Fast luffing control of flexible boom of mobile elevated work platform with uncertain control input gain

Abstract

In this paper, the dynamic model between input electric signal of hydraulic valve and boom luffing angle is established, and luffing control of flexible boom of mobile elevated work platform (MEWP) is studied. The control input gain for the luffing motion of the MEWP is extremely difficult to solve and varies with boom length, work platform load, and luffing angle. For the variable control input gain, all low-order state quantities, rigid-flexible coupling quantity, and control input are included as the “total disturbance” to be observed. Based on the third-order linear tracking differentiator and the first-order ordinary differential linear equation, an active disturbance rejection controller without control input gain is proposed. With the bandwidth-based parameter settings, the controller only needs to adjust two parameters. As a comparison, this paper analyzes in detail the proposed controllers, the fixed input gain active disturbance rejection controller and the PID controller in terms of tracking error, overshoot, setting time, tracking stability, and robustness.

Keywords

Flexible boom control input gain extended state observer tracking differentiator MEWP ADRC

Introduction

Mobile elevated work platform (MEWP) is used to transport operators, tools and materials to the workplace, including self-propelled MEWP and vehicle mounted MEWP. As the demand for large heights increases in the market, the boom lengths of MEWP are becoming larger and larger. For example, for self-propelled MEWP, such as JLG 1850SJ, the platform height exceeds 56 m, and for vehicle mounted MEWP, such as Bronto S104HLA, the working height exceeds 100 m. To ensure the safety and comfort of operators on the work platform, the vibration of the boom must be strictly controlled and reduced. At present, the most common method is to reduce the speed, for example, the standard stipulates that the raising and lowering speed of the working platform should not exceed 0.4 m/s.¹ However, as a large workspace service robot, it is very important to increase the speed of MEWP, especially for rescue, firefighting, and other use cases. At present, MEWP is mostly operated by open-loop control structure, and the control accuracy depends heavily on the experience of operators.

MEWP’s boom luffing motion can be categorized as flexible linkage motion control, for which there has been much literature on modeling and control design. The control strategy of flexible manipulator system can be divided into feedforward control and feedback control. Feedforward control technology is mainly used to suppress vibration caused by input commands, mainly including filtering technology^2–4 and input shaping.^5,6 The filter is used for preprocessing the input signal and can be realized by using a low-pass filter or a band stop filter. Input shaping is to convolute a desired command with an impulse sequence called an input shaper, and then drive the system with the shaping command generated by the convolution. Compared with filtering technology, input shaping can achieve higher vibration reduction level and robustness.^2,7 Li et al.⁸ proposed a hybrid controller combining input shaping technology with adaptive active disturbance rejection controller. Ju et al.⁹ introduced the full-state feedback controller on the basis of the ZV input shaper to control the elastic vibration of a flexible manipulator with variable loads. However, the design of the input shaper needs to obtain the natural frequency and damping ratio of the system in advance, and the effect of the shaper mainly depends on whether the modal parameters of the system can be accurately identified. Therefore, adaptive input shaping based on modal parameter identification is widely used for systems with varying modal parameters during operation.^10–13İlman et al.¹⁴ combined machine learning with input shaping to produce more accurate shaping design.

From the above analysis, we can see that although feedforward control can reduce the vibration caused by improper operation of the operator. However, the increased number of sensors and identification algorithms for accurate identification of modal parameters undoubtedly increase the cost and complexity of the control system. For example, for MEWP, in order to identify the modal parameters accurately during luffing, it is necessary to know the length of the boom, the angle of luffing and the load capacity of the working platform. More importantly, smoothing of input commands can suppress vibration caused by operator mishandling, but cannot do anything about vibration caused by external disturbing forces applied directly to the boom. Disturbances such as wind and unexpected force on the platform cannot be compensated for by feedforward control. Hence, a feedback loop is needed to stabilize the system around the reference trajectory to actively damp the oscillations.

Among various feedback control methods, because the structure of the single-link flexible manipulator is relatively simple, model-based control methods have been widely studied, such as singular perturbation method,^15–17 sliding mode control,^18–20 robust control,^21,22 LQR,^23,24 etc. In these literatures, the mathematical model of flexible manipulator is established by homogeneous beam or concentrated mass. For the multi-section telescopic boom structure of MEWP, Kharitonov et al.²⁵ proposed a model based on a uniform Euler-Bernoulli beam. The uniform beam is a strong simplification, and to achieve higher model accuracy, Pertsch et al.²⁶ and Zimmert et al.²⁷ chose segmental Euler-Bernoulli beams to model the telescopic boom. However, in the above literature, the luffing drive system of the boom is not modeled in combination with the hydraulic drive system. Instead, the luffing drive system of the boom is simplified to a drive hub with rotational inertia, and it is assumed that the boom rotates around the luffing drive hinge point. However, this is not the case. The boom rotates around the hinge point connected with the turntable, and the luffing driving torque is nonlinear. With the changes of boom length, luffing angle and working platform load, it is difficult to obtain an accurate mathematical model.

There are also many model-free controllers used for the control of flexible manipulators, such as PID and improved PID,^28–30 model free predictive control,^31,32 neural network,^33–35 ADRC,^36,37 etc. Among those control methods, the ADRC proposed by Han³⁸ has simple structure and excellent disturbance rejection performance, forming a new paradigm for feedback control systems.³⁹ Although ADRC requires few models, the order of the system and the control input gain are necessary. In addition, although the nominal control input gain of the controller design does not need to be the same as the ideal control input gain, its error has a certain range.⁴⁰ However, during the luffing motion of MEWP, the control input gain varies as the boom length and work platform load changes. Even for a single luffing process with a fixed boom length and working platform load, the control input gain still varies due to the nonlinearity of the hydraulic system and the drive torque. It is difficult to obtain satisfactory control performance with a fixed control input gain. Morales et al.⁴¹ proposed an algebraic parameter identification technique to extract unknown time-varying input gain. Considering the uncertainty of control input gain, Fareh et al.³⁶ improved the PD control law of conventional ADRC to sliding mode control. Chen et al.⁴² designed a new ADRC control system without input gain by designing an extended state observer and observing the control input as “total disturbance.” When the input gain is changed, the new ADRC control system can achieve better track tracking effect than applying fixed input gain. However, the designed controller needs to refer to the differential signal of each order of the reference signal.

