Robust motion control of live-line maintenance robot mechanical arm for high-voltage transmission line based on H∞ theory

Abstract

To overcome the low operation efficiency, high labour-intensiveness and high risk in the artificial live-line replacement of insulator strings, a robot for overhead transmission line maintenance was developed. In order to suppress effectively the influences of disturbance signals and uncertainties on tracking precision and stability of the robot mechanical arm motion under high voltage and strong electromagnetic interference, this paper proposed a H∞ control theory-based robust trajectory tracking control method for the robot mechanical arm. Through layering robot control architecture, a dynamic model of mechanical arm basic motion was established by the Lagrange method combined with an armature voltage equation of the joint motor, and the unified dynamic model of mechanical arm different motion was obtained. On this basis, the state-space model of mechanical arm motion error was deduced under disturbances and uncertainties, and thus an H∞ control model for mechanical arm motion was constructed. Subsequently, the H∞ controller for the mechanical arm trajectory tracking control system was solved by linear matrix inequality (LMI) based on the established model, and the asymptotic stability of the mechanical arm motion control system was verified by selecting the appropriate Lyapunov function. The proposed method for such a controller was proved to be of good versatility, strong adaptability and sound expansibility. Finally, simulation results verified the effectiveness of the H∞ controller and field operation tests further validated the engineering practicability of such a control method in macro and micro aspects.

Keywords

H∞ control live-line maintenance robot LMI mechanical arm trajectory tracking

Introduction

A live-line maintenance robot is a special electric power operation robot that walks along a high-voltage transmission line to replace or assist artificial power maintenance. It plays an important role in the power industry and has very broad application prospects (e.g. Aracil et al., 2012; Lu et al., 2003; Qi, 2003; Takaoka et al., 2001). Replacement of insulators is an important sector of live-line maintenance operations (Alexandru et al., 2011; Cherney et al., 2013; Xi et al., 2014; Zareinia et al., 2010). From the perspective of motion control, effective and stable control of the mechanical arm trajectory tracking motion is a prerequisite for normal maintenance operations. However, the accuracy and stability of robot mechanical arm motion control may be affected by uncertainties such as high voltage, strong electromagnetic interference, high-altitude wind load, wire galloping, flexible operating environments, vibrations and shocks caused by insulator clamping action, robot collisions, and location suspension clamps in the process of insulator replacement. In some cases, it may cause motion interference or collision between the mechanical arm and insulators, suspension clamps, hammers or other line fittings, not only leading to economic loss but also damage to the transmission lines and robot. Therefore, research on the robust motion control of the robot mechanical arm is of important theoretical value and practical significance.

To solve effectively the influence of disturbances and uncertainty factors on the accuracy and stability of robot mechanical arm motion control, domestic and foreign research institutions and scholars in related fields have proposed different methods, which may be mainly divided into adaptive control and robust control. Lewis et al. (2009) conducted an online study of the manipulator’s unknown parameters using an artificial neural network (ANN) adaptive control method, to compensate for the effects of system disturbances and parameter uncertainties in real time; Kim et al. (2005) obtained the optimal control law of the manipulator system under uncertainties by solving the Riccati equation. Although both of these achieved a sound control effect, their methods exhibited poor scalability and versatility, which led to compulsory redesign of the controller upon any changes in the mechanical arm structure. Zhao et al. (2007) and Liu et al. (2010) studied underwater robot robust control based on the mixed sensitivity and μ methods, respectively. Both methods showed a good inhibitory effect against external disturbance and variable parameters, but also suffered inferior adaptability to arm structural changes. Moreover, most research remained theoretical. Tang et al. (2011) proposed robust control of a dual-arm space robot based on the backstepping method. Through selecting reasonable controller parameters, it may eliminate some non-linear terms, to avoid the singular value problems in controller design using the traditional backstepping method. Considering trajectory tracking control, Khaloozadeh and Homaeinejad (2014) proposed a variable structure compensation controller based on bounded uncertainties and equivalent control law, which gave a real-time solution to the tracking problem, Bouakrif and Zasadzinski (2016) studied trajectory tracking control of perturbed robot manipulators using the iterative learning method and the effectiveness of the proposed method was verified mainly by simulation analysis. Galeani et al. (2012), Huang and Chiang (2015), Kayacan et al. (2015) and Lai et al (2016), proposed different manipulator robust tracking control methods on different types of robots under different uncertain factors. All solved the influences of various disturbances and uncertainties on robot operation and improved the quality of robot control system to a certain extent. However, the scalability, versatility and adaptability of the manipulator robust motion controllers proposed in above methods need to be further enhanced. Moreover, due to the diversity of robots and the relative independence of service objects, the robots proposed in the above studies were not or even may never be applied as live-line maintenance robots operating under high-voltage and strong electromagnetic interference. Concerning the transmission line robot, Wang et al. (2010, 2011) proposed an HJI inequality-based robust control method based on establishing an inspection robot manipulator motion model that studied the robust control of a transmission line inspection robot under high voltage and strong electromagnetic interference. It is easy to see that all the above studies only focused on robot control at a macro level and disturbances are mostly sourced from outside of the system. Therefore, a key point is to improve the adaptive capability by analysing the coordination motion of mechanical arm joints and robust control of a live-line maintenance robot, especially at a micro level under the current specific operating environments of high-voltage transmission lines.

As the motion state of the mechanical arm is the sum of all joint movements, to improve further the robustness of the robot mechanical arm motion and versatility–scalability of the robot controller, this paper established a dynamical model of mechanical arm motion control by combining the armature voltage equation of the joint motor, decomposing the structure of the robot control system, analysing the dynamical model of the mechanical arm basic motions from the micro level, and considered the influences of internal and external disturbances (including high voltage and strong electromagnetics etc.) and uncertainties on the mechanical arm motion. Compared with the traditional control methods, H∞ robust control theory, which regards the system infinite norm as the performance index, is a relatively more mature theoretical system for solving disturbances and uncertainties at present. Therefore, this paper proposed the H∞ theory-based robust trajectory tracking control method for a robot mechanical arm for live-line maintenance of a high-voltage transmission line. In addition, the H∞ controller was solved by LMI, the stability of the system was proved by selecting the appropriate Lyapunov function, and the validity and engineering practicability of the method were verified by simulation results and field operation tests, so that a solid guarantee may be provided for actual robot maintenance operations.

Structure and operation principle of the robot

Robot structure

The configuration and entity structure of the live-line maintenance robot is shown in Figure 1. The robot mainly comprised a control box (part 1), double manipulators (parts 2, 3 and 4), double mobile mechanical arms (parts 5 and 6), the walking and holding mechanism (parts 7, 8, 9 and 10), and several other parts. The mechanical arm 1 (part 5) fixed on the body has three degrees of freedom, including rotation, stretch out and draw back, and vertical move, whereas mechanical arm 2 (part 6) has a degree of freedom of horizontal move, in addition to the three degrees of freedoms of mechanical arm 1. Both mechanical arms in total have seven degrees of freedom. The robot manipulator constituted a socket eyes gripper mechanism (part 3) and W pin pushing mechanism (part 4) on mechanical arm 1, as well as an insulator string clamping mechanism (part 2) on mechanical arm 2. The socket eyes gripper mechanism may clamp insulator socket eyes by a mobile joint, so the insulator string is fixed. The W pin pushing mechanism contains two mobile joints, which may be used to realize the pushing out of the insulator connection component W pin. The insulator string clamping mechanism, which comprises a crank rocker mechanism, clamps the insulator steel cap with a clamping jaw, and then realizes the push-out or push-in of the insulator string through forward and backward movements of the horizontal joint.

Operation object and operation principle

The operation object of the live-line maintenance robot is a suspension insulator string, the connection structure and operation environment of which is shown in Figure 2. The insulator string (part 1) is connected to the transmission line (part 2) through suspension clamps (part 3). The insulator and insulator are connected through the bowl head hanging broad (part 4). In Figure 3, the key to replacing the insulator string is to roll out the joint connection component W pin (part 2) using the W pin pushing block (part 1); the W pin is located in the bowl head notch, which is shown in Figure 3(a). The final step is to push the new insulator string ball head (part 3) into the bowl head notch and push-in W pin, so that the free state insulator string (part 4) is fixed and the entire replacement is completed. In the entire process, the robot is mainly responsible for clamping the insulator socket eyes, pushing the W pin in and out, pushing the ball head in and out, whereas artificial work mainly includes tightening or loosening the wire, and insulator replacement. By way of man–machine coordination, the replacement operation of the insulator strings may be achieved.

