotic torso rotating around the the ankle variable impedance model, and the ankle dynamic ankle can be expressed as torque is calculated by constructing an inverted pendulum model of the bipedal robot. By combination of anti- F l sin θ − F l cos θ � τ . (1) y x r disturbance torque and dynamic torque, the expected ankle torque for standing balance control is estimated. *e main *e robotic CoM horizontal force (F ) can be expressed contributions of this work are threefold: (1) it proposes a as variable impedance control for bipedal robot standing balance, (2) it develops an impedance parameter sets update (2) F � m (l sin θ), approach based on artiﬁcial muscle activation, and (3) the dt proposed work was validated and evaluated by both sim- that is, ulation and prototype experimentation approach. € _ F � mlθcos θ − θ sin θ. (3) 2. Methods *e robotic CoM vertical force (F ) can be described as *e proposed variable impedance control method is used to estimate the ankle desired torque (τ ) for bipedal robotic (4) F � mg + m (l cos θ), standing balance control. Accordingly, the proposed vari- dt able impedance control method is composed of three vital that is, components: a dynamic model to estimate the dynamic torque (τ ), an ankle impedance model to calculate the 2 € _ F � mg − ml θsin θ + θ cos θ . (5) disturbance torque (τ ), and an impedance adjust compo- nent to update the impedance parameters based on artiﬁcial According to equations (1), (3), and (5), the bipedal muscle activation model. robotic standing balance ankle dynamic model can be de- scribed as 2.1. Control Method Overview. *e framework of the pro- (6) τ � Iθ − mglsin θ, posed variable impedance control for bipedal robot standing balance is illustrated in Figure 1, with the vehicle platform acceleration and deceleration in this work to simulate where I represents the rotational inertia, and I �ml . various perturbations of a bipedal robot standing balance. *ere are two control loops in parallel, with the dynamic model and the impedance model being the main compo- 2.3. Disturbance Torque Estimation. *e impedance model nents of the two loops. *e inputs of the dynamic model are refers to the dynamic relationship between the input ﬂow the robotic ankle angle and its change rate [θ θ ], and and the output eﬀort at the interaction port between a foot foot the output is the dynamic torque (τ ). Likewise, the inputs of manipulator and its environment [17]. *is paper regarded the impedance adjust component are the same as those of the the robotic ankle joint as an impedance model and used it to dynamic model. *e model ﬁrst calculates the muscle ac- estimate the ankle disturbance torque. *e bipedal robotic tivation (a) based on the artiﬁcial muscle activation model ankle impedance model’s schematic diagram is shown in and, subject to the parameters update operation, inputs to Figure 3. Journal of Robotics 3 External θ θ foot foot disturbance Bipedal robot Muscle activation Parameter update Impedance adjust component K B M Impedance model Dynamic model τ τ q r Ankle torque actuator The hardware The control system Figure 1: *e framework of variable impedance control for bipedal robot standing balance control. Equilibrium Robotic CoM position torso foot mg foot BK Figure 3: *e bipedal robot ankle impedance model’s schematic diagram. *e disturbance torque can be an estimation based on the ankle impedance model, which can be expressed as τ � Jθ kx + bx _ + mx € . (9) e foot e e e *e kinematic relationship between the bipedal robot CoP ankle Cartesian coordinates system and the joint coordinates system is m g x � l sin θ , (10) foot Figure 2: *e bipedal robotic inverted pendulum model. where l is the mass of center vertical height of the bipedal robot. In the Cartesian coordinate system, the external dis- For the task of bipedal robot standing balance control, turbance (f ) can be estimated by the ankle impedance the ankle angle θ , that is, the robotic swing amplitudes, is foot model, which can be expressed as usually small; that is, sin θ ≈ θ . According to equa- foot foot _ € tion (3), the disturbance torque