# Numerical Quantification of Controllability in the Null Space for Redundant Manipulators
## 1 Introduction  
Robot manipulators, which generally consist of six degrees of freedom (DoF), are widely adopted in industrial fields [1]. However, six DoF manipulators can only perform a single task at a single time in three-dimensional space, due to lack of excess DoF. Hence, redundancy is necessary to resolve this issue. Robots are redundant when it has more DoF than the required to perform the given task (e.g., seven DoF for six direction tasks). Redundancy is a crucial feature of robotic manipulators that can be utilized to execute a wide range of tasks with flexibility, dexterity, and versatility. Due to these unique characteristics, redundant manipulators can execute several tasks simultaneously with null space projection [2; 3; 4]. Applications of null space projection includes, obstacle avoidance [5; 6; 7], impedance control [8; 9; 10], the joint saturation [11; 12; 13], and the posture control [14; 15]. Not only does the null space projection enable the robot's capability to execute tasks simultaneously, it also has the advantage of prioritizing the tasks. When null space projection is used, excess DoF is utilized to prioritize the tasks into (1) primary task and (2) secondary task. By doing so, the robotic manipulator can execute both tasks simultaneously without disturbing the primary task (higher priority). While executing the tasks, the null space projector plays a key role to minimize the interference. This interference is mainly caused by the multi-DoF dynamics of the robot. Thus, a null space projector was examined with respect to the robot dynamics to stabilize the system [2; 16; 17]. Although the studies were focused on minimizing the interference of the primary task, a further study must be conducted to consider the controllability of the secondary task. In the past, numerous studies using self-motion as the secondary task were conducted within the joint space [18; 19; 20]. However, when the self-motion is conducted within the taskspace, singularity of robot joints become an issue. Thus, controllability of a robot manipulator must be evaluated while controlling the manipulator within the task space. In a general situation, evaluation of the controllability in task space can be substituted by evaluating the robot manipulability [21]. The robot manipulability refers to the manipulating ability of a robot within the task space, and it is used as an indicator to avoid robot singularity [22]. However, robot manipulability does not account for the changed movable direction of motion due to null space projection. For example, once the null space projector is applied to the robot motion, a new type of singularity occurs. Therefore, a new indicator that deals with this type of singularity is required to evaluate the controllability of a robot within the null space. In order to address this issue, we propose a new indicator named 'null space quality', which evaluates the manipulability of a robot with null space projection. Conventional manipulability measures the capacity of a robot to move in a certain direction while considering every single joint. However, it is not possible to evaluate the manipulability in its standard form when the null space projection is used due to the priority of the tasks. This is because the movable direction of motion of a robot changes with the null space projector, in contrast to the movable direction of motion of a robot that considers every single joint. Thus, it is crucial to consider both the null space projector and the Jacobian matrix to deal with the extra capacity of joints. When the null space projection method is used, it is crucial to consider the controllable direction of a robot. Generally, when the tasks controlled within the null space are under a relatively low DoF situation (i.e., under-actuated), the DoF in task space are usually coupled to one another. Thus, in order to control the robot to the desired motion, it is necessary to find and utilize the controllable direction to acquire better control performance.