Time-delay estimation based computed torque control with robust adaptive RBF neural network compensator for a rehabilitation exoskeleton

ISA Transactions xxx (xxxx) xxx

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Time-delay estimation based computed torque control with robust adaptive RBF neural network compensator for a rehabilitation exoskeleton✩ ∗

Shuaishuai Han a , Haoping Wang a , , Yang Tian a , Nicolai Christov b a

School of Automation, Nanjing University of Science & Technology, Nanjing, 210094, China Research Center in Computer Science, Signal and Automatic Control (CRIStAL), University of Lille 1, Batiment P2, 59655 Villeneuve d’Ascq Cedex, France b

highlights • • • • •

A TDE based CTC method is proposed with a robust adaptive RBF neural networks. To enhance CTC method, TDE is used to compensate unknown dynamics and disturbance. To reduce TDE error, a robust adaptive RBF neural networks compensator is designed. The asymptotic stability of exoskeleton control is guaranteed with Lyapunov theory. Compared to CTC, SMC/TDE-CTC, high performances of the proposed method are validated.

article

info

Article history: Received 2 October 2018 Received in revised form 26 July 2019 Accepted 30 July 2019 Available online xxxx Keywords: Computed torque control Time-delay estimation Robust adaptive RBF neural networks Robotics toolbox 12 DOF lower limb exoskeleton

a b s t r a c t A new approach to gait rehabilitation task of a 12 DOF lower limb exoskeleton is proposed combining time-delay estimation (TDE) based computed torque control (CTC) and robust adaptive RBF neural networks. In addition to the conventional advantages of the CTC, TDE technique is integrated to estimate unmodeled dynamics and external disturbance. To realize more accurate tracking, a robust adaptive RBF neural networks compensator is designed to approximate and compensate TDE error. The final asymptotic stability is guaranteed with Lyapunov criteria. To validate the proposed approach, cosimulation experiments are realized using SolidWorks, SimMechanics and MATLAB/Robotics Toolbox. Compared to CTC, sliding mode based CTC and TDE based CTC, the higher performances of the proposed controller are demonstrated by co-simulation. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction With the increasing of the aging population and nursing cost for elderly and injured patients, many researchers have been investigating lower limb exoskeletons for rehabilitation or walking assistance. During the last decades, impressive achievements have been realized [1]. The exoskeletons Ekso [2], ALEX [3] and LOKOMATE [4] provide medical rehabilitation assistance for patients with neurological injuries. Developed by Cyberdyne, HAL [5] is focused on assistance for elderly or disabled people through complex sensing and human exoskeleton interaction ✩ This work was partially supported by International Science & Technology Cooperation Program of China (2015DFA01710), by the Natural Science Foundation of China (61773212, 61304077), by the Natural Science Foundation of Jiangsu province, China (BK20170094) and by the 11th Jiangsu Province Six talent peaks of high level talents, China (2014_ZBZZ_005). ∗ Corresponding author. E-mail address: [email protected] (H. Wang).

control. Equipped with two canes, ReWalk [6] aims to help the patients with lower limb paraplegia regain walking abilities. Similarly, eLEGS [7], Ekso [2], CUHK-EXO [8], and Vanderbilt exoskeleton [9] use canes for order generation or balance support. The lower limb exoskeletons are highly nonlinear coupled dynamic systems and need efficient control strategies to operate coordinately with the human lower limb [10]. Various strategies have been proposed to realize stable and efficient repetitive motion control [11], which is the basis and most important function in lower limb exoskeleton systems. The conventional computed torque control (CTC) [12] realizes feedback linearization of the nonlinear dynamic model and is applied to various robotic systems, like serial manipulators in [13], parallel manipulators in [14, 15], service robot in [16], exoskeletons in [17] and etc.. CTC can efficiently utilize the robot dynamics to ensure rapid tracking control. However, with the increasing requirements for tracking accuracy, CTC can hardly achieve satisfying control performance because of the dynamic parametric variations and existing disturbances.

https://doi.org/10.1016/j.isatra.2019.07.030 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: S. Han, H. Wang, Y. Tian et al., Time-delay estimation based computed torque control with robust adaptive RBF neural network compensator for a rehabilitation exoskeleton. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.030.

