4.1.1
Background
Mechanisms are widely used to improve the production efficiency and accuracy of repetitive processes in industries, such as automobile and aerospace manufacturing, various types of as- sembling plants and computer-aided medical surgery.
A mechanism consists of several links and joints to transmit the motion or force from one link to another. The links and the connections at the joints are not perfectly accurate due to the manufacturing tolerances, material deformation and wearing over the service life of the mech- anism. The uncertain variations affect the positional and directional control (or kinematics) of the motion performed by the mechanism. As a result, actual motion output of the mechanism deviates from the target output required by the design. This deviation between actual and target performance of the mechanism is referred to as output error in the study.
High precision is often required in mechanism applications (Hirschhorn, 1962). A mech- anism consists of several rigid links and joints to transmit the motion or force from one point to another. The connections at the joints are not perfectly accurate due to the manufacturing tolerances, material deformation and wearing over the service life of the robot. The uncertain variations in dimensions of links and joint clearances affect the positional and directional control (or kinematics) of the motion performed by the mechanism.
Randomness in dimensions and clearance in joints will lead to unacceptable performance of mechanism. As a result, actual motion output of the manipulator deviates from the target output
required by the design. This deviation between actual and intended positions of the end-effector is referred to as positional error. Consequently, Dhande and Chakraborty(1973) proposed an optimization procedure to allocate the random tolerances so that the output error is within a spec- ified limit (Faik and Erdman,1991). In the contributions (Dubowsky et al.,1975;Freudenstein,
1954;Lee and Freudenstein,1976a,b), an identification procedure was developed to identify the sources of poor mechanism performance due to the random clearance. Choi et al. (1998) em- ployed the clearance vector model to analyze the effectiveness of tolerance on the performance of system output.
The reliability of manipulator kinematics is defined as the probability that the manipulator realizes its required motion path or trajectory within a specified tolerance range (Rao and Bhatti,
2001). The shape and size of specified range (or safety region) depend on the intended use of the manipulator. The reliability can be defined in two ways. The point reliability means that reliability is evaluated with reference to a particular point on the trajectory of the output motion, whereas the system reliability considers the reliability over a range of output motion.
In the point reliability analysis, Kim et al. (2010) applied the first-order reliability method (FORM) to compute the reliability of an open-loop manipulator with six degrees of freedom. Here, all geomantic dimensions and joint angles were considered as normally distributed. By combining the Monte Carlo simulations with Kriging method, Lai and Duan (2011) analyzed the reliability of a turning machine with a random coefficient of friction. Wu and Rao (2007) discussed optimal allocation of tolerances to joint angles by modeling them as interval variables.
Huang and Zhang(2010) applied the Taguchi method to determine optimal specifications for a function generation mechanism under the constraints of positional accuracy and assembly cost.
Mechanism system reliability analysis is computationally demanding, because the failure region is represented by multiple performance functions. Literature in this area is rather limited.
Zhang et al.(2011a) utilized a stochastic process to describe the positional error along the whole range of mechanism outputs. Considering a mechanism with Normal variables, the linearized limit state function also follows the Gauss distribution. The first-order second moment (FOSM) method, then, can be employed to perform the point reliability analysis. As consider the whole range of outputs, the Poisson process is further assumed to describe the events of output error up-crossing and down-crossing of a design limit. The corresponding counted numbers for the occurrences of failure event in disjoint intervals are independent (Cox and Isham,1980;Snyder,
developed in literature by using a general parallel system (Andrieu-Renaud et al.,2004;Rackwitz and Flessler, 1978;Sudret, 2008a). This approach was applied to analyze the system reliability of a four-bar function generator.
4.1.2
Motivation and Approach
Since a mechanism is designed for repetitive work, it may be necessary in most cases that error is controlled in the entire trajectory, instead of a point. This provides motivation for cumulative or system reliability analysis of the mechanism. Approximate methods, such as FORM, is not adequate to analyze a large system reliability problem (Kim et al.,2010). The first-order second moment (FOSM) method is of limited applicability, as it is based on a spurious assumption of the normality of the positional error.
The key objective of this study is to develop a computationally efficient and accurate method for the cumulative (or system) reliability analysis of mechanisms. The developed method should be able to deal with a large number of implicit and correlated performance functions defining the system reliability.
In the study, the cumulative reliability analysis is formulated as a series system reliability problem. The study shows that the series system reliability is equivalent to the probability that the maximum positional error in the entire trajectory is less than a specified limit.
The distribution of maximum positional error is derived using the maximum entropy (Max- Ent) principle, widely used in probabilistic analysis. A novel feature of the study is the use of fractional moments as constraints, instead of integer moments commonly used in the entropy lit- erature. To compute fractional moments of the positional error, a small sample of positional error is simulated based on the model of manipulator kinematics. The Chapter shows that proposed method is highly efficient, and it achieves the same accuracy as that obtained by a large scale Monte Carlo simulation method.
4.1.3
Organization
The Chapter has been organized as follows. Section4.2describes the kinematic model and joint clearance analysis of a six degrees of freedom elbow manipulator. Section4.3summarizes basic information for the system reliability analysis of a manipulator. A method based on the extreme event distribution is formulated to evaluate the mechanism reliability by using the maximal po-
sitioning error along the entire output trajectory of the end-effector. To estimate the maximal output error distribution, the principle of maximum entropy (MaxEnt) with fractional moment (ME-FM) proposed in Chapter3is employed. Numerical result on the system reliability analysis of an elbow manipulator is presented in Section4.4. Section4.5further examines the proposed method by the examples of slider-crank mechanism and a four-bar linkage. And conclusions are summarized in the last Section.