Configuration Space

Suppose we have a robotic arm that mimics the movements of a person’s arm. The arm exists in the familiar 3-D physical space. Consider a simplification of the robotic arm that assumes just three connected parts: upper arm, forearm, and hand, all in the shape of straight rods that are connected. The body of the robot, including the shoulder, remains fixed in position. We wish to have the robotic arm touch a particular pointlike object in the room. How many numbers are required in a computer program to describe the arm position?

Answer

There are many ways to approach this problem of describing the arm position in physical 3-D space. We consider one approach only. In all approaches, the end of the rod like hand must touch the specified point, so three numbers define the end point of the hand.

Let’s start at the fixed shoulder position. Two numbers will describe the upper arm position, the angle in the vertical plane measured from a fixed vertical axis through the shoulder, and an angle about this vertical axis. Two more numbers describe the forearm position, an angle in the vertical plane measured from a vertical axis through the end of the upper arm, and an angle about this vertical axis. Likewise, two more angles are needed for the hand. At least six numbers are necessary for the robot to locate the particular point in the room. The program will calculate the extent of the arm to determine its end point distance, thus requiring three more numbers, the lengths of the three parts. The space of operation is nine-dimensional and is called a 9-D configuration space to distinguish between physical space and coordinate space. Of course, one could have determined this result by realizing that each rod end point requires three coordinate values to be specified.

The movement of the arm to touch the point is the next challenge. If feedback exists in the robot, such as visual feedback of the hand position and the desired point location, the movement algorithm can use a correction procedure that becomes finer and finer as the fingertip approaches the point, as we humans tend to operate. If there is no continual feedback mechanism, then the algorithm must move the arm to the point directly, somehow knowing where the fingertip is at all times. A systematic error cannot correct itself if no feedback exists. Many robotic arms operate in both modes, first without feedback for rapid deployment and then with feedback for fine adjustment. As humans, we learn to perform many tasks and do many of them several times daily. As a result, we often forget how we learned a particular procedure and how much practice was required. To relive that learning experience, try using the “other hand” to punch in data in a calculator, or some similar task. The learning curve is sometimes very steep!