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To improve CTC performances, various approaches have been proposed to compensate model inaccuracy and uncertainties. Disturbance observer based (DOB) [18] CTC is a common choice to reject dynamic uncertainties and improve robustness [19]. In [20], a high-order disturbance observer was used to estimate unknown compound disturbance and an adaptive sliding mode controller (ASMC) was combined to achieve final Lyapunov stability. The disturbance dynamics was assumed to be a linear high order model, which can only express a specific kind of disturbance. Another way to improve CTC is PD parameters tuning in CTC structure. In [14], neural networks were used for parameter tuning in the CTC scheme and make CTC more adaptive. In [21], Gaussian process regression has been used to achieve variable CTC adaption, without dealing directly with unmodeled dynamics or uncertain parameters. Sliding mode control (SMC) is also a popular choice for uncertainty elimination. However, in humanexoskeleton system, the magnitude of uncertainty is relatively large and one has to choose much larger SMC gains to cover all the possible uncertainty range. Time-delay estimation (TDE) based control was proposed in [22,23] and then applied to various plants [24–26]. With the assumptions of continuity and small time interval, TDE directly estimates the effects of uncertainties and can be easily implemented. In [24], TDE based control was applied to a brush DC motor and closed-loop stability was analyzed according the results in [22,23]. Combined with other control techniques, TDE has valuable applications to robotic systems [25,27,28]. In [25], the TDE has been combined with ideal velocity feedback to cancel nonlinear friction of robotic manipulators and used for compliant motion control. In our previous work [28], the TDE technique has been combined with terminal sliding mode control (TSMC) theory to achieve fast convergence for trajectory tracking. Successful applications to robotic systems have been realized combining TDE with adaptive control [29,30] and fuzzy logic control [27] as well. Those applications demonstrate great feasibility of TDE to eliminate robotic unknown dynamics and disturbances. Though TDE technique helps compensate unknown dynamics and external disturbance, the time-delay estimation error may cause instability and robustness reduction. Besides, the interaction effects between human and exoskeleton are piecewise continuous and periodic, which limits the application of TDE. With its high adaptability and learning ability, neural networks (NNs) have been applied to various areas. Compared to other types of NNs (Markovian NNs [31,32], Hopfield NNs [33], etc.), the radial basis function NNs (RBFNN) are easier to be implemented and analyzed because of their simple and smooth characteristics. In literature [31,32], the authors given stability analysis or states estimation for analysis of NNs, instead of using NNs for control. The perfect approximation ability of RBFNN can contribute to improvement of control system robustness [34–36] and handle periodic disturbances [37]. To improve the formation tracking performance of multiple unnamed helicopters, the authors in [38] design an adaptive RBF estimator to estimate unknown dynamics and then compensate for it. In [39], the neural networks (NN) are utilized to a robotic exoskeleton control as well. Until now, few researchers utilize RBFNN to lower limb exoskeleton control and usually tracking errors are driven to a relatively small region [40, 41] in stead of asymptotic convergence. RBFNN still have great potential to improve rehabilitation motion tracking of lower limb exoskeleton. In this paper, RBFNN are utilized to compensate TDE error caused by the time delay and human-exoskeleton interaction. Considering uncertain discontinuous interaction effects with large magnitude, sliding mode control (SMC) [28] is not so practical to directly eliminate TDE error and may still cause chattering problem. Comparing to the previous research in [42], RBFNN are

modified with a robust term and strictly asymptotic stability is achieved. Besides, for such a complex human-exoskeleton interaction system, obtained dynamic information releases TDE burden and the final controller is applied to another virtual prototype with interaction effects which are not considered in [42]. The trajectory tracking errors are set to be the input of RBFNN, which reflect the effect of TDE error to exoskeleton system. The RBFNN output will be a compensative input in entire controller scheme. The other parts are organized as following: In Section 2, a lower limb exoskeleton is briefly introduced and corresponding dynamic model is formulated. In Section 3, a TDE based CTC with robust adaptive RBF neural network compensator is proposed with closed-loop stability analysis. Section 4 shows some cosimulation results to verify the proposed controller. Section 5 gives some conclusion and remarks. 2. Lower limb exoskeleton description and dynamic model The following Fig. 1 shows the developed lower limb exoskeleton. For each leg, there are five actuated DOFs and one passive DOF in ankle. Detailed mechanical design description is given in [28]. The length range of thigh link is 387 mm to 523 mm and shank link is 300 mm to 419 mm, respectively. For the model analysis and co-simulation research, an average size was chosen to fit a man about 170 cm height. The operator will be fixed by straps on links and exoskeleton can lead human limbs to move during rehabilitation training. With Lagrangian equation, the dynamics of an exoskeleton limb can be described as M(q)q¨ + C (q, q˙ )q˙ + G(q) = τ + τhe

