Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 8790575, 13 pages
https://doi.org/10.1155/2018/8790575

Research Article

Research on Kinematics and Pouring Law of a Mobile Heavy Load Pouring Robot

Long Li,1,2 Chengjun Wang,² and Hongtao Wu¹

¹College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China
²College of Mechanical Engineering, Anhui University of Science and Technology, Huainan, China

Correspondence should be addressed to Chengjun Wang; 67155946@qq.com

Received 13 December 2017; Revised 7 March 2018; Accepted 3 April 2018; Published 23 May 2018

Academic Editor: Aki Mikkola

Copyright © 2018 Long Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In order to meet the requirement of continuous pouring in many varieties and small batches in casting production, a mobile heavy load pouring robot is developed based on a new 4-UPU parallel mechanism due to its strong carrying capacity. Firstly, the instantaneous motion characteristics of the novel 4-UPU parallel mechanism with four degrees of freedom (DOF) are analyzed using screw theory. By using the geometric method, both the forward and inverse kinematic solutions of the proposed robot system are given out. Secondly, based on a common pouring ladle, the volume change of pouring liquid in pouring process and the relationship between tilting angular velocity and flow rate are analyzed, and the results show that the shape of the ladle and the design of the pouring mouth have great influence on the tilting model. It is an important basis for the division of the sectional model. Finally, a numerical example is given to verify the effectiveness of the developed tilting model. The mapping relation between the tilting model and the parallel mechanism shows that the pouring flow can be adjusted by controlling the movement of parallel manipulator. The research of this paper provides an important theoretical basis for the position control of mobile heavy load pouring robot and the research of pouring speed control.

1. Introduction

Pouring is the process of pouring the molten metal with high temperature from the ladle into the mould cavity. The adverse operating environment, such as dust, vibration, and noise, causes potential adverse effects on the personal security of workers. In order to solve this problem, various types of automatic pouring equipment have been developed for the pouring process [1, 2].

There are several commonly used pouring methods, such as bottom gating type, tilting type, and air pressure type [3]. Among them, the tilting pouring equipment is the most widely used because of its simple principle and convenient control. Back in the 1908, Benjamin invented the earliest hand cranked pouring machine [4]. By the second half of the 20th century, the United States, Japan, China, and other countries have developed various types of semiautomatic and fully automatic pouring machines [5–9]. At present, the structure of the title-typed automatic pouring equipment is still not changed greatly, and the movement mode is in accordance with the given walking route and tilting mode, which is suitable for mass production. However, in the face of more varieties and small quantities of castings, the production demand is increasing day by day. It is urgent to have automatic pouring equipment with movability, high flexibility, and high carrying capacity.

The parallel mechanism has the advantages of compact structure, large rigidity, strong bearing capacity, and high accuracy [10]. Thus, it is suitable for the tilting-typed automatic pouring equipment of actuators. HEXEL, Delta, Tricept, and many other parallel mechanisms have been extensively used in machine tools, steel, medical, aviation, and other fields [11]. For the flow control of tilting pouring equipment, early automatic pouring system was finished by preteaching. However, this method has many problems in inappropriate speed and pouring, for example, the teaching process needs a lot of time and manpower and the accuracy is not high [12]. In order to solve these problems, a lot of pouring flow control methods have been proposed. Noda et al. put forward a complete set of flow control and position control method based on the liquid metal flow model. The molten steel sputtering mechanism and vibration suppression were studied, which achieved a good effect in practice [13–16]. Sun et al. discretized the fluid and simulated the tilting process, revealing the general law of fluid dumping [17]. Pan et al. used the same idea and gave a reasonable trajectory planning for the dumping process [18]. In the research of fixed point tilting pouring machine, Zhu established the relationship between flow and tilting angle [19]. However, the instantaneous model of pouring flow and angular velocity is studied at the present stage, especially in the relationship between the instantaneous flow state of pouring ladle and the shape of ladle. The influence of melt in pouring mouth on pouring flow is too simplistic and fuzzy, and the model is usually effective only for specially designed ladle; the type of ladle commonly used in casting industry is rarely involved. These problems not only hinder the development and promotion of products, but also increase the difficulty and complexity of the flow control.

