Description:FIELD OF INVENTION
The traditional cam mechanism in mechanical reciprocating motion is constrained by the profile curve and lacks flexibility, which will be replaced by a new, flexible and controllable electronic cam.
BACKGROUND OF INVENTION
As a combination of the cam and linkage mechanisms, the cam-linkage mechanism is one of the most popular transmission mechanisms in mechanical science. By integrating the merits of both kinds of mechanisms, the cam-linkage mechanism can achieve superior kinematic performance while maintaining high reliability and compact structures. Due to such prominent abilities in realizing complex motion laws, the cam-linkage mechanism has been widely used in various machinery and automatic production equipment, such as textile machinery, packaging machines, rehabilitation devices, bionic horse robots, and parallel manipulators. In view of the significant role of the cam-linkage mechanism in mechanical transmission, lots of studies have been proposed to investigate the design optimization of the cam and connecting rod.
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LITERATURE SURVEY
For example, Rybansky et al studied the topological optimization of the internal shape of the biaxial spring cam mechanism. The weight/stiffness trade-off in the cam design was investigated during the topology optimization. Abderazek et al. discussed the motion law of the disk cam mechanism with a roller follower. Li et al. generated the design of a mold substructure with cams in the form of an assembly. In addition to the cam optimization, Zhang et al. designed a 1-DOF (degree of freedom) cam-linked bi-parallelogram mechanism. This mechanism was applied to a double-deck parking system. Wu et al. presented a new robot with a five-bar spatial linkage design form. The robot has the advantage of a larger working space. For the cam profile that people are generally concerned about, Arabaci et al. proposed a dimensionless design method for a double-arc cam mechanism. The motion equation of the cam profile was obtained during the design process. Moreover, Xia et al. and Ouyang et al. constructed the optimization design model of the new cam profile. Chen et al. investigated the X- and Y-shaped cam profiles. It was found that the cam linkage polishing device has a bicircular polishing trajectory with zero velocity deviation.

SUMMARY
Although these studies have provided a number of impressive techniques for contour design, motion analysis, pressure angle calculation, and overall dimensional optimization of the traditional cam-linkage mechanism, most of them focus mainly on design analysis in the slow-speed environment. The influence of component dimensions is not taken into account. However, when oriented to high-speed operation, the kinematics performance of the cam-linkage mechanism differs significantly from low- or normal-speed scenarios. Moreover, the high-speed conditions bring significant challenges to the stable operation and fatigue life of the mechanism. The parameter optimization of the large-size high-speed cam-linkage mechanism remains to be resolved. There is scant research discussing the multi-objective optimization of the cam-linkage mechanism under such challenging environments.
To bridge this gap, this paper proposes a parameter optimization method for a large-size high-speed cam-linkage mechanism considering kinematic performance. Specifically, the cam five-bar mechanism is introduced as an example. First of all, the modeling analysis of the cam five-bar mechanism is presented. Then, the multi-objective optimization of the cam five-bar mechanism is investigated under high-speed scenarios. Finally, the reliability and sensitivity analysis is conducted to investigate the kinematic performance of the optimized structure. The main contributions of this paper are as follows.
(1) A mathematical model is constructed to determine the performance parameters of the cam five-bar mechanism. The motion characteristics of the mechanism are obtained by resolving the mathematical model.
(2) A multi-objective optimization method for a large-size cam-linkage mechanism is proposed. The optimal kinematic parameters are identified by solving the optimization problems.
(3) A computer-aided platform is developed for the design analysis of the cam five-bar mechanism. The parameter calculation, optimization, and reliability analysis are well-handled with the aid of the software package.
(4) A real-world case study of the transverse device is put forward to demonstrate the effectiveness of the proposed method. The productivity of the transverse device is substantially improved.
DETAILED DESCRIPTION OF INVENTION
Compared with traditional cam-linkage mechanisms, the compactness of the large-size cam-linkage mechanism is poor. The fatigue damage and wear also differ significantly. In this study, we take the cam five-bar mechanism as an example to investigate the design optimization of the large-size cam-linkage mechanism.

