**1. Introduction**

Modern large-scale manipulators are increasingly used in industrial, construction, and other fields. With the increase in the operational speed requirements of large-scale manipulators and the demand for lightweight design, the flexibility of these manipulators have gradually increased. Compared to the conventional heavy and bulky manipulators, flexible link manipulators have the potential advantage of lower cost, larger working volume, faster operational speed, larger payload-to-manipulator-weight ratio, smaller actuators, lower energy consumption, better maneuverability, better transportability, and safer operation due to reduced inertia [1–4]. The concepts of modern light-weight construction enable the large-scale manipulators like mobile concrete pump manipulators with extended operating range and less static load. However, due to the reduced weight, the elasticity of the construction elements has a significant influence on the precise positioning of the endpoint [5,6]. The influence of large-scale manipulators' static deformation on the precise control of the endpoint cannot be ignored anymore [6,7].

The research on the motion control of the engineering manipulator has made good progress [8–17]. Many researchers have studied the control of the endpoint trajectory of the mobile concrete pump manipulator. Most of them considered the manipulator as rigid. Although the control of the endpoint

trajectory was realized, there are still positioning errors (deviation of the endpoint) due to static deformation [18–20]. Despite the length of the mobile concrete pump manipulator ranging from 24 m to 53 m that are widely used now, the demand for the length of the mobile concrete pump manipulator will be higher with the development of the construction level and the influence of static deformation will be more serious.

In the literature, there are many types of research on the modeling of flexible multi-body systems. The existing methods are well developed and are presented in several textbooks, e.g., Bremer, Shabana, and many others [21]. For large-scale manipulators like mobile concrete pump manipulators, hydraulic actuators comprising hydraulic cylinders and valves are commonly used. Their dynamic behavior and the nonlinear characteristics have to be considered in the controller design. Henikl et al. and Lambeck et al. studied the combination of flexible multi-body systems and hydraulic actuators [22–24]. Zimmert et al. presented a control design considering the infinite-dimensional model of a flexible turntable ladder [25].

Many researchers have studied different schemes for modeling flexible link manipulators. Links are subjected to torsion, bending, and compression. The main concern is bending. For bending one may often use the Euler–Bernoulli equation, which ignores shearing and rotary inertia effects. These two effects may be incorporated using a Timoshenko beam element, which always is used if the beam is short relative to its diameter. However, since links may be considered as being rigid [26], in most models of flexible manipulators Euler–Bernoulli beams are used. In the literature [27,28], there are many well-established dynamic models in which three main modeling methods of the flexible link manipulators are the assumed mode method (AMM), the finite element method (FEM), and the lumped parameter model. AMM and FEM use either the Lagrangian formulation or the Newton–Euler recursive formulation.

The flexibility of the link is usually represented by a truncated finite modal series in terms of spatial mode eigenfunctions and time-varying mode amplitudes in assumed mode model (AMM) formulation. This method's main disadvantage is the difficulty in finding modes for links with non-regular cross sections and multi-link manipulators [29]. Using the law of conservation of momentum, the Lagrangian principle was utilized to model the dynamic function of the space flexible manipulator incorporating the assumed modes method in Deng-Feng's research [30]. Subudhi and Morris [31] presented a dynamic modeling technique for a manipulator with multiple flexible links and flexible joints based on a combined Euler–Lagrange formulation and assumed modes method. Then, they controlled the system by formulating a singularly perturbed model and used it to design a reduced-order controller. In the finite element method (FEM), the elastic deformations are analyzed by assuming a known rigid body motion and later superposing the elastic deformation with the rigid body motion [32–37]. In order to solve a large set of differential equations derived by the finite element method, a lot of boundary conditions have to be considered, which are, in most situations, uncertain for flexible manipulators [38]. Using the assumed mode method to derive the equations of motion of the flexible manipulators, only the first several modes are usually retained by truncation and the higher modes are neglected. The lumped parameter model is the simplest one for analysis purposes; the manipulator is modeled as a spring and mass system, which does often not yield sufficiently accurate results [39–41]. Zhu et al. [42] employed a lumped model to simulate the tip position tracking control of a single-link flexible manipulator. Raboud et al. [43] showed the existence of multiple equilibrium solutions under a given load condition by studying the stability of very flexible cantilever beams.

