Terminal super-twisting sliding mode based on high order sliding mode observer for two DOFs lower limb system

Abstract

In this paper, a novel backstepping terminal super-twisting sliding mode (TSTSM) with high order sliding mode observer (HOSMO) is proposed to control the two degrees of freedom (DOFs) Serial Elastic Actuator (SEA), inspired by a lower limb of humanoid robots. First, the dynamic model, extended from our previous study, is presented for developing the control algorithm. Secondly, the backstepping technique is utilized to separate the overall system into two subsystems. One of the challenges of SEA is to deal with the evident oscillations caused by the elastic element, which might lead to the degrading performance of load position control. In order to reduce this adverse effect, a TSTSM is proposed to control the position tracking of two subsystems. The advantages of TSTSM are the finite-time convergence despite the bounded perturbation and the dramatic reduction of the chattering phenomenon. To construct and implement the TSTSM controller, it requires the knowledge of all states, which is not available in the current lower limb system setup. Therefore, a HOSMO is utilized to estimate the unknown states. Finally, experiment results are carried out to assess the effectiveness of the proposed controller and compare it with those of different control schemes.

Keywords

Super twisting algorithm sliding mode SEA humanoid robot super-twisting observer terminal sliding mode backstepping algorithm

Introduction

Ever since the past, the art of replicating artificial human body parts has been developed for various purposes: physiology research, missing body parts for human replacement, humanoid robot manufacture, and many more. As the technology and health-care system advance, the demand for more biologically accurate artificial limbs for prosthetic and humanoid robots is growing extremely fast. One of the most critical aspects of making artificial limbs is to mimic the soft, flexible characteristics of a human’s joint. Even though numerous choices of actuators are available such as electric motors, hydraulic actuators, and pneumatic actuators, the stiffness property of the actuators prevents these actuators from imitating the human joint in reducing the contact force. One of the solutions to address this drawback is to incorporate an elastic element into the mechanism, see Refs.^1–5 and the references therein. The combination of an elastic element with a stiff actuator is also known as the compliant actuator design. An alternative approach is to utilize an impedance control scheme to reduce the impact of the external forces, as proposed in some research about humanoid robots and manipulators.^6–9 Among those solutions, the Serial Elastic Actuator (SEA) design that was initially proposed by Pratt and Williamson¹⁰ has gained much attention in the robotics field over the year.^11–16 This innovative design enables the actuator to be more flexible in humanoid robot applications since it replicates the elastic tendon in the human body. By utilizing the SEA design, many humanoid robots and bio-inspired robots have successfully demonstrated their smooth motion capabilities.^17–23 Nevertheless, the development of SEA still remains a considerable challenge.

Traditionally, a simple control scheme such as PID is implemented to control the SEA system.^24–26 However, the stability of the PID control scheme might be degraded due to the large amplitude of friction and backlash. This problem can be alleviated by utilizing the feedback of positions and velocities.^27–31 To further enhance the robustness, disturbance observer (DOB) is employed in many research.^32–34 Since the SEA is a nonlinear system, the sliding mode control (SMC) is a suitable control scheme because of its robustness and simple design.^35,36 Despite the asymptotic stability in the traditional SMC, it is well-known that as the system dynamic gets closer to equilibrium, the rate of convergence becomes slower, which means the states of the system may never reach equilibrium in finite-time. To solve the aforementioned problem, terminal sliding mode control (TSMC) was first proposed in 1992.³⁷ The controller was based on the concept of a terminal attractor introduced in the study of neural networks.³⁸ From then on, it has been utilized significantly in various literature^39–42 and even in SEA applications.⁴³

In our previous study,⁴⁴ an event-trigger sliding mode control was proposed for two DOFs humanoid robot’s lower limbs consisting of two SEA links connected in series. The experimental results have demonstrated that the elastic element induces evident oscillation. The adverse oscillation of the primary SEA joint affects the position adjustment of the secondary SEA joint and vice versa. Consequently, the lumped oscillation of the two SEA links eventually leads to the control system’s instability. Moreover, the utilization of the sign function for the SMC gave rise to the chattering phenomenon. As a result, this phenomenon triggers the lumped oscillation caused by the two SEA links, which can cause severe instability. Therefore, a different control scheme based on the sliding mode is required to attenuate the chattering phenomenon and provide finite-time convergence. So far, many methodologies have been proposed in the literature to reduce the chattering phenomenon. One of the traditional solutions is to replace the discontinuous sign function with the saturation and sigmoid function,^45–47 which is known as the boundary layer method. However, the control system may lose robustness to the disturbances as the sliding trajectories are constrained to the vicinity.⁴⁸ Another drawback is that it also increases the tracking errors in the steady-state.⁴⁹ Another alternative method, such as the disturbance estimation method, can be implemented to reduce the chattering phenomenon. For instance, Shtessel et al.⁵⁰ developed a control scheme with an asymptotic disturbance observer to achieve continuous control input. In Kuang et al.,⁵¹ Du et al.⁵² an equivalent control based on non-smooth control without using any switching term is chosen to control the discrete-time system. This control scheme demonstrated that it is able to alleviate the chattering problem. Recently, one of the most important approaches in SMC is to utilize the super-twisting algorithm (STA).^53–59 The ability to generate a continuous control signal is one of the prominent features of STA, thus reducing the overall chattering phenomenon. The STA is also a powerful tool in providing finite-time convergence to the origin of the sliding variables $σ$ and $\overset{\cdot}{σ}$ . Moreover, the terminal sliding mode control is well-known for its finite-time convergence of the state variables, which has been discussed in various research. Therefore, by combining the TSMC and STA, one can get the finite-time convergence of the state variables while alleviating the chattering phenomenon.

Since the implementation of the STA requires the knowledge of all states, whereas our current humanoid robot’s lower-limb system only provides information regarding the position, it is crucial to obtain the other states through an observer. Real-time differentiators (observers) have played a vital role in many output feedback control designs,^60,61 such as the backward Euler method and spline interpolation technique.^62,63 Among those, a high-gain observer is probably the most well-known approach to obtain the unknown states because of its simplicity and capability.^64,65 Although the advantages of the high-gain observer are obvious, it is also sensitive to small high-frequency noises and has a drawback called the peaking effect: as the gains rise to infinity, the maximal output value during the transient also grows infinitely.⁶⁶ To deal with the bounded uncertainties, a type of observer which is based on sliding mode (SMO) was introduced in Edwards and Spurgeon⁶⁷ and is widely used in further research because of its finite-time convergence.^68–70 In recent years, a new generation of observers based on STA has been developed,^66,71,72 which is actually a second-order case of the high-order observer. This type of observer inherits the properties of the STA, providing robustness and convergence in finite time in the presence of disturbance.^73–75

The main objective of this paper is to propose the backstepping TSTSM to control the two DoFs SEA inspired by a lower limb of humanoid robots. The proposed controller can deal with the perturbation caused by the elastic elements while guaranteeing that the states will converge in finite time and offering chattering attenuation. A high-order sliding mode observer (HOSMO) is also utilized to provide the required full-states feedback.

This paper is organized as follows. Section 2 gives the dynamic model of the 2 DOFs SEA-inspired humanoid robot’s leg. Section 3 shows the design process of the backstepping TSTSM based on HOSMO for the two SEA links. Section 4 details the experimental setup and demonstrates the obtained experimental results. Finally, further analysis and discussion are given to clarify the experimental results.

System modeling

The SEA inspired by the lower limb system is taken from our previous study.⁴⁴ The dynamical modeling for the two SEA links in Figure 1 is given as follows:

{\begin{matrix} \overset{\cdot\cdot}{θ} = f_{x} + g_{x} (Δ + d) \\ \overset{\cdot\cdot}{Δ} = F_{R} + δ \overset{\cdot}{r} - U_{v} \end{matrix}

(1)

Figure 1.

Kinematic diagram of humanoid robot’s lower limb.

where $δ$ is a positive constant, the joint angle vector $θ = {[\begin{matrix} θ_{1} & θ_{2} \end{matrix}]}^{T}$ , the function $f_{x} = - M^{- 1} [\begin{matrix} N_{1} \\ N_{2} \end{matrix}],$ the inertia matrix $M = [\begin{matrix} M_{11} & - M_{12} \\ - M_{21} & M_{22} \end{matrix}],$ the function $g_{x} = M^{- 1} [\begin{matrix} - k_{1} d_{4} \sin γ_{1} & 0 \\ 0 & - k_{2} d_{9} \sin γ_{2} \end{matrix}],$ the spring’s deformation $Δ = {[\begin{matrix} Δ_{1} & Δ_{2} \end{matrix}]}^{T},$ the disturbance $d = [\begin{matrix} - \frac{d_{1}}{k_{1} d_{4} \sin γ_{1}} \\ - \frac{d_{2}}{k_{2} d_{9} \sin γ_{2}} \end{matrix}],$ acceleration of the end effector point of the SEA $F_{R} = [\begin{matrix} F_{R 1} \\ F_{R 2} \end{matrix}],$ the nut’s displacement $r = [\begin{matrix} r_{1} \\ r_{2} \end{matrix}],$ and the input voltage $U_{v} = [\begin{matrix} U_{v 1} \\ U_{v 2} \end{matrix}]$ .

