This is a kind of motion called Inverse Kinematics and it is a quite popular way to make rag dolls in games. It uses a design pattern called the Composite Pattern where one object adds another object as a child of its and then when it's update() function if called, it calls all of its (usually one) child's update() functions. The most common example of this is of a snake. The snake's head follows your mouse, and the rest of the snake's body pieces move with the snake, and it looks immensely realistic. This exact example is explained and build here although it does not include joint restrictions at all.
This example is in the middle of a book, and so may be hard to start reading, but if your somewhat familiar with design patterns and/or have some intermediate experience with programming, then i'm sure you can understand it. I advise that you, after reading and understanding the example, scratch what you have now because it is not very elegant coding. You may feel that this example uses too many classes, but trust me, its worth it as it allows you to very easily edit your code, if you decide to change it in the future, with no drawbacks.
Also, i know that this snake is not what you want, but if you understand the concept then you can apply it to your own specific needs.
I hope this helps.