(1)

]T

[

q1 · · · qi · · · q5 is composed of the five where q = actuated joint angles and q˙ , q¨ are the velocity and acceleration. M(q) is a nonsingular and invertible matrix composed of inertial parameters. C (q, q˙ ) denotes Coriolis and centrifugal forces and G(q) is the gravitational force. τ represents the actuation torque vector exerting on joints. τhe is a vector of interaction torques between the human limbs and exoskeleton, which is an external disturbance for the exoskeleton. 3. TDE based computed torque control with robust adaptive RBFNN compensator The proposed controller is composed of two components: a time-delay estimation based CTC (TDECTC) and a robust adaptive RBFNN compensator, see Fig. 2. In TDECTC, the TDE technique is utilized to compensate unknown dynamics and external disturbance effects. To compensate the time-delay estimation errors, a robust adaptive RBFNN compensator is utilized with online updated node weights. Thus a fast gait tracking for rehabilitation exoskeleton can be realized. 3.1. Computed torque control As described in [12,43], the CTC controller is designed as

τ = M(q)[¨qd + Kd (q˙ d − q˙ ) + Kp (qd − q)] + C (q, q˙ )q˙ + G(q)

(2)

where qd ∈ R is the desire joint trajectories and q¨ d , q˙ d ∈ R5 are acceleration and velocity vectors. Kp = diag {Kp1 , . . . , Kp5 } and Kd = diag {Kd1 , . . . , Kd5 } are positive gain matrixes. If τhe = 0, from Eqs. (1) and (2), one obtains 5

q¨ = q¨ d + Kd (q˙ d − q˙ ) + Kp (qd − q).

(3)

Please cite this article as: S. Han, H. Wang, Y. Tian et al., Time-delay estimation based computed torque control with robust adaptive RBF neural network compensator for a rehabilitation exoskeleton. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.030.

S. Han, H. Wang, Y. Tian et al. / ISA Transactions xxx (xxxx) xxx

3

Fig. 1. Exoskeleton mechanical configuration in SolidWorks.

Fig. 2. Scheme of TDE based CTC with robust adaptive RBFNN.

Define the error vector e = derivatives as

[

e1

e = qd − q, e˙ = q˙ d − q˙ , e¨ = q¨ d − q¨ .

···

ei

···

e5

]T

and its (4)

Then Eq. (3) can be rewritten as e¨ + Kd e˙ + Kp e = 0.

(5)

The gain matrices Kd , Kp are determined by the pole placement method in order to realize exponentially convergence. In this paper, Kpi , Kdi of joint i(i = 1, 2, . . . , 5) are determined by (s + ωi ) = 0 2

(6)

where s is Laplace variable and ωi is the pole set by practitioner. Therefore, the corresponding parameters are Kpi = ωi 2 , Kdi = 2ωi .

(7)

Thus a convergent trajectory tracking with decoupled torque calculation is realized. However, inaccurate model information may cause instability and tracking failure.

3.2. Time-delay estimation based computed torque control The results presented in Section 3.1 are obtained with unrealistic assumption that accurate dynamic model is known and the term τhe in (1) is ignored. Besides, the matrices M(q), C (q, q˙ ), G(q) in (1) are functions of physical parameters like masses and link lengths. When model inaccuracy and external influence τhe are considered, the dynamic equation (1) can be modified as

¯ q¨ + C¯ (q, q˙ )q˙ + G(q) ¯ +F =I +F τ = M(q)

(8)

¯ q¨ + C¯ (q, q˙ )q˙ + G(q), ¯ ¯ , C¯ (q, q˙ ), G(q) ¯ where I = M(q) M(q) denote the calculation results of M(q), C (q, q˙ ), G(q) in SolidWorks, and ¯ (q) q¨ + C (q, q˙ ) − C¯ (q, q˙ ) + G (q) − G¯ (q) −τhe F = M (q) − M

(

)

(

) (

)

(9) includes modeling deviation and human-exoskeleton interaction effects.

Please cite this article as: S. Han, H. Wang, Y. Tian et al., Time-delay estimation based computed torque control with robust adaptive RBF neural network compensator for a rehabilitation exoskeleton. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.030.

