Integrated Observer-based Fixed-time Control with Backstepping Method for Exoskeleton Robot

Gao-Wei Zhang Peng Yang Jie Wang Jian-Jun Sun Yan Zhang

Gao-Wei Zhang, Peng Yang, Jie Wang, Jian-Jun Sun and Yan Zhang. Integrated Observer-based Fixed-time Control with Backstepping Method for Exoskeleton Robot. International Journal of Automation and Computing, vol. 17, no. 1, pp. 71-82, 2020 doi:  10.1007/s11633-019-1201-z
Citation: Gao-Wei Zhang, Peng Yang, Jie Wang, Jian-Jun Sun and Yan Zhang. Integrated Observer-based Fixed-time Control with Backstepping Method for Exoskeleton Robot. International Journal of Automation and Computing, vol. 17, no. 1, pp. 71-82, 2020 doi:  10.1007/s11633-019-1201-z

doi: 10.1007/s11633-019-1201-z

Integrated Observer-based Fixed-time Control with Backstepping Method for Exoskeleton Robot

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    Author Bio:

    Gao-Wei Zhang received the B. Sc. degree in automation from Hebei University of Technology, China in 2014. Currently, he is a Ph. D. degree candidate in control theory and control engineering in Hebei University of Technology, China.His research interests include nonlinear control theory, sliding mode control and wearable exoskeleton. E-mail: 201612501004@stu.hebut.edu.cn ORCID iD: 0000-0002-2884-1325

    Peng Yang recieved the M. Sc. degree in automation from Harbin Institute of Technology, China in 1988, and recieved the Ph. D. degree in electrical engineering from Hebei University of Technology, China in 2001. Since 1982, he has been in Hebei University of Technology where his present position is professor. From January to May in 2005, he has been in University of Munich as a visiting scholar. He has published more than 100 journal and conference papers. His awards and honors include the Natural Science Award of Hebei Province, Science and Technology Progress Award of Hebei Province, Natural Science Award of Chongqing. His research interests include complex system modeling and control, robot control and prosthetics.E-mail: yangp@hebut.edu.cn ORCID iD: 0000-0003-3006-2184

    Jie Wang received the M. Sc. degree in basic mathematics from Northeastern University, China in 2010, and the Ph. D. degree in control science and engineering from Tianjin University, China in 2014. Since 2014, she has been in Hebei University of Technology where her present position is associate professor. She has published about 20 papers and one of them is high cited paper in ESI. Her research interests include nonlinear control theory, sliding mode control and observation with applications to hypersonic vehicle, quadrotor aircraft and wearable exoskeleton. E-mail: wangjie@hebut.edu.cn (Corresponding author) ORCID iD: 0000-0002-8613-3976

    Jian-Jun Sun received the B.Sc. degree in measurement and control technology and instruments from Taiyuan University of Technology, China in 2016. He is currently a Ph. D. degree candidate in control theory and control engineering in Hebei University of Technology, China.His research interests include nonlinear control theory and sliding mode control. E-mail: 201812501002@stu.hebut.edu.cn ORCID iD: 0000-0002-8142-1928

    Yan Zhang received the M. Sc. degree in automation from Hebei University of Technology, China in 1999, and the Ph. D. degree in control theory and control engineering from Nankai University, China in 2004. She has been Hebei University of Technology since 1999 where her present position is professor. She has hosted and participated in several National Natural Science Foundations of China and the Natural Science Foundations of Hebei Province projects, China. Her research interests include nonlinear system control, intelligent algorithm and prosthetics. E-mail: yzhangz@163.com ORCID iD: 0000-0002-9727-0212

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出版历程
  • 收稿日期:  2019-03-15
  • 录用日期:  2019-09-17
  • 网络出版日期:  2019-11-07
  • 刊出日期:  2020-02-01

Integrated Observer-based Fixed-time Control with Backstepping Method for Exoskeleton Robot

doi: 10.1007/s11633-019-1201-z
    作者简介:

    Gao-Wei Zhang received the B. Sc. degree in automation from Hebei University of Technology, China in 2014. Currently, he is a Ph. D. degree candidate in control theory and control engineering in Hebei University of Technology, China.His research interests include nonlinear control theory, sliding mode control and wearable exoskeleton. E-mail: 201612501004@stu.hebut.edu.cn ORCID iD: 0000-0002-2884-1325

    Peng Yang recieved the M. Sc. degree in automation from Harbin Institute of Technology, China in 1988, and recieved the Ph. D. degree in electrical engineering from Hebei University of Technology, China in 2001. Since 1982, he has been in Hebei University of Technology where his present position is professor. From January to May in 2005, he has been in University of Munich as a visiting scholar. He has published more than 100 journal and conference papers. His awards and honors include the Natural Science Award of Hebei Province, Science and Technology Progress Award of Hebei Province, Natural Science Award of Chongqing. His research interests include complex system modeling and control, robot control and prosthetics.E-mail: yangp@hebut.edu.cn ORCID iD: 0000-0003-3006-2184

    Jie Wang received the M. Sc. degree in basic mathematics from Northeastern University, China in 2010, and the Ph. D. degree in control science and engineering from Tianjin University, China in 2014. Since 2014, she has been in Hebei University of Technology where her present position is associate professor. She has published about 20 papers and one of them is high cited paper in ESI. Her research interests include nonlinear control theory, sliding mode control and observation with applications to hypersonic vehicle, quadrotor aircraft and wearable exoskeleton. E-mail: wangjie@hebut.edu.cn (Corresponding author) ORCID iD: 0000-0002-8613-3976

    Jian-Jun Sun received the B.Sc. degree in measurement and control technology and instruments from Taiyuan University of Technology, China in 2016. He is currently a Ph. D. degree candidate in control theory and control engineering in Hebei University of Technology, China.His research interests include nonlinear control theory and sliding mode control. E-mail: 201812501002@stu.hebut.edu.cn ORCID iD: 0000-0002-8142-1928

    Yan Zhang received the M. Sc. degree in automation from Hebei University of Technology, China in 1999, and the Ph. D. degree in control theory and control engineering from Nankai University, China in 2004. She has been Hebei University of Technology since 1999 where her present position is professor. She has hosted and participated in several National Natural Science Foundations of China and the Natural Science Foundations of Hebei Province projects, China. Her research interests include nonlinear system control, intelligent algorithm and prosthetics. E-mail: yzhangz@163.com ORCID iD: 0000-0002-9727-0212

English Abstract

Gao-Wei Zhang, Peng Yang, Jie Wang, Jian-Jun Sun and Yan Zhang. Integrated Observer-based Fixed-time Control with Backstepping Method for Exoskeleton Robot. International Journal of Automation and Computing, vol. 17, no. 1, pp. 71-82, 2020 doi:  10.1007/s11633-019-1201-z
Citation: Gao-Wei Zhang, Peng Yang, Jie Wang, Jian-Jun Sun and Yan Zhang. Integrated Observer-based Fixed-time Control with Backstepping Method for Exoskeleton Robot. International Journal of Automation and Computing, vol. 17, no. 1, pp. 71-82, 2020 doi:  10.1007/s11633-019-1201-z
    • Recently, power-assisted wearable exoskeleton, including passive and active exoskeletons, for recovering and enhancing human motion ability have attracted growing attention due to their research and application significance[1, 2]. It has been widely investigated in medical rehabilitation and the military. Generally, active exoskeleton is known as a robot with actuated joints providing assisting torque such that the muscle torque from human can be reduced. Unlike other robotics, wearable exoskeletons are paralleled to the human body, and have stricter demands in dynamic response[3, 4]. In addition, the inevitable model uncertainties and disturbances in the presence of the human-robot system cause difficulties in robust controller design processes, such as censoring errors and unmodeled dynamics[5].