In this paper, a more concise ADRC without input gain control is proposed. Although a detailed mathematical model is not required, the order of the control system is necessary. Therefore, first, a flexible boom luffing dynamics equation containing a hydraulic drive system is developed. Then, starting from the tracking differentiator, the differential signal of each order of the tracking differentiator is taken as the ideal state vector of the system, and then the ideal control output is derived. By constructing the first-order linear differential equations of the actual control output and the ideal control output, the fast following of the actual control output to the ideal control output is realized. Further, an extended state observer is built to observe the low-order state quantities, the rigid-flexible coupling and the control output as “total perturbations.” Finally, the observed values are used to replace the state vectors of the tracking differentiator, and the control law without control input gain is obtained. The final control law does not need the differential signal of each order of the reference signal. The main contributions of the paper are presented as follows.

(1) The dynamic equation of boom luffing motion including hydraulic drive system is established, instead of just simplifying the drive mechanism as a drive hub with rotational inertia. Based on the newly established dynamic equation, it is obvious that the control input gain has nonlinear uncertainty.

(2) Based on a linear tracking differentiator and a first-order linear ordinary differential equation, an ADRC without control input gain is proposed. The signals of each order of the tracking differentiator are used to construct the ideal control input, and the linear differential equation is used to make the actual control input converge asymptotically to the ideal one.

(3) The tracking and disturbance rejection performance of the proposed ADRC is analyzed. Despite the large range of model parameter changes, the proposed ADRC has good tracking consistency, and for a variable control input gain plant, it has better tracking stability than conventional ADRC.

The rest of this paper has the following organization. In Section 2, the dynamic model of boom luffing motion is established. In Section 3, a new ADRC without control input gain is proposed. In Section 4, the numerical verification is given. The conclusion is presented in Section 5.

Dynamic models

Dynamic models of boom luffing motion

In the modeling, the boom system is simplified as follows: (1) the area consisting of the rotating hinge point $O$ of the boom around the turntable, the hinge point $A$ of the connection between the luffing cylinder and the turntable, and the hinge point $O'$ of the connection between the luffing cylinder and the boom is a rigid area. (2) The boom is regarded as an Euler-Bernoulli beam with a stepped cross section. (3) The working platform at the head of the boom and the operators and equipment on board are regarded as the concentrated mass $m_{c}$ at the head of the boom, and because the working platform to rotate against the boom to reach parallel to the ground, so it will produce a concentrated rotational inertia $J_{c}$ in the head of the boom. Due to the use of master-slave cylinder tandem leveling, the rotation angle of the working platform is equal to the luffing angle of the boom.

The system of the MEWP in the luffing plane is shown in Figure 1. Taking the articulation point between the boom and the turntable as the coordinate origin O, the X-axis parallel to the ground over the point O, the Y-axis perpendicular to the ground, $XOY$ constitutes the inertial coordinate system. Taking the articulation point between the luffing cylinder and the boom as the coordinate origin $O'$ , the mid-axis of the undeformed boom as the Z-axis, the direction perpendicular to the mid-axis of the boom as the $ω$ -axis, $Z O' ω$ constitutes the floating coordinate system. $ω (x, t)$ is the neutral axis deflection of the boom, $θ_{m}$ is the luffing angle of the boom, $θ_{z}$ is the angle of the boom after deformation, and $α$ is the angle between the cylinder and the boom. The total length of the telescopic boom is l, the length of the rigid region of the telescopic boom is $z_{0}$ , and each section of the boom of the remaining length is regarded as a section of Euler-Bernoulli beam. Each section of the boom is represented by the left and right endpoints $Z_{i - 1}$ and $Z_{i}$ (i = 1, 2, 3) on the neutral axis, whose linear density $ρ A_{i}$ and bending stiffness $E I_{i}$ are constant and still satisfy the bending vibration equation of a homogeneous equal-section beam.

Figure 1.

Sketch map of boom luffing.

The vector diameter of the deformed Z at any point on the telescopic boom can be expressed in the inertial coordinate system $XOY$ as:

r = [\begin{matrix} \cos θ_{m} & - \sin θ_{m} \\ \sin θ_{m} & \cos θ_{m} \end{matrix}] [\begin{matrix} z_{0} + z \\ ω (z, t) \end{matrix}]

(1)

Considering that the boom is a small deformation and $z_{0}$ is much smaller than the length of the boom, the length of the OZ point $‖ r ‖_{2}$ is:

‖ r ‖_{2} = \sqrt{{(z_{0} + z)}^{2} + ω^{2} (z, t)} = z_{0} + z

(2)

The angle $θ_{z}$ between the line of point Z and the hinge point O and the horizontal plane is

θ_{z} = θ_{m} + \frac{ω (z, t)}{z_{0} + z}

(3)

Then the kinetic energy $T$ of the boom is:

\begin{array}{l} T = \frac{1}{6} ρ A_{1} {z_{0}}^{3} {\dot{θ}}_{m}^{2} + \frac{1}{2} \sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} {[(z_{0} + z) \dot{θ_{m}} + \dot{ω} (z, t)]}^{2} d z + \frac{1}{2} m_{c} {[l \dot{θ_{m}} + \dot{ω} (l, t)]}^{2} \\ + \frac{1}{2} J_{c} {\dot{θ_{m}}}^{2} = T_{θ θ} + T_{q q} + T_{θ q} \end{array}

(4)

\begin{array}{l} T_{θ θ} = \frac{1}{6} ρ A_{1} {z_{0}}^{3} {\dot{θ}}_{m}^{2} + \frac{1}{2} \\ \sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} {[(z_{0} + z) \dot{θ_{m}}]}^{2} d z + \frac{1}{2} m_{c} {(l \dot{θ_{m}})}^{2} + \frac{1}{2} J_{c} {\dot{θ_{m}}}^{2} = (\frac{1}{2} J_{B} + \frac{1}{2} m_{c} l^{2} + \frac{1}{2} J_{c}) {\dot{θ_{m}}}^{2} \end{array}

(5)

T_{qq} = \frac{1}{2} \sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} \overset{\cdot}{ω} {(z, t)}^{2} dz + \frac{1}{2} m_{c} \overset{\cdot}{ω} {(l, t)}^{2}

(6)

T_{θ q} = \dot{θ} \sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} (z_{0} + z) \dot{ω} (z, t) d z + m_{c} l {\dot{θ}}_{m} \dot{ω} (l, t)

(7)

Where $T_{θ θ}$ , $T_{qq}$ , and $T_{θ q}$ are the rigid body kinetic energy, flexible body kinetic energy, and rigid-flexible coupling kinetic energy of the boom, respectively. $J_{B}$ is the rotational inertia of the boom rotating around point O. $m_{c}$ is the mass of the working platform and the operators and equipment. $J_{c}$ is the rotational inertia of the working platform rotating around the end point of the boom. $l$ is the total length of the boom.

