Sampled-data extended state observer-based backstepping control of two-link flexible manipulator

Abstract

Currently, space robots such as planetary robots and flexible-link manipulators (FLMs) are finding specific applications to reduce the cost of launching. However, the structural flexible nature of their arms and joints leads to errors in tip positioning owing to tip deflection. The internal model uncertainties and disturbance are the key challenges in the development of control strategies for tip-tracking of FLMs. To deal with these challenges, we design a tip-tracking controller for a two-link flexible manipulator (TLFM) by developing a sampled-data extended state observer (SD-ESO). It is designed to reconstruct uncertain parameters for accurate tip-tracking control of a TLFM. Finally, a backstepping (BS) controller is designed to attenuate the estimation error and other bounded disturbances. Convergence and stability of the proposed control system are investigated by using Lyapunov theory. The benefits (control performance and robustness) of the proposed SD-ESO-based BS controller are compared with other similar approaches by pursuing both simulation and experimental studies. It is observed from the results obtained that SD-ESO-based BS Controller effectively compensates the deviation in tip-tracking performance of TLFM due to non-minimum phase behavior and model uncertainties with an improved transient response.

Keywords

Backstepping control sampled-data extended state observer tip-tracking control two-link flexible manipulator

Introduction

Nowadays, space robots, planetary robots and flexible robots have been developed and utilized in satellite, aerospace and space industries. This flexible-link manipulator (FLM) is very advantageous in terms of lightweight, reduced overall cost and low energy consumption in transportation, the higher payload carrying capacity, more maneuverable and operational speed. However, the structural flexible nature of the FLM’s arms and joints lead to inaccuracy in tip positioning as compared with a rigid manipulator (Subudhi and Pradhan, 2016).

Over the last three decades, control of FLM has been an active research area. A good review on control of FLMs is found in Lochan et al. (2016); Sayahkarajy et al. (2016); and Castillo-Berrio et al. (2015). Owing to a nonlinear and non-minimum phase behavior coupled with model uncertainties and disturbances, control of such manipulator posses a challenging task.

In the flexible manipulator, improving tip-tracking performance and an accurate dynamic model of the system is necessary, which implies that dynamics of both rigid and flexible nature of the links should be represented accurately (Tokhi et al., 2000). It may be noted that due to distributed link flexure, infinite number of flexible modes are necessary to describe the dynamics of an FLM, that is, its dynamic model is infinitely dimensional( Khairudin et al., 2010). However, it is necessary to obtain approximate dimensional model of FLM for designing a controller with finite dimension.

Dynamics of a two-link flexible manipulator (TLFM) exhibit non-minimum phase behavior and design of control strategy for tip-tracking control of this system is very difficult due to unstable internal dynamics (Chen et al., 2012). Many control schemes have been developed to solve this difficulty of tip-tracking control of the TLFM. Generally, tip-tracking control schemes are divided into model-based and non-model- based methods. Non-model-based methods are simpler and easier to implement than the model-based methods. In recent years, many non-model-based control methods have been developed to solve the tip-tracking control problem of the TLFM, for example, composite control (Lochan et al., 2018a), sliding mode control (Lochan et al., 2018b), robust control (Mohamed et al., 2016), adaptive control (Pradhan and Subudhi, 2012, 2014; Subudhi and Pradhan, 2016), intelligent control (Agee et al., 2014; Cheng, 2015), observer-based control (Zhang et al., 2017), and so forth. The above controllers handle tip-tracking issue under both kinematic and dynamics uncertainties. However, owing to the link flexure, the dynamics of FLM is a distributed parameter system, that is, an infinite number of flexible modes are needed for the modeling. However, for realization of the controller, a finite dimensional model is necessary. Hence, it is necessary to truncate the higher order flexible modes. However, this neglect of higher order flexible modes gives rise to uncertainties in the dynamics. We consider these as disturbances and un-modeled dynamics that are the major source of uncertainty in the dynamics of TLFM. Hence, designing a suitable controller for real-time application for a TLFM that exhibits high nonlinearity, uncertainties and disturbances is still a challenging task in real-time, which motivates us to attempt this in this paper.

In order to handle the model uncertainties and disturbances associated with TLFM, a linear quadratic regulator (LQR) controller (model-based) has been used by Xu and Ritz (2009) and Etxebarria et al. (2005) for the flexible dynamics. However, in Etxebarria et al.’s (2005) technique, a larger overshoot occurs that causes inaccurate tip-tracking performance in FLM. Furthermore, backstepping (BS) control (non-model based) is adopted by many researchers, to handle uncertainties and disturbances in nonlinear systems. BS control is one of the most powerful control techniques that ensures the stability of the system as each control law is derived by employing Lyapunov criterion. In addition, the backstepping control approach shows several advantages such as follows. In the situation when uncertainty information is unavailable, a controller can be designed to reduce on-line calculation (Jamil et al., 2014). The backstepping approach is also used for tip-tracking control of a single-link flexible robot in Zhu et al. (1997). In Jong and Lee (1998), the backstepping control provides excellent tracking of tip position for a flexible joint manipulator. A robust regulator for accurate tip-tracking for flexible joint robots using integrator backstepping approach is proposed by Abouelsoud (1998). Adaptive sliding control for a single-link flexible-joint robot with external noise and modeling uncertainties is developed in Huang and Chen (2004). Backstepping control using a genetic algorithm for single link flexible joint manipulator is proposed in Sahab and Modabbernia (2011). Backstepping control is applied for trajectory tracking control of TLFM in Lochan and Roy (2016). Control of two-link flexible manipulator using a backstepping controller with an extended state observer has been presented in Yang et al. (2015). However, most of the literature deal with only tip-tracking control of a single-link flexible manipulator (SLFM). Only a few works have been reported on trajectory tracking control of TLFMs. In view of successful applications of backstepping control of SLFM, we extend the approach to design a tip-position tracking controller of a TLFM in this work.

Further, nonlinearities, unmodeled dynamics, unmeasurable noise and so forth, are most often considered as major uncertainties in TLFM. Due to these uncertainties, partial state vector or a non-uniform signal is usually received by the controller (Tian et al., 2016). In view of estimating uncertain parameters, a state observer needs to be designed. The extended state observer (ESO) is designed to estimate uncertainties by lumping these into as an extended state. In Yang et al. (2015) and Sahu and Patra (2016), a continuous-time ESO has been developed on the basis of unmodeled dynamics of FLM. However, uncertain parameters approximated by a continuous-time ESO are more complex and erroneous. Moreover, an ESO often needs to be implemented digitally in various computer-based control applications. Hence, it would be appropriate to use digitally implemented ESO. The digital implementation of ESO has also received considerable attention. In Miklosovic et al. (2006), various digital approximation techniques of ESO are studied and compared. The discrete-time ESO is implemented for an uncertain single-input-single-output (SISO) system in Huang et al. (2015). The discrete ESO-based repetitive control for servo motor is presented in Sayem et al. (2016). The discrete ESO has been developed to approximate the uncertain parameters for a servo system (Wang et al., 2017) and non-linear system (Tian et al., 2016). In most of the existing works, discrete ESO is designed based on a direct discretization of the plant model. Inspired by this line of research, we propose design of a sampled-data extended state observer (SD-ESO) to estimate the uncertain parameters that will be used for designing tip-tracking controller for a TLFM to achieve accuracy.

