A composite position control of flexible lower limb exoskeleton based on second-order sliding mode

Aiming at the problem that the trajectory tracking error of flexible lower limb exoskeleton robot is too large under the condition of external disturbance and parameters uncertainty, a composite position control method based on second-order sliding mode control was proposed. Firstly, the flexible lower limb exoskeleton robot is modeled by Lagrange function. Secondly, considering that the system is affected not only by matched disturbance but also by unmatched disturbance, two finite time state observers are used to observe and compensate the two disturbances in real time. The super-twisting method is employed in the position control section to ensure that the trajectory tracking error of the knee joint converges to zero in finite time. Finally, the Lyapunov function proves the stability of the suggested control technique. The experimental findings reveal that the suggested control approach has a better trajectory tracking effect and resilience than classic PD control and sliding mode control, indicating its superiority.


Introduction
With the aging of World's population becoming more and more serious, lower limb movement disorders caused by stroke and other diseases have become an increasingly common problem in the elderly. However, these patients have a high chance to stand up again through rehabilitation training [1]. Traditional rehabilitation training needs to be accompanied by medical staff, which consumes a lot of manpower and material resources. Exoskeleton robots have made significant advances in military, medical, rehabilitative, and other domains in recent decades. Exoskeleton robots with rehabilitative capabilities have the potential to minimize the conventional use of human and material resources.
Rehabilitation training is mainly divided into two stages. The first stage is passive training, which means that human legs are driven by the exoskeleton to move in a certain trajectory. The second stage is active training, when the passive training reaches a certain level and the lower limbs can move autonomously, the exoskeleton will assist the lower limbs in training. Compared with rigid exoskeleton robot, flexible lower limb exoskeleton robot has the advantages of easy human-machine interaction and easy to deal with external disturbance and impact. Because the flexible lower exoskeleton is impacted by external disturbances and structural parameter uncertainties, designing an appropriate controller to enable accurate trajectory tracking of the flexible lower exoskeleton system is a critical challenge. Domestic and international researchers have made significant advances in the trajectory tracking of exoskeleton robots. Because of its high resilience, sliding mode control (SMC) is commonly employed in robot motion control. For example, the integrated terminal sliding mode controller is designed for the exoskeleton robot with external disturbance and model uncertainty. The simulation results show that the controller has good tracking performance [2]. In [3], a robust adaptive integral terminal sliding mode controller is proposed for the upper limb exoskeleton robot. Experiments show that the controller has good robustness. In [4], in view of the ankle joint robot, combining sliding mode control with the extended state observer. The uncertainties and external disturbances of the system are observed by the observer, experiments show that with good anti-disturbance capability. But it is undeniable that the traditional sliding mode control has the phenomenon of shaking and cannot effectively deal with the mismatched disturbance [5,6].
In addition to sliding mode control, many other nonlinear control methods are also used in this field. In [7], a method combining adaptive impedance control and time-delay disturbance observer is proposed for the upper limb rehabilitation robot. Experiments show that the proposed method has good anti-interference ability. An adaptive controller is designed to solve the disturbance and uncertainty in system [8]. In [9], the backstopping method is used to design the controller. The flexible lower limb exoskeleton has the advantages of easily handling external shocks and suitable for human-computer interaction. However, the model of the system is becoming more and more complex due to the addition of flexible elements. Traditional control methods, such as PD control, has been unable to obtain a good trajectory tracking control effect [10][11][12][13][14][15][16].
In this paper, we propose a second-order sliding mode composite position control method(SOSMCC). The novelty and contributions of this work can be organized into three aspects: (1) a composite position control method based on second-order sliding mode control is designed to solve the path tracking control problem of the flexible lower limb exoskeleton robot which can sig-nificantly weaken the adverse effects of chattering brought by the traditional SMC. (2) By combining the second-order sliding mode controller (SOSMC) and finite time state observer techniques (FTO), a composite control method is proposed to avoid high gains in the designed SOSMC controller. In fact, the lumped disturbance is directly estimated by using FTO technique, and it can synchronously compensate the lumped disturbance and reduce the gains of the derived SOSMC controller. (3) Two finite-time state observers are used to observe disturbances in the system, both match and mismatch disturbances are simultaneously considered in the control laws design, which makes the designed control methods more maneuverable in real control implementation. The remaining part of the paper is organized as follows: the dynamic model and the objective are stated in Sect. 2. Section 3 gives the design of the composite position control. Section 4 gives the experimental based on the single-joint flexible lower limb exoskeleton system. Section 5 summarizes the paper briefly.

