Abstract
As society uses computers as the catalyst to progress through the greatest technological revolution ever known to mankind, the field of robotics has had enormous advancements in recent years.  We have started a trend of the assigning of what use to be strictly human workload to non-human entities.  Not only have our metal and silicon friends proven to be effective, but, they are fast and efficient.  Not to destroy the spirits of blue-collar workers, but global society is becoming more automated every day.

The simultaneous revolution of computing power and robotic applications is not a coincidence.  Manipulating a robot takes enormous computing power.  In fact, lack of powerful enough computers is usually the limiting factor of practical robotic applications.  What I have done in this paper is attempt to compute the inverse kinematics of two specific models (PUMA robot and the Tetrahedron Tripod) using Mathematica.

The PUMA algorithm would take a four by four matrix (called the Denavit-Hartenberg matrix) as input parameters and it would animate a PUMA robot.  Since all calculations are done symbolically, the parameters and orientation of the robot are at the discretion of the operator.  The Tetrahedron Tripod only needs three parameters to do a complete kinematic analysis:  initial point, end point, and the time elapsed to complete the path.  A much more in depth analysis was done on the Tripod.  The length, velocity, and acceleration was computed and graphed for each of the three legs.  Also, straight and circular trajectories were computed and animated.

Introduction
With the pressing needs for increasing productivity and quality end products, industry is turning more and more toward computer-based automation, specifically the use of computer-controlled robots.  Most automated manufacturing tasks are done by special purpose machines designed to  perform predetermined functions in a manufacturing process.  The inflexibility of these machines makes the computer-controlled manipulators more attractive and cost effective in various manufacturing and assembly tasks.

An industrial robot is a general purpose manipulator consisting of several rigid links connected in series by revolute or prismatic joints.  One end of the chain is attached to a supporting base while the other end is free and attached with a tool to manipulate or perform assembly tasks.  The motion of the joints results in relative motion of the links.  Mechanically, a robot is composed of an arm (or main frame) and a wrist subassembly plus a tool.  It is designed to reach a workpiece located within its work volume.  The work volume is the sphere of influence of a robot whose arm can deliver the wrist subassembly unit to any point within the sphere. The arm subassembly consists of a three degrees of freedom movement.  The combination of the movements will place or position the wrist unit at the workpiece.  The wrist subassembly unit usually consists of three rotary motions.  The combination the these motions will orient the tool according the configuration of the object for ease in pickup.  These first three motions are often labeled as pitch, yaw, and roll.  Hence for a six-joint robot, the arm subassembly is the positioning mechanism, while the wrist subassembly is the orientation mechanism.

 I am exploring the inverse kinematics of parallel and serial linked robot systems.   Robotics is defined as the science of automated devices used to replace live workers. You tell a worker what to do and they do it.  You tell a robot what to do and they will do it.  Sounds pretty easy.  Essentially, that is all it is.  The major difference is how the input is processed to render an output.  The human brain is almost infinitely more complex then any robotic processing unit.  Telling another human what to do is very trivial because we use the same form of communication.  Robotic programmers realize how much we take this for granted.  Robots are very dumb.  Every moving part on the robot must be explicitly told what to do;  And communication between the robot and the programmer must take place in the language of the robot.  The robot cannot think, so everything that the robot does is a direct result of the programmer.

Project Overview
The project was funded by the NSF and the department of mechanical and industrial engineering at the University of Illinois and was supervised by Wolfram Research (creators of Mathematica).  All of the projects shared the common theme of symbolic computing in engineering.  I chose the project that dealt with developing software that that will compute the trajectory of a robot given a starting and ending points (often referred to as the inverse kinematics).

Inverse kinematics is the process of computing the joint parameters by knowing the position of the end effector.  The inverse kinematics of a robot manipulator demands the transformation of Cartesian coordinates into joint coordinates.  The desired motion of the end-effector is transformed into joint trajectories since these are the variables under control.  The common approach for solving this problem is to obtain a closed-form solution to the inverse transformation.  However, the closed-form solution to the inverse kinematics problem exists only for certain classes of robot mechanisms.  Such mechanisms contain special arrangements of degrees of freedom.  For example, for any six-derees-of-freedom mechanism with an arbitrarily chosen major linkage and minor linkage, in which the joint axes intersect in a point, the closed-form inverse solution exists.  The mechanisms for which the closed-form solution can be found normally contain parallel revolute joints or more than one translation, or the have less than six degrees of freedom.  But, for the PUMA the solution to the inverse kinematics problem is multivalued.  In general, it requires iterative, time-consuming numerical procedures, which may not find all the solutions and may only give approximate values for those they do find.

Because of its great ability to work well with matrices and symbolic computations, Mathematica has proven to be very useful in this project.  The program that I wrote on Mathematica will take the position and orientation of the end effector as an input, and animate a PUMA robot arm as the output.  The PUMA robot is a 6-degree of freedom robot with three joints in the wrist and the other 3 as part of the arm.  An advantage to having a wrist with all three z-axises intersecting at a common point is that the problem can be broken down into two parts.  The first part solves.  All six of the joints are revolute joints, so each link of the robot can only rotate about its joint.  Most general purpose robots have 6-degrees of freedom, because you need all six degrees to insure that any vector can be reached in 3-dimensional space.

The PUMA is a fairly sturdy and reliable robot.  It is probably the most widely studied robot because of its great versatility.  But sometimes even the PUMA or any serial linked robot won't meet the demands of some specific task.  Sometimes it is impractical to build a serial linked big and heavy enough to handle heavy loads or repetitive precision.  This is when the exciting field of parallel linked robotics comes to use.
 
I use the word "exciting" because this a very new and unexplored field.  Relatively little research and publications have been contributed to this field.  As you may have guessed, the obvious difference between the serial linked robots and the parallel linked robots is the relative positioning of the links.  Serial linked robots (like the PUMA) have joints and links connected in series to one another.  These robots have proven to be fast and efficient.  But, they traditionally have not been able to handle heavy loads.  Also, the accuracy can easily diminish with excessive use.  Parallel linked robots are designed to pick up where serial link robots left off.  Although they do not have the speed of the serial linked robots, accuracy and heavy load havkling are strong points of parallel linked robots.

The specific model of analysis has been named the tetrahedron tripod.  It was designed at the University of Illinois by Professor Placid M. Ferreira and graduate student Bashar S. El-Khasawneh.  The mechanism looks like a tripod with a equilateral triangular platform connected by the three legs which are connected to the base.  The three legs give it its three degrees of freedom.  It moves spatially with no rotations.  The parallel linked manipulator mechanism generates a controlled three-dimensional translational motion.  The platform always remains parallel to the base and each side of the platform remains parallel to its corresponding side of the base.  The mechanism consists of a base holing three non-parallel cylindrical joints.  The traveling part of the cylindrical joint is connected to the prismatic joint.  The other part of the prismatic joint is connected to the moving polatform via a revolute joint.  The three prismatic joints are the only controlled joints.  The other joints are free to move to retain the mechanisms constraints.

The inverse kinematics of the tripod, like its relatives of the parallel linked manipulator family, does usually have a unique real solution defined by the workspace of the manipulator.  To analyze the kinematics of a path of the manipulator one only really needs the initial point and the end point of the path.

The major product of this project was the development of Mathematica code that could successfully compute and simulate the trajectory of the tetrahedron tripod given the initial and ending points.  The tetrahedron tripod is currently in used for high-precision manufacturing.

last modified 3 august 2000