Dynamic Underactuated Manipulation:

 

Here we consider the problem of controlling an underactuated dynamical system with only one single input. An example of such a system is a ball rolling inside an elliptical bowl. The elliptical bowl can move in only one direction which is not aligned with any of its principal axes. A schematic diagram of the system is shown below:

 

Note the direction of motion of the bowl is not aligned with any of its principal axes.

 

Goal:

Initially we aim at showing that the underactuted dynamical system is controllable. After that we wish to control the motion of the ball by controlling the single input (which is the motion of the bowl). More specifically, the planning problem which we are solving is:

 

Given an initial configuration of the ball, plan a set of input motions of the bowl which will take the ball to a desired goal configuration

 

Results:

Controllability:

Using results from differential geometry and nonlinear control theory we have shown that the above dynamical system is globally controllable with a single input. More details regarding this paper are available in ICRA 2000 paper.

Planning:

To validate our planning experiments we have build an experimental setup. The orienation of the ball is determined with a vision system which consists of a overhead camera. A snapshot of the setup is shown below:

 

The set of motions required to take the ball to the desired goal configuration is shown below. The change in orientation of the ball after each input step is shown with an arrow. An example is shown in the figure below:

 

 

Here (a) Initial configuration of ball given by quaternion q = (1.0, 0.0, 0.0, 0.0); (b) Configuration of ball after first step, q = (0.94, -0.22, 0.15, -0.17); (c) Configuration of ball after second step, q = (0.71, -0.45, 0.35, -0.39); (d) Final configuration of ball reached after third step, q = (0.55, -0.55, 0.37, -0.46)

 

All the details related to the planning (shown above) is available in the CCA 2001 paper.

 

Keywords: robotics, nonholonomic planning, nonlinear control, underactuated systems, motion planning