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Nonlinear disturbance observer based sliding mode control of a human-driven knee joint orthosis
Accepted Manuscript Nonlinear disturbance observer based sliding mode control of a human-driven knee joint orthosis Samer Mohammed, Weiguang Huo, Jian Huang, Hala Rifa¨ı, Yacine Amirat PII: DOI: Reference:
S0921-8890(14)00234-6 http://dx.doi.org/10.1016/j.robot.2014.10.013 ROBOT 2382
To appear in:
Robotics and Autonomous Systems
Please cite this article as: S. Mohammed, W. Huo, J. Huang, H. Rifa¨ı, Y. Amirat, Nonlinear disturbance observer based sliding mode control of a human-driven knee joint orthosis, Robotics and Autonomous Systems (2014), http://dx.doi.org/10.1016/j.robot.2014.10.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Nonlinear Disturbance Observer Based Sliding Mode Control of a Human-Driven Knee Joint Orthosis Samer Mohammeda,∗, Weiguang Huoa , Jian Huangb , Hala Rifa¨ıa , and Yacine Amirata a Laboratory
of Images, Signals and Intelligent Systems (LISSI), University of Paris-Est Cr´ eteil, Vitry-Sur-Seine, France b School of Automation, Huazhong University of Science and Technology, Wuhan, China
Abstract The present paper deals with the control of a knee joint orthosis intended to be used for rehabilitation and assistive purposes. A model, integrating human shank and orthosis, is presented. To reduce the influence of the uncertainties in muscular torque modeling on the system control, a nonlinear observer is proposed to estimate the muscular torque developed by the wearer. Additionally, a robust terminal sliding mode control approach combined with the nonlinear observer is presented. To illustrate the effectiveness of the proposed control method, a comparison with two control methods, basic sliding mode and sliding mode with nonlinear observer, are also given. The asymptotic stability of the presented approaches and observer convergence are proved by means of a Lyapunov analysis. Furthermore, the proof of advantage of the robust terminal sliding mode control method with the nonlinear observer (improving the tracking precision and reducing the required time for eliminating external disturbances) is proposed as well. The experiment results show that the robust terminal sliding mode control approach combined with the nonlinear observer has a significant advantage with respect to the position tracking and robustness regarding to the modeling identification errors and external disturbances. Keywords: Robust Terminal Sliding Mode Control, Nonlinear Disturbance ∗ Corresponding
author Email address: [email protected] (Samer Mohammed)
Preprint submitted to Elsevier
September 23, 2014
Observer, Knee Joint Orthosis
1. Introduction Wearable robots are mechatronic devices that are embodied by the human upper and/or lower limbs to enhance the ability to achieve a given task. Their structure should suit perfectly the embodying limbs such that the set moves 5
synchronously and can be considered as one unit. These exoskeletons, also called orthoses, have drawn a large amount of interest related to the increase of the elderly and dependant populations. The domains of application concern the assistance of the user during daily activities, the rehabilitation of the impaired joints or limbs and the muscles strengthening.
10
Lower limb exoskeletons act at the hip, knee and/or ankle joint levels. They are equipped with sensors gathering information about the orientation of the joints, position of the limbs, ground reaction force, muscular activities, etc. as well as actuators delivering the necessary power to drive the exoskeletons and contribute to, or even induce, the movement of the limbs. The amount of power
15
that should be provided to ensure a given task is determined by means of an appropriate control law. The task can be characterized by a desired trajectory, determined by a therapist in post-operation for rehabilitation cases, or following the intention of the user to improve his/her performance. This intention is estimated relying on data issued from on-board sensors, detecting the brain or
20
muscle activities, posture, etc. Exoskeletons, having several degrees of freedom, allow to tackle deficiencies in multiple joints such as the robot suit HAL [1] and ensure gait restoration using some treadmill-based devices as the Lokomat [2] and LOPES [3]. Knee orthoses are mainly used to treat musculoskeletal impairments at the knee joint
25
level. Two main purposes are targeted: the rehabilitation and assistance, to strengthen or aid damaged limb functions. For example, they are used by osteoarthritis patients, whose ligaments or cartilage are affected, to strengthen the muscles, reduce the stiffness and provide joint stabilization. A patient-directed
2
orthosis is used after total knee arthroplasty to reduce the knee stiffness and 30
increase its range of motion by performing a static progressive stretch in [4]. They are equally used by patients who have been subject to a stroke, spinal cord or traumatic injuries to help them regain control of their limbs and ensure daily activities. A study has shown that the use of an untethered knee joint orthosis has enhanced the gait speed of post stroke patients; the level of patient
35
intervention during training is fixed by a physical therapist [5, 6]. So far, many researches on the control strategies of the wearable exoskeletons have been presented [7]. Some works dealing with the control of knee joint orthoses will be presented in the following. Roboknee aims to enhance the wearer performance by assisting the thigh muscles during flexion and extension
40
of the knee joint using a proportional controller [8]. The user’s contribution is evaluated by measuring the ground vertical reaction force using two load cells placed in the shoes. The user’s effort is determined in [9] using Electromyogram (EMG) electrodes placed at the thigh and linked to the knee joint torque through the muscle model. Proportional control of the orthosis linear DC motor
45
is also used to amplify the weak torque generated by the knee joint. Adaptability of the exoskeleton’s parameters has been used to define a desired behavior of the system composed of the lower limb and the exoskeleton. The system’s stiffness is reduced by adding, in parallel to the knee joint, an elastic brace aiming to increase running abilities [10]. Damping and inertia parameters of the sys-
50
tem has been modulated in [11],[12],[13] with respect to the wearer’s intention. PID and Linear Quadratic (LQ) controllers have been used respectively to track the desired trajectory. Previous works of the authors [14, 15] have considered the orthosis EICOSI (Exoskeleton Intelligently COmmunicating and Sensitive to Intention) with a full motor actuation for knee joint passive rehabilitation
55
purposes. For very weak muscles, hybrid exoskeletons associated to functional electrical stimulation of the muscles can be used [16]. For example, the cyberthosis is dedicated for muscle training and knee joint flexion/extension movement execution by means of a proportional, integral, derivative (PID) controller [17]. The present paper deals with the control of the orthosis EICOSI (Exoskele3
60
ton Intelligently COmmunicating and Sensitive to Intention), which has one revolute joint degree of freedom at the knee joint level. EICOSI has a simple structure, easy to don and doff and practical for people suffering from knee joint impairments. In this paper, the EICOSI is mainly considered to assist patients to strengthen theirs muscles in a sitting position. A model integrat-
65
ing the human lower-limb and orthosis is proposed. Considering the different characteristics of the joints, the human muscular effort is taken into account in the control strategy. To obtain satisfied control performances, a nonlinear observer is presented for estimating the muscular torque developed by the wearer, and the uncertainty in muscular torques estimation is considered as an exter-
70
nal disturbance. Furthermore, a robust terminal sliding mode control method combined (RTSMC) with a nonlinear disturbance observer (NDO) is presented. Compared with the basic sliding mode, NDO based sliding mode control(SMC) can enhance the tracking precision, estimate the external disturbance and extend the control bandwidth, which is a major disadvantage of the basic sliding
75
mode. Furthermore, since the terminal sliding mode control can guarantee the tracking error to converge to zero in finite time [18], a NDO based RTSMC is proposed to reduce the required time for eliminating external disturbances and improve the robustness at the same time. The asymptotically analysis of the NDO based SMC and NDO based RTSMC are given in this paper. The advan-
80
tage of NDO based RTSMC with respect to the convergence time is analyzed as well. For comparison, experiments conducted with different subjects within three control methods (basic SMC, NDO based SMC and NDO based RTSMC) have been performed respectively to show the efficiency and advantages of the NDO based RTSMC in terms of tracking precision, required time for eliminating
85
disturbances and robustness to external disturbances. The paper is organized as follows. The shank-orthosis model is presented in section 2. The NDO based SMC and NDO based RTSMC approaches and analysis of their stability by means of Lyapunov analysis are presented in section 3. Experimental setup and the parametric identification of the shank-orthosis
90
system are addressed in section 4. Experimental results are presented in section 4
5. Finally, conclusions and future works are outlined in section 6. 2. System Modeling A subject wearing the robotic orthosis with the shank freely moving around the knee joint is illustrated in Fig.1. This system is designed to carry out the 95
flexion-extension movement of human knee joint. The robotic orthosis consists of two segments attached to the thigh and shank separately. It is fixed to the wearer leg by appropriate braces. The robotic orthosis has one DOF at the knee joint, which is driven by both the actuator and human thigh muscles. According to the Lagrange formulation, the dynamic model of the shank-
100
orthosis system can be written as: J θ¨ = −τg cos θ − Asignθ˙ − B θ˙ − K(θ − θr ) + τ + τh
(1)
with: • J = Js + Jo : the sum of inertia of shank and orthosis, • τg = ms ls g + mo lo g: the gravitational term in which ms and li (i ∈ {s, o}) are the masses and lengthes of the shank and orthosis respectively,
105
• A = As + Ao , B = Bs + Bo : the solid and viscous friction parameters of shank and orthosis,
• K: the stiffness of the knee joint, • θr : the rest position of the knee joint, • τ , τh : the input torques of the actuator and human effort respectively.