In which $F_{R 1}, F_{R 2}, N_{1}, N_{2},$ and the inertia matrix $M$ are calculated as follows:

\begin{matrix} F_{R 1} = {\overset{\cdot\cdot}{θ}}_{1} d_{4} \sin γ_{1} - {\overset{\cdot}{θ}}_{1}^{2} d_{4} \cos γ_{1} \\ F_{R 2} = {\overset{\cdot\cdot}{θ}}_{1} [L_{1} \sin (α_{2} - θ_{2} - γ_{2}) - d_{9} \sin γ_{2}] + {\overset{\cdot\cdot}{θ}}_{2} d_{9} \sin γ_{2} \\ - {\overset{\cdot}{θ}}_{1}^{2} L_{1} \cos (α_{2} - θ_{2} - γ_{2}) - {({\overset{\cdot}{θ}}_{1} - {\overset{\cdot}{θ}}_{2})}^{2} d_{9} \cos γ_{2} \end{matrix}

(2)

\begin{matrix} M_{11} = m_{1} R_{1}^{2} + I_{1} + m_{2} L_{1}^{2} + m_{2} R_{2}^{2} + 2 m_{2} L_{1} R_{2} \cos θ_{2} \\ M_{12} = m_{2} R_{2}^{2} + m_{2} L_{1} R_{2} \cos θ_{2}, \\ M_{21} = m_{2} R_{2}^{2} + m_{2} L_{1} R_{2} \cos θ_{2} \\ M_{22} = m_{2} R_{2}^{2} + I_{2} \\ N_{1} = m_{1} g R_{1} \sin θ_{1} + m_{2} g L_{1} \sin θ_{1} - m_{2} g R_{2} \sin (θ_{2} - θ_{1}) \\ + m_{2} L_{1} R_{2} \sin θ_{2} ({\overset{\cdot}{θ}}_{2}^{2} - 2 {\overset{\cdot}{θ}}_{1} {\overset{\cdot}{θ}}_{2}) + B_{1} {\overset{\cdot}{θ}}_{1} \\ N_{2} = m_{2} g R_{2} \sin (θ_{2} - θ_{1}) + m_{2} L_{1} R_{2} {\overset{\cdot}{θ}}_{1}^{2} \sin θ_{2} + B_{2} {\overset{\cdot}{θ}}_{2} \end{matrix}

(3)

Varied angles $γ_{1}$ and $γ_{2}$ are calculated as follows:

\begin{matrix} γ_{1} = \sin^{- 1} (\frac{d_{3} \sin (θ_{1} - α_{1} + \frac{π}{2} - σ_{1})}{L_{1}}), \\ γ_{2} = \sin^{- 1} (\frac{d_{8} \sin (π + θ_{2} - α_{2} - σ_{2})}{L_{2}}) \end{matrix}

The explanation of all the above symbols is shown in Table 1.

Table 1.

List of symbols.

Symbol	Description	Units
$θ_{1}, θ_{2}$	Absolute rotation angle of the L-shape link	$rad$
$α_{1}, α_{2}, σ_{1}, σ_{2}$	Fixed angles in lower limb model	$rad$
$L_{1}, L_{2}$	SEA link length	$m$
$d_{1}, d_{2}$	System’s perturbation	$N . m$
$d_{i} (i = 3, \dots, 9)$	Geometry dimension of the SEA	$m$
$R_{1}, R_{2}$	Distances from the knee and hip joint axis to the center of mass	$m$
$I_{1}, I_{2}$	Inertia of moment	$kg . m^{2}$
$B_{1}, B_{2}$	Damping coefficient of the joint	$N . m . s / rad$
$k_{1}, k_{2}$	Stiffness coefficient of the spring	$N / m$
$m_{1}, m_{2}$	Lumped mass of SEA	$kg$
$g$	Gravitational acceleration	$m / s^{2}$
$r_{1}, r_{2}$	Nut’s displacement	$m$
$Δ_{1}, Δ_{2}$	Spring’s deformation	$m$
$U_{v 1}, U_{v 2}$	Input voltage of DC motor	$V$

By defining $x_{1} \overset{Δ}{=} θ, x_{2} \overset{Δ}{=} \overset{\cdot}{θ}, x_{3} \overset{Δ}{=} Δ, x_{4} \overset{Δ}{=} \overset{\cdot}{Δ},$ the system of equation (1) is further divided into two subsystems as follows:

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = f_{x} + g_{x} (x_{3} + d) \end{matrix}

(4)

{\begin{matrix} {\overset{\cdot}{x}}_{3} = x_{4} \\ {\overset{\cdot}{x}}_{4} = F_{R} + δ \overset{\cdot}{r} - U_{v} \end{matrix}

(5)

Backstepping terminal super-twisting SMC based on HOSMO

Backstepping process

Considering the system of equation (1), the motor voltage $U_{v}$ is defined as the control input, and the joints’ angle $θ$ is defined as the system output. The objective is to build a suitable control input $U_{v}$ for the system of equation (1) to track the reference trajectories $θ_{d} = {[\begin{matrix} θ_{1 d} & θ_{2 d} \end{matrix}]}^{T}$ . To achieve this objective, the controller design process is illustrated as follows:

Step 1: A virtual control input $u_{v} \overset{Δ}{=} {[\begin{matrix} u_{v 1} & u_{v 2} \end{matrix}]}^{T}$ is designed for the system of equation (4) using a terminal super-twisting SMC based on HOSMO to drive $θ$ to $θ_{d}$ .

Step 2: The difference between $x_{3}$ and $u_{v}$ is defined as the new error variables. Then the control input $U_{v}$ is built for the system of equation (5) to drive the new error variables to zero.

Super-twisting observer design

It can be seen from equations (4) and (5) that only the angle displacement $x_{1}$ and the spring’s deformation $x_{3}$ can be measured in practice. However, full-states feedback is required to successfully implement the TSTSM. Therefore, the utilization of the HOSMO is necessary to provide the prediction of the unknown states effectively.^71,75 The observer used to estimate the state variables of equations (4) and (5) is inspired by the observer in the literature.⁷⁵ To obtain the observer, equations (4) and (5) can be rewritten as follows:

\begin{matrix} Φ_{1} : {\begin{matrix} {\overset{\cdot}{y}}_{1} = y_{2} \\ {\overset{\cdot}{y}}_{2} = f_{x 1} (y_{1}, y_{5}, y_{2}, y_{6}) + g_{x 1} (y_{1}, y_{5}) (u_{v 1} + d_{1}) \\ {\overset{\cdot}{y}}_{3} = y_{4} \\ {\overset{\cdot}{y}}_{4} = F_{R 1} (y_{5}, y_{2}, y_{6}, {\overset{\cdot}{y}}_{2}, {\overset{\cdot}{y}}_{6}) + δ {\overset{\cdot}{r}}_{1} (y_{1}, y_{5}, y_{2}, y_{6}) - U_{v 1} \end{matrix} \\ Φ_{2} : {\begin{matrix} {\overset{\cdot}{y}}_{5} = y_{6} \\ {\overset{\cdot}{y}}_{6} = f_{x 2} (y_{1}, y_{5}, y_{2}, y_{6}) + g_{x 2} (y_{1}, y_{5}) (u_{v 2} + d_{2}) \\ {\overset{\cdot}{y}}_{7} = y_{8} \\ {\overset{\cdot}{y}}_{8} = F_{R 2} (y_{5}, y_{2}, y_{6}, {\overset{\cdot}{y}}_{2}, {\overset{\cdot}{y}}_{6}) + δ {\overset{\cdot}{r}}_{2} (y_{1}, y_{5}, y_{2}, y_{6}) - U_{v 2} \end{matrix} \end{matrix}

(6)

where $Φ_{1}$ and $Φ_{2}$ are the subsystems for the femur and the tibia SEA, respectively. $y_{1} \overset{Δ}{=} θ_{1}, y_{2} \overset{Δ}{=} {\overset{\cdot}{θ}}_{1}, y_{3} \overset{Δ}{=} Δ_{1}, y_{4} \overset{Δ}{=} {\overset{\cdot}{Δ}}_{1}$ are the state variables for the femur SEA system interconnected to the hip joint, whereas $y_{5} \overset{Δ}{=} θ_{2}, y_{6} \overset{Δ}{=} {\overset{\cdot}{θ}}_{2}, y_{7} \overset{Δ}{=} Δ_{2}, y_{8} \overset{Δ}{=} {\overset{\cdot}{Δ}}_{2}$ are the state svariables for the tibia SEA system interconnected to the knee joint.

