A Robust Adaptive Admittance Control Scheme for Robotic Knee Prosthesis Using Human-Inspired Virtual Constraints

A robust adaptive admittance control scheme using human-inspired virtual constraints was presented for a robotic knee prosthesis. The controller is able to deal with the wearer’s motion intention as well as the partial unknown parameter values of the prosthesis dynamics. The desired trajectory of the prosthetic knee joint is parameterized by the amputee-driven thigh phase variable and implemented by human-inspired virtual constraints rather than the preprogrammed time-dependent trajectory or human data replaying. A reference admittance model was set up to make the prosthesis be more compliant in the presence of ground reaction forces impact. A composite reaching law is proposed to cope with the chattering phenomenon and the finite-time convergence problem. The controller is designed by the back-stepping method based on Lyapunov. The tracking performance and the stability of the closed-loop system are proven via Lyapunov stability analysis. The feasibility and effectiveness of the proposed control scheme are proved in simulation results.


I. INTRODUCTION
Amputee's ambulation is usually slower, less stable, and more energy expenditure than able-bodied person [1]. Highperformance powered prosthesis, compared to passive prosthesis' not being able to provide active net power around prosthetic knee and/or ankle, could significantly restore the lost function of joints of lower limb amputee. Advances in mechanical design present new opportunities in prosthetic control systems, but current challenges are to develop efficient and smart controllers [2].
In powered lower limb prosthesis field, the prevailing control strategy is by far the finite state impedance control (FSIC) [3]- [8]. In FSIC scheme, a gait cycle is usually divided into four or five discrete stages according to foot events (heel stride, foot flat, mid-stance, toe off), and in each stage a passive spring-damp system that contains three impedance parameters (stiffness, damping, and equilibrium point) are set up. As long as these impedance parameter values are set appropriately and switch at a right time to The associate editor coordinating the review of this manuscript and approving it for publication was Aysegul Ucar . inject energy, this control strategy could enforce a desired gait motion patterns. Owing to FSIC have human-like impedance and passive characteristics, powered prosthesis based on FSIC will have an inherent stability when interacting with the amputee and environment. [9], [10].
Despite FSIC in [3]- [8] have aforementioned advantages, one shortcoming of such strategy is its relying on discontinuous representations of walking gait. The peoblem which brought is that there are dozens of impedance parameters need to be tuned in just one single locomotion mode. Moreover, they have to be retuned as locomotion modes changed and need to be personalized to individual user differences, such as different weight, height, and physical ability. Currently, prosthesis impedance parameters are usually tuned personalized manually with a time-consuming process [6]. In order to overcome obstacles of the parameters tuning process, techniques such as fuzzy logic-based expert systems [11] and reinforcement learning based method [12] have been proposed. Although they are all worked well, the dependence of switching rules is not fundamentally solved. That is the second disadvantage of FSCI: highly rely on a precise estimation of the switching time from one stage to another. VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ An estimation error of the switching time would cause the prosthesis to enact a wrong control model at a wrong time, that would potentially cause the wearer to fall. Another drawback of FSIC is that it can only control each joint independently without the ability of coordinating multi-joints of amputee and prosthesis [13]. An alternative strategy is continuous cotroller that employing unified representations of the entire human gait cycle. Unified gait continuous controllers do not suffer the aforementioned shortcomings of the non-unified gait control scheme. In [14], a CPG-based controller applying biological concepts was proposed for lower limb prosthesis. Nevertheless, the prosthesis is position controlled, that departs from the actuation type of humans. In [15]- [20], the continuous control method using virtual constraints was employed. Virtual constraints control was originally a biped robot control method using a monotonic and unactuated phase variable to parameterize the gait cycle in a time-independent manner [21]- [23]. Holgate et al. [15] are the first to use this method on a robotic prosthesis to parameterize the gait cycle across different stride lengths, and Robert D. Gregg group have developed a serious of robotic prosthesis with this control scheme [16]- [20]. By utilizing virtual constraints, trajectories of prosthetic joints would have a kind of humanlike characteristics of continuous and time-independent, that would potentially enable amputee subjects to walk at different gait speeds with the same controller.