In this paper, we intend to develop a unified tip-tracking control algorithm for flexible manipulator by employing backstepping control technique together with designing an SD-ESO for estimating disturbances. The contribution of the paper is as follows. A SD-ESO is developed for simple, fast and accurate estimation of uncertainties present in TLFM ensuring its convergence, and SD-ESO-based BS controller is designed for tip-tracking control of TLFM. Also, the tip-tracking performance of the proposed SD-ESO-based BS controller is compared with that of linear quadratic regulator (LQR) controller (model based) (Xu and Ritz, 2009), BS controller (non-model based) (Lochan and Roy, 2016) and ESO-based BS controller (Yang et al., 2015).

The rest of the paper is organized as follows. In the next section, the dynamic model of TLFM is provided, and the control problem is formulated. Then, the design and convergence analysis of the proposed SD-ESO-based BS controller are presented. Simulation and experimental results with discussion are presented, followed by conclusions.

Preliminaries and problem formulation

Dynamic model of TLFM

In a FLM, due to distributed link flexure, the positioning and tracking of the tip are very difficult. We assume that motion of the TLFM in the horizontal plane. The links are assumed to have uniform material properties and constant cross-sectional area (Morris and Madani, 1997). The schematic diagram of a TLFM is shown in Figure 1, where $τ_{i}$ represents the actual applied torque of the $i^{th}$ link, $θ_{i}$ represents the joint angle of the $i^{th}$ joint, and $y_{i} (l_{i}, t)$ denotes the deflection along $i^{th}$ link. The tip of the TLFM is attached with a payload of mass.

Figure 1.

Schematic diagram of a TLFM.

The dynamics of a flexible manipulator system exhibits a non-minimum phase characteristics when the tip position is taken as the output. The actual output vector $y_{pi}$ is the output for the $i^{th}$ link. Hence, the elastic displacement can be redefined as

\begin{matrix} y_{pi} = θ_{i} + (\frac{y_{i} (l_{i}, t)}{l_{i}}) \end{matrix}

(1)

where, $l_{i}$ is the $i^{th}$ link length.

The dynamics of TLFM can be derived by applying the Euler-Lagrangian principle using Assumed Mode Method (AMM). The dynamics of TLFM given in Pradhan and Subudhi (2014) is given by

\begin{matrix} M (θ_{i}, δ_{i}) [\begin{matrix} {\overset{\cdot\cdot}{θ}}_{i} \\ {\overset{\cdot\cdot}{δ}}_{i} \end{matrix}] + [\begin{matrix} c_{1} (θ_{i}, δ_{i}, {\overset{\cdot}{θ}}_{i}, {\overset{\cdot}{δ}}_{i}) \\ c_{2} (θ_{i}, δ_{i}, {\overset{\cdot}{θ}}_{i}, {\overset{\cdot}{δ}}_{i}) \end{matrix}] + K [\begin{matrix} 0 \\ δ_{i} \end{matrix}] + D [\begin{matrix} 0 \\ {\overset{\cdot}{δ}}_{i} \end{matrix}] + [\begin{matrix} Δ d \\ 0 \end{matrix}] = [\begin{matrix} τ_{i} \\ 0 \end{matrix}] \end{matrix}

(2)

In (2), M is a positive definite symmetric inertia matrix, $c_{1}$ and $c_{2}$ are the Coriolis and Centrifugal force vectors, respectively, K is the stiffness matrix, D is the Damping Matrix, and $Δ d \in ℜ^{n}$ is the disturbance or un-modeled dynamics. The details of these matrices are given in the Appendix.

The dynamics of TLFM (2) can be rewritten in state space form as follows

\begin{matrix} \begin{matrix} \overset{\cdot}{x} (t) = Ax (t) + Bu (t) \\ y (t) = Cx (t) \end{matrix} \end{matrix}

(3)

where, $x (t) \in ℜ^{2 n}$ , $y (t) \in ℜ^{m}$ and $u (t) \in ℜ^{n}$ denote state vector, tip position and control input, respectively. A, B and C are matrices with appropriate dimensions.

\begin{array}{l} A = [\begin{matrix} 0_{m} & I_{m} \\ - M^{- 1} K & - M^{- 1} (D + Δ d) \end{matrix}], B = [\begin{matrix} 0_{m \times 1} \\ M^{- 1} e_{1} \end{matrix}], \\ C = [\begin{matrix} 0_{m} & I_{m} \end{matrix}], u = {[\begin{matrix} τ_{i} & 0 \end{matrix}]}^{T}, x = {[\begin{matrix} θ_{i} & {\dot{θ}}_{i} & δ_{i} & {\dot{δ}}_{i} \end{matrix}]}^{T} . \end{array}

The discrete time approximation of $x (t)$ can be obtained using forward difference approximation as

\begin{matrix} \begin{matrix} \overset{\cdot}{x} (t) ≅ \frac{1}{T} [x (kT + T) - x (kT)] \end{matrix} \end{matrix}

(4)

where, T is the sampling time and $x (kT)$ is the state vector. Similarly, the discrete-time representation of (3) using (4) can be expressed as

\begin{matrix} \begin{matrix} x (k + 1) = A_{d} x (k) + B_{d} u (k) \\ y (k) = C_{d} x (k) \end{matrix} \end{matrix}

(5)

where, $x (k)$ , $u (k)$ and $y (k)$ are a state vector, input and output, respectively, $A_{d} = e^{AT}$ , $B_{d} = \int_{0}^{T} e^{As} dsB$ and $C_{d} = C$ . The dynamics of the TLFM (3) and (5) are used for designing the controller. The control problem is formulated as follows.

The structure of the proposed BS with SD-ESO-based control is shown in Figure 2. Given the nonlinear FLM system with uncertainties, the objective is to find a control input $u (k)$ such that, the output $y (k)$ should track the desired output trajectory, $y_{d} (k)$ .

Figure 2.

Structure diagram of the proposed BS with SD-ESO.

It is necessary to design a control scheme for a closed-loop system (3) such that the output trajectory $y (k)$ should track the desired output trajectory $y_{d} (k)$ as close as possible.