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Using Eq. (8), F can be estimated by TDE technique [22,26,29] as Fˆ = Ft −L = (τ − I )t −L

(10)

where the Ft −L denotes the delayed F with a delay interval L which can be chosen according to hardware conditions. Under the assumptions that L is small enough and F is continuous, TDE can achieve satisfying approximation of the unknown F [22,23,44]. Using the estimate Fˆ , a TDE based CTC can be proposed as

¯ + Fˆ . ¯ τ = M(q)( q¨ d + Kd e˙ + Kp e) + C¯ (q, q˙ )q˙ + G(q)

(11)

From (11) and (8) one obtains

( ) ¯ q¨ + F = M ¯ q¨ d + Kd e˙ + Kp e + Fˆ M

(12)

and thus the error dynamics can be described as

Fig. 3. RBFNN structure.

e¨ + Kd e˙ + Kp e = ∆H

(13)

(

¯ −1 Fˆ − F where ∆H = M

)

=

[

∆H1

∆Hi

···

···

∆H5

]T

¯ is invertible. Besides, to eliminate the potential chattering and M caused by the fast change of F , a filter in the form of 1 1 is added f

s+1

n > 1 is the number of networks nodes and the RBFNN have only one hidden layer. Here

∥Ei − cj ∥2

)

(19)

for Fˆ . Here s is the Laplace variable and f is the cut-off frequency.

hj (Ei ) = exp(−

3.3. Robust adaptive RBFNN compensator and stability analysis

is a Gaussian function with the center of receptive fields cj = [ ]T c1j c2j and width bj . The RBFNN are able to approximate any continuous function to arbitrary accuracy [46] and a lemma is given as following.

From Eqs. (9)–(13), one can find that ∆H includes TDE error, filter error and discontinuity of the human-exoskeleton coupled dynamics. ∆H can be efficiently approximated by RBFNN. A robust adaptive RBFNN compensator is designed and the corresponding control law is finally expressed as

¯ ˆ + u) + C¯ (q, q˙ )q˙ + G(q) ¯ + Fˆ (14) τ = M(q)( q¨ d + Kd e˙ + Kp e + ∆H [ ]T ˆ = ∆Hˆ 1 · · · ∆Hˆ i · · · ∆Hˆ 5 where ∆H is the approx[ ]T u1 · · · ui · · · u5 imation of ∆H and u = denotes a robust term to be designed. The robust control part ui of i − th joint aims at realizing asymptotic trajectory tracking and detailed description will be discussed in Eq. (25) later. With the proposed controller, the error dynamics are described as

ˆ −u e¨ + Kd e˙ + Kp e = ∆H − ∆H

(15)

ˆ → ∆H, the motion controller makes it possible to obtain If ∆H substantial improvement[ of system ]T performances. For joint i(i = 1, 2, . . . , 5), define Ei = ei e˙i . Then the error dynamics can be rewritten as E˙ i =

[

e˙ i e¨ i

]

[ =

0 −Kpi

1 −Kdi

][

ei e˙ i

]

[ +

0 1

]

(∆Hi − ∆Hˆ i − ui )

= ΛEi + B(∆Hi − ∆Hˆ i − ui ) (16)

2b2j

Lemma 1 ([46–48]). For the piecewise continuous unknown term ∆Hˆ i , there exists a piecewise constant vector W ∗ of optimal weights defined as

{

⏐ ⏐

( )⏐} ⏐

ˆi W ˆ ⏐ W ∗ := arg min sup ⏐∆Hi − ∆H

(20)

with the approximation error

( ) σ := ∆Hi − ∆Hi W ∗ .

(21)

For any arbitrary given small constant σ¯ , there exists a limited number of RBFNN nodes n to make |σ | ≤ σ¯ . That is

∆Hi = W ∗ h (Ei ) + σ , |σ | ≤ σ¯ .

(22)

Remark 1. In the existing literatures, ∆Hi is assumed to be a continuous function to be approximated and W ∗ is a constant vector. In human-exoskeleton systems, ∆Hi may not be always continuous because of the discontinuous interaction forces. However, it can be assumed to be continuous between each fast change of interaction and thus the RBFNN can achieve optimal approximation during such period with a piecewise constant vector W ∗ .

with

Λ=

[

0 −Kpi

1 −Kdi

] [ ] 0 ,B = . 1

For a scalar ∆Hi , RBFNN approximation law is designed as

(17)

The RBFNN are utilized to approximate the unknown term ∆H. Inspired from [34,45], the adopted RBFNN for joint-i can be illustrated as following Fig. 3. ˆ i denotes In Fig. 3, ei , e˙i are the network input data and ∆H the approximation result. The input–output relationship of the considered neural networks is

∆Hˆ i = Wh(Ei ) (18) [ ] where W = W1 · · · Wn is a weight vector to be tuned and [ ]T h(Ei ) = h1 (Ei ) · · · hn (Ei ) is the activation function vector.

