## 1 Introduction  
Nowadays, the modern industry in all sectors is facing a new revolution known as Industry 4.0 [1, 2], where many challenges and requirements are taken into consideration with the aim of building smart factories that combine flexibility and ability concepts [3, 4] by developing a new paradigm based on the latest technologies, where automation and network systems present the efficient keys for realizing the new industrial revolution [5]. Recently, industrial robotics has become an important solution used in different sectors due to the advantages guaranteed by industrial robots [6] as manipulator arms and parallel robots represented with higher precision and higher productivity. This optimizes the lead time of the production process [7]. Especially with technological developments, the manipulator arm, for example, presents the most often used tool in the production sector [8], where it can cooperate with its environment and work safety [9, 10]. In addition to the ability to pick huge products and control itself in a flexible and smart way, the physical structure of the manipulator arm regroups two essential parts [11]. These parts are the serial links articulated as an arm and the end-effector, which can be reconfigurable according to the task [12; 13]. The motion planning of a manipulator arm is always based on the degree of freedom characterized by the joints placed in each link, where the number of the degree of freedom limits the workspace and defines the redundancy of the robot. The control of manipulator robots can be studied in two directions, depending on the requirements needed to achieve it [14]: (1) task execution, where the process is based on pick and place, welding, and painting; (2) path planning execution, where the process is based on the trajectory of the end effector, depending on each joint connected to the link of the robot arm. In the area of robotics, the trajectory planning of manipulator arms represents an essential field for focus. The execution of a robot arm's defined task optimizes its trajectory, which can guarantee many benefits such as a reduced cycle time and energy consumption, as well as increased productivity. Basically, the main objective in the trajectory planning field is to compute the desired points that represent the reference input data for the controller of a robot using mathematical techniques [15; 16]. The motion executed from the reference inputs always represents two categories known as forward and inverse kinematics: (1) in free space based on the joint angles, where the motion is limited by the structure constraints, i.e., velocity, torque, and workspace limits; or (2) in task space based on the position and the orientation of the end-effector, where it depends on precision and avoiding obstacles [17]. The approaches generally used are polynomial interpolation function, the bang-bang law, the trapezoid law, etc. [18]. Over the years, researchers studied this field deeply by proposing many methods and solutions to solve trajectory problems for industrial robots [19; 20; 21; 22]. The definition of the optimality concept is divided in many directions. Some scientists focus on a time-optimal trajectory to increase productivity [23; 24], while others work on the smoothness of trajectories [25; 26], taking into account reducing cycle time by implementing fast trajectories combined with optimal jerk values in order to reduce the excitation of the resonant frequencies and limit the vibrations of the mechanical system [27; 28; 29]. From the literature, a basic approach is known for generating a trajectory using splines [30; 31], where the virtual points are required to ensure the continuity of the trajectory from the starting point to the endpoint. The development of this approach motivated the authors of this study to apply an improved technique in the aspect of motion optimality using B-spline interpolation, based on the calculation of inverse of Jacobian matrix.