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Abstract

A mine bolter is a fast, safe and economical support equipment, in which the manipulator is the key component for its parallel operation. In this study, the kinematics of mine bolter manipulator is selected as the research object. Aiming at the problem of the dynamic system response estimation of positive solution matrix with uncertain parameters, an interval estimation analysis method based on Chebyshev polynomial is proposed. The response envelope interval of the forward kinematics solution matrix of the manipulator with uncertain parameters is investigated using three types of surrogate models: scanning method, tensor product and collocation method. The feasibility and efficiency of the proposed algorithm are proved by a prototype test. The study of response interval with uncertain parameters has become a new method for predicting the spatial trajectory tracking and location of the manipulator to ensure that its prediction error is minimal.

Keywords

Uncertainty estimation interval algorithm kinematic error estimation

Introduction

To meet the needs of coal enterprises for bolt support speed, operation safety, people reduction and increased efficiency, research and development of a mine bolter were conducted to track the international advanced level of support equipment. Developing the equipment can reduce the labour intensity of operators, improve the working environment and greatly increase the support efficiency and safety factor.

A mine bolter can realize the automatic support of roof and sidewall in rows. Two manipulators are used to achieve the spatial positioning of the automatic drilling frame. In this research, the right manipulator of a mine bolter was selected as the research object. On the basis of screw theory, the mathematical model of the space position and attitude of the manipulator was established, and its forward solution expression was solved. To predict and evaluate the errors of the manipulator during motion at the design stage, the reasons affecting the high precision positioning of the manipulator end were analysed. The main reasons included the manufacturing errors of mechanical parts and components, the assembly and installation errors of the entire machine, the deformation of the manipulator caused by temperature and force effects, transmission mechanism errors and control system errors (e.g. interpolation, servo-system and detection element errors). On the basis of the error sources, numerous references propose the methods of error analysis and reduction. Li¹ used differential method^2–5 to derive the differential motion formula of the object coordinate system. They applied matrix and perturbation methods to establish the static pose error model of robots, but no error compensation model was proposed. P Guo⁶ utilized the differential method to study the variation of pose errors. On the basis of the characteristics of the input parameters of robots, the angle of the moving joint was set as the compensation link, and the error compensation model was established. The positioning accuracy was relatively improved. However, the shortcoming of large calculations was not overcome. In addition, the calculation process was cumbersome and complex, thereby reducing calculation efficiency and accuracy. To improve computational efficiency, L Wang et al.⁷ applied the extreme learning machine algorithm to the mathematical model of the research object. They used the spatial network sampling method to analyse the variation law of errors. Thus, the error compensation model and variation law were obtained. J Mei et al.⁸ adopted the loop increment method to model the error of the research object. The method improved the identification efficiency of the error model and the rapidity of location. These proposed methods aim to analyse and suggest solutions to the motion errors of mechanisms. However, errors of most research objects are not limited to the motion of mechanisms. Errors caused by the original machining assembly and variable load on the space trajectory tracking and positioning of the manipulator should also be considered. Y Zhao et al.⁹ considered the influence coefficient and virtual displacement principle of parallel robots. A new general method of static error modelling and analysis was used to investigate the independent influence of various original machining and assembly error sources on the position and posture of the end of parallel robots. X Hou et al.¹⁰ applied simulation and experiment methods to simulate the study object in ADAMS environment. They analysed the dynamic errors caused by variable load, improved the positioning accuracy of robots and designed and developed an error compensation interface for human–computer interaction. S Wan et al.¹¹ examined the formation errors from the perspective of information theory, which provides a new idea for formation errors that may occur in multi-robot systems. The abovementioned references have contributed to the analysis of robot motion errors, but they were all aimed at a certain error source and thus had great limitations. The research object is also focused on the model of determining parameters for calculation and analysis. A large gap exists between the system response obtained by the traditional deterministic model and the test results. The working environment of the mine bolter is poor, and the positioning accuracy must be high. In addition, the errors caused by manufacturing, assembly and the physical parameters affected by environmental factors can produce several uncertain factors. Uncertain parameters can affect the spatial trajectory tracking and final positioning of the end of the mine bolter manipulators. Small uncertain parameters greatly affect the response of non-linear motion systems.^12–14 Therefore, analysing the kinematic errors of manipulators with uncertain parameters in this study is necessary. On the basis of the kinematics mathematical model of the manipulator, this research comprehensively considers error sources, including mechanism movement, production and assembly and variable load. Thus, the high precision and fast positioning of manipulators are obtained. An interval algorithm is proposed to analyse the kinematics of the mine bolter manipulator with uncertain parameters and estimate errors.

Interval algorithm has also become an important tool in the field of uncertainty analysis.^15–18 Interval method is a vital supplement to the probability method because it only requires the upper and lower bound information of uncertain parameters. The method requires neither probability distribution information nor the fuzzy membership function of uncertain parameters. Common interval algorithms include sampling¹⁹ and fast modelling methods. Fast modelling method is also called proxy model method, which includes polynomial proxy model, mobile least squares proxy model, radial basis function proxy model, Gauss process (or kriging) proxy model, neural network and support vector regression. In interval algorithm research, numerous proxy models^20–24 have been proposed to construct interval algorithms. Agent model, also known as approximation model, is an important tool in approximation modelling. As an approximate replacement model of complex systems, agent model can reduce the excessive occupation of resources, simplify the complex process and effectively promote the implementation of rapid calculation, design optimization and real-time display in various engineering fields. In the implementation process of interval algorithms, an appropriate agent model can assist the implementation of interval analysis and improve the generality and robustness of interval algorithms. To minimize random errors, the sample points should cover the entire design area.²⁰ The construction of proxy models depends on Taylor, Fourier and Chebyshev sequences. The Taylor series and perturbation theory are developed by Z Qiu et al.^25,26 for small-scale uncertainties only. By contrast, the interval method based on Fourier sequence²⁷ and the polynomial of trigonometric function is a general non-embedded method, but its sample demand increases with the uncertain variable dimension and approximate order. The Chebyshev polynomial has a good orthogonal property, and the power function term in the expansion formula can easily calculate the variable fractional differential. The Chebyshev polynomial has the basis of function approximation and the condition of forming operator matrix.²⁸ When using a polynomial to approximate an unknown function, the Chebyshev polynomial has good convergence because it is the eigensolution of the Sturm–Liouville problem and can be used to expand any function in finite interval.²⁹ In addition, the Chebyshev polynomial has high approximation accuracy in approximation theory. A Chebyshev interval expansion function based on Chebyshev series expansion is proposed in this study. The Chebyshev expansion function can more effectively compress the wrapping effect of interval algorithm than the traditional Taylor expansion function, especially when calculating the variation interval of non-monotone functions. The Chebyshev interval expansion function only needs to calculate the output of the original function at the interpolation point. This requirement is easier to achieve than that of calculating the derivative of the original function when the Taylor interval expansion function is established. On the basis of Chebyshev expansion function theory, our study performs the uncertainty error analysis of the kinematics of mine bolter manipulators by combining scanning method (SAS), collocation method (CCM) and tensor product (CTP) proxy models. Response envelope interval and calculation time under different proxy models are also compared. The response study with uncertain interval parameters is used as a new method for predicting the end trajectory interval of the manipulator. This method ensures that the error of the predicted kinematics response is minimal and improves the accuracy of the end positioning of the manipulator. Finally, a good uncertainty interval algorithm can provide theoretical support for the kinematic error analysis of the two manipulators of the mine bolter, including its multi-manipulators, in the future. The high-precision operation of the mine bolter manipulator also expands the breadth and depth of applying the interval algorithm theoretical model.