can be rewritten as f � kx + bx + mx , (7) e e e e where k, b, and m stand for the stiﬀness, damping, and _ € τ � Kθ + Bθ + Mθ , (11) e e e e inertia in the ankle impedance model, respectively, and x represents displacement errors. where θ � θ − θ denotes the robotic tilt angle and θ e foot ref ref *e Jacobi matrix J(θ ) is deﬁned as foot is the ankle angle as the robot at the equilibrium position. K, B, and M stand for the target stiﬀness, damping, and inertia dx � Jθ dθ , (8) foot foot value of the robotic ankle impedance model, and the fol- where θ denotes the bipedal robot ankle angle. lowing equations were met: foot 4 Journal of Robotics 2Il ⎧ ⎪ K � J θ kl, foot ⎪ 0 ⎪ ⎛ ⎜ ⎞ ⎜ ⎟ ⎜ ⎟ ⎨ ⎜mg ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ T ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ B � J θ bl, (12) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ foot ⎜ ⎟ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎜ ⎟ ⎜ ⎟ ⎪ P � ⎜ ⎟, ⎜ ⎟ ⎜ ⎟ ⎪ ⎜ ⎟ ⎜ ��� ⎟ ⎜ ⎟ ⎩ T ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ M � J θ ml. ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ foot ⎜ ⎟ ⎜ 2I Il ⎟ ⎝ ⎠ (19) mg mg Let u � τ ; the robotic tilt angle and its time derivative are the state variables for a model system x � (θ , θ ); then the ��� e e Il equation can be written as u � 2mglθ + 2mg θ . e e mg x _ � Ax + Bu, ⎧ ⎪ ⎪ In order to avoid the noise eﬀect caused by the quadratic 0 1 ⎪ 0 diﬀerentiation of ankle angle, the inertia term in the im- ⎪ ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎨ ⎜ ⎟ ⎜ ⎟ pedance model is ignored in this paper, and the analytical ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A � ⎜ ⎟, B � ⎜ ⎟, ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ optimal solution of the ankle impedance model is ⎪ ⎝ ⎠ ⎝ ⎠ mgl 1 K � 2mgl, ⎪ I ���� (20) ⎪ B � 2 Imgl . x _ (0) � 0, x (∞) � 0, x (0) � θ , x (∞) � 0. 1 1 2 ref 2 When the bipedal robot suﬀers from external distur- (13) bances, the state variables deviate from the equilibrium point, and the ankle impedance parameters will be changed. According to the Zero Moment Point (ZMP) stability Taking the robotic ankle like the human ankle, the ankle criterion theory, if ZMP is inside of the support polygon, the muscle contraction causes human ankle impedance pa- bipedal robot can maintain a stable and upright state; if ZMP rameter sets to be diﬀerent at diﬀerent ankle states. In other is outside of the supporting polygon (including the words, ankle muscle contractions are closely related to the boundary), the bipedal robot is in an upright unstable state. regulation of human ankle impedance. *e variable im- A cost function can be constructed by the square value of pedance characteristics of the human ankle make that have ZMP deviation, which can be deﬁned as the advantages of low energy consumption and fast stability. ∞ ∞ 2 r V � x dt � dt. (14) c ZMP 0 0 (mg) 2.4. Parameter Update. *e neuromuscular research sug- gests muscle activation linked to humanoid joint mechanical According to the above cost function based on ZMP, the impedance. In this paper, the impedance parameters were linear quadratic regulator (LQR) can be used to address the updated by applying the muscle stretch reﬂex model which is bipedal robotic standing balance control problem. *e a fast muscle contraction generation mechanism. *e optimal solution is determined by Behrman’s optimal muscle stretch reﬂex model sensory information is moti- principle, and the form of the optimal solution can be vated by the signals based on the muscle spindle length expressed as change and its contraction velocity [18]; and the muscle spindle length can be gained using the ankle angle θ . In foot u(t) � −K x(t), (15) particular, the ankle muscle spindle length l can be where K represents the appropriate gain parameters, and expressed as following equation holds: l (t) � r ρsinθ (t) − θ − sinθ − θ + l , m foot foot max ref max opt − 1 T (21) (16) K � R B P, where r stands for the attachment radius of the ankle muscle, foot where P is a unique positive-deﬁnite matrix and satisﬁes the ρ denotes the scaling factor