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Biomimetic and Biologically Inspired Control 405

Comparing with supervised learning, the self-organization approach does not depend on
any external teacher. It focuses on the spatial order of the input data and organizes the learning
system so that the neighbor nodes have the similar outputs (Amari, 1980; Kohonen, 1982). By
considering spatial characteristics of motor learning, self-organization algorithm has also been
extended to generate the topology conserving sensory-motor map (Ritter et al., 1989). In this
approach, we first construct a three-dimensional lattice, and specify the sensory input vectors, the
corresponding inverse Jacobian matrixes, and the joint angle vectors to each node within the lattice.
The lattice then outputs desired joint angles for the arm to perform many physical trial motions. For
each trial motion, a visual system is used to input the end-effector position of the arm in the task
space. The algorithm is then used to search for a winner node with its sensory vector closest to the
visual input. After that, the sensory vector, the inverse Jacobian matrix as well as the joint angle
vector of the winner node, together with that in its neighbor nodes, are adjusted, respectively. The
neighbor region of adjustment decreases as the learning proceeds. As a result, the vectors (or a
matrix) in one node are similar to that of its neighbor nodes. That is, a topology conserving map is
self-organized without any supervisor’s command. In this algorithm, for every adjustment step, the
arm has to perform the real physical trial motions. Since it is still within the learning process,
sometimes these trial motions are dangerous or may be impossible due to the incorrectness of the
map. In addition, both in searching the winner node as well as when adjusting the neighbor nodes,
the approach requires a centralized gating network to interact with all nodes, which makes the
learning algorithm centralized and not parallel as seen from the computation point of view. Finally,
besides the fact of topology conserving, we could not obtain any information about the map’s
spatial optimality.

16.2.3 Diffusion-Based Learning

Researches on motor learning of biological system are not limited to the two learning approaches
in above subsection. In order to overcome their drawbacks, we presented a diffusion-based motor
learning approach, in which each neuron only interacts with its neighbor neurons and generates a
sensory-motor map with some spatial optimality.
In detail, we consider the spatial optimality of the coordination: to minimize the motor control
error of the system as well as the differentiation of the motor control with respect to the sensory
input overall the bounded task space. By using variational calculus, we derive a partial differential
equation (PDE) of the motor control with respect to the task space. The equation includes a
diffusion term. For the given boundary conditions and the initial conditions, this PDE can be solved
uniquely and the solution is a well-coordinated map (Luo and Ito, 1998).
From the motor learning point of view, our approach contains both the aspects of supervised
learning and self-organization. Firstly, we assumed that the forward many-to-one relation from the
hand system’s motor control to the task space sensory input can be obtained using supervised
learning, and at the boundary, the supervisor can provide correct motor teacher information. Then,
by evolving the diffusion equation, we can obtain the sensory-motor coordination overall the
bounded task space.

16.2.3.1 Robotic Researches of Kinematic Redundancy

Before describing diffusion-based learning, we first briefly review the redundancy problem
and summarize previous robotic approaches. Without losing generality, we only consider the
kinematic nonlinear relation between the work space and the joint space which is represented as

x ¼ f(u) (16:1)