I used this exercise as a quiz for Kinesiology because it is a prime example of a sagittal plane exercise. Flexion and extension are performed at the shoulder, trunk (namely the cerival and lumbar spine), hip, and knee.
The most challenging portion of this exercise is when the force (in this case gravity) is applied at a perpendicular angle to the body. That means fully stretched out and initally bringing your knees to your chest will be more diffcult than any other part of the exercise.
The agonist (this word mean the muscle that is the prime mover and responsible for the action) depends on the joint. At the trunk, the rectus abdominis is the agonist, both concentrically contracting (muscle shortens as it develops force) during the sit up, and eccentrically contracting (muscle lengthens as it develops force) during the lowering phase. Both internal and external obliques will be active to provide additional force and stability.
The hip flexors (psoas major, psoas minor, pectineus, rectus femoris) will be eccentrically and concentrically controlling the flexion and extension of the hip.