Some researchers have paid attention to the static deformation of large-scale manipulators. Most of them are based on the application of finite element method (FEM) in the static deformation of flexible manipulators. Lee et al. reduced the endpoint deviation of the mobile concrete pump manipulator by 30% compared with steel structures by using carbon fiber material in the last link [44]. In order to better improve the trajectory tracking accuracy of the working platform at high altitude, Qing Hui Yuan considered the elastic deformation of the manipulator and the influence of vibration on the trajectory tracking control and introduces the deformation compensation strategy to eliminate the influence [45]. Xia Jijun et al., based on finite element deformation analysis under the actual load conditions of the mobile concrete pump manipulator system, established an expert database of full position and orientation deformation compensation of the manipulator and applied it to the trajectory control of the endpoint. The position deviation of the endpoint could be controlled within ±15 cm [46]. Zhao Xin et al. obtained the deformation compensation model of the manipulator and the vehicle body through the deformation analysis of the whole concrete pump truck under the full working condition, established the kinematics model of the concrete pump truck after the deformation compensation and used the control method of the cerebellar model neural network in the motion control, and well solved the dynamic detection and trajectory control of the manipulator position and orientation [47]. Wang Xiaoming et al. used finite element simulation to establish the data model of the deformation of the manipulator during the trajectory control process. Then, the BP neural network model was used to establish the deformation compensation algorithm, and the deformation law of the manipulator was obtained [48]. Pan Daoyuan et al. used FEM to analyze the variation of the acceleration of the manipulator and the force applied on the manipulator in different positions and orientations, and determined their influence on the deformation [49]. These studies have made some achievements and have made great progress in the deformation compensation of mobile concrete pump manipulators. However, these studies are based on the prototype studied by them, such as Xia Jijun et al. with Zoomlion 52m-6RZ concrete pump truck (RZ manipulator folding structure) as the research prototype [46]. Wang Xiaoming, Zhao Xin et al., and Pan Daoyuan et al. all built finite element simulation results databases based on their respective experimental prototypes [47–49]. All of the above studies needed to establish an accurate simulation model and needed to consider a lot of boundary conditions. The amount of calculation was very large, and the methods used to develop the prototype were not universal and thus are difficult to use in practical applications.

In this paper, we propose a static deformation-compensation method for large-scale manipulators based on inclination sensor feedback. Compared with the finite element method, this method does not need to consider many boundary conditions that are uncertain for flexible manipulators in most situations. It has appropriate accuracy and is universal for large-scale manipulators of different sizes and working under different loads.

#### **2. The Structure and the Forward Kinematic Model of the Mobile Concrete Pump Manipulator**

#### *2.1. Structure of the Mobile Concrete Pump Manipulator*

The mobile concrete pump manipulator is a multi-degree of freedom manipulator system. This paper takes the mobile concrete pump manipulator with three links as an example to explain the static deformation-compensation method. Its main components are rotation base, 1st joint, 1st link, 2nd joint, 2nd link, 3rd joint, 3rd link, and the concrete discharge hose. Given that the concrete discharge hose has little capacity to bear the load of concrete, it is only used as the concrete discharge guide, ignoring its degree of freedom. It is a manipulator composed of four joints, each link of which is a long-scale flexible link.

During the process of concrete placing during construction, the manipulator produces a large elastic deformation caused by complex stress due to the influence of gravity on the manipulator, the pumping concrete load, the concrete flow impact, and other factors. Due to the large span of the mobile concrete pump manipulator, the deformation accumulation to the end outlet has seriously affected the concrete placing accuracy. The elastic deformation varies in different positions and orientations. Figure 1 shows a structural diagram of the mobile concrete pump manipulator.

**Figure 1.** Structure diagram of the mobile concrete pump manipulator. 1. Rotation base; 2. 1st joint; 3. 1st link; 4. 2nd joint; 5. 2nd link; 6. 3rd joint; 7. 3rd link; 8. End outlet; 9. Concrete discharge hose.

#### *2.2. Forward Kinematics Model of the Mobile Concrete Pump Manipulator*

In order to study the influence of the joint compensation angle on the position of the endpoint, the relationship between the joint angle change of each joint and the position and orientation of the endpoint is established based on the D–H matrix (Denavit–Hartenberg Matrix) method. As shown in Figure 2, D–H coordinate system is established for the mobile concrete pump manipulator.

