# Valkyrie--Design and Development of Gaits for Quadruped Robot Using Particle Swarm Optimization
## 1 Introduction  
Recent developments in society show the use of robots for different kinds of applications. Robots have automated various sectors of society, reduced manpower, and assisted humans in places where they are restricted, or it is hazardous to work. A robot needs locomotion mechanisms to make it move through its environment, which can be accomplished by wheels, legs, etc. In recent years, the utilization of leg-based walking robots has turned out to be very efficient in the robotics field. These robots possess the ability to deal with irregular terrains, particularly when it's compared with wheeled systems [1, 2, 3]. Legged robot locomotion mechanisms, which are often inspired by biological systems, are very successful in moving through a wide area of rough and harsh environments. Leggedrobots are classified based on the number of legs used, such as biped, quadruped, hexapod, and so on. Quadruped robots are preferred among the other legged robots because of their less complex structure, giving the fullest requirements for the statically stable gaits during walking [4]. Stability is a very important issue of a robot that is maintained as long as the center of mass (CoM) is within the support polygon which is a polygon with the vertices described by the position of legs in contact with the ground. Stability can be divided into static and dynamic stability criteria. Static stability is given when the CoM is within the support polygon and the polygon's area is greater than zero. Therefore, static stability requires at least three points of ground contact [1]. In general, it is extremely difficult to design gait controllers for legged robots. Various hand-crafted methods have been proposed to construct gait controllers, in which human supervision is needed in most cases [5]. [PERSON] and [PERSON] defined gait as something that is defined by the time and the location of the placing and lifting of each foot, coordinated with the motion of the body in its six degrees of freedom, to move the body from one place to another [6]. The observation, comprehension, and mathematical formulation of bio-inspired gaits were the main focus on legged locomotion, and it was found that many of these motions are periodic in nature. Gaits are generally classified as periodic and non-periodic, and it was found that the motion pattern of the quadruped robots is periodic [7]. Many researchers have developed mathematical formulations of various biological gaits of quadruped robots that ensure maintaining static stability. [PERSON] et al. [8] designed and implemented a mixture of free and discontinuous gaits to negotiate uneven terrain with a real machine. The fusion of these two main gaits plus the addition of extra constraints to avoid leg-transfer deadlocking produced a new free-crab gait, a free-spinning gait, and a free-turning gait. The developed gaits were tested on a SIL04 walking robot, and their algorithms were proved to be efficient in real-time and adaptive to irregular terrain. [PERSON] et al. [9] adopted a statically stable crawling gait as their main gait and proposed a fuzzy multi-model switching control based on friction compensation to achieve smooth switching of force and position. Their method was found to offer a solution for stable passage in a slope environment by realizing smooth switching of force/position, precise positioning in the swing process, and soft control of force in the supporting phase. [PERSON] et al. [10] explored the benefits of using multiple gaits in a single robot and analyzed how increasing speed affects the choice of gait and how the choice of gait influences optimal speed. The authors used optimal control as a tool to identify motions that minimize the cost of transport of two detailed models: a planar biped and a planar quadruped. It was found that changing gaits as speed varies leads to the greatly increased energetic economy and concluded that for quadrupeds, the optimal gaits were four-beat walking at low speeds, trotting at intermediate speeds, and galloping at high speeds.