      To ensure the tracking behaviour, various practical control strategies have been proposed depending on modern control approaches, including proportion integral derivative (PID), impedance control and adaptive control[6-8]. However, the control strategies mentioned above cannot handle large uncertainties and disturbances. To further extend the stability constraints, a sliding mode control (SMC) method is utilized in the exoskeleton, which rejects the disturbances by switching items so that the robustness can be obtained[9-12]. In [13], the architecture and dynamics of the upper-limb exoskeleton are constructed and SMC is adopted to enable the tracking performance of the exoskeleton under disturbances. It is generally known that an essential disadvantage of traditional SMC methods is the chattering phenomenon resulting from the switch items[14]. The most common way to tackle the problem is estimating the exact value of disturbances and compensating the controller[15]. Furthermore, intelligent algorithms and observers are always employed to estimate the disturbances[16-20]. In [16], fuzzy logic systems (FLS) are utilized to estimate exact values of disturbances and compensate the backstepping sliding mode controller online to reject the chattering phenomenon.

      For the trajectory tracking problem in the exoskeleton, one significant requirement is fast responses[21]. From the perspective of convergence speed, terminal and high-order SMC possess the finite-time property[22], which means that the ultimate convergence time can be resolved as a function of initial system states. Therefore, a better convergence performance and tracking precision than asymptotic methods can be obtained[23, 24]. In [25], a non-singular terminal sliding mode control (NTSMC) strategy is developed and allocated to a knee orthosis where the tracking errors of the closed-loop system are proved to remain in a small vicinity of zero. In addition, a second order sliding mode control (SOSMC) strategy is proposed for the same orthosis in [26] and the tracking errors are forced to zero in finite-time. However, the problem in the presence of the finite-time control theory is that the convergence time function is heavily associated with initial system conditions, which may result a slow dynamic response when the initial error is large enough[27, 28].

      To further promote the dynamic response, a fixed-time control theory is proposed and extensively researched. Unlike finite-time convergence, fixed-time can provide a certain convergence time which is bounded and independent of initial system states[29]. To achieve the fixed-time stability, many approaches have been obtained. Two types of nonlinear control algorithms are proposed by Polyacov and the global fixed-time stability is obtained[30]. However, it is limited in its use due to the complicated forms. A concise non-singularity terminal SMC is constructed in [31] and the fixed-time stability is proved through Lyapunov theory. In order to suppress the disturbances, a comprehensive control scheme composed of the fixed-time controller and disturbance observer is proposed for a second-order nonlinear system and the tracking errors are proved to be converged to the origin within fixed-time in [32]. Additionally, a novel exponential unified form was interpreted in [33], which provides a direct functional relationship between the designed parameters and convergence time.

      Furthermore, fixed-time theory and methods have been investigated and employed in many control tasks and applications, including multi-agent systems, hypersonic gliding vehicles, spacecraft[34, 35] due to its bounded settling time regardless of the initial states. However, to the best of the authors' knowledge, little research on fixed-time control strategies has been executed in exoskeletons. Therefore, a fixed-time control strategy is proposed for a 5 DoF (degrees of freedom) upper-limb exoskeleton[36] and a novel sliding mode surface is introduced to guarantee the fixed-time convergence. To achieve the robustness, multivariable fixed-time disturbance observers are constructed and utilized. The key contributions are interpreted as follows:

      1) A comprehensive multivariable sliding mode strategy based on a backstepping method is proposed for an upper-limb exoskeleton by synthesizing an SOSMC and fixed-time disturbances observer (FTDO). Not only the tracking errors but also estimation errors are proved to converge to zero in a bounded time independent of initial conditions. Additionally, the chattering phenomenon is attenuated by allocating the FTDOs.

      2) An exponential sliding mode surface is introduced, which provides a fast convergence and direct relationship between control gains and convergence time. To further improve the response speed, SOSMC is employed to achieve the fixed-time stabilization.

      3) The applications of the proposed control strategy are not only limited to exoskeletons but also to a class of multiple-input multiple-output (MIMO) nonlinear systems with bounded uncertainties and disturbances.

      The organisation of this paper can be summarized as: Dynamics of a 5 DoF upper-limb exoskeleton is constructed and processed for control purposes in Section 2. In Section 3, a comprehensive fixed-time control scheme synthesized of SOSMC and FTDO is developed, where a novel sliding mode surface is introduced. The fixed-time stability is analyzed by Lyapunov theory. In Section 4, simulation results are demonstrated and the conclusion of this work is interpreted in Section 5.

    • For clear and concise description, some useful symbols employed in following process are illustrated first. Throughout the paper, $ {\bf R}^+ $, $ {\bf R}^n $ and $ {\bf R}^{m \times n} $ indicate the sets of positive real number, $ n $ dimensional vectors and $ m \times n $ real matrices respectively. $ \left\| B \right\| $ represents the Euclidean norm of vector $ B \in {\bf R}^n $, i.e., $ \left\| B \right\| = \sqrt{B^{\rm T}B} $. For a $ n $-order square matrix A, A−1 is the inverse matrix of A satisfied A−1 $ A = E $, where E is unit matrix.

    • The dynamics of the 5 DoF upper-limb exoskeleton employed in this work is initially proposed in [7], which is composed of five essential actuated joints to ensure the similar kinematics and dynamics with the human body. In details, the five joints include: shoulder abduction/adduction, shoulder flexion/extension, elbow flexion/extension, wrist flexion/extension and forearm rotation. The specific architecture and the disposition of the 5 DoF can be shown as Fig. 1.

      Figure 1.  Schematic and architecture of exoskeleton

      The dynamic model describes the relationship between assisting torques provided by actuators and the exoskeleton's joint angles. One of the most common methods to construct the dynamic model of rigid body robotics is the Lagrange equation, which can be illustrated as

      $$ \frac{\rm d}{{{\rm d}t}}\frac{{\partial Q}}{{\partial \dot \theta}} - \frac{{\partial Q}}{{\partial \theta}} = \tau $ $ (1)

      where $ Q $ is the difference of kinetic energy and potential energy. $ \theta $ and $ \dot \theta $ indicate state parameters of the system, which are the joint angles and velocities in this study.

      Then the nominal dynamics of the exoskeleton can be expressed as (2) with Lagrange method.

      $$ M(\theta)\ddot \theta + C(\theta,\dot \theta)\dot \theta + G(\theta) = \tau $ $ (2)

      where $ \theta,\dot \theta,\ddot \theta \in {{\bf R}^5} $ indicate angle position, velocity and acceleration of active joints respectively. $ \tau \in {{\bf R}^5} $ is the auxiliary torques from drive elements. The coefficients $ M(\theta) \in {{\bf R}^{5 \times 5}},\,C(\theta,\dot \theta) \in {{\bf R}^{5 \times 5}}, \,G(\theta) \in {{\bf R}^{5 \times 1}} $ represent inertia matrix, carioles/centripetal matrix and gravity vector. To make the description clear, the specific form and parameters are shown in Appendix.