{

ˆ h(Ei ) ∆Hˆ i = W ˙ˆ W = γ Ei T PBh(Ei )

(23)

ˆ is an vector which tries to approximate the ideal weight where W W ∗ . γ is a positive constant and determine the adaption speed. P is the solution of Lyapunov equation ΛT P + P Λ = −Q , P =

[

p1 p2

p2 p3

] (24)

where Q > 0 is a positive definite.

Please cite this article as: S. Han, H. Wang, Y. Tian et al., Time-delay estimation based computed torque control with robust adaptive RBF neural network compensator for a rehabilitation exoskeleton. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.030.

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To achieve final asymptotically stable tracking, an additional robust term ui in Eq. (14) is designed as

( ) ⎧ ⎪ ⎨ ui = ξˆ + η sgn (s) ∫ ˙ˆ ˆ ⎪ ⎩ ξ = |s| , ξ = |s| dt s = p2 ei + p3 e˙i

(25)

bound ξ , i.e., ⏐∆Hi − ∆Hˆ i ⏐ ≤ ξ . s is a combined error to be determined by Lyapunov parameters. η is a positive constant for stability consideration. With Eq. (22), the error dynamics equation (16) can be written as

⏐

⏐

E˙i = ΛEi + BN ˆ i (W ∗ ) − ∆Hˆ i + σ − ui = (W ∗ − W ˆ )T h(Ei ) + σ − ui . N = ∆H (26)

Theorem 1. Consider human-exoskeleton dynamics described as Eq. (1) with the proposed TDE-based CTC controller in Eq. (11), adaptive RBFNN laws in Eq. (23), and the robust term in Eq. (25). Given suitable control parameters Kpi , Kdi , γ , η, Q , the joint-i of human-exoskeleton system can track the given desire trajectory qdi and the closed-loop system can achieve asymptotic stability. Proof. A Lyapunov function V is designed as

= =

1 2 1 2

1

(27)

2

2

T

E˙i PEi +

Ei Λ PEi + T

T

1 2 1 2

Ei T P E˙i = T

N PEi +

Ei (Λ P + P Λ)Ei + T

T

1 2

1 2 1 2 T

(Ei T ΛT + N T )PEi + Ei P ΛEi + T

N PEi +

1 2

1 2

1 2

Ei T P(ΛEi + N)

T

Ei PN

T

Ei PN

1

= − Ei T Q Ei + Ei T PN 2

(28)

ˆ )T Ei T PBh(Ei ), one has ˆ )T h(Ei ) = (W ∗ − W Since Ei T PB(W ∗ − W 1 ˆ )T h(Ei ) + σ − ui ] V˙ 1 = − Ei T QEi + Ei T PB[(W ∗ − W 2 1 ˆ )T Ei T PBh(Ei ) + Ei T PB(σ − ui ) = − Ei T QEi + (W ∗ − W 2

(29)

1 ˙ˆ ˆ )T (Ei T PBh(Ei )− 1 W V˙ 1 +V˙ 2 = − Ei T QEi +s(σ −ui )+(W ∗ −W ) (30) 2 γ With the designed adaptive RBF approximation law in Eq. (23) and robust control term ui in Eq. (25), one obtains

( ( ) ) 1 V˙ 1 + V˙ 2 = − Ei T QEi + s σ − ξˆ + η sgn (s) 2

(31)

As

(

)

( ) = ssign (s) ξˆ − ξ

(33)

Then with Eqs. (31) and (33), one has

( ( ) ) ( ) 1 = − Ei T Q Ei + s σ − ξˆ + η sgn (s) + s sin (s) ξˆ − ξ 2 1

= − Ei T QEi + s (σ − ηsgn (s) − ξ sgn (s)) 2 1

= − Ei T QEi − sηsgn (s) + s (σ − ξ sgn (s)) 2 1

= − Ei T QEi − η |s| + s (σ − ξ sgn (s)) 2

(34)

ˆ i (W ∗ ) and |∆Hi − ∆Hˆ i | ≤ ξ , σ ≤ ξ . Thus As σ = ∆Hi − ∆H s (σ − ξ sgn (s)) = sσ − |s| ξ ≤ 0

(35)

and therefore from Eq. (34), one obtains 1 (36) V˙ ≤ − Ei T QEi − η |s| ≤ 0 2 Thus the Lyapunov function V will continuously decrease until Ei = 0. Then the human-exoskeleton system with the proposed controller will be stable with convergent trajectory tracking.

