Position control for a two-joint robot finger system driven by pneumatic artificial muscles

Abstract

In this paper, position control is addressed for a two-joint robot finger system driven by pneumatic artificial muscles. It is hard to obtain high precision control for a two-joint robot finger system due to coupling and nonlinearities. A two-input and two-output decoupling problem is solved via active disturbance rejection control without complicated calculations. An extended state observer is designed to estimate the nonlinearities. Furthermore, the stability of the two-joint robot finger system is shown by a back-stepping method. Results from experiments are demonstrated to show the effectiveness of the proposed control approach.

Keywords

Position control pneumatic artificial muscles two-joint finger robot systems decoupling control back-stepping control

Introduction

Anthropomorphic pneumatic robots have drawn increasing attention in the biorobotic fields in recent years. Pneumatic artificial muscles (PAMs) are widely used as muscle-like actuators in hominine robotics due to their large output force-weight/force-volume ratio, high-tension force, long durability, low cost, cleanliness, and compliance (Ahn et al., 2007). There exists an obvious advantage for pneumatic actuators over rigid actuators when robots need to interact with human beings (Park et al., 2002; Shin et al., 2014). However, the compressibility of air and nonlinear elasticity of the rubber tube cause unwanted properties for PAMs, such as nonlinearities and time-varying behaviour (Chang et al., 2011). Hence, it is difficult to get a satisfactory control performance for a system driven by PAMs. Many control methods for systems actuated by PAMs have been designed recently. Some conventional proportion-integration-differentiation (PID) controllers are introduced for certain mechanisms driven by PAMs. A fuzzy self-tuning PID controller is designed to achieve desired dynamic performance targets under different conditions (Nuchkrua and Leephakpreeda, 2013). Moreover, a nonlinear PID controller based on neural networks is introduced for a 2-axes PAM manipulator to address the nonlinearities and time-varying problems in Thank and Ahn (2006). These controllers show a latent conflict between the fast nonlinear dynamics and the slow parameters updating, controllers give certain degrees of compensation for nonlinearities (Shen, 2010). Sliding mode control can guarantee the performance of trajectory tracking for a PAM driven joint system which is robust enough (Xing et al., 2010). A high order sliding mode controller is employed to assure the convergence of the electro-pneumatic systems (Girin et al., 2009). It is shown that sliding mode controllers guarantee a good transitional process and acceptable tracking accuracy, but it is hard to address the chattering in the tracking process (Zhang et al., 2007). Adaptive robust controllers are designed for unmeasurable uncertainties and disturbances (Zhu et al., 2008; Yang et al., 2014b). In Slotine and Li (1985), an adaptive controller is designed to deal with uncertain loads. An adaptive extended active observer is proposed to estimate the force and states in nonlinear robot systems (Chan et al., 2013). A robot arm system is convergent in finite time by adaptive control and sliding mode control methods (Malyuddin et al., 2014). An adaptive recurrent neural networks method is introduced to control the PAM manipulator under load varying situations (Ahn and Ahn, 2009). Controllers based on neural networks are good at fast learning, but they are limited by an inherent slow convergence rate. A decentralized single-input single-output (SISO) controller is introduced for a dexterous robot hand which is implemented in a sophisticated mechanism (Grossard et al., 2015). Therefore, to overcome the drawbacks of the aforementioned control methods, an active disturbance rejection controller is introduced in this paper.

Active disturbance rejection control (ADRC) technology was proposed by Jingqing Han in the late 1980s. A Tracking differentiator (TD), an extended state observer (ESO) and a nonlinear state error feedback controller (NSEFC) make up three main parts of the ADRC framework. There exist two properties in ADRC which are independence on the concrete model and strong anti-disturbance. Hence, it has been employed more and more for nonlinear control systems in recent years (Han, 1998, 2009). ADRC is applied in the fields of aeronautics and astronautics to tackle highly nonlinear, coupling, and uncertainty problems (Yang et al., 2016b). The ADRC approach tracks given input signals swiftly with reducing noise signals by TD (Xia et al., 2007). Severe uncertain nonlinearities are estimated by the ESO of the ADRC approach (Yang et al., 2015). The NSEFC realizes good static and dynamic performances of systems (Yang et al., 2014a). ADRC achieves a good control output performance for nonlinear disturbance problems and coupling problems. There is too much calculation involved when using a control method which needs a concrete system model. Note that ADRC is suitable to solve the multi-input and multi-output decoupling problem without too many complicated calculations. An ADRC controller shows its robustness for estimating and compensating uncertainties (Han, 2008). A nonlinear robust controller for horizontal motor-tendon driven joints and a PID controller for vertical cylinder-tendon driven joints are given in Beytullah et al. (2015) and Takahiro et al. (2005). To the best of our knowledge, there are few researches on two-joint robot finger driven by PAMs in a coupling power transmission way via ADRC, which motivates us to do this work.

In this paper, a two-joint robot finger system with a specific tendon transmission route driven by PAMs is designed. For the robot finger system with 2 degrees of freedom (DOFs), control shortcomings not only lie in the nonlinearities of PAMs but also lie in the unmodelled dynamics and coupling which is caused by tendon transmission route. That is, there exist severe uncertain nonlinearities in the two-joint robot finger system. An active disturbance rejection nonlinear controller is proposed based on a back-stepping method, which is used to solve the nonlinearities in the trajectory tracking process of the 2 DOF two-joint robot finger system. Some comparison results with this proposed method is presented in the experiment section.

Experiment setup and dynamical model

Experiment setup

In this paper, the research aims at emulating the movement of the distal index finger joint (DIFJ) and the proximal index finger joint (PIFJ) of the index finger, which is demonstrated in Figure 1.

Figure 1.

Movement of the PIFJ and DIFJ.

Note that fingers are driven by muscles in the forearm via tendons, which is shown in Figure 2.

Figure 2.

Anatomy of finger tendon.

Tendons are elastic tissues which connect the muscles and bones like a bridge. It has the tenacity to transmit large power. Some tendons are responsible for the extension of fingers, the other tendons control bending. In recent years, many biomechanically-designed robots’ dexterous hands have applied the tendon bionic principle, such as DIST hand (Caffaz and Cannata, 1998), it is demonstrated in Figure 3.

Figure 3.

DIST dexterous hand.

The movement of the two joints is realized by two cables and four PAMs. In the two joint fingers, each joint is driven by a pair of artificial muscles and their cables. When the PAM diastoles, the cables will not diastole. To emulate the synergetic principle of agonist-antagonist muscles of a human being, an artificial muscle will inflate while another artificial muscle deflates in the same pair of actuators. If the joint needs to rotate counter clockwise, the PAMs need to deflate/inflate reversely as in the situation of clockwise. It doesn’t matter if the PAMs are inflated or deflated, the cables always are tensed. Hence, the cables are fixed on each joint by screws so that the cables can not skid on the joints. Figure 4 shows the diagrammatic drawing of the detail of a 2 DOF two-joint tendon driven finger actuated by four PAMs.

Figure 4.

Detail diagram of a 2 DOF two-joint tendon driven finger.

A 3D model is established via SolidWorks software according to the principle of Figure 4, and an experiment setup is manufactured following the 3D model. The 3D model and manufactured setup are depicted in Figure 5.

Figure 5.

Design and realization of two joint mechanism. (a) 3D design model. (b) Realization of 3D design.

The two-joint control experiment facility is depicted in Figure 6. Figure 6(a) demonstrates the work principle of Figure 6(b), which is a picture of the overall facility.

Figure 6.

Principle and actual experiment setup. (a) Principle of the two joint system. (b) Two joint control experiment setup.