110
3. Nonlinear disturbance observer based sliding mode control In a human-in-the-loop robotic control system, the human-robot interaction force is mixed with the actuator control input, which makes the control much
5
~zf
F ~y f ~xf S
~xs
~zs ~y s
θ = 0 rad
θ
l mg θ = − π2 rad
Figure 1: A subject wearing the robotics orthosis (F : Base Frame; S: Coordinate System of Shank).
difficult. If the human effect is considered as an external disturbance, the Sliding Mode Control (SMC) might be an effective solution due to its robustness to 115
uncertainties and disturbances. Note that a conventional SMC often has to face the trade-off between the control bandwidth and tracking precision because of the chattering phenomenon. Recently, some researches indicate that disturbance observer based SMC has a better control performance than conventional one [19, 20]. The disturbance observer based SMC can improve the control bandwidth
120
and tracking precision effectively, and reduce the control chattering. Thus, a nonlinear disturbance observer based sliding mode control method is developed in the present work. (A) Nonlinear disturbance observer ˙ Therefore, Define the state variable x = [x1 , x2 ]T with x1 = θ and x2 = θ. the dynamic model (1) can be rewritten in the following vector form: x˙ = F1 (x) + G1 (x)u + G2 (x)d
6
(2)
where:
F1 (x) =
x2 1 J (−τg
cos x1 − Asignx2 − Bx2 − K(x1 − θd )) 0 G1 (x) = G2 (x) =
1 J
u=τ
d = τh 125
Consider the human input torque τh as a disturbance d, and assume τˆh is the estimation of human input torque τh and dˆ = τˆh . A nonlinear disturbance observer is designed as follows: dˆ =
z + p(x)
(3)
z˙
L(−F1 (x) − G1 (x)u − G2 (x)(z + p(x)))
(4)
=
where p(x) is chosen as p(x) = k1 x1 + k2 x2 , while k1 and k2 are two positive constants. L satisfies: L=
∂p(x) = [k1 k2 ]. ∂x
(5)
From (3), it follows that: ˙ ˆ dˆ = LG2 (d − d)
(6)
ˆ Normally, there is no prior information about the Let’s define d˜ = d − d.
derivative of disturbance d. When the disturbance varies slowly relative to the
observer dynamics, it is reasonable to suppose that d˙ = 0. Then, from (6) it follows that: k2 d˜˙ = −LG2 d˜ = − d˜ J
(7)
Then the residual of disturbance estimation approaches zero since k2 is a positive constant. 130
(B) NDO based sliding mode control To design the NDO Based SMC, first we consider the following sliding surface: s = c1 e + e˙ with 7
e = x1 − θd
(8)
where, θd is the desired trajectory. The NDO Based SMC controller is then given by: u = − Jc1 e˙ + τg cos x1 + Asignx2 + Bx2 + K(x1 − θr ) − dˆ + J θ¨d − λs − σsigns
(9)
where c1 , λ > 0, σ > |d˜max |. |d˜max | is used to denote the upper bound of
estimation error of the disturbance. The whole closed-loop control system is depicted on Fig.2.
Figure 2: Nonlinear disturbance observer based sliding mode control diagram for knee-joint robotic orthosis system.
135
The stability of the knee-joint robotic orthosis system with NDO Based SMC is analyzed in the following. Proposition 1. Consider the knee-joint robotic orthosis system (1) with a NDO Based SMC controller (9). Assume the disturbance d varies slowly relative to the observer dynamics. Then, the trajectory of the system (1) can be driven
140
onto the sliding surface (8).
8
Proof. Choose the Lyapunov function V as: V =
1 2 1 ˜2 s + d 2 2
(10)
By differentiating (10) one can notice that: V˙
= =
ss˙ + d˜d˜˙ 1 s (−τg cos x1 − Asignx2 − Bx2 − Ke + u + d) J ˜ 2 d˜ +c1 e˙ − sθ¨d − dLG
Substituting (9) into the derivative of the Lyapunov function, we obtain: V˙
d˜ − λs − σsigns
− LG2 d˜2
=
s
=
1 ˜ 1 s(d − σsigns) − λs2 − LG2 d˜2 ≤ 0 J J
J
Therefore, the Lyapunov function is decreasing. This completes the proof. (C) NDO-based robust terminal sliding mode control Furthermore, a NDO based robust terminal sliding mode control method is also presented to enhance performance of the control system. Compared to the basic SMC, the TSMC algorithm can guarantee the tracking error convergence to zero in finite time [18]. This means that the tracking error converges to zero in a setting time, even if there is an initial error between the reference trajectory and the real position of the shank-orthosis. Additionally, a robust TSMC method is proposed to reduce the required time for eliminating the influence of large external disturbance after the system reaches the sliding surface. Finally, the parametrical uncertainty of the system is taken into account in RTSMC method as well. In RTSMC, the sliding surface is defined as follows: s = c1 e + e˙ − w
with
w = c1 v + v˙
(11)
where v is defined as a second continuously differentiable function, and v ∈ C[0, ∞], v, v˙ ∈ L∞ , and ∀t > T, v = v˙ = 0, T > 0; the initial values of v and v˙
9
are chosen as: v(0) = e(0), v(0) ˙ = e(0). ˙ In the present study, a cubic spline is used as the function:
145
a 0 + a 1 t + a 2 t2 + a 3 t3 v= 0
0≤t≤T
(12)
t>T
where, the coefficients of the cubic spline are defined as: a0 = e(t = 0), a1 = e(t ˙ = 0), a2 = 3(e(t = T ) − e(t = 0))/T 2 − 2e(t ˙ = 0)/T − e(t ˙ = T )/T ,
a3 = −2(e(t = T ) − e(t = 0))/T 3 + (e(t ˙ = 0)/T + e(t ˙ = T ))/T 2
For convenience, f1 and g1 are employed to denote the nonlinear dynamics
and control gain of the shank-orthosis system separately. According to equation(2), f1 and g1 are given as: f1 = F1 (2, 1), g1 = G1 (2, 1)
(13)
Hence, the dynamic equation (1) can be rewritten as follows: x˙ 2 = f1 + g1 (u + d)
(14)
Additionally, the controller of NDO Based RTSMC is shown as: u = gˆ1−1 [¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨ − k · sign(s) − λs] − dˆ
(15)
where, gˆ1 and fˆ1 are the estimation of g1 and f1 respectively. λs ( λ > 0) is the robust item used to reduce the required time for eliminating the external diturbance. The proof is given in equation (20). By considering the parametrical uncertainty of the identified system, k is defined as follows: k = (1 − Dm )−1 (Fm + Dm |¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨| + T + (1 + Dm )ˆ g1 σ) (16) where, 1 > Dm > △ = |g1 − gˆ1 |, Fm > |f1 − fˆ1 | and T > 0.
The analysis of stability of the NDO based RTSMC is given in the following:
150
Proposition 2.
10
Proof. Choose the Lyapunov function V as: V =
1 2 1 ˜2 s + d 2 2
(17)
Differentiating (17) along the equation (7), (11) - (16), one can notice that:
V˙
=
˙ ss˙ + d˜d˜
=
s[f1 + g1 (u + d) − x ¨d + c1 e˙ − v¨ − c1 v] ˙ + (−LG2 d˜2 )
=
s[f1 + g1 [(ˆ g1−1 [¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨ − k · sign(s)
ˆ + d] − x −λs] − d) ¨d + c1 e˙ − v¨ − c1 v] ˙ + (−LG2 d˜2 ) =
s[f1 + (1 + △)gˆ1 [ˆ g1−1 [¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨ − k · sign(s)
−λs] − dˆ + d] − x¨d + c1 e˙ − v¨ − c1 v] ˙ + (−LG2 d˜2 ) =
s[f1 − fˆ1 + △(¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨) − (1 + △)k · sign(s)
˜ − (1 + △)λs2 − LG2 d˜2 +(1 + △)ˆ g1 d] ≤ ≤ ≤
s[f1 − fˆ1 + △(¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨) + (1 + △)ˆ g1 d˜
−(1 − Dm )k · sign(s)] − (1 − Dm )λs2 − LG2 d˜2
˜ s[(Fm + Dm |¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨| + (1 + Dm )ˆ g1 d)sign(s)
−(1 − Dm )k · sign(s)] − (1 − Dm )λs2 − LG2 d˜2 −T |s| − (1 − Dm )λs2 − LG2 d˜2 ≤ 0
Therefore, the Lyapunov function of NDO Based RTSMC system is decreasing. This completes the proof. Using the proof procedure mentioned above, we also have: ss˙ ≤ −T |s| − (1 − Dm )λs2 ≤ −T |s| According to the proof about the TSMC proposed by K-B, Park [18], we notice that s(t) ≡ 0 and e(t) ≡ v(t). Thus, e(t) can converge to zero in setting time T , i.e. e(t) = 0, ∀t ≥ T . Furthermore, it can be known: ss˙ =
1 d 2 s ≤ −T |s| − (1 − Dm )λs2 2 dt 11
(18)
155
Assume for instance that s(t = Ts ) > 0,(Ts is the beginning time of some point which is not on the sliding surface). Tr is defined as the time when the system reaches on the sliding surface. Integrating the equation (18) between t = Ts and t = Tr leads to:
0 − s(t = Ts ) = s(t = Tr ) − s(t = Ts ) ≤ −T (Tr − Ts ) − (1 − Dm )λ
Z
Tr
sdt (19)
Ts
which implies that
Tr − Ts ≤
s(t = Ts ) − (1 − Dm )λ T
R Tr Ts
sdt
(20)
If s(t = Ts ) < 0, a similar result can be obtained, Hence, the equation (20) can be rewritten as: Tr − Ts ≤ 160
|s(t = Ts )| − (1 − Dm )λ T
R Tr Ts
|s|dt
(21)
If Ts = 0, one can notice that the |s(t = 0)| will be equal to zero in NDO
based RTSMC system. This means that the control system is always on the sliding surface. Hence, the tracking error can converge to zero in finite time (T ) [18]. On the other hand, if Ts is a certain time when the control system breaks away from the sliding surface because of the external disturbance (Ts > T ), 165
one can find that w = 0, ∀t > T . In this case, the w has no effect on reducing
the disturbance. However, the convergence time (Tr − Ts ) can also be reduced RT by the item, (1 − Dm )λ Tsr |s|dt, which is introduced by the robust item λs.
Therefore, it is clear that NDO based RTSMC can reduce the required time for eliminating external disturbances effectively during the whole control process.
170
4. Experimental Setup and Parametric Identification (A) Experimental Setup All experiments are implemented using the EICOSI prototype, as shown in Fig.3. The orthosis has a single degree of freedom at the knee joint level and is 12
attached to the shank and thigh by means of straps. It is driven by an on-board 175
brushless DC motor. The orthosis is equipped with an incremental encoder that measures the knee joint angle. Note that a forth-order Butterworth filter is used to process the raw measured knee joint angle to solve the oversampling problem. All computations are performed using a dSPACE board, equipped with an IBM processor (PowerPC 604th) that has a frequency of 400 M Hz.
180
The controller is based on a sliding mode control law (Basic SMC, RTSMC) (9) coupled to an observer (3) to estimate the human muscular torque. The control law is computed over the angle measurement delivered by the incremental encoder and the angular velocity obtained by a simple derivation. The dSPACE board delivers the pulse width modulation (PWM) level necessary to control the
185
actuator’s velocity. The control loop runs at 1 kHz, fixed due to the onboard sensors constraints. The experiments are conducted with three healthy subjects. Some of their relevant information such as sex, age, weight and height are listed in Table1. Subject 1 is one of the researchers, while the other subjects are totally naive
190
to the experimentation. Additionlly, a plastic mannequin (Fig.1) is used in the expriments to compare the effectiveness of the above-mentioned three control approaches . Three different control methods are used in the experiments: Basic SMC, NDO Based SMC and NDO Based RTSMC. Note that all experiments are implemented in the sitting position, as shown in Fig.4. The parameters of these
195
three controllers used in the experiments are chosen based on the experimental performance conducted by Subject 1. To compare the robustness and the ability for estimating disturbances of the three methods, these parameters are also used in the experiments conducted by other subjects and the mannequin. Note that some common parameters are chose to retain consistency as far as possible. The
200
parameters are given as follows: • Basic SMC: c1 = 5, and σ = 15; • NDO Based SMC: c1 = 5, k1 = 1.5, k2 = 0.5, and σ = 15; • NDO Based RTSMC: c1 = 5, k1 = 1.5, k2 = 0.5, σ = 7.5, λ = 5, 13
Dm = 0.1, Fm = 1, and T = 0.5;
Figure 3: The structure of the orthosis EICOSI.
