# Models for Performing Robot Reliability and Maintenance Studies

There are many mathematical models that can be used, directly or indirectly, to conduct various types of robot reliability and maintenance studies. Three of these models are presented below.

**5.6.1 Model I**

This model is concerned with determining the economic life of a robot. More clearly, the time limit beyond this is not economical to conduct robot repairs. Thus, the robot economic life is defined by [16-19]:

where

*REL* is the robot economic life.

*RIC* is the robot initial cost (installed).

*RSV* is the robot scrap value.

*RRC _{ai}* is the robot’s annual increase in repair cost.

**Example 5.9**

Assume that an electrical robot costs $150,000 (installed) and its estimated scrap value is $3,000. The estimated annual increase in its repair cost is $200. Estimate the time limit beyond which the robot-related repairs will not be beneficial.

By substituting the given data values into Equation (5.24), we obtain

Thus, the time limit beyond which the robot-related repairs will not be beneficial is 38.34 years.

**5.6.2 Model II**

This model represents a robot system that can fail either due to a human error or other failures (e.g., hardware and software) and the failed robot system is repaired to its operating state. The robot system state-space diagram is shown in Figure 5.9.

FIGURE 5.9 Robot system state-space diagram.

The numerals in the diagram circle and rectangles denote system states. The model is subjected to the following assumptions [9,19]:

- • Human error and other failure rates are constant.
- • Failed robot system repair rates are constant.
- • Human error and other failures are statistically independent and the repaired robot system is as good as new.

The following symbols are associated with the diagram shown in Figure 5.9 and its associated equations:

*P,* (r) is the probability that the robot system is in state у at time /; for / = 0 (operating normally),у = 1 (failed due to a human error), у = 2 (failed due to failures other than human errors).

A is the robot system constant non-human error failure rate.

*X/,* is the robot system constant human error rate. *в* is the robot system constant repair rate from failed state 2. *вi,* is the robot system constant repair rate from failed state 1.

Using the Markov method presented in Chapter 4, we write down the following equations for Figure 5.9 [9,19]:

At time *t* = 0, *P _{0}* (0) =

*,P*(0) =

_{X}*0,and,P*(0) = 0.

_{2}Solving Equations (5.25)-(5.27) using Laplace transforms, we obtain where

The robot system availability, *AV _{rs}* (/), is given by

As time *t* becomes large in Equations (5.33)—(5.35), we obtain the following steady- state probability equations:

where

*A V _{rs}* is the robot system steady-state availability.

*P _{x}* is the steady-state probability of the robot system being in state 1.

*P _{2}* is the steady-state probability of the robot system being in state 2.

For *0 =0 _{h} =* 0, from Equations (5.28), (5.33), and (5.34) we obtain

The robot system reliability from Equation (5.39) is where

*R _{rs}* (/) is the robot system reliability at time

*t.*

By inserting Equation (5.42) into Equation (5.5), we obtain the following equation for *MTTRF:*

Using Equation (5.42) in Equation (5.10), we obtain the following equation for the robot system hazard rate:

It is to be noted that the right-hand side of Equation (5.44) is independent of time *t, *which means the failure rate of the robot system is constant.

Example 5.10

Assume that a robot system can fail either due to a human error or other failures, and its human errors and other failure rates are 0.0002 errors per hour and 0.0006 failures per hour, respectively. The robot system repair rate from both the failure modes is 0.004 repairs per hour. Calculate the robot system steady-state availability.

By inserting the given data values into Equation (5.36), we get

Thus, the robot system steady-state availability is 0.8333.

**5.6.3 Model III**

This mathematical model can be used for calculating the optimum number of inspections per robot facility per unit time [18,19]. This information is very useful to decision makers but inspections are quite often disruptive; however, such inspections generally reduce the robot downtime because they lower breakdowns. In this model, the total downtime of the robot is minimized to get the optimum number of inspections.

The total downtime of the robot, *TDTR,* per unit time is expressed by [20]
where

*m* is the number of inspections per robot facility per unit time. *c* is a constant for a specific robot facility.

*T _{di}* is the downtime per inspection for a robot facility.

*T _{Jh}* is the downtime per breakdown for a robot facility.

By differentiating Equation (5.45) with respect to m and then equating it to zero, we obtain

where

*m* is the optimum number of inspections per robot facility per unit time.

By inserting Equation (5.46) into Equation (5.45), we obtain where

*TDTR.’* is the minimum total downtime of the robot.

**Example 5.11**

Assume that for a robot facility, the following data values are given:

- • c = 2
- •
*T*= 0.1 months_{db} - •
*T*= 0.02 months_{d!}

Calculate the optimum number of robot inspections per month and the minimum total robot downtime.

By inserting the above specified values into Equations (5.46) and (5.47), we obtain

and

Thus, the optimum number of robot inspections per month and the minimum total robot downtime are 3.16 and 0.12 months, respectively.

# Problems

- 1. Discuss classifications of robot failures and their causes.
- 2. What are the robot effectiveness dictating factors?
- 3. Write down the general formula for obtaining time-dependent robot reliability.
- 4. Write down formula for calculating mean time to robot problems.
- 5. Compare an electric robot with a hydraulic robot with respect to reliability.
- 6. Assume that an electric robot costs $200,000 (installed) and its estimated scrap value is $4,000. The estimated annual increase in its repair cost is $250.

Estimate the time limit beyond which the robot-related repairs will not be beneficial.

- 7. Assume that a robot system can fail either due to a human error or other failures, and its human error and other failure rates are 0.0004 errors per hour and 0.0008 failures per hour, respectively. The repair rate of the robot system from both the failure modes is 0.005 repairs per hour. Calculate the robot system steady-state unavailability.
- 8. Prove Equation (5.47) by using Equation (5.45).
- 9. Prove that the sum of Equations (5.28), (5.33), and (5.34) is equal to unity.
- 10. Assume that for a robot facility, the following data values are given:
- •
*T*= 0.2 months_{db} - •
*T*= 0.01 months_{dj} - •
*c*= 3

- •

Calculate the optimum number of robot inspections per month and the minimum total robot downtime.

# References

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