Robot control architecture

The control system structure of the live-line maintenance robot is shown in Figure 4. In the process of insulator replacement, the robot control system structure may be divided into three layers from top to bottom, including an intelligent decision-making layer, task coordination layer and action execution layer. As the top layer, the intelligence decision-making layer is the centre of the robot, responsible for giving all control commands and monitoring the status information of the task coordination layer and action executive layer. The task coordination layer is responsible for several key operations, including robot online walking, suspension clamp location, bowl head hanging broad clamping, insulator clamping, W pin pushing out and pushing in. These operations are the embodiment of the total basic actions of all joints of the executive layer. The action executive layer is the bottom layer, which controls the basic movements of each joint, such as double arm rotation, stretch, vertical and horizontal movement, walking wheel and manipulator control. All layers are connected by the human–machine interactive (HMI) system. Different operations may be realized through planning and coordination of the basic movements of all joints in the robot action execution layer. From the perspective of control, live-line maintenance robot control may be categorized into macro and micro levels. At the macro level, three layers are combined as a whole. At the micro level, different movements of the robot mechanical arm and manipulator are focused. In this paper, the extensibility of the robot control system was enhanced by modelling the basic movements of the robot mechanical arm at the micro level.

Dynamic model of mechanical arm joint motion

Dynamic model method for n joint robot

The general robot joint may be divided into two categories – movement joint and rotation joint; set the ground as the reference plane, set the vertical to the ground as the Y-axis, then the motion characteristics of the movement joint may be divided into two categories according to whether changes in potential energy and joint motion model may be established by the Lagrange method. Regarding the rotation joint, the joint motion model may be established by the relation equation between the rotational inertia and torque. Therefore, suppose m_i is the mass of mechanical arm, the centroid displacement of the mechanical arm to which the movement joint i belongs is s_i, $E_{A i}$ is the mechanical arm kinetic energy at joint i movement, $E_{P i}$ is the mechanical arm potential energy at joint i movement and $τ_{i}$ is the sum of external forces on the mechanical arm movement at joint i. The kinetic equations of the two categories of movement joints are obtained as Equation (2) by the Lagrange Equation (1), wherein Equation (2a) is the joint kinetic equation when the potential energy is kept constant and Equation (2b) is the joint kinetic equation when the potential energy is variable.

\frac{d}{d t} (\frac{\partial E_{Ai}}{\partial {\overset{\cdot}{s}}_{i}}) - \frac{\partial E_{A i}}{\partial s_{i}} + \frac{\partial E_{P i}}{\partial s_{i}} = τ_{i}

(1)

{\begin{matrix} m_{i} {\overset{\cdot\cdot}{s}}_{i} = τ_{i} (a) \\ m_{i} {\overset{\cdot\cdot}{s}}_{i} + m_{i} g = τ_{i} (b) \end{matrix}

(2)

Supposing the rotation angle of the rotation joint i is $θ_{i}$ , the mechanical arm rotational inertia is J_i and $τ_{i}$ is the sum of external forces on mechanical arm rotation joint i, then Equation (3) is obtained by the relation equation between the rotational inertia and torque.

J_{i} {\overset{\cdot\cdot}{θ}}_{i} = τ_{i}

(3)

Therefore, the joint motion kinetic model may be directly obtained by Equations (2) and (3) according to the joint motion characteristics, which not only greatly improve the efficiency of robot joint motion modelling but also further demonstrate the versatility, adaptability and scalability of the modelling method.

Establishment of coordinate system

In the process of insulator string replacement, the basic movements of the mechanical arm mainly include rotation, stretch, and vertical and horizontal movement, as mechanical arm 2 has one more horizontal joint than mechanical arm 1. Therefore, mechanical arm 2 was adopted as the research object in this work, and the basic actions of the mechanical arm 2 kinetic model was established. In order to facilitate analysis and modelling of mechanical arm 2, four coordinate systems O_i(X_i,Y_i,Z_i) (i=1,2,3,4) for the four different actions were established, respectively, as shown in Figure 5(a). It may be seen that each joint is set as the coordinate origin, the forward direction of the horizontal moving joint is set as the X-axis in the positive direction, the extending direction of the stretch joint is set as the Y-axis in the positive direction and the inward moving direction of the vertical move joint is set as the Z-axis in the positive direction; a motion direction sketch of the different actions is shown in Figure 5(b).

Horizontal movement

Suppose x is the coordinate of the mechanical arm 2 centroid, m₂ is the total mass of mechanical arm 2 and manipulator 2, and m₂* is the mass of manipulator 2. The horizontal moving velocity of mechanical arm 2 is $v = \overset{\cdot}{x}$ and the kinetic energy while horizontally moving is $E_{A 1} = \frac{1}{2} m_{2} (\overset{\cdot}{x})^{2}$ . During horizontal movement, its potential energy variable is $Δ E_{P 1} = 0$ , as the vertical coordinate of the mechanical arm 2 centroid remains unchanged. $τ_{1}$ is the sum of the external forces on the mechanical arm 2 horizontal joint. By substituting $E_{A 1}$ , $Δ E_{P 1}$ and $τ_{1}$ into the Lagrange Equation (1), we may obtain kinetic Equation (4a) for mechanical arm 2 in horizontal movement. If k₁ represents the angular velocity of the horizontal motor of mechanical arm 2 as well as the displacement conversion constant, we may obtain Equation (4b) according to the relationship between the displacement and screw rod angular velocity $θ_{1}$ .

{\begin{matrix} m_{2} \overset{\cdot\cdot}{x} = τ_{1} (a) \\ m_{2} k_{1} {\overset{\cdot\cdot}{θ}}_{1} = τ_{1} (b) \end{matrix}

(4)

Combining the armature voltage balance equation of the horizontal joint motor, we may obtain Equation (5) as follows.

{\begin{matrix} m_{2} k_{1} {\overset{\cdot\cdot}{θ}}_{1} = K_{M 1} i_{1} \\ L_{1} {\overset{\cdot}{i}}_{1} + R_{1} i_{1} + U_{a 1} = U_{1}, (U_{a 1} = K_{A 1} {\overset{\cdot}{θ}}_{1}) \end{matrix}

(5)

During vertical movement, as the vertical coordinate of the mechanical arm 2 centroid remains unchanged and the potential energy variable is 0, the derivation process and the result of the kinetic equation is exactly the same as that when mechanical arm 2 is in horizontal movement, so we not repeat this here.

Movement of stretch

The only difference between stretch and horizontal movement is that the vertical coordinate of the mechanical arm 2 centroid is changed during the stretch movement, which will lead to a change of potential energy. Supposing that y is the vertical coordinate of the mechanical arm 2 centroid, then the kinetic energy is $E_{A 2} = \frac{1}{2} m_{2} (\overset{\cdot}{y})^{2}$ , the potential energy is $E_{P 2} = m_{2} g y$ and $τ_{2}$ is the sum of external forces on the stretch joint of mechanical arm 2. By substituting $E_{A 2}$ , $E_{P 2}$ and $τ_{2}$ into the Lagrange Equation (1), we may obtain kinetic Equation (6a) for mechanical arm 2 in stretch movement. If k₂ is the angular velocity of the stretch movement motor of mechanical arm 2 and the displacement conversion is constant, we may obtain Equation (6b) according to the relationship between the displacement and screw rod angular velocity $θ_{2}$ .

{\begin{matrix} m_{2} \overset{\cdot\cdot}{y} + m_{2} g = τ_{2} (a) \\ m_{2} k_{2} {\overset{\cdot\cdot}{θ}}_{2} + m_{2} g = K_{M 2} i_{2} (b) \end{matrix}

(6)

Combining the armature voltage balance equation of the stretch joint motor, we may obtain Equation (7).