Figure 4: The human lower limb and the orthosis EICOSI.
205
(B) Identification of the parameters. The inertial parameters of the system composed of the wearer and the orthosis are A, B, J and τg . The paramters, J and τg , can be considered to be composed of the human shank’s interia Js and gravity τgs and the interia Jo and gravity τgo of the orthosis respectively. The shank’s interia Js and gravity τgs
210
can be identified based on the weight and height of wearers [21]. The interia Jo 14
Table 1: Subjects’ Sex, Age, Weight and Height
Subject
Sex
Age(yr)
W eight(kg)
Height(cm)
S1
M
27
66
170
S2
F
24
60
172
S3
M
23
72
173
Table 2: System’s identified parameters
J(Jo + Js )(Kg · m2 )
0.2639(0.0117+0.2522)
A(N · m)
4.4289
K(N · m · rad−1 )
0.3382
τg (τgo + τgs )(N · m)
8.5724(0.2424+8.33)
B(N · m · s · rad−1 )
0.5950
and gravity τgo of the orthosis can be obtained by the methods presented in our previous study [22]. The other paremeters are identified by using the passive pendulum test [21]. During the identification process, the wearer is passive; he/she is not delivering any muscular effort: τh = 0. This is monitored by a 215
null electromyographical activity of the thigh muscles spanning the knee joint. Hence, the equation (1) can be rewritten as: τg cos θ = −J θ¨ − τg cos θ − Asignθ˙ − B θ˙ − K(θ − θr ),
(22)
The knee joint angle is measured by means of an incremental encoder; the angular velocity and acceleration are derived numerically. Therefore, one can notice that only the parameters A, B, K have to be identified in equation (22). 220
The least squares optimization algorithm is used to process the measured data. The identification experiments are conducted by subject 1 and the identified parameters are presented in Table 2.
15
5. Experimental Results To evaluate the performance of the control methods proposed in this study, 225
three groups of experiments were implemented to compare the tracking precision, required time for eliminating the disturbance and robustness respectively. Additionally, a step-response experiment was employed to verify the stability of the NDO Based RTSMC system as well. (A) Comparison of tracking precision.
230
In this experiment, a given sinusoidal trajectory with a frequency of 0.5Hz was used as the reference trajectory and three different control methods (Basic SMC, NDO Based SMC, NDO Based RTSMC) were used to control the shank-orthosis system to track the reference trajectory respectively. Subject 1 participated in this experiment. He was asked to sit on a table while wearing
235
the orthosis, and keep his knee joint relaxed during the experiment as shown in Fig.4. The average tracking errors are shown in Table 3. The current and desired knee joint angles are plotted in Fig.5. The tracking errors and the control torques are shown in Fig.6 and Fig.7 respectively. The comparative results
240
shows that NDO Based RTSMC can reduce the tracking error effectively, compared to the basic SMC. At the same time, the chattering of the control torque of NDO Based RTSMC is the lowest among these three approaches. During the experimets, the comfortability of the user is often associated with the chattering of control torques; the lower the frequency and amplitude of chattering,
245
the higher the comfortability. Additionally, although the tracking error and chattering of control torque were not decreasing significantly when using NDO Based SMC compared to the use of basic SMC; the peak error values are kept at a relatively low level, as shown in Fig.6. (B) Comparison of required times for eliminating disturbance.
250
As mentioned above, the NDO Based RTSMC method has the advantages of having the tracking error converging to zero in finite time and eliminating the 16
Ref
0
SMC NDOBSMC
−0.2
NDOBRTSMC
Position (rad)
−0.4 −0.6 −0.8 −1 −1.2 −1.4
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 5: Comparison of tracking precision (conducted by subject 1): the desired knee joint angles (black line) and the tracking control results of shank-orthosis system by employing the basic SMC, NDO Based SMC and NDO Based RTSMC methods respectively.
0.12 SMC NDOBSMC NDOBRTSMC
0.1 0.08
Error (rad)
0.06 0.04 0.02 0 −0.02 −0.04 −0.06
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 6: Comparison of tracking precision (conducted by subject 1): the tracking errors of different control methods:|Error|= {SMC, mean:0.03576; std:0.0259},{NDO Based SMC, mean:0.03197; std:0.01635},{NDO Based RTSMC, mean:0.02486; std:0.01492}
17
SMC NDOBSMC NDOBRTSMC
15
Torque (Nm)
10
5
0
−5
−10
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 7: Comparison of tracking precision (conducted by subject 1): the control torques of different control methods.
external disturbance faster than the other two control methods. To verify these advantages, two experiments are designed. In one experiment, a setting error was given at the beginning time. In the other one, a given external disturbance 255
was added to the shank-orthosis after the control system reached the sliding surface. In these experiments, convergence times to the sliding surface was calculated and compared. The sinusoidal trajectory with frequency of 0.5 Hz was also used as reference trajectory. Additionally, to guarantee the accuracy of these comparison experiments, a plastic mannequin was used to replace the
260
human being subject. This is because with the human being, it is difficult to keep the same experimental conditions during each experiment. Note that the associated tracking errors were changed because of the subject changes. Here, we just focus on the convergence time. The robustness will be discussed in section (C).
265
According to the first experiment, the current and desired knee joint angles are plotted in Fig.8. The tracking errors and the control torques are shown in Fig.9 and Fig.10 respectively. The current and desired knee joint angles corresponding to the second experiment are plotted in Fig.11. The tracking errors
18
and the control torques are shown in Fig.12 and Fig.13 respectively. According 270
to the tracking errors, when the system reaches the sliding surface, ±0.15rad
are chosen as the boundaries for evaluating the convergence time as shown in Fig.8 and Fig.11.
One can observe that the convergence time of NDO Based RTSMC method is relatively smaller than that of the basic SMC, and NDO Based SMC does 275
not show an advantage on the convergence time, compared to the basic SMC as shown in Fig.8. This is due to the fact that NDO increases the control torque quickly (Fig.10) to track the desired trajectory at the beginning time and the sudden increase of the torque makes a bad influence on the follow-up convergence. However, this problem is avoided in the NDO Based RTSMC
280
methods effectively. NDO Based RTSMC controller can response to sudden changes of the position at the beginning time due to the item v, and avoid the influence of the sudden torque after time T , because of the use of the item λs. In the second experiment, a given external disturbance is added to the system (between t=4s and t=5.2s), as shown in Fig.11 and Fig.13. The comparative
285
results show that the NDO Based RTSMC approach can significantly reduce the required time for eliminating the external disturbances. Additionally, one can observe that the basic SMC almost cannot compensate the external disturbances as shown in Fig.11 and Fig.13. (C) Robustness comparison.
290
To verify the robustness of the control methods, two other subjects (Subject 2 and Subject 3) and a plastic mannequin with 1 kg load on its foot were chosen instead of Subject 1 to conduct the same experiment as mentioned in Section 5.(A). The identified parameters of the shank-orthosis model for Subject 1 and parameters of three control methods were still used in this experiment, although
295
the parameters of Subject 2, Subject 3 and the mannequin are obviously different from those of the Subject 1(especially the parameter of the mannequin: K is equal to zero, the weight of its shank is larger than that of Subject 1). Three groups of experiments were conducted by each subject or mannequin, and each 19
Ref
0
SMC NDOBSMC NDOBRTSMC
−0.2
Position (rad)
−0.4 −0.6 −0.8 −1 −1.2 −1.4
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 8: Comparison of the required times for eliminating disturbance (conducted by the mannequin) - the convergence time at the beginning time: the desired trajectory and the tracking trajectories of the controllers: basic SMC, NDO Based SMC and NDO Based RTSMC respectively.
1 SMC NDOBSMC NDOBRTSMC 0.15 rad (15%) −0.15 rad (15%)
0.8
Error (rad)
0.6 0.4 0.2 0 −0.2 −0.4
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 9: Comparison of the required times for eliminating disturbance (conducted by the mannequin) - the convergence time at the beginning time: the tracking errors for the controllers: basic SMC, NDO Based SMC and NDO Based RTSMC respectively.
20
SMC NDOBSMC NDOBRTSMC
20
15
Torque (Nm)
10
5
0
−5
−10
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 10: Comparison of the required times for eliminating disturbance (conducted by the mannequin) - the convergence time at the beginning time: the control torques for the controllers: basic SMC, NDO Based SMC and NDO Based RTSMC respectively.
Ref SMC
0
NDOBSMC NDOBRTSMC
−0.2
Position (rad)
−0.4 −0.6 −0.8 −1 −1.2
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 11: Comparison of the required times for eliminating disturbance (conducted by the mannequin) - the convergence time for eliminating the external disturbance: the desired trajectory and the tracking trajectories for the controllers: basic SMC, NDO Based SMC and NDO Based RTSMC respectively.
21
0.8 SMC NDOBSMC NDOBRTSMC 0.15 rad (15%) −0.15 rad (15%)
0.6
Error (rad)
0.4 0.2 0 −0.2 −0.4 −0.6
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 12: Comparison of the required times for eliminating disturbance (conducted by the mannequin) - the convergence time for eliminating the external disturbance: the tracking errors for the controllers: basic SMC, NDO Based SMC and NDO Based RTSMC respectively.
20
SMC NDOBSMC NDOBRTSMC
15
Torque (Nm)
10
5
0
−5
−10
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 13: Comparison of the required times for eliminating disturbance (conducted by the mannequin) - the convergence time for eliminating the external disturbance: the control torques for the controllers: basic SMC, NDO Based SMC and NDO Based RTSMC respectively.