Mathematical modelling of manipulator kinematics

On the basis of screw theory,^30–34 the kinematics analysis and modelling of the mine bolter manipulator are investigated in this research. The specific method is used to establish the calculation formula of forward kinematics in accordance with the D-H coordinates and parameters of the right manipulator.

Screw theory

The rigid body transformation can be expressed by Euclidean group $SO (3)$ .³⁵ If the rigid body inertial coordinate system is S and the tool coordinate system is T, then the set of the rigid body posture transformation on S can be expressed as follows

SE (3) = {[\begin{matrix} R & p \\ 0 & 1 \end{matrix}] : R \in SO (3), p \in R^{3}}

(1)

In formula (1), $SO (3)$ is 3D rotating group; $R \in SO (3)$ represents the attitude matrix of the T system relative to the S system; $p \in R^{3}$ denotes the position and attitude vector of the T system relative to the S system; $R^{3}$ refers to a group of 3D column vectors as unit elements.

If a rigid body rotates at a uniform speed around axis $ω$ at angle $θ$ and assumes that $ω = (ω_{x}, ω_{y}, ω_{z})^{T} \in R^{3}$ , then the transformation of the rigid body from the initial to final state can be described by the following matrix exponents

R (ω, θ) = e^{\overset{\land}{ω} θ} \in SO (3)

(2)

where $\overset{\land}{ω} = [\begin{matrix} 0 & - ω_{z} & ω_{y} \\ ω_{z} & 0 & - ω_{x} \\ - ω_{y} & ω_{x} & 0 \end{matrix}] \in SO$ is the antisymmetric matrix of $ω$ ; $ω \in R^{3}$ denotes the unit vector on the rotation axis; $θ$ indicates the angle of rotation (°); $e^{\overset{\land}{ω} θ}$ refers to the matrix index corresponding to $\overset{\land}{ω} θ$ .

Unit matrix I is defined as $3 \times 3$ . The matrix index can be calculated using Rodrigues’ formula

e^{\hat{ω} θ} = I + \hat{ω} \sin θ + {\hat{ω}}^{2} (1 - \cos θ)

(3)

Chasles proved that rigid body attitude transformation $g$ can be obtained by the complex motion of a rigid body rotating around $ω$ and $θ$ and translating v along the $ω$ axis

g = e^{\hat{ξ} θ} \in SE (3)

(4)

where $\hat{ξ} = [\begin{matrix} \hat{ω} & ν \\ 0 & 0 \end{matrix}] \in se (3)$

$ω, ν \in R^{3}$ represents the spinors of motion and infinitesimal quantities of the spiral motion of rigid bodies; $ν = - ω \times q$ is the unit vector in the direction of the rigid body movement; $q \in R^{3}$ indicates the coordinate of any point on the rotation axis; $se (3)$ refers to a Lie algebra of $SE (3)$ ; $e^{\hat{ξ} θ}$ denotes the matrix index corresponding to $\hat{ξ} θ$ .

On the basis of formulas (3) and (4), we can obtain the following

e^{\hat{ξ} θ} = {\begin{matrix} [\begin{matrix} e^{\hat{ω} θ} & (I - e^{\hat{ω} θ}) (ω \times ν) + ω ω^{T} ν θ \\ 0 & 1 \end{matrix}], ω \neq 0 \\ [\begin{matrix} I & ν θ \\ 0 & 1 \end{matrix}], ω = 0 \end{matrix}

(5)

Let $g_{ST} (θ)$ and $g_{ST} (0)$ be the initial and post-rotation positions of reference system T relative to reference system S, respectively. Thus, the rotation of a rigid body in space can be expressed as follows

g_{ST} (θ) = e^{\hat{ξ} θ} g_{ST} (0)

(6)

For a given robot, let its degree of freedom be n, its joint variable be $θ = {[θ_{1}, θ_{2}, \dots, θ_{n}]}^{T}$ , the end pose be $g_{ST} (θ) \in SE (3)$ and the initial pose be $g_{ST} (0)$ .

For each joint i, a corresponding motion screw $ξ_{i}$ exists. For the rotating pair, $ω_{i} \in R^{3}$ is the unit vector on the axis of the motion screw. For the moving pair, $ν_{i} \in R^{3}$ denotes the unit vector corresponding to the moving direction. $q_{i} \in R^{3}$ indicates the point arbitrarily selected on the joint axis. For the rotating and moving joints, the rotational coordinates of $ξ_{i}$ are expressed as $ξ_{i} = [\begin{matrix} ω \\ q_{i} \times ω_{i} \end{matrix}]$ and $ξ_{i} = [\begin{matrix} 0 \\ ν_{i} \end{matrix}]$ , respectively. The kinematics of each joint is combined; thus, the exponential product equation of the robots’ forward kinematics can be expressed as follows

g_{ST} (θ) = e^{{\hat{ξ}}_{1} θ_{1}} e^{{\hat{ξ}}_{2} θ_{2}} \dots e^{{\hat{ξ}}_{n} θ_{n}} g_{ST} (0)

(7)

Kinematic model based on screw theory

The right manipulator of the mine bolter is the research object. The positive expression of the space position and posture of the manipulator are derived. Figure 1 shows the structure diagram of the right manipulator of the mine bolter, and Figure 2 presents its coordinate system definition. The manipulator has six degrees of freedom, including rotating $θ_{1}$ around the $z_{1}$ axis, rotating $θ_{2}$ around the $z_{2}$ axis, stretching $d_{3}$ along the $z_{3}$ axis, rotating $θ_{4}$ around the $z_{4}$ axis, rotating $θ_{5}$ around $z_{5}$ the axis and rotating $θ_{6}$ around the $z_{6}$ axis.