{\begin{matrix} m_{2} k_{2} {\overset{\cdot\cdot}{θ}}_{2} + m_{2} g = K_{M 2} i_{2} \\ L_{2} {\overset{\cdot\cdot}{θ}}_{2} + R_{2} i_{2} + U_{a 2} = U_{2}, (U_{a 2} = K_{A 2} {\overset{\cdot}{θ}}_{2}) \end{matrix}

(7)

Movement of rotation

Supposing that the rotary inertia of mechanical arm 2 is J when performing rotation movement, the rotation angle of rotation motor is $θ_{3}$ and $τ_{3}$ is the sum of the external forces on the rotation joint of mechanical arm 2, we may obtain Equation (8) according to the relationship between the rotation torque and rotary inertia.

J {\overset{\cdot\cdot}{θ}}_{3} = τ_{3}

(8)

Combining the armature voltage balance equation of the rotation joint motor, we may obtain Equation (9).

{\begin{matrix} J {\overset{\cdot\cdot}{θ}}_{3} = K_{M 3} i_{3} \\ L_{3} {\overset{\cdot}{i}}_{3} + R_{3} i_{3} + U_{a 3} = U_{3}, (U_{a 3} = K_{A 3} {\overset{\cdot}{θ}}_{3}) \end{matrix}

(9)

L_x, R_x, i_x, U_x, U_ax, K_Mx and K_ax represent the armature circuit inductance, resistance, current, voltage, reverse potential and related parameters of the joint motor, respectively (the motors may be the horizontal movement motor, stretch movement motor, rotation movement motor and vertical movement motor when x=1, 2, 3 and 4, respectively).

Unified dynamical model

In order to facilitate modelling and analysis of the mechanical arm motion control, Equations (5), (7) and (9) may be described by a unified expression as equation (10).

{\begin{matrix} P \overset{\cdot\cdot}{θ} + Q = K_{M} i (a) \\ L \overset{\cdot}{i} + R i + K_{A} \overset{\cdot}{θ} = U (b) \end{matrix}

(10)

Supposing $ω = \overset{\cdot}{θ}$ , where $ω$ represents rotation velocity of joint motor, the unified dynamical model of the basic movements of the mechanical arm joint may be turned into Equation (11), which is divided into four cases and discussed in Table 1 regarding the constant P and Q.

{\begin{matrix} P \overset{\cdot}{ω} + Q = K_{M} i (a) \\ L \overset{\cdot}{i} + R i + K_{A} ω = U (b) \end{matrix}

(11)

Table 1.

Discussion of mechanical arm 2 kinetic equation.

	Parameter value	Meaning
Case 1	$P = m_{2} k_{1}, Q = 0, K_{M} = K_{M 1}, L = L_{1}, R = R_{1}, K_{A} = K_{A 1}$	Horizontal movement kinetic equation
Case 2	$P = m_{2} k_{2}, Q = m_{2} g, K_{M} = K_{M 2}, L = L_{2}, R = R_{2}, K_{A} = K_{A 2}$	Stretch movement kinetic equation
Case 3	$P = J, Q = 0, K_{M} = K_{M 3}, L = L_{3}, R = R_{3}, K_{A} = K_{A 3}$	Rotation movement kinetic equation
Case 4	$P = {m_{2}}^{*} k_{4}, Q = 0, K_{M} = K_{M 4}, L = L_{4}, R = R_{4}, K_{A} = K_{A 4}$	Vertical movement kinetic equation

H∞ trajectory tracking control of mechanical arm

Structure and process

H∞ tracking control structure of the live-line maintenance robot mechanical arm is shown in Figure 6. u is the input signal for system control, d is the sum of the internal–external disturbance input signal and uncertainty factors, z is the modulated output signal, y is the measurement output signal, X is the controlled object and K is the feedback control. The goal of H∞ control is to design a state feedback control, to eliminate the influences of interference and uncertainty factors on the controlled object, and thus to guarantee a normal operation of the controlled object even under disturbances and uncertainties. The transfer function from d to z is $T_{d z} (s)$ . If there is K that may minimize the value of $‖ T_{d z} (s) ‖_{\infty}$ , then K is called the optimal H∞ control of X. The H∞ controller may be directly solved by the MATLAB LMI toolbox.

The H∞ trajectory tracking control process of the live-line maintenance robot mechanical arm is shown in Figure 7. The four joints are controlled by the H∞ controller and d is the sum of the internal–external disturbances and uncertainty factors. By outputting the difference between the actual angular velocity and the expected angular velocity, we may obtain the motion errors of the four joints, which may be adopted as the input signal for the controller. By adjustment of the H∞ controller, as the system error approaches 0, mechanical arm 2 shifts from actual posture tracking to the expected posture, and the manipulator gradually locates to the operation object, so that the whole control process is finished.

Establishment of H∞ motion control mode

(a) Ideal model

By ignoring the influences of disturbances and uncertainties on mechanical arm movement, according to the unified expression (11a) of the basic movement kinetic equation of the robot executive layer, we may obtain the joint armature current expressed by Equation (12).

i = \frac{P}{K_{M}} \overset{\cdot}{ω} + \frac{Q}{K_{M}}

(12)

By substituting Equation (12) into Equation (11b), we may obtain Equation (13).

\frac{L P}{K_{M}} \overset{\cdot\cdot}{ω} + \frac{R P}{K_{M}} \overset{\cdot}{ω} + K_{A} ω = U - \frac{R Q}{K_{M}}

(13)

If $ω$ is the actual angular velocity of joint motor, $ω_{d}$ is the expected angular velocity of the joint motor, and by substituting $ω_{d}$ for $ω$ in Equation (13), we may obtain Equation (14).

\frac{L P}{K_{M}} {\overset{\cdot\cdot}{ω}}_{d} + \frac{R P}{K_{M}} \dot{ω_{d}} + K_{A} ω_{d} = U - \frac{R Q}{K_{M}}

(14)

The angular velocity error of the joint is defined as $e = ω - ω_{d}$ , and we may obtain the joint motion error in Equation (15) by subtracting Equation (13) from Equation (14).

\frac{L P}{K_{M}} \overset{\cdot\cdot}{e} + \frac{R P}{K_{M}} \overset{\cdot}{e} + K_{A} e = 0

(15)

By defining the joint state variable as $x = [\begin{matrix} e \\ \overset{\cdot}{e} \end{matrix}]$ , Equation (15) may be turned into the state-space form shown as in Equation (16).

[\begin{matrix} \overset{\cdot}{e} \\ \overset{\cdot\cdot}{e} \end{matrix}] = [\begin{matrix} 0 & 1 \\ - \frac{K_{A} K_{M}}{L P} & - \frac{R}{L} \end{matrix}] [\begin{matrix} e \\ \overset{\cdot}{e} \end{matrix}]

(16)

Suppose $x_{m n} = [\begin{matrix} e_{m n} \\ {\overset{\cdot}{e}}_{mn} \end{matrix}]$ (where m represents the number of mechanical arms and n represents the number of mechanical arm different actions), the mechanical arm motion state-space mode of an m arms and n actions robot system may be established as formula (17), wherein $Φ_{i (2 \times 2)} (i = 1 \cdot \cdot \cdot m \times n)$ is the robot mechanical arm m×n joint action state matrix.

[\begin{matrix} \overset{\cdot}{x_{11}} \\ \dots \\ \overset{\cdot}{x_{1 r}} \\ \overset{\cdot}{x_{21}} \\ \dots \\ \overset{\cdot}{x_{2 v}} \\ \dots \\ \overset{\cdot}{x_{m 1}} \\ \dots \\ \overset{\cdot}{x_{m n}} \end{matrix}] = [\begin{matrix} Φ_{1 (2 \times 2)} & 0 & \dots & \dots & \dots & \dots & 0 \\ 0 & Φ_{2 (2 \times 2)} & \dots & \dots & \dots & \dots & 0 \\ . & \dots & . & \dots & \dots & \dots & . \\ . & \dots & \dots & . & \dots & \dots & . \\ . & \dots & \dots & \dots & . & \dots & . \\ 0 & \dots & \dots & \dots & \dots & Φ_{mn - 1 (2 \times 2)} & 0 \\ 0 & \dots & \dots & \dots & \dots & 0 & Φ_{m n (2 \times 2)} \end{matrix}] [\begin{matrix} x_{11} \\ \dots \\ x_{1 r} \\ x_{21} \\ \dots \\ x_{2 v} \\ \dots \\ x_{m 1} \\ \dots \\ x_{mn} \end{matrix}]

(17)

As the research object is robot mechanical arm 2, then Equation (17), with m=2, n=1, 2, 3 and 4, represents the state variable of mechanical arm 2 horizontal movement, stretch movement, rotation movement and vertical movement, respectively. Therefore, the state-space model of mechanical arm 2 motion control is shown as formula (18).