Sampled-data ESO design and convergence analysis

The dynamics of the TLFM (2) is used in the design of the SD-ESO. Here, SD-ESO is developed to estimate uncertainty such as disturbance and un-modeled dynamics. In this section, the subscripts of the variables in (2) are neglected for simplification. From the Appendix, (2) can be rewritten as

\begin{matrix} M_{11} \overset{\cdot\cdot}{θ} + M_{12} \overset{\cdot\cdot}{δ} + c_{11} \overset{\cdot}{θ} + c_{12} \overset{\cdot}{δ} + Δ d = τ \end{matrix}

(6)

\begin{matrix} M_{21} \overset{\cdot\cdot}{θ} + M_{22} \overset{\cdot\cdot}{δ} + c_{21} \overset{\cdot}{θ} + c_{22} \overset{\cdot}{δ} + K δ + D \overset{\cdot}{δ} = 0 \end{matrix}

(7)

From (7), one obtains

\begin{matrix} \overset{\cdot\cdot}{δ} = - M_{22}^{- 1} [M_{21} \overset{\cdot\cdot}{θ} + c_{21} \overset{\cdot}{θ} + c_{22} \overset{\cdot}{δ} + K δ + D \overset{\cdot}{δ}] \end{matrix}

(8)

Substituting (8) into (6), gives

\begin{matrix} \begin{matrix} (M_{11} - M_{12} M_{22}^{- 1} M_{21}) \overset{\cdot\cdot}{θ} + (c_{11} - M_{12} M_{22}^{- 1} c_{21}) \overset{\cdot}{θ} + (c_{12} - M_{12} M_{22}^{- 1} c_{22} \\ - M_{12} M_{22}^{- 1} D) \overset{\cdot}{δ} - M_{12} M_{22}^{- 1} K δ + Δ d = τ \end{matrix} \end{matrix}

(9)

Equation (9) can be written as

\begin{matrix} P \overset{\cdot\cdot}{θ} + Q \overset{\cdot}{θ} + S = τ \end{matrix}

(10)

The discrete-time representation of (10) is rewritten by considering $x_{1} (k) = θ$ , and $x_{2} (k) = \overset{\cdot}{θ}$ as

\begin{matrix} {\begin{matrix} x_{1} (k + 1) = x_{2} (k) \\ x_{2} (k + 1) = P^{- 1} (τ - (Q x_{1} (k) + S)) \end{matrix} \end{matrix}

(11)

In (11), considering $P^{- 1} (Q x_{1} (k) + S)$ as the disturbance and uncertain term, which is assumed as an extra state $x_{3} (k)$ for convenience of SD-ESO design, (11) can be rewritten as

\begin{matrix} {\begin{matrix} y (k) = x_{1} (k) \\ x_{1} (k + 1) = x_{2} (k) \\ x_{2} (k + 1) = P^{- 1} τ + x_{3} (k) \\ x_{3} (k + 1) = x_{3} (k) + w (k) \end{matrix} \end{matrix}

(12)

Using (12), SD-ESO is designed, in the similar manner as adopted for continuous-time ESO development

\begin{matrix} {\begin{matrix} γ_{1} (k) = z_{1} (k) - y (k) \\ z_{1} (k + 1) = z_{2} (k) - α_{1} [z_{1} (k) - y (k)] \\ z_{2} (k + 1) = z_{3} (k) - α_{2} f (γ_{1} (k)) + P^{- 1} τ \\ z_{3} (k + 1) = - α_{3} f (γ_{1} (k)) \end{matrix} \end{matrix}

(13)

where, $z_{1} (k), z_{2} (k)$ and $z_{3} (k)$ are the estimated states of $x_{1} (k), x_{2} (k)$ and $x_{3} (k)$ , respectively. $f (γ_{1})$ is a nonlinear function, $α_{1}, α_{2}$ and $α_{3}$ are observer gains. $γ_{1} (k)$ , $γ_{2} (k)$ and $γ_{3} (k)$ are observation error variables, which can be defined as

\begin{matrix} {\begin{matrix} γ_{1} (k) = z_{1} (k) - x_{1} (k) \\ γ_{2} (k) = z_{2} (k) - x_{2} (k) \\ γ_{3} (k) = z_{3} (k) - x_{3} (k) \end{matrix} \end{matrix}

(14)

The observer gains $α_{1}, α_{2}$ and $α_{3}$ are must be selected such that following conditions are satisfied

\begin{matrix} {\begin{matrix} α_{3} > 0 \\ α_{3} - 2 α_{2} + 2 α_{1} < 2 \\ | α_{3} - α_{2} | < 1 \\ | {(α_{3} - α_{2})}^{2} - 1 | > | α_{1} α_{3} - α_{1} α_{2} - α_{3} + α_{1} | \end{matrix} \end{matrix}

(15)

Assumption 1: The solution of the discrete-time dynamic system, that is, $ϕ_{T}^{x} (k, k_{0}, x_{0})$ is assumed as uniform globally bounded (UGB), If $k \in k_{\infty}$ for all $Δ > 0$ and $k_{0} \geq 0$ , $x (k_{0}) = x_{0}$ , with $| x_{0} | ⩽ Δ$ and $x_{0} \in ℜ^{n}$ it holds that

\begin{matrix} | ϕ_{T}^{x} (k, k_{0}, x_{0}) | ⩽ k (| x_{0} |) \end{matrix}

(16)

for all $k \geq k_{0}$

Theorem 1: Assume that the SD-ESO (13) is applied to the system (11), the observer gains ( $α_{1}$ , $α_{2}$ and $α_{3}$ ) are selected to satisfy (15), and the considered uncertain and disturbance term $x_{3} (k)$ satisfies Assumption 1, then the observation error is UGB (Wang et al., 2017).

Proof: From (12) and (13), the error function can be defined as

\begin{matrix} {\begin{matrix} γ_{1} (k) = z_{1} (k) - y \\ γ_{1} (k + 1) = - α_{1} γ_{1} (k) + γ_{2} (k) \\ γ_{2} (k + 1) = - α_{2} f (γ_{1} (k)) + γ_{3} (k) \\ γ_{3} (k + 1) = - α_{3} γ_{1} (k) + γ_{3} (k) + w (k) \end{matrix} \end{matrix}

(17)

where, $w (k) = x_{3} (k + 1) - x_{3} (k)$

Using (17), the observation error dynamics is given by

\begin{matrix} E (k + 1) = GE (k) + Hw (k) \end{matrix}

(18)

where

G = [\begin{matrix} - α_{1} & 1 & 0 \\ - α_{2} & 0 & 1 \\ - α_{3} & 0 & 1 \end{matrix}], H = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}], E (k) = [\begin{matrix} γ_{1} (k) \\ γ_{2} (k) \\ γ_{3} (k) \end{matrix}]

The eigen polynomial of G is given by

\begin{matrix} | λ I - G | = λ^{3} + (α_{1} - 1) λ^{2} + (α_{2} - α_{1}) λ + (α_{3} - α_{2}) \end{matrix}

(19)

where, $λ$ is the eigenvalue of matrix G. All eigen values $λ_{i} (i ⩽ 3)$ are bounded as $| λ_{i} | < 1$ , if the observer gains $α_{1}$ , $α_{2}$ and $α_{3}$ satisfy (15). The maximum eigen value of of matrix G is defined as $λ_{max} = max {λ_{i} (i ⩽ 3)}$ with $| λ_{\max} | < 1$ .