The planned gait for straight walking rehabilitation and corresponding forward/inverse kinematics have been analyzed in our previous work [28]. The planned foot trajectory in Y–Z plane is drawn in Fig. 4 which includes beginning motion and one gait cycle to repeat. 4.2. Model information Robotics Toolbox (RTB) [49] is very useful for simulation of real robots. The considered lower limb exoskeleton can be described using MATLAB m-files containing the exoskeleton physical parameters, which are analyzed in SolidWorks software. After determining the exoskeleton material (aluminum alloy in our research), the physical parameters given in Table 1 have been obtained. In co-simulation, the weight of each actuation motor is set to be 1.1 kg with moment of inertia 25.4 kg ∗ cm2 , which is equal to the real weight of the selected actuation part (Maxon motor EC flat 60–100 W and Harmonic reducer SHD-17-2SH1:100) in practical hardware system for better simulation. Based on these parameters, RTB can calculate the required CTC torque. 4.3. Co-simulation experiments

˙ˆ ˆ )W As V˙ 2 = − γ1 (W ∗ − W ,

˙ ˙ V˙ 3 = ξ˜ ξ˜ = ξˆ − ξ ξˆ

)

4.1. Gait definition

Calculate the derivative of first term V1 , one obtains V˙ 1 =

(

V˙ 3 = |s| ξˆ − ξ

4. Co-simulation experiments

⎧ V = V1 + V2 + V3 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ V = E T PEi ⎪ ⎨ 1 2 i 1 ˆ )T (W ∗ − W ˆ) ⎪ V2 = (W ∗ − W ⎪ ⎪ 2 γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V3 = 1 ξ˜ 2 = 1 (ξˆ − ξ )2 2

According to the defined robust term in Eq. (25),

V˙ = V˙ 1 + V˙ 2 + V˙ 3

where ξˆ is the ⏐estimate of ⏐the unknown RBF approximation error

{

5

(32)

With SimMechanics, the exoskeleton prototype in SolidWorks can be imported to MATLAB/Simulink and the co-simulation process is shown in Fig. 5 The time delay interval in the TDE component is 0.001s, the same as the sample time interval. The hidden layer in the RBFNN contains five nodes. The parameters of the TDECTC and the robust adaptive RBFNN compensator are summarized in Table 2. Both legs have the same parameters and for the robust adaptive RBFNN compensator, each single joint share the same parameters. Besides, for better demonstration of the proposed method, sliding mode control (SMC) is used to eliminate uncertainties as well and comparative co-simulation results will be given.

Please cite this article as: S. Han, H. Wang, Y. Tian et al., Time-delay estimation based computed torque control with robust adaptive RBF neural network compensator for a rehabilitation exoskeleton. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.030.

6

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Fig. 4. Desired gait cycle.

Fig. 5. Co-simulation process. Table 1 Exoskeleton parameters from SolidWorks software. Link (L)

Weight (kg)

Center of mass (m)

Moment of inertia (kg ∗ cm2 )

1

0.555

[−0.867, −0.0476, 0.472]

[

2

0.209

[0, −0.0121, −0.0615]

3

0.736

[0.284, 0, −0.0348]

4

0.685

[0.159, 0, 0.0386]

5

0.148

[0.029, 0, 0]

6

0.472

[−0.0723, −0.0396, 0]

13.5 [ 45.8 [ 45.6 [ 0.595 [ 35.1

Link (R) 1

Weight (kg) 0.555

90

[

60

13.9

775

284

269

2.89 62.3

1.1

0

3.39

−30

0

44.3

0

[−0.867, −0.0476, −0.472]

40 13.5

13.9

45.8

789

0.209

[0, 0.0121, −0.0615]

3

0.736

[0.284, 0, 0.0348]

[

45.6

4

0.685

[0.159, 0, 0.0386]

[

5

0.148

[0.029, 0, 0]