It realizes angle measure, control algorithm computation, and desired position tracking by PAMs and cables. Angles are measured by the optical-electricity encoders (OMRON, E6B2-CWZ3E 2000P/R. HENGXIANG, S18-J3N 1600P/R), a counting card (ADVANTECH, PCI-1784U) in the control computer (ADVANTECH, 610H) which receives the angle pulse signals sent by the encoders. Control signals are processed by the computer, sent from a D/A output card (ADVANTECH, PCL-726) to four electric proportional valves (SMC, ITV0050-3MS). These four electric proportional valves regulate the air pressure of the four related PAMs according to the control signals, then the PAMs will drive the joints to desired position.

Dynamic model

Based on Yang et al. (2014c) and Karayiannidis and Doulgeri (2012), the dynamic expression of the 2 DOF two-joint finger system is expressed as

H (θ) \overset{\cdot\cdot}{θ} + C (θ, \overset{\cdot}{θ}) \overset{\cdot}{θ} + g (θ) = τ

(1)

where $θ$ is the joint angle vector with

θ = {[\begin{matrix} θ_{1} & θ_{2} \end{matrix}]}^{T}

in which $θ_{i}$ with $i = 1, 2$ is the angle of the ith joint as depicted in Figure 7. Then $H (θ)$ denotes a $2 \times 2$ inertia matrix as follows

H (θ) = [\begin{matrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{matrix}]

(2)

with

\begin{matrix} h_{11} = I_{zz 1} + I_{zz 2} + m_{1} r_{c 1}^{2} + m_{2} (l_{1}^{2} + r_{c 2}^{2}) \\ + m_{2} l_{1} l_{2} \cos θ_{2} \\ h_{12} = I_{zz 2} + m_{2} r_{c 2}^{2} + 0.5 m_{2} l_{1} l_{2} \cos θ_{2} \\ h_{21} = I_{zz 2} + m_{2} r_{c 2}^{2} + 0.5 m_{2} l_{1} l_{2} \cos θ_{2} \\ h_{22} = I_{zz 2} + m_{2} r_{c 2}^{2} \end{matrix}

where $I_{zzi}$ with $i = 1, 2$ denotes the moment of inertia around the ith joint, $l_{i}$ with $i = 1, 2$ represents the length of the ith link, $r_{ci}$ with $i = 1, 2$ means the centroid vector of the ith link, $r_{ci} = - 0.5 l_{i}$ , $m_{i}$ with $i = 1, 2$ is the mass of the ith link. Note that $C (θ, \overset{\cdot}{θ})$ is a $2 \times 2$ centrifugal and Coriolis matrix as below

C (θ, \overset{\cdot}{θ}) = [\begin{matrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{matrix}]

(3)

with

\begin{matrix} c_{11} = - 0.5 m_{2} l_{1} l_{2} {\overset{\cdot}{θ}}_{2} \sin θ_{2} \\ c_{12} = - 0.5 m_{2} l_{1} l_{2} ({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2}) \sin θ_{2} \\ c_{21} = 0.5 m_{2} l_{1} l_{2} {\overset{\cdot}{θ}}_{1} \sin θ_{2} \\ c_{22} = 0 \end{matrix}

Figure 7.

Details of executive mechanism.

Moreover, $g (θ)$ is a $2 \times 1$ gravitational force vector which is shown as

g (θ) = [\begin{matrix} g_{1} \\ g_{2} \end{matrix}]

(4)

with

\begin{matrix} g_{1} = (r_{c 2} + l_{2}) m_{2} g \cos (θ_{1} + θ_{2}) + l_{1} m_{2} g \cos θ_{1} \\ + (r_{c 1} + l_{1}) m_{1} g \cos θ_{1} \\ g_{2} = (r_{c 2} + l_{2}) m_{2} g \cos (θ_{1} + θ_{2}) \end{matrix}

Furthermore, $τ$ is a $2 \times 1$ input torque vector

τ = J^{T} K (θ) p

(5)

in which J is a Jacobian matrix to describe the tendon transmission routing mechanism and this routing is reasonable and controllable (Ozawa et al., 2014). It is written as

J = {[\begin{matrix} r_{1} & - r_{1} & - r_{1} & r_{1} \\ 0 & 0 & r_{2} & - r_{2} \end{matrix}]}^{T}

(6)

where $r_{i}$ denote the radius of the ith joint. Note that $K (θ)$ is a matrix which is demonstrated as

K (θ) = [\begin{matrix} k_{1} & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & k_{4} \end{matrix}]

(7)

where $k_{j}$ with $j = 1, 2, 3, 4$ , is the equivalent coefficient of sectional area of all four pneumatic artificial muscles (Klute et al., 1999; Tondu and Lopez, 2000). It is written as

k_{j} = ζ \frac{3 l_{j}^{2} - b^{2}}{4 π N^{2}}

(8)

where b denotes the length of wire mesh fibre, N is the winding number of the fibre, $ζ$ is treated as a compensation factor of the structure parameters, $l_{j}$ denotes the real time length of each pneumatic muscle. Since $l_{j}$ is an element of length vector $l (θ)$ , we get $l_{j}$ by calculating the vector

l (θ) = l_{0} - J θ

(9)

where $l_{0}$ is the initial length vector as

l_{0} = {[\begin{matrix} l_{b} & l_{b} & l_{b} & l_{b} \end{matrix}]}^{T}

where $l_{b}$ means the initial length of each PAM. Moreover, $p$ denotes the air pressure vector of each pneumatic artificial muscle. It is given as

\begin{matrix} p = {[\begin{matrix} k_{u} u_{1} & - k_{u} u_{1} & k_{u} u_{2} & - k_{u} u_{2} \end{matrix}]}^{T} \end{matrix}

(10)

where $k_{u}$ is the gain of the proportional control valve, $u_{i}$ is the control signal for each pneumatic artificial muscle with $i = 1, 2$ . Thereby, $u_{1}$ is the control signal changing for Muscle 1 and Muscle 2, $u_{2}$ is the control signal changing for Muscle 3 and Muscle 4 in Figure 6(a). The control input signal changing for Muscle 1 is positive and the input signal variation of Muscle 2 is negative to realize the inflation of Muscle 1 and the deflation of Muscle 2 in the same pair of PAMs, similarly for both Muscle 3 and Muscle 4. A coupling problem has to be considered for the reason of transmission routing for the two tendons in Figure 6. In order to solve the coupling problem, system (1) is rewritten as follows

H (θ) \overset{\cdot\cdot}{θ} + C (θ, \overset{\cdot}{θ}) \overset{\cdot}{θ} + g (θ) = [\begin{matrix} A & B \\ 0 & C \end{matrix}] [\begin{matrix} k_{u} u_{1} \\ k_{u} u_{2} \end{matrix}]

(11)

where

\begin{matrix} A = r_{1} ζ (\frac{3 l_{b}^{2} - b^{2}}{2 π N^{2}} + \frac{3 r_{1}^{2} θ_{1}^{2}}{2 π N^{2}}) \\ B = - r_{1} ζ (\frac{3 l_{b}^{2} - b^{2}}{2 π N^{2}} + \frac{3 {(r_{1} θ_{1} - r_{2} θ_{2})}^{2}}{2 π N^{2}}) \\ C = r_{2} ζ (\frac{3 l_{b}^{2} - b^{2}}{2 π N^{2}} + \frac{3 {(r_{1} θ_{1} - r_{2} θ_{2})}^{2}}{2 π N^{2}}) \end{matrix}

Setting

D = [\begin{matrix} A & B \\ 0 & C \end{matrix}]

and $x_{1} = θ$ , system (11) is rewritten as follows

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = τ_{u} + f (θ, \overset{\cdot}{θ}, t) \\ y = x_{1} \end{matrix}

(12)

where $τ_{u}$ is regarded as a vector which consists of elements corresponding to $u_{1}$ and $u_{2}$ , it is written as follows

τ_{u} = H (θ)^{- 1} D k_{u} U

(13)

where

U = {[\begin{matrix} u_{1} & u_{2} \end{matrix}]}^{T}

The nonlinear function $f (θ, \overset{\cdot}{θ}, t)$ indicates a complicated uncertain nonlinear part vector for each joint. The nonlinear item will be treated as an interior disturbance for the design controller in the following section.

