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The paper presents a neuro-fuzzy model of the inverse kinematics of 4 DOF robotic arm employing the relevance vector learning algorithm. Although the direct kinematics of the robotic arm can be modeled with ease by the same approach, the paper focuses on the much more interesting kinematic task, since its solution presents a basis for robot control design. The presented model is of a Takagi-Sugeno type, but its parameters and number of fuzzy rules are automatically generated and optimized through the adopted learning algorithm based on M. E. Tipping's relevance vector machine. The presented model illustrates the effectiveness of the adopted neuro-fuzzy modeling approach.

The paper presents several unconventional models of residuary resistance based on fuzzy logic and neural network techniques. First, two fuzzy models are built based on different hull parameters and different Froude numbers. These models are identified by a modification of Sugeno and Yasukawa identification algorithm. Next, a neuro-fuzzy model of residuary resistance is build, based on statistical learning theory. The model presents a fuzzy inference system of Takagi and Sugeno type that uses an extended relevance vector machine for learning its parameters and number of fuzzy rules. Finally, a neural network approach is applied to build four different models of residuary resistance. Two of the neural models apply classic extreme learning machine, and the other two implement incremental extreme learning machine philosophies. The obtained models are validated for their generalization and approximation performance, and although they all possess excellent approximation capabilities, our neural models based on extreme learning machine have shown the best simulation results.

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