Considering the high bearing capacity of parallel mechanism, a mobile casting robot is proposed in our research. It can realize the flexibility of pouring equipment with many varieties and small batches with the ability of continuous pouring. This paper focuses on analyzing of the kinematic characteristics of the proposed 4-UPU parallel mechanism and giving the kinematic closed solution to the mechanism. At the same time, a pouring stage model is built on the basis of a commonly ladle in casting production. The influences of the shape of the ladle and the angle of the pouring nozzle on the dumping model are emphatically investigated. Numerical simulations are given to verify the validity of the proposed model. Moreover, the mapping relation between the kinematic model of parallel manipulator and the pouring model of ladle is given.

2. Structure of Mobile Heavy Load Pouring Robot

As shown in Figure 1, a hybrid truss type mobile heavy load pouring robot is proposed [20]. The robot is composed of a 4WD mobile platform, a slewing device, a hoisting device, a forward device, a backward device, a counterweight device, a parallel manipulator, and a ladle. The 4WD mobile platform adopts four-wheel drive plus four-angle stationary support system to realize long distance flexible and stable walking and stationary self-balancing support, so as to improve the stability of operation support. The robot body has 6 DOF in space, three translations, and three rotations. The slewing device and the hoisting device can realize the rotation and the lifting adjustment, respectively. The 4 DOF parallel manipulator can be adjusted to the ladle. The design dimension of the pouring robot is 2150 mm × 1200 mm × 2000 mm. The maximum pouring quantity is 60 kg. Under normal pouring condition, the maximum forward and backward movement range of ladle is about 100 mm, right and left movement is 50 mm, and the vertical elevator height is about 400 mm.

Figure 1: Hybrid truss type movable pouring robot. 4WD mobile platform. Slewing device. Hoisting device. Forward device. Backward device. Counterweight device. Parallel manipulator. End-effector.

Because the ladle in the process of pouring is around a certain axis rotation, and in order to adapt to the complex pouring environment, we must ensure that the ladle will not slosh in any other direction. Therefore, the parallel unit is designed to be a 4-DOF parallel mechanism with 1-axis rotation and 3-axis translation. Based on the analysis of 3-UPU parallel mechanism (3-DOF) [21], a new type of 4-UPU parallel mechanism is proposed to meet the requirements of the pouring operation.

3. Analysis of Instantaneous Motion Characteristics of 4-UPU Parallel Mechanism

3.1. Description of Parallel Mechanisms

As shown in Figure 2, the parallel manipulator of the heavy load pouring robot can be simplified as 4-UPU ( on behalf of a universal joint, on behalf of a shifting pair) parallel mechanism. The upper platform is the fixed platform, installed in the heavy pouring robot beam; the lower platform is a moving platform and is connected with the ladle. The 4 branches are UPU structural kinematic chains. Initial assembly time, the plane ABab formed by chains 1 and 2, and the upper and lower platforms are vertical to the fixed platform ABCD, and the angle between the moving platform abcd and the plane ABab is γ. The plane CDdc formed by chains 3 and 4 and the upper and lower platforms, the angle between the fixed platform ABCD and the plane CDdc is Ψ, and the angle between the moving platform ABCD and the plane CDdc is ψ. The fixed coordinate system O-XYZ is located at the midpoint AB edge, Z axis is perpendicular to the fixed platform, Y axis and AB edge overlap, C points in X axis extension line, the moving coordinate system o-xyz is located at the midpoint ab edge, z axis is perpendicular to the moving platform, y axis and ab edge overlap, c points in x axis extension line, and the y axis is parallel to the Y axis. ob length is m, oc length is n, dc length is k, OB length is M, OC length is N, DC length is K. Aa, Bb, Cc, and Dd are , , , and , respectively.

Figure 2: Schematic diagram of 4-UPU parallel manipulator for pouring robot.

The pair is composed of pairs perpendicular to the two axes ( represents revolute joint), while kinematic chains UPU are equivalent to combinations of RRPRR joints. The axes of each chain kinematic pair are successively represented as  , , , , , ( = 1, 2, 3, 4, for the simple diagram, only the axis of the chain 1 is marked). is consistent with the direction of   axis,   consistent with the direction of the   axis, the angle between   and   for , , , .

3.2. Instantaneous Motion Analysis

The screw theory is used to analyze the motion characteristics. According to the literature [22], the screw is usually expressed as

where   represents the   axis vector,    represents the vector of any point on the axis to the reference coordinate system, and h is parallel to , representing the axial pitch of . When h is 0, the screw is reduced to a line vector that can be used to represent a revolute joint or a binding force. When h is infinite, $ = , the screw degenerates into a couple, which can be used to represent a moving pair or a constraint couple.