Considering the elastic potential energy and gravitational potential energy of the boom, the total potential energy V is:

\begin{matrix} V = \frac{1}{2} \sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} E I_{i} {[ω^{″} (z, t)]}^{2} dz + \frac{1}{2} g ρ A_{1} {z_{0}}^{2} \sin θ_{m} \\ + g \sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} [(z_{0} + z) \sin θ_{m} + ω (z, t) \cos θ_{m}] dz \\ + m_{c} g (lsin θ_{m} + ω (l, t) \cos θ_{m}) = V_{θ θ} + V_{qq} + V_{θ q} \end{matrix}

(8)

V_{θ θ} = m_{B} g l_{B} \sin θ_{m} + m_{c} glsin θ_{m}

(9)

V_{qq} = \frac{1}{2} \sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} E I_{i} {[ω^{″} (z, t)]}^{2} dz

(10)

V_{θ q} = gcos θ_{m} \sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} ω (z, t) dz + m_{c} g ω (l, t) \cos θ_{m}

(11)

Where $V_{θ θ}$ , $V_{qq}$ , and $V_{θ q}$ are the rigid potential energy, elastic potential energy, and rigid-flexible coupling potential energy of the boom, respectively. $m_{B}$ is the total mass of the boom, and $l_{B}$ is the length of the center of mass of the boom from point O.

According to the hypothetical modal method, the deformation $ω (z, t)$ of the boom at z at moment t during the motion can be described as:

ω (z, t) = \sum_{j = 1}^{n} φ_{j} (x) q_{j} (t) = Φ q

(12)

Where $φ_{j} (x)$ denotes the jth order oscillation function; $q_{j} (t)$ denotes the corresponding modal coordinates. $Φ = [φ_{1}, φ_{2}, \dots, φ_{n}] \in R^{1 \times n}, q = {[q_{1}, q_{2}, \dots, q_{n}]}^{T} \in R^{n \times 1}$ .

Then $T_{qq}$ , $T_{θ q}$ , $V_{qq}$ , $V_{θ q}$ can be rewritten in the following form:

\begin{matrix} T_{qq} = \frac{1}{2} {\overset{\cdot}{q}}^{T} (\sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} Φ^{T} Φ dz + m_{c} {[Φ (l)]}^{T} [Φ (l)]) \\ \overset{\cdot}{q} = \frac{1}{2} {\overset{\cdot}{q}}^{T} (M_{B} + M_{c}) \overset{\cdot}{q} \end{matrix}

(13)

T_{θ q} = \overset{\cdot}{θ} [\sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} (z_{0} + z) Φ dz + m_{c} l Φ (l)] \overset{\cdot}{q}

(14)

V_{qq} = \frac{1}{2} q^{T} (\sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} E I_{i} {Φ^{″}}^{T} Φ^{″} dz) q = \frac{1}{2} q^{T} K_{B} q

(15)

V_{θ q} = gcos θ_{m} (\sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} Φ dz + m_{c} Φ (l)) q

(16)

Where $M_{B} = \sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} Φ^{T} Φ dz \in R^{n \times n}$ is the boom mass matrix, $M_{c} = m_{c} [Φ (l {)]}^{T} [Φ (l)] \in R^{n \times n}$ is the mass matrix of working platform and load, $K_{B} = \sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} E I_{i} {Φ^{″}}^{T} Φ^{″} dz \in R^{n \times n}$ is the boom stiffness matrix.

Introducing the Lagrangian function $L = T - V$ , with $θ_{m}$ and $q$ as generalized coordinates, from the Lagrangian equation, we get:

\frac{d}{dt} (\frac{\partial L}{\partial {\overset{\cdot}{θ}}_{m}}) - \frac{\partial L}{\partial θ_{m}} = τ

(17)

\frac{d}{dt} (\frac{\partial L}{\partial \overset{\cdot}{q}}) - \frac{\partial L}{\partial q} = 0

(18)

The differential equation of motion of the system is obtained after the arithmetic collation.

m_{θ θ} {\overset{\cdot\cdot}{θ}}_{m} + m_{θ q} \overset{\cdot\cdot}{q} = τ + ω_{1}

(19)

m_{qq} \overset{\cdot\cdot}{q} + {m_{θ q}}^{T} {\overset{\cdot\cdot}{θ}}_{m} + K_{B} q = ω_{2}

(20)

Where $m_{θ θ}$ is the moment of inertia of the rigid body, $m_{θ θ} = J_{B} + m_{c} l^{2} + J_{c}$ . $m_{θ q} \in R^{1 \times n}$ is inertial array, $m_{θ q} = \sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} (z_{0} + z) Φ dz + m_{c} l Φ (l)$ . $m_{qq} \in R^{n \times n}$ is mass matrix, $m_{qq} = M_{B} + M_{c}$ , $M_{B}$ is the boom mass matrix, $M_{c}$ is the mass matrix of working platform and load. $K_{B}$ is the boom stiffness matrix. $ω_{1}$ , $ω_{2}$ are the disturbance moments respectively, $ω_{1} = - G_{θ} - C_{1} - Δ$ , $ω_{2} = - C_{2}$ . $G_{θ}$ is the weight moment, $G_{θ} = (m_{B} g l_{B} + m_{c} gl) \cos θ_{m}$ $C_{1}$ , $C_{2}$ are the coupling moments respectively, $C_{1} = - gsin θ_{m} (\sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} Φ dz + m_{c} Φ (l)) q$ , $C_{2} = gcos θ_{m} (\sum_{i = 1}^{3} \int_{z_{i - 1}}^{z_{i}} ρ A_{i} Φ dz + m_{c} Φ (l))$ . $τ$ is the driving moment of the cylinder to the boom. $Δ$ is the unmodeled term including system frictional moment, Coriolis force, centripetal force and other disturbance terms.

