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Biomimetic and Biologically Inspired Control 409

Figure 16.5 Resultant map of the 3 D.O.F. robot reaching its end-effector onto different positions of x space with
different configurations. For the smooth change of the end-effector’s position, the robot’s joint angles are also
changed smoothly.

16.2.3.3 Diffusion-Based Generalization of Optimal Control
Diffusion-based learning can also be effectively applied to generalize an optimal control for a robot
manipulator (Luo et al., 2001).
Generally, in optimal control we have to solve a two-point boundary value problem with respect
to increase and decrease of time. However, it is very difficult to solve it analytically, especially for a
nonlinear system like a robot.
By now, there are many numerical approaches to solving the optimal control problem for a
given set of initial and terminal conditions. However, these approaches require enormous
computations. For every change in the initial and terminal conditions, they have to perform the
complex computation again, which make it difficult to realize the optimal control for the robot in
real time.
In our approach, we assume that, for some initial and terminal conditions, we already obtained
the optimal solutions. Then, by using the diffusion-based learning algorithm, these optimal solu-
tions can be generalized overall the bounded task space. For example, as shown in Figure 16.6, we
assume that if for the initial S and four terminal conditions of T 1 to T 4 , the optimal control inputs