# Enhanced Trajectory Tracking via Disturbance-Observer-Based Modified Sliding Mode Control
## 1 Introduction  
Accurately tracking trajectories is a challenging task for control engineers. Designing controllers for real-life problems is demanding due to the nonlinear nature of the system [1; 2; 3; 4; 5; 6; 7; 8; 9]. Real systems are affected by perturbations, which are the sum of system uncertainties, modeling errors, and external disturbances. The controller's task is to achieve the desired state in the presence of perturbations. Multi-degree of freedom (DOF) robot manipulators are nonlinear systems, and accurately modeling them mathematically is not an easy task. Researchers have proposed various methods to formulate the mathematical model of the system, including mathematical derivation and system-identification techniques [10; 11; 12; 13]. Typically, identification techniques estimate the linear parameters of the system, whereas the nonlinear parameters, uncertainties, and external disturbances are considered as perturbations. Nonlinear control techniques, such as sliding mode control (SMC), are robust against perturbations. The SMC technique has been designed and implemented for desired-position tracking in various practical applications, including underwater vehicles, motor drives, and robot-position control [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. In SMC, the main concept is to design the desiredsliding surface and then force the system state to move towards the designed sliding surface. The switching control moves the system state to the sliding surface [27]. There are two phases in SMC: the reaching phase and the sliding phase. The sliding-surface dynamics are affected by perturbations during the reaching phase; therefore, the system's stability is not guaranteed. To maintain stability while avoiding chattering, the magnitude of the switching gain must be greater than the upper bound of the perturbation. However, determining the upper bound of the perturbation can be difficult, and large switching gains can introduce unacceptable levels of chattering in real-world control problems. Several solutions have been proposed in the literature to mitigate these issues associated with SMC. High-order sliding mode control (HOSMC) was introduced to improve system stability, while integral sliding mode control (ISMC) enhanced the dynamics of the actual sliding variable by incorporating an auxiliary sliding surface [28; 29]. Nonetheless, ISMC still exhibits chattering due to the switching function. Researchers have proposed the use of a smooth switching function to reduce chattering, but this may increase the system's position error [29]. Researchers have proposed a disturbance observer (DO)-based SMC for accurate position tracking [30; 31; 32; 33]. The DO estimates system states and perturbations/disturbances. The control input's design requires information about the system states, and the estimated disturbance information is used along with the SMC control input to cancel the effects of actual perturbation. The compensation of perturbation in DO-based SMC affects the system dynamics through the estimation error of the perturbation. However, this error is much smaller than the actual perturbation [30], which ensures the system's stability due to disturbance rejection. The DO-based SMC is also effective in reducing chattering without compromising the system's performance in terms of position error. [PERSON] et al. [30] proposed the sliding perturbation observer (SPO)-based SMC for accurate trajectory tracking. The SPO is a nonlinear observer. It estimates the system state and perturbation. The estimated perturbation was used in the control feedback to cancel the actual perturbation. The perturbation compensation reduced the magnitude of the switching gains (reduced chattering). The design of the above-mentioned disturbance observer requires the nominal model of the system, which is itself a difficult task. [PERSON] et al. [31] proposed a disturbance observer (DO)-based sliding mode control (SMC) technique for stabilizing a system that is subjected to mismatched disturbances and time delays.