Schemes discussed above are the two main types of control methods in the prosthetic legs field [2]. And the main difference between them is how to represent a gait cycle(continuous or discontinuous). As the above discussion, continuous control (e.g, implemented by virtual constraint control) seems to be more attractive, and it is also closer to the characteristics of people walking in continuously manner. However, the continuous control scheme also brings many challenges that need to be solved. Such as i) how to generate a reference trajectory that can reflect the wearer's motion intention; ii) how to achieve the flexibility or compliance of the prosthetic leg when it interacting with ground; iii) how to realize the adaptability and robustness of the controller so that the prosthesis can adapt to different wearers and adapt to more complex environments.
The prosthetic system should not interfere with the person and should be convenient for the wearer, that requires sensors should be installed on the body of the prosthesis, namely ''on board''. It brings great obstacles to human motion intention detection and trajectory generation. On the other hand, at present, continuous control based on virtual constraints is often implemented by feedback linearization method which depends on an accurate mathematical model of the prosthetic leg and the precise interaction force between the prosthesis and the amputee. That potentially present barrier to clinical viability when components like prosthetic feet are interchangeable. References [16]- [20] adopt impedance controller(essentially is similar to proportion differentiation(PD) controller) to approximately enforce virtual constraints, but one condition of this approximation is that the robotic dynamics and the interaction dynamics are required. however, the fact is that the dynamic nature of the prosthetic leg may vary from different wearer in that it is a human-in-the-loop system, and even for a single wearer it would be changed in cases with load carrying or not. An alternative is set the impedance gain to be large enough, but the side effects of this methods is that it would lead to a compliance-losing of prosthesis.
As the dual form of the impedance control, admittance control has also been widely used in the applications of pHRI [24] and the dancing robot [25]. Admittance control accepts a force as input and reacts with the robot's motions. The appropriate choice of the mass, damper, and spring coefficients can make the admittance control conform to the required effect. Compared with impedance control, one advantage of admittance control is that the robotic dynamics and the interaction dynamics are not required [26]. Another advantage of admittance control is that the expected trajectory of the robot is reshaped into a reference trajectory under the action of the interaction force with the external environment, and the compliance of the robot is completely determined by the admittance model. The low-level high-gain controller that designed to track the reference trajectory does not affect the flexibility or compliance of the prosthetic system. This feature brings great freedom to our controller design, and it also make many advanced robust adaptive control methods [48]- [55] are suitable for the prosthetic controller. According to our current knowledge, we have not yet seen a literature that systematically discusses the application of admittance control to prosthrtic legs.
Based on the above discusstion, the motivation of this paper is to provide a feasible control scheme to promote the development of continuous controllers for powered lower limb prostheses. The objective of this paper is that using the human-inspired virtual constraints for prosthesis so that it can reflect the wearer's motion intention, and introducing admittance model to make the prosthesis be more compliance, and designing a robust adaptive controller in lower-level controller so that the prosthesis can be more adaptability and robustness. The block diagram of the proposed robust adaptive admittance control scheme using human-inpired virtual constraints(RAAVCC) that composed of an inner and a outer loop is shown in Fig.1. Main contributions of this research work are summarized as following: 1) A novel RAAVCC scheme that can unify the entire gait cycle was proposed. In which the dividing of gait cycle and switching rules are not needed. 2) The desired trajectory of the prosthetic knee joint is parameterized by the amputee-driven phase variable and implemented by human-inspired virtual constraints. It is time-independent and can coordinate the amputee's residual hip joint and prosthetic knee joint.
