*3.3. Compensation Angle Solution Based on Cantilever Model Combined Deformation of Compression and Bending*

For the mobile concrete pump manipulator, its main feature is that the links are slender and have a large scale, especially the 1st link, which is the longest and has the largest deformation and compensation amount. The deformation caused by insufficient joint stiffness (piston–cylinder oil compression at the joint) is not considered at present, and each link is simplified to a cantilever beam model that is deformed by compression and bending.

The deformation analytical formulae of three links are as follows:

$$w\_1 = \frac{\rho g A c\_1}{24EI} \mathbf{x}^4 - \frac{\rho g A c\_1 l\_1}{6EI} \mathbf{x}^3 + \left(\frac{\tau\_1}{2EI} + \frac{\rho g A c\_1 l\_1^2}{4EI}\right) \mathbf{x}^2 \tag{15}$$

$$w\_2 = \frac{\rho g A c\_{12}}{24EI} \mathbf{x}^4 - \frac{\rho g A c\_{12} l\_2}{6EI} \mathbf{x}^3 + \left(\frac{\tau\_2}{2EI} + \frac{\rho g A c\_{12} l\_2^2}{4EI}\right) \mathbf{x}^2 \tag{16}$$

$$w\_3 = \frac{\rho g A c\_{123}}{24EI} \mathbf{x}^4 - \frac{\rho g A c\_{123} l\_3}{6EI} \mathbf{x}^3 + \left(\frac{\tau\_3}{2EI} + \frac{\rho g A c\_{123} l\_3^2}{4EI}\right) \mathbf{x}^2 \tag{17}$$

where

*w* is Deflection of the corresponding point of the corresponding link;

ρ is Equivalent density;

*A* is Equivalent cross-sectional area.

Deriving the deformation analytical formula to obtain the tangential deformation angle (the angle between the tangent and the x-axis) analytical formula yields

$$\gamma\_1 = \arctan\left(\frac{\rho g A c\_1}{6EI} \mathbf{x}^3 - \frac{\rho g A c\_1 l\_1}{2EI} \mathbf{x}^2 + \left(\frac{\pi\_1}{EI} + \frac{\rho g A c\_1 l\_1^2}{2EI}\right) \mathbf{x}\right) \tag{18}$$

$$\gamma\_2 = \arctan\left(\frac{\rho g A c\_{12}}{6EI} \mathbf{x}^3 - \frac{\rho g A c\_{12} l\_2}{2EI} \mathbf{x}^2 + \left(\frac{\pi\_2}{EI} + \frac{\rho g A c\_{12} l\_2^2}{2EI}\right) \mathbf{x}\right) \tag{19}$$

$$\gamma\_3 = \arctan\left(\frac{\rho g A c\_{123}}{6EI} \mathbf{x}^3 - \frac{\rho g A c\_{123} l\_3}{2EI} \mathbf{x}^2 + \left(\frac{\tau\_3}{EI} + \frac{\rho g A c\_{123} l\_3^2}{2EI}\right) \mathbf{x}\right) \tag{20}$$

Take the 1st link as an example; the endpoint deflection is as follows:

$$w\_1 = \frac{\rho g A c\_1}{8EI} l\_1^4 + \frac{\tau\_1}{2EI} l\_1^2. \tag{21}$$

The endpoint tangential deformation angle is

$$\gamma\_1 = \arctan\left(\frac{\rho g A c\_1}{6EI} l\_1^3 + \frac{\tau\_1}{EI} l\_1\right). \tag{22}$$

The compensation angle is

$$
\Delta\Theta\_1 = \frac{w\_1}{l\_1} = \frac{\rho g A c\_1}{8EI} l\_1^{\;3} + \frac{\tau\_1}{2EI} l\_1. \tag{23}
$$

According to the engineering practice, there are three methods to solve the compensation angle:


Take the 1st link as an example.

The compensation angle is calculated by Formula (1).