22
Table 3: Average Tracking Errors of Three Control Methods
|Error|(rad)
SMC
NDO Based SMC
NDO Based RTSMC
mean
std
mean
std
mean
std
Subject 1
0.0358
0.0259
0.0320
0.0164
0.0249
0.0149
Subject 2
0.2450
0.2028
0.0653
0.0479
0.0532
0.0392
Subject 3
0.3647
0.1221
0.0525
0.0299
0.0450
0.0268
M annequin
0.0584
0.0421
0.0401
0.0451
0.0328
0.0328
group lasted 15 seconds. The experimental results (tracking errors) are shown 300
in Table.3. According to the tracking errors, one can notice that the mean errors (|Error|) of NDO based SMC and RTSMC are always smaller than those of the basic SMC for each subject and the mannequin. Additionlly, the robustness of the NDO based RTSMC is the best among these three control methods. The tracking errors of NDO based RTSMC do not increase significantly, with the
305
changes of the system parameters, while there are relatively significant changes in errors when using the two other approaches, particularly the basic SMC. (D) Stability analysis. To evaluate the stability of the NDO Based RTSMC approach proposed in this paper, a step response experiment was implemented. In this experiment,
310
the plastic mannequin with a 1 Kg load was used. The experimental results are shown in Fig.14, Fig.15 and Fig.16. The results show that the shank-orthosis system can response quickly and the tracking error can converge to zero in relative short time. We can observe that there is an overshoot of this controller in this experiment.
315
6. Conclusions The present work addressed the development of a sliding mode control of a knee joint exoskeleton taking into account the estimation of the wearer’s muscular torque. The model of the shank-orthosis was shown and its parameters 23
−0.4 −0.5 Ref NDOBRTSMC
Position (rad)
−0.6 −0.7 −0.8 −0.9 −1 −1.1
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 14: Analysis of the stability: the desired trajectory and the tracking trajectories by employing the controller NDO Based RTSMC.
0.7 NDOBRTSMC 0.6
Error (rad)
0.5 0.4 0.3 0.2 0.1 0 −0.1
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 15: Analysis of the stability: the tracking errors by employing the controller NDO Based RTSMC.
24
16 NDOBRTSMC 14 12 Torque (Nm)
10 8 6 4 2 0 −2
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 16: Analysis of the stability: the control torques by employing the controller NDO Based RTSMC.
were estimated using least squares optimization. A basic sliding mode controller 320
coupled to an observer and a robust terminal sliding mode control method combined with a nonlinear disturbance obserber were proposed to drive the orthosis along some desired trajectories. The human muscular torque was considered as an external disturbance of the system in this work, so the NDO based SMC and NDO based RTSMC were designed to estimate the human torque and enhance
325
the performance of the shank-orthosis system. The asymptotic stability of the system is proved by means of a Lyapunov analysis. The experiment results confirmed the advantages of NDO Based RTSMC: improving the tracking precision and reducing the required time for eliminating the disturbance. At the same time, the robustness and stability of NDO Based RTSMC method were verified
330
by the experiments as well. The current paper does not deal with the misalignment problem that was considered in the literature, chiefly in exoskeletons with multiple degrees of freedom. In this study, this problem has been considerably reduced by adjusting manually the orthosis to every wearer’s morphology using adaptable straps.
335
Special cares have been taken during experiments in order to avoid reaching of
25
the full knee joint flexion which considerably reduce the joints misalignment. Although the system model is built based on the sitting position, the control strategy proposed in this paper is considered to be suitable for other daily living activities, for instance, the gait walk. In these situations, the muscular effort 340
can also be considered as the disturbance like that in the sitting position. In this case, a rational reference trajectory given by therapists can also be tracked by users’ lower limb with the assistance of the EICOSI, if the associated models are built based on these situations. Hence, more system models based on different activities during daily living will be studied in the future work, such as normal
345
gait walk, sit-to-stand and stairs climbing,etc. The control methods for the EICOSI will be further studied to assist the wearer to accomplish more activities during daily living as well. Acknowledgements This work was supported in part by the Fundamental Research Funds for
350
the Central Universities, huazhong University of Science and Technology under Grant 2013ZZGH007. References [1] A. Tsukahara, R. Kawanishi, Y. Hasegawa, Y. Sankai, Sit-to-stand and stand-to-sit transfer support for complete paraplegic patients with robot
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suit hal, Journal of Advanced Robotics 24 (1) (2010) 1615–1638. [2] G. Colombo, M. Joerg, R. Schreier, V. Dietz, Treadmill training of paraplegic patients using a robotic orthosis, Journ. of Rehabilitation Research and Development 37 (6) (2000) 693–700. [3] R. Veneman, J. Kruidhof, E. Hekman, R. Ekkelenkamp, E. Van Asseldonk,
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H. Van Der Kooij, Design and evaluation of the lopes exoskeleton robot for interactive gait rehabilitation, IEEE Transactions on Neural Systems and Rehabilitation Engineering 15 (3) (2007) 379–386. 26
[4] P.-M. Bonutti, G.-A. Marulanda, M.-S. McGrath, M.-A. Mont, M.-G. Zywiel, Static progressive stretch improves range of motion in arthrofibro365
sis following toal knee arthroplasty, Knee surgery, sports traumatology, arthroscopy 18 (2) (2010) 194–199. [5] N. Byl, Mobility training using a bionic knee orthosis in patients in a apoststroke chronic state: a case series, Journal of medical case reports 6 (1) (2012) 216. doi:10.1186/1752-1947-6-216.
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[6] C.-K. Wong, L. Bishop, J. Stein, A wearable robotic knee orthosis for gait training: a case series of hemiparetic stroke survivors, Prosthetics and orthotics international 36 (1) (2011) 113–120. [7] S. Mohammed, Y. Amirat, H. Rifai, Lower-limb movement assistance through wearable robots:
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state of the art and challenges, Advanced
Robotics 26 (1-2) (2012) 1–22. [8] J.-E. Pratt, B.-T. Krupp, C.-J. Morse, S.-H. Collins, The roboknee: An exoskeleton for enhancing strength and endurance during walking, in: Porceedings of the International Conference on Robotics and Automation, New Orleans, USA, 2004, pp. 2430–2435.
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[9] C. Fleischer, G. Hommel, A human-exeskeleton interface utilizing electromyography, IEEE Transactions on Robotics 24 (4) (2008) 872–882. [10] M.-S. Cherry, D.-J. Choi, K.-J. Deng, S. Kota, D.-P. Ferris, Design and fabrication of an elastic knee orthosis: Preliminary results, in: Int. design engineering technical conf. & Conputers and information in engineering
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conf., USA, 2006, pp. 565–573. [11] G. Aguirre-Ollinger, J.-E. Colgate, M.-A. Peshkin, A. Goswami, Inertia compensation control of a one-degree-of-freedom exoskeleton for lower limb assistance: initial experiments, IEEE Transactions on neural systems and rehabilitation engineering 20 (1) (2012) 68–77.
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[12] G. Aguirre-Ollinger, J.-E. Colgate, M.-A. Peshkin, A. Goswami, Design of an active one-degree-of-freedom lower-limb exoskeleton with inertia compensation, International Journal of Robotics Research 30 (4) (2011) 486– 499. [13] G. Aguirre-Ollinger, J.-E. Colgate, M.-A. Peshkin, A. Groswami, A 1-dof
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assistive exoskeleton with virtual negative damping: Effects on the kinematic response of the lower limbs, in: Proceedings of International Conference on Intelligent Robots and Systems, San Diego, CA, USA, 2007, pp. 1938–1944. [14] S. Mefoued, S. Mohammed, Y. Amirat, Knee joint movement assistance
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through robust control of an actuated orthosis, in: Proceedings of International Conference on Intelligent Robots and Systems, San Francisco, California, USA, 2011, pp. 1749–1754. [15] H. Rifai, W. Hassani, S. Mohammed, Y. Amirat, Bounded control of an actuated lower limb orthosis, in: Proceedings of the IEEE Conference on
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Decision and Control and European Control Conference, Orlando, Florida, USA, 2011, pp. 873–878. [16] A.-J. Del-Ama, A.-D. Koutsou, J.-C. Moreno, A. De-los Reyes, A. GilAgudo, J.-L. Pons, Review of hybrid exoskeletons to restore gait following spinal cord injury, Journal of rehabilitation robotics and development 49 (4)
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(2012) 497–514. [17] C. Schmitt, P. M´etrailler, A. Al-Khodairy, R. Brodard, J. Fournier, M. Bouri, R. Clavel, A study of a knee extension controlled by a closed loop functional electrical stimulation, in: Proceedings, 9th An. conf. of the IFESS, Bournemouth, 2004.
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[18] K.-B. Park., T. Tsuji, Terminal sliding mode control of second-order nonlinear uncertain systems, International Journal of Robust and Nonlinear Control 9 (11) (1999) 769–780. 28
[19] K. X. Xing, J. Huang, Y. J. Wang, Tracking control of pneumatic artificial muscle actuators based on sliding mode and nonlinear disturbance observer, 420
IET Control Theory & Applications 4 (10) (2010) 2058–2070. [20] J. Wu, J. Huang, Y. J. Wnag, K. X. Xing, Nonlinear disturbance observerbased dynamic surface control for trajectory tracking of pneumatic muscle system, IEEE Transactions on Control Systems technology PP (99) (2013) 1. doi:10.1109/TCST.2013.2262074.
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[21] P. Leva., Adjustments to zatsiorsky-seluyanov’s segment inertia parameters, Journal of Biomechanics 29 (9) (1996) 1223–1230. [22] H. Rifa¨ı, S. Mohammed, W. Hassani, Y. Amirat, Nested saturation based control of an actuated knee joint orthosis, Mechatronics 23 (8) (2013) 1141– 1149.