Figure 1.

Structural diagram of manipulator.

Figure 2.

Coordinate diagram of manipulator.

Figure 1 exhibits the origin of each coordinate system. The positions of $O_{O}$ and $O_{1}$ are the base surface, corresponding to coordinate systems $S_{0}$ and $S_{1}$ , respectively. $O_{2}$ represents the outer sleeve revolving pin centre, corresponding to coordinate system $S_{2}$ . $O_{3}$ and $O_{4}$ denote the turning centres of the levelling seat, corresponding to coordinate systems $S_{3}$ and $S_{4}$ , respectively. $O_{5}$ is located on the surface of the levelling seat and corresponds to coordinate system $S_{5}$ . $O_{6}$ is located on the mass centre of the rotary cylinder, corresponding to coordinate system $S_{6}$ . $O_{7}$ is located on the mass centre of the drill rig clamp, corresponding to coordinate system $S_{7}$ . Among them,

$O_{2}$ : coordinates $(x_{1}, 0, z_{1}) = (230, 0, 139)$ in the $S_{1}$ coordinate system.

$O_{3}$ / $O_{4}$ : coordinates $(x_{2}, y_{2}, 0) = (2180, 0, 0)$ in the $S_{2}$ coordinate system.

$O_{5}$ : coordinates (x₄, y₄, 0) = (220, −191, 0) in the $S_{3}$ / $S_{4}$ coordinate system.

$O_{6}$ : coordinates (x₅, 0, z₅) = (642.5, 0, −220) in the $S_{5}$ coordinate system.

$O_{7}$ : coordinates (x₆, y₆, z₆) = (−1385, 131, 435) in the $S_{6}$ coordinate system.

The positive solution of space pose matrix can be obtained by establishing a transformation matrix among coordinate systems and a position matrix of the origin coordinate system as shown in formula (8)

\begin{matrix} T_{7}^{0} = T_{1}^{0} \cdot T_{2}^{1} \cdot T_{3}^{2} \cdot T_{4}^{3} \cdot T_{5}^{4} \cdot T_{6}^{5} \cdot T_{7}^{6} \\ = (\begin{matrix} c θ_{1} \cdot c θ_{5} \cdot c (θ_{2} + θ_{4}) - s θ_{1} \cdot s θ_{5} \\ s θ_{1} \cdot c θ_{5} \cdot c (θ_{2} + θ_{4}) + c θ_{1} \cdot s θ_{5} \\ - c θ_{5} \cdot s (θ_{2} + θ_{4}) \\ 0 \end{matrix} \\ \begin{matrix} - c θ_{1} \cdot s θ_{5} \cdot c θ_{6} \cdot c (θ_{2} + θ_{4}) - s θ_{1} \cdot c θ_{5} \cdot c θ_{6} + c θ_{1} \cdot s θ_{6} \cdot s (θ_{2} + θ_{4}) \\ - s θ_{1} \cdot s θ_{5} \cdot c θ_{6} \cdot c (θ_{2} + θ_{4}) + c θ_{1} \cdot c θ_{5} \cdot c θ_{6} + s θ_{1} \cdot s θ_{6} \cdot s (θ_{2} + θ_{4}) \\ s θ_{5} \cdot c θ_{6} \cdot s (θ_{2} + θ_{4}) + s θ_{6} \cdot c (θ_{2} + θ_{4}) \\ 0 \end{matrix} \\ \begin{matrix} c θ_{1} \cdot s θ_{5} \cdot s θ_{6} \cdot c (θ_{2} + θ_{4}) + s θ_{1} \cdot c θ_{5} \cdot s θ_{6} + c θ_{1} \cdot c θ_{6} \cdot s (θ_{2} + θ_{4}) \\ s θ_{1} \cdot s θ_{5} \cdot s θ_{6} \cdot c (θ_{2} + θ_{4}) - c θ_{1} \cdot c θ_{5} \cdot s θ_{6} + s θ_{1} \cdot c θ_{6} \cdot s (θ_{2} + θ_{4}) \\ - s θ_{5} \cdot s θ_{6} \cdot s (θ_{2} + θ_{4}) + c θ_{6} \cdot c (θ_{2} + θ_{4}) \\ 0 \end{matrix} \\ \begin{matrix} \begin{matrix} [- c θ_{1} \cdot s θ_{5} \cdot s θ_{6} \cdot c (θ_{2} + θ_{4}) - s θ_{1} \cdot c θ_{5} \cdot s θ_{6} - c θ_{1} \cdot c θ_{6} \cdot s (θ_{2} + θ_{4})] \cdot x_{6} \\ + [- c θ_{1} \cdot s θ_{5} \cdot c θ_{6} \cdot c (θ_{2} + θ_{4}) - s θ_{1} \cdot c θ_{5} \cdot c θ_{6} + c θ_{1} \cdot s θ_{6} \cdot s (θ_{2} + θ_{4})] \cdot y_{6} \\ + [c θ_{1} \cdot c θ_{5} \cdot c (θ_{2} + θ_{4}) - s θ_{1} \cdot s θ_{5}] \cdot (z_{6} + x_{5}) + c θ_{1} \cdot s (θ_{2} + θ_{4}) \cdot (z_{5} - y_{4}) \\ + c θ_{1} \cdot c (θ_{2} + θ_{4}) \cdot x_{4} + c θ_{1} \cdot c θ_{2} \cdot (d_{3} + x_{2}) - c θ_{1} \cdot s θ_{2} \cdot y_{2} + c θ_{1} \cdot x_{1} \end{matrix} \\ \begin{matrix} [- s θ_{1} \cdot s θ_{5} \cdot s θ_{6} \cdot c (θ_{2} + θ_{4}) + c θ_{1} \cdot c θ_{5} \cdot s θ_{6} - s θ_{1} \cdot c θ_{6} \cdot s (θ_{2} + θ_{4})] \cdot x_{6} \\ + [- s θ_{1} \cdot s θ_{5} \cdot c θ_{6} \cdot c (θ_{2} + θ_{4}) + c θ_{1} \cdot c θ_{5} \cdot c θ_{6} + s θ_{1} \cdot s θ_{6} \cdot s (θ_{2} + θ_{4})] \cdot y_{6} \\ + [s θ_{1} \cdot c θ_{5} \cdot c (θ_{2} + θ_{4}) + c θ_{1} \cdot s θ_{5}] \cdot (z_{6} + x_{5}) + s θ_{1} \cdot s (θ_{2} + θ_{4}) \cdot (z_{5} - y_{4}) \\ + s θ_{1} \cdot c (θ_{2} + θ_{4}) \cdot x_{4} + s θ_{1} \cdot c θ_{2} \cdot (d_{3} + x_{2}) - s θ_{1} \cdot s θ_{2} \cdot y_{2} + s θ_{1} \cdot x_{1} \end{matrix} \\ \begin{matrix} [s θ_{5} \cdot s θ_{6} \cdot s (θ_{2} + θ_{4}) - c θ_{6} \cdot c (θ_{2} + θ_{4})] \cdot x_{6} + [s θ_{5} \cdot c θ_{6} \cdot s (θ_{2} + θ_{4}) + s θ_{6} \cdot c (θ_{2} + θ_{4})] \cdot y_{6} \\ - c θ_{5} \cdot s (θ_{2} + θ_{4}) \cdot (z_{6} + x_{5}) + c (θ_{2} + θ_{4}) \cdot (z_{5} - y_{4}) - s (θ_{2} + θ_{4}) \cdot x_{4} - s θ_{2} \cdot (d_{3} + x_{2}) - c θ_{2} \cdot y_{2} + z_{1} \end{matrix} \\ 1 \end{matrix}) \end{matrix}

