#### Abstract

This paper proposes a novel 6-DOF robotic crusher that combines the performance characteristics of the cone crusher and parallel robot, such as interparticle breakage and high flexibility. Kinematics and dynamics are derived from the no-load and crushing parts in order to clearly describe the whole crushing process. For the no-load case, the kinematic and dynamic equations are established by using analytical geometry and Lagrange equation. Analytical geometry is mainly used to solve the inverse kinematics and then establish the velocity relationship between generalized coordinates and actuators. Lagrange equation which takes into account the weight of the mantle and actuators is used to solve driving forces of actuators. For the crushing case, crushing pressure is related to the compression ratio and particle size distribution, but the selection and breakage functions should be established first. Because the trajectory model of the mantle is difficult to be established by using analytical method, it can be obtained by an eccentric simulation. The results of input velocities and driving forces of actuators are distinctive due to the eccentric angle and selection of the initial position. Finally, the proposed approach is verified by a numerical example and then the energy consumption is calculated.

#### 1. Introduction

Crushers are commonly used in the mining, construction, and recycling industries to crush a variety of raw materials [1]. Many different types of crushers have been developed over the years, which play a vital role in reducing the particle size of granular solids [2]. As one of the typical crushers, cone crusher is an indispensable piece of equipment [3]. It is typically used in secondary and tertiary crushing stages in minerals processing plants [4, 5]. The mantle and concave are the two main crushing parts. The main shaft of the mantle is suspended on a spherical radial bearing at the top and in an eccentric at the bottom [6]. The crushing action of the mantle around the pivot point is an oscillating motion which can be described with a cyclic function of the eccentric angle. Previous research of scholars has made the performance experience a significant improvement, but cone crusher is inevitably accompanied by high power consumption and low flexibility due to its own structural characteristics. Meanwhile, the parallel robot has received a great concern from many researchers. Compared with serial robot, the parallel robot is a closed-loop mechanism presenting very good potential in terms of high stiffness, large payload, and high speed capability [7–10]. It has been widely used in many fields, such as medical equipment, entertainment, and factory automation [11]. The forward kinematic solution is more complicated than inverse kinematic solution because of the coupling among actuators. The mantle motion of cone crusher is usually set in advance and then the motors are adjusted, which is similar to the inverse kinematic solution. Contemporary crushers are developing towards intelligence. This paper proposes a novel 6-DOF robotic crusher which has their respective advantages through combining the performance characteristics of the cone crusher and parallel robot.

A novel 6-DOF robotic crusher has achieved both interparticle breakage of a cone crusher and high flexibility of a parallel robot. In order to systematically describe the performance characteristics of the whole crushing process, modeling and analysis would be performed from the no-load and crushing parts. Kinematics and dynamics are essential research issues in evaluating the performance. For the no-load case, the inverse kinematic solution which describes the velocity relationship between generalized coordinates and actuators is established. It plays a vital role in the design and component selection [12]. The dynamics of a 6-DOF robotic crusher are complicated by the existence of multiple closed-loop chains, which have several effects caused by inertia, centripetal, and gravity forces [13]. Dynamic modeling can be used for computer simulation without the need of a real system to test various specified tasks, and it plays an important part in system control [14, 15]. Dynamic equations accounting for the parallel configuration of a 6-DOF robotic crusher can be derived in the task-space through the modeling approach of Lagrange equation which provides a well analytical and orderly structure. For the crushing case, the crushing process can be described by a number of crushing zones. The output from the previous crushing zone is the input for the next crushing zone. Crushing pressure is generated on the surfaces of the mantle and concave, and it is related to the compression ratio and particle size distribution.

The trajectory model of the mantle is an essential element for establishing the kinematic and dynamic equations. But it is very difficult to be established by using analytical method. Taking into account the motion characteristics of cone crusher, a small-scale cone crusher is created and the model is obtained by an eccentric simulation. Then, the mathematical calculation tools, MATLAB and Maple, can be employed to solve the input velocities and driving forces of the actuators.

#### 2. Mathematical Foundation

##### 2.1. Principle of a 6-DOF Robotic Crusher

3D geometric model of a novel 6-DOF robotic crusher is shown in Figure 1, which consists of a fixed unit (CFU) and a drive unit (CDU). The CDU has six actuators. Each actuator is made up of a cylinder and a piston which are connected together by a prismatic joint. The upper and lower ends of each actuator are both spherical joint. A coordinate frame O(X,Y,Z) is attached to the fixed base and the other coordinate frame O_{1}(X_{1},Y_{1},Z_{1}) is attached to the mantle.

Particles are squeezed and crushed between the mantle and concave. The transition of the closed side setting (CSS) and open side setting (OSS) is achieved by the extension and contraction of six actuators. The motion of the mantle can be described by a cyclic function of the eccentric angle* θ* which represents the angle between the eccentric axis and vertical axis. The final crushed material is excluded from the OSS due to gravity.