[\begin{matrix} \overset{\cdot}{x_{21}} \\ \overset{\cdot}{x_{22}} \\ \overset{\cdot}{x_{23}} \\ \overset{\cdot}{x_{24}} \end{matrix}] = [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ - \frac{K_{A 1} K_{M 1}}{R_{1} m_{2} k_{1}} & - \frac{R_{1}}{L_{1}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & - \frac{K_{A 2} K_{M 2}}{R_{2} m_{2} k_{2}} & - \frac{R_{2}}{L_{2}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & - \frac{K_{A 3} K_{M 3}}{R_{3} J} & - \frac{R_{3}}{L_{3}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{K_{A 4} K_{M 4}}{R_{4} {m^{*}}_{2} k_{4}} & - \frac{R_{4}}{L_{4}} \end{matrix}] [\begin{matrix} x_{21} \\ x_{22} \\ x_{23} \\ x_{24} \end{matrix}]

(18)

From the modelling process of mechanical arm motion, it may be shown that this method is of good versatility, strong adaptability and sound expansibility. If more mechanical arms or different types of mechanical arm actions are added, a multi-arm multi-action robot system motion control model may be easily obtained by extending the recurrence of the state-space model in formula (17) layer by layer, which may not only further shorten the development cycle of the robot control system but also may increase the application scope of the proposed method.

Actual mode

In order to eliminate influences of disturbances and uncertainties on the mechanical arm motion control performance, an external control variable must be introduced to offset the internal–external disturbances and uncertainties, to enhance the robustness of the mechanical arm motion control. As compared with error Equation (15) of the mechanical arm joint under ideal conditions, it is more general to set u as the system input and set d as the sum of the disturbance signals and uncertainties, then we may obtain error Equation (19) under disturbances and uncertainty factors, wherein the role of the system input u is to suppress the influences of disturbances and uncertainty factor d on the system stability through the appropriate control variable.

\frac{L P}{K_{M}} \overset{\cdot\cdot}{e} + \frac{R P}{K_{M}} \overset{\cdot}{e} + K_{A} e = u + d

(19)

Equation (19) is rewritten in the form of state-space, as shown as formula (20).

[\begin{matrix} \overset{\cdot}{e} \\ \overset{\cdot\cdot}{e} \end{matrix}] = [\begin{matrix} 0 & 1 \\ - \frac{K_{A} K_{M}}{R P} & - \frac{R}{L} \end{matrix}] [\begin{matrix} e \\ \overset{\cdot}{e} \end{matrix}] + [\begin{matrix} 0 \\ \frac{K_{M}}{L P} \end{matrix}] u + [\begin{matrix} 0 \\ \frac{K_{M}}{L P} \end{matrix}] d

(20)

Therefore, under the disturbances and uncertainties, the mechanical arm motion state-space model of m arms and n actions robot system is shown in formula (21), wherein $Φ_{i (2 \times 2)} (i = 1 \cdot \cdot \cdot m \times n)$ is the action state matrix of the robot mechanical arm with m×n joints, $Ω_{i (2 \times 1)}$ is the input matrix of different joint actions and the m×n type robot H∞ controller may be directly designed based on the established mode as formula (21).

Similarly, Equation (21) with m=2, n=1, 2, 3 and 4 represents the state variable of mechanical arm 2 horizontal movement, stretch movement, rotation movement and vertical movement, respectively. Therefore, the state-space model of mechanical arm 2 movement under the disturbances and uncertainties is shown as formula (22).

[\begin{matrix} \overset{\cdot}{x_{11}} \\ \dots \\ \overset{\cdot}{x_{1 r}} \\ \overset{\cdot}{x_{21}} \\ \dots \\ \overset{\cdot}{x_{2 v}} \\ \dots \\ \overset{\cdot}{x_{m 1}} \\ \dots \\ \overset{\cdot}{x_{m n}} \end{matrix}] = [\begin{matrix} Φ_{1 (2 \times 2)} & 0 & \dots & \dots & \dots & \dots & 0 \\ 0 & Φ_{2 (2 \times 2)} & \dots & \dots & \dots & \dots & 0 \\ . & \dots & . & \dots & \dots & \dots & . \\ . & \dots & \dots & . & \dots & \dots & . \\ . & \dots & \dots & \dots & . & \dots & . \\ 0 & \dots & \dots & \dots & \dots & Φ_{m n - 1 (2 \times 2)} & 0 \\ 0 & \dots & \dots & \dots & \dots & 0 & Φ_{m n (2 \times 2)} \end{matrix}] [\begin{matrix} x_{11} \\ \dots \\ x_{1 r} \\ x_{21} \\ \dots \\ x_{2 v} \\ \dots \\ x_{m 1} \\ \dots \\ x_{m n} \end{matrix}] + [\begin{matrix} Ω_{1 (2 \times 1)} \\ Ω_{2 (2 \times 1)} \\ \dots \\ \dots \\ \dots \\ Ω_{(\frac{m n - 1}{2}) (2 \times 1)} \\ Ω_{(\frac{m n}{2}) (2 \times 1)} \end{matrix}] u + [\begin{matrix} Ω_{1 (2 \times 1)} \\ Ω_{2 (2 \times 1)} \\ \dots \\ \dots \\ \dots \\ Ω_{(\frac{m n - 1}{2}) (2 \times 1)} \\ Ω_{(\frac{m n}{2}) (2 \times 1)} \end{matrix}] d

(21)

[\begin{matrix} \overset{\cdot}{x_{21}} \\ \overset{\cdot}{x_{22}} \\ \overset{\cdot}{x_{23}} \\ \overset{\cdot}{x_{24}} \end{matrix}] = [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ - \frac{K_{A 1} K_{M 1}}{R_{1} m_{2} k_{1}} & - \frac{R_{1}}{L_{1}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & - \frac{K_{A 2} K_{M 2}}{R_{2} m_{2} k_{2}} & - \frac{R_{2}}{L_{2}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & - \frac{K_{A 3} K_{M 3}}{R_{3} J} & - \frac{R_{3}}{L_{3}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{K_{A 4} K_{M 4}}{R_{4} {m^{*}}_{2} k_{4}} & - \frac{R_{4}}{L_{4}} \end{matrix}] [\begin{matrix} x_{21} \\ x_{22} \\ x_{23} \\ x_{24} \end{matrix}] + [\begin{matrix} 0 \\ \frac{K_{M 1}}{L_{1} m_{2} k_{1}} \\ 0 \\ \frac{K_{M 2}}{L_{2} m_{2} k_{2}} \\ 0 \\ \frac{K_{M 3}}{L_{3} J} \\ 0 \\ \frac{K_{M 4}}{L_{4} {m^{*}}_{2} k_{4}} \end{matrix}] u + [\begin{matrix} 0 \\ \frac{K_{M 1}}{L_{1} m_{2} k_{1}} \\ 0 \\ \frac{K_{M 2}}{L_{2} m_{2} k_{2}} \\ 0 \\ \frac{K_{M 3}}{L_{3} J} \\ 0 \\ \frac{K_{M 4}}{L_{4} {m^{*}}_{2} k_{4}} \end{matrix}] d

(22)

Supposing that:

A = [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ - \frac{K_{A 1} K_{M 1}}{R_{1} m_{2} k_{1}} & - \frac{R_{1}}{L_{1}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & - \frac{K_{A 2} K_{M 2}}{R_{2} m_{2} k_{2}} & - \frac{R_{2}}{L_{2}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & - \frac{K_{A 3} K_{M 3}}{R_{3} J} & - \frac{R_{3}}{L_{3}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{K_{A 4} K_{M 4}}{R_{4} {m^{*}}_{2} k_{4}} & - \frac{R_{4}}{L_{4}} \end{matrix}], B = [\begin{matrix} 0 \\ \frac{K_{M 1}}{L_{1} m_{2} k_{1}} \\ 0 \\ \frac{K_{M 2}}{L_{2} m_{2} k_{2}} \\ 0 \\ \frac{K_{M 3}}{L_{3} J} \\ 0 \\ \frac{K_{M 4}}{L_{4} {m^{*}}_{2} k_{4}} \end{matrix}]

(23)

And

X = [\begin{matrix} x_{21} \\ x_{22} \\ x_{23} \\ x_{24} \end{matrix}]

(24)

Under the conditions of disturbance and uncertainties, the mechanical arm motion control model may be simplified as formula (25).