Model description
The flexible lower limb exoskeleton system adopted in this paper includes flexible transmission mechanism, leg connecting rod, motor and reducer, etc. The servo motor's output torque is transferred to the torsion spring at the joint through the double lasso transmission system, and the torsion spring forces the joint to move. Combined with motor motion equation and lasso force analysis, the dynamics model of the whole flexible lower extremity exoskeleton was obtained through Lagrange function: where q = [q 1 , q 2 , ..., q n ] T , θ = [θ 1 , θ 2 , ..., θ n ] T is the rotation angle at the joint and the rotation angle at the side of the motor. J, B, K ∈ R 5x5 , respectively, represents the moment of inertia on the side of the motor, the damping and the elastic coefficient matrix of the torsion spring. w, F, r, μ, κ, respectively, represents angular velocity of the noose transmission system, the preload of the noose, the radius of the executing wheel, the friction coefficient and the curvature of the noose. M(q) is the moment of inertia matrix of the exoskeleton, V (q,q) is the Matrix of Coriolis force and centripetal force, G(q) is the gravity matrix of the exoskeleton robot, τ is the control torque vector. To quickly verify the advantages of the algorithm, only the knee joint was tested in this paper. To facilitate the design of the controller, let Transforming Eq. (1) into the following state space equation: where and w 1 , w 2 is the external disturbance on the motor side and joint side, d 1 , d 2 is the unmatched disturbance and the matched disturbance in the system, including the unmolded dynamics of the system and external disturbances.

Design process of controller
Before designing the controller, the following definitions and lemmas are given: Definition 1 sig(·) α = sign(·)|.| α α > 0. Lemma 1 [17] If there exists a continuously positive definite function V (t) for any t > t 0 satisfy: Then, the system converges to the equilibrium point in finite time, and the convergence time satisfies: where t 0 is initial time, η > 0, λ > 0, 0 < α < 1. Lemma 2 [18] Considering system: Giving the following finite time state observer (FTSO): where γ =− p q ∈ (−1 (n + m), 0) , p and q represent, respectively, positive even and positive odd. When choosing the appropriate to make the roots of following characteristic polynomials, all lie on the left half plane of the complex plane S which makes the state of the detector converges in finite time to the true state of system (6).
Considering the existence of matched and unmatched disturbances in the system, the sliding mode control is not robust to the unmatched disturbances. According to Lemma 2, two finite time state observers (FTSO) are given for systems (2) and (3), respectively, to observe the disturbances and their derivatives, and a feedforward compensation is made for the controller. The two observers are shown as follows: Equation (8) is the form of unmatched disturbance observer Equation (9) is the form of matched disturbance observer wherex 1 ,x 2 ,ẑ 0 ,ẑ 1 ,ẑ 2 represent the estimate of x 1 , x 2 , d 1 ,ḋ 1 ,d 1 , respectively;x 3 ,x 4 ,η 0 ,η 1 ,η 2 represent the estimate of x 3 , x 4 , d 2 ,ḋ 2 ,d 2 , respectively. According to Lemma 2, choosing appropriate values of λ 1,i λ 2,i (i=1, 2, · · · , 5), the state estimation errors of (8) and (9) will approach zero in finite time. In order to enable the exoskeleton to quickly track the desired reference curve and reduce the impact of chattering, the improved super-twisting algorithm proposed in literature [19] is adopted.
Considering the influence of disturbance in the system, the following sliding mode surface with mismatched disturbance estimation is given in combination with the observer of Equations (8): where r − x 4 −ẑ 1 Then, we can get derivative of Eq. (11) with respect to time: According to the super-twisting algorithm proposed in reference [20], by combining Eqs. (2), (3) and (10), the composite controller based on the super-twisting algorithm is obtained as follows: where k 1 , k 2 and k 3 are controller gains greater than 0. The controller block diagram is shown in Fig. 1.

Stability analysis
Substituting Eq. (13) into Eq. (12):ṡ Let w = z m +η 0 −d 2 +c 2 (ẑ 0 −d 1 )+c 3 (ẑ 1 −ż 0 )−ż 1 we can get: According to the hypothesis in literature [12], we can get φ = r (t)sign(s), |r | ≤ φ max . We assume X T = x 1 x 2 = |s| 0.5 sign(s) w and define a subspace v i = (s i , w i ) ∈ R 2 |s i = 0 the time derivative of X except in the space of v i can be given as follows: where A= The Lyapunov function can be given as follows: where P = k 2 1 + 1 −k 1 −k 1 k 2 + 1 is a symmetric positive definite matrix. The time derivative of (17) can be yielded as follows: Making (18) split as follows: wherẽ By satisfying the following conditions, it can makė V (X ) < 0.
For (20) and (21), we can get Equation (22) holds obviously. Equation (23) can be written as follows: is a monotonic increasing function. When f (0) ≥ 0, −a 2 2a 1 ≥ 0, there exists a condition make >0 hold always. So, the following conditions which we should to satisfy Equation (28) holds always. By (25) and (27), we can get If Eqs. (24), (25), and (29) are satisfied, which tells us thatV (X ) is negative definite almost everywhere expect on the space. We can divideV (X ) into the following form: where

Fig. 2 Experiment platform
According to Formula (30), it can be obtained: where λ min is the minimum eigenvalue of the matrix. ||.|| is the Euclidean norm. Because of λ min (H )||X || 2 ≤ X T H X ≤ λ max (H )||X || 2 , we can get x 1 | ≤ ||x|| ≤ V 0.5 (X ) λ 0.5 min (P) .(31) can be transform as follows: where γ 1 =λ 0.5 min (P)λ min (H 1 ) λ max (P) γ 2 = λ min (H 2 ) λ max (P) According to Lemma 1, the system state variable converges to the sliding mode surface in finite time, and the convergence time is t s : When the tracking error of the system reaches the sliding mode surface, it gradually converges to the equilibrium point, and the proof is completed.