(8)

Among them, the manipulator has six degrees of freedom, including the following: $θ_{1}$ : Degree of freedom 1, rotate around $Z_{1}$ axis; $θ_{2}$ : Degree of freedom 2, rotate around $Z_{2}$ axis; $d_{3}$ : Degree of freedom 3, Telescoping along $Z_{3}$ axis; $θ_{4}$ : Degree of freedom 4, rotate around $Z_{4}$ axis; $θ_{5}$ : Degree of freedom 5, rotate around $Z_{5}$ axis; $θ_{6}$ : Degree of freedom 6, rotate around $Z_{6}$ axis; $T_{1}^{0}$ : The transformation from coordinate system $S_{1}$ to coordinate system $S_{0}$ ; $T_{2}^{1}$ : The transformation from coordinate system $S_{2}$ to coordinate system $S_{1}$ ; $T_{3}^{2}$ : The transformation from coordinate system $S_{3}$ to coordinate system $S_{2}$ ; $T_{4}^{3}$ : The transformation from coordinate system $S_{4}$ to coordinate system $S_{3}$ ; $T_{5}^{4}$ : The transformation from coordinate system $S_{5}$ to coordinate system $S_{4}$ ; $T_{6}^{5}$ : The transformation from coordinate system $S_{6}$ to coordinate system $S_{5}$ ; $T_{7}^{6}$ : The transformation from coordinate system $S_{7}$ to coordinate system $S_{6}$ .

Chebyshev interval function algorithms

The current focus of interval method research is how to control the expansion of the result response interval caused by the wrapping effect (i.e. overestimation phenomenon). In interval algorithm research, numerous proxy models^20–23,36 have been proposed to construct interval algorithms. The previous Taylor expansion function method involves the first- and second-order differentials of independent variables with respect to uncertain parameters. However, the derivation process is complex. For the problems of large uncertain ranges or strong non-linear systems, JL Wu et al.³⁷ proposed the Chebyshev expansion function.

The continuous function $f (x)$ in interval $[a, b]$ can be approximated as a polynomial in accordance with the expected accuracy. Therefore, JL Wu et al.³⁷ used the truncated Chebyshev polynomial to approximate the original function and then replace it within the allowable range of accuracy. The n-th Chebyshev polynomial is defined in $x \in [- 1, 1]$ as follows

C (x) = \cos n θ

(9)

where $θ = \arccos x, θ \in [0, π]$

The Chebyshev expansion function of original function $f (x)$ is defined as follows

[f_{Cn}] ([x]) = \frac{1}{2} f_{0} + Σ_{i = 1}^{n} f_{i} C_{i} ([x]) = \frac{1}{2} f_{0} + Σ_{i = 1}^{n} f_{i} \cos i [θ]

(10)

where $f_{i}$ is the constant coefficient of the Chebyshev polynomial, which is related to the interpolation point.

The basic concepts, models and general solutions of interval analysis are mature. In this study, the input–output proxy models of CCM,^38–40 SAS⁴¹ and CTP⁴² are constructed on the basis of the characteristics of Chebyshev polynomials. The response envelope interval of the forward kinematics solution matrix of the manipulator with uncertain parameters is analysed using three agent models. Finally, an interval algorithm with high efficiency and accuracy is obtained.

H Peng et al.⁴³ suggested the necessity of establishing a proxy model and the advantages of Chebyshev series. After defining that accuracy depends on the elected approximate model, three methods are obtained by selecting the series as the proxy model. The construction process of the proxy model is independent of the design of the interval algorithm. The proxy model is used to construct the functional relationship between output and input variables. Although the construction process of different agent models is different, the final form is composed of relatively fixed basis functions and correlation coefficients containing input variables. By using $p (x)$ to represent the constructed proxy model, the Chebyshev sequence is approximated as follows

\begin{matrix} p (x) = \sum_{i_{1} = 0}^{d} {\dots \sum_{i_{r} = 0}^{d} \dots \sum_{i_{n}}^{d} (\frac{1}{2})}^{β} \\ f_{i_{1, \dots i_{r, \dots i_{n}}}} \cos (i_{1} α_{1}) \dots \cos (i_{r} α_{r}) \dots \cos (i_{n} α_{n}) \end{matrix}

(11)

