Part 2: Problem Identification

It is evident that there are at least two planes of movement as it is built. 

Figure: x-z plane of the bear

Figure: y-z plane of the bear

Intuitively, there are two approaches I explored :

1. Find an equivalent mechanism setup that places both crank and slider mechanism and the hands movement in the same plane
2. Observe the kinematics of the components in two discrete planes and combine the results at the end 

First, the #1 approach is explored:

The easiest approach to simplify the problem would have been to put the crank on the same plane as the hands' movement plane. If the crank is connected to the head in a manner that can be rotated in the z-axis without changing any motion profiles, this would be a viable option.

Figure: Spine of Peek-a-Bear (Saka ,2010) 

It can be observed that there is a "spine" mechanism on the x-z axis is a four-link slider-crank mechanism where the output is the bear's head moving in a linear path. The orientation of the slider-crank linkage to the hands' movement made it just as if not more difficult as modelling the system in two different planes. 

Second, the #2 approach is explored:

By simplifying the kinematics of the hands and the head as simple joints with a single degree of freedom, we can analyze the motion of the head and the motion of the hands separately. 

Figure: x-z plane of the bear 

Figure: y-z plane of the bear 

An important detail here is that the shoulders of the Peek-a-Bear were simplified to a slider joint. This was possible because observation of the shoulders' movement showed that they approximately follow a linear path. 

Figure: Overlay of initial and final position of the shoulder

In consideration of the hand positions, the angle of the link that connects the shoulders to the must also be considered because the Peek-a-Bear's arms are rotating with the shoulder links.

Figure: Overlay of initial and final angles of the shoulder (red) and hands (black)