29
*Biography of each author
Samer Mohammed received the Diploma degree in electrical engineering from Lebanese University, Tripoli, Lebanon, the M.S. degree in automatic and microelectronic systems from the University of Montpellier II, Montpellier, France, and the Ph.D. degree in computer science from the Laboratory of Computer Science, Robotic and Microelectronic of Montpellier (LIRMM/CNRS), Montpellier. He is currently an Assistant Professor of computer science with the Laboratory of Image, Signal and Intelligent Systems, University of Paris-Est Créteil, Créteil, France. His current research interests include modeling, identification, and control of robotic systems (wearable robots), artificial intelligence, and decision-support theory. His current research involved the applications for the functional assisting of dependent people. Dr. Mohammed was the recipient of the Fellowship Award from the Japan Society for the Promotion of Science, the French Ministerial Doctoral Research Scholarship Award, and the École Polytechnique Fédéral de Lausanne Scholarship Award. He is actively involved in organizing committees of some national and international workshops in the automatic and robotic domains. Weiguang Huo is a PhD student in LISSI laboratory (UPEC), under the supervision of Prof. Amirat and Dr. Samer Mohammed. His PhD is supported by the French high education institution. He finished his Master study at the Department of Control Science and Engineering, Huazhong University of Science and Technology (HUST), China. He obtained his bachelor degree at 2009 from the Department of Measurement and Control Technology, Wuhan University of Technology WHST), China. Jian Huang graduated from Huazhong University of Science & Technology (HUST), China in 1997 and received the Master of Engineering degree from HUST in 2000. He received his Ph.D from HUST in 2005. From 2006 to 2008, he was a postdoctoral researcher in the Department of Micro-Nano System Engineering and Department of Mechano-Informatics and Systems, Nagoya University, Japan. He is currently an associate professor at School of Automation, HUST. His main research interests include rehabilitation robot, robotic assembly, networked control systems and bioinformatics. Hala RIFAI graduated in electronic-electrical engineering from the Lebanese University in 2004. She received a master diploma in automatic control in 2005 and a Ph.D. in non-linear control in 2008 from the Institut Polytechnique de Grenoble, GIPSA-Lab (UMR CNRS), Grenoble, France. Between 2008 and 2010, she had a post-doctoral scholarship in I3S-Lab (UMR CNRS), Sophia Antipolis, France. Currently, she has a post-doctoral scholarship in the Laboratory of Signals, Systems and Intelligent Systems (LSSI) of the University of Paris-Est Créteil (UPEC). Her research domains include the modeling, control and bilateral teleoperation of robotic systems. Applications concern the wearable robots and unmanned aerial vehicles.
Yacine Amirat received the Ph.D. degree in computer science and robotics from the University of Pierre et Marie Curie, Paris, France, in 1989, and the Habilitation degree in artificial intelligence and control of complex systems from the University of Paris 12-France, Paris, in 1996. He co-created the Laboratory of Computer Sciences and Robotics, Paris 12 University
(newly UPEC), Créteil, France, in 1990. He is currently the Director of the Laboratory of Image, Signal and Intelligent Systems (LISSI) – UPEC, and the Scientific Director of several research projects. He has authored or co-authored more than 100 papers in scientific journals, books, and conference proceedings. His current research interests include artificial intelligence, soft computing, knowledge processing, and control of complex systems. Application fields are intelligent and robotic systems such as ubiquitous and service robots, wearable robots, and ambient intelligence. Prof. Amirat is a reviewer of many international journals and conférences and a Technical Committee Member, the Session Chair, a Session Organizer, or an Associate Editor of several IEEE-RAS conferences. He is actively collaborating with industrial and academic partners in various European projects.
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S0921-8890(14)00234-6 http://dx.doi.org/10.1016/j.robot.2014.10.013 ROBOT 2382
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Please cite this article as: S. Mohammed, W. Huo, J. Huang, H. Rifa¨ı, Y. Amirat, Nonlinear disturbance observer based sliding mode control of a human-driven knee joint orthosis, Robotics and Autonomous Systems (2014), http://dx.doi.org/10.1016/j.robot.2014.10.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Nonlinear Disturbance Observer Based Sliding Mode Control of a Human-Driven Knee Joint Orthosis Samer Mohammeda,∗, Weiguang Huoa , Jian Huangb , Hala Rifa¨ıa , and Yacine Amirata a Laboratory
of Images, Signals and Intelligent Systems (LISSI), University of Paris-Est Cr´ eteil, Vitry-Sur-Seine, France b School of Automation, Huazhong University of Science and Technology, Wuhan, China
Abstract The present paper deals with the control of a knee joint orthosis intended to be used for rehabilitation and assistive purposes. A model, integrating human shank and orthosis, is presented. To reduce the influence of the uncertainties in muscular torque modeling on the system control, a nonlinear observer is proposed to estimate the muscular torque developed by the wearer. Additionally, a robust terminal sliding mode control approach combined with the nonlinear observer is presented. To illustrate the effectiveness of the proposed control method, a comparison with two control methods, basic sliding mode and sliding mode with nonlinear observer, are also given. The asymptotic stability of the presented approaches and observer convergence are proved by means of a Lyapunov analysis. Furthermore, the proof of advantage of the robust terminal sliding mode control method with the nonlinear observer (improving the tracking precision and reducing the required time for eliminating external disturbances) is proposed as well. The experiment results show that the robust terminal sliding mode control approach combined with the nonlinear observer has a significant advantage with respect to the position tracking and robustness regarding to the modeling identification errors and external disturbances. Keywords: Robust Terminal Sliding Mode Control, Nonlinear Disturbance ∗ Corresponding
author Email address: [email protected] (Samer Mohammed)
Preprint submitted to Elsevier
September 23, 2014
Observer, Knee Joint Orthosis
1. Introduction Wearable robots are mechatronic devices that are embodied by the human upper and/or lower limbs to enhance the ability to achieve a given task. Their structure should suit perfectly the embodying limbs such that the set moves 5
synchronously and can be considered as one unit. These exoskeletons, also called orthoses, have drawn a large amount of interest related to the increase of the elderly and dependant populations. The domains of application concern the assistance of the user during daily activities, the rehabilitation of the impaired joints or limbs and the muscles strengthening.
10
Lower limb exoskeletons act at the hip, knee and/or ankle joint levels. They are equipped with sensors gathering information about the orientation of the joints, position of the limbs, ground reaction force, muscular activities, etc. as well as actuators delivering the necessary power to drive the exoskeletons and contribute to, or even induce, the movement of the limbs. The amount of power
15
that should be provided to ensure a given task is determined by means of an appropriate control law. The task can be characterized by a desired trajectory, determined by a therapist in post-operation for rehabilitation cases, or following the intention of the user to improve his/her performance. This intention is estimated relying on data issued from on-board sensors, detecting the brain or
20
muscle activities, posture, etc. Exoskeletons, having several degrees of freedom, allow to tackle deficiencies in multiple joints such as the robot suit HAL [1] and ensure gait restoration using some treadmill-based devices as the Lokomat [2] and LOPES [3]. Knee orthoses are mainly used to treat musculoskeletal impairments at the knee joint
25
level. Two main purposes are targeted: the rehabilitation and assistance, to strengthen or aid damaged limb functions. For example, they are used by osteoarthritis patients, whose ligaments or cartilage are affected, to strengthen the muscles, reduce the stiffness and provide joint stabilization. A patient-directed
2
orthosis is used after total knee arthroplasty to reduce the knee stiffness and 30
increase its range of motion by performing a static progressive stretch in [4]. They are equally used by patients who have been subject to a stroke, spinal cord or traumatic injuries to help them regain control of their limbs and ensure daily activities. A study has shown that the use of an untethered knee joint orthosis has enhanced the gait speed of post stroke patients; the level of patient
35
intervention during training is fixed by a physical therapist [5, 6]. So far, many researches on the control strategies of the wearable exoskeletons have been presented [7]. Some works dealing with the control of knee joint orthoses will be presented in the following. Roboknee aims to enhance the wearer performance by assisting the thigh muscles during flexion and extension
40
of the knee joint using a proportional controller [8]. The user’s contribution is evaluated by measuring the ground vertical reaction force using two load cells placed in the shoes. The user’s effort is determined in [9] using Electromyogram (EMG) electrodes placed at the thigh and linked to the knee joint torque through the muscle model. Proportional control of the orthosis linear DC motor
45
is also used to amplify the weak torque generated by the knee joint. Adaptability of the exoskeleton’s parameters has been used to define a desired behavior of the system composed of the lower limb and the exoskeleton. The system’s stiffness is reduced by adding, in parallel to the knee joint, an elastic brace aiming to increase running abilities [10]. Damping and inertia parameters of the sys-
50
tem has been modulated in [11],[12],[13] with respect to the wearer’s intention. PID and Linear Quadratic (LQ) controllers have been used respectively to track the desired trajectory. Previous works of the authors [14, 15] have considered the orthosis EICOSI (Exoskeleton Intelligently COmmunicating and Sensitive to Intention) with a full motor actuation for knee joint passive rehabilitation
55
purposes. For very weak muscles, hybrid exoskeletons associated to functional electrical stimulation of the muscles can be used [16]. For example, the cyberthosis is dedicated for muscle training and knee joint flexion/extension movement execution by means of a proportional, integral, derivative (PID) controller [17]. The present paper deals with the control of the orthosis EICOSI (Exoskele3
60
ton Intelligently COmmunicating and Sensitive to Intention), which has one revolute joint degree of freedom at the knee joint level. EICOSI has a simple structure, easy to don and doff and practical for people suffering from knee joint impairments. In this paper, the EICOSI is mainly considered to assist patients to strengthen theirs muscles in a sitting position. A model integrat-
65
ing the human lower-limb and orthosis is proposed. Considering the different characteristics of the joints, the human muscular effort is taken into account in the control strategy. To obtain satisfied control performances, a nonlinear observer is presented for estimating the muscular torque developed by the wearer, and the uncertainty in muscular torques estimation is considered as an exter-
70
nal disturbance. Furthermore, a robust terminal sliding mode control method combined (RTSMC) with a nonlinear disturbance observer (NDO) is presented. Compared with the basic sliding mode, NDO based sliding mode control(SMC) can enhance the tracking precision, estimate the external disturbance and extend the control bandwidth, which is a major disadvantage of the basic sliding
75
mode. Furthermore, since the terminal sliding mode control can guarantee the tracking error to converge to zero in finite time [18], a NDO based RTSMC is proposed to reduce the required time for eliminating external disturbances and improve the robustness at the same time. The asymptotically analysis of the NDO based SMC and NDO based RTSMC are given in this paper. The advan-
80
tage of NDO based RTSMC with respect to the convergence time is analyzed as well. For comparison, experiments conducted with different subjects within three control methods (basic SMC, NDO based SMC and NDO based RTSMC) have been performed respectively to show the efficiency and advantages of the NDO based RTSMC in terms of tracking precision, required time for eliminating
85
disturbances and robustness to external disturbances. The paper is organized as follows. The shank-orthosis model is presented in section 2. The NDO based SMC and NDO based RTSMC approaches and analysis of their stability by means of Lyapunov analysis are presented in section 3. Experimental setup and the parametric identification of the shank-orthosis
90
system are addressed in section 4. Experimental results are presented in section 4
5. Finally, conclusions and future works are outlined in section 6. 2. System Modeling A subject wearing the robotic orthosis with the shank freely moving around the knee joint is illustrated in Fig.1. This system is designed to carry out the 95
flexion-extension movement of human knee joint. The robotic orthosis consists of two segments attached to the thigh and shank separately. It is fixed to the wearer leg by appropriate braces. The robotic orthosis has one DOF at the knee joint, which is driven by both the actuator and human thigh muscles. According to the Lagrange formulation, the dynamic model of the shank-
100
orthosis system can be written as: J θ¨ = −τg cos θ − Asignθ˙ − B θ˙ − K(θ − θr ) + τ + τh
(1)
with: • J = Js + Jo : the sum of inertia of shank and orthosis, • τg = ms ls g + mo lo g: the gravitational term in which ms and li (i ∈ {s, o}) are the masses and lengthes of the shank and orthosis respectively,
105
• A = As + Ao , B = Bs + Bo : the solid and viscous friction parameters of shank and orthosis,
• K: the stiffness of the knee joint, • θr : the rest position of the knee joint, • τ , τh : the input torques of the actuator and human effort respectively.