Figure 1. Schematic diagram of the cam five-bar mechanism.
In general, the objective of the cam five-bar mechanism is to achieve a controlled change of the rotational angular speed with the rotation angle. As illustrated the cam five-bar mechanism is composed of one cam and five bars. Rod 1 is the prime mover, whose angular velocity is constant. Rod 4 is the output member, whose angular velocity is supposed to meet working requirements. In addition, rods 2 and 5 are fixedly connected. Rods 2 and 3 generate the triangle BCD. During the system operation, the angle BCD would be changed if the motion state of rod 2 is adjusted through the cam. Correspondingly, the distance between B and D is also changed. As a result, the movement of the output member (i.e., rod 4) can be determined by adjusting the distance between B and D.
Theoretically, the output angular velocity and cam shape can be uniquely determined if the angular velocity of rod 1 and the length of each rod are known. This process of deriving the output velocity and cam shape from the given input angular velocity and rod structure is known as a positive solution. By contrast, the input and output angular velocities are often given in real applications, whereas the linkage mechanism characteristics and cam shape need to be resolved. This process is known as inverse solving. Compared with the positive solution, reverse solving may produce multiple solutions in which the motion characteristics of different structures can vary significantly. For example, some cam shapes may generate certain points where the pressure angle is too large, resulting in uneven motion and poor forces. Similarly, some linkage structures may produce points where the transmission angle is too small or even dead in motion. Thus, it is quite challenging to identify excellent motion characteristics while performing inverse solving.
Mathematical Model
As shown in Figure 1, a right-angle coordinate system Oxy is established to facilitate the theoretical analysis. The center of rotation of the prime mover (point A) is set as the origin (i.e., O). Suppose l1, l2, l3, l4, and l5 are the lengths of the corresponding rods. ?1 is the angular velocity of the prime mover. ?1 is the angle of rotation. ?4 and ?4 are the angular velocity and angle of rotation to be satisfied by the follower, respectively. The vector equation is obtained as:

where l1, l2, l3, and l4 are vectors of magnitude l1, l2, l3, and l4, respectively.
By expressing Equation (1) in complex form and expanding it according to Euler’s formula, it can be written as:

Then, the rotation angle of each rod is obtained as:
where
To find the final solution, i.e., the trajectories of points A, B, C, D, and E, the values of l1, l2, l3, l4, l5, ?, ?4, and ?1 and the initial value of ?14 (the angle between rod 1 and rod 4) need to be determined. In real practice, ?4 is usually given as ?41 and ?42, which is shown. The connection curve between ?41 and ?42 can have many choices. The angular velocity ?4 is completely determined only after selecting the connection curve concerning the actual working condition requirement.

Figure 2. The desired motion law of the output member
Thus, the coordinates of each point in the motion of the cam five-bar mechanism are determined as:

where xi and yi (i = b,c,d,e) are the horizontal and vertical coordinates of points B, C, D, and E, respectively.
By solving Equation (6), the following mechanism motion characteristics can be obtained.
(1) Cam theoretical profile curve:
From the above analysis, it can be concluded that the trajectory of point E is the theoretical contour curve of the cam.
(2) Acceleration at point E during the motion of the mechanism

where xe? and ye? are the horizontal and vertical accelerations of point E, respectively. ae is the acceleration at point E.
(3) Transmission angle:

where f1 is the angle between rods 2 and 3.
(4) Cam pressure angle:

After determining the motion characteristics and the cam profile curve, a particle swarm-based multi-objective optimization method is proposed to determine the optimal solution for the inverse solving, which is conducted as follows.
Selection of Optimization Variables
In light of the mathematical analysis of the cam five-bar mechanism, the initial values of ?14, ?4, and rod lengths l1, l2, l3, and l5 are selected as optimization variables to enhance the motion characteristics of the mechanism.
First of all, a combination of five straight lines and four curves is introduced for the representation of ?4, which is depicted. Specifically, the angular velocity of the 0~T1, T4~T6, and T9~T10 periods are given in advance. Therefore, the remaining curve is divided into two parts. The first part is the T2~T3 and T7~T8 periods, while the second is the period of T1~T2, T3~T4, T6~T7, and T8~T9. In particular, the angular velocity keeps constant in the first part, whereas the second part (i.e., the buffer section) introduces a motion law curve as the corresponding angular velocity curve. The integration of ?4 in the 0 to T10 period is 360°.

Figure 3. Angular velocity curve of the output component
As illustrated in Figure 3, the first part of the curve can be considered symmetric about t = T5. Then, the time parameters are determined as:
Constraint Establishment
When rods 2 and 3 are in a straight line, the mechanism is in the limit state. At this time, the angle between rods 1 and 4 is expressed as ?max. The following constraint should be satisfied:

Moreover, the following constraint should be satisfied for the large-size cam-linkage mechanism:

where ?14i is the initial value of the angle between rods 1 and 4
Model for the Optimal Design
Based on the above analysis, the objective of the optimization is to maximize the transmission angle of the five-bar mechanism, minimize the cam pressure angle, and minimize the acceleration at point E, which is represented as:

Then, the optimization model is obtained as:

Finally, the particle swarm optimization (PSO) algorithm is introduced to resolve the optimization problem, which is described as:

As a population-based optimization method, the optimal solution can be determined by updating the position and velocity of the particles. Once the optimization model is constructed, the solving process can be handled by the POS module in MATLAB.
Reliability Analysis of Kinematic Accuracy
In the process of reliability analysis, the mathematical model is generally established according to the reliability design principle and practical problems. Finally, a suitable algorithm is used to solve the problem. The general solution process is shown in Figure 4.

