A few years ago when i was toying with an interest in writing/illustrating children's books, I reached out to an illustrator whom I have long admired, Joe Mathieu. I can't emphasize enough how important Joe's role was during the early days of Sesame Street in capturing the essense of those characters in drawings. As a child, I could instantly recognize Joe's work. His illustrations looked and felt exactly like the Muppet characters. He gave me an insight as to why his work was so infused with the personalities of each character.
Nowadays, there are style guides for the characters which guide aritists in a variety of mediums as to how the characters should be rendered. When Joe started working for Sesame, no such guides existed. He only had the actual puppets to work from. He would visit the set on a regular basis to watch the Muppet performers at work and to see the life they brought to their characters. He explained that often, he'd see others trying to illustrate the puppets while they were on a stationary stand. But it wasn't until the puppeteer put the puppet on their hand that the characters really came to life.
I brought along my favorite childhood book, "Cookie Monster and the Cookie Tree," which Joe illustrated, for him to sign. What a treasure!