When , and are called reciprocal, and the reciprocal screw is denoted as . The symbol “” means reciprocal product. The physical meaning of the reciprocal product reflects the work done by the force screw in the kinematics screw. The reciprocal product is zero, at which point the force screw acts as a constraint on the moving object, whereas the force screw is equivalent to doing work on the moving object.

The kinematic screw of each chain of the parallel mechanism can be obtained from Figure 2:

The reciprocal screw of each chain is

From formula (3), it can be seen that the parallel mechanism of the design has 2 over constraints; the kinematic screw system of the 4-UPU parallel mechanism can be obtained by solving the reciprocal screw of (3):

Formula (4) shows that when the initial assembly is performed, the parallel mechanism has three degrees of freedom and a rotational degree of freedom along the direction.

The degree of freedom of the parallel mechanism can be obtained by using the modified G-K formula [23]:

In (5), is the order of the mechanism, , is the number of common constraint mechanisms, the number of common constraints parallel mechanism in the paper is 0, is the number of components, is the number of kinematic pairs, and is the degree of freedom of the component, is the total number of over constraints of the mechanism. The over constraints number of the parallel mechanism in this paper is 2.

It can be seen from formulas (4) and (5) that the analysis results of instantaneous motion characteristics are consistent with the number of degrees of freedom of the mechanism. At the same time, it can be obtained that, under the established coordinate system, the axis relation of each branch chain is invariable, and the number of public constraints and redundant constraints of the mechanism is fixed, so the characteristics of instantaneous motion mechanism are global, and parallel mechanism is designed to meet the needs of pouring operation.

3.3. Judgment of Drive Joint

Scenario 1 (​4P). Selecting the shifting pair in the 4 chains as the drive joint, according to the literature [24] described in the drive subrule, four mobile pairs are rigidized to get a new 4-UU mechanism, and the kinematic screw of the parallel mechanism is calculated as

According to formula (6), when the 4 shifting pairs are rigid, the new 4-UU mechanism holds a rotational degree of freedom along the axis, and the preselected 4 shifting pairs cannot be the drive pairs of the parallel mechanism at the same time.

Scenario 2 (3P + 1R). The shifting pairs of chains 1, 2, and 3 are selected as drive joints. The axis of the chain 4 is the drive pair, the 4 drive pairs are rigid, and the new parallel mechanism is 3UU-UPR. The kinematic screw of the parallel mechanism is calculated as follows:

According to formula (7), the new 3UU-UPR mechanism has a degree of freedom of 0, and the preselected 3 shifting pairs and 1 revolute joint can simultaneously be used as the drive pairs of the parallel mechanism. Therefore, Scenario 2 is chosen as the parallel mechanism drive arrangement mode, which makes the designed 4-UPU parallel mechanism obtain the forward and inverse kinematics, which is beneficial to the mechanism control.

4. Position Solution of 4-UPU Parallel Mechanism

First of all, for the further analysis of the chain 4, Section 3.3 of this paper takes the axis as the driving pair. Therefore, the axis is rigid and the chain 4 is degenerated into UPR structure. The kinematic screw of the chain is analyzed:

When the chain is rigid, the chain has 4 DOFs, including the freedom of movement of the moving platform, indicating that the chain 4 does not interfere with the moving motion of the moving platform, but to control the rotation of the moving platform. The of the chain 4 is only forced motion. Therefore, the chain 4 can be regarded as the aided chain. In the process of solving the positive kinematics, the chain 4 is omitted, and a virtual rotation along the direction is added at the point only. As shown in Figure 3. The forward kinematics of the 4-UPU mechanism is finally equivalent to the coordinates of the midpoint of the AB edge of the three-prism ABCD-abcd.

Figure 3: 4-UPU equivalent kinematic model.

4.1. Forward Kinematics

When 4-UPU has three translational and one rotational degrees of freedom, the equivalent kinematic model of Figure 3 can be divided into two planes along and , respectively, ABba and OCco in the XZ plane projection plane O′C′co, as shown in Figure 4.

(a) Planar ABba

(b) Projective plane O′C′co

Figure 4: The segmentation plane of equivalent kinematic model.