[

0.595

6

0.472

[0.0723, −0.0396, 0]

[

35.1

Item

Value

Kp

diag {400, 400, 400, 100, 100}

Kd

diag {40, 40, 40, 20, 20}

b

[

2

[

−2 −2

2

2

2

−1 −1

0 0

1 1

1000

Q

diag {500, 500}

η

1

] 2 2

]

60

284 2.89 62.3

]T

0

0.8 2.41 775 269 3.39 44.3

0

0

−0.9 0.132

0

0 0

−15.4

]T

0

0

0.844

]T

]T

8

−1.1

0

]T

0

−15.4

Moment of inertia (kg ∗ cm2 ) 90

]T

−0.132 0 −63.6 ]T 0.844 0 −23.3 ]T

[

2

c

−2

2.41

789

[

2

−3

Center of mass (m)

Table 2 Control parameters.

γ

40

0 0

63.6

]T

−60 ]T

]T

0

]T

4.3.1. Setpoint tracking co-simulation For primary test of the proposed controller, setpoint tracking of each joint (left leg) is conducted, which shows the step response characteristics of the controlled exoskeleton. The setpoint tracking results are shown as Fig. 6. From the step response above, the proposed method (red solid line in Fig. 6) achieves best performances in tracking accuracy, overshoot and tracking speed compared to CTC, SMC base CTC and TDE based CTC. Large unknown model dynamics and disturbance limits the use of CTC and SMC based CTC. Combining TDE technique to CTC scheme can obviously improve entire performance but it is still not accurate enough to satisfy high accuracy requirement in human-exoskeleton rehabilitation tasks. With the designed robust adaptive RBFNN compensator, the improvement of tracking accuracy and speed is obvious, as RBFNN can efficiently estimate and eliminate unknown dynamics

Please cite this article as: S. Han, H. Wang, Y. Tian et al., Time-delay estimation based computed torque control with robust adaptive RBF neural network compensator for a rehabilitation exoskeleton. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.030.

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Fig. 6. Setpoint tracking results of each joint with different control methods.

Fig. 7. Gait tracking in operation space (foot position).

and disturbances. Take joint-3 as an example, the settling time of proposed CTC-TDE-RBF is about 1.8 s and for CTC-TDE without RBFNN, the settling time is about 2.3 s with larger overshoot. The other two controller cannot even achieve efficient tracking because of model inaccuracy. 4.3.2. Co-simulation without piecewise continuous interaction torques In this case the exoskeleton dynamics is simulated without considering the discontinuity of interaction torques. The foot trajectory tracking results in operation space are shown in Fig. 7. From Fig. 7, the exoskeleton can perfectly track the planned trajectory with the smallest error when CTC-TDE-RBF method is applied. For other controllers, they can achieve effective tracking but the convergence and robustness cannot be guaranteed. As time goes on, other controllers may not complete satisfying walking gait. For further illustration, the joint trajectory tracking errors and torques are given as Fig. 8. It can be observed from Fig. 8 that TDE technique considerably improves the exoskeleton tracking performance compared to the traditional CTC, though strict convergence is not achieved. Because of the parameter uncertainties and modeling error, the control performance of CTC will get worse, especially for joint-3 and joint-4. The tracking performance is substantially improved and convergence is achieved when the robust adaptive RBFNN compensator is added to the TDECTC. The results for the right leg are similar to those for the left one and are not presented here. For better demonstration of tracking performance, the integral square error (ISE) of each joint is calculated with following definition t

∫

e2 dt

JISE =

(37)

Table 3 ISE of joint trajectory tracking of left and right exoskeleton leg. Left leg