**Figure 2.** Denavit–Hartenberg (D–H) coordinate system of the mobile concrete pump manipulator.

The distance from the rotation axis of the rotation base to the rotation axis of the 1st link is negligible compared with the length of the whole manipulator (24 m), so let the origin of the rotation base coordinate system T1 coincide with the origin of the 1st link coordinate system T2. The base coordinate system T0, the rotation base coordinate system T1, the 1st link coordinate system T2, the 2nd link coordinate system T3, the 3rd link coordinate system T4, and the end coordinate system T5 is established. According to the D–H matrix method, the homogeneous coordinate transformation matrix is calculated. The joint angle θ0θ1θ2θ<sup>3</sup> is positive clockwise and negative counterclockwise:

$$\begin{aligned} \begin{aligned} \;^0\_1\text{T} &= \left[ \begin{array}{ccc} c\_0 & -s\_0 & 0 & 0\\ s\_0 & c\_0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \;^1\_2\text{T} = \left[ \begin{array}{ccc} c\_1 & -s\_1 & 0 & 0\\ 0 & 0 & -1 & 0\\ s\_1 & c\_1 & 0 & 0\\ 0 & 0 & 0 & 1 \end{array} \;^1\_3\text{T} \right] \\ \end{aligned} \\ \end{aligned} \\ \begin{aligned} \;^2\_3\text{T} &= \left[ \begin{array}{ccc} c\_2 & -s\_2 & 0 & l\_1\\ s\_2 & c\_2 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \;^3\_3\text{T} = \left[ \begin{array}{ccc} c\_3 & -s\_3 & 0 & l\_2\\ s\_3 & c\_3 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \;^1\_3\text{T} = \left[ \begin{array}{ccc} 1 & 0 & 0 & l\_3\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \;^1\_3\text{T} \right] \end{aligned}$$


#### **3. Joint Angle Independent Compensation Method Based on Inclination Sensor Feedback**

#### *3.1. Principle of Joint Angle Independent Compensation*

The elastic deformation varies in different positions and orientations. Therefore, the elastic deformation of the entire mobile concrete pump manipulator is decomposed into the elastic deformation of each link. The position recovery of the endpoint of each link is achieved by compensation of the joint angle of each joint. Since the deformation angle is very small, the following simplification is made in the compensation angle solution: The deflection of the endpoint of each link with respect to the horizontal axis of the corresponding coordinate system is regarded as the compensation angle arc length. The basic principle of joint angle independent compensation can be simplified as the angle compensation of each link coordinate system around the z-axis rotation, as shown in Figure 3.

**Figure 3.** The basic principle of joint angle independent compensation.

The deviation of the 1st link' s endpoint can be compensated by the 1st joint angle. The compensation angle of the 1st joint is as follows:

$$
\Delta\theta\_1 = \frac{w\_1}{l\_1} \tag{1}
$$

⎞

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where

*w*<sup>1</sup> is deflection of the endpoint of the 1st link;

*l*<sup>1</sup> is length of the 1st link;

γ<sup>1</sup> is tangential deformation angle of the 1st link.

The compensation angle of the 2nd joint and the 3rd joint can be calculated in the same way.

#### *3.2. Joint Torque Solution Based on Jacobian Matrix*

The deviation of the endpoint caused by gravity is often ignored in the existing static analysis of the manipulator, and most of the studies regard the manipulator as a rigid body. The relationship between the force on the endpoint of the manipulator and the torque of each joint has been established by the Jacobian matrix.

$$
\boldsymbol{\pi} = \mathbf{J}^{\mathsf{T}} \boldsymbol{\mathcal{F}} \tag{2}
$$

where

F is force vector and moment vector acting on the end actuator;

T is joint torque.

The gravity effect is simplified as follows: The Jacobian matrix of the three-link model and the two-link model is established separately without considering the rotation of the rotation base. When considering the joint torque, the gravity of the rear link of the joint is regarded as the load acting on the centroid of the latter link. The joint torque is calculated by the Jacobian matrix.

The manipulator's kinematic equation is as follows:

$$\begin{cases} \mathbf{x} = l\_1 \cos \theta\_1 + l\_2 \cos(\theta\_1 + \theta\_2) \\ \mathbf{y} = l\_1 \sin \theta\_1 + l\_2 \sin(\theta\_1 + \theta\_2) \end{cases} \tag{3}$$

Then, the two sides are respectively derived from t as follows:

$$\begin{cases}
\dot{\mathbf{x}} = -l\_1 \sin \theta\_1 \dot{\theta}\_1 - l\_2 \sin(\theta\_1 + \theta\_2)(\dot{\theta}\_1 + \dot{\theta}\_2) \\
\dot{y} = l\_1 \cos \theta\_1 \dot{\theta}\_1 + l\_2 \cos(\theta\_1 + \theta\_2)(\dot{\theta}\_1 + \dot{\theta}\_2)
\end{cases}.
\tag{4}$$

