Flexible materials

In some applications there is motivation to move beyond linear transformations. Imagine trying to warp a flexible material, such as a mattress, through a doorway. The mattress could be approximated by a 2D array of links; however, the complexity and degrees of freedom would be too cumbersome. For another example, suppose that a snake-like robot is designed by connecting $100$ revolute joints together in a chain. The tools from Section 3.3 may be used to transform it with 100 rotation parameters, $\theta_1$, $\ldots$, $\theta_{100}$, but this may become unwieldy for use in a planning algorithm. An alternative is to approximate the snake with a deformable curve or shape.

For problems such as these, it is desirable to use a parameterized family of curves or surfaces. Spline models are often most appropriate because they are designed to provide easy control over the shape of a curve through the adjustment of a small number of parameters. Other possibilities include the generalized-cylinder and superquadric models that were mentioned in Section 3.1.3.

One complication is that complicated constraints may be imposed on the space of allowable parameters. For example, each joint of a snake-like robot could have a small range of rotation. This would be easy to model using a kinematic chain; however, determining which splines from a spline family satisfy this extra constraint may be difficult. Likewise, for manipulating flexible materials, there are usually complicated constraints based on the elasticity of the material. Even determining its correct shape under the application of some forces requires integration of an elastic energy function over the material [577].