3) The prosthetic leg would have a compliant characteristic consistent with the prescribed admittance model by reshaping the desired trajectory to the reference trajectory in the presence of ground reaction forces (GRFs) impact. 4) A novel composite reaching law(CRL) combing of a power reaching law (PRL) and a variable rate reaching law(VRRL) was proposed. That would make the controller have the ability to deal with uncertain parameters and estimation errors of external force and torque, and effectively suppress chattering and ensure finite time convergence. The rest of this paper is organized as follows: Section 2 establish the prosthetic leg model and GRFs model. Section 3 presents the design of the controller. Section 4 shows the simulation results and discussion. Conclusion and future work are drawn in Section 5.

II. PROSTHETIC LEG MODEL AND GROUND REACTION FORCES MODEL A. PROSTHETIC LEG MODEL
In this paper we consider a case that an unilateral aboveknee amputee wear a powered knee prosthesis (the ankle is passive), and the planar prosthetic leg model and GRFs model corresponding to this case are shown in Fig.2. The prosthetic leg model is composed of human subsystem (the grey thick line) and prosthesis subsystem and they are assumed to be rigidly attached to each other through a prosthetic socket. Based on the fact that amputee intrinsically tends to control the dynamics of the residual thigh on which effects of the prosthesis subsystem would also be directly executed, it is reasonable to represent the dynamic of the human subsystem by dynamics of the residual thigh and hip [27], [28]. Another fact is that when the prosthesis is at the end of the support period (only the prosthetic toes is in contact with the ground), the system is stay in an under-actuated stage. However, considering that the stability of the humna-robot system is mostly borne by the wearer and the duration is relatively short, so we assume that the whole system is fully actuated.  Fig.2, the human subsystem consists of the residual thigh and hip and a point mass which represents the upper body. The prosthesis subsystem consists of a prosthetic thigh, a prosthetic shank and a prosthetic foot. The full planar model has four active degree of fredom(DoF): the horizontal and vertical DoF of the human hip and the rotational DoF of the residual thigh and the prosthetic knee(in this paper the DoF of prosthetic ankle is passive). The dynamic equation is expressed as following:

As shown in
where q = q T h , q k T is the vector of generalized coordinates of the full model and q h = p x , p y , q t T is the configuration vector of human subsystem(p x and p y is the horizontal and vertical displacement of the human hip respectively, q t is VOLUME 8, 2020 the absolute angle of the residual thigh relative to the world frame, and q k is the prosthetic knee angle). M (q) ∈ R 4×4 , C (q,q) ∈ R 4×4 , G (q) ∈ R 4 and f dis (q,q) ∈ R 4 are the inertia matrix, coriolis and centripetal matrix, gravity vector, and friction and damping vector respectively. τ ext is the external force and torque vector resulted from GRFs. u = F x F y τ h τ k T is the applied force and torque as the input item. For a practical systems, they always suffer from issues such as unmodeled dynamics, velocity unavailability, parametric uncertainties, etc. In this paper, we only consider the parametric uncertainties (M (q) ,C (q,q), and,G (q)), estimation errors of friction and damping force(f dis (q,q)) and estimation errors of the external force and torque vector (τ ext ) resulted from GRFs. In section iii, we will design an adaptive controller that is responsible for handling parametric uncertainties, and design a robust controller to cope with estimation errors. Regarding problems of unmodeled dynamics and speed uncertainty, literatures like [51] and [52] have provided effective solutions to them.
There are the following properties.

Property 3 ([31]):
There exists a parameter vector θ with components depending on robotic prosthesis parameters (masses, moments of inertia, etc.), such that for the prosthetic leg dynamics, we have where Y (q,q, v,v) is called the dynamic regressor matrix and v is a vector of differentiable function. The specific forms of Y (q,q, v,v) and θ are given in Appendix B.

B. GROUND REACTION FORCES MODEL
Ground reaction forces (GRFs) are generated during the gait as the result of interaction between the foot and ground. Such reaction forces are then transferred up to the ankle, knee, and hip joints with the effect of altering the joint angular positions. GRFs shouldn't be ignored during the study of the human gait. The reason why the prosthesis needs to be compliant is just to prevent excessive ground impact response, especially when the heel strikes the ground.
