Underactuated robotics: A review <s> Closed-loop controls <s> Abstract This paper addresses the problem of trajectory tracking control for underactuated autonomous underwater vehicles (AUVs) in 3 dimensions space. Given a smooth, inertial, 3D reference trajectory, in order to make the position error to stay in the neighborhood of zero, the control algorithm uses vehicle׳s kinematic equation and linear system stability theory to compute the desired body-fixed velocities. Using these methods, the velocity error dynamics are obtained. Backstepping techniques are used, forcing the tracking error to an arbitrarily small neighborhood of zero. Simulation of a spiral trajectory is performed. The result shows that the controller can realize 3 dimensions trajectory tracking effectively. <s> BIB001 </s> Underactuated robotics: A review <s> Closed-loop controls <s> Abstract This paper proposes a simplified nonlinear fuzzy controller integrating an improved three-dimensional (3D) guidance law, in order to address the problem of path following for an underactuated autonomous underwater vehicle (AUV) exposed to unknown environmental disturbances. First, an improved 3D line-of-sight guidance law that makes full use of the essentially equivalent coordinate transformation is derived to transform 3D path following position errors into controlled guidance speeds, which also reduces the path following system form second-order to first-order. The side-slip angle and angle of attack are integrated into 3D guidance design to account for the underactuated configuration in sway and heave. Second, a nonlinear single-input fuzzy controller is designed to reduce computation complexity resulting from square rules in a double-input fuzzy controller, and to force the steerable speeds of an AUV to attain their guidance profiles. Subsequently, sensitivity analysis is adopted to suggest that the nonlinear fuzzy design with the convergent distribution and small slope for the single input have better robustness against unknown disturbances than the linear design. Finally, numerical examples with quantitative comparison are provided to illustrate the performance of the nonlinear single-input fuzzy controller for 3D path following of an underactuated AUV exposed to unknown environmental disturbances. <s> BIB002 </s> Underactuated robotics: A review <s> Closed-loop controls <s> Underactuated systems represent a class of mechanical systems with fewer independent control actuators than degrees of freedom. The control problem of underactuated systems is one of the most chall... <s> BIB003
The closed-loop control system is also called the feedback control system. The relationship between input and output is established through feedback. The controller could implement predetermined system functions by determining control strategies based on actual conditions of the inputs and outputs. Feedback is one of the most important basic concepts in cybernetics. It is characterized by adjusting future behavior based on past conditions. The working principle of a typical UR closed-loop control system: The control system sends commands to the comparator, and the sensors are mounted on the terminal tools (tachometer, photoelectric encoder, resolver, highprecision potentiometer, proximity sensor, force sensor, pressure sensor, sliding sensor, contact sensor, etc.). The error value is obtained by comparing the feedback signal of the actual state of the robot and amplifying it. The driving source is used to make the UR control a certain link to perform corresponding action. The new motion state of the driving robot is sent to the comparator for comparison again. The newly generated error signal continues to adjust the motion of the UR. Then the process continues until the error information is zero. According to the different functions and characteristics in the UR system, the feedback could be divided into negative feedback and positive feedback. The feedback signal of negative feedback is opposite to the polarity of the input signal or the direction of change is opposite (inverted). The result of the superposition would weaken the net input signal. The feedback signal of positive feedback is the same as the polarity of the input signal or the direction of change is in phase. The result of signal mixing would cause the amplifier's net input signal to be greater than the output signal. Negative feedback causes the output to act in the opposite direction to the input, so that the error between the system output and the system target is reduced. So the system tends to be stable. The positive feedback causes the output to function similarly to the input, causing the system deviation to increase and the system to oscillate. Positive feedback is mainly used for signal generation circuits. Automatic control systems typically stabilize the system's operating state by negative feedback techniques. Closedloop control is studied from the following aspects, which is shown in Figure 12 . Control methods based on fuzzy systems. Takagi and Sugeno proposed a fuzzy model to describe complex systems. There are three kinds of fuzzy methods: Takagi-Sugeno (T-S) fuzzy system, Mamdani fuzzy inference system, and type-II fuzzy system. BIB003 Fuzzy control. Fuzzy control is an early form of intelligent control. It absorbs the ambiguity of human thinking and controls objects by using the control experience of human experts. Fuzzy control does not require a mathematical model. Fuzzy control is an effective way to solve the uncertain system control. Fuzzy control also has its own shortcomings: the control precision of fuzzy control is low. The membership function and control rules of the fuzzy controller are pre-summarized based on experience. It could not be corrected during the control process and does not have the ability to learn and adapt. Many scholars have made a lot of improvements to traditional fuzzy control. Various forms of fuzzy control have been developed. Fuzzy models and identification, adaptive fuzzy control, and so on have emerged. And progress has been made in stability analysis and robust design. Yu et al. BIB002 proposed a simplified nonlinear fuzzy controller integrating an improved 3-D guidance law. In order to address the problem of path following for an underactuated autonomous underwater vehicle (AUV) exposed to unknown environmental disturbances, Li et al. BIB001 address the problem of trajectory tracking control for underactuated AUVs in three dimensions space. Fuzzy control based on model and analysis method has become a modern fuzzy control, which brings new vitality to fuzzy control. The process for fuzzy inference of an output Y in Figure 13 requires several steps. The input X is fuzzified into sets. The output sets V are generated by the inference engine based on the fuzzy rules. Y is the "real world" output which is retrieved through the defuzzification of V. For fuzzy systems, there are three methods such as model-based fuzzy control, model-free fuzzy control, and hybrid fuzzy control.