In the model, $β$ is the statistical number of zero-order Chebyshev polynomials $(i_{r} = 0)$ , and r denotes the component index of input $x = {x_{1}, \dots, x_{r}, \dots, x_{n}}$ and intermediate variable $α = {α_{1}, \dots, α_{r}, \dots, α_{n}}$ . If $\bar{x_{r}}$ and $\underline{x_{r}}$ are used to represent the upper and lower boundaries of input component $x_{r}$ , respectively, then component $α_{r}$ of the intermediate variable can be calculated using the following formula

α_{r} = \arccos (\frac{2 x_{r} - (\bar{x_{r}} + \underline{x_{r}})}{\bar{x_{r}} - \underline{x_{r}}}) \in [0, π]

(12)

Coefficient $f_{i_{1}, \dots i_{r}, \dots i_{n}}$ is calculated using Chebyshev’s orthogonality and the Mehler integral. The formula is as follows

\begin{matrix} {f_{i_{1, \dots i_{r, \dots i_{n}}} \approx (\frac{2}{\tilde{d}})}}^{n} \sum_{q_{1} = 1}^{\tilde{d}} \dots \sum_{q_{s} = 1}^{\tilde{d}} \dots f (x_{q_{1}}, \dots, x_{q_{s}}, \dots, x_{q_{n}}) \\ \cos (i_{1} θ_{q_{1}}) \dots \cos (i_{r} θ_{q_{s}}) \dots \cos (i_{n} θ_{q_{n}}) \end{matrix}

(13)

For convenience, s equally represents the component index of input variables. The integral point (also the zero of the Chebyshev polynomial) is as follows

θ_{q_{s}} = \frac{2 q_{s} - 1}{\tilde{d}} \frac{π}{2}

(14)

$q_{s}$ takes values from 1 to $\tilde{d}$ , which is the approximate calculation order of the Chebyshev sequence of order d, which is often greater than or equal to $d + 1$ . $x_{q_{x}}$ can be calculated using the following formula

x_{q_{s}} = \frac{\bar{x_{q_{s}}} + \underline{x_{q_{s}}}}{2} + \frac{\bar{x_{q_{s}}} - \underline{x_{q_{s}}}}{2} \cos (θ_{q_{s}})

(15)

The provided proxy model shows that the display expression can be expressed in the form of coefficients and input variables (or input variable correlation basis functions). When the agent model is constructed, the interval boundaries of input $x$ are determined, and the corresponding interval expansion can be obtained using formula $y^{I} = [f] (ξ^{I})$ . For example, the interval expansion function of an interval algorithm using response surface proxy model is as follows

y^{I} = [p] (x^{I}) = β_{0} + \sum_{k = 1}^{n} β_{k} x_{k}^{I} + \sum_{k = 1}^{n} {β_{k} (x_{k}^{I})}^{2} + \dots

(16)

By using the Chebyshev sequence approximation to implement the interval algorithm, the interval expansion is as follows

\begin{matrix} y^{I} = [p] (x^{I}) = \sum_{i_{1} = 0}^{d} \dots \sum_{i_{r} = 0}^{d} \dots \sum_{i_{n} = 0}^{d} {(\frac{1}{2})}^{β} \\ f_{i_{1}, \dots, i_{r} \dots, i_{n}} . \cos (i_{1} [α_{1}]) \dots \cos (i_{r} [α_{r}]) \dots \cos (i_{n} [α_{n}]) \end{matrix}

(17)

y^{I} = [p] (x^{I}) = {(\frac{1}{2})}^{n} f_{i_{1} = 0, \dots, i_{r} = 0 \dots, i_{n} = 0} + \sum_{i_{1} = 0}^{d} \dots \sum_{i_{r} = 0}^{d} \dots \sum_{i_{n} = 0}^{d} | {(\frac{1}{2})}^{β} f_{i_{1}, \dots, i_{r} \dots, i_{n} (\exists i_{r} \neq 0)} | ϕ^{I}, ϕ^{I} = [- 1, 1]

(18)

On the basis of the expansion functions in formulas (16) and (18), the intervals of output variables can be obtained through the interval operations defined in the four-rule operation of intervals.

Algorithm flow

Initializing algorithm parameters. The interval boundary ${[\underline{ξ}, \bar{ξ}]}^{k}$ of uncertain input is determined, and the computational resources required for ordinary differential equation (ODE) numerical solution of uncertain input are recorded. Furthermore, the number of samples is determined.

Sampling. The sample is mapped from original space ${[0, 1]}^{k}$ A to uncertainty interval ${[\underline{ξ}, \bar{ξ}]}^{k}$ .

ODE equation solution. The dynamic system equation corresponding to each sample in the sample set must be solved by the ODE numerical method. If the ODE calculation is 0 ∼ t, and discrete $Ω$ is an output variable, then the corresponding response sequence with length $Ω$ is obtained.

Constructing the design matrix. The exponential vector set ${η^{(j)}}_{j = 1,,, N_{b}}$ of corresponding parameters $η_{m}$ and k is calculated by the algorithm. The regression term of the Chebyshev polynomial basis function is calculated. Finally, the generalized design matrix $ϕ$ for regression analysis is established.

Coefficient estimation. On the basis of the known design matrix $ϕ$ and dynamic response system response combination ${y^{(p)} (t, ξ)}_{p = 1,,,, n}$ , the coefficient path is obtained.

Model selection. The optimal model is obtained after the coefficient path is determined.

Interval analysis. After establishing the model, the Chebyshev interval expansion function can be obtained. Finally, the upper and lower bounds of the expansion function output can be calculated.

Figure 3 shows the flow chart of the algorithm. The process of solving kinematics response is similar to a ‘black box’ model. Only by transforming uncertain parameters into interpolation point data of the Chebyshev expansion function can the corresponding time-varying response data be obtained. The Chebyshev expansion function is constructed by using the Chebyshev polynomial coefficients obtained from the process, and the dynamic response interval is acquired. Among them, h is the iteration step, $t_{0}$ represents the initial time, $t_{end}$ indicates the termination time and K denotes the iteration number. $I_{1}$ and $I_{2}$ refer to the discrete expressions of generalized velocities and coordinates.³³

Figure 3.

The Chebyshev interval expansion function is used to solve the algorithm flow of kinematics system with interval uncertain parameters.

Uncertainty analysis of interval parameters of manipulator system

Figure 1 illustrates that the dynamic matrix nonlinearity of the right manipulator of a mine bolter is high. The attitude of the manipulator supporting the roof vertically upwards is selected as the analysis condition, that is, the coordinate value of the end positioning is R11 (3400, −2143,2894). Table 1 presents the corresponding joint angles and rod lengths. The positive solution expression of the manipulator matrix is shown in formula (1).