To better simulate the real situation, GRFs model is created as follwing [32], [33]: where f GRFy and f GRFx are vertical and horizontal force components; y pen andẏ pen are the penetration and penetration rate; c 1 is the spring coefficient, e is the spring exponent and c 2 is the damp coefficient and µ is the friction coefficient. In order to smoothly ramps up the damping force as contact penetration increases, the damp coefficient c 2 is desgined as a function that varies linearly between 0 and c max (maximum damping coefficient) as y pen varies between 0 and y max (maximum penetration). The c 2 is 0 for y pen < 0, and c max for y pen > 0. Effects of GRFs on each joint is τ ext = J T f GRFx , f GRFy and given as follows: where J is the jacobian matrix which reflex the relationship between GRFs and joint torques of the robotic prosthesis.
Assumption 1: The external force and torque vector τ ext result from GRFs have a upper bound. namely, there exist inequation τ ext τ m , where τ m denotes the maximum joint force and torque vector result from GRFs.

III. CONTROLLER DESIGN A. HUMAN-INSPIRED VIRTUAL CONSTRAINTS FOR A POWERED KNEE PROSTHESIS
This section is to obtain the desired trajectory of the prosthetic knee joint through implementing human-inspired virtual constraints. When an able-body people walking in a certain motion pattern, there is an accordingly specific relationship h (q t , q k ) = 0 which is called virtual constraints between the angle of the thigh (or hip) and the angle of the knee (and the ankle). Inspired from human walking, if prosthetic leg can reproduce virtual constraints similar to those of ablebody people, it would reproduce the motor characteristics of healthy people. In order to get an explicit expression from the virtual constraints h (q t , q k ) = 0 as shown in equation (12), we use the thigh angle q t and the thigh angle speedq t to get the phase variable. We divide the process into two steps: the first is to obtain the phase variable from residual thigh angle and speed, and the second is acquire the desired joint trajectory of knee form the explicit joint trajectory function which is fited from the data of able-body walking.

1) DERIVING THE THIGH PHASE VARIABLE
Phase variable should be monotonicity, under-actuation, and boundedness [34]. Monotonicity over time helps distinguish precisely where the process is, under-actuation makes the phase variable independent from the controlled process itself, and boundedness makes it realizable and meaningful. Prosthesis, unlike bipedal robot of which all states can be measured, is a human in the loop system of which the human subsystem states are usually unknown. That once brought obstacles to the construction of phase variable of prosthesis. Thanks to the cosine-like property of the angular of the residual thigh(q t shown in Fig.1), such that the phase variable can be constructed by the phase angle from the phase plane of the thigh angle vs. thigh velocity [35].
The phase variable could be computed using the four-quadrant atan2(y, x) function as where w is used to make the phase portrait more circular so that ϕ (t) approximates a linearity function of time. It can be computed from the range of motion of the phase portrait.
In practice the derivate of the thigh angle would bring huge noise. Consider the fact that the derivative of a cosine function has a linear relationship to its integral, given as follows: So we can compute a phase angle utilizing angular position and its integral: where Φ (t) = t 0 q t (τ )dτ is the thigh angle integral. The phase variable is normalized between 0 and 1 using:

2) IMPLEMENTING OF HUMAN-INSPIRED VIRTUAL CONSTRAINTS
Taking advantage of the periodic kinematics observed in human gait in [36], the method of discrete Fourier transform (DFT) could be used to construct virtual constraints for a powered prosthesis controller, and desired trajectories of prosthetic joint could be obtained from virtual constraints when parameterized by phase variable. Let x[n] be evently distributed discrete data of the joint trajectory which is sampled from able-body walking. The discrete frequency components X [k], as a sequence of complex numbers, is transformed from x[n] by the DFT [35]: where N is the finite number of samples, K is the running index for the finite sequence of k, and W kn N = e −j(2knπ/N ) is the complex quantity. The number of discrete frequencies of X [k] is finite when the time-domain signal x[n] is periodic.