This is a joint angle compensation method based on the basic principle. In this method, direct measurement of the endpoint deflection of each link can be used to solve the joint compensation angle. Direct measurement of deflection reduces the error caused by deflection calculation using other methods. Nowadays, in the engineering practice, laser is adapted to measure the endpoint deflection, which has a high cost.


The tangential deformation angle γ at the endpoint can be obtained by the difference of inclination sensor installed at both ends of the link.

Compensation angle:

$$
\Delta\Theta\_1 = \tan\gamma\_1 - \frac{1}{24} \frac{\rho g A c\_1}{EI} l\_1^3 - \frac{\pi\_1}{2EI} l\_1. \tag{24}
$$

This method has the error of theoretical modeling and calculation, with medium accuracy and low cost. However, considering the subtraction term and the deformation is small, when *l* is not large, γ can be used to replace Δθ directly. The simulation verification in the following text has adopted this method for the 2nd link and the 3rd link and achieved good results.


The analytical formula can be solved by the feedback value from four inclination sensors. Figure 4 shows the inclination sensors installed at three positions on the upper surface of the link and the inclination sensor installed at the head end of the link.

The three inclination sensor (x = a, b, c) measures the tangential deformation angle γ as γ*a*, γ*b*, γ*c*, which are the differences between the feedback value from each inclination sensor and the feedback value from the inclination sensor at the head end of the link (reference inclination sensor) respectively.

According to Cramer's Rule, the coefficient matrix is as follows:

$$\mathbf{A} = \begin{bmatrix} a^3 & a^2 & a \\ b^3 & b^2 & b \\ c^3 & c^2 & c \end{bmatrix} \tag{25}$$

and the constant term matrix is

β = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ *tan*γ*<sup>a</sup> tan*γ*<sup>b</sup> tan*γ*<sup>c</sup>* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ . (26)

**Figure 4.** Schematic diagram of installation positions of inclination sensors.

As long as the coefficient matrix A is non-singular, the unique coefficient solution of the analytical formula of the tangential deformation angle can be obtained as follows:

$$\mathbf{A}^{-1}\begin{bmatrix}\mathbf{A}^{-1}\begin{bmatrix}-\frac{\text{tutry}\_{d}}{a^{2}b+a^{2}c-c^{3}-abc}-\frac{\text{tutry}\_{b}}{ac^{2}+bc^{2}-c^{3}-abc}-\frac{\text{tutry}\_{c}}{ab^{2}+b^{2}c-b^{3}-abc}\\\-\frac{(b+c)\text{tutry}\_{a}}{a^{2}b+a^{2}c-a^{3}-abc}+\frac{(a+b)\text{tutry}\_{b}}{ac^{2}+bc^{2}-c^{3}-abc}+\frac{(a+c)\text{tutry}\_{c}}{ab^{2}+b^{2}c-b^{3}-abc}\\-\frac{\text{tutry}\_{c}}{a^{2}b+a^{2}c-a^{3}-abc}-\frac{\text{tutry}\_{b}}{ac^{2}+bc^{2}-c^{3}-abc}-\frac{\text{tutry}\_{c}}{ab^{2}+b^{2}c-b^{3}-abc}\end{bmatrix}.\tag{27}$$

Then the compensation angle is obtained as follows:

$$
\Delta\theta\_1 = \frac{w\_1}{l\_1} = K\_3 l\_1^{\;3} + K\_2 l\_1^{\;2} + K\_1 l\_1 \tag{28}
$$

where

$$K\_3 = \frac{1}{4} \left( -\frac{\tan \gamma\_a}{a^2 b + a^2 c - a^3 - abc} - \frac{\tan \gamma\_b}{ac^2 + bc^2 - c^3 - abc} - \frac{\tan \gamma\_c}{ab^2 + b^2 c - b^3 - abc} \right) \tag{29}$$

$$K\_2 = \frac{1}{3} \left( \frac{(b+c)\tan\gamma\_a}{a^2b + a^2c - a^3 - abc} + \frac{(a+b)\tan\gamma\_b}{ac^2 + bc^2 - c^3 - abc} + \frac{(a+c)\tan\gamma\_c}{ab^2 + b^2c - b^3 - abc} \right) \tag{30}$$

$$K\_{1} = \frac{1}{2} \Big( -\frac{b \text{ct} \text{any} \text{\textdegree}\_{a}}{a^{2}b + a^{2}c - a^{3} - abc} - \frac{ab \text{ct} \text{any} \text{\textdegree}\_{b}}{ac^{2} + bc^{2} - c^{3} - abc} - \frac{\text{act} \text{any} \text{\textdegree}\_{c}}{ab^{2} + b^{2}c - b^{3} - abc} \Big). \tag{31}$$