Consider the dynamic system for the hip joint of the femur SEA $Φ_{1}$ :

{\begin{matrix} {\overset{\cdot}{y}}_{1} = y_{2} \\ {\overset{\cdot}{y}}_{2} = f_{x 1} (y_{1}, y_{5}, y_{2}, y_{6}) + g_{x 1} (y_{1}, y_{5}) (u_{v 1} + d_{1}) \end{matrix}

(7)

{\begin{matrix} {\overset{\cdot}{y}}_{3} = y_{4} \\ {\overset{\cdot}{y}}_{4} = F_{R 1} (y_{5}, y_{2}, y_{6}, {\overset{\cdot}{y}}_{2}, {\overset{\cdot}{y}}_{6}) + δ {\overset{\cdot}{r}}_{1} (y_{1}, y_{5}, y_{2}, y_{6}) - U_{v 1} \end{matrix}

(8)

The HOSMO dynamic systems to estimate ${\overset{\cdot}{y}}_{1}$ and ${\overset{\cdot}{y}}_{3}$ are given by the following form:

{\begin{matrix} {\dot{\hat{y}}}_{1} = {\hat{y}}_{2} + z_{1} \\ {\dot{\hat{y}}}_{2} = f_{x 1} (y_{1}, y_{5}, {\hat{y}}_{2}, {\hat{y}}_{6}) + g_{x 1} u_{v 1} + z_{2} + {\hat{y}}_{α} \\ {\dot{\hat{y}}}_{α} = z_{α} \end{matrix}

(9)

\begin{matrix} {\begin{matrix} {\dot{\hat{y}}}_{3} = {\hat{y}}_{4} + z_{3} \\ {\dot{\hat{y}}}_{4} = F_{R 1} (y_{5}, {\hat{y}}_{2}, {\hat{y}}_{6}, {\dot{\hat{y}}}_{2}, {\dot{\hat{y}}}_{6}) + δ {\overset{\cdot}{r}}_{1} (y_{1}, y_{5}, {\hat{y}}_{2}, {\hat{y}}_{6}) \\ - U_{v 1} + z_{4} + {\hat{y}}_{β} \\ {\dot{\hat{y}}}_{β} = z_{β} \end{matrix} \end{matrix}

(10)

where ${\hat{y}}_{i}$ is the estimated state variable for the hip joint system $(i = 1, . ., 4)$ .

${\hat{y}}_{α}, {\hat{y}}_{β}$ are additional state variables.

${\hat{y}}_{6}$ is the estimated state variable of the knee joint’s angular velocity.

$z_{1}, z_{2}, z_{3}, z_{4}, z_{α}, z_{β}$ are the correction terms.

Let us define the error variables as follows:

ε_{1} \overset{Δ}{=} y_{1} - {\hat{y}}_{1}, ε_{2} \overset{Δ}{=} y_{2} - {\hat{y}}_{2}, ε_{3} \overset{Δ}{=} y_{3} - {\hat{y}}_{3}, ε_{4} \overset{Δ}{=} y_{4} - {\hat{y}}_{4}

(11)

The correction terms are chosen in the same manner given in the literature⁷⁵:

\begin{matrix} z_{1} = k_{1} {| ε_{1} |}^{\frac{2}{3}} sign (ε_{1}), z_{2} = k_{2} {| ε_{1} |}^{\frac{1}{3}} sign (ε_{1}), z_{α} = k_{α} sign (ε_{1}) \\ z_{3} = k_{3} {| ε_{3} |}^{\frac{2}{3}} sign (ε_{3}), z_{4} = k_{4} {| ε_{3} |}^{\frac{1}{3}} sign (ε_{3}), z_{β} = k_{β} sign (ε_{3}) \end{matrix}

(12)

where $k_{1}, k_{2}, k_{3}, k_{4}, k_{α}, k_{β}$ are positive constants.

Then, the HOSMO error dynamics for the two subsystems can be written as:

{\begin{matrix} {\overset{\cdot}{ε}}_{1} = - k_{1} {| ε_{1} |}^{\frac{2}{3}} sign (ε_{1}) + ε_{2} \\ {\overset{\cdot}{ε}}_{2} = - k_{2} {| ε_{1} |}^{\frac{1}{3}} sign (ε_{1}) - {\hat{y}}_{α} + Δ f_{x 1} + g_{x 1} (y_{1}, y_{5}) d_{1} \\ {\dot{\hat{y}}}_{α} = z_{α} \end{matrix}

(13)

where $Δ f_{x 1} = f_{x 1} (y_{1}, y_{5}, y_{2}, y_{6}) - {\hat{f}}_{x 1} = f_{x 1} (y_{1}, y_{5}, y_{2}, y_{6}) - f_{x 1} (y_{1}, y_{5}, {\hat{y}}_{2}, {\hat{y}}_{6})$

\begin{matrix} {\begin{matrix} {\overset{\cdot}{ε}}_{3} = - k_{3} {| ε_{3} |}^{\frac{2}{3}} sign (ε_{3}) + ε_{4} \\ {\overset{\cdot}{ε}}_{4} = - k_{4} {| ε_{3} |}^{\frac{1}{3}} sign (ε_{3}) - {\hat{y}}_{β} + Δ F_{R 1} + δ Δ {\overset{\cdot}{r}}_{1} \\ {\dot{\hat{y}}}_{β} = z_{β} \end{matrix} \end{matrix}

(14)

where $Δ F_{R} = F_{R 1} (y_{5}, y_{2}, y_{6}, {\overset{\cdot}{y}}_{2}, {\overset{\cdot}{y}}_{6}) - {\hat{F}}_{R 1} = F_{R 1} (y_{5}, y_{2}, y_{6}, {\overset{\cdot}{y}}_{2}, {\overset{\cdot}{y}}_{6}) - F_{R 1} (y_{5}, {\hat{y}}_{2}, {\hat{y}}_{6}, {\dot{\hat{y}}}_{2}, {\dot{\hat{y}}}_{6})$

Δ {\dot{r}}_{1} = {\dot{r}}_{1} (y_{1}, y_{5}, y_{2}, y_{6}) - {\dot{\hat{r}}}_{1} = {\dot{r}}_{1} (y_{1}, y_{5}, y_{2}, y_{6}) - {\dot{r}}_{1} (y_{1}, y_{5}, {\hat{y}}_{2}, {\hat{y}}_{6})

Let us define $ρ_{α} = Δ f_{x 1} + g_{x 1} (y_{1}, y_{5}) d$ , $ρ_{β} = Δ F_{R 1} + δ Δ {\overset{\cdot}{r}}_{1}$ , $ε_{α} = - {\hat{y}}_{α} + ρ_{α}, ε_{β} = - {\hat{y}}_{β} + ρ_{β}$ , where $ρ_{α}$ and $ρ_{β}$ are considered as perturbation terms.

Assumption 1: $ρ_{α}, ρ_{β}$ are Lipschitz and satisfy the inequalities $| {\overset{\cdot}{ρ}}_{α} | < Δ_{1}, | {\overset{\cdot}{ρ}}_{β} | < Δ_{2}$ .

The STO error dynamics in equations (13) and (14) can be further rewritten as:

\begin{matrix} {\begin{matrix} {\overset{\cdot}{ε}}_{1} = - k_{1} {| ε_{1} |}^{\frac{2}{3}} sign (ε_{1}) + ε_{2} \\ {\overset{\cdot}{ε}}_{2} = - k_{2} {| ε_{1} |}^{\frac{1}{3}} sign (ε_{1}) + ε_{α} \\ {\overset{\cdot}{ε}}_{α} = - k_{α} sign (ε_{1}) + {\overset{\cdot}{ρ}}_{α} \end{matrix} \end{matrix}

(15)

\begin{matrix} {\begin{matrix} {\overset{\cdot}{ε}}_{3} = - k_{3} {| ε_{3} |}^{\frac{2}{3}} sign (ε_{3}) + ε_{4} \\ {\overset{\cdot}{ε}}_{4} = - k_{4} {| ε_{3} |}^{\frac{1}{3}} sign (ε_{3}) + ε_{β} \\ {\overset{\cdot}{ε}}_{β} = - k_{β} sign (ε_{3}) + {\overset{\cdot}{ρ}}_{β} \end{matrix} \end{matrix}

(16)

Equations (15) and (16) are proven to be finite-time stable under Assumption 1 in literature^76,77 by choosing appropriate gains $k_{1}, k_{2}, k_{3}, k_{4}, k_{α}, k_{β}$ as the procedure given in.⁶⁶ This means that $ε_{1}, ε_{2}, ε_{3}, ε_{4}, ε_{α},$ and $ε_{β}$ will converge to zero in finite time. As a result, it can be concluded that $θ_{1} = {\hat{θ}}_{1}, {\dot{θ}}_{1} = {\dot{\hat{θ}}}_{1}, Δ_{1} = {\hat{Δ}}_{1},$ and ${\dot{Δ}}_{1} = {\dot{\hat{Δ}}}_{1}$ in finite time. By applying the same manner for the knee joint of the tibia SEA system $Φ_{2}$ , it can also conclude that $θ_{2} = {\hat{θ}}_{2}, {\dot{θ}}_{2} = {\dot{\hat{θ}}}_{2}, Δ_{2} = {\hat{Δ}}_{2}, {\dot{Δ}}_{2} = {\dot{\hat{Δ}}}_{2}$ .

