Robot and Machine Kinematics

Kinematics is the study of motion without consideration of the forces that cause those motions. Robots can have many joints, sometimes called degrees-of-freedom or DOF, which makes their kinematics quite interesting. Robot kinematics falls into two categories: forward and inverse. The solution to the forward kinematics problem is not difficult. It can be found with common matrix transformations. The solution to the inverse kinematics problem can be quite difficult. In fact, the general solution to the inverse problem is known as the "Mt. Everest of kinematics problems."

We can often use examples with our own bodies when describing concepts in robotics. This makes them easy to grasp for most anyone. The forward kinematics problem can be stated as follows: Given all of the robot's joint displacements (angles and lengths), find the position and orientation of the robot's hand." Using the human arm as an example the problem can be stated as: "Given the angles at the shoulder, elbow and wrist; where is the hand?" The solution to this problem mathematically can be found using standard six-by-six transformation matrices. Denavit and Hartenberg used screw theory to show that there was a more efficient solution using only four-by-four transformation matrices. Denavit's and Hartenberg's parameters have become the standard for describing robot kinematics.

We seldom concern ourselves with the forward kinematics problem during day-to-day activities. The inverse kinematics problem is much more interesting. Here is a statement of the inverse kinematics problem: "Given the position and orientation of the robot's hand, what are all of the joint displacements (angles and lengths)?" As we said previously, the general mathematical solution to this problem is the Mt. Everest of kinematics problems. The really interesting part is that your brain solves this problem all the time. When you go to pick up your fork you just stick your hand out and it goes where you want it. You don't think "I've got to move my shoulder this much and my elbow that much, etc." Even more interesting is that we actually solve an even more complex form of the inverse kinematics problem known as the redundant inverse problem. The problem becomes redundant when the robot has more joints than constraints (typically more than six joints). This gives the robot the ability to perform the same task in more than one way. The robot can use it's extra joints to reach around obstacles and such.

 

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