The original signal can be reconstructed using Fourier Interpolation after obtaining the frequency content terms X [k]: where X [k] and W −kn N = e j(2knπ/N ) are standard complex form. In x[n], only the real part is remained in that the joint kinematic signals are real numbers.
Using Euler's relationship, Eq.(11) can then be decomposed as a summation of sinusoids and the desired trajectory function of the prosthetic knee joint is given as:

B. ADMITTANCE MODEL
A powered prothesis is essentially a robot that has interaction force with its human user and external environment (usually the ground). Huge ground impact forces resulted from the heel striking on the ground will cause the prothesis system to deviate from the desired stable manifold (or desired trajectories). Large control actions are demanded if only rely on a controller to prevent the deviation caused by the gigantic ground impact forces. That may lead to the compliance-losing and damage the prosthetic motor and transmission and cause unpleasant shock forces transfer to the human body. The consideration of interactions between a robot and its external environment motivated the development of impedance and admittance control [37]- [42], its idea is to regulate both position and force by specifying a dynamic relationship between them. Considering the prosthesis dynamic (1) is a second order system, we thus define a second order target admittance model [43] that have characteristics similar to able-body walking [44].
where D r ∈ R 4×4 , B r ∈ R 4×4 and K r ∈ R 4×4 are desired inertia, damping, and stiffness of the target admittance model with diagonal matrix form, respectively. Desired trajectories q d = p xd (t) , p yd (t) , q td (t) , q kd (σ h ) T would be reshaped to reference trajectories q r = [p xr (t) , p yr (t) , q tr (t) , q kr (σ h )] T in the presence of GRFs impact. The two poles of the admittance are given as follows: Selecting appropriate values of D r , B r and K r so that both poles are in the left half-plane, the admittance model will become stable and would make the prosthesis be compliant especially when the heel hits the ground. Note that, in practical applications, the damping parameters of the motor can be used as part of the damping coefficient of the admittance model.

C. ROBUST ADAPTIVE CONTROLLER DESIGN
The proposed RAAVCC control framework is shown in Fig.1. In this two-loop control scheme, reference trajectories are obtained from the outer loop. And the robust adaptive controller was designed in the inner loop to deal with unknown VOLUME 8, 2020 dynamics of the robotic prothesis, external force and torque estimation errors, and the finite time convergence.
According to Eq.(1), let x 1 = q, x 2 =q, then the dynamic of the prosthetic leg can be rewritten in the following form: We design the controller with the backstepping method which was first proposed in [43]. Tracking errors z 1 = [z 11 , z 12 , z 13 , z 14 ] T and z 2 = [z 21 , z 22 , z 23 , z 24 ] T are defined as follows: where a 1 ∈ R 4 is the virtual control to z 1 , we havė Considering a Lyapunov function candidate V 1 = 1 2 z T 1 z 1 , the time derivative of V 1 iṡ Let a 1 =ẋ r − K 1 z 1 with K 1 = diag (K 11 , K 12 , K 13 , K 14 ) and λ min (K 1 ) > 0, the Lyapunov function can be written in the following form:V Design the sliding mode surface s = [s 1 , s 2 , s 3 , s 4 ] T as: where K 2 = diag (K 21 , K 22 , K 23 , K 24 ) and λ min (K 2 ) > 0. Combining Eq.(18) and Eq.(16), we havė (23) Consider a Lyapunov function V 2 = V 1 + 1 2 s T Ms, so the time derivative of V 2 iṡ Substituting Eq.(23) into Eq.(24), we havė As discuss in section II, M , C and G all include uncertain parameters. We use the Property 3 to deal with unknow dynamics. Where parameter vector θ have many realizations and in this paper θ ∈ R 10 . Let: Applying Property 1 and Property 3, then Eq.(25) can be written aṡ G+f dis is the regressor matrix. Definingθ = θ −θ andτ ext = τ ext −τ ext as the parameters estimate errors and external force and torque estimated errors. Rewrite Eq.(28) as: Considering a Lyapunov function V 3 = V 2 + 1 2θ T γθ, where λ min (γ ) > 0 and λ max (γ ) <∝. we havė In order to have an adaptation mechanism, the item γθ + Y T (q,q, v,v) s is constrained to be zero and in turn we obtaine the adaptive law aṡ Substitute Eq.(31) and Eq. (21) into Eq.