For planar ABba, the AB is always parallel to the ab because the mechanism does not have the rotation along the and axes. The relationship can be obtained by trapezoid geometry:

O′o and C′c in the projection plane O′C′co of OCco can be obtained by trigonometric function, respectively:

At this time, O′C′, C′c, co, Oo, and are known in the projection plane ; in order to find the moving platform of position and attitude, and can only demand. Connect and , and the can be obtained by the triangle cosine theorem:

From (13), (14) to angle ,

By (9), (12), and (15), we obtain the positive solutions of the 4-UPU parallel mechanism at last:

4.2. Inverse Kinematics

The inverse kinematics is given structural parameters and moving platform pose (, , , ); then the solution of the 4 inputs (, , , ) is obtained. According to the coordinate transformation relation, the following equations can be established:

Among them, , b, c. is the homogeneous transform matrix:

When the structural parameters of the moving platform and the fixed platform are determined, the coordinate of the point in the fixed coordinate system O-XYZ can be obtained by formula (17). The input equations of the mechanism are

In formula (19), is the vector representation of the point in the fixed base coordinate system, and is the vector representation of the length of the projection rod in the fixed coordinate system.

4.3. Numerical Examples of Positive and Inverse Solutions

Set the fixed platform  mm,  mm, the moving platform  mm,  mm; then the corresponding kinematics inverse kinematics solution is shown in Table 1, and the corresponding forward kinematic solutions are shown in Table 2.

Table 1: Inverse kinematics solution of 4-UPU.

(mm)

(mm)

(mm)
θ
(°)

0
0
−625
−28

−50
−50
−625
−28

−50
−50
−625
62

50
50
−625
62

(mm)

(mm)

(mm)

(mm)

626.547
626.547
872.906
1.149

627.026
634.004
855.101
1.202

627.026
634.004
551.715
0.584

634.004
627.026
637.245
0.497

Table 2: Kinematics positive solution of 4-UPU.

(mm)

(mm)

(mm)

(mm)

626.547
626.547
872.906
1.149

627.026
634.004
855.101
1.202

627.026
634.004
551.715
0.584

634.004
627.026
637.245
0.497

(mm)

(mm)

(mm)

(°)

8.7047 × 10⁻¹²
0
−625
−27.983

−50
−50
−625
−27.983

−50
−50
−625
61.968

50
50
−625
61.968

As can be seen from Tables 1 and 2, the kinematics forward and inverse solution equations are closed, which shows that the algorithm in this paper is correct and feasible. Compared with algebraic method, the unique solution in a given workspace can be obtained directly.

4.4. Inverse Dynamics

The dynamics of the 4-UPU parallel mechanism of the pouring robot is shown in Figure 5 Since each leg, only a cylinder, can be assumed to be composed of two segments, two parallel local systems, -frame and -frame attached to the upper segment 1 and lower segment 2, respectively, of the leg , are defined with their corresponding origins and . and are the center of gravity of the upper and lower segments. The position vector of the center of mass of each part is written as

Figure 5: Coordinate systems for a leg and moving platform.

, are the position vectors of and in the local coordinate system, respectively, is the position vector of the point and the moving platform centroid, , , are all constant vectors, , is the direction vector of the shifting axis, and is the length of the shifting axis. The absolute velocities of the masses on each body are given by differentiating (20) through (22) as

can be obtained by solving two differentials of the “” of the vector-loop equation of the parallel mechanism.

The total kinetic energy and potential energy of the new 4-UPU parallel mechanism can be written as

In (24) and (25), is the gravitational acceleration vector, and , , and are the mass of the moving platform, the upper segment, and the lower segment, respectively.

Using the Lagrange equation, the dynamic equation of the new 4-UPU parallel mechanism can be written as

where , generalized coordinates , and the driving force and driving torque .

5. Whole Cycle Tilting Prediction Model for Ladle

The goal of the mobile heavy load pouring robot is to complete the continuous casting of many varieties and small batches. The pouring flow rate is one of the key factors affecting the quality of the castings and also the basis for the movement of the new 4-UPU parallel mechanism. Therefore, it is necessary to establish the pouring model of the ladle to realize the dumping movement of the parallel mechanism. Noda and Terashima established the flow rate model and feedback control in the study of the flow control of the pouring robot [25]. This paper focuses on the volume change of the ladle in the pouring process and then establishes the model of pouring angular velocity and pouring flow.