      Considering the inevitable unmatched and matched disturbances existing in human-robot system and introducing a new state vector $ x = {[x_1^{\rm T},x_2^{\rm T}]^{\rm T}} $, which satisfies $ [{x_1},{x_2}] = [\theta, \dot \theta] $, (2) can be transformed into a state space described as follows:

      $$ \left\{ \begin{aligned}& {{\dot x}_1} = {x_2} + {d_1}\\ & {{\dot x}_2} = - {M^{ - 1}}({x_1})[C(x){x_2} + G({x_1}) - \tau ] + {d_2} \end{aligned} \right. $ $ (3)

      where $ {d_1},{d_2} \in {{\bf R}^5} $ are the unmatched and matched disturbance vectors, respectively. It can be shown that the exoskeleton system is a complicated coupled, nonlinear and multi-degree system.

    • The position errors are defined as $ {e_1} = {x_1} - {x_d} $, where $ {x_d} \in {{\bf R}^5} $ is the expected trajectory provided by a human. For the aspect of disturbance observer, the observer errors can be described as $ {e_{o1}} = {d_1} - {\hat d_1} $ and $ {e_{o2}} = d_2-{\hat d_2} $ by assuming the estimated values as $ {\hat d_1},\,{\hat d_2} $. Afterwards, the control objectives of the fixed-time control strategy can be demonstrated as:

      1) By utilizing the FTDOs, the estimation errors can be forced to the origin within a certain convergence time $ t_o $, which means

      $$ \mathop {\lim }\limits_{t \to {t_{o1}}} \left\| {{e_{o1}}} \right\| = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop {\lim }\limits_{t \to {t_{o2}}} \left\| {{e_{o2}}} \right\| = 0 . $ $

      2) The tracking errors $ {e_1} $ of closed-loop system could converge to zero in a short time bounded with a certain value $ {t_1} $, i.e.,

      $$ \mathop {\lim }\limits_{t \to {t_1}} \left\| {{e_1}} \right\| = 0 . $ $
    • In this section, an integrated fixed-time control strategy is designed for the aforementioned upper-limb exoskeleton. For the purpose of handling with unmatched disturbance in position subsystem and matched disturbance in velocity subsystem, the dynamics of exoskeleton is divided into two control subsystems, shown as Fig. 2. Subsequently, the stability of the closed-loop system is ensured when all the subsystems is stable.

      Figure 2.  Block diagram of fixed-time control scheme

      Before the design process of fixed-time controller, basic definitions and lemmas used in the proof are illustrated firstly:

      Definition 1.[30] The equilibrium point of system (3) is said to be fixed-time stable if it is globally stable and the settling time function $ T({x_0}) $ is bounded by some positive value independent of initial system conditions, which means $ \exists {T_{\max }} > 0 $, such that $ T({x_0}) \le {T_{\max }},\,\forall {x_o} \in {{\bf R}^n} $.

      Assumption 1. The 2nd norm of disturbances $ {d_1},\,{d_2} $ and their derivation are bounded with positive values $ {D_1},\,{D_2} $, which means, $ \left\| {{{\dot d}_1}} \right\| \le {D_1},\,\,\left\| {{{\dot d}_2}} \right\| \le {D_2} $, where $ {D_1},\,{D_2} $ are positive constants.

      Lemma 1.[37] Considering the second-order system

      $$ \left\{ \begin{aligned} & {{\dot l}_1}(t) = - {\lambda _1}\frac{{{l_1}(t)}}{{{{\left\| {{l_1}(t)} \right\|}^{\frac1 2}}}} - {\lambda _2}{l_1}(t){\left\| {{l_1}(t)} \right\|^{p - 1}} + {l_2}(t)\\ & {{\dot l}_2}(t) = - {\lambda _3}\frac{{{l_1}(t)}}{{\left\| {{l_1}(t)} \right\|}} + \iota (t) \end{aligned} \right. $ $ (4)

      where $ {l_1}({t_0}) = {l_{10}},\,{l_2}({t_0}) = 0 $. The designed value $ p > 1 $ is a constant relative to the speed of convergence when the states are far away from equilibrium point. $ \left\| {\iota (t)} \right\| \le D,\, D $ is a positive constant. The state vectors $ {z_1}(t),\,{\dot z_1}(t) $ converge to the origin within a fixed-time $ {T_{\max }} $ if $ {\lambda _1} > \sqrt {2{\lambda _3}},\,{\lambda _2} > 0,\,{\lambda _3}> 4D $ are satisfied, and

      $$ \begin{split} {T_{\max }} =\; & \left( {\frac{1}{{{\lambda _2}(p - 1){\varepsilon ^{p - 1}}}} + \frac{{2{{(\sqrt n \epsilon )}^{\frac 1 2}}}}{{{\lambda _1}}}} \right) \times\\ & \left( { 1 + \frac{M}{{m\left(1 - \dfrac{\sqrt {2{\lambda _3}} }{\lambda _1}\right)}}} \right) \end{split} $ $ (5)

      where $ \epsilon > 0 $, $ Z={\lambda _3} + D,z = {\lambda _3} - D $. $ n $ is the number of degrees existing in the system. According to (5), it can be drawn that the minimum value of the fixed-time $ {T_{\max }} $ is achieved when $ \epsilon = {\left({n^{\frac 1 4}}\dfrac{\lambda _1}{\lambda _2}\right)^{\frac {1}{p + \frac 1 2}}} $.

    • Supposing a typical 1-order multivariable nonlinear system described as

      $$ \dot \theta = f(\theta ,t) + g(\theta ,t)u + \Delta $ $ (6)

      where $ f(\theta ,t) \in {{\bf R}^n},\; g(\theta ,t) \in {{\bf R}^{n \times m}} $ and $ \Delta $ is the uncer- tainties and disturbances which satisfy $ || {\dot \Delta } || \le L $.

      Afterwards, the form of FTDO can be described as

      $$ \dot \omega = f(\theta ,t) + g(\theta ,t)u + v . $ $ (7)

      Selecting an auxiliary sliding mode surface as

      $$ {s_o} = \theta - \omega . $ $ (8)

      Then, the derivative of (8) along with time can be described as follows by (6) and (7).

      $$ {\dot s_o} = \dot \theta - \dot \omega = \Delta - v. $ $ (9)

      It can be concluded that the exact value of $ \Delta $ can be indicated by $ v $, if $ \dot s_o $ can be reached and remain zero ultimately. Motivated by the approach in [38] and Lemma 1, the auxiliary sliding surface and its time derivative $ s_o,\,\dot s_o $ can converge to origin in fixed-time under the designed item $ v $ selected as

      $$ v = {r_1}\frac{{{s_o}}}{{{{\left\| {{s_o}} \right\|}^{\frac 1 2}}}} + {r_2}{s_o}{\left\| {{s_o}} \right\|^{p - 1}} - \gamma ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \dot \gamma = - {r_3}\frac{{{s_o}}}{{\left\| {{s_o}} \right\|}} . $ $ (10)

      Combining (9) and (10) yields

      $$ \left\{ \begin{aligned} & {{\dot s}_o} = - {r_1}\frac{{{s_o}}}{{{{\left\| {{s_o}} \right\|}^{\frac 1 2}}}} - {r_2}{s_o}{\left\| {{s_o}} \right\|^{p - 1}} + \gamma \\ & \dot \gamma = - {r_3}\frac{{{s_o}}}{{\left\| {{s_o}} \right\|}} + \dot \Delta \end{aligned} \right. $ $ (11)

      where $ p>1 $. $ {r_1},{r_2},{r_3} $ are control parameters which satisfy $ r_1>\sqrt{2r_3}, \,r_2>0,\, r_3>4L $.

    • By adopting the backstepping method, the virtual control law of the position subsystem is designed as $ {x_{2c}} $ such that the position tracking errors are forced to zero in fixed-time.