Figure 4. The general process of reliability analysis.
Mathematical Model of Reliability Analysis
Compared with the traditional cam-linkage mechanism, the geometric shape error of the cam five-bar mechanism with a large size has a more significant impact on motion accuracy, especially in the high-speed scenario. To investigate the kinematic accuracy of the optimized structure, a reliability analysis of motion accuracy is presented based on the Monte Carlo methodology. Firstly, motion analysis is conducted before the reliability analysis. Specifically, the errors in the machining process of the rod are investigated in the motion analysis. Assuming l1', l2', l3', l4', and l5' are the actual lengths of the corresponding rod shown in the intersection point with the theoretical contour line of the cam is E'(xE', yE'). Then, the vector equation of the linkage mechanism considering the errors is built as:

By the transformation operation similar to Equation (1), Equation (20) is also written as:
Accordingly, ?3 and ?4 are obtained as:

In addition to the linkage mechanism, the cam and the roller errors are also investigated. As shown in Figure 5, ?l is the cam’s shape error. rg is the roller radius error.

Figure 5. Error diagram of the swing cam mechanism.
Ideally, the cam and the roller are in tangential contact at point C. However, the point would change from C to C', when ?l and rg are taken into account. Then, the angular error ??5 of rod 5 is calculated as:
where d indicates the combined error of roller radius and cam geometry, d = ?l + rg.
The rotational angle ?5 of rod 5 is obtained as:

Therefore, the actual output angle of the mechanism at any given moment can be obtained from the above motion analysis. In addition, the dimensional errors of the components can be considered mutually independent and normally distributed random variables [13]. The probability distribution function N(µ,s) of each rod size can be obtained according to the “3s principle”.
Reliability Analysis
After the mathematical model is established, the Monte Carlo strategy is introduced to conduct the reliability analysis, which is illustrated in Figure 6.

Figure 6. Flow chart of Monte Carlo reliability analysis.
In Figure 6, I(X) indicates the number of samples that meet the requirements. g(X) is the limit state function, by which the product is judged to be failing or not. In the process of reliability analysis by Monte Carlo strategy, first set the sampling number, input the probability distribution function of the variables to generate N samples, and then substitute each sample into the limit state function for calculation. The Monte Carlo method is a numerical calculation technique guided by probabilistic statistical theory. It estimates the overall probability of failure by the failure frequency of the sample, which is described as:

In Equation (27), the number of samples N should be chosen with a balance of computational speed and accuracy. The appropriate number of samples is determined based on the failure probability of the mechanism. Thus, the calculated failure probability will not be seriously distorted and large errors can be avoided.
DETAILED DESCRIPTION OF DIAGRAM
Figure 1. Schematic diagram of the cam five-bar mechanism.
Figure 2. The desired motion law of the output member
Figure 3. Angular velocity curve of the output component
Figure 4. The general process of reliability analysis.
Figure 5. Error diagram of the swing cam mechanism.
Figure 6. Flow chart of Monte Carlo reliability analysis. , Claims:1. Robust Cam Mechanism for High-Speed Industrial Applications claims that the motion characteristics and kinematic accuracy of the cam-linkage mechanism under high-speed scenarios, this paper proposes a series of methods for the kinematic modeling, optimization, and reliability analysis of the large-size cam-linkage mechanism considering kinematic performance. A mathematical model is constructed to determine the performance parameters of the cam-linkage mechanism, such as E-point acceleration, transmission angle, and cam pressure angle.
2. A multi-objective optimization methodology is proposed for the parameter optimization of the large-size cam-linkage mechanism. The co-linear of rods 2 and 3 is set as constraint, while the maximum drive angle, minimum cam pressure angle, and minimum E-point acceleration are taken as the objective function.
3. The kinematic model of the cam-linkage mechanism is presented. Reliability analysis is conducted to evaluate the kinematic accuracy of the optimized mechanisms.
4. A computer-aided platform is developed to assist the parameter calculation, optimization, and reliability analysis of the cam-linkage mechanism.
5. The effectiveness of the proposed method is validated by a real-world case study. The productivity of the transverse device is increased from 600 PPM to 1200 PPM