Table 1.

A list of parameters of the manipulator.

Member	1	2	3	4	5	6
Quality (kg)	50	475	200	40	100	2000
Range	[0°,50°]	[–5°,55°]	[0,1100 mm]	[–5°,55°]	[0°,50°]	[0°,120°]
Stance point: R11	43.3955°	26.5454°	950.4343	26.5454°	43.3955°	1.9481°

In the matrix, $θ_{1}, θ_{2}, θ_{4}, θ_{5}, θ_{6}$ represent the angles of the bar, and $d_{3}$ indicates the length of the telescopic arm. $θ_{1}, θ_{2}, d_{3}, θ_{4}, θ_{5}, θ_{6}$ are assumed as the uncertain input parameters in the positive solution of the manipulator matrix. This method is different from the parameters mentioned in Cheng and Sandu,¹⁴ which proved the rapidity of the previously mentioned method. Table 2 shows the range and median of uncertain parameters. The input and output codes and the three proxy model algorithms are detailed in the Appendix 1.

Table 2.

The interval uncertainty of the positive solution of the manipulator.

Parameter name	Mean value	Range
$θ_{1}$ (rad)	0.7574	[0.7498, 0.7650]
$θ_{2}$ (rad)	0.4633	[0.4587, 0.4679]
$d_{3}$ (mm)	950.4034	[940.9003, 959.9083]
$θ_{4}$ (rad)	0.4633	[0.4587, 0.4679]
$θ_{5}$ (rad)	0.7574	[0.7498, 0.7650]
$θ_{6}$ (rad)	0.0314	[0.0311, 0.0317]

The calculation time of the matrix is 10 s, and the discrete time step of the response output of the dynamic system is 1000. In the CTP and CCM methods, the order of calculation is 3, and the truncation order is 2. The interval of each uncertain parameter is uniformly selected at 16 time points. The order $η_{m}$ of the largest Chebyshev polynomial in CCM, CTP and SAS methods is set to 4. For the forward solution of the manipulator matrix, the original calculation method and the mentioned three interval methods are used to calculate the solution. The central processor (CPU) time consumption of the four methods is also compared. Table 3 shows the comparison results.

Table 3.

Comparison of CPU time of positive solution of manipulator.

Method	Original calculation method	SAS	CTP	CCM
CPU time (s)	69.257082	16.3973111214409	57.136721507906740	1.630121614900830
Time-consuming reduction (%)	−	76.32%	17.50%	97.65%

SAS: scanning method; CPU: central processor; CTP: tensor product; CCM: collocation method.

Table 3 confirms that in the statistics of computing resource consumption, CCM needs the least CPU time, whereas CTP needs the highest CPU time. In addition, SAS and CCM have more advantages in CPU time consumption than CTP. In comparison with the original calculation method, the CPU time consumption of SAS is reduced by 76.32% (Figure 4) and that of CTP by 17.50% (Figure 5).

Figure 4.

Comparison between the estimated value of the upper boundary of the three interval algorithms and the theoretical upper boundary value.

Figure 5.

Comparison between the estimated value of the lower boundary of the three interval algorithms and the theoretical lower boundary value.

Figures 4 and 5 illustrate that the three interval methods can obtain good interval envelope results. The results of CCM and SAS are close to the theoretical values. Both methods can produce the least conservative estimations in the error estimation, with a small estimation range. The CTP method can obtain a wide range of estimations, but it takes a long time. In displacement calculation, the accuracy of SAS and CCM can maintain the same level, but the CPU time of CCM is short. This finding highlights the advantages of CCM in less conservative estimation and high computational efficiency. Tables 4 and 5 further show the comparison of numerical results at certain time points. The approximation accuracy of the proxy model based on polynomial only depends on the sampling points but is independent of the regression basis function vector. The zero of the Chebyshev polynomial is used as a sampling point to construct a high-order polynomial regression model. SAS, CCM and CTP can locally obtain a good conservative estimation. By comparison, CCM has the least computation time, and its estimated value is close to the theoretical values. On the contrary, CTP can only obtain good estimation from local values, and its CPU takes the longest time. The following are the main reasons: (1) The sampling points of CTP are interpolation points formed by the CTP of 1D polynomial zeros, whereas those of CCM are selected from these interpolation points. (2) CTP uses the highest polynomial as the regression model, whereas the least polynomial is utilized to construct the regression model. The reason is that CTP converges fast enough only when the number of its sampling points is large and when that of CCM sampling points is small.

Table 4.

Comparison of the theoretical value of the upper boundary and three upper bounds estimates under the positive solution matrix.

(Time / s)	Theoretical value	SAS	CCM	CTP
1	−0.2	−0.13888	−0.13883	−0.13095
2	−0.8	−0.77315	−0.77306	−0.77177
3	0.6	0.614085	0.614149	0.616417
4	2924.7	2971.61	2971.803	2991.778
5	0.8	0.838743	0.83887	0.844267
6	0.3	0.27733	0.277462	0.283556
7	0.5	0.549952	0.550014	0.550527
8	3801.9	3835.219	3835.54	3839.28
9	−0.6	−0.57278	−0.57272	−0.57049
10	0.6	0.576082	0.57614	0.576594
11	0.6	0.590827	0.590872	0.594697
12	−1268.8	−1231.32	−1231.18	−1214.69
13	0	0	0	0
14	0	0	0	0
15	0	0	0	0
16	1	1	1	1

SAS: scanning method; CTP: tensor product; CCM: collocation method.

Table 5.

Comparison of the theoretical value of the lower boundary and three lower bounds estimates under the positive solution matrix.

(Time / s)	Theoretical value	SAS	CCM	CTP
1	−0.2	−0.17091	−0.17103	−0.17198
2	−0.8	−0.78749	−0.78767	−0.79033
3	0.6	0.597471	0.597411	0.59512
4	2924.7	2877.675	2877.392	2871.686
5	0.8	0.816732	0.816658	0.81602
6	0.3	0.245841	0.245803	0.243254
7	0.5	0.532809	0.532746	0.528591
8	3801.9	3768.329	3768.055	3753.57
9	−0.6	−0.58919	−0.58924	−0.59151
10	0.6	0.559743	0.559681	0.555688
11	0.6	0.575144	0.575143	0.574711
12	−1268.8	−1306.43	−1306.47	−1310.42
13	0	0	0	0
14	0	0	0	0
15	0	0	0	0
16	1	1	1	1

SAS: scanning method; CTP: tensor product; CCM: collocation method.