Terminal super-twisting sliding mode controller

Let us define the error variable $e_{1} = y_{1} - θ_{1 d}, {\hat{e}}_{2} = {\hat{y}}_{2} - {\overset{\cdot}{θ}}_{1 d}, e_{3} = y_{3} - Δ_{1 d}, {\hat{e}}_{4} = {\hat{y}}_{4} - {\overset{\cdot}{Δ}}_{1 d}$ , then the sliding surfaces based on the fast terminal sliding mode introduced in⁷⁸ for controlling the hip joint of the femur SEA system in equations (7) and (8) are given as:

\begin{matrix} {\hat{s}}_{1} = {\hat{e}}_{2} + μ_{1} e_{1} + η_{1} {| e_{1} |}^{γ_{1}} sign (e_{1}) \\ {\hat{s}}_{2} = {\hat{e}}_{4} + μ_{2} e_{3} + η_{2} {| e_{3} |}^{γ_{2}} sign (e_{3}) \end{matrix}

(17)

Taking the first time-derivative of equation (17) and using equation (7), equations (9–11), it yields:

\begin{matrix} {\overset{\cdot}{\hat{s}}}_{1} = f_{x 1} (y_{1}, y_{5}, {\hat{y}}_{2}, {\hat{y}}_{6}) + g_{x 1} u_{v 1} + z_{2} + {\hat{y}}_{α} - {\overset{\cdot\cdot}{θ}}_{1 d} \\ + μ_{1} ({\hat{y}}_{2} - {\overset{\cdot}{θ}}_{1 d} + ε_{2}) + η_{1} γ_{1} {| e_{1} |}^{γ_{1} - 1} ({\hat{y}}_{2} - {\overset{\cdot}{θ}}_{1 d} + ε_{2}) \\ {\overset{\cdot}{\hat{s}}}_{2} = F_{R 1} (y_{5}, {\hat{y}}_{2}, {\hat{y}}_{6}, {\overset{\cdot}{\hat{y}}}_{2}, {\overset{\cdot}{\hat{y}}}_{6}) + δ {\overset{\cdot}{r}}_{1} (y_{1}, y_{5}, {\hat{y}}_{2}, {\hat{y}}_{6}) \\ - U_{v 1} + z_{4} + {\hat{y}}_{β} - {\overset{\cdot\cdot}{Δ}}_{1 d} + μ_{2} ({\hat{y}}_{4} - {\overset{\cdot}{Δ}}_{1 d} + ε_{4}) \\ + η_{2} γ_{2} {| e_{3} |}^{γ_{2} - 1} ({\hat{y}}_{4} - {\overset{\cdot}{Δ}}_{1 d} + ε_{4}) \end{matrix}

(18)

Theorem 1: Under Assumption 1, the following control inputs of equation (19) lead to the establishment of the second-order sliding mode on ${\hat{s}}_{1}$ and ${\hat{s}}_{2}$ in finite-time. Once ${\hat{s}}_{1} = 0, {\hat{s}}_{2} = 0$ , the error dynamics will reach zero in finite time $T_{1}$ and $T_{2}$ by a properly chosen $γ_{1}$ and $γ_{2}$ .

\begin{matrix} u_{v 1} = g_{x 1}^{- 1} (- f_{x 1} (y_{1}, y_{5}, {\hat{y}}_{2}, {\hat{y}}_{6}) - z_{2} - {\hat{y}}_{α} + {\overset{\cdot\cdot}{θ}}_{1 d} - μ_{1} {\hat{e}}_{2} - η_{1} γ_{1} {| e_{1} |}^{γ_{1} - 1} {\hat{e}}_{2} + u_{1}) \\ U_{v 1} = F_{R 1} (y_{5}, {\hat{y}}_{2}, {\hat{y}}_{6}, {\dot{\hat{y}}}_{2}, {\dot{\hat{y}}}_{6}) + δ {\overset{\cdot}{r}}_{1} (y_{1}, y_{5}, {\hat{y}}_{2}, {\hat{y}}_{6}) + z_{4} + {\hat{y}}_{β} - {\overset{\cdot\cdot}{Δ}}_{1 d} + μ_{2} {\hat{e}}_{4} + η_{2} γ_{2} {| e_{3} |}^{γ_{2} - 1} {\hat{e}}_{4} - u_{2} \end{matrix}

(19)

in which $u_{1}, u_{2}$ is proposed as follows:

\begin{matrix} u_{1} = - α_{1} {| {\hat{s}}_{1} |}^{\frac{1}{2}} sign ({\hat{s}}_{1}) - \int_{0}^{t} β_{1} sign ({\hat{s}}_{1}) d τ \\ u_{2} = - α_{2} {| {\hat{s}}_{2} |}^{\frac{1}{2}} sign ({\hat{s}}_{2}) - \int_{0}^{t} β_{2} sign ({\hat{s}}_{2}) d τ \end{matrix}

(20)

The convergence time is given as the procedure given in the literature⁷⁸:

\begin{matrix} T_{1} = μ_{1}^{- 1} {(1 - γ_{1})}^{- 1} (\ln (μ_{1} {| e_{1} (0) |}^{1 - γ_{1}} + η_{1}) - \ln (η_{1})) \\ T_{2} = μ_{2}^{- 1} {(1 - γ_{2})}^{- 1} (\ln (μ_{2} {| e_{2} (0) |}^{1 - γ_{2}} + η_{2}) - \ln (η_{2})) \end{matrix}

(21)

Proof of Theorem 1. Substituting the control inputs of equation (19) into equation (18), it yields:

\begin{matrix} {\dot{\hat{s}}}_{1} = μ_{1} ε_{2} + η_{1} γ_{1} {| e_{1} |}^{γ_{1} - 1} ε_{2} - α_{1} {| {\hat{s}}_{1} |}^{\frac{1}{2}} sign ({\hat{s}}_{1}) \\ - \int_{0}^{t} β_{1} sign ({\hat{s}}_{1}) d τ \\ {\dot{\hat{s}}}_{2} = μ_{2} ε_{4} + η_{2} γ_{2} {| e_{3} |}^{γ_{2} - 1} ε_{4} - α_{2} {| {\hat{s}}_{2} |}^{\frac{1}{2}} sign ({\hat{s}}_{2}) \\ - \int_{0}^{t} β_{2} sign ({\hat{s}}_{2}) d τ \end{matrix}

(22)