5.1. Ladle Model

In order to control the moving platform movement of the parallel pouring robot and realize constant current control of pouring flow, it is necessary to establish the relationship between the pouring angle of the ladle and the instantaneous change of the pouring flow rate, which can maintain basic constant outflow of ladle pouring liquid flow.

In this paper, pouring ladle selects the steel ladle which is common in the casting foundry [26]. The shape of the inner cavity of the ladle is shown in Figure 6, similar to frustum of a cone. The o-xyz is the moving platform coordinate system of the parallel mechanism, located in the middle of the inner cavity,  o′-x′y′z′ is the ladle coordinate system, located at the center of the ladle bottom,  o′-x′y′z′ can be obtained by translating  o-xyz along   axis-zz′ lengths, and here the pouring mouth and the cone surface two intersection points are marked as , . In order to build the relationship between ladle volume and tilting angle, the section of the ladle is made along the plane as shown in Figure 7.

Figure 6: Inner cavity model of ladle.

Figure 7: Sectional view of surface in ladle inner cavity.

Figure 7 is the flowing state of the pouring liquid in the ladle at the instant when the ladle is poured. is the volume of the pouring liquid in the frustum of the cone below the pouring mouth. is the volume of the pouring liquid in the pouring mouth below the pouring mouth. is the volume of the pouring liquid above the pouring mouth, is the height of the pouring liquid above the pouring mouth, is the pouring flow, and the is the tilting angle.

5.2. Tilting Model

In order to simplify the model, it is assumed that the pouring fluid is an incompressible ideal fluid. Since the shape of the ladle is different in different angle intervals when the pouring angle changes, the segmentation model is established according to the divided angle interval.

Case 1. In Figure 8, the EFGHIJ represents the section of the ladle along the plane, β indicates the angle between IJ and JE, δ indicates the angle between EG and JE, and indicates the intersection point between the level of the pouring mouth I and the EJ extension line. indicates the intersection point between the level of the pouring mouth I and the GF, represents the bottom radius of the ladle, and represents the top radius of the ladle, and the lengths of JE, IJ, and GF are , , and , respectively. Figure 8 is the flow state of the casting liquid at the beginning of casting, , , and all exist, and the ladle does not show the bottom. The solution of the volume of each part is as follows.

Figure 8: Position of liquid level at Case 1.

In order to show the difference, , , , , and are labeled as , , , , and , respectively, at the time of Case 1.

. Divide into two parts for the upper and lower through point , is wedge cone, is a frustum of a cone body, and then the liquid level equation of the pouring mouth is

In (27), ,

EJ equation:

GF equation:

By the simultaneous equations (27) and (29) and (27) and (30), the intersection points and coordinates can be obtained:

Surface equation of frustum cone is

By formulas (31) and (32) and volume integral methods, can be obtained:

In (33),

The and represent the coordinates of the intersection points and .

can be obtained by the volume formula of truncated cone:

In (35),

Add formulas (33) and (35) to get the expression of .

. Because the ladle body taper is generally 1 : 10 ~ 1 : 15, pouring mouth angle within , and after applying the lining, the size of the nozzle is smaller, so the pouring mouth can be regarded as a triangular body. The face of the triangular body is shown in Figure 9. is the angle of .

Figure 9: The face of the pouring mouth triangle.

According to the principle of similar triangle, the wetted area of the liquid in in the triangular can be obtained:

It can be obtained by formula (37) and volume formula of triangular:

where is in the perpendicular line of , .

can be approximately written in the form of (39). is the area of the pouring surface of the over pouring mouth , composed of pouring liquid surface area of and of . . is obtained by (31) and (32) and the area projection methods:

In (40), and represent the coordinates of intersection points and ,

Because the pouring fluid is incompressible ideal fluid, the flow rate of qa can be obtained by Bernoulli equation

In (42), is the gravitational acceleration, and is ladle outflow of the cross section area, which is equal to the projection of the flow velocity direction of the pouring liquid at the exit of the . Its solution is similar to the wetted area of the solution at with no more detailed description.

In (43), stands for side length and HI stands for over point midline length. Simultaneous equations (42), (43) can be obtained:

The equations (40), (44) and are brought in (39) to obtain .

Case 2. Figure 10 shows that the pouring water in the ladle continues to decrease as the pouring occurs, and the bottom of the ladle has been exposed. At this point, , , and all exist, which is similar to the solution of Case 1. Therefore, Case 2 is solved and only a brief introduction is given.