110
3. Nonlinear disturbance observer based sliding mode control In a human-in-the-loop robotic control system, the human-robot interaction force is mixed with the actuator control input, which makes the control much
5
~zf
F ~y f ~xf S
~xs
~zs ~y s
θ = 0 rad
θ
l mg θ = − π2 rad
Figure 1: A subject wearing the robotics orthosis (F : Base Frame; S: Coordinate System of Shank).
difficult. If the human effect is considered as an external disturbance, the Sliding Mode Control (SMC) might be an effective solution due to its robustness to 115
uncertainties and disturbances. Note that a conventional SMC often has to face the trade-off between the control bandwidth and tracking precision because of the chattering phenomenon. Recently, some researches indicate that disturbance observer based SMC has a better control performance than conventional one [19, 20]. The disturbance observer based SMC can improve the control bandwidth
120
and tracking precision effectively, and reduce the control chattering. Thus, a nonlinear disturbance observer based sliding mode control method is developed in the present work. (A) Nonlinear disturbance observer ˙ Therefore, Define the state variable x = [x1 , x2 ]T with x1 = θ and x2 = θ. the dynamic model (1) can be rewritten in the following vector form: x˙ = F1 (x) + G1 (x)u + G2 (x)d
6
(2)
where:
F1 (x) =
x2 1 J (−τg
cos x1 − Asignx2 − Bx2 − K(x1 − θd )) 0 G1 (x) = G2 (x) =
1 J
u=τ
d = τh 125
Consider the human input torque τh as a disturbance d, and assume τˆh is the estimation of human input torque τh and dˆ = τˆh . A nonlinear disturbance observer is designed as follows: dˆ =
z + p(x)
(3)
z˙
L(−F1 (x) − G1 (x)u − G2 (x)(z + p(x)))
(4)
=
where p(x) is chosen as p(x) = k1 x1 + k2 x2 , while k1 and k2 are two positive constants. L satisfies: L=
∂p(x) = [k1 k2 ]. ∂x
(5)
From (3), it follows that: ˙ ˆ dˆ = LG2 (d − d)
(6)
ˆ Normally, there is no prior information about the Let’s define d˜ = d − d.
derivative of disturbance d. When the disturbance varies slowly relative to the
observer dynamics, it is reasonable to suppose that d˙ = 0. Then, from (6) it follows that: k2 d˜˙ = −LG2 d˜ = − d˜ J
(7)
Then the residual of disturbance estimation approaches zero since k2 is a positive constant. 130
(B) NDO based sliding mode control To design the NDO Based SMC, first we consider the following sliding surface: s = c1 e + e˙ with 7
e = x1 − θd
(8)
where, θd is the desired trajectory. The NDO Based SMC controller is then given by: u = − Jc1 e˙ + τg cos x1 + Asignx2 + Bx2 + K(x1 − θr ) − dˆ + J θ¨d − λs − σsigns
(9)
where c1 , λ > 0, σ > |d˜max |. |d˜max | is used to denote the upper bound of
estimation error of the disturbance. The whole closed-loop control system is depicted on Fig.2.
Figure 2: Nonlinear disturbance observer based sliding mode control diagram for knee-joint robotic orthosis system.
135
The stability of the knee-joint robotic orthosis system with NDO Based SMC is analyzed in the following. Proposition 1. Consider the knee-joint robotic orthosis system (1) with a NDO Based SMC controller (9). Assume the disturbance d varies slowly relative to the observer dynamics. Then, the trajectory of the system (1) can be driven
140
onto the sliding surface (8).
8
Proof. Choose the Lyapunov function V as: V =
1 2 1 ˜2 s + d 2 2
(10)
By differentiating (10) one can notice that: V˙
= =
ss˙ + d˜d˜˙ 1 s (−τg cos x1 − Asignx2 − Bx2 − Ke + u + d) J ˜ 2 d˜ +c1 e˙ − sθ¨d − dLG
Substituting (9) into the derivative of the Lyapunov function, we obtain: V˙
d˜ − λs − σsigns
− LG2 d˜2
=
s
=
1 ˜ 1 s(d − σsigns) − λs2 − LG2 d˜2 ≤ 0 J J
J
Therefore, the Lyapunov function is decreasing. This completes the proof. (C) NDO-based robust terminal sliding mode control Furthermore, a NDO based robust terminal sliding mode control method is also presented to enhance performance of the control system. Compared to the basic SMC, the TSMC algorithm can guarantee the tracking error convergence to zero in finite time [18]. This means that the tracking error converges to zero in a setting time, even if there is an initial error between the reference trajectory and the real position of the shank-orthosis. Additionally, a robust TSMC method is proposed to reduce the required time for eliminating the influence of large external disturbance after the system reaches the sliding surface. Finally, the parametrical uncertainty of the system is taken into account in RTSMC method as well. In RTSMC, the sliding surface is defined as follows: s = c1 e + e˙ − w
with
w = c1 v + v˙
(11)
where v is defined as a second continuously differentiable function, and v ∈ C[0, ∞], v, v˙ ∈ L∞ , and ∀t > T, v = v˙ = 0, T > 0; the initial values of v and v˙
9
are chosen as: v(0) = e(0), v(0) ˙ = e(0). ˙ In the present study, a cubic spline is used as the function:
145
a 0 + a 1 t + a 2 t2 + a 3 t3 v= 0
0≤t≤T
(12)
t>T
where, the coefficients of the cubic spline are defined as: a0 = e(t = 0), a1 = e(t ˙ = 0), a2 = 3(e(t = T ) − e(t = 0))/T 2 − 2e(t ˙ = 0)/T − e(t ˙ = T )/T ,
a3 = −2(e(t = T ) − e(t = 0))/T 3 + (e(t ˙ = 0)/T + e(t ˙ = T ))/T 2
For convenience, f1 and g1 are employed to denote the nonlinear dynamics
and control gain of the shank-orthosis system separately. According to equation(2), f1 and g1 are given as: f1 = F1 (2, 1), g1 = G1 (2, 1)
(13)
Hence, the dynamic equation (1) can be rewritten as follows: x˙ 2 = f1 + g1 (u + d)
(14)
Additionally, the controller of NDO Based RTSMC is shown as: u = gˆ1−1 [¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨ − k · sign(s) − λs] − dˆ
(15)
where, gˆ1 and fˆ1 are the estimation of g1 and f1 respectively. λs ( λ > 0) is the robust item used to reduce the required time for eliminating the external diturbance. The proof is given in equation (20). By considering the parametrical uncertainty of the identified system, k is defined as follows: k = (1 − Dm )−1 (Fm + Dm |¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨| + T + (1 + Dm )ˆ g1 σ) (16) where, 1 > Dm > △ = |g1 − gˆ1 |, Fm > |f1 − fˆ1 | and T > 0.
The analysis of stability of the NDO based RTSMC is given in the following:
150
Proposition 2.
10
Proof. Choose the Lyapunov function V as: V =
1 2 1 ˜2 s + d 2 2
(17)
Differentiating (17) along the equation (7), (11) - (16), one can notice that:
V˙
=
˙ ss˙ + d˜d˜
=
s[f1 + g1 (u + d) − x ¨d + c1 e˙ − v¨ − c1 v] ˙ + (−LG2 d˜2 )
=
s[f1 + g1 [(ˆ g1−1 [¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨ − k · sign(s)
ˆ + d] − x −λs] − d) ¨d + c1 e˙ − v¨ − c1 v] ˙ + (−LG2 d˜2 ) =
s[f1 + (1 + △)gˆ1 [ˆ g1−1 [¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨ − k · sign(s)
−λs] − dˆ + d] − x¨d + c1 e˙ − v¨ − c1 v] ˙ + (−LG2 d˜2 ) =
s[f1 − fˆ1 + △(¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨) − (1 + △)k · sign(s)
˜ − (1 + △)λs2 − LG2 d˜2 +(1 + △)ˆ g1 d] ≤ ≤ ≤
s[f1 − fˆ1 + △(¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨) + (1 + △)ˆ g1 d˜
−(1 − Dm )k · sign(s)] − (1 − Dm )λs2 − LG2 d˜2
˜ s[(Fm + Dm |¨ xd − fˆ1 − c1 e˙ + c1 v˙ + v¨| + (1 + Dm )ˆ g1 d)sign(s)
−(1 − Dm )k · sign(s)] − (1 − Dm )λs2 − LG2 d˜2 −T |s| − (1 − Dm )λs2 − LG2 d˜2 ≤ 0
Therefore, the Lyapunov function of NDO Based RTSMC system is decreasing. This completes the proof. Using the proof procedure mentioned above, we also have: ss˙ ≤ −T |s| − (1 − Dm )λs2 ≤ −T |s| According to the proof about the TSMC proposed by K-B, Park [18], we notice that s(t) ≡ 0 and e(t) ≡ v(t). Thus, e(t) can converge to zero in setting time T , i.e. e(t) = 0, ∀t ≥ T . Furthermore, it can be known: ss˙ =
1 d 2 s ≤ −T |s| − (1 − Dm )λs2 2 dt 11
(18)
155
Assume for instance that s(t = Ts ) > 0,(Ts is the beginning time of some point which is not on the sliding surface). Tr is defined as the time when the system reaches on the sliding surface. Integrating the equation (18) between t = Ts and t = Tr leads to:
0 − s(t = Ts ) = s(t = Tr ) − s(t = Ts ) ≤ −T (Tr − Ts ) − (1 − Dm )λ
Z
Tr
sdt (19)
Ts
which implies that
Tr − Ts ≤
s(t = Ts ) − (1 − Dm )λ T
R Tr Ts
sdt
(20)
If s(t = Ts ) < 0, a similar result can be obtained, Hence, the equation (20) can be rewritten as: Tr − Ts ≤ 160
|s(t = Ts )| − (1 − Dm )λ T
R Tr Ts
|s|dt
(21)
If Ts = 0, one can notice that the |s(t = 0)| will be equal to zero in NDO
based RTSMC system. This means that the control system is always on the sliding surface. Hence, the tracking error can converge to zero in finite time (T ) [18]. On the other hand, if Ts is a certain time when the control system breaks away from the sliding surface because of the external disturbance (Ts > T ), 165
one can find that w = 0, ∀t > T . In this case, the w has no effect on reducing
the disturbance. However, the convergence time (Tr − Ts ) can also be reduced RT by the item, (1 − Dm )λ Tsr |s|dt, which is introduced by the robust item λs.
