# A Novel Approach for a Inverse Kinematics Solution of a Redundant Manipulator
## 1 Introduction  
Kinematically-redundant manipulators are mechanisms which have more degrees of freedom (DOF) than is required for the execution of a given task. The advantage of kinematically-redundant manipulators in comparison with conventional manipulators is in the utilization of redundant manipulator joints for optimization tasks [1; 2]. These optimization tasks are secondary tasks of the inverse kinematic or dynamic model. Manipulator redundancy is used for tasks such as avoidance of collision with obstacles, avoidance of kinematic singularities, maintenance of the admissible joint ranges, increasing of manipulability in specified directions, optimization of execution time, minimizing energy consumption, etc. [3; 4]. On the other hand, kinematic redundancy causes disadvantages, such as the requirements of greater structural complexity of manipulator construction (higher number of actuators, sensors, costs, etc.). It is additionally important to note that control algorithms for inverse kinematic and dynamic model are considerably more complicated [5]. This study investigates kinematically-redundant mechanisms moving in an environment with obstacles. The investigated mechanisms additionally have to deal with a joint limit avoidance task and a kinematic singularity avoidance task. There are several methods to solve the mentioned problems, namely Jacobian-based methods, null-space methods, and task augmentation methods [6; 7]. Many approaches have been used for the kinematic control of manipulators with secondary tasks. The gradient projection method (GPM) is one of them. It was first used in [8] to deal with joint limitavoidance. A later study [9] additionally introduced an iterative approach for joint limit avoidance. These approaches used null space or enlarged space. A problem occurs when the number of all tasks exceeds the number of DOF of manipulators. The weighted least-norm (WLN) method is a method which deals with constraints all the time. Considering the joint limit avoidance task, WLN uses self-motion only when it is necessary in comparison with GPM [10]. An approach with consideration of WLN solution was suggested by [PERSON] [11], and has been used for minimizing energy by using inertia matrix as the weighting matrix. This approach was also used in [12] for minimizing joint torques and in [13] for minimizing joint velocities. [PERSON] and [PERSON] [14,15] describe the role of weight matrices and the priority of its choice for emphasizing or de-emphasizing the role of some components in the computing process. Another algorithm, namely the clamping loop algorithm, ensures the avoidance of joint limits, however is fairly time consuming [16]. Earlier research attempted to assign the priority of the particular tasks by weight matrices [17,18]. However, in some cases the task requirements cannot be achieved [19,20]. In [21], the authors assigned a lower priority to obstacle avoidance task. Problems occurred when the secondary task was not compatible with the main task and the numerical simulations failed. Some works allow the activation or deactivation of secondary tasks by continuous inverse of Jacobian multiplied by the activation matrix [22]. In [23], the authors proposed a new task-regulation framework based on a hierarchy of quadratic program. Within their framework it is possible to forward the constraint task separation across priority levels, eliminating the need for converting inequality constraints into equalities. Our developed approach deals with Jacobian-based method using weight matrices to set the priority of primary as well as all secondary tasks. Our approach is based on changing value of task priority during numerical simulation. The behavior of priority changing is based on numerical computing smoothness. During computing, all secondary tasks are active. This paper is organized as follows. First, the paper deals with path-planning for end-effector of a manipulator moving in an environment with obstacles.