      Firstly, FTDO for the position subsystem is constructed by the aforementioned method.

      Considering the desired trajectory $ x_d $, the tracking error can be defined as

      $$ {e_1} = {x_1} - {x_d}. $ $ (12)

      Differentiating the tracking errors yields

      $$ {\dot e_1} = {x_1} + {d_1} - {\dot x_d} .$ $ (13)

      According to the design process of FTDO, the observer can be designed as

      $$ {\dot \omega_1} = {x_1} + {v_1} - {\dot x_d} . $ $ (14)

      Introducing the auxiliary sliding surface $ {s_{o1}} = {e_1} - {\omega_1} $, then the time derivative of the auxiliary valuable can be illustrated as

      $$ {\dot s_{o1}} = d_1-v_1 . $ $ (15)

      Selecting the control term $ {v_1} $ as

      $$ \left\{\begin{aligned} & {v_1} = {k_o}_1\frac{{{s_{o1}}}}{{{{\left\| {{s_{o1}}} \right\|}^{\frac 1 2}}}} + {k_{o2}}{s_{o1}}{\left\| {{s_{o1}}} \right\|^{p - 1}} - {\gamma _1}\\ & {\dot \gamma _1} = - {k_{o3}}\frac{{{s_{o1}}}}{{\left\| {{s_{o1}}} \right\|}} {\text{.}}\end{aligned}\right. $ $ (16)

      Substituting (16) into (15) yields

      $$ \left\{\begin{aligned} & \dot s_{o1} = -{k_o}_1\frac{{{s_{o1}}}}{{{{\left\| {{s_{o1}}} \right\|}^{\frac 1 2}}}} - {k_{o2}}{s_{o1}}{\left\| {{s_{o1}}} \right\|^{p - 1}} + {\gamma _1}\\ & {\dot \gamma _1} = - {k_{o3}}\frac{{{s_{o1}}}}{{\left\| {{s_{o1}}} \right\|}}+\dot d_1 {\text{.}}\end{aligned}\right. $ $ (17)

      According to the conclusion of Lemma 1, the estimated errors $ {e_{o1}} = {d_1} - {v_1} $ are forced to zero in fixed-time $ {t_{o1}} $ by selecting suitable $ {k_{o1}},{k_{o2}},{k_{o3}} $, and

      $$ \begin{aligned} {t_{o1}} =\; &\left( \frac{1}{{{k_o}_2(p - 1){\varepsilon ^{p - 1}}}} + \frac{{2{{(\sqrt n \varepsilon )}^{\frac 1 2}}}}{{{k_{o1}}}} \right) \times \\ & \left( {1 + \frac{{{M_{o1}}}}{{{m_{o1}}(1 - \frac{ \sqrt {2{k_{o3}}}} {k_{o1}})}}} \right) \end{aligned} $ $ (18)

      where $ {M_{o1}} = {k_{o3}} + {L_1},\,{m_{o1}} = {k_{o3}} - {L_1} $.

      To guarantee the tracking performance of the position subsystem, an exponential fixed-time sliding surface is constructed as

      $$ {s_1} = {e_1} + \int_{{t_0}}^t {\frac{n}{{{t_{c1}}}}{{\rm e}^{\frac{{\left\| {{e_1}} \right\|}}{n}}}\frac{{{e_1}}}{{\left\| {{e_1}} \right\|}}} {\rm d}\tau $ $ (19)

      where $ {t_{c1}} \ge 0 $ is a designed parameter. The convergence time of the settling stage is bounded with $ t_{c1} $ and the specific proof will be given in a subsequent description.

      Hence, the virtual control law $ x_{2c} $ can be designed as

      $$ \left\{ \begin{aligned} {x_{2c}} = &\;\;{{\dot x}_d} - {v_1} - \frac{n}{{{t_{c1}}}}{{\rm e}^{\frac{{\left\| {{e_1}} \right\|}}{n}}}\frac{{{e_1}}}{{\left\| {{e_1}} \right\|}} - {k_1}\frac{{{s_1}}}{{\left\| {{s_1}} \right\|}^{\frac 1 2}}-\\ & \;\;{k_2}{s_1}{\left\| {{s_1}} \right\|^{p - 1}} + {\varpi _1}\\ {{\dot \varpi }_1} = & - {k_3}\frac{{{s_1}}}{{\left\| {{s_1}} \right\|}} \end{aligned} \right. $ $ (20)

      where $ n = 5 $ is the freedoms of exoskeleton; $ t_{c1}>0, $ $ p_1>1 $ are designed parameters. $ k_1,k_2,k_3 \in {\bf R}^+ $ can be selected according to Lemma 1.

      Theorem 1. For the position subsystem of exoskeleton indicated by (3), there exit certain values of control parameters $ {k_1},{k_2},{k_3}, $ $ {k_{o1}},{k_{o2}},{k_{o3}},{t_{c1}} $, such that the tracking errors $ e_1 $ of the position subsystem can be fixed-time stable with controller (20) and FTDO (14) and (16).

      Proof. Differentiating the sliding surface (19) yields

      $$ \begin{split} {{\dot s}_1}\;& = {{\dot e}_1} + \frac{n}{{{t_{c1}}}}{{\rm e}^{\frac{{\left\| {{e_1}} \right\|}}{n}}}\frac{{{e_1}}}{{\left\| {{e_1}} \right\|}}=\\ & \quad{x_2} - {x_{2c}} + {x_{2c}} + {d_1} - {{\dot x}_d} + \frac{n}{{{t_{c1}}}}{{\rm e}^{\frac{{{e_1}}}{{\left\| {{e_1}} \right\|}}}}\frac{{{e_1}}}{{\left\| {{e_1}} \right\|}} {\text{.}}\end{split} $ $ (21)

      Substituting (20) into (21), then

      $$ \begin{split} {\dot s_1} =\; & - {k_1}\frac{{{s_1}}}{{{{\left\| {{s_1}} \right\|}^{\frac 1 2}}}} - {k_2}{s_1}{\left\| {{s_1}} \right\|^{p - 1}} + \\ & \int_{{t_0}}^t {{k_3}\frac{{{s_1}}}{{\left\| {{s_1}} \right\|}}} {\rm d}\tau + {x_2}\! - {x_{2c}} + {e_{o1}} {\text{.}}\end{split} $ $ (22)

      Define a new variable $ \rho = e_2 + {e_{o1}} $, where $ e_2 = x_2-x_{2c} $. Then, (22) can be written as

      $$ \left\{ \begin{split} & {{\dot s}_1} = - {k_1}\frac{{{s_1}}}{{{{\left\| {{s_1}} \right\|}^{\frac 1 2}}}} - {k_2}{s_1}{\left\| {{s_1}} \right\|^{p - 1}} + {\varpi _1}\\ & {{\dot \varpi }_1} = - {k_3}\frac{{{s_1}}}{{\left\| {{s_1}} \right\|}} + \dot \rho {\text{.}}\end{split} \right. $ $ (23)

      Supposing that $ \rho $ satisfies $ \left\| {\dot \rho } \right\| \le {K_1} $, then the conclusion can be achieved that $ {s_1} $ can converge to zero in fixed-time $ {t_{r1}} $ in accordance with Lemma 1. Besides, $ {t_{r1}} $ can be described as

      $$ \begin{split} {t_{r1}} =\; & \left( {\frac{1}{{{k_2}(p - 1){\varepsilon ^{p - 1}}}} + \frac{{2{{(\sqrt n \varepsilon )}^{\frac 1 2}}}}{{{k_1}}}} \right) \times \\ & \left( {1 + \frac{{{M_{r1}}}}{{{m_{r1}}(1 - \frac{\sqrt {2{k_3}}} {k_1})}}} \right) \end{split} $ $ (24)

      where $ M_{r1} = k_3+K_1,\, m_{r1} = k_3-K_1 $.