The functional relationship between the compensation angle and the inclination sensor installation positions *a*, *b*, *c*, and the tangential deformation angles γ*a*, γ*b*, γ*<sup>c</sup>* is established.

This method has high accuracy and reasonable cost. The inclination sensor can be installed at any position on the upper surface of the link and only the accurate position data needs to be provided without the consideration of making the inclination sensor be close to the joint hinge. It is especially suitable for the link with a large scale and has large deformation.

### **4. Deformation Compensation Verification Based on ANSYS Workbench and MATLAB Co-Simulation**

The finite element parametric simulation of a 24m-3R mobile concrete pump manipulator was carried out. The parameters of the mobile concrete pump manipulator are shown in Table 1.

**Table 1.** Parameters of 24m-3R mobile concrete pump manipulator.


The mobile concrete pump manipulator has strong structural strength, light weight, and a no-load equivalent density of 7000 kg/m3. Since there is no simple method for directly measuring the tangent slope of a point after deformation in ANSYS Workbench, in this paper, in order to simulate the inclination angle reading value measured by the inclination sensor in the actual experiment, the idea of limit was adopted.

Take two points that are very close to each other on the upper surface of one link (the distance between the two points is known), then the tangent angle of the midpoint of the two points is obtained by taking the deflection at these two points relative to the corresponding coordinate system. The approximate tangent obtained by this method has some errors. As shown in Figure 4, the measurement area in ANSYS is a rectangular area with a length of 200 mm. Compared with the length of each link of about 8000 mm, the distance between the two ends of the rectangular area is very small. The slope of the secant passing through the two ends of the small rectangle is equal to the tangent slope at the center of the small rectangle.

Use the maximum and minimum values of the deflection on the rectangle, as shown in Figure 5; then the tangent angle of the point in this coordinate system can be solved.

$$\beta = \arcsin\frac{MAX - MIN}{200} \tag{32}$$

**Figure 5.** The tangent angle solution method.

The tangential deformation angle γ is obtained by making a difference with the tangent angle of the point at the head end in the coordinate system.

$$\gamma = \arcsin \frac{MAX\_l - MIN\_l}{200} - \arcsin \frac{MAX\_0 - MIN\_0}{200} \tag{33}$$

In the simulation, method -<sup>3</sup> is applied to the 1st link, and the compensation angle values are calculated from the three inclination sensor readings at a, b, and c. For the 2nd link and 3rd link, method -<sup>2</sup> is directly adopted due to their small deformation, and γ is directly used to replace Δθ.

#### *4.1. ANSYS Workbench Parametric Simulation Verification*

#### 4.1.1. Parametric Simulation of Compensation Angle

Since θ<sup>0</sup> is the rotation angle of the rotation base, its influence on the deformation of the manipulator is little; take θ<sup>0</sup> = 0 in the simulation.

The range of joint angles is θ<sup>1</sup> ∈ [0, 90]; θ<sup>2</sup> ∈ [−180, 0]; θ<sup>3</sup> ∈ [−180, 40], A total of 216 positions and orientations are simulated.


⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦

The relationship between the endpoint's deviation of the mobile concrete pump manipulator and θ<sup>1</sup> θ<sup>2</sup> θ<sup>3</sup> is obtained.

As shown in Figure 6, they are continuous three-dimensional relational surface diagrams between the endpoint's deviation and θ<sup>1</sup> θ<sup>2</sup> when θ<sup>3</sup> takes −175◦, −133◦, −91◦, −49◦, −7◦, 35◦ respectively.

