## 1 Introduction  
Humanoids are robots designed in the shape of humans to carry out a variety of tasks. They are built to receive data from their environment through suitable sensors and to take actions by moving their joints in a certain way to accomplish the required task (Figure 0(a)). Their sophisticated control system is partitioned into smaller dedicated-purpose controllers. The most crucial part of the overall control system is the one that is responsible for generating the motors' commands to maintain balance and stability. For this control goal, a triple inverted pendulum system is used to develop, test, and enhance the control design. Mechanically, the torso, thighs, and shanks of the humanoid are represented by three links connected by joints that are driven by electric direct current (DC) motors, as shown in Figure 0(b). Hence, the triple inverted pendulum system describes the basic dynamics of a humanoid. The angles that are made by the three links with the vertical axis are to be controlled by the torques produced by two DC motors. The torques are provided by the drivers to the upper two joints through two belt pulleys. The angles are measured by three potentiometers. The horizontal bars are added to ease the balance by increasing the moment of inertia [1]. The triple inverted pendulum is an unstable system which is difficult to control. Besides being an unstable system, it is subject to perturbations, due to uncertainties in the system's friction parameters and moments of inertia. In addition, external disturbances like wind gusts affect the system response. Noise in measurement readings is another source of uncertainty that affects the system response. In such systems, where perturbations occur, the controller used must take the uncertainties into account during its design to provide robustness against them. The other challenge of controlling multiple degrees of freedom (DOFs) by fewer control signals is encountered practically. This may occur by the failure of one actuator during operation or may even be a goal in itself for the sake of minimizing the consumed energy, cost, or size of the system. The underactuation makes disturbance rejection more difficult, because the applied disturbance to the unactuated joint would not fulfill the matching condition. In the last few years, both optimal and non-optimal control methods have been suggested and developed to control the inverted pendulums and robotic systems. Among the researchers who utilized non-optimal control methods are [PERSON] et al. [3]; they presented a decoupling sliding mode algorithm to control a single inverted pendulum on a cart. The poles of the reduced order system have been placed once near the imaginary axis and in another case away from the imaginary axis, and in the dominant region. It has been shown that locating the poles in the dominant region reduces steady state error and produces better disturbance rejection. [PERSON] et al. [4] used the Attractive Ellipsoid Method (AEM) to stabilize the triple inverted pendulum and compared it to the sliding mode control (SMC) system. It was found that AEM-based control rejects external perturbations more smoothly, while the SMC consumes less energy. [PERSON] [5] developed an SMC system for the two-link fully actuated robot arm. In system response, the angles of the arms reach the desired value within 1.5 s. [PERSON] et al. [6] proposed a hybrid sliding mode/H-infinity control approach using a fuzzy neural network weighting method to  
Figure 1: Resemblance of humanoid structure to triple inverted pendulum: (**a**) humanoid structure [2]; (**b**) triple inverted pendulum structure [1]. reduce nonlinearity, provide precise trajectory tracking, and to the enhance noise rejection capability of a single-link flexible manipulator system. The proposed hybrid controller reduces the total system nonlinearity, leading to higher performance with respect to noise cancellation, compared to sliding mode controller alone.