Test verification

Experimental research on uncertainties can investigate the kinematics characteristics of the manipulator system. Therefore, conducting such research can solve practical engineering problems.

Figure 6 shows that experiments are performed on the newly developed manipulator to achieve the same trajectory tracking and end-point positioning. Such experiments verify the validity and efficiency of the interval method investigated in this research. Three interval algorithms are used individually to test the kinematics of the mine bolter manipulator with uncertain parameters. The difference between the positive solution of the manipulator matrix and the upper and lower bounds estimated by three interval algorithms is obtained, as shown in Tables 6 and 7.

Figure 6.

Test prototype of mechanical arm interval algorithm for manipulator.

Table 6.

The difference between the theoretical upper boundary value and the upper boundary value estimated by three intervals.

Time(s)	$Δ SAS$	$Δ CCM$	$Δ CTP$
1	0.0161	0.0161	0.024
2	0.0073	0.0072	0.0086
3	0.0084	0.0083	0.0106
4	47.1423	46.9491	67.1173
5	0.0111	0.011	0.0165
6	0.0158	0.0157	0.0219
7	0.0086	0.0086	0.0091
8	33.6527	33.3316	37.3927
9	0.0083	0.0082	0.0105
10	0.0082	0.0082	0.0087
11	0.0079	0.0078	0.0117
12	37.6627	37.5197	54.1559
13	0	0	0
14	0	0	0
15	0	0	0
16	0	0	0

Table 7.

The difference between the theoretical lower boundary value and the lower boundary value estimated by three intervals.

Time(s)	$Δ SAS$	$Δ CCM$	$Δ CTP$
1	0.0161	0.016	0.017
2	0.0073	0.0071	0.0099
3	0.0084	0.0083	0.0107
4	47.2686	46.9855	52.974
5	0.0111	0.0111	0.0118
6	0.0158	0.0158	0.0184
7	0.0086	0.0086	0.0128
8	33.8321	33.5577	48.3168
9	0.0082	0.0082	0.0105
10	0.0082	0.0082	0.0122
11	0.0079	0.0079	0.0083
12	37.6289	37.5902	41.5778
13	0	0	0
14	0	0	0
15	0	0	0
16	0	0	0

Sixteen array time points are used to highlight the difference among the three interval methods and theoretical values. The minimum error is also obtained. Tables 6 and 7 show that the results of the compact package validate the accuracy and validity of the proposed algorithm. In most time discrete points, SAS and CCM have the smallest conservative estimations, and CTP method can obtain good local estimations.

Conclusion

On the basis of the Chebyshev expansion function algorithm, the response envelope interval of the forward kinematics solution matrix of the manipulator with uncertain parameters is investigated by establishing three proxy models: SAS, CTP and CCM. Through the research and comparison of three interval algorithms, this method can effectively reduce the error of the predicted value of kinematic response. The response interval of the kinematic system of the manipulator can also be quickly and steadily obtained. The study concludes with the following points.

On the basis of screw theory, the mathematical model of the forward kinematics solution of mine bolter manipulator is established. In accordance with the process of the Chebyshev interval algorithm and the range of uncertain parameters, three interval algorithms based on interval mathematics and Chebyshev theory are applied to the kinematics analysis of the manipulator. The change range of the positive solution matrix value of the mine bolter manipulator is solved. Interval algorithm has more important engineering practical value than deterministic method.

In the CPU consumption statistics of computing resources, the three interval algorithms have less value than the original method. In comparison with the original method, SAS, CTP and CCM reduce the CPU time consumption by 76.32%, 17.50% and 97.65%, respectively.

By comparing the results and theoretical values of the upper and lower boundaries of the three interval proxy models, our results reveal that the CCM and SAS are close to the theoretical values and can produce the least conservative approximations in error estimation. The CCM proxy model consumes the least CPU time, and the calculated value is close to the theoretical values. Therefore, the CCM proxy model is the most ideal among the three proxy models.

Finally, experiments indicate that the interval algorithm is effective and feasible. However, the experimental research involves numerous uncertain and uncontrollable factors. Thus, realizing the characteristic analysis of the system kinematics with uncertain parameters is difficult. Given this problem, follow-up work must further investigate and discuss uncertain problems to obtain further experimental results.

The study of the interval boundaries of kinematic system responses with uncertain parameters has become a new method for predicting the trajectory intervals and terminal positioning errors of manipulators. This approach has provided a theoretical support for the analysis of the kinematic errors of the two manipulators of mine bolter, including its multi-manipulators, in the future. The high-precision operation of manipulators has also expanded the breadth and depth of applying the interval algorithm theoretical model.

Supplemental Material

AEM-18-2686-Supplement2019-1-17 – Supplemental material for Kinematics uncertainty analysis of mine bolter manipulator based on Chebyshev interval algorithms

Supplemental material, AEM-18-2686-Supplement2019-1-17 for Kinematics uncertainty analysis of mine bolter manipulator based on Chebyshev interval algorithms by Jun Zhang, Qing-xue Huang, Wenjun Meng, Xiangdong Yu, Guoying Meng and Lifeng Ma in Advances in Mechanical Engineering

Footnotes

Appendix 1

Handling Editor: Jia-Jang Wu

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the National Key Research and Development of China (Grant number: 2016YTC0600900); The Yue Qi Distinguished Scholar Project of China University of mining & Technology (Beijing) (Grant number: 800015Z1145); Shanxi ‘1331 Project’ Key Subjects Construction (Grant number:20181101017) and CCTEG Project (Grant number:2018-TD-QN038).

ORCID iD

Qing-xue Huang

References

Design and motion error analysis of spraying robot. Wuhan, China: Huazhong University of Science and Technology, 2012.

Cheng

Analysis and optimization of 6R industrial robot terminal space time error coupling. Chongqing, China: Chongqing Jiaotong University, 2016.

Guo

Research on the compensation method of location error based on KUKA industrial robot. Changchun, China: Changchun University of Technology, 2016.

Hou

Zhu

Jiang

Error analysis of 6R robot posture based on rigid-flexible coupling modeling. Mod Mach Tool Autom Process Technol 2018; 1: 51–55.