Define a candidate Lyapunov function as

\begin{matrix} V_{λ} (k) = E^{T} (k) E (k) \end{matrix}

(20)

Then

\begin{matrix} \begin{matrix} Δ V_{λ} (k) = V_{λ} (k + 1) - V_{λ} (k) \\ ⩽ - (1 - λ_{max}^{2}) {‖ E (k) ‖}^{2} + 2 λ_{max} ‖ E (k) ‖ | x_{3} (k) | + {| x_{3} (k) |}^{2} \end{matrix} \end{matrix}

(21)

and $| x_{3} (k) | < μ$ , then (21) becomes

\begin{matrix} Δ V_{λ} (k) < - (1 - λ_{max}^{2}) {‖ E (k) ‖}^{2} + 2 λ_{max} ‖ E (k) ‖ μ + μ^{2} \end{matrix}

(22)

Then, $Δ V_{λ} (k) < 0$ once $‖ E (k) ‖ ⩾ μ / (1 - λ_{max})$ , so $E (k)$ will asymptotically converge to the set $Ω_{1}$ defined by

\begin{matrix} Ω_{1} = {E (k) | | | E (k) | | < \frac{μ}{1 - λ_{max}}} \end{matrix}

(23)

So, the observation error $E (k)$ is UGB.

Convergence analysis of SD-ESO

The Self-Stable Region (SSR) method proposed for the uncertain system in Huang and Han (1999) is used for convergence analysis of the closed-loop error system (17).

Assumption 2: Assume that R is any region in the state space with the origin $(0, 0)$ as its vertex. So, except the origin, there is no equilibrium point in the region R. Then, the region is said to be SSR of the system, if after a finite time, each convergent trajectories, eventually approach to the origin. Assume $R_{1}$ and $R_{2}$ are the two regions of the error system (17) as follows

\begin{matrix} \begin{matrix} R_{1} = {γ_{1} (k), γ_{2} (k), γ_{3} (k) : | h_{2} | < g_{1}} \\ R_{2} = {γ_{1} (k), γ_{2} (k), γ_{3} (k) : | h_{3} | < g_{2}} \end{matrix} \end{matrix}

(24)

where

\begin{matrix} h_{2} = h_{2} (γ_{1} (k), γ_{2} (k)) = γ_{2} (k) - α_{1} γ_{1} (k) + C_{1} g_{1} \\ h_{3} = h_{3} (γ_{1} (k), γ_{2} (k), γ_{3} (k)) \\ = γ_{3} (k) - α_{2} f (γ_{1} (k)) - α_{1} (γ_{2} (k) - α_{1} γ_{1} (k)) \\ + C_{2} g_{2} (γ_{1} (k) + γ_{2} (k)) \end{matrix}

and $g_{1} = g_{1} (γ_{1} (k))$ and $g_{2} = g_{2} (γ_{1} (k), γ_{2} (k))$ are continuous positive definite functions with $g_{1} = 0$ and $g_{2} = 0$ , $C_{1}$ and $C_{2}$ are constant that must satisfy the conditions $C_{1} > 1$ and $C_{2} > 1$ .

Theorem 2: In the error system (17), after a finite time, the trajectories of $γ_{1} (k)$ and $γ_{2} (k)$ converge to the origin, if $(γ_{1} (k), γ_{2} (k)) \in R_{1}$ , that is, the observed states $z_{1} (k)$ and $z_{2} (k)$ converge to the actual system states $x_{1} (k)$ and $x_{2} (k)$ , respectively.

Proof: Letting $V_{h_{2} g_{1}} = (h_{2}^{2} - g_{1}^{2}) / 2$ , for $(γ_{1} (k), γ_{2} (k)) \in R_{1}$ , we get

\begin{matrix} h_{2} - γ_{1} (k) < 0 \end{matrix}

(25)

and

\begin{matrix} - g_{1} - C_{1} g_{1} < γ_{2} (k) - α_{1} γ_{1} (k) < g_{1} - C_{1} g_{1} \end{matrix}

(26)

Defining a continuous positive definite function as $V_{0} = γ_{1}^{2} (k)$ , the derivative of $V_{0}$ is given as

\begin{matrix} {\overset{\cdot}{V}}_{0} = γ_{1} (k) (γ_{2} (k) - α_{1} (γ_{1} (k)) \\ < γ_{1} (k) (g_{1} - C_{1} g_{1}) \\ < - | γ_{1} (k) | (C_{1} - 1) g_{1} \\ < 0 \end{matrix}

(27)

Hence, the trajectories $γ_{1} (k)$ and $γ_{2} (k)$ approach to the origin, that is, $z_{1} (k) \to x_{1} (k)$ and $z_{2} (k) \to x_{2} (k)$ , respectively.

Theorem 3: In the error system (17), after a finite time the trajectories of $γ_{1} (k)$ , $γ_{2} (k)$ and $γ_{3} (k)$ , which remain in $R_{2}$ and outside $R_{1}$ converge to the origin, if the condition holds

\begin{matrix} g_{2} > [\frac{{(C_{1} + 1)}^{2}}{(C_{2} - 1)}] | \frac{d g_{1}}{d γ_{1}} | | h_{2} | \end{matrix}

(28)

that is, the observed states $z_{1} (k), z_{2} (k)$ and $z_{3} (k)$ converge to the actual states $x_{1} (k), x_{2} (k)$ and $x_{3} (k)$ , respectively.

Proof: Letting $V_{h_{3} g_{2}} = (h_{3}^{2} - g_{2}^{2}) / 2$ , for $(γ_{1} (k), γ_{2} (k), γ_{3} (k)) \in R_{2}$ , we get

\begin{matrix} h_{3} < g_{2} \end{matrix}

(29)

Taking the time derivative of $V_{h_{2} g_{1}}$ , we obtain

\begin{matrix} {\overset{\cdot}{V}}_{h_{2} g_{1}} = h_{2} (h_{2} - C_{2} g_{2} + C_{1} \frac{d g_{1}}{d γ_{1}} h_{2}) - g_{1} \frac{d g_{1}}{d γ_{1}} h_{2} \end{matrix}

(30)

As $V_{h_{3} g_{2}} < 0, V_{h_{2} g_{1}} \geq 0$ , that is, there exists

\begin{matrix} \begin{matrix} {\overset{\cdot}{V}}_{h_{2} g_{1}} \leq - | h_{2} | ((C_{2} - 1) g_{2} + C_{1} | h_{2} |) + g_{1} | \frac{d g_{1}}{d γ_{1}} h_{2} | \\ \leq - | h_{2} | ((C_{2} - 1) g_{2} + h_{2}^{2} {(C_{1} + 1)}^{2}) | \frac{d g_{1}}{d γ_{1}} h_{2} | < 0 \end{matrix} \end{matrix}

(31)

So, the trajectories of $γ_{1}$ and $γ_{2}$ converge into region $R_{1}$ for the error system (29), if $γ_{1} (k) \to 0$ and $γ_{2} (k) \to 0$ , then we have $γ_{3} (k) \to 0$ , when $z_{1} (k) \to x_{1} (k), z_{2} (k) \to x_{2} (k)$ and $z_{3} (k) \to x_{3} (k)$ eventually.