Active disturbance rejection controller

An ADRC technique is proposed to control the angles of two joints to desired ones in this paper. The ADRC controller consists of a TD, an ESO, and a nonlinear controller based on a back-stepping method.

Tracking differentiator

In this paper, a continuous smooth signal and its differential signal are obtained by TD. This is a practical algorithm to mediate the conflict between rapidity and overshoot. A second order TD is designed for system (12). Which gives an error vector as follows

η = {[\begin{matrix} η_{1} & η_{2} \end{matrix}]}^{T}

(14)

where $η_{1}$ and $η_{2}$ are the angle errors assigned for joints. The error vector $η$ is calculated from $η = v_{1} - v_{0}$ , where

v_{0} = {[\begin{matrix} θ_{1 d} & θ_{2 d} \end{matrix}]}^{T}

(15)

is the desired angle signal vector. The TD is expressed as follows

{\begin{matrix} {\overset{\cdot}{v}}_{1} = v_{2} \\ {\overset{\cdot}{v}}_{2} = F_{han} \end{matrix}

(16)

with

\begin{matrix} v_{1} = {[\begin{matrix} v_{11} & v_{12} \end{matrix}]}^{T}, v_{2} = {[\begin{matrix} v_{21} & v_{22} \end{matrix}]}^{T} \\ F_{han} = {[\begin{matrix} f_{han} (η_{1}, v_{21}, r_{01}, h_{01}) & f_{han} (η_{2}, v_{22}, r_{02}, h_{02}) \end{matrix}]}^{T} \end{matrix}

where $v_{1}$ is an output signal vector tracking $v_{0}$ and $v_{2}$ is the differential signal vector of $v_{1}$ . That is, $v_{2}$ is the angular velocity vector. The function $f_{han} (*)$ is established as follows

f_{h a n} (η_{i}, v_{2 i}, r_{0 i}, h_{0 i}) = {\begin{matrix} d = r_{0 i} h_{0 i}^{2}, a_{0} = h_{0 i} v_{2 i}, a_{1} = η_{i} + a_{0} \\ a_{2} = \sqrt{d (d + 8 | a_{1} |)} \\ a_{3} = a_{0} + s i g n (a_{1}) (a_{2} - d) / 2 \\ f s g_{1} = s i g n (a_{1} + d) / 2 - s i g n (a_{1} - d) / 2 \\ a = (a_{0} + a_{1}) f s g_{1} + a_{3} (1 - f s g_{1}) \\ f s g_{2} = f s g (a + d) / 2 - s i g n (a - d) / 2 \\ f h a n = - r_{0 i} a f s g_{2} / d - r_{0 i} s i g n (a) (1 - f s g_{2}) \end{matrix}

where $η_{i}$ and $v_{2 i}$ are the variables, $r_{0 i}$ and $h_{0 i}$ are two parameters, with $i = 1, 2$ .

Extended state observer

In this section, an ESO is designed to deal with the nonlinear interior disturbance part of the 2 DOF two-joint finger system. The nonlinear term is extended as an extended system state, i.e. $f (θ, \overset{\cdot}{θ}, t) = x_{3}$ , for system (12). There may be some finite non-differentiable parts in $f (θ, \overset{\cdot}{θ}, t)$ in practice. In this case, some differentiable curves can be used to fit the non-differentiable parts (Yang et al., 2016a). So function $f (θ, \overset{\cdot}{θ}, t)$ is given to be continuously differentiable and bounded in this paper. System (12) is rewritten as follows

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = τ_{u} + x_{3} \\ {\overset{\cdot}{x}}_{3} = ϱ (t) \end{matrix}

(17)

where $ϱ (t)$ is the derivative vector of $x_{3}$ . Define an error $e_{1} = {[\begin{matrix} e_{11} & e_{12} \end{matrix}]}^{T}$ and it is calculated from

e_{1} = z_{1} - y

The concrete expression of the extended state observer is written as follows

{\begin{matrix} {\overset{\cdot}{z}}_{1} = z_{2} - β_{1} e_{1} \\ {\overset{\cdot}{z}}_{2} = z_{3} - β_{2} Fa l_{1} + τ_{u} \\ {\overset{\cdot}{z}}_{3} = - β_{3} Fa l_{2} \end{matrix}

(18)

where $z_{i} = {[\begin{matrix} z_{i 1} & z_{i 2} \end{matrix}]}^{T}$ is the observation value of state $x_{i} = {[\begin{matrix} x_{i 1} & x_{i 2} \end{matrix}]}^{T}$ with $i = 1, 2, 3$ . Note that the subscript 1 and 2 denote the first joint and the second joint, respectively. $β_{1}, β_{2} and β_{3}$ and $Fa l_{1}$ are gain matrices. $Fa l_{1}$ and $Fa l_{2}$ are two different function vectors with

Fa l_{1} = {[\begin{matrix} fa l_{11} (e_{11}, σ_{1}, δ) & fa l_{12} (e_{12}, σ_{1}, δ) \end{matrix}]}^{T}

and

Fa l_{2} = {[\begin{matrix} fa l_{21} (e_{11}, σ_{2}, δ) & fa l_{22} (e_{12}, σ_{2}, δ) \end{matrix}]}^{T}

The concrete expression of function $fa l_{11} (e_{11}, s_{1}, δ)$ , $fa l_{12} (e_{12}, s_{1}, δ)$ , $fa l_{21} (e_{11}, s_{2}, δ)$ and $fa l_{22} (e_{12}, s_{2}, δ)$ are written according to the structure of function $fal (e, σ, δ)$ , which is demonstrated as follows

fal (e, σ, δ) = {\begin{matrix} e / δ^{1 - σ}, | e | \leq δ \\ | e |^{σ} sign (e), | e | > δ \end{matrix}

where $e_{11}$ and $e_{12}$ are variable errors, $δ$ , $σ_{1}$ and $σ_{2}$ are positive, with $σ_{1} \in (0, 1)$ and $σ_{2} \in (0, 1)$ , respectively.

ESO (18) is a vector expression which consists of two sub-ESOs. The two sub-ESOs share a similar expression structure. Only the convergence of one sub-ESO is needed to verify and the other sub-ESO is obtained in the same way. One sub-ESO state reconstructed error system is established based on ESO (18) and system (17). Let $τ_{u}$ in ESO (18) be the same as the one in system (17). The sub-ESO state reconstructed error system is given as follows

{\begin{matrix} {\overset{\cdot}{e}}_{11} = e_{21} - β_{11} e_{11} \\ {\overset{\cdot}{e}}_{21} = e_{31} - β_{21} fa l_{11} (e_{11}) \\ {\overset{\cdot}{e}}_{31} = - ϱ_{1} (t) - β_{31} fa l_{21} (e_{11}) \end{matrix}

(19)

where $e_{11} = z_{11} - x_{11}$ , $e_{21} = z_{21} - x_{21}$ , $e_{31} = z_{31} - x_{31}$ , $fa l_{11}$ and $fa l_{21}$ denote the functions $fa l_{11} (e_{11}, σ_{1}, δ)$ and $fa l_{21} (e_{11}, σ_{2}, δ)$ , respectively. A self stable region (SSR) method is introduced to verify the effectiveness of ESO. That is, $z_{i 1}$ is converge to $x_{i 1}$ with $i = 1, 2, 3$ .