Combining equations (15)–(18), the dynamic of the hip joint of the femur SEA can be deduced as follows:

\begin{matrix} Γ_{1} : {\begin{matrix} {\overset{\cdot}{e}}_{1} = {\hat{s}}_{1} - μ_{1} e_{1} - η_{1} {| e_{1} |}^{γ_{1}} sign (e_{1}) + ε_{2} \\ {\dot{\hat{s}}}_{1} = μ_{1} ε_{2} + η_{1} γ_{1} {| e_{1} |}^{γ_{1} - 1} ε_{2} - α_{1} {| {\hat{s}}_{1} |}^{\frac{1}{2}} sign ({\hat{s}}_{1}) + v_{1} \\ {\overset{\cdot}{v}}_{1} = - β_{1} sign ({\hat{s}}_{1}) \end{matrix} \\ Θ_{1} : {\begin{matrix} {\overset{\cdot}{ε}}_{1} = - k_{1} {| ε_{1} |}^{\frac{2}{3}} sign (ε_{1}) + ε_{2} \\ {\overset{\cdot}{ε}}_{2} = - k_{2} {| ε_{1} |}^{\frac{1}{3}} sign (ε_{1}) + ε_{α} \\ {\overset{\cdot}{ε}}_{α} = - k_{α} sign (ε_{1}) + {\overset{\cdot}{ρ}}_{α} \end{matrix} \\ Γ_{2} : {\begin{matrix} {\overset{\cdot}{e}}_{3} = {\hat{s}}_{2} - μ_{2} e_{3} - η_{2} {| e_{3} |}^{γ_{2}} sign (e_{3}) + ε_{4} \\ {\dot{\hat{s}}}_{2} = μ_{2} ε_{4} + η_{2} γ_{2} {| e_{3} |}^{γ_{2} - 1} ε_{4} - α_{2} {| {\hat{s}}_{2} |}^{\frac{1}{2}} sign ({\hat{s}}_{2}) + v_{2} \\ {\overset{\cdot}{v}}_{2} = - β_{2} sign ({\hat{s}}_{2}) \end{matrix} \\ Θ_{2} : {\begin{matrix} {\overset{\cdot}{ε}}_{3} = - k_{3} {| ε_{3} |}^{\frac{2}{3}} sign (ε_{3}) + ε_{4} \\ {\overset{\cdot}{ε}}_{4} = - k_{4} {| ε_{3} |}^{\frac{1}{3}} sign (ε_{3}) + ε_{β} \\ {\overset{\cdot}{ε}}_{β} = - k_{β} sign (ε_{3}) + {\overset{\cdot}{ρ}}_{β} \end{matrix} \end{matrix}

(23)

The error dynamics $Θ_{1}, Θ_{2}$ were shown to converge to zero in the previous section. Therefore, we can substitute $ε_{2} = ε_{4} = 0$ into $Γ_{1}, Γ_{2}$ since the trajectories of the subsystems $Γ_{1}$ and $Γ_{2}$ cannot escape to infinity in finite time.⁷⁹ Subsequently, $Γ_{1}$ and $Γ_{2}$ can be rewritten as:

\begin{matrix} Γ_{1} : {\begin{matrix} {\overset{\cdot}{e}}_{1} = {\hat{s}}_{1} - μ_{1} e_{1} - η_{1} {| e_{1} |}^{γ_{1}} sign (e_{1}) \\ {\dot{\hat{s}}}_{1} = - α_{1} {| {\hat{s}}_{1} |}^{\frac{1}{2}} sign ({\hat{s}}_{1}) + v_{1} \\ {\overset{\cdot}{v}}_{1} = - β_{1} sign ({\hat{s}}_{1}) \end{matrix} \\ Γ_{2} : {\begin{matrix} {\overset{\cdot}{e}}_{3} = {\hat{s}}_{2} - μ_{2} e_{3} - η_{2} {| e_{3} |}^{γ_{2}} sign (e_{3}) \\ {\dot{\hat{s}}}_{2} = - α_{2} {| {\hat{s}}_{2} |}^{\frac{1}{2}} sign ({\hat{s}}_{2}) + v_{2} \\ {\overset{\cdot}{v}}_{2} = - β_{2} sign ({\hat{s}}_{2}) \end{matrix} \end{matrix}

(24)

In the second step of the proof, a stability analysis of equation (24) is performed. Let us consider a generalized system:

\begin{matrix} \overset{\cdot}{s} = - α {| s |}^{\frac{1}{2}} sign (s) + v \\ \overset{\cdot}{v} = - β sign (s) \end{matrix}

(25)

To analyze the stability of equation (25), a candidate Lyapunov function candidate is chosen as^80,81:

V (ξ_{1}, ξ_{2}) = ξ^{T} P ξ

(26)

where $ξ = {[\begin{matrix} ξ_{1} & ξ_{2} \end{matrix}]}^{T} = {[\begin{matrix} {| s |}^{\frac{1}{2}} sign (s) & v \end{matrix}]}^{T}$

$P = [\begin{matrix} λ + 4 ω^{2} & - 2 ω \\ - 2 ω & 1 \end{matrix}]$ is a positive definite matrix with $λ > 0$ and $ω > 0$ .

Notice that $| ξ_{1} | = {| s |}^{\frac{1}{2}}, {\overset{\cdot}{ξ}}_{2} = \frac{- β}{{| s |}^{\frac{1}{2}}} ξ_{1}$ , by taking the first-order derivative of $ξ$ and representing in the vector form, it can get the following expression:

\overset{\cdot}{ξ} = [\begin{matrix} {\overset{\cdot}{ξ}}_{1} \\ {\overset{\cdot}{ξ}}_{2} \end{matrix}] = \underset{A (ξ_{1})}{\underset{︸}{\frac{1}{2 | ξ_{1} |} [\begin{matrix} - α & 1 \\ - 2 β & 0 \end{matrix}]}} [\begin{matrix} ξ_{1} \\ ξ_{2} \end{matrix}]

(27)

By taking the first-order derivative of the Lyapunov function candidate, it can be obtained as follows:

\begin{array}{l} \dot{V} (ξ_{1}, ξ_{2}) = {\dot{ξ}}^{T} P ξ + ξ^{T} P \dot{ξ} \\ = ξ^{T} (A^{T} P + P A) ξ \leq - \frac{1}{2 | ξ_{1} |} ξ^{T} Q ξ \end{array}

(28)

where $Q = [\begin{matrix} 2 α λ + 8 α ω^{2} - 8 β ω & - λ - 4 ω^{2} - 2 α ω + 2 β \\ - λ - 4 ω^{2} - 2 α ω + 2 β & 4 ω \end{matrix}]$ satisfying the algebraic Lyapunov equation (ALE):

{\bar{A}}^{T} P + P \bar{A} = - Q

(29)

To guarantee that $Q$ is positive definite, $α$ and $β$ must satisfy the following conditions:

\begin{matrix} α > \frac{{(λ + 4 ω^{2})}^{2}}{8 λ ω} \\ β = α ω \end{matrix}

(30)

Recall the standard inequality for quadratic forms:

λ_{\min} {P} ‖ ξ ‖_{2}^{2} \leq ξ^{T} P ξ \leq λ_{\max} {P} ‖ ξ ‖_{2}^{2}

(31)

λ_{\min} {Q} ‖ ξ ‖_{2}^{2} \leq ξ^{T} Q ξ \leq λ_{\max} {Q} ‖ ξ ‖_{2}^{2}

(32)

where $‖ ξ ‖_{2}^{2} = | s | + v^{2}$

By utilizing equation (32), equation (28) becomes:

\overset{\cdot}{V} (ξ_{1}, ξ_{2}) \leq - \frac{1}{2 | ξ_{1} |} ξ^{T} Q ξ \leq - \frac{1}{2 | ξ_{1} |} λ_{\min} {Q} ‖ ξ ‖_{2}^{2}

(33)

Equation (31) gives us the fact that $| ξ_{1} | \leq ‖ ξ ‖_{2} \leq \frac{V^{\frac{1}{2}} (ξ_{1}, ξ_{2})}{λ_{\min}^{\frac{1}{2}} {P}}$ , combining with Equation (33) it yields:

\overset{\cdot}{V} (ξ_{1}, ξ_{2}) \leq - ψ V^{\frac{1}{2}} (ξ_{1}, ξ_{2})

(34)

where ψ = \frac{λ_{\min}^{\frac{1}{2}} {P} λ_{\min} {Q}}{2 λ_{\max} {P}}

From equation (34), it can be concluded that the trajectory of $s$ in equation (25) converges to zero in finite time. By implementing the same manner for equation (24), it can be easily seen that ${\hat{s}}_{1} = {\dot{\hat{s}}}_{1} = {\hat{s}}_{2} = {\dot{\hat{s}}}_{2} = 0$ in finite-time, which further implies that the error dynamics in equation (24) become:

\begin{matrix} {\overset{\cdot}{e}}_{1} = - μ_{1} e_{1} - η_{1} {| e_{1} |}^{γ_{1}} sign (e_{1}) \\ {\overset{\cdot}{e}}_{3} = - μ_{2} e_{3} - η_{2} {| e_{3} |}^{γ_{2}} sign (e_{3}) \end{matrix}

(35)

This implies that both the errors $e_{1}$ and $e_{3}$ will converge to zero in finite-time, as shown in equation (21).

The block diagram for the whole controller is shown in Figures 2 and 3.

Figure 2.

Block diagram for implementation of TSTSM with HOSMO on SEA system (subsystem 1).

Figure 3.

Block diagram for implementation of TSTSM with HOSMO on SEA system (subsystem 2).