Figure 10: Position of liquid level at Case 2.

In order to show the difference, , , , , and are labeled as , , , , and respectively, at the time of Case 2.

. In Case 2, at this point , the intersection point of the pouring mouth and the EF is

Then is

In (46), , other upper and lower solutions and forms are the same as Case 1.

. The solution is exactly the same as Case 1.

. The solution is exactly the same as Case 1.

Case 3. Figure 11 is the end of the tilting phase, disappears, and and remain. After pouring the liquid through the point of J, the IJ falls off a certain distance and then falls. Compared with Cases 1 and 2, this stage is slightly different.

Figure 11: Position of liquid level at Case 3.

In order to show the difference, , , , , and are labeled as , , , , and respectively, at the time of Case 2.

In order to show the difference, , , , , and are labeled as , , , , and respectively, at the time of Case 2.

. The liquid level equation of the overflow pouring mouth is

In (47), , .

Then

In (48),

Other upper and lower solutions and forms are the same as Case 1.

. Because of the in Case 3, .

In Case 3, the outlet of the pouring is , then

other equations solving methods are the same as Case 1.

At this point, the sectional model of the water flow in the whole cycle is established. As can be seen from the model, each stage is divided according to the angle of the pouring mouth. When the angle of the pouring mouth is too small, Case 2 may disappear, the direct transition to Case 3, and the volume of will be increased, but the solution is the same. It is also possible that the original pouring volume of the ladle is relatively small, and the pouring process starts directly from Case 2 or Case 3, but the tilting model established in this paper is still valid, except that the initial flow angle of the ladle is different.

Here are some explanations for the tilting model:

(a)The actual pouring should control the pouring liquid height, avoiding too high pouring liquid to cause safety accidents. The maximum pouring height in the model is related to the length of center lines HJ and HI.(b)The model established in this paper is an ideal fluid model, without considering the influence of viscosity on the flow velocity of the pouring fluid. The viscosity should be considered in the actual pouring and formula (42) should be modified.(c)The model described in this paper is only applicable to a kind of common pouring ladle, and other kinds of ladle should be analyzed separately.

5.3. Angular Velocity Model

The relationship between volume and flow can be obtained:

When (38) is brought into the upper expansion, the tilting velocity can be obtained:

Also by , we can get the total time needed for tilting:

In (54), is the beginning of the tilting angle. is the end of tilting angle.

6. Numerical Verification

6.1. Validation of the Tilting Prediction Model

To verify the correctness of the tilting model, the model is compiled into MATLAB. And the  mm,  mm,  mm,  mm,  mm, , , . When pouring, the total volume of pouring liquid in pouring ladle is  mm³. In order to facilitate verification, the flow rate of the whole pouring process is constant,  mm³/s,  mm³/s,  mm³/s, and is also the fixed value. Before computing, we start with some initialization settings for the model.

(1) Critical Point Judgment. According to the geometric relation, the three phases in Section 5 are easy to obtain. The critical angles are 40.930° and 42.994°, respectively. The initial flow angle of the pouring liquid is , the shed end angle is , and they can be obtained according to the following formula:

The search method is used to get , . Assuming that the dip angle of the ladle at the beginning of the tilting is , the whole tilting stage can be divided into

 Initial stage:  Case 1 stage:  Case 2 stage:  Case 3 stage:

(2) Angular Velocity at Initial Stage. is the initial stage, this stage without pouring liquid outflow, and no angular velocity model. In order to make the pouring out of the initial angular velocity change smoothly when the angle reaches , the initial stage angular velocity is set to the value, and its value is .

(3) Treatment of Differential Terms. In solving the angular velocity, it is necessary to derivate the volume term, but for the volume without analytic solution, only the analytic formula of the second integral in the integral can be solved:

where and contain differential terms . The derivative of formula (56) about is derived by using parametric integral derivative formula:

The integral term in (57) is solved by MATLAB numerical integration. cannot find the analytic solution, and its calculation process is similar to that of .

With the method of (54), the angle and angular velocity change with time, as shown in Figure 12.

(a)

(b)

Figure 12: Numerical simulation results of tilting prediction.