Definition 1.Let the region $R_{e}$ be a self stable region of system (19) and its vertex is the origin. Then all error trajectories in $R_{e}$ will converge to the origin after a certain moment.

Two functions are set based on the following state error reconstructed system as

\begin{matrix} ℏ_{1} (e_{11}) = {\overset{\cdot}{e}}_{11} + ℜ_{1} h_{1} (e_{11}) sign (e_{11}) \\ ℏ_{2} (e_{11}, e_{21}) = {\overset{\cdot}{e}}_{21} - β_{11} {\overset{\cdot}{e}}_{11} + ℜ_{2} h_{2} (e_{11}, e_{21}) sat (\frac{ℏ_{1} (e_{11})}{h_{1} (e_{11})}) \end{matrix}

where both $h_{1} (e_{11})$ and $h_{2} (e_{11}, e_{21})$ are two arbitrary continuous positive definite functions, with $h_{1} (0) = 0$ and $h_{2} (0, 0) = 0$ . Note that $ℜ_{1}$ and $ℜ_{2}$ are two constants, both of which are larger than 1.

Given two regions

\begin{matrix} R_{e 3} = {| ℏ_{2} (e_{11}, e_{21}) | \leq h_{2} (e_{11}, e_{21})} \\ R_{e 2} = {| ℏ_{1} (e_{11}) | \leq h_{1} (e_{11})} \end{matrix}

and two Lyapunov functions

\begin{matrix} V_{ℏ_{1} h_{1}} = \frac{1}{2} {ℏ_{1}^{2} (e_{11}) - h_{1}^{2} (h_{11})} \\ V_{ℏ_{2} h_{2}} = \frac{1}{2} {ℏ_{2}^{2} (e_{11}, e_{21}) - h_{2}^{2} (e_{11}, e_{21})} \end{matrix}

To relax notation, $h_{1} (e_{11})$ , $h_{1} sign (e_{11})$ and $h_{2} (e_{11}, e_{21})$ are simply denoted as $h_{1}$ , $h_{1} s$ and $h_{2}$ . Both $ℏ_{1} (e_{11})$ and $ℏ_{2} (e_{11}, e_{21})$ are simply written as $ℏ_{1}$ and $ℏ_{2}$ , respectively.

Theorem 1.If there exists a error trajectory $(e_{11}, e_{21}, e_{31})$ which is outside region . initially in system (19), it will be drawn by the self stable region $R_{e 3}$ and get into the self stable region $R_{e 3}$ .

Proof 1. The trajectory satisfies that $V_{ℏ_{2} h_{2}} \geq 0$ initially. A Lyapunov function $V_{ℏ_{2} h_{2}}$ is adopted as

V_{ℏ_{2} h_{2}} = \frac{1}{2} {ℏ_{2}^{2} (e_{11}, e_{21}) - h_{2}^{2} (e_{11}, e_{21})}

(20)

where $h_{2}$ in $V_{ℏ_{2} h_{2}}$ is designed as

h_{2} = ℜ_{3} | ℏ_{1} |

(21)

where $| ℏ_{1} |$ is satisfied with $| ℏ_{1} | \geq h_{1}$ , and $ℜ_{3}$ . is a constant which is satisfied with

ℜ_{3} > \frac{{(ℜ_{1} + 1)}^{2}}{(ℜ_{2} - 1)} | \frac{d_{h_{1}}}{d_{e_{11}}} | > 0

(22)

It is obtained that

\begin{matrix} {\overset{\cdot}{e}}_{11} & = e_{21} - β_{11} e_{11} = ℏ_{1} - ℜ_{1} h_{1} s \\ {\overset{\cdot}{e}}_{21} & = e_{31} - β_{21} fa l_{11} (e_{11}) = ℏ_{2} + β_{11} (ℏ_{1} - ℜ_{1} h_{1} s) - ℜ_{2} h_{2} sat \\ h_{2} sat & = h_{2} sat (ℏ_{1} / h_{1}) = ℜ_{3} | ℏ_{1} | \\ \frac{\partial ℏ_{1}}{\partial e_{11}} & = - β_{11} + ℜ_{1} \frac{d h_{1}}{d e_{11}} s \\ \frac{\partial ℏ_{1}}{\partial e_{21}} & = 1 \end{matrix}

The derivative of $V_{ℏ_{2} h_{2}}$ is given as follows

\begin{array}{l} {\dot{V}}_{ℏ_{2} h_{2}} = ℏ_{2} {\dot{ℏ}}_{2} - h_{2} {\dot{h}}_{2} \\ = ℏ_{2} [({\dot{e}}_{21})' - β_{11} ({\dot{e}}_{11})' + ℜ_{2} \frac{\partial h_{2} s a t}{\partial e_{11}} (e_{11})' + ℜ_{2} \frac{\partial h_{2} s a t}{\partial e_{21}} (e_{21})'] \\ - h_{2} [\frac{\partial h_{2}}{\partial e_{11}} (e_{11})' + \frac{\partial h_{2}}{\partial e_{21}} (e_{21})'] \\ + ℜ_{2} \frac{\partial h_{2} s a t}{\partial e_{21}} (e_{21})' - h_{2} [\frac{\partial h_{2}}{\partial e_{11}} (e_{11})' + \frac{\partial h_{2}}{\partial e_{21}} (e_{21})'] \\ = ℏ_{2} {[- ϱ_{1} (t) - β_{31} f a l_{21}] - β_{21} f a l'_{11} (ℏ_{1} - ℜ_{1} h_{1} s) \\ - β_{11} (ℏ_{2} - ℜ_{2} ℜ_{3} | ℏ_{1} |) \\ + ℜ_{2} ℜ_{3} (- β_{11} + ℜ_{1} \frac{d h_{1} s}{d e_{11}}) (ℏ_{1} - ℜ_{1} h_{1} s) + ℜ_{2} ℜ_{3} [ℏ_{2} + β_{11} (ℏ_{1} \\ - ℜ_{1} h_{1} s) - ℜ_{2} ℜ_{3} | ℏ_{1} |]} - h_{2} {\frac{\partial h_{2}}{\partial e_{11}} (ℏ_{1} - ℜ_{1} h_{1} s) + \frac{\partial h_{2}}{\partial e_{21}} [ℏ_{2} \\ + β_{11} (ℏ_{1} - ℜ_{1} h_{1} s) - ℜ_{2} ℜ_{3} | ℏ_{1} |] \\ = ℏ_{2}^{2} (- β_{11} + ℜ_{2} ℜ_{3}) \\ + ℏ_{2} ℏ_{1} [ℜ_{1} ℜ_{2} ℜ_{3} \frac{d h_{1} s}{d e_{11}} - ℜ_{2}^{2} ℜ_{3}^{2} - β_{21} f a l'_{11} + β_{11} ℜ_{2} ℜ_{3}] \\ + h_{1} ℏ_{2} s [β_{31} \frac{| f a l_{21} |}{h_{1}} + ℜ_{1} β_{21} f a l'_{11} - ℜ_{1}^{2} ℜ_{2} ℜ_{3} \frac{d h_{1} s}{d e_{11}}] - ℏ_{2} h_{2} \frac{\partial h_{2}}{\partial e_{21}} \\ - ℏ_{2} ϱ_{1} (t) - ℏ_{1} h_{2} (- ℜ_{2} ℜ_{3} \frac{\partial h_{2}}{\partial e_{21}} + \frac{\partial h_{2}}{\partial e_{11}} + β_{11} \frac{\partial h_{2}}{\partial e_{21}}) \\ + ℜ_{1} h_{1} h_{2} s (β_{11} \frac{\partial h_{2}}{\partial e_{21}} + \frac{\partial h_{2}}{\partial e_{11}}) \end{array}

