Computer Graphics Forum, 2(4), October 1992

8

3.1

Joint motion as a reference model

The basic idea is to consider the initial joint motion as a reference model whose tracking is

enforced through the secondary task while the main task insures the realization of desired

cartesian constraints over

some

specified

end

effector(s).

The

simplest

approach

is

to

minimize a cost function of the squared distance between the current state of the reference

configuration and the corrected configuration. The gradient vector providesthe direction of

the highest slope, which is twice the joint difference vector, and we limit its norm to some

predefined value to insure the small movement hypothesis (figure 2).

Figure 2 : Reference configuration tracking as a secondary task

As the secondary task is performed on the null space of the

linear

transformation,

its

dimension is of first importance in tracking the reference model. The more the user adds

constraints the less he can retrieve from the reference motion. This brings up two issues.

First, the user should work in an incremental manner, beginning with a one-dimensional

constraint, processing it and then adding another constraint over the resulting motion, and so

on. This methodology always assures the high dimension null space necessary for secondary

task performance. Second, local constraints are preferred for small motion deformation and

as such half-space constraints appear to be the best choice.

3.2

Half-space as a convenient class of cartesian constraints

Design of one-dimensional to six-dimensional trajectories for an end effector is particularly

tedious with respect to our objective of joint motion deformation. The user is more interested

in local reshaping of the motion both in time and space. For this reason we prefer to use half-

space constraints as in the example shown on figure 3. Inverse kinematic control is only

enforced when the end effector tries to enter the forbidden zone. Planar half-spaces are very

convenient and will be used throughout this paper but we will also consider cylindrical and

spherical half-space constraints for position or orientation (figure 4).