As a direct consequence of the use of more actuators than those needed to perform a specific task, a robot becomes a highly redundant system which complicates the solution of the inverse kinematics. This work considers a task-space kinematic control using a weighted quadratic optimization solver, which allows for whole-body motion control fully exploiting all the movements that a highly redundant robot can achieve. This represents a useful tool for legged robots in order to display more natural and stable movements. At the cost of higher computational power requirements, the proposed optimization problem is a fast, precise and stable solution to the robot's kinematic redundancy. Compared to other methods such as the pseudo-inverse solution, the weighted inverse kinematics solver is capable of imposing hard constraints, guarantees the avoidance of kinematic singularities in most cases, and allows to solve multiple tasks at the same time. This work presents the simulation and implementation of the proposed model.