**Figure 6.** Relationship of endpoint deviation and each joint angle.

It can be seen that the larger the angle of the 2nd link unfolding, the larger the endpoint's deviation will become, and the sharper the endpoint's deviation will change when the 1st link's corresponding action is performed. The position and orientation of the 3rd link has a certain influence on the total deformation of the end. The effect on the deviation when the 3rd link is in the approximate symmetrical orientation (such as 35◦ and −49◦) is approximately the same.

The relationship between the 1st joint compensation angle and the and θ<sup>1</sup> θ<sup>2</sup> θ<sup>3</sup> is obtained. As shown in Figure 7, they are continuous three-dimensional relational surface diagrams between the joint compensation angles Δθ<sup>1</sup> and θ<sup>1</sup> θ<sup>2</sup> when θ<sup>3</sup> taking −175◦, −133◦, −91◦, −49◦, −7◦, 35◦.

$$
\theta\_3 = -1 \mathcal{T} 5^{\circ} \qquad \qquad \qquad \qquad \theta\_3 = -13 3^{\circ} \text{s}^{\circ}
$$

$$
\theta\_3 = -91^{\circ} \qquad \qquad \qquad \qquad \theta\_3 = -49^{\circ}
$$

**Figure 7.** Relationship of compensation angle of the 1st joint angle and each joint angle.

It can be seen that the orientations of the 1st link and the 2nd link have a great influence on the compensation angle of the 1st joint and the orientation of the 3rd link has less influence. The compensation angle of the 1st joint reaches the maximum when the manipulator is fully extended.

4.1.2. Parametric Simulation to Verify the Reliability of the Compensation Method

Considering the ratio of deflection and link length as the actual value of angle compensation, the accuracy of the compensation method can be verified by comparing the actual value of angle compensation and the angle compensation value obtained by this static deformation-compensation method. In this paper, the accuracy of the compensation method is measured by the error rate (actual value − calculated value)/(actual value).

The relationship between the compensation error rate of the 1st link and the joint angles is obtained. As shown in Figure 8, they are continuous three-dimensional relational surface diagrams between error rate and θ<sup>1</sup> θ<sup>2</sup> when θ<sup>3</sup> takes −133◦, −7◦, 35◦.

**Figure 8.** Relationship of compensation error of the 1st joint angle and each joint angle.

It can be seen that when the manipulator is in various positions and orientations, the deformation compensation error rate is controlled within 14%, which verifies the reliability of the static deformation-compensation method based on the inclination sensor feedback. The orientations of the 1st and the 2nd links have a great influence on the deformation compensation error. It can be seen from the figure that the deformation compensation error is the smallest when the angles of the 1st and 2nd joints are in the middle of the value range, and there is a low error valley, which can be used as the research object in the future.

The main reason for the error is that using the secant slope approximates the tangent slope to obtain the inclination angle approximate value. In engineering practice, errors will be of two types: γi experimental determination and accuracy of the coefficient matrix (Equation (26)). In order to compensate the endpoint deviation of the concrete truck manipulator with a length of 24 m up to 200 mm, it is recommended that the accuracy of the inclination sensor is 0.01◦ to achieve better results.

### *4.2. Verification of Deformation Compensation E*ff*ect Based on ANSYS Workbench and MATLAB Co-Simulation*

The six common orientations and six different loads of the 24m-3R mobile concrete pump manipulator were simulated by ANSYS Workbench and MATLAB co-simulation. The validity of the static deformation-compensation method was verified by comparing the theoretical endpoint position of the manipulator with the actual endpoint position (simulation result) after deformation compensation. The compensation error under different loads was obtained, and the universality of the compensation method to different loads was verified.

#### 4.2.1. Verification of the Static Deformation-Compensation Method's Validity

The six common orientations of the link were simulated by ANSYS Workbench and MATLAB co-simulation. ANSYS Workbench parametric simulation obtained the deformation value before and after deformation compensation and obtained the simulation values of inclination sensors; then the results were input to MATLAB, MATLAB according to the set joint angle and ANSYS Workbench input data real-time calculate the manipulator's endpoint position. As shown in Table 2, the deformation compensation effects of six common orientations were obtained.