Therefore, it is clear that NDO based RTSMC can reduce the required time for eliminating external disturbances effectively during the whole control process.
170
4. Experimental Setup and Parametric Identification (A) Experimental Setup All experiments are implemented using the EICOSI prototype, as shown in Fig.3. The orthosis has a single degree of freedom at the knee joint level and is 12
attached to the shank and thigh by means of straps. It is driven by an on-board 175
brushless DC motor. The orthosis is equipped with an incremental encoder that measures the knee joint angle. Note that a forth-order Butterworth filter is used to process the raw measured knee joint angle to solve the oversampling problem. All computations are performed using a dSPACE board, equipped with an IBM processor (PowerPC 604th) that has a frequency of 400 M Hz.
180
The controller is based on a sliding mode control law (Basic SMC, RTSMC) (9) coupled to an observer (3) to estimate the human muscular torque. The control law is computed over the angle measurement delivered by the incremental encoder and the angular velocity obtained by a simple derivation. The dSPACE board delivers the pulse width modulation (PWM) level necessary to control the
185
actuator’s velocity. The control loop runs at 1 kHz, fixed due to the onboard sensors constraints. The experiments are conducted with three healthy subjects. Some of their relevant information such as sex, age, weight and height are listed in Table1. Subject 1 is one of the researchers, while the other subjects are totally naive
190
to the experimentation. Additionlly, a plastic mannequin (Fig.1) is used in the expriments to compare the effectiveness of the above-mentioned three control approaches . Three different control methods are used in the experiments: Basic SMC, NDO Based SMC and NDO Based RTSMC. Note that all experiments are implemented in the sitting position, as shown in Fig.4. The parameters of these
195
three controllers used in the experiments are chosen based on the experimental performance conducted by Subject 1. To compare the robustness and the ability for estimating disturbances of the three methods, these parameters are also used in the experiments conducted by other subjects and the mannequin. Note that some common parameters are chose to retain consistency as far as possible. The
200
parameters are given as follows: • Basic SMC: c1 = 5, and σ = 15; • NDO Based SMC: c1 = 5, k1 = 1.5, k2 = 0.5, and σ = 15; • NDO Based RTSMC: c1 = 5, k1 = 1.5, k2 = 0.5, σ = 7.5, λ = 5, 13
Dm = 0.1, Fm = 1, and T = 0.5;
Figure 3: The structure of the orthosis EICOSI.
Figure 4: The human lower limb and the orthosis EICOSI.
205
(B) Identification of the parameters. The inertial parameters of the system composed of the wearer and the orthosis are A, B, J and τg . The paramters, J and τg , can be considered to be composed of the human shank’s interia Js and gravity τgs and the interia Jo and gravity τgo of the orthosis respectively. The shank’s interia Js and gravity τgs
210
can be identified based on the weight and height of wearers [21]. The interia Jo 14
Table 1: Subjects’ Sex, Age, Weight and Height
Subject
Sex
Age(yr)
W eight(kg)
Height(cm)
S1
M
27
66
170
S2
F
24
60
172
S3
M
23
72
173
Table 2: System’s identified parameters
J(Jo + Js )(Kg · m2 )
0.2639(0.0117+0.2522)
A(N · m)
4.4289
K(N · m · rad−1 )
0.3382
τg (τgo + τgs )(N · m)
8.5724(0.2424+8.33)
B(N · m · s · rad−1 )
0.5950
and gravity τgo of the orthosis can be obtained by the methods presented in our previous study [22]. The other paremeters are identified by using the passive pendulum test [21]. During the identification process, the wearer is passive; he/she is not delivering any muscular effort: τh = 0. This is monitored by a 215
null electromyographical activity of the thigh muscles spanning the knee joint. Hence, the equation (1) can be rewritten as: τg cos θ = −J θ¨ − τg cos θ − Asignθ˙ − B θ˙ − K(θ − θr ),
(22)
The knee joint angle is measured by means of an incremental encoder; the angular velocity and acceleration are derived numerically. Therefore, one can notice that only the parameters A, B, K have to be identified in equation (22). 220
The least squares optimization algorithm is used to process the measured data. The identification experiments are conducted by subject 1 and the identified parameters are presented in Table 2.
15
5. Experimental Results To evaluate the performance of the control methods proposed in this study, 225
three groups of experiments were implemented to compare the tracking precision, required time for eliminating the disturbance and robustness respectively. Additionally, a step-response experiment was employed to verify the stability of the NDO Based RTSMC system as well. (A) Comparison of tracking precision.
230
In this experiment, a given sinusoidal trajectory with a frequency of 0.5Hz was used as the reference trajectory and three different control methods (Basic SMC, NDO Based SMC, NDO Based RTSMC) were used to control the shank-orthosis system to track the reference trajectory respectively. Subject 1 participated in this experiment. He was asked to sit on a table while wearing
235
the orthosis, and keep his knee joint relaxed during the experiment as shown in Fig.4. The average tracking errors are shown in Table 3. The current and desired knee joint angles are plotted in Fig.5. The tracking errors and the control torques are shown in Fig.6 and Fig.7 respectively. The comparative results
240
shows that NDO Based RTSMC can reduce the tracking error effectively, compared to the basic SMC. At the same time, the chattering of the control torque of NDO Based RTSMC is the lowest among these three approaches. During the experimets, the comfortability of the user is often associated with the chattering of control torques; the lower the frequency and amplitude of chattering,
245
the higher the comfortability. Additionally, although the tracking error and chattering of control torque were not decreasing significantly when using NDO Based SMC compared to the use of basic SMC; the peak error values are kept at a relatively low level, as shown in Fig.6. (B) Comparison of required times for eliminating disturbance.
250
As mentioned above, the NDO Based RTSMC method has the advantages of having the tracking error converging to zero in finite time and eliminating the 16
Ref
0
SMC NDOBSMC
−0.2
NDOBRTSMC
Position (rad)
−0.4 −0.6 −0.8 −1 −1.2 −1.4
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 5: Comparison of tracking precision (conducted by subject 1): the desired knee joint angles (black line) and the tracking control results of shank-orthosis system by employing the basic SMC, NDO Based SMC and NDO Based RTSMC methods respectively.
0.12 SMC NDOBSMC NDOBRTSMC
0.1 0.08
Error (rad)
0.06 0.04 0.02 0 −0.02 −0.04 −0.06
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 6: Comparison of tracking precision (conducted by subject 1): the tracking errors of different control methods:|Error|= {SMC, mean:0.03576; std:0.0259},{NDO Based SMC, mean:0.03197; std:0.01635},{NDO Based RTSMC, mean:0.02486; std:0.01492}
17
SMC NDOBSMC NDOBRTSMC
15
Torque (Nm)
10
5
0
−5
−10
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 7: Comparison of tracking precision (conducted by subject 1): the control torques of different control methods.
external disturbance faster than the other two control methods. To verify these advantages, two experiments are designed. In one experiment, a setting error was given at the beginning time. In the other one, a given external disturbance 255
was added to the shank-orthosis after the control system reached the sliding surface. In these experiments, convergence times to the sliding surface was calculated and compared. The sinusoidal trajectory with frequency of 0.5 Hz was also used as reference trajectory. Additionally, to guarantee the accuracy of these comparison experiments, a plastic mannequin was used to replace the
260
human being subject. This is because with the human being, it is difficult to keep the same experimental conditions during each experiment. Note that the associated tracking errors were changed because of the subject changes. Here, we just focus on the convergence time. The robustness will be discussed in section (C).
265
According to the first experiment, the current and desired knee joint angles are plotted in Fig.8. The tracking errors and the control torques are shown in Fig.9 and Fig.10 respectively. The current and desired knee joint angles corresponding to the second experiment are plotted in Fig.11. The tracking errors
18
and the control torques are shown in Fig.12 and Fig.13 respectively. According 270
to the tracking errors, when the system reaches the sliding surface, ±0.15rad
are chosen as the boundaries for evaluating the convergence time as shown in Fig.8 and Fig.11.
One can observe that the convergence time of NDO Based RTSMC method is relatively smaller than that of the basic SMC, and NDO Based SMC does 275
not show an advantage on the convergence time, compared to the basic SMC as shown in Fig.8. This is due to the fact that NDO increases the control torque quickly (Fig.10) to track the desired trajectory at the beginning time and the sudden increase of the torque makes a bad influence on the follow-up convergence. However, this problem is avoided in the NDO Based RTSMC
280
methods effectively. NDO Based RTSMC controller can response to sudden changes of the position at the beginning time due to the item v, and avoid the influence of the sudden torque after time T , because of the use of the item λs. In the second experiment, a given external disturbance is added to the system (between t=4s and t=5.2s), as shown in Fig.11 and Fig.13. The comparative
285
results show that the NDO Based RTSMC approach can significantly reduce the required time for eliminating the external disturbances. Additionally, one can observe that the basic SMC almost cannot compensate the external disturbances as shown in Fig.11 and Fig.13. (C) Robustness comparison.
290
To verify the robustness of the control methods, two other subjects (Subject 2 and Subject 3) and a plastic mannequin with 1 kg load on its foot were chosen instead of Subject 1 to conduct the same experiment as mentioned in Section 5.(A). The identified parameters of the shank-orthosis model for Subject 1 and parameters of three control methods were still used in this experiment, although
295
the parameters of Subject 2, Subject 3 and the mannequin are obviously different from those of the Subject 1(especially the parameter of the mannequin: K is equal to zero, the weight of its shank is larger than that of Subject 1). Three groups of experiments were conducted by each subject or mannequin, and each 19
Ref
0
SMC NDOBSMC NDOBRTSMC
−0.2
Position (rad)
−0.4 −0.6 −0.8 −1 −1.2 −1.4
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 8: Comparison of the required times for eliminating disturbance (conducted by the mannequin) - the convergence time at the beginning time: the desired trajectory and the tracking trajectories of the controllers: basic SMC, NDO Based SMC and NDO Based RTSMC respectively.