The Jacobian matrix relative to the base coordinates is as follows:

$$\mathbf{J} = \begin{bmatrix} -\mathbf{l}\_1 \mathbf{s}\_1 - \mathbf{l}\_2 \mathbf{s}\_{12} & -\mathbf{l}\_2 \mathbf{s}\_{12} \\\ \mathbf{l}\_1 \mathbf{c}\_1 + \mathbf{l}\_2 \mathbf{c}\_{12} & \mathbf{l}\_2 \mathbf{c}\_{12} \end{bmatrix}. \tag{5}$$

Transposing the Jacobian matrix gives

$$\mathbf{J}^{T} = \begin{bmatrix} -\mathbf{l}\_{1}\mathbf{s}\_{1} - \mathbf{l}\_{2}\mathbf{s}\_{12} & \mathbf{l}\_{1}\mathbf{c}\_{1} + \mathbf{l}\_{2}\mathbf{c}\_{12} \\ -\mathbf{l}\_{2}\mathbf{s}\_{12} & \mathbf{l}\_{2}\mathbf{c}\_{12} \end{bmatrix}. \tag{6}$$

For the two-link model, the distance between the centroid of the 2nd link and the 2nd joint is a2 and the gravity is simplified to be applied to the centroid.

The torque of each joint produced by the gravity of the 2nd link is expressed as

$$
\pi\_{12} = (\mathbf{l}\_1 \mathbf{c}\_1 + a\_2 \mathbf{c}\_{12}) m\_2 \,\mathrm{g};\tag{7}
$$

$$
\pi\_{22} = a\_2 \varsigma\_{12} \mathfrak{m}\_2 \not\subset \tag{8}
$$

where

τ<sup>12</sup> is torque on the 1st joint produced by the gravity of the 2nd link;

τ<sup>22</sup> is torque on the 2nd joint produced by the gravity of the 2nd link.

Similarly, for the three-link model, the distance between the centroid of the 3rd link and the 3rd joint is a3.

The torque of each joint produced by the gravity of the 3rd link is as follows:

$$\mathbf{r}\_{13} = (l\_1\mathbf{c}\_1 + l\_2\mathbf{c}\_{12} + a\_3\mathbf{c}\_{123})m\_3 \,\mathrm{g};\tag{9}$$

$$\mathbf{r}\mathbf{r}\mathbf{z} = (l\_2\mathbf{c}\_{12} + a\mathbf{c}\_{123})m\mathbf{z} \text{ g.} \tag{10}$$

$$
\pi\_{33} = a\_3 \mathbf{c}\_{123} \mathbf{m}\_3 \text{ g.} \tag{11}
$$

where

τ<sup>13</sup> is torque on the 1st joint produced by the gravity of the 3rd link;

τ<sup>23</sup> is torque on the 2nd joint produced by the gravity of the 3rd link;

τ<sup>33</sup> is torque on the 3rd joint produced by the gravity of the 3rd link;

The gravity of the 1st link is simplified to the centroid. In summary, the total torque at each joint is finally obtained as follows:

$$\mathbf{r}\_1 = a\_1 \mathbf{c}\_1 \mathbf{m}\_1 \, \mathbf{g} + (\mathbf{l}\_1 \mathbf{c}\_1 + a\_2 \mathbf{c}\_1 \mathbf{z}) \mathbf{m}\_2 \, \mathbf{g} + (\mathbf{l}\_1 \mathbf{c}\_1 + l\_2 \mathbf{c}\_1 \mathbf{z} + a\_3 \mathbf{c}\_1 \mathbf{z}) \mathbf{m}\_3 \, \mathbf{g},\tag{12}$$

$$
\pi\_2 = a\_2 \mathbf{c}\_{12} m\_2 \text{ g} + (l\_2 \mathbf{c}\_{12} + a\_3 \mathbf{c}\_{123}) m\_3 \text{ g};\tag{13}
$$

$$
\pi\_3 = a\_3 \mathbf{c}\_{123} \mathbf{m}\_3 \text{ g}\_{\prime} \tag{14}
$$

where

τ<sup>1</sup> is torque on the 1st joint;

τ<sup>2</sup> is torque on the 2nd joint;

τ<sup>3</sup> is torque on the 3rd joint,

where *c*<sup>12</sup> = cos(θ<sup>1</sup> + θ2); *c*<sup>123</sup> = cos(θ<sup>1</sup> + θ<sup>2</sup> + θ3). The rest of the abbreviation has the same principle.