(30), we havė The control law was designed as follows: where Y (q,q, v,v)θ is an adaptive control term that is responsible for handing the uncertain parameters, and u R is the robust controller to be designed. Considering that sliding mode control is insensitive to external interference and internal parameter change and has excellent robustness [48]- [55], we adopt it as the robust controller. However, the chattering phenomenon is a problem in sliding mode control. An effective method to reduce chattering is to design the boundary layer [27], but it is at the expense of tracking performance. One of other methods to suppress chattering is by designing reaching law such as power reaching law [45]. The advantage of power reaching law is that it wouldn't sacrifice tracking performance due to there no need to set up a boundary layer. However, the drawback is that the convergence rate is particularly slow when the state is close to the sliding surface [46]. In order to solve this problem, we proposed a composite reaching law (CRL) which combining a power reaching law(PRL) and a variable rate reaching law(VRRL). As is shown in Fig.3, advantages of CRL as following: when |s| 1, PRL guarantees a fast convergence speed of the system. when |s| < 1, VRRL guarantees a fast convergence speed, and the width of the sliding zone is modulated by the tracking error.
Based on the above discussion, the robust controller designed as follows It comprises three different parts. The first part, −hs, is used to compensate tracking errors, and where h = diag (h 1 , h 2 , h 3 , h 4 ). The second part is PRL and has the following forms: Note that the exponentiation operations for |s| in the term |s| β are interpreted element-wise, namely, |s| β = |s 1 | β 1 , |s 2 | β 2 , |s 3 | β 3 , |s 4 | β 4 T . The third part is VRRL and has the following forms: The division and saturation operations for |z 1 | are interpreteted element-wise, namely, sat Theorem 1 (Considering the prosthetic dynamics 1): that satisfies Assumption 1 and assuming the required tracking accuracy is set as |z 1 min |, using the adaptive law proposed in (31) and the control law proposed in (33), the trajectory tracking error will converge to the region: The proof of Theorem 1 is provided in Appendix A.

IV. SIMULATION STUDIES
In this section, the proposed RAAVCC scheme shown in Fig.1 is evaluated using a prosthetic leg model for the task of level ground to prove its feasibility and effectiveness.

A. MODELS AND DESIRED TRAJECTORIES
In order to verify the feasibility of the RAAVCC scheme, we built a simulation model in the Simscape environment corresponding to the experimental platform of our laboratory (as shown in fig.4). As shown in Fig.2, an above-knee    Table 2. In order to simulate the knee-above amputee wear a prosthesis leg with active knee joint and passive ankle joint, trajectories of human subsystem represented by the hip and thigh and prosthesis subsystem represented by the knee are required. These desired trajectories that generate stable level ground gait of this paper are obtained from gait experiment data on human subjects in [36]. Desired trajectories function of time of hip and thigh, q hd (t) = p xd (t) , p yd (t) , q td (t) T , are reconstructed by DFT with K indices of 8, 6, 8 respectively. The desired knee trajectory function of time, q kd (t), whose time parameter will be replaced by the phase variable σ h in virtual constraints, is reconstructed by DFT with K indices of 8. Velocity and acceleration trajectories of them can be obtained using an analytical differentiation of the corresponding function. VOLUME 8, 2020  For better simulate the impedance characteristics of human and make the system be more similarly to an able-bodies leg(avoiding be too soft or too hard), we set up a corresponding admittance model at each DoF. According to Eq.(14), we define all the target admittance model with two real roots at −5 and −107 for the knee, −10 and −109 for the thigh, and −4 and −558 for both horizontal and vertical displacement of hip. They will make the target admittance model be more stable and guarantee the tracking performance of desired trajectories with an error <0.05 rad. The corresponding admittance model matrices are choosen as: Dr = diag (8,8,8,8), Br = diag (4500, 4500, 950, 900), Kr = diag (18000, 18000, 8500, 4500).