In Figures 12(a) and 12(b), respectively, for the instantaneous relationship of tilting angle and angular velocity of the whole period, (a) shows that the changes of tilting cycle angle are relatively smooth, no severe fluctuations; (b) shows that the angular velocity changes are nonlinear during the full tilting period, and the angular velocity changes greatly at the transition of B and C, which is caused by the disappearance of the volume. As can be seen from Table 3, the error of the theory time (, is the total volume of pouring liquid in the ladle. is pouring liquid flow) and simulation time of different pouring flow are only 0.8~2.4%; the error of time was less than 3 S, showing that the model is correct and can be used for real-time control of ladle pouring speed. It is especially suitable for developing tilting control of low cost automatic pouring robot and also can be used as the basic model of tilting closed loop control for high precision automatic pouring robot.

Table 3: Comparison between dumping theoretical time and simulation time.

No.
Theoretical time (s)
Simulation time (s)
Error (%)

290
292.345
0.8

145
148.165
2.1

96.667
99.020
2.4

6.2. Kinematic Simulation of Parallel Mechanism

The pouring process of ladle eventually depends on the designed parallel mechanism, so it is necessary to obtain the change of the driving pair of the parallel mechanism in the whole cycle of the tilting, so as to realize the tilting control of the ladle. In this example, the pouring data of the ladle in Section 6.1 at the flow rate of are brought into the inverse position model of Section 4.2 parallel mechanism, and the structure parameters are the same as in Section 4.3. For calculation and reflecting the driving pair of ladle pouring control will be simplified, a parallel platform to rotate around the fixed position. The coordinates of , , and are 0 mm, 0 mm, and −625 mm, respectively. The numerical results are shown in Figure 13. In order to simplify the calculation and reflect the tilting control of the driving pair, the parallel platform is set to rotate around the fixed position.

Figure 13: Drive pair changed during all the pouring cycle.

As can be seen from Figure 13, the length of and rods is fixed throughout the full cycle of tilting, and rod and the axis vary with time. This shows that when pouring in the fixed position, the ladle rotation is realized by the coupling of the rod and the axis, while and are only used to restrict the space position of the moving platform. The variation of rod length is symmetrical in the whole tilting cycle; that is, the rod is shortened and then extended again, which conforms to the principle of rod length variation in the crank slider mechanism. When the shortest position of rod is 558.594 mm, it is known from the geometrical relation and the mechanical principle that the mechanism is the dead point of the mechanism and the mechanism can continue to move through the dead point with the help of the axis; it also verified the correctness of the driving scheme selection.

7. Conclusion

In this paper, a new type of 4-UPU parallel mechanism with 4 degrees of freedom is proposed and applied to the automatic pouring equipment for the first time. A mobile pouring robot suitable for complex operating environment is exploited and developed. The main work and conclusions are as follows.

(1) For the first time, the parallel mechanism was proposed to be used in heavy-duty continuous pouring operation. The application of screw theory proves that the 4-UPU mechanism is over constrained mechanism with 4 DOF, can realize space three rotation, and rotate around the Y axis, in which 3 chains of shifting pairs and 1 chains of revolute joints can be used as drive pairs; the modified G-K formula to verify the screw theory to calculate degree of freedom is correct.

(2) The application of screw theory demonstrates the rationality of using 4th chains as auxiliary chains, and using geometric method, the closed form equations of the kinematic positive and inverse solutions of a new type of 4-UPU parallel mechanism are given. Compared with algebraic method, the positive solution method presented in this paper has the advantages of unique solution, high accuracy, and fast calculation speed.

(3) Based on the Bernoulli equation and the model of the volume change of the ladle liquid, an instantaneous section model of pouring liquid flow in ladle was established. The numerical example is validated that the pouring model is correct. Numerical examples are given to verify that the model is correct. The change of each driving pair during the tilting of the fixed position is given, and it is concluded that the rotation of the ladle is the coupling motion between the rod and the shaft. The tilting model built in this paper is very suitable for developing an automatic pouring robot with low cost and no closed loop control and guiding the robot’s tilting movement.

Conflicts of Interest

The authors declare no conflicts of interest related to this paper.

Acknowledgments

The authors acknowledge the support of the Major Science and Technology Projects in Anhui China: The Development of Heavy Load Casting Robot in Complex Operation Environment (Grant no. 16030901012); thanks are due to Mr. Wang Dong of Jiangsu Fangcun Map Information Technology Co. Ltd. in this paper for providing important technical support.

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