      Subsequently, according to $ \dot {s_1} = 0 $, (19) can be obtained as

      $$ \dot s_1 = {\dot e_1} + \frac{n}{{{t_{c1}}}}{{\rm e}^{\frac{{\left\| {{e_1}} \right\|}}{n}}}\frac{{{e_1}}}{{\left\| {{e_1}} \right\|}} = 0 . $ $ (25)

      Select the Lyapunov function as

      $$ {V_{e1}} = \left\| {{e_1}} \right\| .$ $ (26)

      Differentiating (26) along with time, we can obtain

      $$ \begin{split} {{\dot V}_{e1}}\; = & \frac{{e_1^{\rm T}}}{{\left\| {{e_1}} \right\|}}{{\dot e}_1}=\\ & \frac{{e_1^T}}{{\left\| {{e_1}} \right\|}}( - \frac{n}{{{t_{c1}}}}{{\rm e}^{\frac{{\left\| {{e_1}} \right\|}}{n}}}\frac{{{e_1}}}{{\left\| {{e_1}} \right\|}}) =\\ & - \frac{n}{{{t_{c1}}}}{{\rm e}^{\frac{{\left\| {{e_1}} \right\|}}{n}}} = - \frac{n}{{{t_{c1}}}}{{\rm e}^{\frac{{{V_{e1}}}}{n}}}{\text{.}} \end{split} $ $ (27)

      The solutions of (27) is

      $$ {V_{e1}} = n\ln \left( {\frac{1}{{\frac{t - {t_0}}{t_{c1}} + {{\rm e}^{ - \frac{{{V_{e1}}({t_0})}}{n}}}}}} \right) .$ $ (28)

      Note that $ {V_{e1}} = 0 $, only if $ \frac{t - {t_0}}{t_{c1}} + {{\rm e}^{ - \frac{{{V_{e1}}({t_0})}}{n}}} = 1 $. Therefore, the settling time function is

      $$ t = {t_{c1}}\left(1 - {{\rm e}^{^{ - \frac{{{V_{e1}}({t_0})}}{n}}}}\right) + {t_0} $ $ (29)

      where $ t_0 $ is the starting time of settling mode, which is the time when $ s = 0 $ is satisfied. $ t-t_0 $ is the interval time of settling mode. From the solution of (29), it can be shown that the interval time is bounded with a certain value $ {t_{c1}} $ when the initial condition $ V_{e1}(t_0)\rightarrow\infty $.

      Ultimately, the overall convergence time in position subsystem can be illustrated as $ {t_1} = {t_{o1}} + {t_{r1}} + {t_{c1}} $. □

    • In this subsection, the control toque $ \tau $ is designed based on the tracking errors $ e_2 = {x_2}-x_{2c} $. For rejecting the matched disturbance $ d_2 $, FTDO is adopted to estimate the disturbance and compensate the controller online.

      Tracking errors of velocity subsystem can be indicated as

      $$ {e_2} = {x_2} - {x_{2c}} .$ $ (30)

      Differentiating (30) along with the velocity subsystem of exoskeleton indicated by (3) yields

      $$ {\dot e_2} = - {M^{ - 1}}({x_1})[C(x){x_2} + G({x_1}) - \tau ] + {d_2} - {\dot x_{2c}} .$ $ (31)

      It should be noted that the time derivative of $ x_{2c} $ is difficult to achieve or resolve both in theory and practice, which may result in the "complexity explosion", the main disadvantage of backstepping method. In order to overcome the problem, the FTDO is developed to estimate the lumped item composed of $ {d_2} $ and $ {\dot x_{2c}} $.

      In accordance to the design process of the position subsystem, the FTDO of velocity subsystem is demonstrated as

      $$ {\dot w_2} = - {M^{ - 1}}({x_1})[C(x){x_2} + G({x_1}) - \tau ] + {v_2} .$ $ (32)

      Introduce a sliding surface as

      $$ {s_{o2}} = {e_2} - {\omega_2} . $ $ (33)

      Then, the time derivative of the auxiliary variable can be described as

      $$ \dot s_{o2} = {d_2} - {\dot x_{2c}}-v_2 . $ $ (34)

      Note that $ {v_2} $ is the estimation value of $ {d_2} - {\dot x_{2c}} $ which can be expressed as

      $$ \left\{\begin{aligned} & {v_2} = {k_o}_4\frac{{{s_{o2}}}}{{{{\left\| {{s_{o2}}} \right\|}^{\frac 1 2}}}} + {k_{o5}}{s_{o2}}{\left\| {{s_{o2}}} \right\|^{p - 1}} - {\gamma _2}\\ & {\dot \gamma _2} = - {k_{o6}}\frac{{{s_{o2}}}}{{\left\| {{s_{o2}}} \right\|}} {\text{.}}\end{aligned}\right. $ $ (35)

      Then, there exist certain values of control gains $ {k_{o4}},{k_{o5}},{k_{o6}} $ to ensure that $ {e_{o2}} = {d_2} - {\dot x_{2c}} - {v_2} $ converge to zero within fixed-time $ {t_{o2}} $, and

      $$ \begin{split} {t_{o2}} =\; & \left( {\frac{1}{{{k_o}_5(p - 1){\varepsilon ^{p - 1}}}} + \frac{{2{{(\sqrt n \varepsilon )}^{\frac 1 2}}}}{{{k_{o4}}}}} \right) \times \\ &\left( {1 + \frac{{{M_{o2}}}}{{{m_{o2}}(1 - \frac{\sqrt {2{k_{o6}}}} {k_{o4}})}}} \right) {\text{.}}\end{split} $ $ (36)

      To guarantee the fast convergence of the tracking error of velocity subsystem, a sliding surface for velocity subsystem is selected as

      $$ {s_2} = {e_2} + \int_{{t_0}}^t {\frac{n}{{{t_{c2}}}}{{\rm e}^{\frac{{\left\| {{e_2}} \right\|}}{n}}}\frac{{{e_2}}}{{\left\| {{e_2}} \right\|}}{\rm d}\tau } . $ $ (37)

      And the time derivative of the sliding surface can be described as

      $$ \begin{split} {{\dot s}_2} = \;& {{\dot e}_2} + \frac{n}{{{t_{c2}}}}{{\rm e}^{\frac{{\left\| {{e_2}} \right\|}}{n}}}\frac{{{e_2}}}{{\left\| {{e_2}} \right\|}}= \\ & - {M^{ - 1}}({x_1})[C(x){x_2} + G({x_1}) - \tau ]+\\ & {d_2} - {{\dot x}_{2c}} + \frac{n}{{{t_{c2}}}}{{\rm e}^{\frac{{\left\| {{e_2}} \right\|}}{n}}}\frac{{{e_2}}}{{\left\| {{e_2}} \right\|}}{\text{.}} \end{split} $ $ (38)

      Afterward, control torques $ \tau $ can be designed as

      $$ \left\{ \begin{aligned} \tau = & M({x_1})[ - {k_4}\frac{{{s_2}}}{{{{\left\| {{s_2}} \right\|}^{\frac 1 2}}}} - {k_5}{s_2}{\left\| {{s_2}} \right\|^{p - 1}} + \varpi - \\ & \frac{n}{{{t_{c2}}}}{{\rm e}^{\frac{{\left\| {{e_2}} \right\|}}{n}}}\frac{{{e_2}}}{{\left\| {{e_2}} \right\|}}- {v_2}] + C(x){x_2} + G({x_1})\\ \dot \varpi = & - {k_6}\frac{{{s_2}}}{{\left\| {{s_2}} \right\|}} \end{aligned} \right. $ $ (39)

      where $ n = 5 $ is the freedoms of exoskeleton, $ t_{c2}>0, \,p>1 $ are the designed parameters. $ k_4,k_5,k_6 \in {\bf R}^+ $ can be selected according to Lemma 1.