\overset{\cdot}{X} = AX + Bu + Bd

(25)

Therefore, the H∞ motion control model of the robot mechanical arm may be established based on formula (26).

{\begin{matrix} \overset{\cdot}{X} = AX + Bu + Bd (a) \\ z = CX + D_{1} u + D_{2} d (b) \end{matrix}

(26)

Wherein formula (26a) is derived from formula (22) and formula (26b) is the human-controlled modulation signal. During the design of the mechanical arm motion control system formula (22), the state feedback control law u = KX should be considered, so that it may make closed-loop system formula (27) asymptotically stable on the basis of meeting H∞ performance condition formula (28).

{\begin{matrix} \overset{\cdot}{X} = (A + BK) X + Bd \\ z = (C + D_{1} K) X + D_{2} d \end{matrix}

(27)

‖ T_{d z} (s) ‖_{\infty} = ‖ ((C + D_{1} K) {[S I - (A + BK)]}^{- 1} B + D_{2}) ‖_{\infty} < γ

(28)

Solving H∞ motion controller

The solution to the H∞ controller requires the involvements of Theorems 1 and 2, wherein Theorem 1 has been proved by Meng (2013) and Theorem 2 will be proved in this work.

Theorem 1. For the system ${\begin{matrix} \overset{\cdot}{x} = Ax + Bd \\ z = Cx + Dd \end{matrix}$ , suppose $γ$ is a given positive constant, then the following two conditions are equivalent:

The corresponding closed-loop system is asymptotic stable, which meets the H∞ performance of $‖ T_{d z} ‖_{\infty} < γ$ .

There is a positive-definite matrix $ξ$ , which meets the conditions of linear matrix inequality (LMI) (29).

[\begin{matrix} A^{T} ξ + ξ A & ξ B & C^{T} \\ B^{T} ξ & - γ I & D^{T} \\ C & D & - γ I \end{matrix}] < 0

(29)

Theorem 2.For the system ${\begin{matrix} \overset{\cdot}{x} = Ax + B_{1} d + B_{2} u \\ z = C_{1} x + D_{11} d + D_{12} u \end{matrix}$ , after designing the state feedback $u = Kx$ , the closed-loop system may be obtained as ${\begin{matrix} \overset{\cdot}{x} = (A + B_{2} d) x + B_{1} d \\ z = (C_{1} + D_{12} K) x + D_{11} d \end{matrix}$ . According to Theorem 1, the system is stable and may meet the condition $‖ T_{d z} ‖_{\infty} < γ (γ = 1)$ . Only when there is a symmetric positive-definite matrix W, $φ$ , may the LMI (30) be established. If the matrix inequality has a set of feasible solutions such as $W^{*}$ , $φ^{*}$ , then $u = W^{*} (φ^{*})^{- 1} x$ is a state feedback H∞ controller of the system.

[\begin{matrix} A φ + B_{2} W + {(A φ + B_{2} W)}^{T} & B_{1} & {(C_{1} φ + D_{12} W)}^{T} \\ {B_{1}}^{T} & - I & {D_{11}}^{T} \\ C_{1} φ + D_{12} W & D_{11} & - I \end{matrix}] < 0

(30)

Proof. From Theorem 1, it may be shown that if the closed-loop system

{\begin{matrix} \overset{\cdot}{x} = (A + B_{2} K) x + B_{1} d \\ z = (C_{1} + D_{12} K) x + D_{11} d \end{matrix}

satisfies the performance conditions of the H∞ controller and is asymptotically stable, then it should meet the conditions of the LMI formula (29). The A , B , C and D in formula (29) are revised into $A + B_{2} K$ , $B_{2}$ , $C_{1} + D_{12} K$ and $D_{11}$ ; then we may obtain formula (31) as follows.

[\begin{matrix} {(A + B_{2} K)}^{T} ξ + ξ (A + B_{2} K) & ξ B_{1} & {(C_{1} + D_{12} K)}^{T} \\ {B_{1}}^{T} ξ & - I & {D_{11}}^{T} \\ C_{1} + D_{12} K & D_{11} & - I \end{matrix}] < 0

(31)

When the matrix in the left part of the inequality (31) is premultiplied and postmultiplied by $d i a g {ξ^{- 1}, I, I}$ , inequality (32), which is equivalent to formula (31) may be obtained as follows.

[\begin{matrix} A ξ^{- 1} + B_{2} K ξ^{- 1} + {(A ξ^{- 1} + B_{2} K ξ^{- 1})}^{T} & B_{1} & {(C_{1} ξ^{- 1} + D_{12} K ξ^{- 1})}^{T} \\ {B_{1}}^{T} & - I & {D_{11}}^{T} \\ C_{1} ξ^{- 1} + D_{12} K ξ^{- 1} & D_{11} & - I \end{matrix}] < 0

(32)

As the closed-loop system (27) formula is stable, the matrix $ξ$ must exist. In inequality (32), if $φ^{*} = ξ^{- 1}$ and $W^{*} = K ξ^{- 1}$ are defined respectively, inequality (30) may be obtained by inequality (32) and thus Theorem 2 is verified. By comparing inequality (32) and inequality (30), it may be shown that $φ^{*} = ξ^{- 1}$ and $W^{*} = K ξ^{- 1}$ is a set of feasible solutions to inequality (30) and therefore, the H∞ controller of the system (26) may be also obtained.

In order to solve the H∞ controller, system matrix A and B should be obtained first. To simplify the calculation process, the parameters of the joint motor are selected as K_A=0.5 V·s/rad, K_M=1 N·m/A, $R = 1 Ω$ , L=0.1 H, the armature voltage is 24 V, the total mass of mechanical arm 2 and manipulator 2 is m₂=5 kg, the mass of insulator clamp mechanism is m₂^*=1 kg, the length of mechanical arm 2 is 0.47 m, the rotary inertia of the rotation joint of mechanical arm 2 is J =1.105 kg·m², the conversion constant between the angular velocity and displacement of joint motor is k=2.5, and the gravitational constant is g= 9.8 m/s². By substituting the above parameters into formula (23), matrices A and B may be calculated as formula (33).

A = [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ - \frac{1}{25} & - 10 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & - \frac{1}{25} & - 10 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & - \frac{5}{11} & - 10 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{1}{5} & - 10 \end{matrix}], B = [\begin{matrix} 0 \\ \frac{4}{5} \\ 0 \\ \frac{4}{5} \\ 0 \\ 9.1 \\ 0 \\ 4 \end{matrix}]

(33)

The system controllability matrix may be obtained through formula (33), which is shown in formula (34). The determinant of matrix P_c is not equal to 0, which proves that the system controllable and the status feedback controller design is reasonable and practicable.

| P_{c} | = | \begin{matrix} A & AB & A^{2} B & A^{3} B & A^{4} B & A^{5} B & A^{6} B & A^{7} B \end{matrix} | = 6.88 \neq 0

(34)

By substituting matrices A and B into LMI formulae (30) and (31), we may solve a feasible solution $φ^{*}$ and $W^{*}$ through taking reasonable C ₁, D ₁₁ and D ₁₂ using the MATLAB LMI toolbox FEASP solver. From Theorem 2, it may be shown that feedback control matrix K is formula (35), and therefore the motion controller of the robot mechanical arm corresponds to formula (36).