(23)

The derivative of the extended state $x_{3}$ is bounded, i.e. $| ϱ_{1} (t) | \leq Q$ with $Q > 0$ . One has that $| ℏ_{2} | > h_{2} > | ℏ_{1} | > h_{1}$ is deduced from $V_{ℏ_{2} h_{2}}$ and equation (21), then ${\overset{\cdot}{V}}_{ℏ_{2} h_{2}}$ is amplified as follows

\begin{matrix} {\overset{\cdot}{V}}_{ℏ_{2} h_{2}} \leq ℏ_{2}^{2} (- β_{11} + ℜ_{2} ℜ_{3}) + ℏ_{2}^{2} | - \frac{β_{21}}{ℜ_{3}} fa {l'}_{11} + β_{11} ℜ_{2} - ℜ_{2}^{2} ℜ_{3} | \\ + \frac{ℏ_{2}^{2}}{ℜ_{3}} | - β_{31} \frac{fa l_{21}}{h_{1}} - ℜ_{1} β_{21} fa {l'}_{11} | + ℏ_{2}^{2} ℜ_{1} ℜ_{2} | \frac{d h_{1} s}{d e_{11}} | \\ + ℜ_{1}^{2} ℜ_{2} ℏ_{2}^{2} | \frac{d h_{1} s}{d e_{11}} | + | ℏ_{2} Q | + ℏ_{2}^{2} | \frac{\partial h_{2}}{\partial e_{2}} | \\ + \frac{ℏ_{2}^{2}}{ℜ_{3}} | \frac{\partial h_{2}}{\partial e_{11}} + β_{11} \frac{\partial h_{2}}{\partial e_{21}} - ℜ_{2} ℜ_{3} \frac{\partial h_{2}}{\partial e_{21}} | \\ + \frac{ℜ_{1} ℏ_{2}^{2}}{ℜ_{3}} | \frac{\partial h_{2}}{\partial e_{11}} + β_{11} \frac{\partial h_{2}}{\partial e_{21}} | \end{matrix}

(24)

Choose parameters $β_{11}$ , $β_{21}$ and $β_{31}$ such that

\begin{matrix} ℏ_{2}^{2} | - \frac{β_{21}}{ℜ_{3}} fa {l'}_{11} + β_{11} ℜ_{2} - ℜ_{2}^{2} ℜ_{3} | = 0 \\ \frac{ℏ_{2}^{2}}{ℜ_{3}} | - β_{31} \frac{fa l_{2}}{h_{1}} - ℜ_{1} β_{21} fa {l'}_{1} | = 0 \end{matrix}

and ${\overset{\cdot}{V}}_{ℏ_{2} h_{2}} \leq 0$ . hold. Then trajectory $(e_{11}, e_{21}, e_{31})$ outside region $R_{e 3}$ is attracted by $R_{e 3}$ and it will converge into the region eventually.

Theorem 2. If there exists a trajectory $(e_{11}, e_{21}, e_{31})$ which belongs in region $R_{e 3}$ but out of $R_{e 2}$ in system (19), it will converge to the origin directly or get into $R_{e 2}$ in a finite time if the following inequation

h_{2} > \frac{{(ℜ_{1} + 1)}^{2}}{(ℜ_{2} - 1)} | \frac{d_{h_{1}}}{d_{e_{11}}} | | ℏ_{1} |

(25)

holds.

Proof 2. According to the positional information of the trajectory $(e_{11}, e_{21}, e_{31})$ , the trajectory satisfies $V_{ℏ_{2} h_{2}} < 0$ and $V_{ℏ_{1} h_{1}} \geq 0$ . Then an inequation is given as follows

ℏ_{1} (ℏ_{2} - ℜ_{2} h_{2} sat (ℏ_{1} / h_{1})) \leq - | ℏ_{1} | (ℜ_{2} - 1) h_{2} < 0

(26)

A Lyapunov function $V_{ℏ_{1} h_{1}}$ is given as follows

V_{ℏ_{1} h_{1}} = \frac{1}{2} {ℏ_{1}^{2} (e_{11}, e_{21}) - h_{1}^{2} (e_{11})}

The derivative of the Lyapunov function $V_{ℏ_{1} h_{1}}$ is deduced as follows

\begin{matrix} {\overset{\cdot}{V}}_{ℏ_{1} h_{1}} = ℏ_{1} {\overset{\cdot}{ℏ}}_{1} - h_{1} {\overset{\cdot}{h}}_{1} \\ = ℏ_{1} [ℏ_{2} - ℜ_{2} h_{2} sat (ℏ_{1} / h_{1}) + ℜ_{1} \frac{d h_{1} s}{d e_{11}} (ℏ_{1} - ℜ_{1} h_{1} s)] \\ - h_{1} \frac{d h_{1}}{d e_{11}} (ℏ_{1} - ℜ_{1} h_{1} s) \end{matrix}

(27)

According to inequation (26), ${\overset{\cdot}{V}}_{ℏ_{1} h_{1}}$ is amplified as follows

\begin{matrix} {\overset{\cdot}{V}}_{ℏ_{1} h_{1}} \leq - | ℏ_{1} | (ℜ_{2} - 1) h_{2} + ℏ_{1} ℜ_{1} \frac{d h_{1} s}{d e_{11}} (ℏ_{1} - ℜ_{1} h_{1} s) \\ - h_{1} \frac{d h_{1}}{d e_{11}} (ℏ_{1} - ℜ_{1} h_{1} s) \leq - | ℏ_{1} | (ℜ_{2} - 1) h_{2} \\ + ℜ_{1} | ℏ_{1} \frac{d h_{1} s}{d e_{11}} (ℏ_{1} - ℜ_{1} h_{1} s) | + h_{1} | \frac{d h_{1}}{d e_{11}} (ℏ_{1} - ℜ_{1} h_{1} s) | \\ \leq - | ℏ_{1} | (ℜ_{2} - 1) h_{2} + | \frac{d h_{1}}{d e_{11}} | | ℏ_{1} + ℜ_{1} ℏ_{1} | (ℜ_{1} | ℏ_{1} | + | ℏ_{1} |) \\ \leq - | ℏ_{1} | (ℜ_{2} - 1) h_{2} + | ℏ_{1} |^{2} (ℜ_{1} + 1)^{2} | \frac{d h_{1}}{d e_{11}} | \end{matrix}

(28)

There exists ${\overset{\cdot}{V}}_{ℏ_{1} h_{1}} < 0$ when $h_{2}$ satisfies the inequation (25). The trajectory which belongs in region $R_{e 3}$ but out of $R_{e 2}$ will converge into $R_{e 2}$ or it will converge into region $R_{e 2}$ after certain time.

Theorem 3.A trajectory $(e_{11}, e_{21}, e_{31})$ , which belongs in region $R_{e 2}$ , will converges to the origin eventually in system (19).

