## 1 Introduction
The PD control was proposed to replace the equivalent control part of the SMC, and the proposed control scheme does not require a precise dynamic model of the manipulators, which implies that the proposed scheme is model-independent. Although this control scheme exhibits quite good performance, switching between PD and SMC is a problem that needs to be considered carefully in practice. [PERSON] et al. [26; 27; 28] proposed a method based on a PD controller and SMC for linear robotic systems, and the SMC gain was assumed to be greater than the upper limit of the dynamics system. However, in most cases, the gains are over-estimated because it is difficult to obtain an upper bound for the uncertainty of the dynamics [29]. To solve this problem, the adaptive proportional derivative sliding mode control (APD-SMC) technique was developed. This technique uses an adaptive law in PD-SMC to estimate the uncertain system dynamics and proposes the dynamic adaptation of the SMC gain as small as possible to overcome the uncertainties and disturbances [30; 31]. Motivated by the requirement for a simple controller that has strong robustness to model uncertainties, external disturbances, parameter uncertainties, and unknown nonlinear dynamics, we proposed a robust hierarchical controller scheme of multiple Euler-Lagrange systems for an escort mission with obstacle avoidance. An outer-inner loop structure was proposed in the proposed control scheme. NSB control was used in the outer loop to produce a velocity vector as a reference value to the inner loop. The APD-SMC technique was designed for the dynamic structure of the robots to follow the reference velocity in the inner loop. The main contributions of this paper are stated as follows:  
1. NSB control applied in \(p-\) dimensional space (where \(p\geq 2\) is a positive integer) realizes multiple Euler-Lagrange systems to execute the escorting mission while avoiding obstacles. 2. NSB control was extended to combine with APD-SMC, which is robust to model uncertainties, external disturbances, parameter uncertainties, and unknown nonlinear dynamics. The PD control part was based only on the tracking errors and tracking velocity errors, and it is model-free and intensive to model uncertainties; the SMC part has strong robustness for the external disturbances and unknown nonlinear dynamics; the adaptive control part was utilized to estimate the unknown system parameters so that the need for large control gains of PD-SMC was avoided. 3. Both the position control and the behavioral control method were proven to be stable by Lyapunov theory for nonconflicting tasks, while for conflicting tasks, it was shown that conflicts will not occur. Compared with ASMC, the proposed control algorithm can drive the tracking errors and tracking velocity errors to reach small regions at a faster rate in both two-dimensional and three-dimensional space. The rest of this paper is organized as follows. Section 2 introduces the models of multiple Euler-Lagrange systems and four properties. The NSB control architecture is detailed in Section 3. APD-SMC, the main results, and stability analysis are presented in Section 4. Simulation results for different conditions are given in Section 5, and the final section concludes the article.