From equations (19) and (20), we have that:

\begin{matrix} (m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}) {\overset{\cdot\cdot}{θ}}_{m} = m_{θ q} {m_{qq}}^{- 1} K_{B} q - ω_{1} \\ - m_{θ q} {m_{qq}}^{- 1} ω_{2} + τ \end{matrix}

(21)

After increased order, equation (21) can be rewritten as:

\begin{matrix} θ_{m}^{. . .} = \frac{m_{θ q} {m_{qq}}^{- 1} K_{B} \overset{\cdot}{q}}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}} - \frac{{\overset{\cdot}{ω}}_{1}}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}} \\ - \frac{m_{θ q} {m_{qq}}^{- 1} {\overset{\cdot}{ω}}_{2}}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}} + \frac{\overset{\cdot}{τ}}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}} \end{matrix}

(22)

Dynamic models of hydraulic system

From Figure 1, according to the geometric relationship, it can be obtained that:

τ = \frac{F z_{0} asin (θ_{0} + θ_{m})}{\sqrt{{z_{0}}^{2} + a^{2} - 2 z_{0} acos (θ_{0} + θ_{m})}} = H (θ_{m}) F

(23)

Where $H (θ_{m}) = \frac{z_{0} asin (θ_{0} + θ_{m})}{\sqrt{{z_{0}}^{2} + a^{2} - 2 z_{0} acos (θ_{0} + θ_{m})}}$ , F is the cylinder thrust, $F = p_{1} A_{1} - p_{2} A_{2}$ , $p_{1}$ , $A_{1}$ are the pressure and effective area of the piston chamber, respectively. $p_{2}$ , $A_{2}$ are the pressure and effective area of the rod chamber, respectively. $a$ is the length of the lower hinge point $A$ of the cylinder from the rotating hinge point $O$ of the boom, and $θ_{0}$ is the angle between the line of OA and the X-axis.

Neglecting the internal leakage and external leakage of the hydraulic cylinder, the dynamic equation of the pressure flow in the hydraulic cylinder is ⁴³:

\frac{V_{1}}{β_{e}} {\overset{\cdot}{p}}_{1} = - A_{1} \overset{\cdot}{y} + q_{1}

(24)

\frac{V_{2}}{β_{e}} {\overset{\cdot}{p}}_{2} = A_{2} \overset{\cdot}{y} - q_{2}

(25)

Where $V_{1}$ and $V_{2}$ are the volumes of the piston chamber and rod chamber of the hydraulic cylinder, respectively. $y$ , $\overset{\cdot}{y}$ are the stroke of the hydraulic cylinder and the piston rod extension speed, respectively. $\overset{\cdot}{y} = H (θ_{m}) {\overset{\cdot}{θ}}_{m}$ . $β_{e}$ is the bulk modulus of the hydraulic fluid. $q_{1}$ and $q_{2}$ are the flow rate of the piston chamber and rod chamber of the hydraulic cylinder, respectively.

The specific principle of the hydraulic system is shown in Figure 2, the supply pressure is $p_{p}$ , the pressure of the tank is 0, then $q_{1}$ , $q_{2}$ can be expressed as:

\begin{matrix} q_{1} = s (u) C_{d} W x_{v} \sqrt{\frac{2 (p_{p} - p_{1})}{ρ_{e}}} u + s (- u) C_{d} W x_{v} \sqrt{\frac{2 p_{1}}{ρ_{e}}} U \\ = (k_{1} + k_{2}) u \end{matrix}

(26)

\begin{matrix} q_{2} = s (u) C_{d} W x_{v} \sqrt{\frac{2 p_{2}}{ρ_{e}}} u + s (- u) C_{d} W x_{v} \sqrt{\frac{2 (p_{p} - p_{2})}{ρ_{e}}} u \\ = (k_{3} + k_{4}) u \end{matrix}

(27)

Figure 2.

Hydraulic schematic diagram.

Where $s (u) = {\begin{matrix} 1, u > 0 \\ 0, u \leq 0 \end{matrix}$ , $u$ is the input electrical signal to the servo valve. $C_{d}$ and $W$ are the servo valve flow coefficient and the valve port area gradient, respectively. $x_{v}$ is the spool displacement, proportional to the input electrical signal $u$ . $ρ_{e}$ is the density of the hydraulic oil. $k_{1} = s (u) C_{d} W x_{v} \sqrt{\frac{2 (p_{p} - p_{1})}{ρ_{e}}}, k_{2} = s (- u) C_{d} W x_{v} \sqrt{\frac{2 p_{1}}{ρ_{e}}}, k_{3} = s (u) C_{d} W x_{v} \sqrt{\frac{2 p_{2}}{ρ_{e}}}, k_{4} = s (- u) C_{d} W x_{v} \sqrt{\frac{2 (p_{p} - p_{2})}{ρ_{e}}}$ .

Combining equations (23)–(27), the relationship between the boom drive torque $τ$ and the servo valve input electrical signal $u$ can be obtained as follows:

\begin{matrix} \overset{\cdot}{τ} = β_{e} H (θ_{m}) (\frac{A_{1} (k_{1} + k_{2})}{V_{1}} + \frac{A_{2} (k_{3} + k_{4})}{V_{2}}) u \\ - β_{e} H^{2} (θ_{m}) (\frac{A_{1}}{V_{1}} + \frac{A_{2}}{V_{2}}) {\overset{\cdot}{θ}}_{m} \end{matrix}

(28)

Bringing equation (28) into equation (22), the kinetic equation of the boom rotation angle $θ_{m}$ and the valve input electrical signal $u$ is obtained as follows:

\begin{matrix} θ_{m}^{. . .} = - \frac{β_{e} H^{2} (θ_{m}) (\frac{A_{1}}{V_{1}} + \frac{A_{2}}{V_{2}}) {\overset{\cdot}{θ}}_{m}}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}} + \frac{m_{θ q} {m_{qq}}^{- 1} K_{B} \overset{\cdot}{q}}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}} \\ - \frac{{\overset{\cdot}{ω}}_{1}}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}} - \frac{m_{θ q} {m_{qq}}^{- 1} {\overset{\cdot}{ω}}_{2}}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}} \\ + \frac{β_{e} H (θ_{m}) (\frac{A_{1} (k_{1} + k_{2})}{V_{1}} + \frac{A_{2} (k_{3} + k_{4})}{V_{2}})}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}} u \end{matrix}

(29)

ADRC design without control input gain

Based on the formulation in the literature,³⁹ equation (29) is written in the standard form of the conventional ADRC as:

θ_{m}^{. . .} = f + bu

(30)

Where $f = - \frac{β_{e} H^{2} (θ_{m}) (\frac{A_{1}}{V_{1}} + \frac{A_{2}}{V_{2}}) {\overset{\cdot}{θ}}_{m}}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}} + \frac{m_{θ q} {m_{qq}}^{- 1} K_{B} \overset{\cdot}{q}}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}} - \frac{{\overset{\cdot}{ω}}_{1}}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}} - \frac{m_{θ q} {m_{qq}}^{- 1} {\overset{\cdot}{ω}}_{2}}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}}$ is defined as the total disturbance of the system, $b = \frac{β_{e} H (θ_{m}) (\frac{A_{1} (k_{1} + k_{2})}{V_{1}} + \frac{A_{2} (k_{3} + k_{4})}{V_{2}})}{m_{θ θ} - m_{θ q} {m_{qq}}^{- 1} {m_{θ q}}^{T}}$ is the control input gain of the system.