Experimental setup and results

The experimental setup is utilized to validate the proposed controller, as shown in Figure 4. Two SEA links assist the hip joint and knee joint. The SEA mechanism is powered by a DC motor. The rotational motion of the motor shaft is transmitted through a gear drive to establish the prismatic motion of the SEA. This motion, in turn, is transformed into rotational motion in the joints. To measure the angular position of the two joints, linear membrane potentiometers are installed to measure the linear position of the SEA, which is then calculated into angular position by trigonometric constraint of the SEA mechanism. Additionally, another potentiometer is installed onto the nut to measure the deformation of the elastic element – the spring, which is shown in Figure 5.

Figure 4.

SEA-inspired humanoid robot’s leg.

Figure 5.

Detail of the SEA.

First, the desired trajectories are defined as step functions to demonstrate the performance of the controller with $θ_{1 d} = {\begin{matrix} 0, t \leq 1.8 s \\ 0.12, t > 1.8 s \end{matrix}$ and $θ_{2 d} = {\begin{matrix} 0.18, t \leq 1.8 s \\ 0.25, t > 1.8 s \end{matrix}$ . The sampling time is chosen as $t_{s} = 0.005 s$ , the experiment time is processed as 10 $s$ . The control and physical parameters of the system are given in Tables 2 and 3.

Table 2.

Control parameters.

Control parameters	Value	Control parameters	Value
$k_{1}$	40	$k_{3}$	40
$k_{2}$	40	$k_{4}$	60
$k_{α}$	60	$k_{β}$	60
$μ_{1}$	50	$μ_{2}$	15
$η_{1}$	5	$η_{2}$	3
$γ_{1}$	0.965	$γ_{2}$	0.965
$α_{1}$	50	$α_{2}$	20
$β_{1}$	600	$β_{2}$	30

Table 3.

Physical parameters.

Physical parameters	Value	Physical parameters	Value
$d_{1}$	$0.177$	$R_{2}$	$0.054$
$d_{2}$	$0.035$	$B_{1} = B_{2}$	$0.3$
$d_{3}$	$0.1804$	$I_{1}$	$0.028$
$d_{4}$	$0.18$	$I_{2}$	$0.02$
$d_{5}$	$0.028$	$m_{1}$	$1.5$
$d_{6}$	$0.19$	$m_{2}$	$0.5$
$d_{7}$	$0.085$	$k_{1} = k_{2}$	$20000$
$d_{8}$	$0.2081$	$α_{1}$	$0.1543$
$d_{9}$	$0.084$	$α_{2}$	$1.1796$
$d_{10}$	$0.08$	$σ_{1}$	$0.1952$
$g$	$9.8$	$σ_{2}$	$0.4207$
$R_{1}$	$0.043$

The position tracking of the step trajectory is shown in Figure 6. The actual angular positions of the joints are represented by the solid blue line while the red dotted line represents the reference trajectories. It can be seen that both of the joints converged to the reference trajectories. However, because of the oscillation nature of the springs in both joints, the combined oscillation makes the fluctuation in $θ_{2}$ (Figure 6(b)) more prominent than in the case of $θ_{1}$ (Figure 6(a)). It is also noteworthy that the current mechanical setup of the membrane potentiometers caused the feedback to be disturbed heavily because the pressure on the surfaces of the potentiometers caused the surfaces to be worn out in a short amount of time, creating grooves on them. Ultimately, this leads to a heavily disturbed signal even when there are no control inputs which can be seen clearly at between 0 and 1.8 s in Figure 6. Moreover, the backlash of the joints contributed greatly to the fluctuation in $θ_{2}$ , a small force exerted by the spring mechanism in the SEA may cause the joints to move, leading to inaccuracy in position tracking (Figure 6). As for the sliding surfaces, despite some initial overshoot, they managed to reach zero after approximately 3.2 s. The HOSMO successfully estimated the angular velocities in Figure 7 and the estimation errors also converged to zero, as shown in Figure 8. Regarding the control inputs shown in Figure 9, despite the presence of $- α_{1} {| {\hat{s}}_{1} |}^{\frac{1}{2}} sign ({\hat{s}}_{1})$ and $- α_{2} {| {\hat{s}}_{2} |}^{\frac{1}{2}} sign ({\hat{s}}_{2})$ functions in equation (20), they are continuous functions of time as opposed to the traditional control input $- Ksign (s)$ which is not derivable and thus is discontinuous. However, because there are discontinuous functions under the integral, which are $- \int_{0}^{t} β_{1} sign ({\hat{s}}_{1}) d τ$ and $- \int_{0}^{t} β_{2} sign ({\hat{s}}_{2}) d τ$ , therefore, the chattering phenomenon is not eliminated but attenuated.

Figure 6.

Position tracking of the hip and knee joints (step trajectory): (a) tracking of the hip joint and (b) tracking of the knee joint.

Figure 7.

Evolution of the sliding surfaces (step trajectory): (a) sliding surface value of the first subsystem of the hip joint and (b) sliding surface value of the first subsystem of the knee joint.

Figure 8.

Evolution of HOSMO error (step trajectory): (a) estimation error of the hip joint’s angle and (b) estimation error of the knee joint’s angle.

Figure 9.

Voltage inputs for the control system (step trajectory): (a) control input for the hip joint and (b) control input for the knee joint.

To quantitatively evaluate the effectiveness of the proposed controller, we conduct two other different control schemes, traditional SMC and TSMC. The desired trajectories are defined as $θ_{1 d} = 0.1 * \sin (π t) + 0.11$ and $θ_{2 d} = 0.05 * \sin (π t) + 0.25$ . The sampling time is chosen as $t_{s} = 0.005 s$ , the experiment time is processed as 10 $s$ . The control and physical parameters are again taken from Tables 2 and 3.

The experimental results comparison of SMC, TSMC, and the proposed controller are illustrated in Figure 10. The solid blue line represents the actual angular position of the joints, and the red dotted line represents the reference trajectories. The desired trajectory of the hip joint is easily achieved by all three control schemes, as shown in Figure 10(a) to (c). Despite the initial overshoot, the TSTSM successfully alleviated the chattering phenomenon, producing a smoother trajectory compared to the other two control schemes. From the observation, one can see that the tracking of the knee joint is much more difficult to achieve, as shown in Figure 10(d) to (f). As seen in Figure 10(d) and (e), the trajectory of the knee joint suffered significantly from the combined perturbation (the chattering phenomenon and the vibration of the springs), resulting in chattered outputs.

Figure 10.

Position tracking results of three control schemes: (a) tracking of the hip joint under traditional SMC. (b) Tracking of the hip joint under terminal SMC. (c) Tracking of the hip joint under TSTSM. (d) Tracking of the knee joint under traditional SMC. (e) Tracking of the knee joint under terminal SMC. (f) Tracking of the knee joint under TSTSM.

On the other hand, Figure 10(f) shows that the effect of the combined perturbation is reduced dramatically, thus providing the trajectory smoother. The sliding surfaces and the error of HOSMO are shown in Figures 11 to 13, respectively. The error of the HOSMO converges to zero, as shown in Figure 8. The final control input still suffers from slight chattering, as shown in Figure 14.

Figure 11.

Evolution of the sliding surfaces (sine trajectory): (a) sliding surface value of the first subsystem of the hip joint and (b) Sliding surface value of the first subsystem of the knee joint.

Figure 12.

Evolution of the estimated state (sine trajectory): (a) estimated value of the angular velocity of the hip joint and (b) estimated value of the angular velocity of the knee joint.

Figure 13.

Evolution of HOSMO error (sine trajectory): (a) estimation error of the hip joint s angle. and (b) estimation error of the knee joint’s angle.

Figure 14.

Voltage inputs for the control system (sine trajectory): (a) SM control input for the hip joint. (b) SM control input for the knee joint. (c) TSM control input for the hip joint. (d) TSM control input for the knee joint. (e) TSTSM control input for the hip joint. (f) TSTSM control input for the knee joint.

Figure 15.

Response of the state variable $θ_{1}$ with different control parameters: (a) response of $θ_{1}$ with different $μ_{1}$ , (b) Response of $θ_{1}$ with different $η_{1}$ , (c) Response of $θ_{1}$ with different $α_{1}$ , and (d) Response of $θ_{1}$ with different $β_{1}$ .