1 SMC NDOBSMC NDOBRTSMC 0.15 rad (15%) −0.15 rad (15%)
0.8
Error (rad)
0.6 0.4 0.2 0 −0.2 −0.4
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 9: Comparison of the required times for eliminating disturbance (conducted by the mannequin) - the convergence time at the beginning time: the tracking errors for the controllers: basic SMC, NDO Based SMC and NDO Based RTSMC respectively.
20
SMC NDOBSMC NDOBRTSMC
20
15
Torque (Nm)
10
5
0
−5
−10
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 10: Comparison of the required times for eliminating disturbance (conducted by the mannequin) - the convergence time at the beginning time: the control torques for the controllers: basic SMC, NDO Based SMC and NDO Based RTSMC respectively.
Ref SMC
0
NDOBSMC NDOBRTSMC
−0.2
Position (rad)
−0.4 −0.6 −0.8 −1 −1.2
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 11: Comparison of the required times for eliminating disturbance (conducted by the mannequin) - the convergence time for eliminating the external disturbance: the desired trajectory and the tracking trajectories for the controllers: basic SMC, NDO Based SMC and NDO Based RTSMC respectively.
21
0.8 SMC NDOBSMC NDOBRTSMC 0.15 rad (15%) −0.15 rad (15%)
0.6
Error (rad)
0.4 0.2 0 −0.2 −0.4 −0.6
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 12: Comparison of the required times for eliminating disturbance (conducted by the mannequin) - the convergence time for eliminating the external disturbance: the tracking errors for the controllers: basic SMC, NDO Based SMC and NDO Based RTSMC respectively.
20
SMC NDOBSMC NDOBRTSMC
15
Torque (Nm)
10
5
0
−5
−10
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 13: Comparison of the required times for eliminating disturbance (conducted by the mannequin) - the convergence time for eliminating the external disturbance: the control torques for the controllers: basic SMC, NDO Based SMC and NDO Based RTSMC respectively.
22
Table 3: Average Tracking Errors of Three Control Methods
|Error|(rad)
SMC
NDO Based SMC
NDO Based RTSMC
mean
std
mean
std
mean
std
Subject 1
0.0358
0.0259
0.0320
0.0164
0.0249
0.0149
Subject 2
0.2450
0.2028
0.0653
0.0479
0.0532
0.0392
Subject 3
0.3647
0.1221
0.0525
0.0299
0.0450
0.0268
M annequin
0.0584
0.0421
0.0401
0.0451
0.0328
0.0328
group lasted 15 seconds. The experimental results (tracking errors) are shown 300
in Table.3. According to the tracking errors, one can notice that the mean errors (|Error|) of NDO based SMC and RTSMC are always smaller than those of the basic SMC for each subject and the mannequin. Additionlly, the robustness of the NDO based RTSMC is the best among these three control methods. The tracking errors of NDO based RTSMC do not increase significantly, with the
305
changes of the system parameters, while there are relatively significant changes in errors when using the two other approaches, particularly the basic SMC. (D) Stability analysis. To evaluate the stability of the NDO Based RTSMC approach proposed in this paper, a step response experiment was implemented. In this experiment,
310
the plastic mannequin with a 1 Kg load was used. The experimental results are shown in Fig.14, Fig.15 and Fig.16. The results show that the shank-orthosis system can response quickly and the tracking error can converge to zero in relative short time. We can observe that there is an overshoot of this controller in this experiment.
315
6. Conclusions The present work addressed the development of a sliding mode control of a knee joint exoskeleton taking into account the estimation of the wearer’s muscular torque. The model of the shank-orthosis was shown and its parameters 23
−0.4 −0.5 Ref NDOBRTSMC
Position (rad)
−0.6 −0.7 −0.8 −0.9 −1 −1.1
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 14: Analysis of the stability: the desired trajectory and the tracking trajectories by employing the controller NDO Based RTSMC.
0.7 NDOBRTSMC 0.6
Error (rad)
0.5 0.4 0.3 0.2 0.1 0 −0.1
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 15: Analysis of the stability: the tracking errors by employing the controller NDO Based RTSMC.
24
16 NDOBRTSMC 14 12 Torque (Nm)
10 8 6 4 2 0 −2
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Figure 16: Analysis of the stability: the control torques by employing the controller NDO Based RTSMC.
were estimated using least squares optimization. A basic sliding mode controller 320
coupled to an observer and a robust terminal sliding mode control method combined with a nonlinear disturbance obserber were proposed to drive the orthosis along some desired trajectories. The human muscular torque was considered as an external disturbance of the system in this work, so the NDO based SMC and NDO based RTSMC were designed to estimate the human torque and enhance
325
the performance of the shank-orthosis system. The asymptotic stability of the system is proved by means of a Lyapunov analysis. The experiment results confirmed the advantages of NDO Based RTSMC: improving the tracking precision and reducing the required time for eliminating the disturbance. At the same time, the robustness and stability of NDO Based RTSMC method were verified
330
by the experiments as well. The current paper does not deal with the misalignment problem that was considered in the literature, chiefly in exoskeletons with multiple degrees of freedom. In this study, this problem has been considerably reduced by adjusting manually the orthosis to every wearer’s morphology using adaptable straps.
335
Special cares have been taken during experiments in order to avoid reaching of
25
the full knee joint flexion which considerably reduce the joints misalignment. Although the system model is built based on the sitting position, the control strategy proposed in this paper is considered to be suitable for other daily living activities, for instance, the gait walk. In these situations, the muscular effort 340
can also be considered as the disturbance like that in the sitting position. In this case, a rational reference trajectory given by therapists can also be tracked by users’ lower limb with the assistance of the EICOSI, if the associated models are built based on these situations. Hence, more system models based on different activities during daily living will be studied in the future work, such as normal
345
gait walk, sit-to-stand and stairs climbing,etc. The control methods for the EICOSI will be further studied to assist the wearer to accomplish more activities during daily living as well. Acknowledgements This work was supported in part by the Fundamental Research Funds for
350
the Central Universities, huazhong University of Science and Technology under Grant 2013ZZGH007. References [1] A. Tsukahara, R. Kawanishi, Y. Hasegawa, Y. Sankai, Sit-to-stand and stand-to-sit transfer support for complete paraplegic patients with robot
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suit hal, Journal of Advanced Robotics 24 (1) (2010) 1615–1638. [2] G. Colombo, M. Joerg, R. Schreier, V. Dietz, Treadmill training of paraplegic patients using a robotic orthosis, Journ. of Rehabilitation Research and Development 37 (6) (2000) 693–700. [3] R. Veneman, J. Kruidhof, E. Hekman, R. Ekkelenkamp, E. Van Asseldonk,
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H. Van Der Kooij, Design and evaluation of the lopes exoskeleton robot for interactive gait rehabilitation, IEEE Transactions on Neural Systems and Rehabilitation Engineering 15 (3) (2007) 379–386. 26
[4] P.-M. Bonutti, G.-A. Marulanda, M.-S. McGrath, M.-A. Mont, M.-G. Zywiel, Static progressive stretch improves range of motion in arthrofibro365
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*Biography of each author
Samer Mohammed received the Diploma degree in electrical engineering from Lebanese University, Tripoli, Lebanon, the M.S. degree in automatic and microelectronic systems from the University of Montpellier II, Montpellier, France, and the Ph.D. degree in computer science from the Laboratory of Computer Science, Robotic and Microelectronic of Montpellier (LIRMM/CNRS), Montpellier. He is currently an Assistant Professor of computer science with the Laboratory of Image, Signal and Intelligent Systems, University of Paris-Est Créteil, Créteil, France. His current research interests include modeling, identification, and control of robotic systems (wearable robots), artificial intelligence, and decision-support theory. His current research involved the applications for the functional assisting of dependent people. Dr. Mohammed was the recipient of the Fellowship Award from the Japan Society for the Promotion of Science, the French Ministerial Doctoral Research Scholarship Award, and the École Polytechnique Fédéral de Lausanne Scholarship Award. He is actively involved in organizing committees of some national and international workshops in the automatic and robotic domains. Weiguang Huo is a PhD student in LISSI laboratory (UPEC), under the supervision of Prof. Amirat and Dr. Samer Mohammed. His PhD is supported by the French high education institution. He finished his Master study at the Department of Control Science and Engineering, Huazhong University of Science and Technology (HUST), China. He obtained his bachelor degree at 2009 from the Department of Measurement and Control Technology, Wuhan University of Technology WHST), China. Jian Huang graduated from Huazhong University of Science & Technology (HUST), China in 1997 and received the Master of Engineering degree from HUST in 2000. He received his Ph.D from HUST in 2005. From 2006 to 2008, he was a postdoctoral researcher in the Department of Micro-Nano System Engineering and Department of Mechano-Informatics and Systems, Nagoya University, Japan. He is currently an associate professor at School of Automation, HUST. His main research interests include rehabilitation robot, robotic assembly, networked control systems and bioinformatics. Hala RIFAI graduated in electronic-electrical engineering from the Lebanese University in 2004. She received a master diploma in automatic control in 2005 and a Ph.D. in non-linear control in 2008 from the Institut Polytechnique de Grenoble, GIPSA-Lab (UMR CNRS), Grenoble, France. Between 2008 and 2010, she had a post-doctoral scholarship in I3S-Lab (UMR CNRS), Sophia Antipolis, France. Currently, she has a post-doctoral scholarship in the Laboratory of Signals, Systems and Intelligent Systems (LSSI) of the University of Paris-Est Créteil (UPEC). Her research domains include the modeling, control and bilateral teleoperation of robotic systems. Applications concern the wearable robots and unmanned aerial vehicles.
Yacine Amirat received the Ph.D. degree in computer science and robotics from the University of Pierre et Marie Curie, Paris, France, in 1989, and the Habilitation degree in artificial intelligence and control of complex systems from the University of Paris 12-France, Paris, in 1996. He co-created the Laboratory of Computer Sciences and Robotics, Paris 12 University
(newly UPEC), Créteil, France, in 1990. He is currently the Director of the Laboratory of Image, Signal and Intelligent Systems (LISSI) – UPEC, and the Scientific Director of several research projects. He has authored or co-authored more than 100 papers in scientific journals, books, and conference proceedings. His current research interests include artificial intelligence, soft computing, knowledge processing, and control of complex systems. Application fields are intelligent and robotic systems such as ubiquitous and service robots, wearable robots, and ambient intelligence. Prof. Amirat is a reviewer of many international journals and conférences and a Technical Committee Member, the Session Chair, a Session Organizer, or an Associate Editor of several IEEE-RAS conferences. He is actively collaborating with industrial and academic partners in various European projects.
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