representing the muscle ﬁber famous Riccati diﬀerential equations: pennation angle, l describes the optimal length of the muscle opt spindle, l , at which the muscle can provide the maximum T − 1 T isometric force, θ is the ankle reference angle at which −Q − A P − PA + PBR B P � 0, (17) ref l � l , and θ is a constant ankle angle value. From this, the m opt max muscle spindle contraction velocity, v , can be computed via where Q is a symmetric positive-deﬁnite matrix. From the the time derivative of muscle spindle length value l . cost function equation (14), R can be expressed as *e muscle activation value, a, can be computed using the positive feedback reﬂex scheme. *e muscle activation is R � . (18) equal to the preactivation a plus a feedback component, (mg) which can be expressed as a(t) � a + k l (t) − l + k v (t), (22) 0 l m 0 d m From equations (15), (17), and (18), we can get Journal of Robotics 5 where k denotes the feedback gain for muscle spindle length oﬀset, d expresses the feedback gain for the muscle spindle contraction velocity, and l represents the muscle spindle length under muscle relaxation. *e muscle activation is constrained to the range between 0 and 1. According to [19], the joint stiﬀness K can be estimated by multiplication of the intrinsic constant stiﬀness K and the coordinated muscle cocontraction α, and it can be expressed as K � α(t)K . (23) *e optimal stiﬀness value in equation (20) was seen as the intrinsic constant stiﬀness in this paper, and the coor- dinated muscle cocontraction α can be depicted as − β p(t) β 1 − e α(t) � 1 + , (24) −β p(t) 1 + e Figure 4: *e simulated bipedal robot on the moving vehicle. where β and β are constant coeﬃcients and p (t) stands for 1 2 stiﬀness index which can be identiﬁed based on the moving Table 1: *e parameters of the bipedal robot module. average of the ankle joint muscle activation, and it can be Parameters Foot Shank *igh Haunch Torso Total illustrated as follows: Mass (kg) 0.5 0.7 0.8 0.5 1.0 3.5 Center of mass (m) 0.02 0.23 0.45 0.5 0.6 1.8 p(t) � w a (t), (25) i i i�1 where w is the weight of the muscle activation which is i impedance model parameters were calculated by equations deﬁned as (20), (23), and (27). For the robotic body tilt angle, the counterclockwise swing is deﬁned as the negative direction, a (t) w � . (26) i and the clockwise swing is deﬁned as the positive direction. n a (t) i�1 i *e simulation results including robotic body tilt angle, the ankle torque, stiﬀness, and damping value curve are shown In previous work, the study suggested a linear rela- in Figure 5. tionship between square-root joint stiﬀness and damping Before the 5 s time point, the vehicle is stationary, value. *erefore, joint damping value can be described as √�� there is no external interference, the robot can quickly B � v K, (27) adjust to a standing balanced state, and the stabilized equilibrium position is about 2.0 . *e vehicle applies where v is a constant coeﬃcient. 0.5 m/s acceleration at 5 s time point, and the robotic body tilt angle leans forward to about −1.7 and then leans 3. Experimentation backward to about 2.6 . *e bipedal robot gradually stabilized at around 2.0 . *e ankle torque presents a *e proposed variable impedance control was applied to a tendency to increase ﬁrst and then decrease, as illustrated bipedal robot standing balance control on a moving vehicle in Figure 5(b). During the simulation implementation for system validation and evaluation through the simulation process, the stiﬀness and damping values of the ankle and prototype experimentation approach. impedance model are varying depending on the robotic stationary balance, as shown in Figures 5(c) and 5(d). In 3.1. Simulation. *e simulation platform was constructed the antidisturbance range, the stiﬀness and damping using the OpenSim software as shown in Figure 4, and all the amplitudes variation is quite large. data processed was using Matlab. In this simulation part, the robustness of the proposed *e parameters of the bipedal robot module in this variable impedance control method was veriﬁed by simulation are listed