# Robust Multi-Performances Control for Four-Link Manipulator Arm
## 1 Introduction  
The automation of manipulator arms has gained tremendous attention in recent decades due to their wide range of engineering applications, such as agricultural mobile robotics [1], mining mobile robotics [1], space exploration mobile robotics [2], lower limb rehabilitation robotics [3], biped walking robotics [4], unmanned ground vehicle [5] and underwater robotics [6]. There are two types of manipulator arms: the underactuatedone [7; 8; 9], which is a kind of a nonlinear system with fewer control inputs than degrees of freedom, and the fully actuated one [5; 6; 10; 11], which is not. In general, underactuated manipulator arms are grouped into vertical types, which are controlled by gravity, and planar types, which are not. For the vertical underactuated manipulator, the linear Jacobian approximated model of the inverted equilibrium position is completely controllable because the controllability matrix of the linear Jacobian approximated model is full rank. For most traditional approaches to solving the complexity of a vertical underactuated manipulator, the operation space is mainly separated into a swing-up space and a balance space. In the swing-up space, the control methods include the posture-energy approach [9], the energy-based approach [12], the direct fuzzy control approach [11], the fuzzy model reference learning control approach [11], and the adaptive fuzzy control approach [11], and the linear quadratic regulator (LQR) optimal method [13] is applied in the balance space. However, to be quickly captured in the swing-up space for the manipulator arm, the control needs an exact combination of energy and posture to be built, and this property is difficult to be accomplished due to the complexity of the system's dynamics [9]. Moreover, some torque chattering changes occur in the swing-up space and then the energy is quickly pumped into the system [9]. In general, in order to control the nonlinear dynamics well, disturbance decoupling and global stability should be simultaneously met [14]. To achieve these requirements, some significant approaches, such as model predictive control [15], deep learning [16], multi-objective control [17], backstepping control [18] and preview control [19], have been adopted for complex nonlinear systems. However, in the aforementioned approaches, a serious common drawback is that the considered nonlinear systems should be approximated via linear dynamics by the Taylor expansion for small-effect operating ranges. This serious drawback may be impractical for the FLMA. To solve the nonlinear serious drawback of the FLMA, nonlinear function approximators, such as the neural network optimal approach [20] and famous fuzzy method [21], have been adopted to reduce the caused errors [22]. The main drawback of the famous fuzzy method is that constructing the fuzzy rule base relies on a lot of past accumulated knowledge, and the system's performance is almost determined by the constructed experience rule base [23]. The neural network optimal approach is an intelligent supervised learning approach that requires the operating network to offer many sample points [24]. The performance of controller design for the neural network method is completely limited by only applying the current state value. Moreover, complicated interconnecting structures and digital computing loads mean that the physical realization of nonlinear function approximators is impractical. LQR is a common method that calculates the weighting matrices Q and R for a Jacobian linearized system via a trial and error operation. Some improved approaches of calculating the weighting matrices, such as the genetic algorithm approach and [PERSON]'s pole-assignment approach, have been proposed in recent decades. However, their main serious drawbacks including their high computing effort and slow convergent rate of the globally optimal solution limit their performances [25].