K = W^{*} (φ^{*})^{- 1}

(35)

u (t) = KX = {[\begin{matrix} - 4.2516 \\ 0.1794 \\ - 0.3148 \\ 0.6791 \\ 1.7549 \\ 1.3199 \\ - 3.7955 \\ - 1.2718 \end{matrix}]}^{T} X

(36)

System stability analysis

Ideal system

In order to analyse the stability of state Equation (16) of the mechanical arm motion control system matrix,

M = [\begin{matrix} 0 & 1 \\ - \frac{K_{A} K_{M}}{R P} & - \frac{R}{L} \end{matrix}]

and matrix

N = [\begin{matrix} 1 & 0 \\ - \frac{L K_{A} K_{M}}{R^{2} P} & 1 \end{matrix}]

are ordered. By applying the contragradient transformation for matrix M , we may obtain matrix F , described as formula (37). From the nature of the matrix contragradient transformation, it may be shown that matrices M and F share the same characteristic root. According to the formula (37), it may be shown that the characteristic root of matrix F is significantly at the left side of complex plane, and therefore, M is a stable matrix. Thus, regarding the mechanical arm motion control system shown as Equation (16) under ideal conditions, it is a linear time-invariant system without an input of controlled quantity and the movements of each joint are stable.

F = N^{T} MN = [\begin{matrix} - \frac{L K_{A} K_{M}}{R^{2} P} & 1 + \frac{K_{A} K_{M}}{R P} \\ 0 & - \frac{R}{L} \end{matrix}]

(37)

Actual system

Under the condition of disturbances and uncertainties, the error equation of mechanical arm motion is shown as Equation (19). The disturbance compensation PD control u is adopted and shown as formula (38), wherein $k_{p}$ and $k_{d}$ are PD controlled proportional and differential coefficients, respectively.

u = - k_{d} \overset{\cdot}{e} - k_{p} e - d

(38)

As $k_{p}$ and $k_{d}$ are diagonal matrices under normal circumstances, the selections of $k_{p}$ and $k_{d}$ are important to the system stability. The system response speed may be accelerated by increasing $k_{p}$ ; however, the system oscillates and keeps a long adjustment time, if $k_{p}$ is set too high, whereas $k_{d}$ may be able to reduce the overshoot and shorten the adjustment time. Therefore, in the initial phase of control, a relatively larger $k_{p}$ may be used to obtain a fast response speed and then $k_{d}$ may be used to make the system much more stable under the premise of a fast response system. The actual values of $k_{p}$ and $k_{d}$ are mainly dependent on the human experiences or adjusted by an intelligent algorithm. Taking one joint of mechanical arm 2 as a research object to obtain the proper PD control parameter, the proportional gain and differential gain adjustment performance of the mechanical arm 2 joint motor is shown in Tables 2 and 3, respectively.

Table 2.

Proportional gain adjustment of mechanical arm joint motor.

$k_{p 1}$	$k_{d 1}$	Average torque (N·m)	Average velocity (m/s)	Adjustment time (s)	Overshoot
4000	−50	86.4881	0.5613	0.326	21.05%
3000	−50	84.2943	0.4988	0.265	19.28%
2000	−50	81.7575	0.4265	0.306	16.69%
1000	−50	76.7588	0.3445	0.305	12.06%
500	−50	74.7129	0.3178	0.435	7.71%
100	−50	69.5230	0.3657	3.465	0.36%
50	−50	61.4972	0.3917	3.990	0

Table 3.

Differential gain adjustment of mechanical arm joint motor.

$k_{p 1}$	$k_{d 1}$	Average torque (N·m)	Average velocity (m/s)	Adjustment time (s)	Overshoot
500	−550	52.5113	0.2055	2.825	0
500	−450	56.1571	0.2068	2.565	0
500	−350	59.8855	0.2083	2.254	0
500	−250	63.7338	0.2095	1.735	0
500	−150	69.1871	0.2240	0.595	0.23%
500	−50	74.7125	0.3172	0.435	7.71%
500	−20	81.2558	0.5466	1.944	22.98%

It is seen in Table 2 that $k_{d 1}$ remains unchanged and the system appears severe shocked when $k_{p 1} > 4000$ . If the adjustment time is defined as the requirement time of the system response curve to reach and maintain a 5% error of the steady-state value, then the system adjustment time when $k_{p 1} = 500$ is 13% of the adjustment time when $k_{p 1} = 100$ , and the other performance parameters have no distinct disadvantage. Therefore, the proportional gain is selected as $k_{p 1} = 500$ .

It is seen in Table 3 that $k_{p 1}$ remains unchanged and the system overshoot values are all zero when $k_{d 1} < - 150$ , which verify that the control variables converge to the target in a very smooth way, but with a larger adjustment time of more than 1.73 s. Therefore, the differential gain is selected as $k_{d 1} = - 150$ and, although the overshoot is 0.23%, the adjustment time is only 0.595 s. With the same principle, the expected proportional gain and differential gain of mechanical arm 2’s additional three joints may be selected, and the PD control parameter of mechanical arm 2 four joints can be expressed as formula (39).

{\begin{matrix} k_{p} = [\begin{matrix} k_{p 1} & 0 & 0 & 0 \\ 0 & k_{p 2} & 0 & 0 \\ 0 & 0 & k_{p 3} & 0 \\ 0 & 0 & 0 & k_{p 4} \end{matrix}] = d i a g (k_{p 1}, k_{p 2}, k_{p 3}, k_{p 4}) \\ k_{d} = [\begin{matrix} k_{d 1} & 0 & 0 & 0 \\ 0 & k_{d 2} & 0 & 0 \\ 0 & 0 & k_{d 3} & 0 \\ 0 & 0 & 0 & k_{d 4} \end{matrix}] = d i a g (k_{d 1}, k_{d 2}, k_{d 3}, k_{d 4}) \end{matrix}

(39)

By substituting the joint PD control parameter formula (39) into formula (38), and by substituting formula (38) into formula (19), we may obtain Equation (40).

\frac{LP}{K_{M}} \overset{\cdot\cdot}{e} + (\frac{RP}{K_{M}} + k_{d}) \overset{\cdot}{e} + (K_{A} + k_{p}) e = 0

(40)

The Lyapunov function is shown as Equation (41).

V = \frac{1}{2} (\overset{\cdot}{e})^{T} (\frac{LP}{K_{M}}) \overset{\cdot}{e} + \frac{1}{2} e^{T} (K_{A} + k_{p}) e

(41)

By taking the derivative of the Lyapunov function, we may obtain Equation (42).

\begin{matrix} \overset{\cdot}{V} = (\overset{\cdot}{e})^{T} (\frac{LP}{K_{M}}) \overset{\cdot\cdot}{e} + (\overset{\cdot}{e})^{T} (K_{A} + k_{p}) \overset{\cdot}{e} \\ = (\overset{\cdot}{e})^{T} [(\frac{LP}{K_{M}}) \overset{\cdot\cdot}{e} + (K_{A} + k_{d}) \overset{\cdot}{e}] \\ = - (\overset{\cdot}{e})^{T} (\frac{RP}{K_{M}} + k_{d}) \overset{\cdot}{e} \\ \leq 0 \end{matrix}

(42)

According to Lyapunov theory, by selecting the appropriate controlled quantity, it may ensure that the robot mechanical arm tracks from an arbitrary initial pose to the desired pose with the motion tracking error converging to zero under the PD compensation control law, and ensures the global asymptotic stability of the mechanical arm motion control system. Therefore, the system may be certainly remain stable under H∞ control.