      Theorem 2. For the velocity subsystem of exoskeleton indicated by (3), there exit certain values of control parameters $ {k_4},{k_5},{k_6}, $ $ {k_{o4}},{k_{o5}},{k_{o6}},{t_{c2}} $, such that the tracking errors $ e_2 $ of velocity subsystem is fixed-time stable with control toque (39) and FTDO (32) and (35).

      Proof. Substituting control torque $ \tau $ into the time derivation of sliding surface (37), we can obtain

      $$ \left\{ \begin{aligned} & {{\dot s}_2} = - {k_4}\frac{{{s_2}}}{{{{\left\| {{s_2}} \right\|}^{\frac 1 2}}}} - {k_5}{s_2}{\left\| {{s_2}} \right\|^{p - 1}} + \varpi - {v_2} + {d_2} - {{\dot x}_{2c}}\\ & \dot \varpi = - {k_6}\frac{{{s_2}}}{{\left\| {{s_2}} \right\|}}{\text{.}} \end{aligned} \right. $ $ (40)

      It can be achieved from the design process of FTDO that the observer error $ {e_{o2}} $ can converge to the origin in fixed-time $ {t_{o2}} $, which means $ d_2-\dot x_{2c} = v_2 $. Therefore, (40) can be rewritten as

      $$ \left\{ \begin{aligned} & {{\dot s}_2} = - {k_4}\frac{{{s_2}}}{{{{\left\| {{s_2}} \right\|}^{\frac 1 2}}}} - {k_5}{s_2}{\left\| {{s_2}} \right\|^{p - 1}} + \varpi\\ & \dot \varpi = - {k_6}\frac{{{s_2}}}{{\left\| {{s_2}} \right\|}}{\text{.}} \end{aligned} \right. $ $ (41)

      By utilizing Lemma 1, it can be concluded that the tracking error $ e_2 $ of velocity subsystem and its time derivation $ \dot e_2 $ can reach zero within a certain time $ t_{r2} $ and remain there in all subsequent time, where

      $$ \begin{split} {t_{r2}} = \; &\left( {\frac{1}{{{k_5}(p - 1){\varepsilon ^{p - 1}}}} + \frac{{2{{(\sqrt n \varepsilon )}^{\frac1 2}}}}{{{k_4}}}} \right)\times\\ & \left( {1 + \frac{{{M_{r2}}}}{{{m_{r2}}(1 - \frac{\sqrt {2{k_6}}} {k_4})}}} \right) {\text{.}}\end{split} $ $ (42)

      Subsequently, the sliding surface (37) can be rewritten as

      $$ {\dot s_2} = {\dot e_2} + \frac{n}{{{t_{c2}}}}{{\rm e}^{\frac{{\left\| {{e_2}} \right\|}}{n}}}\frac{{{e_2}}}{{\left\| {{e_2}} \right\|}} = 0 . $ $ (43)

      Select the positive defined Lyapunov function as

      $$ V_{e2} = \left \| e_2 \right \| . $ $ (44)

      Differentiating (44), we can obtain

      $$ \begin{split} {{\dot V}_{e2}} \; & = \frac{{e_2^{\rm T}}}{{\left\| {{e_2}} \right\|}}{{\dot e}_2} =\\ &\quad \frac{{e_2^{\rm T}}}{{\left\| {{e_2}} \right\|}}( - \frac{n}{{{t_{c1}}}}{{\rm e}^{\frac{{\left\| {{e_2}} \right\|}}{n}}}\frac{{{e_2}}}{{\left\| {{e_2}} \right\|}}) =\\ &\quad - \frac{n}{{{t_{c2}}}}{{\rm e}^{\frac{{\left\| {{e_2}} \right\|}}{n}}} = - \frac{n}{{{t_{c2}}}}{{\rm e}^{\frac{{{V_{e2}}}}{n}}}{\text{.}} \end{split} $ $ (45)

      The solution of the differential (45) can be described as

      $$ {V_{e2}} = n\ln \left( {\frac{1}{{\frac{t - {t_0}}{t_{c2}} + {{\rm e}^{ - \frac{{{V_{e2}}({t_0})}}{n}}}}}} \right) .$ $ (46)

      Note that $ V_{e2} = 0 $ can be obtained only if $ \dfrac{t - {t_0}}{t_{c2}} + {\rm e}^{ - \frac{{{V_{e2}}({t_0})}}{n}} = 1 $ was satisfied. Then, we can obtain that

      $$ t = t_0+t_{c2}\left(1-{\rm e}^{- \frac{{{V_{e2}}({t_0})}}{n}}\right) .$ $ (47)

      Similar to the conclusion in the position subsystem, the interval time of the settling mode is bounded with $ t_{c2} $.

      Finally, the convergence time of the velocity subsystem can be illustrated as $ {t_2} = {t_{o2}} + {t_{r2}} + {t_{c2}} $. □

    • By utilizing the proposed fixed-time control strategy, simulations are implemented to validate the effectiveness of the integrated scheme for the upper-limb exoskeleton. Two parts of simulations are established including: comparison of different existing sliding mode surfaces and simulations of a human-robot system.

    • For the purpose of demonstrating the advantages of proposed exponential fixed-time sliding surfaces, following common used sliding surfaces are introduced.

      $$ \begin{split} & {s_1} = e + \int_0^t {ce} {\rm d}\tau \\ & {s_2} = e + {k_1}\int_0^t {(si{g^{{r_1}}}(e) + si{g^{{r_2}}}(e))} {\rm d}\tau \\ & {s_3} = e + \int_0^t {\alpha {{\rm e}^{{\frac 1 2} + {\frac m {2n}} + ({\frac m {2n}} - { \frac 1 2}){\rm sgn}(\left| e \right| - 1)}} - \beta {{\rm e}^{\frac p q}}{\rm d}\tau } \\ & {s_4} = e + \int_0^t {\frac{{{n_s}}}{{{t_c}}}{{\rm e}^{\frac{{\left| e \right|}}{{{n_s}}}}}{\rm sgn}(e){\rm d}\tau } \end{split} $ $ (48)

      where $ si{g^{r}}(e) = {\left| e \right|^{r}}{\rm sgn}(e) $, $ 0 < {r_1} < 1,\, {r_2}>1 $.

      The simulation parameters of different sliding surfaces are chosen as: c = 10; k1 = 10; ${r_1} = 0.5$; ${r_2} $ = 1.5; m = 9; n = 5; $\alpha = 10 $; $\beta = 1 $; p = 5; q = 9; tc = 0.2; ns = 1. In order to illustrate the merits of proposed surfaces, initial values of errors are selected as $ e = 10 $. Subsequently, the results are shown in Fig. 3.