Through the analysis of Theorem 2 and Theorem 3, $R_{2}$ is the self-stable region of the error system (17), that is, after a certain time ( $γ_{1}$ , $γ_{2}$ and $γ_{3}$ ) approach to the origin. Therefore, if parameters $α_{1,} α_{2,}$ and $α_{3}$ are properly chosen such that (15) is satisfied, then the observed states $z_{1} (k), z_{2} (k)$ and $z_{3} (k)$ converge to $x_{1} (k), x_{2} (k)$ and $x_{3} (k)$ , respectively.

Design of BS controller and stability analysis

This section presents the design of three-stage BS controller for accomplishing effective tip-tracking of TLFM. In the BS design, the uncertain parameters of (11) are identified repeatedly as pseudo-control inputs of lower dimension subsystem of the entire system. The structure of the proposed three-stage backstepping control algorithm is shown in Figure 3.

Figure 3.

Structure of three-step BS control algorithm.

The control objective is set as follows. The states, $z_{1} (k)$ , $z_{2} (k)$ and $z_{3} (k)$ should track $x_{1} (k)$ , $x_{2} (k)$ and $x_{3} (k)$ , respectively, that is

z \to x

(32)

where

\begin{matrix} \begin{matrix} x = {[\begin{matrix} x_{1} (k) & x_{2} (k) & x_{3} (k) \end{matrix}]}^{T} \\ z = {[\begin{matrix} z_{1} (k) & z_{2} (k) & z_{3} (k) \end{matrix}]}^{T} \end{matrix} \end{matrix}

that is, $γ_{1} (k) \to 0, γ_{2} (k) \to 0$ and $γ_{3} (k) \to 0$ .

To achieve the above control objective, the error variables can be defined as

\begin{matrix} {\begin{matrix} ε_{1} (k) = x_{1} (k) - x_{1 d} (k) \\ \begin{matrix} ε_{2} (k) = x_{2} (k) - σ_{1} (k) \\ ε_{3} (k) = x_{3} (k) - σ_{2} (k) \end{matrix} \end{matrix} \end{matrix}

(33)

where, $x_{1 d} (k)$ is the control input, $σ_{1} (k)$ and $σ_{2} (k)$ are virtual control variables. From (33), one obtains

\begin{matrix} {\begin{matrix} ε_{1} (k + 1) = x_{1} (k + 1) - x_{1 d} (k + 1) \\ \begin{matrix} ε_{2} (k + 1) = x_{2} (k + 1) - σ_{1} (k + 1) \\ ε_{3} (k + 1) = x_{3} (k + 1) - σ_{2} (k + 1) \end{matrix} \end{matrix} \end{matrix}

(34)

The design steps for BS controller for the system (12) are given below.

Step 1: For $ε_{1} (k) \to 0$ , Define a candidate Lyapunov function as

\begin{matrix} V_{1} (k) = \frac{1}{2} ε_{1}^{2} (k) \end{matrix}

(35)

Then

\begin{matrix} \begin{matrix} V_{1} (k + 1) = ε_{1} (k) ε_{1} (k + 1) \\ = ε_{1} (k) σ_{1} (k) + ε_{1} (k) ε_{2} (k) - ε_{1} (k) x_{1 d} (k + 1) \end{matrix} \end{matrix}

(36)

Virtual control is designed as $σ_{1} (k) = x_{1 d} (k + 1) - k_{1} ε_{1} (k); k_{1} > 0$ is the constant design factor. Substituting $σ_{1} (k)$ into (36) gives

\begin{matrix} V_{1} (k + 1) = - k_{1} ε_{1}^{2} (k) + ε_{1} (k) ε_{2} (k) \end{matrix}

(37)

If the $ε_{2} (k) = 0$ holds, then the first subsystem is stable.

Step 2: For $ε_{2} (k) \to 0$ and $ε_{1} (k) \to 0$ , Define a candidate Lyapunov function as

\begin{matrix} V_{2} (k) = V_{1} (k) + \frac{1}{2} ε_{2}^{2} (k) \end{matrix}

(38)

Then

\begin{matrix} \begin{matrix} V_{2} (k + 1) = - k_{1} ε_{1}^{2} (k) + ε_{1} (k) ε_{2} (k) + ε_{2} (k) ε_{2} (k + 1) \\ = - k_{1} ε_{1}^{2} (k) + ε_{2} (k) (P^{- 1} τ + x_{3} (k) + ε_{1} (k) - σ_{1} (k)) \end{matrix} \end{matrix}

(39)

To realize $V_{2} (k + 1) < 0$ , assume

ε_{1} (k) + x_{3} (k) + P^{- 1} τ - σ_{1} (k + 1) = - k_{2} ε_{2} (k), k_{2} > 0

Then

\begin{matrix} V_{2} (k + 1) = - k_{1} ε_{1}^{2} (k) - k_{2} ε_{2}^{2} (k) \end{matrix}

(40)

Step 3: For $ε_{3} (k) \to 0$ , $ε_{2} (k) \to 0$ and $ε_{1} (k) \to 0$ , Lyapunov candidate function is defined as

\begin{matrix} V_{3} (k) = V_{2} (k) + \frac{1}{2} ε_{3}^{2} (k) \end{matrix}

(41)

Then

\begin{matrix} \begin{matrix} V_{3} (k + 1) = - k_{1} ε_{1}^{2} (k) + ε_{1} (k) ε_{2} (k) + ε_{2} (k) (x_{3} (k) + P^{- 1} τ - σ_{1} (k + 1)) + ε_{3} (k) ε_{3} (k + 1) \\ = - k_{1} ε_{1}^{2} (k) + ε_{2} (k) (ε_{1} (k) + ε_{3} (k) + σ_{2} (k) + P^{- 1} τ - σ_{1} (k + 1)) + ε_{3} (k) ε_{3} (k + 1) \end{matrix} \end{matrix}

(42)

Virtual control is designed as

\begin{matrix} σ_{2} (k) = σ_{1} (k + 1) - ε_{1} (k) - ε_{3} (k) - P^{- 1} τ \\ - k_{2} ε_{2} (k); k_{2} > 0 \end{matrix}

(43)

is the constant design factor. Rearranging (42) using (43) gives

\begin{matrix} \begin{matrix} V_{3} (k + 1) = - k_{1} ε_{1}^{2} (k) + ε_{2} (k) (ε_{1} (k) + σ_{2} (k) + P^{- 1} τ - σ_{1} (k + 1)) + ε_{2} (k) ε_{3} (k) \\ + ε_{3} (k) ε_{3} (k + 1) \\ = - k_{1} ε_{1}^{2} (k) - k_{2} ε_{2}^{2} (k) + ε_{3} (k) (ε_{2} (k) + x_{3} (k) + w (k) - σ_{2} (k + 1)) \end{matrix} \end{matrix}

(44)

To realize $V_{3} (k + 1) <$ 0, assume $ε_{2} (k) + x_{3} (k) + w (k) - σ_{2} (k + 1) = - k_{3} ε_{3} (k), k_{3} > 0$ , Then

\begin{matrix} V_{3} (k + 1) = - k_{1} ε_{1}^{2} (k) - k_{2} ε_{2}^{2} (k) - k_{3} ε_{3}^{2} (k) \end{matrix}

(45)

from (43), control law can be expressed as

\begin{matrix} τ = P (σ_{2} (k) - σ_{1} (k + 1) + ε_{1} (k) + k_{2} ε_{2} (k)) \end{matrix}

(46)

assume $ε_{3} (k) \approx 0$ so that SD-ESO can approximate the system state efficiently. If the value of design factor $k_{1}$ , $k_{2}$ and $k_{3}$ will be chosen such that, $V_{3} (k + 1) < 0$ can be easily realized.