The phase plane of the thigh angle q t (t) vs. its intergral during five complete gait cycles is shown in Fig.5 (a). It shows   that the phase plane is a closed approximately circular curve. The normalized phase variable (PV) computed from phase plane is shown in Fig.5 (b), we can see that during a entired gait cycle the phase variable is monotonic and approximately linear with respect to time or gait percentage. With these features the prosthetic knee joint angle can be parameterized by the phase variable, and it is shown in Fig.5 (c). It can be seen that the prosthetic joint angle is driven by the amputee rather than a predetermined time trajectory used in [27]. The physical meaning of this result is that the reference trajectory VOLUME 8, 2020  of the prosthetic joint is completely driven by the thigh of the residual limb, so that the prosthesis can reflect the person's movement intention.
States of the closed-loop system and the desired trajectories are shown in Fig.6 (a ∼ d), and the GRFs are shown in Fig.6 (e ∼ f )). It can be seen that the vertical force of GRFs has a double-peak characteristics which is similar to able-body walking. The controller is demonstrated to be robustness and the walking behavior of the prosthesis is also proved to be similar to human-like walking. Notice that in order to walk normally and make the amputee model be more flexibility in the presence of the GRF impacts, the hip vertical position is not perfectly follow its desired trajectory and also should not perfectly follow it. In order to simulate the real situation as much as possible in the simulation, this paper simulates the human body subsystem, and the corresponding control of the human body subsystem is also provided, but this is not necessary in practical applications.
The horizontal direction is only affected by the horizontal GRF, which is far less than vertical GRF and gravity, such that the hip horizontal displacement controller would easily handle it. So next we would not discuss the hip horizontal displacement controller.
The hip force and thigh torque acting on the human subsystem and the knee torque acting on the prosthesis are shown in Fig.7. It can be seen that they all have similar magnitude as able-body force or torque. Moreover the control magnitudes of the controller vary little when parameters have an ±50% deviations. This shows that the proposed controller has great potential in real-world prosthesis applications. To a certain extent, the force received by the hip joint represents the impact of dynamic prosthetics on the disabled, and is closely related to the secondary injury caused by the disabled after wearing the prosthesis. Fig.8 shows the estimated parameter vector θ when the system parameters vary by ±50%. We can see that the estimated parameter values don't perfectly match its true but are bounded. They periodically change when there exist impacts result from GRFs. However, our goal is to have a good tracking performancen(shown in Fig.11) not the accurate parameter estimation. In addition, in the simulation, friction forces are set for the horizontal and vertical driving devices of the hip joints, and damping forces are set for the rotational actuator of the hip joints and knee joints. However, considering that the prosthetic leg is directly worn by an amputee in practical applications, there will be no so-called hip joint friction. At the same time, the damping force of the prosthetic knee joint can also be used as the damping force term of the admittance model, so Fig.8 does not discuss the friction and damping force of the joint.
Estimation errors of external force and torque which are resulted from parameter deviations are shown in Fig.9. They are external disturbances to the system and couldn't be directly compensated in controller. In addition to, when the heel hits the ground, system trajectories would deviate from their desired trajectories. That's why the controller needs to have robustness. Fig.10 shows the tracking perfoemance of the proposed composite reaching law(CRL) which is composed of the power reaching law(PRL) and the variable rate reaching law(VRRL). It can be seen that the proposed CRL has an ability to make tracking error converge to zero quickly in a finite time. And VRRL has a faster rate of convergence when |s| < 1, that is in good agreement with the previous theoretical analysis. As is shown Fig.11, tracking errors of the prosthetic system is quite small, which demonstrate the proposed controller is insensitive to external interference and internal parameter change and has an excellent robustness.