      Figure 3.  Simulations of different sliding surfaces

      The proposed sliding surface is indicated by the solid line in Fig. 3 where the conclusion can be obtained that the proposed exponential sliding surface can ensure the fixed-time stable and provide a faster convergence speed when the system state is far away from equilibrium point. Furthermore, the convergence time of proposed sliding surface is bounded with 0.2 s, which is the value of $ t_c $. Hence, an obviously significant feature of the proposed method is that the convergence time is bounded with a certain value which can be tuned easily by selecting different $ t_c $

    • To validate the effectiveness of the proposed fixed-time control method, a simulation of a human-robot system is established in Matlab/Simulink. The simulation conditions are selected including the initial position value of exoskeleton $ {x_1}(0) = [ - 1,1, - 2,2, - 3] $; the expected trajectory $ {x_d} = [{x_{d1}}, {x_{d2}}, {x_{d3,}} {x_{d4}}, {x_{d5}}] $ where $ {x_{di}} = 5\sin (t), $ i = 1, 2, 3, 4, 5; the initial estimated values of FTDOs $ {w_1}(0) = {w_2}(0) = [0.1,0.1,0.1,0.1,0.1] $ and the unmatched and matched disturbances $ {d_{1i}} = {d_{2i}} = (1 + \sin (1.2t)), $ i = 1, 2, 3, 4, 5. To ensure the stability of the closed-loop system, gains in presence of the proposed control scheme are shown in Table 1.

      Table 1.  Controller gains of the closed-loop system

      Symbol Value Symbol Value
      $k_{o1}$ 5 $k_1$ 20
      $k_{o2}$ 4.5 $k_2$ 5
      $k_{o3}$ 5 $k_3$ 10
      $k_{o4}$ 15 $k_4$ 30
      $k_{o5}$ 10 $k_5$ 10
      $k_{o6}$ 20 $k_6$ 20
      $p$ 1.5 $t_{c1}$ 0.5
      $n$ 5 $t_{c2}$ 0.5

      Simulation results of the proposed control strategy are given as Figs. 4-8. The tracking performance of each joint is shown in Fig. 4, where we can obtain that all the desired trajectories can be tracked after a short dynamic adjustment. It can be concluded from Fig. 5 that all the control torques $ \tau $ are bounded and smooth, which means the proposed controller can attenuate the chattering phenomenon in SMC.

      Figure 4.  Position tracking performance of 5 joints

      Figure 5.  Control torques of the 5 DoF exoskeleton

      Figure 8.  Observer errors of velocity subsystem

      To demonstrate the dynamic performance of the fixed-time control scheme with FTDOs, the tracking and observer errors in 0.5 s are shown as Figs. 6-8. In Fig. 6, the position errors' curve are given to illustrate the tracking performance of the fixed-time strategy where the tracking errors are converged to a considerably small neighbourhood after a short time (less than 0.5 s) adjustment. To illustrate the behaviour of FTDOs, the curves of $ e_{o1} $ and $ e_{o2} $ are shown in Figs. 7 and 8. It can be concluded that the observer errors of position and velocity subsystems can be forced to a small range within 0.2 s and 0.5 s respectively.

      Figure 6.  Position tracking errors of the exoskeleton system

      Figure 7.  Observer errors of position subsystem

      A comparison simulation without FTDOs is estab-lished to interpret the superiority of the integrated fixed-time control strategy. Simulation results of the position tracking curves are given in Fig. 9. The desired trajectories $ x_d $ are described by the solid line while the position tracking curves $ x_1 $ under the conditions of with and without FTDOs are indicated by the dash line and dash dot line respectively. The convergence time of the integrated control scheme is less than 0.5 s while it is about 1.8 s in the control scheme without FTDOs, and a faster dynamic adjustment can be acquired by integrating the fixed-time controller and FTDOs.

      Figure 9.  Comparison of the position tracking performance with and without FTDOs

    • In this paper, a comprehensive backstepping fixed-time control strategy has been investigated for a 5 DoF upper-limb exoskeleton under unmatched and matched disturbances. For the purpose of handling with the disturbances, the dynamics of exoskeleton are divided into position and velocity subsystems while multivariable FTDOs are developed to estimate the exact values of disturbances in fixed-time and compensate the controller online. Furthermore, SOSMCs for position and velocity subsystems are designed with novel exponentially sliding surfaces and the fixed-time stable of the closed-loop system are analysed through Lyapunov theory. For the purpose of avoiding the "complexity explosion" existing in traditional backstepping schemes, the matched disturbances and time derivative of virtual control between the two subsystems are combined as lumped uncertainties handled by FTDO. The simulation results verify the advantages of the novel sliding surface and validity of the proposed control scheme.

    • The specific formulas and values are cited from [7], the inertial matrix $ M(\theta) $ is

      $$ M(q) = \left[ {\begin{array}{*{20}{c}} {{M_{11}}}&{{M_{12}}}&{{M_{13}}}&0&0\\ {{M_{21}}}&{{M_{22}}}&{{M_{23}}}&0&0\\ {{M_{31}}}&{{M_{32}}}&{{M_{33}}}&0&{{M_{35}}}\\ 0&0&0&{{M_{44}}}&0\\ 0&0&{{M_{53}}}&0&{{M_{55}}} \end{array}} \right] $ $

      where

      $$ \begin{split} {M_{11}} =\; & {I_2} + {I_{19}} + {I_3}{\cos ^2}({\theta_2}) + {I_4}\sin ({\theta_2} + {\theta_3}) + \\ & {I_5}\sin ({\theta_2} + {\theta_3})\cos ({\theta_2} + {\theta_3}) + {I_6}\sin ({\theta_2})\cos ({\theta_2}) + \\ & {I_7}{\sin ^2}({\theta_2} + {\theta_3}) + 2 + {I_8}\cos ({\theta_2})\sin ({\theta_2} + {\theta_3}) + \\ & {I_9}\cos ({\theta_2})\cos ({\theta_2} + {\theta_3}) + {I_{10}}{\sin ^2}({\theta_2} + {\theta_3}) + \\ & {I_{11}}\cos ({\theta_2})\sin ({\theta_2} + {\theta_3}) + \\ & {I_{12}}\sin ({\theta_2} + {\theta_3})\cos ({\theta_2} + {\theta_3})\\ {M_{12}} =\; & {M_{21}} = {I_{13}}\sin ({\theta_2}) + {I_{14}}\cos ({\theta_2} + {\theta_3}) + {I_{15}}\cos ({\theta_2}) + \\ & {I_{16}}\sin ({\theta_2} + {\theta_3}) - {I_{17}}\cos ({\theta_2} + {\theta_3})\\ {M_{13}} =\; & {M_{31}} = {I_{14}}\cos ({\theta_2} + {\theta_3}) +{I_{16}}\sin ({\theta_2} + {\theta_3}) - \\ & {I_{17}}\cos ({\theta_2} + {\theta_3})\\ {M_{22}} =\; & {I_{18}} + {I_{19}} + {I_{20}} + 2{I_8}\sin ({\theta_3}) + {I_9}\cos ({\theta_2}) + \\ & {I_{10}} + {I_{11}}\sin ({\theta_3}) \\ {M_{23}} =\; & {M_{32}}\! =\! {I_{20}} \!+\! {I_8}\sin ({\theta_3}) \!+\! {I_9}\cos ({\theta_3})\! +\!2{I_{10}} \!+\! {I_{11}}\sin ({\theta_3})\\ {M_{33}} = \; & {I_3} + 2{I_{10}} + {I_{20}}, {M_{35}} = M_{53} = I_{10}+I_{21} \\ {M_{44}} = \; & I_{22}+I_{23}, {M_{55}} = I_{24}+I_{21} {\text{.}}\end{split}$ $