Experiment

Simulation

To verify the effectiveness of trajectory tracking controller for the robot mechanical arm designed in this paper, the centroid trajectory tracking of mechanical arm 2 moving from the initial pose to the insulator clamping pose during the insulator replacement operation was simulated under a MATLAB environment. In the simulated motion track, the motion track of the insulator centroid clamp may be categorized into two classes. When rotation movement is performed, the motion curve approximates a parabola, whereas when the motions of stretch, horizontal or vertical movements are performed, the curve approximates a straight line. In the simulation process, the actual working conditions should be simulated as much as possible, and they may approximately simulate high current under a high-voltage strong electromagnetic environment as well as the disturbance factors such as shocks and vibrations caused by clamping movements of the mechanical arm. Taking the unit step $d = 1 (t)$ as disturbance signal $d_{1} = 1 (t)$ , then at the zero initial condition, the ideal position of manipulator 2 is given as a typical sinusoidal track $θ_{d} (t) = \sin (4 π t)$ , which is more complex than the actual track, and therefore the ideal velocity track for manipulator 2 is $ω_{d} (t) = 4 π \cos (4 π t)$ . At time t=0, the system is applied with disturbance $d_{1}$ and the first simulation research on H ∞ control is conducted. The simulation results for position tracking and velocity tracking of manipulator 2 is shown in Figures 8 and 9, respectively. For all simulation images, the red ‘*’ dotted line is the ideal position and velocity curve, whereas the black solid line is the tracking curve for the actual position and velocity, and the blue solid line is the tracking error curve, which is the same in the following simulation results.

In order to embody the superiority of the H∞ control method in dealing with the robot mechanical arm trajectory tracking under disturbance and uncertainties, the conventional PD, PID and H∞ control, sliding mode control (Ngo et al., 2012) and backstepping control methods (Li et al., 2012) were used in the simulation test under the above-mentioned conditions. Table 4 shows a comparison of the control performances of the different control methods.

Table 4.

Performance comparison of different control methods.

	Control method PD	PID	H∞	Sliding mode control	Backstepping
Error convergence speed (s)	∞	∞	0.6	2.2	1.8
Accuracy	72%	85%	98%	95%	96%
Stability	−	−	Maintains stability	Maintains stability	Maintains stability

According to the results shown in Table 4, that conventional PD and PID cannot suppress the influences of disturbances and uncertainties on the system, and the tracking error cannot converge to zero. However, H∞ control, sliding mode control and backstepping control methods show a sound capability of interference suppression. Compared with sliding mode control and backstepping control, H∞ control not only enjoys fast convergence but also shows a high tracking accuracy; therefore, H∞ control is selected as a preferred method for dealing with the problem proposed in this paper.

By analysing the first set of simulation results in Figures 8 and 9, it may be shown that the error of the velocity tracking of the manipulator shows a slight oscillation due to the shocks of the joint motor in the staring process within the initial 0.5 s, whereas after 0.5 s, when the motor turns into normal operation, such an oscillation naturally disappears. At t=0, there is a certain deviation between the actual and expected values of position and velocity of mechanical arm 2 due to the disturbance effect. Under H∞ control, the deviations of position and velocity have already approached from the 1 rad and 6.88 rad/s to the expected values at about t=0.6 s with the tracking error converging to zero. A sound position and velocity tracking performance has been maintained after t=0.6 s.

In addition to the vibrations and shock disturbance in the first set of simulations, the periodic disturbance $d_{2} = 0.1 \sin (t)$ can also be used to simulate approximately the high altitude wind load and the external disturbance factors under flexible operation environments, to compare and verify further the H∞ control trajectory tracking performance. Suppose that the ideal position signal of manipulator is given as $θ_{d} (t) = \cos (4 π t) + \sin (2 π t)$ and such a track is established by accumulating the triangular wave tracks of different frequencies. It is not only more complex than the typical sinusoidal track in the first set of experiments but also contains related characteristics of both trigonometric function and parabolic function. Under the same conditions, second simulation research is performed by applying d₁, d₂ and H∞ control. The simulation results of the trajectory tracking are shown in Figures 10 and 11, respectively.

By analysing the second set of simulation results shown in Figures 10 and 11, it may be shown that the error of the velocity tracking of the manipulator shows a slight irregular oscillation due to the shocks of the joint motor in the staring process within the initial 0.5 s, whereas after 0.5 s, when the motor turns into normal operation, such an oscillation naturally disappears, which is consistent with the situations in the first simulation research. Under H∞ control, the tracking deviations of the initial position and velocity approach from 1 rad and 13 rad/s to the expected values. The errors of position and velocity meet the requirements of the mechanical arm motion. Table 5 shows a performance comparison of the two sets of experiments when using the H∞ control method.

Table 5.

Performance comparison of the two sets experiment using H∞ control method.

Group	(I)						(II)
(Ideal trajectory/Disturbance)	$\begin{matrix} θ_{d} (t) = \sin (4 π t) \\ d = 1 (t) \end{matrix}$						$\begin{matrix} θ_{d} (t) = \cos (4 π t) + \sin (2 π t) \\ d = 1 (t) + 0.1 \sin (t) \end{matrix}$
Performance index	Position			Velocity			Position			Velocity
Performance index	Max	Min	RMS	Max	Min	RMS	Max	Min	RMS	Max	Min	RMS
Trajectory tracking error	0.99	0	0.21	7.9	−0.5	0.35	0.98	−0.1	0.28	9.8	−2	0.56
Convergence speed (s)	0.6			0.5			1.0			1.2
Accuracy	99%			97%			96%			95%
Stability	Maintains stability			Maintains stability			Maintains stability			Maintains stability

Combining the simulation results of the two sets of simulation experiments, it becomes apparent that H∞ control may suppress the influences of strong impact, periodic disturbance signal and its superimposed signal on the trajectory tracking performance, to realize the tracking complex curves such as typical trigonometric function and trigonometric superposition of different frequencies. The H∞ control method proposed in this paper is effective and may meet design requirements such as fast response speed, accurate tracking and sound stability in control system design.

Field operation test

In order to further verify the engineering practicability of the robot mechanical arm H∞ motion control method, a robot insulator replacement operation was tested on a 220-kV actual high-voltage transmission line under the administration of Hunan electric power company live working centre in Hunan Province, China. Mechanical arm 2 moved from the initial position to the insulator clamping pose through joint coordination motion. The field operation test is shown in Figure 12. In order to make a contrast between H∞ control and conventional control, the insulator replacement operation tests were carried out twice, before and after using H∞ control, respectively. The robustness of the robot mechanical arm motion was tested at macro and micro levels, in order to monitor the macro robust stability dynamically during the insulator replacement process, in which different motions were performed, and the angle ranges, which were measured by a tilt sensor carried by robot itself after and before H∞ control, are recorded in Table 6. The macro stability simulation test of the robot arm motion before and after H∞ control is shown in Figure 13.

Table 6.

Tilt sensor value scope of mechanical arm 2 different actions.

Action mechanical arm 2	Before H∞ control	After H∞ control
Rotation movement	7–12°	9–11°
Stretch movement	10°	10°
Location clamps	7–12°	9–11°
Horizontal movement	10–11°	10°
Vertical movement	10–11°	10°

Figure 13 shows that the measured values of the tilt sensor before and after the H∞ control is maintained at 10° after 15 and 10 min, respectively, indicating that the time consumed for finishing the task before H∞ control is about 15 min (not including online and offline time), and the time consumed for finishing the task after H∞ control is about 10 min (not including online and offline time). Without H∞ control, there are significant errors of trajectory tracking and location of the manipulator due to the influences of disturbances and uncertainty factors, and therefore, it requires more artificial intervention for adjustment and control of the mechanical arm so that more time will be consumed. However, after applying H∞ control, the robustness of the mechanical arm increases, and the trajectory tracking and location are more accurate, which decreases artificial intervention, shortens the operation time by about a third and greatly improve the operation efficiency of the robot. In addition, according to the inclination angle range of different movements of the mechanical arm before and after H∞ control shown in Table 6 and Figure 13, it may be seen that the inclination angle of different movements of the mechanical arm measured after H∞ control are uniformly smaller than that measured before H∞ control; therefore, we may conclude that the mechanical arm moves more stably at a macro level after using H∞ control.

To realize dynamic monitoring for the micro robust stability of the robot arm motion during the operation, as the robot is at high altitude, with periodic wind load and flexible operating environments for live working, the actual velocities of the four joint motors may be measured in real time by an encoder installed on the joint motor of mechanical arm 2, wherein the maximum rotation velocity of the joint motor is 1 krpm (1000 rotations per minute). To secure the safety of the operation, the rotation velocity is normally maintained within 0.8 krpm. As the rotation joint will overcome gravity during the motion process and the vertical movement joint only moves at the end of the mechanical arm, the desired velocity of the rotation joint requires that the maximum velocity vertical shift joint requires the minimum velocity, the stretch joint and horizontal shifting joint require a medium velocity, and the ideal velocity for rotation joint, stretch joint, horizontal shift joint and vertical shift joint may be set as 0.8, 0.68, 0.65 and 0.6 krpm, respectively. Under H∞ control, the actual velocity tracking curve for the joint motor of robot mechanical arm 2 in field operation tests is shown in Figure 14.