From equation (30), we can see that it is difficult to solve for the control input gain b, and b varies with the boom length, work platform load, and boom variation angle, making it impossible to derive an exact value.

Dynamic design of control outputs

First, with reference to the third-order linear tracking differentiator,⁴⁴ the differential signals of each order of the reference input signal are designed, and the differential signals of each order are used as ideal state vectors.

{\overset{\cdot}{x}}^{*} = A^{*} x^{*} + B^{*} r, y^{*} = C x^{*}

(31)

Where $x^{*} = {[\begin{matrix} x_{1}^{*} & x_{2}^{*} & x_{3}^{*} \end{matrix}]}^{T} = {[\begin{matrix} θ_{m}^{*} & {\overset{\cdot}{θ}}_{m}^{*} & {\overset{\cdot\cdot}{θ}}_{m}^{*} \end{matrix}]}^{T}$ is the ideal state vector, $r$ is the reference input signal, $y^{*}$ is the ideal output. $A^{*} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ - k_{1} & - k_{2} & - k_{3} \end{matrix}]$ , $B^{*} = {[\begin{matrix} 0 & 0 & k_{1} \end{matrix}]}^{T}$ . $K = [\begin{matrix} k_{1} & k_{2} & k_{3} \end{matrix}]$ is control gain. To make the ideal output converge asymptotically to the reference signal, the poles of the controller are configured uniformly to the left half real axis of the s-plane $- ω_{c}$ according to Gao,⁴⁵ and K can be expressed as $K = [\begin{matrix} k_{1} & k_{2} & k_{3} \end{matrix}] = [{ω_{c}}^{3} 3 {ω_{c}}^{2} 3 ω_{c}]$ , with $ω_{c}$ denoting the controller bandwidth.

Bringing equation (31) into equation (30), the ideal control signal is obtained as follows:

u^{*} = \frac{k_{1} (r - {x_{1}}^{*}) - k_{2} {x_{2}}^{*} - k_{3} {x_{3}}^{*} - f}{b}

(32)

Assume that the actual control signal $u$ and the ideal control signal $u^{*}$ satisfy the following first-order linear differential equation:

\overset{\cdot}{u} = - a (u - u^{*})

(33)

According to equation (33), if $a > 0$ , $u$ will converge asymptotically to $u^{*}$ and the convergence rate is determined by $a$ .

ESO design

Write equation (30) in the form of the equation of state:

\overset{\cdot}{x} = Ax + B (f + bu), y = Cx

(34)

Where $x = {[x_{1} x_{2} x_{3}]}^{T} = {[θ_{m} {\dot{θ}}_{m} {\overset{\cdot\cdot}{θ}}_{m}]}^{T}$ , $A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}]$ , $B = {[\begin{matrix} 0 & 0 & 1 \end{matrix}]}^{T}$ , $C = [\begin{matrix} 1 & 0 & 0 \end{matrix}]$ .

Since both the control input gain $b$ and the perturbation $f$ are unknown, the new state $x_{4} = f + bu$ is defined, and the equation of state after expanding equation (34) is:

{\overset{\cdot}{x}}_{e} = A_{e} x_{e} + B_{e} {\overset{\cdot}{x}}_{4}, y_{e} = C_{e} x_{e}

(35)

Where $x_{e} = {[x_{1} x_{2} x_{3} x_{4}]}^{T} = [{θ_{m} {\overset{\cdot}{θ}}_{m} {\overset{\cdot\cdot}{θ}}_{m} x_{4}]}^{T}$ , $A_{e} = [\begin{matrix} \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix} \\ \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \end{matrix}]$ , $B_{e} = {[\begin{matrix} 0 & 0 \end{matrix} \begin{matrix} 0 & 1 \end{matrix}]}^{T}$ , $C_{e} = [\begin{matrix} 1 & 0 & \begin{matrix} 0 & 0 \end{matrix} \end{matrix}]$ .

For equation (35), the following state observer is designed to estimate the state vector:

\begin{matrix} {\dot{\hat{x}}}_{e} = A_{e} {\hat{x}}_{e} + L_{e} (y_{e} - C_{e} {\hat{x}}_{e}) = (A_{e} - L_{e} C_{e}) {\hat{x}}_{e} \\ + L_{e} y_{e}, {\hat{y}}_{e} = I_{e} {\hat{x}}_{e} \end{matrix}

(36)

Where ${\hat{x}}_{e} = {[{\hat{x}}_{1} {\hat{x}}_{2} {\hat{x}}_{3} {\hat{x}}_{4}]}^{T}$ is the estimated value of state vector. ${\hat{y}}_{e}$ is the output of the observer. $I_{e} = [\begin{matrix} \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} \\ \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \end{matrix}]$ . $L_{e}$ is the observer gain. The design of $L_{e}$ needs to satisfy $A_{e} - L_{e} C_{e}$ as Hurwitz matrix in order to make the estimation error dynamics asymptotically stable. According to Gao,⁴⁵ the poles of the observer are uniformly assigned to the left half real axis of the s-plane $- ω_{0}$ , $L_{e}$ can be expressed as $L_{e} = {[4 ω_{0} 6 {ω_{0}}^{2} 4 {ω_{0}}^{3} {ω_{0}}^{4}]}^{T}$ , $ω_{0}$ represents the observer bandwidth.

