In this work, direct kinematic formulations for a three link continuum robot, with constant and variable lengths, are derived. Continuum robot is a robotic manipulator with the ability of continuously bending over all parts of its body, like an octopus arm. Unlike robots with rigid links, continuum robot has infinite virtual degrees of freedom, which lead to many challenges in their design, modeling, control, and fabrication. In practical applications, in which soft and dexterous tools are required, robots with rigid links have some limitations. This problem leads to the development of continuum robots. These robots have a wide range of applications, from searching in rescue operations to minimal invasive surgeries. Since continuum robots are naturally highly redundant and dexterous, they can be used to move along narrow channels by applying many motion control strategies. To have a better control over this motion, kinematic formulation for these robots must be derived. Continuous nature and redundancy of robots with high maneuverability lead to different definitions for their kinematics. Two major assumptions lead to deriving the direct kinematic formulation instead of using joint angles and lengths of the links: each link bends along its length, and every link is a part of a circular arc. In line with this approach, curvature and length of the arc would be used, which give a more compatible model. The shape of a continuum robot is in a way that minimizes the potential energy of its arm. This shape is achieved due to a function, which is the output of a system of partial differential equations. Since there is no general closed form solution for this system of equations, to achieve the approximate shape of each link, some simplifying assumptions like ignoring the weight of each link, ignoring the external forces and assuming the shape of each link as a continuous function of the curvature’s radius, are needed. Since the continuum robot links bend continuously, their mobility is more than the rigid link robots. Moreover, due to the kinematic redundancy of a continuum robot, it can be used as a multi-task tool. Generally, the secondary tasks are used as performance criteria to optimize the robot motion during its motion. In case of variable length links, dexterity of the robot increases. In most applications, the continuum robot is used to move in places where there are some limitations on manipulator’s movement. In this case, it is assumed that the overall length of each link varies while all of the secondary backbones of each link have the same variations in their length. As the dynamic of this robot is complicated for motion planning, closed-form inverse kinematics, which gives suitable solutions, is used. From natural characteristics of bending in all sections of the arm, capability of robot to move in thin channels with different shapes is analyzed by using optimization methods like obstacle avoidance, in which the control algorithm avoids collision between from channel walls and manipulator, and the whole arm shape is changed according to the shape of the channel. In medical applications the continuum robot is pushed into patient’s body manually. If the robot hits the internal organ with its tip or body, this process would become harmful. To solve this problem, a case in which the base of the robot is not stationary is modeled. Keywords: Continuum Robot, Redundancy, Closed-Loop Inverse Kinematics, Optimization, Obstacle Avoidance, Artificial Potential Field, Whole Arm Grasping.