The output of the encoder installed on the mechanical arm joint motor is connected to a counter of multi-function data processing boards, PM511P. By setting the counter on incremental mode, it may calculate the number of rotations of the joint motor (count). According to the ideal velocity and moving time period of each joint, we may obtain the actual number of rotations of the rotation movement motor, stretch movement motor, horizontal movement motor and vertical movement motor as 800, 620, 1200 and 400 rpm, respectively, to obtain the tracking performance of the actual position of the joint motor in field operation tests, as shown in Figure 15.

According to Figure 14, it may be shown that the actual velocities of the four joint motors (rotation, stretch, horizontal and vertical) converge to the given ideal velocity at t=2.5, 2.2, 2.3 and 2 s, respectively, and after that a sound tracking performance may be maintained. According to Figure 15, it may be shown that the actual positions of the four joint motors (rotation, stretch, horizontal and vertical) converge to the given ideal positions at t=2.5, 2, 3 and 2.2 s, respectively, and after that a sound tracking perform may be maintained. Therefore, we may conclude that the joint motor of mechanical arm 2 may achieve sound H∞ trajectory tracking performance from a microcosmic perspective. In conclusion, engineering applications of H∞ robust motion control for a robot mechanical arm may be verified at both macro and micro levels through field operation tests. From the whole process, when mechanical arm 2 moves from its initial state to a working state and then to an insulator clamping posture, all joints of mechanical arm 2 are in smooth, continuous and stable operation status. The robot may successfully finish the task of replacing an insulator under uncertainty factors and disturbances such as high voltage and strong electromagnetic interference. With a live-line maintenance robot, the operation efficiency may be improved and the intelligence of robot operation is also reflected to a certain degree.

Conclusion

This study described the development of an experimental prototype of a live-line maintenance robot for replacing insulators on overhead high-voltage transmission lines, which greatly improved the operation efficiency and solved safety risks in manual replacement.

In this paper, the structure of the robot control system was decomposed into three layers. In addition, we established a unified kinetic equation for different motions of the mechanical arm joints, deduced a state-space expression of the mechanical arm motion control system under uncertainties and disturbances, constructed an H∞ control model of mechanical arm joint motions, obtained a robust control law of the mechanical arm control system by solving an LMI inequality and proved the stability of system by selecting the appropriate Lyapunov function. The method proposed in this paper enhanced the robot control system scalability and shortened the development cycle of the controller.

Through a mechanical arm tracking simulation experiment, we verified that the H∞ control method may effectively suppress the influences of disturbances and uncertainties on the robot stability, and meet the requirements of fast response, high tracking accuracy and good stability in designing the control system. Field operation tests further validated the engineering practicability of the H∞ control method both in macro and micro aspects.

Footnotes

Declaration of Conflicting Interests

The authors declare that there is no conflict of interest.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National High Technology Research and Development Program of China (2006AA04Z202), the National Natural Science Foundation of China (51105281), Special Fund of the Central Universities People’s Republic of China (2104005) and State Grid Hunan Electric Power Company of China (5216A01400B1).

References

Alexandru

Oltean

Fagarasan

. (2011) Live replacement of 400 kV busbar insulators in Brasov substation. In: 2011 IEEE PES 12th International Conference on Transmission and Distribution Construction, Operation and Live-Line Maintenance (ESMO), pp. 1–7.

Aracil

Pinto

Ferre

(2012) Robots for live power lines maintenance and inspection task. In: 15th Triennial Work Congress Conference, Barcelona, Spain, pp. 118–120.

Bouakrif

Zasadzinski

(2016) Trajectory tracking control for perturbed robot manipulators using iterative learning method. International Journal on Advances in Manufacturing Technology12(3): 211–220.

Cherney

Baker

Kuffel

(2013) Evaluation of and replacement strategies for aged high-voltage toughened glass suspension insulators. In: IEEE Overhead Lines Subcommittee Working Group on Insulator Performance and Applications, vol. 30, no. 3, June2014, pp. 275–282.

Galeani

Menini

Potini

(2012) Robust trajectory tracking for a class of hybrid systems. IEEE Transactions on Automatic Control57(2): 344–359.

Huang

Chiang

(2015) Robust trajectory tracking design of unmanned aerial vehicles via nonlinear mixed H2/H∞ control method. In: Journal of Mechanical Science and Control Conference (ASCC), 10th Asian, pp. 1–6.

Kayacan

Ramon

Saeys

(2015) Robust trajectory tracking error-based model predictive control for unmanned ground vehicles. IEEE/ASME Transactions on Mechatronics, 1–5.

Khaloozadeh

Homaeinejad

(2014) Real-time regulated sliding mode controller design of multiple manipulator space free-flying robot. Journal of Mechanical Science and Technology24(6): 1337–1351.

Kim

Lewis

Dawson

(2005) Intelligent optimal control of robotic manipulators using neural networks. Automatica36(9): 1355–1364.

10.

Lai

Zhou

(2016) Droop-based distributed cooperative control for microgrids with time-varying delays. IEEE Transactions on Smart Grid 7(4): 879–891.

11.

Lewis

Jagannathan

Yeşildirek

(2009) Neural Network Control of Robot Manipulators and Nonlinear Systems. Melbourne: Taylor & Francis.

12.

Tong

(2012) Adaptive fuzzy output feedback control for a single-link flexible robot manipulator driven DC motor via backstepping. Nonlinear Analysis-Real World Applications14: 483–494.

13.

Liu

Guo

Wang

. (2010) Research on the structure singular value robust control of underwater vehicle. In: Proceedings of the 8th World Congress on Intelligent Control and Automation, 6–9 July, Jinan, China.

14.

. (2003) Research on high voltage electric power live line working robot. Automation of Electric Power Systems27(17): 56–58.

15.

Meng

Jia

(2015) H∞ approach to monotonically convergent ILC for uncertain time-varying delay systems. International Journal of Systems Science46(2): 209–217.

16.

Ngo

Wang

Mai

(2012) Robust adaptive neural-fuzzy network tracking control for robot manipulator. International Journal of Computers, Communications & Control7: 341–352.

17.

Hui

(2003) Study on hot-line robot and its assembly system of tracking. Dissertation, Tientsin University, Tianjin, China, pp. 3–4.

18.

Takaoka

Yokoyama

Wakisako

. (2001) Development of the fully automatic live line maintenance robot phase III. In: Proceedings of the 4th IEEE International Symposium on Assembly and Task Planning, Soft Research Park, Fukuoka, Japan, pp. 28–29.

19.

Tang

X T

Tang

H Z

(2011) A backstepping robust control method for free-floating space robot system with dual-arms. In: IEEE International Conference on Robotics and Biomimetics, Phuket, Thailand. IEEE Computer Society, pp. 761–765.

20.

Wang

Liu

Mei

S W

. (2010) Transmission lines inspection robot. In: Proceedings of the 29th Chinese Control Conference, 29–31 July, Beijing, China.

21.

Wang

Liu

Mei

. (2011) Control system construction and action controller of high voltage transmission line inspection robot. Journal of Tsinghua University (Natural Science)51(7): 921–927.

22.

Zhang

Jin

(2014) Breakdown process experiments of 110–500 kV insulator strings under short tail lightning impulse. IEEE Transactions on Power Delivery5(29): 2394–2401.

23.

Zareinia

Sepehri

Olson

. (2010) Haptic-enabled control of hydraulic manipulators applied to power line maintenance: concept and implementation. Proceeding of the 1th International Conference on Applied Robotics for the Power Industry, 5–7 October, Delta Centre-Ville, Montreal, Canada.

24.

Zhao

Feng

Liu

(2007) Robust control of underwater robot based on mixed sensitivity. Instrument Technology (Natural Science)28(8): 606–609.