Proof 3. Let a trajectory $(e_{11}, e_{21}, e_{31})$ stay in $R_{e 2}$ after time T, i.e. $V_{ℏ_{1} h_{1}} < 0, \forall t > T$ . The following inequation is obtained from region $R_{e 2}$ as

- ℜ_{1} h_{1} s - h_{1} < e_{21} - β_{11} e_{11} < - ℜ_{1} h_{1} s + h_{1}

Design a Lyapunov function

V_{e_{11}} = \frac{1}{2} e_{11}^{2}

(29)

then the derivative of $V_{e_{11}}$ is shown as

\begin{matrix} {\overset{\cdot}{V}}_{e_{11}} = e_{11} {\overset{\cdot}{e}}_{11} \\ \leq - (ℜ_{1} s - 1) e_{11} h_{1} \end{matrix}

(30)

There exist ${\overset{\cdot}{V}}_{e_{11}} \leq - (ℜ_{1} - 1) e_{11} h_{1} < 0$ when $e_{11} > 0$ and ${\overset{\cdot}{V}}_{e_{11}} \leq - (ℜ_{1} + 1) | e_{11} | h_{1} < 0$ when $e_{11} < 0$ , then ${\overset{\cdot}{V}}_{e_{11}} < 0$ for all $e_{11} \neq 0$ . It is shown that $e_{11}$ converges to zero from ${\overset{\cdot}{V}}_{e_{11}} < 0$ , then $e_{21}$ and $e_{31}$ will also converge to zero, which is obtained from regions $R_{e 2}$ and $R_{e 3}$ . So the trajectory $(e_{11}, e_{21}, e_{31})$ belonging in region $R_{e 2}$ will eventually converge to the origin.

Theorem 4.If a trajectory arrives a point of the boundary of one certain region in system (19), such as $R_{e 2}$ or $R_{e 3}$ , after some time, then that point is the origin.

Proof 4.Taking the boundary of region $R_{e 2}$ for example, if the trajectory $(e_{11}, e_{21})$ arrives a point of the boundary of region $R_{e 2}$ , then this point is called the utmost point $({\bar{e}}_{11}, {\bar{e}}_{21})$ of $R_{e 2}$ . The utmost point is satisfies ${\overset{\cdot}{V}}_{ℏ_{1} h_{1}} = 0$ . It is known that only the origin satisfies ${\overset{\cdot}{V}}_{ℏ_{1} h_{1}} = 0$ . Hence, the utmost point is the origin. The similar conclusion is also obtained for the boundary of region $R_{e 3}$ .

Taking Theorems 1 to 4 into consideration comprehensively, no matter which region a trajectory belongs to initially, every error trajectory will converge to the origin directly, or it will be attracted by a certain region and go into it, or it will arrive to the utmost point after some time. It is obtained that $R_{e 3}$ is a SSR of the error reconstructed system (19).

Nonlinear controller

In this paper, a nonlinear controller is designed to make the rotary movement of both joints of the robot finger track the desired angles precisely. The error variable system is shown as

{\begin{matrix} s_{1} = v_{1} - x_{1} \\ s_{2} = ϕ_{1} - x_{2} \end{matrix}

(31)

where $s_{1}$ and $s_{2}$ are error variables, $v_{1}$ is the given reference angle vector, $ϕ_{1}$ is a virtual control variable as

ϕ_{1} = v_{2} + k_{1} Fa l_{s 1}

(32)

with

Fa l_{s 1} = {[\begin{matrix} fal (s_{11}, σ_{3}, δ) & fal (s_{12}, σ_{3}, δ) \end{matrix}]}^{T}

Taking the derivative of the error variable system (31), it is rewritten as follows

{\begin{matrix} {\overset{\cdot}{s}}_{1} = {\overset{\cdot}{v}}_{1} - x_{2} \\ {\overset{\cdot}{s}}_{2} = {\overset{\cdot}{ϕ}}_{1} - x_{3} - H^{- 1} (θ) D k_{u} U \end{matrix}

(33)

The ADRC controller is designed as

U = \frac{1}{k_{u}} D^{- 1} H (θ) ({\overset{\cdot}{ϕ}}_{1} - z_{3} + s_{1} + k_{2} Fa l_{s 2})

(34)

where

{Fal}_{s 2} = {[\begin{matrix} fal (s_{21}, σ_{4}, δ) & fal (s_{22}, σ_{4}, δ) \end{matrix}]}^{T}

The nonlinear controller (34) is substituted into system (33), the close-loop system is obtained as follows

{\begin{matrix} {\overset{\cdot}{s}}_{1} = {\overset{\cdot}{v}}_{1} - x_{2} \\ {\overset{\cdot}{s}}_{2} = z_{3} - x_{3} - s_{1} - k_{2} Fa l_{s 2} \end{matrix}

(35)

Theorem 5.The close-loop two-joint robot finger control system (35) is stable if the elements of the gain matrices $k_{1}$ and $k_{2}$ in the nonlinear controller (34) are tuned as sufficiently large as possible. i.e. ${\overset{\cdot}{v}}_{1} - x_{2} \to 0$ , then the movement of both joints track the input angles precisely.

Proof 5. Step 1: Design a Lyapunov function $V_{1}$ as follows

V_{1} = \frac{1}{2} s_{1}^{T} s_{1}

(36)

Taking the derivative of the Lyapunov function $V_{1}$ , the following equation is obtained as

\begin{matrix} {\overset{\cdot}{V}}_{1} = \frac{1}{2} {\overset{\cdot}{s}}_{1}^{T} s_{1} + \frac{1}{2} s_{1}^{T} {\overset{\cdot}{s}}_{1} \\ = ({\overset{\cdot}{v}}_{1} - {\overset{\cdot}{x}}_{1})^{T} (v_{1} - x_{1}) \\ = (v_{2} - x_{2})^{T} (v_{1} - x_{1}) \\ = (v_{2} + s_{2} - ϕ_{1})^{T} (v_{1} - x_{1}) \\ = [- k_{1} Fa l_{s_{1}}]^{T} s_{1} + {s_{2}}^{T} s_{1} \end{matrix}

(37)

To relax notation, $fal (s_{ij}, σ_{o}, δ)$ with $i, j = 1, 2, o = 3, 4$ is noted as $fal (*)$ . Note that $fal (*)$ is an odd function, so there exists $(Fa l_{s_{i}})^{T} s_{i} \geq 0$ $(Fa l_{s_{i}})^{T} s_{i} = 0$ , if and only if $s_{i} = 0$ holds. If the first system is stable, there exists ${\overset{\cdot}{V}}_{1} \leq 0$ when $s_{2} = 0$

Step 2: A Lyapunov function is given as follows

V_{2} = V_{1} + \frac{1}{2} {s_{2}}^{T} s_{2}

(38)

Taking the derivative of $V_{2}$ , one has that

\begin{matrix} {\overset{\cdot}{V}}_{2} = {\overset{\cdot}{V}}_{1} + \frac{1}{2} {\overset{\cdot}{s}}_{2}^{T} s_{2} + \frac{1}{2} {s_{2}}^{T} {\overset{\cdot}{s}}_{2} \\ = {\overset{\cdot}{V}}_{1} + {\overset{\cdot}{s}}_{2}^{T} s_{2} \\ = {\overset{\cdot}{V}}_{1} + ({\overset{\cdot}{ϕ}}_{1} - {\overset{\cdot}{x}}_{2})^{T} s_{2} \\ = - [k_{1} Fa l_{s_{1}}]^{T} s_{1} + {s_{2}}^{T} s_{1} + ({\overset{\cdot}{ϕ}}_{1} - H (θ)^{- 1} D k_{u} U - x_{3})^{T} s_{2} \end{matrix}

(39)

Considering controller (34), there exists

{\overset{\cdot}{V}}_{2} = - [k_{1} Fal (s_{1})]^{T} s_{1} - [k_{2} Fal (s_{2})]^{T} s_{2} + Δ^{T} s_{2}

(40)

where $Δ = z_{3} - x_{3}$ which is small enough because $z_{3}$ converges to $x_{3}$ . Both $k_{1}$ and $k_{2}$ are positive gains matrices. There exists ${\overset{\cdot}{V}}_{2} < 0$ if we tune the parameters in $k_{1}$ and $k_{2}$ to the appropriate ones. That is, the closed-loop system (35) is convergent and stable by letting $k_{1}$ and $k_{2}$ be large enough.

