To control robots many compromises are made. For example, the simplicity of the control technique is quite often put ahead of efficiency and speed. The computational requirements for complexcontrol schemes require either extensive computation time, or prohibitively expensive computers. Many of the problems encountered with exact methods can be overcome using approximate techniques.
Some exact methods for robot control have been explored. For example, a paper by Rajan  describes a method where robot paths are modelled in joint space using splines. He then uses Bobrow’s algorithm  to determine the optimal trajectories along that path, and thus the optimal path time. Then, to obtain the optimal motion (path and trajectories together), Bobrow adjusts the path splines using a gradient descent technique. This approach will produce good results if a good initial path is suggested. But, this method is plagued by calculation times that are to long, and with varying computation time. This means that while the method is valid, and produces good results, it is not suitable for real-time control in non-repetitive motions. The same problem occurs with other iterative methods.
Miller et. al.  mentions that methods exist for learning to improve a path every time it is traversed. The methods can be useful when attempting long-term optimal control, without exect mathematical models, but are not suited to non-repetitive tasks. Jack  used a feedforward neural network to approximate the optimal motion. The training data for the neural network was developed using numerical optimization of robot joint positions modelled with splines. This method showed great promise, but the neural network solution was not able to extrapolate outside the provided training examples.
The non-linear nature of robots makes it difficult to develop exact techniques for control. Linear control systems are well understood, and can be applied for robot control, but a worst-case dynamic configuration must be used to linearize the controllers. This leads to robot controllers that overshoot in some regions of the workspace, and are overdamped in others. To compensate for this the modern approach is to use non-linear control schemes, such as MRAC (Model Reference Adaptive Control) to adjust the control parameters over the robot workspace. This results in a form of locally-optimal control. While the robot follows the given path in an optimal method, there is an assumption that the path is optimal. This will produce globally sub-optimal motions.
The case in point is seen in Rajan . He develops optimal motions for the robot, as pictured below in Figure 1. Notice that the only one joint angle is changed in global terms, but the optimal solution also bends the elbow joint to reduce the inertial effects of the arm. This results in a path time reduction from 7.38 to 4.37 seconds. Rajan’s results are important in indicating the optimal path, but this method is hindered by the computational time required.
The robot control problem chosen for this research is the 2 degree of freedom planar manipulator, as done by Rajan . This provides results for comparison to numerically optimal values. The 2 degree of freedom manipulator is complicated enough to incorporate the non-linear dynamics of the arm, as well as allowing future addition of other factors such as gravity, and variable payloads. The robot model uses the parameters as given below, with joint torque control (with maximum torque limits).
When possible, mathematical methods will always provide excellent solution. In some cases there are non-exact properties that make a problem impractical with exact mathematical techniques. The original concept of fuzzy logic was formalized by Lofti Zadeh in the mid-1960s . The application of fuzzy logic to control problems was first tried in the mid-1970s to control of cement manufacturing kilns. This application was tried with mathematical rules with varying success, but when Mamandani et. al. [1985?] applied fuzzy logic the results we immediate and successful. Since then there have been a number of equally succesful applications. The main application characteristic is that the problem should have non-crisp rules that are easy to understand, but hard to express mathematically. This feature makes fuzzy logic well suited to the optimal control problem described above. This is obvious when trying to describe the optimal motion of the robot. In particular, a computer would require a few minutes to develop the path above, but all the optimal motion is doing is “bending the elbow to reduce the inertial effects”.
A set of rules was developed to describe the optimal motion of a robot arm. In summary these rules capture the ideas that joint torques are turned on fully for large moves. The smaller joint motions result in lesser torques. In addition. Over large motions the elbow will be bent forward or back for a majority of the motion to reduce inertia. The decision variables for the rules are,
Each of these states calls for different movement strategies. For example, when the elbow is the only joint to be moved, the robot motion can use a simple linear control scheme. When the arm is near steady state, the linear control laws schemes are still useful, but the should joint parameters must be tuned to adapt for various elbow angles. Long moves involving the soulder of the robot should bend the elbow to move it to a lower dynamic position during the move. By reducing the dynamic effects, the arm can move faster.
Miller, W.T., Hewes, R.P., Glanz, F.H., and Kraft L.G., “Real-Time Dynamics Control of an Industrial Manipulator Using a Neural-Network-Based Learning Controller”, IEEE Transactions on Robotics and Automation, Vol. 6, No. 1, 1990.