Stability analysis

To prove the boundedness of closed-loop system, the following assumption and theorems are useful.

Assumption 3: The tracking error variables (33) is bounded.

Theorem 4: Assume that control law (46) is applied to the system (11) and the design factor is chosen such that $| k_{2} | < 1$ , then the tip-tracking error is considered as UGB.

Proof: By applying (46) into (11), the closed-loop system can be obtained as

\begin{matrix} ε (k + 1) = k_{2} ε (k) + δ (k) \end{matrix}

(47)

where, $δ (k) = x_{3} (k) - z_{3} (k)$ . Define a candidate Lyapunov function as

\begin{matrix} V_{4} (k) = ε^{2} (k) \end{matrix}

(48)

Then, we have

\begin{matrix} \begin{matrix} Δ V_{4} (k) = V_{4} (k + 1) - V_{4} (k) \\ = - (1 - {k_{2}}^{2}) ε^{2} (k) + δ^{2} (k) + 2 k_{2} ε (k) δ (k) \\ \leq - (1 - k_{2}^{2}) {| ε (k) |}^{2} + {| δ (k) |}^{2} + 2 | k_{2} | | ε (k) | | δ (k) | \end{matrix} \end{matrix}

(49)

and, $| δ (k) | = | x_{3} (k) - z_{3} (k) | < μ$

\begin{matrix} Δ V_{4} (k) \leq - (1 - {k_{2}}^{2}) {| ε (k) |}^{2} + μ^{2} + 2 μ | k_{2} | | ε (k) | \end{matrix}

(50)

Then $Δ V_{4} (k)$ remains bounded as long as

\begin{matrix} | ε (k) | \geq \frac{μ}{1 - | k_{2} |} \end{matrix}

(51)

This means that the tracking error $ε (k)$ is converges to $Ω_{2}$ , if it remains outside the given set $Ω_{2}$ .

\begin{matrix} Ω_{2} = {ε (k) | | ε (k) | < \frac{μ}{1 - | k_{2} |}} \end{matrix}

(52)

So, the obtained tracking error is UGB.

Consider uncertain term $x_{3} (k)$ as the input of closed loop system, also the system states $x_{i} (k); (i \leq 3)$ and observer states $z_{i} (k); (i \leq 3)$ as the state of the closed-loop system. Let

\begin{matrix} \bar{x} = {[\begin{matrix} x_{i} (k) & z_{i} (k) \end{matrix}]}^{T} \end{matrix}

(53)

\begin{matrix} F (\bar{x}) = [\begin{matrix} A_{d} x_{m} (k) + B_{d} u (k) \\ GE (k) + Hw (k) \end{matrix}] \end{matrix}

(54)

The boundedness of TLFM is established by applying the following theorem.

Theorem 5: The closed-loop system (5) under the proposed SD-ESO (13) with BS control law (46) is bounded if the following conditions are satisfied: (i) Assumptions 1, 2 and 3 are satisfied; (ii) observer gain is selected from (15) such that observer error system (17) is UGB; (iii) the control parameters $k_{i} (i \leq 3)$ are selected such that $V_{3} (k + 1) < 0$ is satisfied.

Proof: Taking into account SD-ESO dynamics (13) and control law (46) into observer error system (17), an augmented closed-loop system is obtained described by

\begin{matrix} {\begin{matrix} x_{i} (k + 1) = X (x (k), γ (k)) \\ z_{i} (k + 1) = Z (z (k), γ (k), w (k)) \end{matrix} \end{matrix}

(55)

Combining (53), (54) and (55), the closed loop system is given

\begin{matrix} \overset{\cdot}{\bar{x}} = F (\bar{x}) + [\begin{matrix} 0 \\ H \end{matrix}] w (k) \end{matrix}

(56)

with the condition given in $(i)$ , $(ii)$ and $(iii)$ , it can be shown that $\overset{\cdot}{\bar{x}} = F (\bar{x})$ is bounded.

Simulation results

The simulations have been carried out in MATLAB/SIMULINK® simulation platform to verify the performance of the proposed SD-ESO-based BS controller applied to TLFM. The physical parameters of TLFM considered for simulation and experimental studies of TLFM are given in Table 1. The Mean Absolute Errors (MAE) and the Mean Square of Errors (MSE) are used as quantitative measures for comparing the performance of the proposed scheme, which is defined in (57) and (58), respectively. Also, Standard Error of Mean (SEM) (59) is used for statistical verification of estimation errors.

\begin{matrix} MAE = \frac{1}{N} \sum_{i = 1}^{N} | ε (t) | \end{matrix}

(57)

\begin{matrix} MSE = \frac{1}{N} \sum_{i = 1}^{N} {(ε (t))}^{2} \end{matrix}

(58)

\begin{matrix} SEM = σ / \sqrt{N} \end{matrix}

(59)

Table 1.

Physical parameter of link-1 and link-2.

Parameter	Link-1	Link-2
Mass of link	$m_{1} = 0.15268 kg$	$m_{2} = 0.0535 kg$
Link length	$L_{1} = 0.201 m$	$L_{2} = 0.2 m$
Armature resistance	$R_{m 1} = 11.5$ $Ω$	$R_{m 2} = 2.32$ $Ω$
Armature inductance	$L_{m 1} = 3.16 mH$	$L_{m 2} = 0.24 mH$
Gear ratio	$K_{g 1} = 100$	$K_{g 2} = 50$
Torque constant	$K_{t 1} = 0.0119 Nm / A$	$K_{t 2} = 0.0234 Nm / A$
Back-EMF constant	$K_{m 1} = 0.119 V . s / rad$	$K_{m 2} = 0.0234 V . s / rad$
Torsional stiffness	$K_{s 1} = 22 Nm / rad$	$K_{s 2} = 2.5 Nm / rad$
Young’s modulus	$E_{1} = 2.0684 \times 10^{11} N / m^{2}$	$E_{2} = 2.0684 \times 10^{11} N / m^{2}$
Rotor MI*	$J m_{1} = 6.28 \times 10^{- 6} kg . m^{2}$	$J m_{2} = 1.03 \times 10^{- 6} kg . m^{2}$
Drive MI*	$J_{1} = 7.361 \times 10^{- 4} kg . m^{2}$	$J_{2} = 44.55 \times 10^{- 6} kg . m^{2}$
Link MI*	$I_{1} = 0.17043 kg . m^{2}$	$I_{2} = 0.0064387 kg . m^{2}$

MI: Moment of Inertia.