The deformation compensation error of the mobile concrete pump manipulator in different orientations under no-load operation can be controlled within 50 mm. Compared with the endpoint deviation of 177 mm before compensation, the effect of deformation compensation is obvious. The validity of this method was verified.

#### 4.2.2. Verification of the Static Deformation-Compensation Method's Universality for Different Loads

Six different loads of the 24m-3R mobile concrete pump manipulator in a certain orientation were simulated by ANSYS Workbench and MATLAB co-simulation. In engineering practice, in order to make the concrete pumpable, the concrete in the pipeline is a fluid liquid. In the simulation analysis of concrete pumping, there is a consensus on the treatment of the liquid concrete in the pipeline. In the analysis and calculation, the rigidity of the liquid concrete is not considered, and it is directly applied to the pipeline arch rib as an external load. This method is supported by many engineering application backgrounds, and the calculated results are consistent with the corresponding measured results. Considering the concrete load has the same uniformity as gravity when verifying the static deformation-compensation method is universal for different loads, the equivalent density of the manipulator is changed in the parametric modeling. The simulation results are shown in Table 3, and the selected orientation was (5, −5, 35).

By changing the equivalent density, the deformation compensation effect of the manipulator under different materials and loads was obtained.

The compensation error increased with the increase of the equivalent density, but the endpoint deviations were all reduced to about 20% of the original endpoint deviations after the compensation. The static deformation-compensation method's universality for different loads was verified.


19646

 496.9285

 (0.573, 0.509, 0.117)

 (23042, 3834)

 (23146.45, 3347.72)

 (23082.54, 3694.66)

 (40.54, 139.34)


**Table**

#### **5. Conclusions and Future Work**

In this paper, we propose a static deformation-compensation method based on inclination sensor feedback for large-scale manipulators with hydraulic actuation like mobile concrete pump manipulators, maritime crane systems, and so on to reduce the deviation of the endpoint. 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.

Based on a 24m-3R mobile concrete pump manipulator, the parametric simulation based on ANSYS is carried out. The relationship between the endpoint's deviation of the mobile concrete pump manipulator and θ<sup>1</sup> θ<sup>2</sup> θ<sup>3</sup> and the relationship between the compensation error rate of the 1st link and the joint angles are obtained. When the manipulator is in various positions and orientations, the deformation compensation error rate is controlled within 14%. The reliability of the static deformation-compensation method is verified, and the error of this method is analyzed.

Based on the ANSYS and MATLAB co-simulation, we compared the theoretical endpoint position with the actual endpoint position after deformation compensation. The deformation compensation error of the mobile concrete pump manipulator in different orientations under no-load operation can be controlled within 50 mm. Compared with the endpoint deviation of 177 mm before compensation, the effect of deformation compensation is obvious. The validity of this method is verified.

Besides, the compensation error under different loads is obtained. The compensation error increases with the increase of the equivalent density, but the endpoint deviations are all reduced to about 20% of the original endpoint deviations after the compensation. The universality of the compensation method for different loads is verified.

In the future, experiments based on this method will be performed to verify the feasibility of the method and to evaluate the deformation-compensation effectiveness of this method in practical application.

**Author Contributions:** J.Q., Q.S., and B.X. conceived and designed the study. J.Q., F.Z., Y.M. and Z.F. performed the simulations. J.Q. wrote the paper. Q.S., and F.Z. reviewed and edited the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by [Fundamental Research on Hydraulic Stepping Drive Technology for Multi-joint Hydraulic Manipulators] grant number [51905473] and [the National Natural Science Foundation of China] grant number [91748210]. And the APC was funded by [the National Natural Science Foundation of China].

**Acknowledgments:** This work was supported by Fundamental Research on Hydraulic Stepping Drive Technology for Multi-joint Hydraulic Manipulators (grant number 51905473) and the National Natural Science Foundation of China (grant number 91748210).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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