Experimental results
In this paper, an experimental platform of flexible lower limb exoskeleton robot as shown in Fig. 2 was built. The  As for the safety, two stop blocks are fixed on the chassis, so the orthosis's range of motion is the safe moving range of human knees. The control system's hardware design comprises primarily of an industrial personal computer as the host computer, a data collection card, a potentiometer contained in the active knee joint to measure the angle and velocity of the knee joint, and a servo controller. In the experiment, the control period is set to 1ms. Experimental verification was carried out on MATLAB2019b by Simulink algorithm. Table 1 shows the parameters of flexible lower exoskeleton system. A. PD Control and SMC To verify the effectiveness of the proposed algorithm, PD control and traditional SMC are selected as the comparison algorithm. PD control is divided into two cascade PD controllers, the inner ring PD controller to get the desired motor rotation angle, and then through the outer ring PD controller to get the motor output torque. SMC adopts the sliding mode surface of Eq. (11) and the approaching law of ksign(s). The knee joint reference curve is taken as q r = 20 sin(0.5 * t − 0.5 * pi) + 20. Experimental parameters of PD control without disturbance are shown as follows: inner and outer loop controller parameters, respectively, are k p1 = 0.48, k d1 = 0.005, k p2 = 0.36, k d2 = 0.01. PD control parameters under external disturbance are shown as follows: k p1 = 0.54, k d1 = 0.008, k p2 = 0.42, k d2 = 0.025.It is stipulated that the sliding mode surfaces shown in the selection formula (11) in this paper should meet the following polynomials: p 1 (s) = (s + w p ) 2 (s + z), so we can get: c 1 = zw 2 p , c 2 = (2w p z + w 2 p ), c 3 = (z + 2w p ) . In this condition, tradition SMC with no load, we choose z = 2, w p = 12, k s = 30. Under load, we choose the same parameters about sliding mode surface, but choose k s = 50 . Figure 3 shows the trajectory tracking diagram, error curve diagram and control input diagram of PD control without load on the end of knee joint. Figure 3a, b shows that PD control cannot accurately track the trajectory reference curve due to the existence of disturbances, and the error range is within (±1.5 0 ). Figure 4 shows the curve at load. Figure 4b shows that the error curve reaches in the rising stage. Figures 3a 4a show that when the exoskeleton robot runs to the top and bottom, there is a flat-top phenomenon, which is caused by the time delay of the double lasso transmission system. Figure 5 shows the trajectory tracking diagram, error curve diagram and control input diagram of traditional SMC without load. Figure 5a, b shows that due to the  Fig. 6a, b, we can get that there is a large trajectory tracking error in the case of load. This is because sliding mode control is not robust to mismatched disturbances, and considering the impact of chattering on the system, the value of k cannot be selected too large.
B. Second-order sliding mode composite control (SOSMCC) Under the proposed method, the parameter selection of the two observers should satisfy the polynomial in Lemma 2. Select the following controller parameters, z = 2, w p = 12, k 1 = 15k 2 = 24, k 3 = 1 . The parameters of the unmatched disturbance observer are shown below:  The parameters of the matched disturbance observer are shown below: λ 2,1 = 81, λ 2,2 = 108, λ 2,3 = 164, λ 2,4 = 66, λ 2,5 = 16, γ 2 = −2 15 Figure 7 shows the trajectory tracking curve and control input diagram of the proposed method when no external disturbance is applied. Figure 7a shows the tracking curve of the knee joint without disturbance by using the second-order sliding mode composite control. It can be seen that the proposed method is effective and can track the reference trajectory at about 0.6 seconds. Figure 8 is the trajectory tracking curve under mismatched disturbances. Figure 8b shows that the error is basically within (−1 0 ∼ 1 0 ), indicating that the proposed method has good anti-interference performance. Figure 9 shows the trajectory tracking diagram and error curve diagram under matching disturbance. Figure 9a, b shows that the proposed method has a good tracking performance under matching disturbance. Since matching disturbance cannot be effectively given in the actual experiment, the following time-varying matching disturbance form is given at the control input: Compared with tracking error of PD (10.3%) and SMC (6.6%), it can be found that the proposed composite algorithm (1.2%) has performed a higher tracking accuracy.

Conclusion
In this paper, a compound position control method based on second-order sliding mode control is proposed to solve the trajectory tracking problem of flexible lower limb exoskeleton robot under disturbance and parameter uncertainty. In this method, two finite time state observers are used to observe the matched and unmatched disturbances of the system, respectively, and do a real-time feedforward compensation for the system. By designing a sliding mode surface with mismatched disturbance estimation information and adopting the super-twisting algorithm in secondorder sliding mode control, the influence of disturbance on the system is effectively reduced. Experimental results show that compared with the traditional PD control and sliding mode control, the proposed method has better tracking performance and anti-jamming ability, and weakens the shaking phenomenon in the traditional sliding mode control.