Model validation

In this section, the simulations are carried to validate the dynamic model of the studied TLFM given in (3). Open loop responses of the TLFM are obtained through both simulation and experiments. Sinusoidal torque inputs with 20° amplitude and 0.1 Hz frequency are applied to the TLFM model. Figure 4 and Figure 5 show the simulation and experimental hub angle responses, and error between these for link-1 and link-2, respectively. MAE and MSE are calculated from error curves obtained from Figure 4 and Figure 5, and the same are listed in Table 2. It is observed from Figure 4 that the hub angle response of link-1 has a maximum and minimum joint position amplitudes are 20° and −10°, respectively. Link-2 hub angle response is shown in Figure 5, from which it is observed that the hub angle response has a maximum amplitude of 20° and minimum amplitude of −19°. It is also observed from Figure 4 and Figure 5 that the error between the simulation and experimental response of the dynamic model is maintained within a small range in dynamic process. The results indicate that the derived dynamic model of TLFM proves to be an efficient model to describe the physical TLFM.

Figure 4.

Link-1 hub angle response and error.

Figure 5.

Link-2 hub angle response and error.

Table 2.

MAE and MSE for hub angle response.

Response	Link	Simulation		Experimental
Response	Link	MAE	MSE	MAE	MSE
Hub angular	Link-1	0.9831	1.1520	0.7550	0.6460
Response $(\deg)$	Link-2	0.4696	0.2884	0.9924	1.1526

Tip-tracking performance

Sinusoidal signals given in (60) are considered as desired trajectories for both the links to investigate the tip-tracking performance. Table 3 listed the control parameters with their values.

θ_{di} (t) = A \sin (0.2 π t)

(60)

where, $θ_{di} (t) = {[\begin{matrix} θ_{d 1} & θ_{d 2} \end{matrix}]}^{T}$ , $θ_{d} (0) = {0, 0}$ are the initial positions of the links and $A = 20^{°}$ is the joint position amplitude for both the link.

Table 3.

Controller parameters for different controllers.

Controller	Control law	Control parameter	Control parameter values
			$α_{1} = diag [0.78, 0.34];$
SD-ESO	Eq. (17)	$α_{1}, α_{2}, α_{3}$	$α_{2} = diag [0.9, 0.48];$
			$α_{3} = diag [1.72, 0.68]$
			$k_{1} = diag [10, 10];$
BS	Eq. (45, 46)	$k_{1}, k_{2}, k_{3}$	$k_{2} = diag [17.2, 11]$ ;
			$k_{3} = diag [18.7, 9]$
			$β_{1} = diag [5, 120]$ ;
ESO	Eq. (7)	$β_{1}, β_{2}, β_{3}$	$β_{2} = diag [1000, 1000];$
	Yang et al. (2015)		$β_{3} = diag [10, 10]$
LQR	Eq. (22)	$K_{1}, K_{2}$	$K_{1} = [15.81; - 9.22; 1.68; 0.8];$
	Xu and Ritz (2009)		$K_{2} = [14.14; - 6.65; 0.93; 0.06]$

The tip-tracking performance of link-1 is shown in Figure 6. Similarly, Figure 7 shows the tip-tracking performance of link-2. Quantitative metrics (MAE and MSE) are calculated form the response curves obtained from simulation studies of LQR, BS, BS with ESO and BS with SD-ESO, and the same are listed in Table 4 and Table 5. It is observed from Table 4 and Table 5 that MAE and MSE for BS with SD-ESO are lower than that obtained for BS with ESO.

Figure 6.

Tip-tracking and estimation of link-1.

Figure 7.

Tip-tracking and estimation of link-2.

Table 4.

MAE (simulation result).

Response	Link	LQR	BS	BS with ESO	BS with SD-ESO
Hub angle	Link-1	2.4297	2.0515	1.1745	0.8148
Response $(\deg)$	Link-2	2.3550	2.0282	1.1949	0.8505
Tip acceleration	Link-1	1.0404	0.0600	0.0472	0.0447
Response $(m / s^{2})$	Link-2	0.898	0.0865	0.0320	0.0212

Table 5.

MSE (simulation result).

Response	Link	LQR	BS	BS with ESO	BS with SD-ESO
Hub angular	Link-1	9.6921	7.0069	1.9839	1.0441
Response $(\deg)$	Link-2	10.6587	7.3756	2.2951	1.2659
Tip acceleration	Link-1	1.8692	0.0125	0.0055	0.0043
Response $(m / s^{2})$	Link-2	0.0120	0.0100	0.0017	0.0008

Table 6.

Tip-tracking (estimation) error (simulation result).

Response	Link	LQR	BS	BS with ESO	BS with SD-ESO
Hub angular	Link-1	0.0535	0.0455	0.0242	0.0176
Response $(\deg)$	Link-2	0.0563	0.0468	0.0261	0.0193
Tip acceleration	Link-1	0.0235	0.0019	0.0013	0.0011
Response $(m / s^{2})$	Link-2	0.0019	0.0017	0.0007	0.0005

Figure 8 shows the tip-tracking error profiles for link-1, from which it is observed that there is a maximum tip deflection error of 6.5° in case of LQR, 5.4° in case of BS controller, 2.7° in case of BS with ESO, but BS with SD-ESO yields the minimum tip deflection error, that is, 1.6°. The tip acceleration error of link-1 is 4.6 $m / s^{2}$ for LQR, 0.84 $m / s^{2}$ for BS controller, 0.8 $m / s^{2}$ for BS with ESO, and yields the minimum tip acceleration error of 0.33 $m / s^{2}$ for the proposed BS with SD-ESO.

Figure 8.

Tip-tracking (estimation) error of link-1.

Figure 9 shows the tip-tracking error profiles for link-2. From Figure 9, it is observed that the maximum tip deflection error is 9.2° in case of LQR, 7.8° in case of BS controller, 3.9° in case of BS with ESO, whereas in case of BS with SD-ESO yields minimum tip deflection error amplitude of 2.5°. The tip acceleration error of link-2 is 0.32 $m / s^{2}$ for LQR, 0.30 $m / s^{2}$ for BS controller, 0.12 $m / s^{2}$ for BS with ESO and 0.055 $m / s^{2}$ for the proposed BS with SD-ESO. Tip-tracking (estimation) error, that is, SEM are calculated statistically from error curves obtained from Figure 8 and Figure 9, and same are listed in Table 6. It is shown from Table 6, that the BS with SD-ESO yields minimum SEM.

Figure 9.

Tip-tracking (estimation) error of link-2.

Figure 10 and Figure 11 show the torque profiles generated by LQR, BS, BS with ESO, and BS with SD-ESO for hub 1 and 2, respectively. It is observed from Figure 10 and Figure 11 that the torque generated by BS with SD-ESO is smooth compared with LQR, BS, and BS with ESO with maximum amplitude of 0.5 Nm and 0.3 Nm, respectively. It is observed from Figure 6 –Figure 11 that, the proposed BS with SD-ESO enables tracking of the desired trajectories with minimum tracking error resulting in better tip-tracking performance.

Figure 10.

Simulation results for torque profiles of hub-1.

Figure 11.

