Daily life would be affected if the upper extremity is injured, especially for people with spinal cord injury, stroke and other neuromuscular disorders. In order to assist in the rehabilitation of these movement disorders, exoskeleton robots are developed. Based on the muscular length relationship and the Hill-type muscle model, a lightweight, soft exosuit is designed for elbow flexion and extension. The exosuit is composed of an actuation box and a wearing sleeve. The power is transmitted from the actuation box to the sleeve utilizing an artificial tendon transmission system. The detailed mechanism design is introduced, and the mathematical double artificial tendon transmission model is developed. Experiments and related simulations are followed to validate the model. The prototype of the soft exosuit is tested on a male subject, and the results indicate that the soft exosuit is capable and effective in helping the elbow joint with flexion and extension.