Chapter 2. Prehension

Chapter 5 - Movement Before Contact 133
above. A training set is presented and joint angles computed for the
current weights. Using the Model Network, a manipulator end point is
computed for this joint configuration (the weights in the Model
Network stay fixed during this second phase). The computed end
point location is compared to the desired goal location, and if there is a
difference, the weights in the Sequence Network are modified, again
using the generalized delta rule. This sequence learning phase took 77
cycles to find a solution for a sequence that involved touching four
goal locations in a counterclockwise order. By shifting the second
goal and asking the manipulator to perform the task again, only 5 trials
were needed.
Learning using the generalized delta rule involves updating the
weights by subtracting the actual output from the desired output and
adjusting the weights so that the error between the two will be
reduced. Jordan added intelligence, so that weights are adjusted
according to various rules and constraints. If the result of the
computation satisfies the constraint, the normal error propagation is
performed; if the constraints are not met, the weight is not changed.
Jordan used two types of constraints. Configurational constraints
defined regions in which an output vector must lie. These included
don’t care conditions, inequalities and ranges, linear constraints,
nonlinear constraints, and other optimality criteria. If it doesn’t matter
that an error exists, the weight is not changed. With inequality
constraints, the weight was changed only when a specific inequality
existed between the computed value and the desired value, for
example, the computed value is larger than the desired value. Range
constraints were used when the error could lie within a defined range
of two values. Linear constraints added two or more units in a linear
fashion, apportioning the error between them. Temporal constraints
defined relationships between outputs produced at different times.
Whereas configurational constraints arise from structural relations in
the manipulator and/or the environment, temporal constraints arise
from dynamical properties of the system. An important one is a
smoothness constraint which allowed configuration choices to be
made in the context of the arm’s location. For this, units in the
Sequential Network stored their activation value at the previous time
step so that inverse kinematic solutions for points nearby in time
differed minimally.
As mentioned, a problem with inverse computations is that there
are non-unique solutions. Jordan turns this problem into a feature by
using these constraints to maintain multiple inverse kinematic
solutions. In Figure 5.9a, the network has learned a sequence of four