A generalized coordinate vector which describes the position and orientation of a 6-DOF robotic crusher is defined as . In Figure 2, the matrix denotes the translation vector of the mantle frame O_{1},X_{1},Y_{1},Z_{1} with respect to the reference frame . defines an Euler angles system representing orientation of the mantle frame O_{1},X_{1},Y_{1},Z_{1} in regard to the reference frame O,X,Y,Z [16].

##### 2.2. Kinematic Constraint Equations

Inverse kinematic is to solve the lengths and velocities of six actuators through the trajectory model of the mantle. The rotation matrix of frame O_{1},X_{1},Y_{1},Z_{1} relative to the reference frame O,X,Y,Z is given by [14]where

From the geometric model of the 6-DOF robotic crusher, vector can be expressed as [16–18]where (i=1,2,…,6) denotes the length vector of each actuator, which is* B*_{i} to* A*_{i} in O-XYZ. (i=1,2,…,6) represents the coordinates of* A*_{i}(i=1,2,…,6) in O_{1}-X_{1}Y_{1}Z_{1}. (i=1,2,…,6) denotes the coordinates of* B*_{i}(i=1,2,…,6) in O-XYZ.

Differentiate , , and with respect to time:where

The velocity vector of six actuators and the corresponding upper points is given bywhere

The derivations of (4) and (6) are in the appendix.

Differentiate (3) with respect to time:

Equation (8) can be rewritten aswhere

Substituting (11) into (6) yieldswhere presents a Jacobian matrix, which can be described as

#### 3. Dynamic Modeling

Lagrange equation is used to derive the driving forces of six actuators for the 6-DOF robotic crusher, which can be written as [19]

where* L* is the kinetic energy* E*_{k} minus the potential energy* E*_{p}.* q*_{i} denotes the generalized coordinate, and* τ*_{i} is the generalized force.

According the principle of virtual work, the generalized force which is projected along the variation of the generalized coordinates can be derived as follows [14, 20]:where** F** denotes the matrix of six driving forces. Equation (13) has been employed, and then (16) can be rewritten as

##### 3.1. Dynamic Model Components

The kinetic energy of the mantle includes its translational kinetic energy and rotational kinetic energy with respect to its center of mass, which can be written aswhere* M*_{u} denotes the mass of the mantle, is the angular velocity vector of the mantle with respect to the mantle frame, and is the rotational inertia matrix in regard to mass center of the mantle.

After simplifying deriving, the relation between the angular velocity and the derivatives of Euler angles with respect to time can be given by

Substituting (19) into (18) yieldswhere

The potential energy of the mantle relative to the horizontal plane passing by the reference frame O,X,Y,Z is

In Figure 3,* l*_{i} denotes the length of the* i*th actuator;* g*_{i} represents the length between the lower spherical joint and the center of mass.* S*_{1} is the length between the lower spherical joint and the center of mass of a cylinder.* S*_{2} is the length between the upper spherical joint and the center of mass of a piston [19]. Then, the length* g*_{i} can be denoted aswhere

The kinetic energy of six actuators can be driven bywhere denotes the matrix of velocity vectors for the center of mass and can be given by

The derivation of (26) is in the appendix.

Substituting (26) into (25), then (25) can be denoted aswhere

The unit vectors along the axes OX, OY, and OZ are defined as

The potential energy of six actuators relative to the horizontal plane passing by the reference frame O,X,Y,Z is

Equation (23) has been employed, and then (30) can be rewritten aswhere

##### 3.2. Dynamic Equations

Using (20) and (27), the kinetic energy of the 6-DOF robotic crusher can be given bywhere

The derivation of (33) is in the appendix.

Using (22) and (31), the potential energy of the 6-DOF robotic crusher can be obtained as follows:

Considering (15) and (33), the equations can be derived aswhere

The derivation of (36) and (37) is in the appendix.

Using (15) and (35), the following equations can be given bywhere

##### 3.3. Compressive Breakage Behavior

As shown in Figure 4, the crushing process can be described by a number of different crushing zones. The feed material is crushed by the interparticle breakage and flows through each crushing zone in the crushing chamber. The material is transformed to the product by a repeated crushing process and crushed once in each crushing zone between the mantle and concave.

In Figure 5, crushing pressure* p* is generated on the surfaces of the mantle and concave [21]. It is related to the compression ratio* ε* and particle size distribution

*. Compression ratio represents the proportional relationship between compression length and height of crushing zone. Particle size distribution describes the uniformity of the particle size distribution. The compressive ratio is the largest value when the material moves to the closed side. Meantime, the corresponding pressure*

*σ**p*is also the largest value of the same horizontal cross section. Crushing pressure

*p*can be represented as

A process model of consecutive crushing events is presented, as shown in Figure 6. The selection function* S*_{i} describes particles of all sizes which enter a crushing process have some probability of being broken, and the probability is constantly changing as the particle size changes. A certain proportion of particles in each size range are selected for breakage and the remainder passes through the process unbroken during the crushing events. The breakage function* B*_{i} reflects the particle size distribution of each size range after particles are broken into smaller fragments.

The process model uses the output from the previous crushing event as input for the next crushing event. Each crushing zone corresponds to a crushing event, and the size-reduction process can be described aswhere* P* represents the product size distribution and* F* is the feed size distribution. The total number of crushing events is denoted as* n*.

