Mobile Robotics has been evolving as one of the most promising domains in the field of Robotics. The ability of these robots to explore and maneuver in complex environments without human intervention attracts the attention of researchers across the globe. The mobile robots are classified into three different areas viz. wheeled robots, tracked robots, and legged robots. Robot locomotion system is an essential characteristic of mobile design, which depends not only on working space but also on technical measures like maneuverability, controllability, terrain condition, efficiency, and stability. Applications involving locomotion over rough terrains or disaster management where the robot is needed to access the remote areas within the debris demand the use of legged robots. Legged robots are further classified depending on the number of legs the robot has. Hence the types of legged robots are pogo-stick robots or one-legged robots, bi-pedal or two-legged robots, quadrupedal or four-legged robots, six-legged or hexapod robots, and eight-legged robots. Each of the types has unique applications and special locomotion mechanisms. The GAIT behavior of the quadrupedal robots is inspired by the quadruped animals like horses, dogs, etc. This project is focused on the Simulation & Control of a Quadrupedal Robot, using trajectory generation for the locomotion and describing three types of GAIT behaviors for the robot, viz. Walking, Trotting and Galloping. These are based on the speed and leg-movement patterns of the robots. These behaviors can be transitioned depending on the application and the terrain pattern. All the mechanisms are designed and simulated in MATLAB, and Simulink.