Now, by replacing each state vector with the observation of equation (36), equation (33) can be expressed as:

\overset{\cdot}{u} = - \frac{a}{b} (bu - b u^{*}) = \frac{a}{b} (k_{1} (r - {\hat{x}}_{1}) - k_{2} {\hat{x}}_{2} - k_{3} {\hat{x}}_{3} - {\hat{x}}_{4})

(37)

Finally, according to the literature, a is expressed as $a = | b | {ω_{0}}^{k} (0 < k < 1)$ , $ω_{0}$ is the bandwidth of the observer.⁴² Detailed analysis can be found in the literature,⁴² which is omitted here. The equation (37) can be further expressed as:

\dot{u} = - \frac{a}{b} (b u - b u^{*}) = s i g n (b) {ω_{0}}^{k} (k_{1} (r - {\hat{x}}_{1}) - k_{2} {\hat{x}}_{2} - k_{3} {\hat{x}}_{3} - {\hat{x}}_{4})

(38)

Where $sign (b) = {\begin{matrix} 1, b > 0 \\ 0, b = 0 \\ - 1, b < 0 \end{matrix}$ .

Numerical verification

In the previous literature, the plant is often written in the form of transfer function, but for complex plant, the transfer function is difficult to be derived accurately. From the perspective of engineers, this paper focuses on the design of the physical system itself rather than the tedious mathematical modeling, builds a complex system model in Simcenter AMESim, and carries out simulation calculation and in-depth analysis on this basis.

Model design

The system model is constructed in Simcenter AMESim by combining equations (36) and (37) as well as the boom luffing structure diagram in Figure 1 and the hydraulic schematic diagram in Figure 2, as shown in Figure 3.

Figure 3.

Overall model of luffing system.

The structure of the flexible boom in the model is shown in Figure 4, which consists of three sections of booms, the first section is the fixed boom, and the second and third sections are the telescopic booms. The model of the flexible boom can be obtained by importing the data obtained from the modal analysis of the finite element software into Simcenter AMESim.

Figure 4.

Sketch map of boom.

The settings of each parameter in the model are shown in Table 1.

Table 1.

Model parameters.

Description	Notation	Value	Units
Elastic modulus of boom material	$E$	2.06 × 10¹¹	Pa
Boom material density	$ρ$	7850	kg/m³
Cross-sectional area of the first boom	$A_{1}$	0.011	m²
Cross-sectional area of the second boom	$A_{2}$	0.0071	m²
Cross-sectional area of the third boom	$A_{3}$	0.006	m²
Boom length	$l$	8.8–22.4	m
Weight of working platform and load	$m_{c}$	110–410	kg
Piston diameter of luffing cylinder	$D$	0.125	m
Rod diameter of luffing cylinder	$d$	0.08	m
Hydraulic oil density	$ρ_{e}$	850	kg/m³
Hydraulic oil bulk modulus	$β_{e}$	1.7 × 10⁹	Pa
Valve maximal cross sectional area	$A_{v}$	5 × 10^-5	m²
Valve rated current	$u$	1000	mA
Valve flow coefficient	$C_{d}$	0.7	–
Oil supply pressure	$p_{p}$	2.5 × 10⁷	Pa
Pump flow rate	$q_{p}$	100	L/min

Performance analysis

Track tracking capability analysis

In this paper, we design the reference input signal $r$ as shown in Figure 5, raise the boom angle from 0° to 70° within the specified time $Δ t = t_{2} - t_{1}$ . Design different $Δ t$ (65, 35, 15 s) to check the trajectory following performance of the controller, and compare the proposed ADRC (hereinafter referred to as P_ADRC) with the conventional fixed control input gain ADRC (hereinafter referred to as C_ADRC) and PID. The parameter settings of each controller are shown in Table 2, where the parameters of the PID are rectified according to the method of the literature to make it have active anti-interference capability.⁴⁶

Figure 5.

Luffing movement time.

Table 2.

Controller parameters.

Controller	Parameters
P_ADRC	$ω_{c} = 3, ω_{o} = 14, k = 0.5$
C_ADRC	$ω_{c} = 3, ω_{o} = 10, b_{o} = 0.5$
PID	$k_{p} = 30, k_{i} = 30, k_{d} = 7$

The analyzed working condition is full extension of the boom and maximum load capacity ( $l$ = 22.4 m, $m_{c}$ = 410 kg). At this time, the boom has the maximum flexibility, the lowest natural frequency, and the most difficult to control. Under this working condition, the following ability of the luffing angle and the acceleration value of the working platform under the three controllers are compared.

From Figure 6, PID has the fastest tracking speed, but its overshoot is greater than C_ ADRC and P_ ADRC. P_ ADRC has a lag in tracking. The faster the luffing speed, the more obvious the lag is, but the time to reach the stable state is faster than that of PID and C_ ADRC. In addition, it is obvious (Figure 6(c) is the most obvious) that the control input gain b changes in the process of luffing, resulting in the uneven tracking speed of PID and C_ADRC. At the beginning of tracking, the lag is the largest, and then the tracking speed is accelerated. Once the overshoot occurs, the tracking speed is reduced. This situation will lead to rapid changes in the speed of the work platform and affect the comfort and safety of the operator. However, P_ADRC has the best tracking stability, which can also be seen from Figure 7.

Figure 6.

Angle tracking capability of three controllers at different speeds: (a) Δt = 65 s, (b) Δt = 35 s, and (c) Δt = 15 s.

Figure 7.

Vertical speed of working platform (Δt = 15 s).

Figure 8 shows the acceleration of the working platform at the time of tracking stop. It is obvious that the acceleration value using PID controller greater than P_ADRC and C_ADRC.

Figure 8.

Vertical acceleration of working platform at the stop time of track tracking: (a) Δt = 65 s, (b) Δt = 35 s, and (c) Δt = 15 s.

Table 3 shows the comparison of P_ADRC, C_ADRC, and PID in terms of maximum absolute error (MAE) obtained, the root mean square error (RMSE), the overshoot (OS), the settling time (ST), and the maximum acceleration of working platform (MA). It has been found that the P_ADRC has a superior performance compared to C_ADRC and PID in terms of OS and ST. On the other hand, PID has the lower MAE and RMSE because the integral action. However, for MEWP, we are more concerned about the comfort of operators, such as acceleration index, both P_ADRC and C_ADRC are significantly better than PID.

Table 3.

Comparison of P_ADRC with C_ADRC and PID at different luffing speeds.

