In the example rule provided above, the coordinate directions are considered separately. However, the coordinate directions should by no means be considered separately — the system is highly coupled and applications of thrust in the x direction will affect motion in all three directions (look back at the equations of motion — x depends on y and y depends on x). However, if rules were written for each possible combination of the controlled actions, the result would be 6⁶ = 66656 rules. In general, the coupling effects need only be considered as the spacecraft approaches the target. Therefore, the coordinate directions are considered separately as long as the spacecraft is more than some predefined distance from the target (requiring 3*36=108 rules — 36 for each coordinate direction). Once the spacecraft is within this predefined distance, a set of “coupled rules” is implemented. These rules are of the form:

where α, β, γ, and Δ are fuzzy terms defined by membership functions like those shown in Figure 4.2 and δ and ε are crisp numeric values. These rules are written for all of the possible combinations of the terms when the four relevant decision variables are in the NEGATIVE-CLOSE or POSITIVE-CLOSE classes. Thus, there are 16 coupling rules that, when combined with the 108 original rules, provide 124 total rules.

At this point a means has been developed for converting a crisp set of conditions existing in the spacecraft rendezvous system to a set of fuzzy conditions, and a set of fuzzy production rules prescribing a real-valued action associated with a particular set of fuzzy conditions. There still remains the task of converting the 124 real-valued actions prescribed by the fuzzy production rules into a single, crisp action to be taken on the rendezvous system. As in previous chapters, the COA method is used to combine all of the applicable rules. In the case of the real-valued consequent approach, the COA method simply yields the average of the action values (real-values) weighted by the product of the confidence levels. It is important to note here that the product of the confidence levels for the six controlled variables were used, not the minimum values. The use of the multiplication operator provided the same type of smooth mappings generated using the minimum operator. However, this is an alternative approach that is sometimes used and thus should be introduced to the reader.

4.4 Details of the Rendezvous Controller

This chapter has introduced two new elements to our fuzzy controllers: a real-valued consequent approach and a multiplier operator. Since these new ideas (especially the real-valued consequent approach) are potentially confusing, it is worthwhile demonstrating the details of an abbreviated example by hand.

To consider the details of a single iteration through the rendezvous controller, we first need to define the membership functions for the condition variables, and this has been done as shown in Figure 4.2. Next, we need a rule set. A complete rule set would consist of 124 rules. However, since the intent here is simply to demonstrate the procedure and explain the details of the operation, only the rules associated with F_x will be considered. These rules consist of the 36 rules that are independent of the y and z coordinate directions, and the 16 coupled rules. The abbreviated rule set is shown in Figure 4.3 in which the values of F_x have been normalized; similar rule sets exist for the other two coordinate directions.

Figure 4.3  The above abbreviated rule set is applicable to the x direction only — there exist similar rule sets for the y and z directions.

Now, consider a specific state of the rendezvous system: x = -1525 and = 0.0. The general procedure is to fuzzify the condition variables, apply the rules, and then defuzzify the actions. For fuzzification, we use the sensory output from the system to determine the appropriate membership function values:

x	μNB = 0.24,μNS = 0.25, μNC = 0.0, and μPC = 0.0, μPS = 0.0, μPB = 0.0;
	μNB = 0.0,μNS = 0.0, μNC = 0.10, and μPC = 0.10, μPS = 0.0, μPB = 0.0;

Next, these membership function values are used to determine the extent to which each rule is applicable (both the independent and the coupling rules). Here, the minimum operator is dismissed in favor of a multiplication operator. The multiplication operator simply takes the product of the membership function values on the condition side of each rule. Figure 4.4 shows the abbreviated rule set with the corresponding weighting factors that are the product of the associated membership function values. Notice that for the above state, there are only four rules that are applicable to a non-zero degree.

Figure 4.4  The abbreviated rule set with the corresponding weighting factors that are the product of the associated membership function values.