Since the TSTSM require the selection of many control parameters, as shown in equations (17) and (20), we have illustrated various responses of hip joint angle $θ_{1}$ with different control parameters as shown in Figure 15 to make the process more intuitive. In the figures, the constant reference trajectory is represented by the red dash line, while the actual trajectories of other scenarios are represented by the continuous lines. Overall, the terminal parameters $μ_{1}$ and $η_{1}$ in the first sliding surface of equation (17) affect the converging rate of the state $θ_{1}$ . Higher $μ_{1}$ makes $θ_{1}$ converge to the reference trajectory quickly as shown in Figure 10(a), the actual trajectory was able to converge to the reference trajectory at around 1.6 s for $μ_{1} = 80$ while it took the case where $μ_{1} = 1$ approximately 2.5 s. However, the state suffers from the overshoot if $μ_{1}$ is too high. As for $η_{1}$ , it can be seen that a high value can reduce the overshoot, but the stability of the system may be compromised ( $η_{1} = 80)$ as shown in Figure 10(b). Regarding the parameters for the control input in equation (20), the parameter $α_{1}$ greatly affects the tracking performance, as shown in Figure 10(c). In the case where $α_{1} = 1$ , it is impossible for the controller to drive the state to the reference trajectory. The tracking performance becomes better as $α_{1}$ increases ( $α_{1} = 50)$ , but if the parameter is too large it will amplify the chattering phenomenon ( $α_{1} = 90)$ . Figure 10(d) shows that a small value of $β_{1}$ increases the time needed for the state to reach the desired trajectory. On the other hand, raising $β_{1}$ to a certain threshold will reduce the convergence time, if $β_{1}$ passes this threshold then the system may suffer from overshooting with no additional benefit as shown in the case where $β_{1} = 600$ .

Conclusion

A backstepping terminal super-twisting sliding mode (TSTSM) controller based on a high-order sliding mode observer (HOSMO) is proposed to control the two DoFs SEA inspired by a humanoid robot’s leg. The HOSMO provides the finite-time convergence of the estimated states, thus eliminating the need for additional sensors. With the obtained estimated states, the main controller successfully drives the states of the overall system to the reference trajectories, while the source of perturbation originating from the combination of the chattering effect and the oscillation of the elastic element is also reduced significantly. The effectiveness of the proposed controller is clearly shown when compared with other control schemes. Experimental results are presented to show the feasibility of the proposed controller. In the future, further studies about how effective the HOSMO is in comparison with other observers will be made. Additionally, the backlash problem in the SEA needs to be addressed as it affects the system heavily.

Footnotes

Acknowledgements

We acknowledge the support of time and facilities from National Key Laboratory of Digital Control and System Engineering (DCSELab), Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for this study.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number TX2022-20b-01.

ORCID iD

Tri Duc Tran

References

Koniuk

. Self-adjusting prosthetic ankle apparatus [Internet]. US Patent App. 09/816,902, http://www.google.com/patents?hl=en&lr=&vid=USPATAPP9816902&id=Zr6DAAAAEBAJ&oi=fnd&dq=Self-adjusting+prosthetic+ankle&printsec=abstract (2001).

Sup

Varol

Mitchell

, et al. Self-contained powered knee and ankle prosthesis: Initial evaluation on a transfemoral amputee. In: 2009 IEEE international conference on rehabilitation robotics, ICORR 2009, 2009, pp. 638–44. New York: IEEE.

Williams

Hansen

Gard

. Prosthetic ankle-foot mechanism capable of automatic adaptation to the walking surface. J Biomech Eng 2009; 131(3): 1–7.

Weber

Herr

. Powered ankle-foot prosthesis improves walking metabolic economy. IEEE Trans Robot 2009; 25(1): 51–66.

Grimmer

Eslamy

Gliech

, et al. A comparison of parallel- and series elastic elements in an actuator for mimicking human ankle joint in walking and running. In: Proceedings – IEEE international conference on robotics and automation, 2012, pp. 2463–2470. New York: IEEE.

Suarez

Heredia

Ollero

. Physical-virtual impedance control in ultralightweight and compliant Dual-Arm aerial manipulators. IEEE Robot Autom Lett 2018; 3(3): 2553–2560.

Jiang

. Impedance control of flexible joint robots based on singular perturbation method. In: Proceedings of the IEEE 2019 9th international conference on cybernetics and intelligent systems and robotics, automation and mechatronics, CIS and RAM 2019, 2019, pp. 548–552. New York: IEEE.

Vorndamme

Schappler

Tödtheide

, et al. Soft robotics for the hydraulic Atlas arms: Joint impedance control with collision detection and disturbance compensation. In: IEEE International conference on intelligent robots and systems, 2016 November, pp. 3360–3367. New York: IEEE.

Hirayama

Hirosawa

Hyon

. Passivity-based compliant walking on torque-controlled hydraulic biped robot. In: IEEE-RAS international conference on humanoid robots, 2019, pp. 552–557. New York: IEEE.

10.

Pratt

Williamson

. Series elastic actuators. In: IEEE international conference on intelligent robots and systems, 1995, pp. 399–406. New York: IEEE.

11.

Suebsomran

. Design and control of a passive compliant actuation with positioning measurement by LED and photodiode detector for medical application. Meas Control 2021; 54(3–4): 216–230.

12.

Stratmann

Lakatos

Ozparpucu

, et al. Legged elastic multibody systems: adjusting limit cycles to Close-To-Optimal Energy Efficiency. IEEE Robot Autom Lett 2017; 2(2): 436–443.

13.

Roozing

Malzahn

Kashiri

, et al. On the stiffness selection for torque-controlled Series-elastic actuators. IEEE Robot Autom Lett 2017; 2(4): 2255–2262.

14.

Verstraten

Furnemont

Beckerle

, et al. A hopping robot driven by a series elastic dual-Motor Actuator. IEEE Robot Autom Lett 2019; 4(3): 2310–2316.

15.

Lee

. Performance Analysis of series elastic actuator based on maximum torque transmissibility. IEEE Trans Control Syst Technol 2022; 30(1): 392–399.

16.

Beckerle

Verstraten

Mathijssen

, et al. Series and parallel elastic actuation: influence of operating positions on design and control. IEEE/ASME Trans Mechatron 2017; 22(1): 521–529.

17.

Pratt

Koolen

de Boer

, et al. Capturability-based analysis and control of legged locomotion, Part 2: Application to M2V2, a lower-body humanoid. Int J Rob Res 2012; 31(10): 1117–1133.

18.

Knabe

Seminatore

Webb

, et al. Design of a series elastic humanoid for the DARPA Robotics Challenge. In: IEEE-RAS International conference on humanoid robots, 2015, pp. 738–743. New York: IEEE.

19.

Park

Kim

Lee

, et al. Mechanical design of the humanoid robot platform, HUBO. Adv Robot 2007; 21(11): 1305–1322.

20.

Ajoudani

Lee

Rocchi

, et al. A manipulation framework for compliant humanoid COMAN: Application to a valve turning task. In: IEEE-RAS International conference on humanoid robots, 2015 February, pp. 664–670. New York: IEEE.

21.

Tsagarakis

Caldwell

Negrello

, et al. WALK-MAN: a high-performance humanoid platform for Realistic Environments. J Field Robot 2017; 34(7): 1225–1259.

22.

Kashiri

Cordasco

Guria

, et al. CENTAURO: a hybrid locomotion and high power resilient manipulation platform. IEEE Robot Autom Lett 2019; 4(2): 1595–1602.

23.

Sharbafi

Yazdanpanah

Ahmadabadi

, et al. Parallel compliance design for increasing robustness and efficiency in legged locomotion – theoretical background and Applications. IEEE/ASME Trans Mechatron 2021; 26(1): 335–346.

24.

Garcia

Arevalo

Sanchez

, et al. Design and development of a biomimetic leg using hybrid actuators. In: 2011 IEEE/RSJ international conference on intelligent robots and systems, 2011, pp. 1507–1512. New York: IEEE.

25.

Sensinger

Weir

. Unconstrained impedance control using a compact series elastic actuator. In: Proceedings of the 2nd IEEE/ASME international conference on mechatronic and embedded systems and applications, MESA 2006, 2006. New York: IEEE.

26.

Hosoda

Takuma

Nakamoto

, et al. Biped robot design powered by antagonistic pneumatic actuators for multi-modal locomotion. Robot Auton Syst 2008; 56(1): 46–53.

27.

Pratt

Willisson

Bolton

, et al. Late motor processing in low-impedance robots: impedance control of series-elastic actuators. Proc Am Control Conf 2004; 4: 3245–3251.

28.

Lagoda

Schouten

Stienen

AHA

, et al. Design of an electric series elastic actuated joint for robotic gait rehabilitation training. In: 2010 3rd IEEE RAS and EMBS international conference on biomedical robotics and biomechatronics, BioRob 2010, 2010, pp. 21–26. New York: IEEE.

29.

Thorson

Caldwell

. A nonlinear series elastic actuator for highly dynamic motions. In: IEEE International conference on intelligent robots and systems, 2011, pp. 390–394. New York: IEEE.

30.

Vallery

Ekkelenkamp

Van Der Kooij

, et al. Passive and accurate torque control of series elastic actuators. In: IEEE international conference on intelligent robots and systems, 2007, pp. 3534–3538. New York: IEEE.

31.

