If your animated characters can't express themselves, it's going to be difficult for the viewer to care about them! This is a step-by-step, frame-by-frame tutorial to help you figure out how to animate a thinking character.
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Step 1: Draw the Character.
Start by drawing the character in its most basic forms. Don't flesh out the drawing until the end! This will keep you from becoming too attached to an animation that has poor movement.
Step 2: Start Anticipated Movement.
The crosshair across the face will help you determine where the character's head is facing. In order to anticipate the movement of the character placing its chin in its hand, it is going to start looking slightly upward and towards the viewer.
Step 3: Continue the Motion.
To make the motion smooth, you'll want several frames that are very similar to each other, but continue moving. As you can see, the little blob of hair is being dragged to the left with the motion. I've started to move the arm forward to anticipate it coming up in the next frame.
Step 4: Start Bringing Up the Arm.
The face is still in the smoothing process of motion, but we can start to bring the arm upwards towards the face. The hand points downwards because it is being dragged upwards from a resting state.
Step 5: Facing Downwards.
We want to exaggerate the movement downwards, so everything should start moving down. The face, the head,and hair should be dragged down. The character is going to begin hunching over a bit, so start to raise the shoulder with the arm moving up. The hand should still be pointed down as well.
Step 6: Blink As You Change Direction.
As a general rule, when people change the direction they are looking in, they will blink. This is to keep the eyes from being overwhelmed by information. It also makes your transition all the more interesting. Start moving the head more downward so it will contact the hand coming up.
Step 7: Facial Squish
It's important to remember that the face is incredibly malleable! When the fingers press into the cheeks, draw a line to show that facial squish being pushed towards the mouth.
Step 8: Let Things Come to Rest.
Let everything that has been moving start to come to a stop, so the final frame isn't a dramatic change. The hair should start to fall into place, the cheek is pushed into a comfortable spot, the hunch is fully complete, and the other arm is coming up to hold the elbow. Note that because the cheek is being pushed forward, we can assume that the head is still being pushed forward, so move that forward just a bit.
To make the rest more lively, you can draw the rest frame, add another frame where they go sliiightly past their intended end, and then repeat the rest frame.
Step 9: Let the Final Frame Stay Still.
Let this final frame go on for several frames to show that the character is thinking for a while. It wouldn't make sense for them to jump back into another motion right away.
If things are moving too fast in your animation, go back in and add more transition frames.