	P_ADRC			C_ADRC			PID
	Δt = 65 s	Δt = 35 s	Δt = 15 s	Δt = 65 s	Δt = 35 s	Δt = 15 s	Δt = 65 s	Δt = 35 s	Δt = 15 s
MAE (°)	0.60	0.80	1.06	0.42	0.58	0.85	0.06	0.15	0.51
RMSE (°)	0.79	1.25	2.18	0.55	0.9	1.79	0.16	0.35	1.11
OS (%)	0.25	0.48	1.17	0.73	1.37	3.65	0.92	1.75	4.0
ST (s)	5.7	6.2	6.6	10.4	10.9	14.2	10.3	10.9	14.3
$\| MA \|$ (m/s²)	0.28	0.51	1.21	0.27	0.48	1.0	1.47	2.3	4.6

Anti disturbance performance analysis

Change working condition: the boom is fully retracted and the working platform is unloaded ( $l$ =8.8 m, $m_{c}$ = 110 kg). Retract the boom completely, and the working platform is empty. At this time, the rigidity of the boom is maximum. It is important to note that the data in the flexible boom sub-model in Figure 3 at this point is the data obtained by the finite element software modal analysis when the boom is fully retracted. Select $Δ t = 15 s$ boom to rise from 0° to 70°, and compare the control effects of the three controllers under the same controller parameters.

It can be seen from Figure 9 and Table 4 that the trajectory following ability of P_ADRC is almost unaffected when the boom length and the working platform load are changed. It means that P_ADRC has better robustness to the parameter changes of the plant.

Figure 9.

Comparison of tracking performance of controllers when the boom length and the working platform load weight change (Δt = 15 s): (a) P_ADRC, (b) C_ADRC, and (c) PID.

Table 4.

Comparison of P_ADRC with C_ADRC and PID at different operating conditions (Δt = 15 s).

	P_ADRC		C_ADRC		PID
	l = 22.4 m, m_c = 410 kg	l = 8.8 m, m_c = 110 kg	l = 22.4 m, m_c = 410 kg	l = 8.8 m, m_c = 110 kg	l = 22.4 m, m_c = 410 kg	l = 8.8 m, m_c = 110 kg
MAE (°)	1.06	1.04	0.85	0.82	0.51	0.32
RMSE (°)	2.18	2.17	1.79	1.64	1.11	0.74

Anti disturbance capability under step disturbance

Solazzi⁴⁷ described an accident of a MEWP during pruning operation “A big branch of three fell down and bumped the basket of MEWP.” This paper analyzes this. The analysis condition is taken as the boom is stabilized in the horizontal position and subjected to a step disturbance of 100 kg at a certain moment.

At this time, in order to suppress the vibration quickly, the boom head angle is used as the feedback control quantity. Figure 10 shows the comparison of the stability capability of the three controllers. When the boom is fully retracted, the boom rigidity is the largest, and the PID suppression effect is the best for the vibration caused by the step disturbance (see Figure 10(a)). However, when the boom is fully extended, the flexibility of the boom is maximum, and the natural frequency of the boom system is low, P_ADRC and C_ADRC still has good suppression on vibration, but PID seems powerless at this time (see Figure 10(b)).

Figure 10.

Stable response to step disturbance under different operating conditions: (a) l = 8.8 m, m_c = 110 kg and (b) l = 22.4 m, m_c = 410 kg.

Conclusion

The control input gain of flexible boom luffing motion of MEWP has great uncertainty and varies with the boom length, working platform load, and luffing angle. A new active disturbance rejection controller without input gain control is proposed for boom luffing motion. The low-order state vector, the rigid-flexible coupling quantity, and the control input are observed as the “total perturbation.” The differential signal of each order of the tracking differentiator is taken as the ideal state, and then the ideal control output is derived. The dynamic tracking of the actual output and the ideal output is realized by constructing the first-order linear differential equation. The research results show that compared with fixed control input gain ADRC and PID in trajectory tracking, the controller proposed in this paper has smaller speed change rate, smaller overshoot and faster stability time, and is almost independent of the change of plant parameters with strong robustness. Of course, it should be noted that the proposed controller requires a larger observer bandwidth than fixed control input gain ADRC. In future, the effectiveness and stability of the controller under more operating conditions will be investigated, such as considering the effects of measurement noise and delay. The application and promotion in the field of MEWP is our future work.

Footnotes

Handling Editor: Chenhui Liang

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the Key Research and Development Project of Rizhao under Grant 2021ZDYF010109 and Rizhao Natural Science Foundation Project under Grant RZ2021ZR35.

ORCID iDs

Xin Fang

Jinsheng Zhang

References

GB/T 25849-2010. Mobile elevating work platforms — design calculations, safety requirements and test method. 2010.

Singhose

Singer

Seering

. Comparison of command shaping methods for reducing residual vibration. In: Proceedings of the 1995 European control conference, 1995, pp.1126–1131. Rome: EUCA.

Tavazoei

. Reduction of oscillations via fractional order pre-filtering. Signal Process 2015; 107: 407–414.

Ling

Wang

Liu

. Adaptive fuzzy tracking control of flexible-joint robots based on command filtering. IEEE Trans Ind Electron 2020; 67: 4046–4055.

Singer

Seering

. Preshaping command inputs to reduce system vibration. J Dyn Syst Meas Control 1990; 112: 76–82.

Shan

Liu

Sun

. Modified input shaping for a rotating single-link flexible manipulator. J Sound Vib 2005; 285: 187–207.

Mohamed

Martins

Tokhi

, et al. Vibration control of a very flexible manipulator system. Control Eng Pract 2005; 13: 267–277.

Luo

Huang

. Active vibration control of flexible joint manipulator using input shaping and adaptive parameter auto disturbance rejection controller. J Sound Vib 2016; 363: 97–125.

Liu

Kan

, et al. Master-slave composite vibration control of a mobile flexible manipulator via synchronization optimization of observation and feedback. Complexity 2019; 2019: 1–13.

10.

Pereira

Trapero

Díaz

, et al. Adaptive input shaping for manoeuvring flexible structures using an algebraic identification technique. Automatica 2009; 45: 1046–1051.

11.

Wei

, et al. Hybrid input shaping control scheme for reducing vibration of robot based on multi-mode control. J Central South Univ 2019; 26: 1649–1660.

12.

Thomsen

Søe-Knudsen

Balling

, et al. Vibration control of industrial robot arms by multi-mode time-varying input shaping. Mech Mach Theory 2021; 155: 104072.

13.