in Table 1. conducting multiple simulations with diﬀerent vehicle acceleration. When the vehicle acceleration is 0.5 m/s , 2 2 Case 1. Variable Impedance Control Simulation 1.0 m/s , and 1.5 m/s , the results of bipedal robotic body During the complete simulation implementation pro- tilt angle and ankle torque are summarized in Figure 6. As cess, the movement of the vehicle includes three states: the vehicle acceleration increases, the bipedal robotic stationary, acceleration, and constant speed. Among them, maximal body tilt angle swing range is increasing. After the vehicle only interferes with the bipedal robot standing the standing balance control is completed, the bipedal balance during acceleration. In this simulation, the accel- robot returns to the consistent balance position: within eration of the vehicle set 0.5m/s as the disturbance; and the the range of 1.7∼2.0 . *e simulation results indicate that 6 Journal of Robotics 3 1 –1 –1 –2 –2 01 5 10 5 01 5 10 5 Time (s) Time (s) (a) (b) 36 4.2 3.8 Antidisturbance range Antidisturbance range 26 3.6 01 5 10 501 5 10 5 Time (s) Time (s) (c) (d) Figure 5: *e simulation results of bipedal robotic standing balance control (0.5m/s ). (a) Tilt angle. (b) Ankle torque. (c) Ankle stiﬀness. (d) Ankle damping. –1 –1 –2 –2 0 510 15 0 5 10 15 Time (s) Time (s) 0.5 m/s 0.5 m/s 1.0 m/s 1.0 m/s 1.5 m/s 1.5 m/s (a) (b) Figure 6: *e results of bipedal robotic standing balance control with diﬀerent acceleration. (a) Tilt angle. (b) Ankle torque. the proposed variable impedance control has a good of the proposed variable impedance control system. In this robust performance against the acceleration interference simulation, the moving process of the vehicle is the same as that of the vehicle. discussed in the ﬁrst simulation; and the impedance model parameters were calculated by equation (20). In order to fa- Case 2. Constant Impedance Control Simulation cilitate the comparison, the robotic body tilt angle and the *is simulation was a continuation of the ﬁrst simulation corresponding ankle torque under the 1.5m/s vehicle accel- which was used as a comparative simulation to verify the power eration are illustrated in Figure 7. θ (°) θ (°) Stiffness (N/m) τ (Nm) Damping (Ns/m) q τ (Nm) q Journal of Robotics 7 –1 –2 –2 –3 05 10 15 05 10 15 Time (s) Time (s) Constant impedance Constant impedance Variable impedance Variable impedance (a) (b) Figure 7: Simulation results of constant impedance and variable impedance control. (a) Tilt angle. (b) Ankle torque. From this ﬁgure, it can be seen that the change trends of the robotic body tilt angle and the ankle torque have some diﬀerences. Compared to the results led by variable im- pedance control, the tilt angle under constant impedance, in reference to the equilibrium position, is larger, and the regulating process to equilibrium position is longer. Cor- respondingly, the ankle torque is also larger and the process is longer under the constant impedance, as shown in Figure 7(b). *is suggests that the proposed variable im- pedance control outperforms the constant impedance control in robotic standing balance control. 3.2. Prototype Experimentation. *e bipedal robotic proto- type was constructed as shown in Figure 8. In this experi- mentation part, the bipedal robotic prototype was standing on a moving vehicle to verify the eﬀectiveness of the pro- posed variable impedance control applied to the robotic prototype. During the complete experimentation implementation process, the movement of the vehicle includes three states: stationary, acceleration, and constant speed. For the vehicle acceleration about 1.0m/s , the bipedal robotic standing balance process is shown in Figure 9. From left to right, the Figure 8: *e photo of bipedal robotic prototype. bipedal robot is tilting backward, tilting forward, and ﬁnally returning to the balanced position. amount to adapt to the swing state of the bipedal robotic *e experimentation results of the bipedal robotic standing balance control based on the proposed variable trunk. *e changing process is shown in Figures 10(c) and 10(d). *e ankle