Selection and breakage functions can be established by the compression ratio and particle size distribution through the analysis of the experimental results. Thus,* S* and* B* can be established aswherewhere* a*_{i} are fitted constants.* x*_{min} represents the minimum particle size of different crushing zones, and* x*_{max} denotes the maximum particle size.* x*_{i} is the particle size distribution of each size range.

#### 4. Numerical Results and Discussion

In this section, a trajectory model of the mantle is established by an eccentric simulation. The main purpose is to solve the input velocities and driving forces of a 6-DOF robotic crusher. At the same time, it demonstrates the suggested approach can solve the dynamic problem effectively. Furthermore, the power of six actuators and energy consumption are calculated.

##### 4.1. Example

The parameters of a 6-DOF robotic crusher are presented in Table 1. The trajectory model of the mantle is an essential element for establishing the kinematic and dynamic equations for the 6-DOF robotic crusher. But it is very difficult to be established by using analytical method.

A small-scale cone crusher is created in a virtual environment by using ADAMS in order to obtain the trajectory model of the mantle, which can be shown in Figure 7. The oscillating motion of the mantle is accomplished by the eccentric simulation. Position and orientation of point O_{1} relative to the fixed point can be extracted and shown in Table 2.

The movement simulation based on ADAMS is carried out to establish the trajectory model of the mantle for the 6-DOF robotic crusher. Then, the model of the mantle frame O_{1},X_{1},Y_{1},Z_{1} relative to the reference frame O,X,Y,Z can be described aswhere* ω*=1.483rad/s.

The proposed approach is used to solve the kinematic and dynamic equations. Input velocities and driving forces of six actuators have the same time period, as shown in Figures 9 and 10. Input velocities of actuators 4 and 5 are greater than others. Negative value indicates that the actuator is contracting. The values of driving forces are in the interval , and the maximum value is found on actuators 3 and 6. They can be used as a basis for the design and component selection. The difference of the peak value is related to the eccentric angle and selection of the initial position.

##### 4.2. Simulation Verification

Settings of connectors and motions of 3D model in ADAMS are shown in Figure 8 [22]. In order to validate the proposed approach, driving forces of six actuators are simulated by using ADAMS, which are represented in Figure 11. Figures 10 and 11 are obtained by executing the simulation for 20s. It can be observed that the calculated and simulated outputs have good agreements, which indicates the suggested approach of dynamic modeling is suitably selected.

Compared with the driving forces, the friction of spherical joints and actuators is negligible. Therefore, power of six actuators can be expressed as follows according to (13) and (17):

The derivation of (52) is in the appendix.

Power of six actuators can be described with a cyclic function of the time, as shown in Figure 12. Power is only related to the payload consumption when six actuators are all expanding, and it has nothing to do with the structure. The energy consumption of the 6-DOF robotic crusher can be mainly divided into two parts: energy consumption during breakage* E*_{1} and no-load mechanical energy* E*_{0}.* E*_{1} is obtained by integrating the pressure* p* over the stroke* s* and multiplying the cross-sectional surface area* A* perpendicular to the compressed volume. Similarly,* E*_{0} can be calculated by integrating the power of six actuators over the time. Therefore, the energy consumption* E* of the 6-DOF robotic crusher can be expressed aswhere* T* represents the crushing period of particles.

#### 5. Conclusions

A novel 6-DOF robotic crusher was proposed which could achieve both interparticle breakage of a cone crusher and high flexibility of a parallel robot. The kinematic and dynamic models were derived from the no-load and crushing parts in order to systematically describe the performance characteristics. For the no-load case, the kinematic model was established by analytical geometry and Jacobian matrix was conducted. The dynamic model which takes into account the weight of the mantle and actuators was derived based on the Lagrange equation. For the crushing case, the crushing process could be described by a number of different crushing zones. The crushing pressure was related to the compression ratio and particle size distribution, and the closed side was the largest location of the same horizontal cross section. In order to establish the trajectory model of the mantle, a small-scale cone crusher was created and the model was obtained by an eccentric simulation. The result showed that the position and orientation functions changed periodically. Then the mathematical calculation tools, MATLAB and Maple, were employed to solve the input velocities and driving forces of actuators. The suggested approach had been verified by using ADAMS. Input velocity and driving force of each actuator were different due to the eccentric angle and selection of the initial position. Finally, the power of six actuators and energy consumption were given.

#### Appendix

Differentiating , , and with respect to time, (4) can be derived as

For any length of an actuator, it can be obtained as

Therefore, the following equation can be deduced aswhere is the unit vector of each actuator.

Using (A.2) and (A.3), (6) can be derived aswhere

The ratio between and* l*_{i} is defined as

Using (A.6), the velocity vectors for the center of mass can be given by

Then, (26) can be obtained as follows:

Using (20) and (27), (33) can be derived as

Equation (36) can be deduced aswhere (15) and (33) have been employed.

Equation (37) can be further derived aswhere

Equation (52) can be deduced as

#### Data Availability

There were no data used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Key Research and Development Program of China (no. 2016YFC0600805).