Remark 1. The ADRC nonlinear controller (34) contains the variable ${\overset{\cdot}{ϕ}}_{1}$ which is obtained by a designed TD as follows

{\begin{matrix} v_{ϕ_{1}} = v_{ϕ_{1}} + h {\overset{\cdot}{ϕ}}_{1} \\ {\overset{\cdot}{ϕ}}_{1} = {\overset{\cdot}{ϕ}}_{1} + h F_{han 1} \end{matrix}

(41)

with

\begin{matrix} v_{ϕ_{1}} = {[\begin{matrix} v_{ϕ_{11}} & v_{ϕ_{12}} \end{matrix}]}^{T}, ϕ_{1} = {[ϕ_{11} ϕ_{12}]}^{T} \\ F_{han 1} = {[\begin{matrix} f_{han} (v_{ϕ_{11}} - ϕ_{11}, {\overset{\cdot}{ϕ}}_{11}, r_{03}, h_{03}) & f_{han} (v_{ϕ_{12}} - ϕ_{12}, {\overset{\cdot}{ϕ}}_{12}, r_{04}, h_{04}) \end{matrix}]}^{T} \end{matrix}

where $v_{ϕ_{1}}$ is used to estimate the tracking signal of $ϕ_{1}$ . Hence, ${\overset{\cdot}{ϕ}}_{1}$ is used to express the differential signal of $v_{ϕ_{1}}$ .

Remark 2.The contribution of this paper is that an active disturbance rejection controller is proposed for a coupling two-joint system actuated by PAMs. A special mechanical design has been proposed to solve a coupling problem of the two-joint system actuated by dc motors in reference (Grossard et al., 2015). Rigid dc motors are chosen in Grossard et al. (2015) as actuators, while the actuators in this paper are compliant PAMs. The dc motors have a faster dynamic performance than PAMs (Shin et al., 2014), so the response time in reference Grossard et al. (2015) is better than this paper. However, PAMs have many additional properties such as high force-to-weight ratios and inherent compliances. Hence, PAMs have been used in many occasions where dc motors can’t be used (Shin et al., 2014). An elaborately designed mechanism of the two-joint system is introduced in Grossard et al. (2015) to tackle the coupling problem. In this paper, the coupling problem is processed by proposing an effective decoupling control algorithm. The decentralized SISO controller in Grossard et al. (2015) is not suitable for the coupling problem in this paper. The controller in this paper can simplify the mechanical design, saving cost in manufacturing and maintenance.

Experiment results

In this section, some experiment results are given to demonstrate the performance of the proposed nonlinear controller under desired step signal. To show the executive mechanism, Figure 7 gives details on the facility.

Note that the experiment is carried out in a maximal air pressure 0.6Mpa, and the input-output function of the electrical proportional valve is expressed as follows

P = 0.09 E + 0.001

where P is the output air pressure(Mpa) and E is input voltage(v).

Firstly, the coupling effect is measured. One joint is controlled to track a step signal while the other one is under passive control, respectively. Coupling effect results are recorded in Figure 8. Figure 8(a) depicts the tracking result of the first joint when it is given a $15^{\circ}$ reference signal, Figure 8(b) shows that the second joint is subject to the disturbance of the first joint in a large extent, the second joint will rotate $8^{\circ}$ clockwise passively when the first joint tracks the reference signal. The first joint gets little effect from the second joint is depicted in Figure 8(c), it will rotate $0 . 05^{\circ}$ counter clockwise when the second joint tracks a $15^{\circ}$ reference signal in Figure 8(d).

Figure 8.

Coupling effect of the two joints. (a) Joint 1’s tracking when Joint 2 is passive. (b) Joint 2’s tracking when Joint 2 is passive. (c) Joint 1’s tracking when Joint 1 is passive. (d) Joint 2’s tracking when Joint 1 is passive.

Set the step length h as 0.01. The gain matrix parameters for ESO are tuned by trial and error method, which are demonstrated as follows

β_{1} = [\begin{matrix} 50 & 0 \\ 0 & 50 \end{matrix}] β_{2} = [\begin{matrix} 150 & 0 \\ 0 & 150 \end{matrix}] β_{3} = [\begin{matrix} 500 & 0 \\ 0 & 600 \end{matrix}]

The gains matrix of the controller $k_{1}$ and $k_{2}$ are given as follows, which is tuned by trial and error method, too.

k_{1} = [\begin{matrix} 510 & 0 \\ 0 & 510 \end{matrix}] k_{2} = [\begin{matrix} 600 & 0 \\ 0 & 600 \end{matrix}]

Other parameters with respect to the two-joint finger system are presented in Tables 1 and 2.

Table 1.

Parameters of the 2 DOF two-joint finger system.

Parameters	First link	Second link
Rotation inertia	86.536 kg mm²	61.23 kg mm²
Mass of joint	0.158 kg	0.027 kg
Length of link	50 mm	41 mm
Radius of joint	14 mm	11.5 mm

Table 2.

Parameters of pneumatic artificial muscles (PAMs).

Constants	Value
$η$	0.75
N	2.7
b	164 mm
$l_{0}$	150 mm

A step reference signal with $15^{\circ}$ is set for each joint. The experiment results of the proposed controller and some PID controller results are demonstrated in Figure 9. Figure 9(a) and Figure 9(b) depict the position tracking results of the two joints, respectively. In the figures, the reference line is the given step signal, the PID line is the joint’s angle which tracks the given signal by the PID controller, the ADRC line shows the situation which the joint is controlled by ADRC controller. The control signals of two control technology is depicted in Figure 10.

Figure 9.

Comparison performance of two joints under two control methods. (a) Position tracking of the first joint. (b) Position tracking of the second joint.

Figure 10.

Control signals of both control methods. (a) Control signals of ADRC method. (b) Control signals of PID method.

As shown in the Figure 9(b), there exists an overshoot of 6°–8° and a response time of 5 s–7 s by using the PID controller in the controlling joints. In contrast, ADRC controller can overcome the coupling effect in controlling the system, which is demonstrated from the low overshoot in Figure 9. A response time of 3 s–4 s and a steady error of $0 . 02^{\circ}$ are depicted in Figure 9, which shows the advantage over PID controller.

Remark 3.As an actuator, a similar tendon-driven two-joint mechanism is designed using DC motors in Beytullah et al. (2015). A nonlinear robust controller is introduced for the system. However, only simulation results are demonstrated in that paper and no experiment results to support the validity of the proposed controller. In Takahiro et al. (2005), a PID feedback control method is used to control a tendon-driven manipulator system. There are some differences in the mechanical design, pneumatic cylinders have been utilized as actuators instead of pneumatic artificial muscles. The experiment results in this paper are compared with the ones in Takahiro et al. (2005) and a PID controller, which is depicted in Table 3.

Table 3.

Compare results with other literature.

Control method	Response time	Overshoot
Controller in Takahiro et al. (2005)	15 s–17 s	18°–20°
PID controller	5 s–7 s	6°–8°
Controller in this paper	3 s–4 s	0°–0.5°

Remark 4.Note that PID control is widely used in control systems. Many researchers apply PID controllers in pneumatic systems, so we choose PID as a comparison control method. Reference Takahiro et al. (2005) is chosen as a comparison due to its similar two-joint coupling mechanism which is powered by pneumatic actuators. PID is adopted to tackle the control problem, however, the results show serious overshoots. In order to verify the validity of PID, the PID controller is introduced in our two-joint coupling system platform. Parameters are tuned by traditional trail and error methods. Experimental results show that PID can’t reduce the overshoots even though we have try our best to make the PID controller optimal. The nonlinear controller in this paper can provide a new solution for two-joint coupling systems actuated by PAMs.