Chen

Research on trajectory planning and pose error analysis of 4-DOF assembly robot. Xi’an, China: Chang’an University, 2016.

Guo

Pose error analysis of multi legged walking robot. Wuhan, China: Huazhong University of Science and Technology, 2008.

Wang

Zhang

et al . Analysis of positioning errors of industrial robots and study of precision compensation based on ELM algorithm. Nanjing, China: Nanjing University of Aeronautics and Astronautics, 2018.

Mei

Sun

Luo

et al . Location error analysis and kinematics calibration of palletizing robot based on one-dimensional wire measuring system. J Tianjin Univ Sci Technol (Nat Sci Eng Ed) 2018; 51: 748–754.

Zhao

Song

Yang

et al . Static error modelling and calibration for parallel robots based on virtual joint pairs. China Mech Eng 2017; 28: 2189–2197.

10.

Hou

Zhu

Wang

Dynamic error analysis of 6-DOF serial robot. J Hefei Univ Technol (Nat Sci Ed) 2017; 40: 299–303.

11.

Wan

Fan

et al . Information theory in formation control: an error analysis to multi-robot formation. Entropy 2018; 20: 618.

12.

Hanss

The transformation method for the simulation and analysis of systems with uncertain parameters. Fuzzy Set Syst 2002; 130: 277–289.

13.

Chen

Stability and chaotic dynamics of a rate gyro with feedback control under uncertain vehicle spin and acceleration. J Sound Vib 2004; 273: 949–968.

14.

Cheng

Sandu

Uncertainty quantification and apportionment in air quality models using the polynomial chaos method. Environ Modell Softw 2009; 24: 917–925.

15.

Wang

Qiu

et al . Novel reliability-based optimization method for thermal structure with hybrid random, interval and fuzzy parameters. Appl Math Model 2017; 47: 573–586.

16.

Wang

Qiu

Yang

Collocation methods for uncertain heat convection-diffusion problem with interval input parameters. Int J Therm Sci 2016; 107: 230–236.

17.

Wang

Qiu

Chen

Uncertainty analysis for heat convection-diffusion problem with large uncertain-but-bounded parameters. Acta Mech 2015; 226: 3831–3844.

18.

Wang

Qiu

An interval perturbation method for exterior acoustic field prediction with uncertain-but-bounded parameters. J Fluid Struct 2014; 49: 441–449.

19.

Feng

Uncertain multi-body dynamics analysis based on surrogate model method. Nanjing, China: Nanjing University of Aeronautics and Astronautics, 2013.

20.

Zhang

Chen

et al . A Chebyshev interval method for nonlinear dynamic systems under uncertainty. Appl Math Model 2013; 37: 4578–4591.

21.

Liu

Wang

JF.

Non-intrusive hybrid interval method for uncertain nonlinear systems using derivative information. Acta Mech Sinica 2015; 32: 170–180.

22.

Liu

Wang

A trigonometric interval method for dynamic response analysis of uncertain nonlinear systems. Sci China Phys Mech 2015; 58: 1–13.

23.

Luo

Zhang

et al . A new sampling scheme for developing metamodels with the zeros of Chebyshev polynomials. Eng Optimiz 2014; 47: 1264–1288.

24.

Luo

Zheng

et al . Incremental modeling of a new high-order polynomial surrogate model. Appl Math Model 2016; 40: 4681–4699.

25.

Qiu

Wang

Non-probabilistic interval analysis method for dynamic response analysis of nonlinear systems with uncertainty. J Sound Vib 2009; 319: 531–540.

26.

Qiu

Wang

Parameter perturbation method for dynamic responses of structures with uncertain-but-bounded parameters based on interval analysis. Int J Solids Struct 2005; 42: 4958–4970.

27.

Qiu

Non-probabilistic convex models and interval analysis method for dynamic response of a beam with bounded uncertainty. Appl Math Model 2010; 34: 725–734.

28.

Liu

Yang

Mao

Solving a class of fractional diffusion equations based on Chebyshev polynomials. J Chifeng Univ (Nat Sci Ed) 2018; 10: 8–10.

29.

Hesthaven

Gottlieb

Spectral methods for time-dependent problems. Cambridge: Cambridge University Press, 2007.

30.

Zhou

The research of RRRP robot inverse kinematics based on screw theory. Mech Sci Technol Aerosp Eng 2014; 33: 194–198.

31.

Kim

Comments on the symmetry of AdS 6, solutions in string/M-theory and Killing spinor equations. Phys Lett B 2016; 760: 780–787.

32.

Yang

Gao

Forward kinematics analysis and experiment of hybrid harvesting robot based on screw theory. Trans Chin Soc Agric Eng 2016; 32: 53–59.

33.

Pei

Dezhang

Wang

Inverse kinematics analyses of 3-finger robot dexterous hand based on screw theory. China Mech Eng 2017; 28: 2975–2980.

34.

Xiuheng

Hongyi

Zhong

. Positional accuracy analysis of polishing robot based on screw theory. Mach Tool Hydraul, 2016; 44: 29–33.

35.

Luo

Zhou

et al . Forward kinematics analysis of 4-R(SS)2 parallel robot based on screw theory. Mach Tool Hydraul 2018; 46: 33–37.

36.

Lin

Stadtherr

MA.

Validated solutions of initial value problems for parametric ODEs. Appl Numer Math 2007; 57: 1145–1162.

37.

Luo

Zhang

et al . Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions. Int J Numer Meth Eng 2013; 95: 606–630.

38.

Tang

Long-term behavior of spectral collocation method for delayed ordinary differential equations and partial integral differential equations. Changsha, China: Hunan Normal University, 2012.

39.

Johson

RW.

A B-spline collocation method for solving the incompressible NavierStokes equations using an ad hoc method: the Boundary Residual method. Comput Fluids 2017; 34: 121–149.

40.

Wang

Meng

Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second-order differential equations. J Comput Appl Math 2017; 313: 185–201.

41.

Chen

Peng

et al . Sparse regression Chebyshev polynomial interval method for nonlinear dynamic systems under uncertainty. Appl Math Model 2017; 51: 505–525.

42.

Alperin

Broue

. Local Methods in Block Theory. Annals of Mathematics 1979; 110: 143–157.

43.

Peng

Shi

Wang

et al . Motion trajectory planning of double pendulum crane considering interval uncertainty. J Mech Eng 2018; 55: 204.

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