torque was calculated based on the ankle impedance model are shown in Figure 10, which shows the experimental results of two arbitrarily selected standing variable impedance model, the maximum ankle torque was balance control processes. During the bipedal robotic 1.1Nm, and the minimum ankle torque was −0.5Nm, as shown in Figure 10(b). *e experimental results demon- standing balance control, the ankle impedance model stiﬀness and damping value are automatically updated strate the power of the proposed control system in the applicability of bipedal robotic standing balance. according to the changes of the virtual muscle activation θ (°) τ (Nm) q 8 Journal of Robotics Figure 9: Standing balance process of the bipedal robot. –1 –2 –1 02 4 6 8 10 12 02468 10 12 Time (s) Time (s) (a) (b) 1.5 30 1.4 1.3 1.2 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Time (s) Time (s) (c) (d) Figure 10: *e experimentation results of the bipedal robotic standing balance control. (a) Tilt angle. (b) Ankle torque. (c) Ankle stiﬀness. (d) Ankle damping. antidisturbance torque is calculated by constructing the 4. Conclusion ankle variable impedance model, and the ankle dynamic Aiming at the problems of poor robustness of commonly torque is obtained by constructing a bipedal robotic inverted used bipedal robotic control methods, this paper proposed a pendulum model. *e simulation and prototype experi- variable impedance control based on an artiﬁcial muscle mentation results based on diﬀerent vehicle acceleration activation model, which was used to generate the desired demonstrated the power of the proposed variable impedance ankle torque for bipedal robotic standing balance. Speciﬁ- control system in improving the robustness of bipedal ro- cally, the desired ankle torque was estimated combination of botic standing balance control. Although the proposed ankle antidisturbance with ankle dynamic torque, the ankle control system in this paper only targets the robotic ankle Stiffness (N/m) θ (°) Damping (Ns/m) τ (Nm) a Journal of Robotics 9 [12] T. Nitish and G. Hartmut, “Toward balance recovery with leg joint, it is readily applicable to the other robotic joints, which prostheses using neuromuscular model control,” IEEE remains as a piece of future work. Transactions on Biomedical Engineering, vol. 63, no. 5, pp. 904–913, 2016. Data Availability [13] N. Hogan, “Impedance control of industrial robots,” Robotics and Computer-Integrated Manufacturing, vol. 1, no. 1, *e data used to support the ﬁndings of this study are in- pp. 97–113, 1984. cluded within the article. [14] P. Song, Y. Yu, and X. Zhang, “A tutorial survey and com- parison of impedance control on robotic manipulation,” Conflicts of Interest Robotica, vol. 37, no. 5, pp. 801–836, 2019. [15] R. Yang, C. Yang, M. Chen, and J. Na, “Adaptive impedance *e authors declare that they have no conﬂicts of interest. control of robot manipulators based on Q-learning and disturbance observer,” Systems Science & Control Engineering, vol. 5, no. 1, pp. 287–300, 2017. Acknowledgments [16] W. He and Y. Dong, “Adaptive fuzzy neural network control *is work was supported by the National Natural Science for a constrained robot using impedance learning,” IEEE Transactions on Neural Networks and Learning Systems, Foundation of China (Grant no. 61903206), the project of vol. 29, no. 4, pp. 1174–1186, 2017. Science and Technology Department of Henan Province, [17] I. Bonilla, M. Mendoza, D. U. Campos-Delgado, and China (Grants nos. 202102310197 and 202102310524), and D. E. Hernandez-Alfaro, ´ “Adaptive impedance control of the Foundation of Henan Educational Committee (Grant robot manipulators with parametric uncertainty for con- no. 19A413011). strained path-tracking,” International Journal of Applied Mathematics and Computer Science, vol. 28, no. 2, pp. 363– References 374, 2018. [18] H. Geyer and H. Herr, “A muscle-reﬂex model that encodes [1] L. Yanfeng and Z. 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Journal of Robotics – Hindawi Publishing Corporation
Published: Jul 2, 2021
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