V. CONCLUSION AND FUTURE WORK
This paper presented a RAAVCC control scheme which consists of two loops for a robotic knee prosthesis. The outer loop was utilized to build virtual constraints and set up admittance models. Virtual constraints are implemented to obtain the desired trajectory of the prosthetic knee and it could reflect the human's motion intention and coordinate joints of human and prosthesis. Admittance models can make the prosthesis leg have a compliant characteristic which is similarly to an able-bodies leg(avoiding being too soft or too hard). The inner loop was used to deal with partial unknown system parameters and the estimated errors of the external force and torque. The proposed composite reaching law makes the controller have the ability of fast convergence. We performed simulations with ±50% parameter deviations. Simulation results show that virtual constraints have unchangeableness under the parameter deviation, which demonstrates the virtual constraints themself also have certain robustness. Although parameter identification did not converge to their true values, the controller is also able to make tracking errors converge to zero quickly in a finite time in the presence of GRFs impacts and estimated errors of the external force and torque. Which demonstrates the robustness and tracking performance of the proposed controller.
In the future work, we would consider other important aspects of the proposed RAAVCC scheme, including the following: designing a better adaptive controller to achieve accurate parameter estimation and make the propose scheme feasible and reliable in real-world prosthesis control; extending the scheme to multi-joints(powered knee-ankle prosthesis) and multi motion modes such as different walking speeds and different slopes and so on; and pursue a design of variable impedance controller based on learning method such as reinforcement learning.
Lemma 1 (Based on [44] for continuous-time systems and [47] for discrete-time systems): A continuously differentiable function V is an exponentially stabilizing control Lyapunov function and will converge to zero in finite time if there exist positive constants η 1 , η 2 , η 3 > 0 and ζ > 0 such that The Lyapunov function of V 3 consists of parameter estimation errors, trajectory tracking errors and velocity tracking errors. Appling the property 2 and λ max (γ ) <∝, it is easily to conclude that V 3 have lower and upper bound. There must exist scalar constants η max and η min that makes the Lyapunov function of V 3 satisfy the following relationship: |s i | τ i max − ρ ai |s i | βi (44) where ξ = ε min 2 η max , when We haveV 3 + ξ V −ρ a dt (48) the reaching time of stage 1 can be calculated as Stage 2: From s = 1 to s = 0. As shown in Fig.3, there approximatly exist relationship S b S a in this stage. The effect of S a can be ignored and we have: the reaching time of stage 2 can be calculated as Let the required tracking accuracy is set as |z 1 min |, assuming trajectories tracking errors in stage 2 have |z 1 | |z 1 min |, the Eq.(52) should be rewritten as follows The total reaching time is the sum of reaching time of stage 1 and stage 2, we have m s + m t + m tor 2 * m s * L 2 sc +m s * L 2 t +2 * m t * L 2 tc +I s +I t 2 * m s * L 2 sc + I s L t * m s + L tc * m t L sc * L t * m s L sc * m s fric px fric py damp qt damp qk where v andv is defined in equation (26) and equation (27) respectively and: Y 14 =v 3 * cos (q t ) −q t * v 3 * sin (q t ) Y 24 =v 3 * sin (q t ) +q t * v 3 * cos (q t ) Y 34 =v 1 * cos (q t ) + g * sin (q t ) +v 2 * sin (q t ) Y 35 = 2 * v 3 * cos (q k ) −v 4 * cos (q k ) −q k * v 3 * sin (q k ) − v 3 * cos (q k −q t ) * (q k −q t )+v 4 * cos (q k −q t ) * (q k −q t ) Y 36 =v 1 * cos (q k −q t )−g * sin (q k −q t )−v 2 * sin (q k − q t ) Y 46 = g * sin (q k −q t )−v 1 * cos (q k −q t )+v 2 * sin (q k −q t )