In this thesis, an new optimal turning gait of four legged walking robot is developed. For 30 years, many topics about walking robot have been researched and introduced. Among them, the gait which is very important in realization of walking of robot is the major. Turning gait which makes possible body turning was developed by Hirose and Z. Bien. But Hirose's called circula gait maximized only the walking speed, and Z. Bien, s maximized only the stability. Due to the drawbacks of above two gaits, author developed an optimal turning gait. This new gait maximize walking speed under the condition ; stability must be larger than required value. Also this gait algorithm is so flexible to make strait forward walking, crab walking and pure rotational walking in special case. In this paper, the availabilty of the proposed gait is shown by many reasonable simulation results.