CTC

CTC-SMC

CTC-TDE

CTC-TDE-RBF

Joint Joint Joint Joint Joint

0.0000653 0.002325 0.126765 0.433215 0.07452

0.000225 0.017505 0.11633 0.183075 0.010011

0.0000295 0.000825 0.0385 0.119088 0.020428

0.00000154 0.000045 0.0000371 0.000314309 0.000846305

1 2 3 4 5

Right leg

CTC

CTC-SMC

CTC-TDE

CTC-TDE-RBF

Joint Joint Joint Joint Joint

0.00036 0.00818 0.22007 0.640572 0.579201

0.00715 0.020726 0.0000432 0.317352 0.735507

0.000587 0.005485 0.05309 0.11609 0.209547

0.00000581 0.000258 0.0000509 0.000107692 0.000364393

1 2 3 4 5

The calculation results is summarized in following Table 3 According to the calculation results above, an intuitive comparing figure can be drawn as following Fig. 13. From the ISE of each joint, the error magnitude of proposed controller is obviously much smaller than others. Take the joint-4 (knee joint) as example, the error magnitude of proposed method is 10−3 rad which means high accuracy in tracking. However for other controllers, their error magnitudes are 10−1 rad. (See Fig. 9.) To identify the robustness of the proposed method, the mass of exoskeleton left foot is increased by 1 kg, 2 kg, 3 kg, 5 kg correspondingly. As the terminal part of the exoskeleton leg, the change of foot mass will have important effects on all joints. The ISE calculation results under different foot mass can be shown as Fig. 10. From the figure above, one can find that until 3 kg is added to exoskeleton foot, the ISE is still in a relatively small range.

0

Please cite this article as: S. Han, H. Wang, Y. Tian et al., Time-delay estimation based computed torque control with robust adaptive RBF neural network compensator for a rehabilitation exoskeleton. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.030.

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Fig. 8. Tracking errors (left) and torques (right) from joint-1 to joint-5.

Until 5 kg is added, the ISE is till stabilized in 10−3 magnitude. The testing results demonstrate good robustness of our proposed controller.

4.3.3. Co-simulation in presence of piecewise continuous interaction torques To improve the human-exoskeleton dynamics model accuracy and testify the efficiency of the proposed method, τhe in Eq. (1) is set to be piecewise continuous. All the control parameters stay

the same. The interaction torque on the left hip joint is shown in Fig. 11. The interaction torques in the other joints are similar but with different magnitude. The tracking results of foot position are given in Fig. 12. The calculation results of ISE are summarized in following Table 4 According to the calculation results above, an intuitive comparing figure can be drawn as following Fig. 13. As can be seen from the above Figs. 12 and 13, discontinuous external torques can considerably affect exoskeleton dynamics.

Please cite this article as: S. Han, H. Wang, Y. Tian et al., Time-delay estimation based computed torque control with robust adaptive RBF neural network compensator for a rehabilitation exoskeleton. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.030.

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Fig. 9. ISE of each joint with proposed control method. Table 4 ISE of joint trajectory tracking of left and right exoskeleton leg. Left leg

CTC

CTC-SMC

CTC-TDE

CTC-TDE-RBF

Joint Joint Joint Joint Joint

0.000469 0.133102 0.1643 2.195345 0.376437

0.000438 0.026548 0.011325 1.220979 0.078293

0.0000851 0.013945 0.048614 0.856426 0.161294

0.00000999 0.000447 0.00037 0.002156542 0.008264375

1 2 3 4 5

Right leg

CTC

CTC-SMC

CTC-TDE

CTC-TDE-RBF

Joint Joint Joint Joint Joint

0.00055 0.129811 0.083946 0.022692 0.494794

0.007507 0.110401 0.0000477 0.001827 0.674513

0.000347 0.049944 0.020523 0.011528 0.225866

0.0000088 0.00048 0.000133 0.001713182 0.00110202

1 2 3 4 5

Compared to other methods, the proposed TDE based computed

torque control with robust adaptive RBFNN compensator is robust to discontinuous uncertainties and can still achieve effective tracking. 5. Conclusion A combination of time-delay estimation (TDE) and robust adaptive RBF neural networks (RBFNN) is utilized to conventional computed torque control (CTC), forming a novel controller for gait tracking of rehabilitation lower limb exoskeleton. TDE has simple structure and it is shown that it can effectively cancel the effects of complex unmodeled human-exoskeleton dynamics. To eliminate TDE errors caused by delay and discontinuous human-exoskeleton interaction torques, a robust adaptive RBFNN compensator is designed and combined to TDE based CTC. The proposed compensator improves exoskeleton’s tracking accuracy and robustness to complex uncertainties and interaction effects. Compared to CTC, sliding mode control base CTC and TDE based

Fig. 12. Gait tracking in operation space (foot position).

Fig. 13. ISE of each joint with proposed control method.

Please cite this article as: S. Han, H. Wang, Y. Tian et al., Time-delay estimation based computed torque control with robust adaptive RBF neural network compensator for a rehabilitation exoskeleton. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.07.030.

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Fig. 10. Robust test under proposed control method.

Fig. 11. Piecewise continuous interaction torque.

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