Simulation results for torque profiles of hub-2.

Experimental results

The experimental study was pursued to verify the performance of proposed SD-ESO-based backstepping control of TLFM. The proposed control scheme has been applied to the TLFM available in the Control and Robotics Laboratory, National Institute of Technology, Rourkela. The photograph of the experimental setup is shown in Figure 12. The hardware and software components for the experiment setup are listed in Table 7 and Table 8, respectively. Figure 13 shows the signal flow diagram of the TLFM setup.

Figure 12.

Experimental setup of TLFM.

Table 7.

Hardware components.

Components	Description
Two-link flexible robot manipulator	Two flexible links, digital optical encoders, strain gauge amplifier boards, dc servo motors are the principal components of TLFM.
AMPAQ power module	The TLFM is powered by a two-channel linear current amplifier from the Quanser AMPAQ-series.
Q4 data acquisition board (DAQ)	AMPAQ and TLFM are connected to the Q4 hardware-In-the-Loop (HIL) Board.
Personal computer	Intel(R) core (TM) 2 DUO E7400 processor operates at 2.8-GHz clock cycle with Q4-DAQ.
Q8 terminal board	TLFM, Q8-DAQ, AMPAQ are connected using Q8 terminal board.

Table 8.

Software components.

Components	Description
MATLAB/Simulink®	MATLAB/Simulink® together combine textual and graphical programming to model-based design of TLFM system in a simulation environment.
QUARC® software	Quanser QURAC® software provides tools and capabilities to Simulink that makes the TLFM control easier by generating direct real-time code from the controller designed in Simulink and runs it in real-time on the Windows target.

Figure 13.

Interfacing of signals for the TLFM.

Experimental results of tip-tracking performance of link-1 and link-2 are shown in Figure 14 and Figure 15, respectively. Tip-tracking error profiles of both the links are shown in Figure 16 and Figure 17. The MAE and MSE obtained from experimental studies are listed in Table 9 and Table 10, from which it is observed that both the errors for BS with SD-ESO are lower than that obtained for BS with ESO.

Figure 14.

Experimental results for tip-tracking performance of link-1.

Figure 15.

Experimental results for tip-tracking performance of link-2.

Figure 16.

Tip-tracking (estimation) error of link-1.

Figure 17.

Tip-tracking (estimation) error of link-2.

Table 9.

MAE (experimental result).

Response	Link	LQR	BS	BS with ESO	BS with SD-ESO
Hub angle	Link-1	0.7206	0.7126	0.7045	0.7031
Response $(\deg)$	Link-2	1.0375	1.0264	0.9796	0.9627
Tip acceleration	Link-1	0.5780	0.5618	0.0131	0.5373
Response $(m / s^{2})$	Link-2	0.0605	0.0553	0.0522	0.0594

Table 10.

MSE (experimental result).

Response	Link	LQR	BS	BS with ESO	BS with SD-ESO
Hub angle	Link-1	0.6005	0.5872	0.5734	0.5718
Response $(\deg)$	Link-2	1.2775	1.2539	1.1376	1.1012
Tip acceleration	Link-1	0.6123	0.5283	0.0002	0.4670
Response $(m / s^{2})$	Link-2	0.0061	0.0055	0.0050	0.0059

Figure 16 shows the tip-tracking error profiles obtained from experimental results for link-1, which shows that deflection error is 1.086° for LQR, 1.076° for BS controller, 1.07° for BS with ESO, whereas it is 1.055° in the case of the BS with SD-ESO. The tip acceleration error of link-1 is 0.0004 $m / s^{2}$ for LQR, 0.00027 $m / s^{2}$ for BS controller, 0.00038 $m / s^{2}$ for BS with ESO and 0.00017 $m / s^{2}$ for the proposed BS with SD-ESO.

Figure 17 shows the tip-tracking error profiles obtained from experimental results for link-2, which show that deflection error is 1.78° for LQR, 1.6° for BS controller, 1.59° for BS with ESO, whereas it is 1.51° in the case of the BS with SD-ESO. The tip acceleration error of link-2 is 0.13 $m / s^{2}$ for LQR, 0.12 $m / s^{2}$ for BS controller, 0.1 $m / s^{2}$ for BS with ESO and 0.06 $m / s^{2}$ for the proposed BS with SD-ESO. Tip-tracking (estimation) error, that is, SEM are calculated statistically from error curves obtained from Figure 16 and Figure 17, and the same are listed in Table 11. It is observed from Table 11, that the BS with SD-ESO yields minimum SEM.

Table 11.

Tip-tracking (estimation) error (experimental result).

Response	Link	LQR	BS	BS with ESO	BS with SD-ESO
Hub angular	Link-1	0.0065	0.0055	0.0058	0.0054
Response $(\deg)$	Link-2	0.0569	0.0552	0.0544	0.0187
Tip acceleration	Link-1	0.0189	0.0129	0.0016	0.0014
Response $(m / s^{2})$	Link-2	0.0474	0.0449	0.0312	0.0030

Figure 18 and Figure 19 show the torque profiles generated by LQR, BS, BS with ESO, and BS with SD-ESO for hub 1 and 2, respectively. From Figure 18 and Figure 19, it is observed that the torque generated by BS with SD-ESO is smooth compared with LQR, BS, and BS with ESO with maximum amplitude of 0.05 Nm and 0.015 Nm, respectively. It is observed from the tip-tracking error profiles shown in Figure 16 and Figure 17 that the minimum tip deflection and acceleration error are obtained when the BS with SD-ESO is applied. Also, tip position converges to its desired trajectory with minimum MAE and MSE, with an improved transient response.

Figure 18.

Experimental results for torque profiles of hub-1.

Figure 19.

Experimental results for torque profiles of hub-2.

Remark 1: It is observed from the comparison results (simulation and experiment) that the LQR and BS controllers exhibit unsatisfactory performance in terms of modeling error and uncertainty. Compared with LQR controller, BS controller and ESO-based BS controllers, the proposed SD-ESO-based BS controller has the following advantages: (1) it provides simple, fast and accurate approximation of uncertain parameters of TLFM, (2) it tracks the desired trajectories with minimum tracking error and provide better tip-tracking performance, and (3) it can be have great potential for the real-time application, because the designed digital controller reduces the hardware complexity and increases the flexibility of the hybrid control system.

Conclusions

We proposed a new tip-tracking control scheme called sampled-data extended state observer-based BS controller for a two-link flexible manipulator to address the problem of uncertainties and disturbances in its dynamics. Tip positioning and link vibration are two of the most important issues that arise in TLFM due to non-minimum phase behavior together with model uncertainties and external disturbance. We design a SD-ESO to estimate the above uncertainties and disturbance. We then designed a BS controller to accomplish tip-tracking of a TLFM in face of the parameter uncertainty and disturbance. Simulation and experimental results demonstrate the effectiveness of the proposed SD-ESO-based backstepping control scheme. The proposed SD-ESO-based BS controller exhibits excellent tip-tracking performance of the TLFM with input constraint and external disturbance.

Footnotes

Appendix

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Bidyadhar Subudhi

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