Boscariol

Richiedei

Tamellin

. Residual vibration suppression in uncertain systems: a robust structural modification approach to trajectory planning. Robot Comput Integr Manuf 2022; 74: 102282.

14.

İlman

Yavuz

Taser

. Generalized input preshaping vibration control approach for multi-link flexible manipulators using machine intelligence. Mechatronics 2022; 82: 102735.

15.

Siciliano

Book

. A singular perturbation approach to control of lightweight flexible manipulators. Int J Rob Res 1988; 7: 79–90.

16.

Hooshmand

Fateh

. Voltage control of flexible-joint robot manipulators using singular perturbation technique for model order reduction. J Electr Comput Eng Innovations 2021; 10: 123–142.

17.

Sun

Zhang

Yang

, et al. Singular perturbation-based saturated adaptive control for underactuated Euler–Lagrange systems. ISA Trans 2022; 119: 74–80.

18.

Chen

Fukuda

. Robust sliding-mode tip position control for flexible arms. IEEE Trans Ind Electron 2001; 48: 1048–1056.

19.

Hamzeh Nejad

Fayazi

Ghayoumi Zadeh

, et al. Precise tip-positioning control of a single-link flexible arm using a fractional-order sliding mode controller. J Vib Control 2020; 26: 1683–1696.

20.

Chen

Dong

Yang

, et al. Fixed-time tracking control for flexible joint manipulator with prescribed performance constraint. IEEE Access 2021; 9: 99388–99397.

21.

Morales

Feliu

Jaramillo

. Position control of very lightweight single-link flexible arms with large payload variations by using disturbance observers. Robot Auton Syst 2012; 60: 532–547.

22.

Nasiri

Fakharian

Menhaj

. Observer-based robust control for flexible-joint robot manipulators: a state-dependent Riccati equation-based approach. Trans Inst Meas Control 2020; 42: 3135–3155.

23.

Jin

Huang

. Active vibration control of large space flexible slewing truss using cable actuator with input saturation. Int J Robust Nonlin 2018; 28: 504–518.

24.

Cristofaro

De Luca

Lanari

. Linear-quadratic optimal boundary control of a one-link flexible arm. IEEE Control Syst Lett 2021; 5: 833–838.

25.

Kharitonov

Zimmert

Sawodny

. Active oscillation damping of the fire-rescue turntable ladder. In: 2007 IEEE International conference on control applications, Singapore, 1–3 October 2007, pp.391–396. New York, NY: IEEE.

26.

Pertsch

Zimmert

Sawodny

. Modeling a fire-rescue turntable ladder as piecewise Euler-Bernoulli beam with a tip mass. In: Proceedings of the 48th IEEE conference on decision and control (CDC) held jointly with 2009 28th Chinese control conference, Shanghai, China, 15–18 December, pp.7321–7326. New York, NY: IEEE.

27.

Zimmert

Pertsch

Sawodny

. 2-DOF control of a fire-rescue turntable ladder. IEEE Trans Control Syst Technol 2012; 20: 438–452.

28.

Chen

. The response analysis and vibration control of flexible arms with two nonlinear factors. Int J Appl Mech 2021; 13: 2150005.

29.

Alandoli

Lee

, et al. Robust P-H∞ integrated controller for flexible link manipulator system in the presence of disturbance. J Appl Comput Mech 2020; 7: 646–654.

30.

Pradhan

Subudhi

. Position control of a flexible manipulator using a new nonlinear self tuning PID controller. IEEE/CAA J Autom Sin 2018; 7: 1–14.

31.

Song

Ding

, et al. Vibration suppression method of humanoid flexible arm based on model-free predictive control. Adv Robot 2021; 35: 450–458.

32.

Jiang

Zhang

, et al. Model-free control of flexible manipulator based on intrinsic design. IEEE/ASME Trans Mechatron 2021; 26: 2641–2652.

33.

Zhang

. Intelligent control of flexible joint manipulator based on dynamic pattern recognition. Int J Pattern Recogn 2020; 34: 2050021.

34.

Zhang

Jiang

. Adaptive boundary control of flexible manipulators with parameter uncertainty based on RBF neural network. Shock Vib 2020; 2020: 1–13.

35.

Shang

Yin

, et al. Rotation angle control strategy for telescopic flexible manipulator based on a combination of fuzzy adjustment and RBF neural network. Chin J Mech Eng 2022; 35: 53.

36.

Fareh

Al-Shabi

Bettayeb

, et al. Robust active disturbance rejection control for flexible link manipulator. Robotica 2020; 38: 118–135.

37.

Jiang

Xing

, et al. Research on vibration suppression of the long flexible arm based on three tilt sensors. Math Probl Eng 2021; 2021: 1–16.

38.

Han

. From PID to active disturbance rejection control. IEEE Trans Ind Electron 2009; 56: 900–906.

39.

Gao

. Active disturbance rejection control: a paradigm shift in feedback control system design. In: 2006 American control conference, Minneapolis, MN, USA, 14–16 June 2006, pp.2399–2405. New York, NY: IEEE.

40.

Chen

Huang

Zhao

. The necessary and sufficient condition for the uncertain control gain in active disturbance rejection control. arXiv preprint arXiv:2006.11731, 2020.

41.

Morales

Sira-Ramírez

Feliu

. Adaptive control based on fast online algebraic identification and GPI control for magnetic levitation systems with time-varying input gain. Int J Control 2014; 87: 1604–1621.

42.

Chen

Huang

, et al. New design of active disturbance rejection control for nonlinear uncertain systems with unknown control input gain. Sci China Inf Sci 2022; 65: 156–168.

43.

Shen

Yuan

Liu

. Event-triggered control for hydraulic position tracking system with extended state observer. J Mech Eng 2022; 58: 274–284.

44.

Yang

. Proof of a third-order linear tracking differentiator’s convergence and simulation of its dynamic characteristics. J Yanshan Univ 2006; 30: 44–47.

45.

Gao

. Scaling and bandwidth-parameterization based controller tuning. In: Proceedings of the 2003 American control conference, Denver, CO, USA, 4–6 June 2003, pp.4989–4996. New York, NY: IEEE.

46.

Han

Tan

. Tuning of linear active disturbance rejection controllers based on PID tuning rules. Control Decis 2021; 36: 1592–1600.

47.

Solazzi

. Experimental and analytical study on elevating working platform. Procedia Eng 2017; 199: 2597–2602.