Wyeth

. Control issues for velocity sourced series elastic actuators. In: Proceedings of the 2006 Australasian conference on robotics and automation, ACRA, 2006. ARAA. https://www.araa.asn.au/

32.

Kong

Bae

Tomizuka

. Control of rotary series elastic actuator for ideal force-mode actuation in human–Robot Interaction Applications. IEEE/ASME Trans Mechatron 2009; 14: 105–118.

33.

Kong

Bae

Tomizuka

; Member S. A compact rotary series elastic actuator for human assistive systems. In: IEEE/ASME transactions on mechatronics, 2010, pp. 1–10. New York: IEEE.

34.

Bae

Kong

Tomizuka

. Gait phase-based smoothed sliding mode control for a rotary series elastic actuator installed on the knee joint. In: Proceedings of the 2010 American control conference, ACC 2010, 2010, pp. 6030–6035. New York: IEEE.

35.

Yin

Sun

Wang

, et al. Position control of a series elastic actuator based on global sliding mode controller design. IEEE/CAA J Automatica Sinica 2019; 6(3): 850–858.

36.

Garofalo

Werner

Loeffl

, et al. Joint-space impedance control using intrinsic parameters of compliant actuators and inner sliding mode torque loop. IEEE Control Syst Lett 2019; 3(1): 1–6.

37.

Venkataraman

Gulati

. Control of nonlinear systems using terminal sliding modes. Proc Am Control Conf 1992; 891–893.

38.

Zak

. Terminal attractors for addressable memory in neural networks. Phys Lett A 1988; 133(1–2): 18–22.

39.

Zhihong

Paplinski

. A robust mimo terminal sliding mode control scheme for rigid robotic manipulators. IEEE Trans Automat Contr 1994; 39(12): 2464–2469.

40.

Truong

Kang

. Implementation of an adaptive neural terminal sliding mode for tracking control of magnetic levitation systems. IEEE Access 2020; 8: 206931–206941.

41.

Feng

Man

. Non-singular terminal sliding mode control of rigid manipulators. Automatica 2002; 38(12): 2159–2167.

42.

Man

. Model reference adaptive control systems with terminal sliding modes. Int J Control 1996; 64(6): 1165–1176.

43.

Sun

Chen

. Trajectory tracking of series elastic actuators using terminal sliding mode control. In: Proceedings of the 33rd Chinese control and decision conference, CCDC 2021, 2021, pp.189–194. New York: IEEE.

44.

Luu

AKL

Duong

Nguyen

, et al. Application of event-triggered sliding mode control of 2-DOF humanoid’s lower-limb powered by series elastic actuator. In: Anh

Koh

S-J

Nguyen

TDL

, et al. (eds) Intelligent systems and networks. Singapore: Springer Nature Singapore, 2022, pp.110–120.

45.

Burton

Zinober

ASI

. Continuous approximation of variable structure control. Int J Syst Sci 1986; 17(6): 875–885.

46.

Slotine

. Applied Nonlinear control. Vol. 8. Englewood, NJ: Prentice hall, 2011.

47.

Utkin

. Sliding modes in control and optimization. Springer. 1992.

48.

Shtessel

Taleb

Plestan

. A novel adaptive-gain supertwisting sliding mode controller: Methodology and application. Automatica 2012; 48(5): 759–769.

49.

Feng

Han

. Chattering free full-order sliding-mode control. Automatica 2014; 50(4): 1310–1314.

50.

Shtessel

Shkolnikov

Brown

MDJ

. An asymptotic second-order Smooth Sliding Mode control. Asian J Control 2003; 5(4): 498–504.

51.

Kuang

Sun

, et al. Contouring control of linear motor direct-drive X-Y table via chattering-free discrete-time sliding mode control. In: IEEE international symposium on industrial electronics, 2018 June, pp. 201–205. New York: IEEE.

52.

Chen

MZQ

, et al. Chattering-free discrete-time sliding mode control. Automatica 2016; 68: 87–91.

53.

Levant

. Sliding order and sliding accuracy in sliding mode control. Int. J. Control 1993; 58(6): 1247–1263.

54.

Levant

. Principles of 2-sliding mode design. Automatica 2007; 43(4): 576–586.

55.

Gonzalez

Moreno

Fridman

. Variable gain super-twisting sliding mode control. IEEE Trans Automat Contr 2012; 57(8): 2100–2105.

56.

Moreno

Osorio

. Strict lyapunov functions for the super-twisting algorithm. IEEE Trans Automat Contr 2012; 57(4): 1035–1040.

57.

Ren

Zhang

. Dynamic analysis and decoupled control of a heavy-duty walking robot with flexible feet based on super twisting algorithm. Meas Control 2021; 54(1–2): 55–64.

58.

Humaidi

Hasan

. Particle swarm optimization–based adaptive super-twisting sliding mode control design for 2-degree-of-freedom helicopter. Meas Control 2019; 52(9–10): 1403–1419.

59.

Zhang

Wang

, et al. Disturbance observer–based super-twisting sliding mode control for formation tracking of multi-agent mobile robots. Meas Control 2020; 53(5–6): 908–921.

60.

Dabroom

Khalil

. Discrete-time implementation of high-gain observers for numerical differentiation. Int J Control 1999; 72(17): 1523–1537.

61.

Dabroom

Khalil

. Output feedback sampled-data control of nonlinear systems using high-gain observers. IEEE Trans. Automat Contr 2001; 46(11): 1712–1725.

62.

Diop

Grizzle

Moraal

, et al. Interpolation and numerical differentiation for observer design. Proc Am Control Conf 1994; 2: 1329–1333.

63.

Pisano

Usai

. Sliding mode control: A survey with applications in math. Math Comput Simul 2011; 81(5): 954–979.

64.

Esfandiari

Khalil

. Output feedback stabilization of fully linearizable systems. Int. J. Control 1992; 56(5): 1007–1037.

65.

Teel

Praly

. Tools for semiglobal stabilization by partial state and output feedback. SIAM J. Control Optim 1995; 33(5): 1443–1488.

66.

Levant

. Higher-order sliding modes, differentiation and output-feedback control. Int J Control 2003; 76(95–610): 924–941.

67.

Edwards

Spurgeon

. On the development of discontinuous observers. Int J Control 1994; 59(5): 1211–1229.

68.

Edwards

Spurgeon

Patton

. Sliding mode observers for fault detection and isolation. Automatica 2000; 36(4): 541–553.

69.

Spurgeon

. Sliding mode observers: A survey. Int J Syst Sci 2008; 39(8): 751–764.

70.

Barbot

Levant

Livne

, et al. Discrete differentiators based on sliding modes. Automatica 2020; 112: 1–12.

71.

Davila

Fridman

Levant

. Second-order sliding-mode observer for mechanical systems. IEEE Trans Automat Contr 2005; 50(11): 1785–1789.

72.

Borkar

Patil

. Super twisting observer based full order sliding mode control. Int J Dyn Control 2021; 9(4): 1653–1659.

73.

Nguyen

Kang

. A non-singular fast terminal sliding mode control based on third-order sliding mode observer for a class of second-order uncertain nonlinear systems and its application to robot manipulators. IEEE Access 2020; 8: 78109–78120.

74.

Van

Kang

Suh

, et al. Output feedback tracking control of uncertain robot manipulators via higher-order sliding-mode observer and fuzzy compensator. J Mech Sci Technol 2013; 27(8): 2487–2496.

75.

Chalanga

Kamal

Fridman

, et al. Implementation of super-twisting control: super-twisting and higher order sliding-mode observer-based approaches. In: IEEE transactions on industrial electronics [Internet], 2016, pp. 3677–85. New York: IEEE. https://ieeexplore.ieee.org/document/7398047

76.

Angulo

Moreno

Fridman

. Robust exact uniformly convergent arbitrary order differentiator. Automatica 2013; 49(8): 2489–2495.

77.

Moreno

. Lyapunov function for Levant’s second order differentiator. In: Proceedings of the IEEE conference on decision and control, 2012, pp. 6448–6453. New York: IEEE.

78.

Feng

Man

. Terminal sliding mode control – an overview. IEEE Open J Industr Electr Soc 2021; 2: 36–52.

79.

Moreno

. A Lyapunov approach to output feedback control using second-order sliding modes. IMA J Math Control Inf 2012; 29(3): 291–308.

80.

Moreno

Osorio

. A Lyapunov approach to second-order sliding mode controllers and observers. In: Proceedings of the IEEE conference on decision and control, 2008, pp. 2856–2861. New York: IEEE.

81.

Moreno

. A linear framework for the robust stability analysis of a generalized super-twisting algorithm. In: 2009 6th International conference on electrical engineering, computing science and automatic control, CCE 2009, 2009. New York: IEEE.