      The carioles/centripetal matrix $ C(\theta, \dot \theta) $ can be described as

      $$ C(\theta,\dot \theta) = \left[ {\begin{array}{*{20}{c}} {{c_1}{{\dot \theta}_2}}&{{c_2}{{\dot \theta}_3} + {c_3}{{\dot \theta}_2}}&{{c_4}{{\dot \theta}_2} + {c_5}{{\dot \theta}_3}}&0&0\\ {{c_6}{{\dot \theta}_1}}&{{c_7}{{\dot \theta}_3}}&{{c_8}{{\dot \theta}_3}}&0&0\\ {{c_5}{{\dot \theta}_1}}&{{c_9}{{\dot \theta}_2}}&0&0&0\\ {{c_{10}}{{\dot \theta}_2}}&0&{{c_{11}}{{\dot \theta}_1}}&0&0\\ 0&{{c_{12}}{{\dot \theta}_2}}&0&0&0 \end{array}} \right] $ $

      where

      $$ \begin{split} {c_1} =& 2[-{I_3}\sin ({\theta_2})\cos ({\theta_2}) + {I_8}\cos (2{\theta_2} + {\theta_3}) + \\ &{I_4}\sin ({\theta_2} + {\theta_3})\cos ({\theta_2}) - {I_9}\sin (2{\theta_2} + {\theta_3}) - \\ &2{I_{10}}\sin ({\theta_2} + {\theta_3}){\kern 1pt} + {I_{11}}\cos (2{\theta_2} + {\theta_3}) +\\ & {I_7}\sin ({\theta_2} + {\theta_3})\cos ({\theta_2} + {\theta_3}) + \\ & {I_{12}}(1 - 2{\sin ^2}({\theta_2} + {\theta_3}))] + \\ & {I_5}(1 - 2\sin ({\theta_2} + {\theta_3}) + {I_6}(1 - 2{\sin ^2}({\theta_2})))\\ {c_2} = & 2[ - {I_{14}}\sin ({\theta_2} + {\theta_3}) + {I_{16}}\cos ({\theta_2} + {\theta_3}) + \\ &{I_{17}}\sin ({\theta_2} + {\theta_3})]\\ {c_3} = & {I_{13}}\cos ({\theta_2}) - {I_{14}}\sin ({\theta_2} + {\theta_3}) - {I_{15}}\sin ({\theta_2}) + \\ &{I_{16}}\cos ({\theta_2} + {\theta_3}) + {I_{17}}\sin ({\theta_2} + {\theta_3})\\ {c_4} = & 2[{I_8}\cos ({\theta_2})\cos ({\theta_2} \!+ \!{\theta_3}) \!+ \!{I_4}\sin ({\theta_2}\! +\! {\theta_3})\cos ({\theta_2} \!+ \!{\theta_3}) - \\ &{I_9}\cos ({\theta_2})\sin (2{\theta_2}) + 2{I_{10}}\sin ({\theta_2} + {\theta_3})\cos ({\theta_2} + {\theta_3}) + \\ &{I_{11}}\cos ({\theta_2})\cos ({\theta_2} \!+\! {\theta_3})+ {I_7}\sin ({\theta_2} \!+\! {\theta_3})\cos ({\theta_2} \!+ \!{\theta_3}) + \\ &{I_{12}}(1 - 2{\sin ^2}({\theta_2} + {\theta_3}))]+ {I_5}(1 - 2{\sin ^2}({\theta_2} + {\theta_3}))\\ c_{5} = & 0.5c_2,\,c_6 = -0.5c_1\\ {c_7} = & 2[ - 2{I_9}\sin ({\theta_3}){\kern 1pt} + {I_8}\cos ({\theta_3}) + {I_{11}}\cos ({\theta_3})]\\ c_8 = & -0.5c_4\\ {c_9} = & \sin ({\theta_2} + {\theta_3})\cos ({\theta_2} + {\theta_3}) - 2{I_{10}}\sin ({\theta_2} + {\theta_3})\cos ({\theta_2} + \\ & {\theta_3}) - {I_{11}}\cos ({\theta_2})\cos ({\theta_2} + {\theta_3}) - {I_{12}}{\cos ^2}({\theta_2} + {\theta_3})\\ c_{10} =& -I_{23}\sin(\theta_2 \!+\!\theta_3)-I_{19}\sin(\theta_2 \!+\!\theta_3)-I_{20}\sin(\theta_2+\theta_3)\\ c_{11} = & -I_{20}\sin(\theta_2+\theta_3)\! +\! I_{23}\sin(\theta_2+\theta_3)\!+ \!I_{19}\sin(\theta_2+\theta_3)\\ c_{12} = &-I_{11}\cos(\theta_3)-I_{12}{\text{.}} \end{split} $ $

      The gravity vector $ G(\theta) $ is

      $$ G(\theta) = {[\begin{array}{*{20}{c}} 0&{{G_2}}&{{G_3}}&0&{{G_5}} \end{array}]^{\rm T}} $ $

      where

      $$ \begin{split} {G_2} = &\; {g_1}\cos ({\theta_2}) + {g_2}\sin ({\theta_2} + {\theta_3}) + {g_3}\sin ({\theta_2}) + \\ &\;{g_4}\cos ({\theta_2} + {\theta_3}) + {g_5}\sin ({\theta_2} + {\theta_3})\\ {G_3} = &\; {g_2}\sin ({\theta_2} + {\theta_3}) + {g_4}\cos ({\theta_2} + {\theta_3}) + {g_5}\sin ({\theta_2} + {\theta_3})\\ G_5 = &\; g_5\sin(\theta_2+\theta_3) {\text{.}}\end{split} $ $

      Values of the inertial and gravity parameters used are shown in Tables 2 and 3.

      Table 2.  Inertial constant

      $I_1 = 1.14$ $I_2 = 1.43$ $I_3 = 1.38$
      $I_4 = 0.298$ $I_5 = -0.021\,3$ $I_6 = -0.014\,2$
      $I_7 = -0.000\,1$ $I_8 = 0.372$ $I_9 = -0.011$
      $I_{10} = 0.001\,25$ $I_{11} = -0.012\,4$ $I_{12} = 0.000\,058$
      $I_{13} = -0.69$ $I_{14} = 0.134$ $I_{15} = 0.238$
      $I_{16} = 0.003\,79$ $I_{17} = 0.000\,642$ $I_{18} = 4.71$
      $I_{19} = 1.75$ $I_{20} = 0.333$ $I_{21} = 0.000\,642$
      $I_{22} = 0.2$ $I_{23} = 0.001\,64$ $I_{24} = 0.179$

      Table 3.  Gravitational constant

      $g_1 = -37.2$ $g_2 = -8.43$ $g_3 = 1.02$
      $g_4 = 0.249$ $g_5 = -0.002\,92$
    • This work was supported by National Natural Science Foundation of China (Nos. 61703134, 61703135, 61773151, 61503118 and 61871173), Natural Science Foundation of Hebei Province (Nos. F2015202150, F2016202327 and F2018202279), Natural Science Foundation of Tianjin (No. 17JCQNJC04400), the Foundation of Hebei Educational Committee (Nos. QN2015068 and ZD2016071), the Colleges and Universities in Hebei Province Science and Technology Research Youth Fund (No. ZC2016020) and the Graduate Innovation Funding Project of Hebei Province (No. CXZZBS2017038).

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