Remark 5.An adaptive controller is introduced for a n-link rigid manipulator with uncertain loads in Slotine and Li (1985). A nonlinear controller is proposed for a two-joint coupling system actuated by PAMs in this paper. The uncertain parameters are assumed to be constant in Slotine and Li (1985). However, the unknown disturbances in our paper are time-varying. Only simulation results are provided in Slotine and Li (1985), but the simulation requires that the two links are identical and the parameters are known. Reliable experiment verifications are provided in this paper. Hence, the adaptive controller in Slotine and Li (1985) is not suitable for the two-joint coupling system in this paper.

Conclusion

In this paper, an ADRC controller has been designed to study the position tracking problem of a 2 DOF two-joint coupling finger system which is actuated by PAMs. A desired trajectory has been tracked via TD with low overshoot by arranging transitional process. All state variables have been estimated via a designed ESO effectively. The close-loop control system is stable if the gain matrices are tuned large sufficiently, which is proved by a back-stepping method. Experiment results show the validity of proposed ADRC method. The response time of the close-loop system is 3 s – 4 s and the steady error is demonstrated as 0.02°.

Footnotes

Acknowledgements

The authors would like to thank the anonymous reviewers for their detailed comments which helped to improve the quality of the paper.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work of Ling Zhao was supported by National Natural Science Foundation of China (grant number 51505413) and the Hebei Provincial Natural Science Fund (grant number E2014203122). The work of Tao Wang was supported by National Natural Science Foundation of China (grant number 51375045).

References

Ahn

(2009) Design and implement of an adaptive recurrent neural networks (ARNN) of the pneumatic artificial muscle (PAM) manipulator. Mechatronics 19(6): 816–828.

Ahn

Mailah

Nguyen

(2007) Intelligent switching control of a pneumatic muscle robot arm using learning vector quantization neural network. Mechatronics 17: 255–262.

Beytullah

Orhan

Erkan

. (2015) Nonlinear robust control of tendon-driven robot manipulators. Journal of Intelligent Robotic Systems 80(1): 3–14.

Caffaz

Cannata

(1998) The Design and development of the DIST-Hand dexterous gripper. In: Proceedings of the 1998 IEEE international conference on robotics and automation, (ICRA 1998), Leuven, Belgium, May 1998, pp.2075–2080.

Chan

Naghdy

Stirling

(2013) Extended active observer for force estimation and disturbance rejection of robotic manipulators. Robotics and Autonomous Systems 61(12): 1277–1287.

Chang

Liou

Chen

(2011) T-S fuzzy model -based tracking control of a one-dimensional manipulator actuated by pneumatic artificial muscles. Control Engineering Practice 19(12): 1442–1449.

Girin

Plesten

Brun

. (2009) High order sliding-mode controllers of an electropneumatic actuator: Application to an aeronautic benchmark. IEEE Transaction Control Systems Technology 17(3): 633–645.

Grossard

Martin

Cruz Pacheco

(2015) Control-oriented design and robust decentralized control of the CEA dexterous robot hand. IEEE/ASME Transactions on Mechatronics 20(4): 1809–1821.

Han

(1998) Auto-disturbance rejection control and its applications. Control and Decision 13: 19–23.

10.

Han

(2008) Active Disturbance Rejection Control Technique-The Technique for Estimating and Compensating the Uncertainties. Beijing: National Defense Industry Press, pp.288–296.

11.

Han

(2009) From PID to active disturbance rejection control. IEEE Transactions on Industrial Electronics 56(3): 900–906.

12.

Karayiannidis

Doulgeri

(2012) Model free robot joint position regulation and tracking with prescribed performance guarantees. Robotics and Autonomous Systems 60(2): 214–226.

13.

Klute

Czerniecki

Hanaford

(1999) Mckibben artificial muscles: Pneumatic actuators with biomechanical intelligence. IEEE/ASME international conference on advanced intelligent mechantronics, (ICAIM 1999), Atlanta, USA, September 1999, pp.221–226.

14.

Malyuddin

Khan

Herrmann

(2014) A novel robust adaptive control algorithm with finite time online parameter estimation of a humanoid robot arm. Robotics Autonomous Systems 62(3): 294–305.

15.

Nuchkrua

Leephakpreeda

(2013) Fuzzy self-tuning PID control of hydrogen-driven pneumatic artificial muscle actuator. Journal of Bionic Engineering 10(3): 329–340.

16.

Ozawa

Kobayashi

Hashirii

(2014) Analysis, classification and design of tendon driven mechanisms. IEEE Transaction on Robotics 30(2): 396–410.

17.

Park

Yang

Park

. (2002) Position/vibration control of two-degree-of-freedom arms having one flexible link with artificial pneumatic muscle actuators. Robotics and Autonomous Systems 40(4): 239–253.

18.

Shen

(2010) Nonlinear model-based control of pneumatic artificial muscle servo systems. Control Engineering Practice 18(3): 311–317.

19.

Shin

Yeh

Khatib

(2014) A new hybrid actuation scheme with artificial pneumatic muscles and a magnetic particle brake for safe human-robot collaboration. The International Journal of Robotics Research 33(4): 507–518.

20.

Slotine

(1985) On the adaptive control of robot manipulators. The International Journal of Robotics Research 6(3): 49–59.

21.

Takahiro

Osamu

Toshihiro

(2005) Development of tendon-driven assistance manipulator system by air pressure control. In: Proceedings of the 6th JFPS international symposium on fluid power (ISFP 2005), Tsukuba, Japan, November 2005, pp.740–744.

22.

Thank

Ahn

(2006) Nonlinear PID control to improve the control performance of 2-axes pneumatic artificial muscle manipulator using neural network. Mechatronics 16(9): 577–587.

23.

Tondu

Lopez

(2000) Modeling and control of mckibben artificial muscle robot actuators. IEEE Control Systemss Magazine 20(2): 15–38.

24.

Xia

Shi

Liu

. (2007) Active disturbance rejection control for uncertain multivariable systems with time-delay. IET Control Theory & Applications 1(1): 75–81.

25.

Xing

Huang

Wang

. (2010) Tracking control of pneumatic artificial muscle actuators based on sliding mode and non-linear disturbance observer. IET Control Theory & Applications 4(10): 2058–2070.

26.

Yang

Fan

Shi

. (2016a) Nonlinear control for tracking and obstacle avoidance of a wheeled mobile robot with nonholonomic constraint. IEEE Transactions on Control Systems Technology 24(2): 741–746.

27.

Yang

Fan

Xia

. (2015) Robust tracking control for wheeled mobile robot based on extended state observer. Advanced Robotics 30(1): 68–78.

28.

Yang

You

Xia

. (2014a) Nonlinear attitude tracking control for spacecraft formation with multiple delays. Advances in Space Research 54(4): 759–769.

29.

Yang

You

Xia

. (2014b) Adaptive control for attitude synchronisation of spacecraft formation via extended state observer. IET Control Theory and Application 8(18): 2171–2185.

30.

Yang

You

Hua

(2016b) Attitude tracking control for spacecraft formation with time-varying delays and switching topology. Acta Astronautica 126: 98–108.

31.

Yang

Yuan

. (2014c) Back-stepping control of two-link flexible manipulator based on an extended state observer. Advances in Space Research 56(10): 2312–2322.

32.

Zhang

Xie

(2007) Adaptive robust control of one-link joint actuated by pneumatic artificial muscles. In: The 1st international conference on bioinformatics and biomedical engineering (ICBBE 2007), Wuhan, China, July 2007, pp.1185–1189.

33.

Zhu

Tao

Yao

. (2008) An adaptive robust posture control of a parallel manipulator driven by pneumatic artificial muscles. Automatica 44(9): 2248–2257.