The model relating robot control signals
to the features we
observe
, can be expressed as a (multidimensional) Taylor expansion,
given n+1 continuous derivatives in an interval about
:
(Usually
is a vector and f is vector valued. In this case derivatives of f
are called Jacobian, Hessian and so on.)
In this thesis we restrict our attention to linear models:
where for a sufficiently "smooth" f the term
is small for
.
Depending on the rank of J and how many control parameters
and feature observations
we have, we call the system exact, over determined,
under constrained or both of the last two (
).
The last case deserves a little further elaboration. In practice this can
have two causes: We may be unfortunate so that all our observation may fall
into a too low dimensional
space. For example if we observe a manipulator controllable in three
dimensions with a camera, and control on the movement of one point
in the image we have no control over the third dimension, and thus arbitrary
drift can occur. The solution is to measure more independent features.
The second case is when the manipulator has more degrees of freedom than the control problem. A common case is that one wants to move a rigid object in 6 DOF with a manipulator with more than 6 DOF. A reason for having more DOF on an arm type manipulator than 6 can be to achieve a larger operating area. In this case the problem can be simultaneously over determined and under constrained. (eg. J is singular when rank(J)<n=m)
The natural approach for redundant manipulators is to explicitly constrain the DOF's. One example is already given. The basis function manipulations of Nelson and Fuentes [Fuentes and Nelson, 1994] is just such a system. In other cases linear constraints may do. To for instance control the Puma robot in a two DOF system, we may want to use all the three first joints to achieve a bigger operating space. One mapping which does this on the Puma robot is:
Sometimes the issue of what is a constraint and what should be controlled is not immediately obvious. Consider for instance the task of balancing a tray of objects on the Puma's end effector. We want to move around without dropping the tray or objects. Either we can constrain manipulation so tilting the end effector is not possible, or we can observe the angle of the tray (preferably with some special sensor on the tray, since it is doubtful that vision would be stable enough for the task), and control it. If we have two separate control loops, the inner keeping the end effector horizontal, and the outer moving the arm around, then we have achieved the constraint with a control loop. (Note the similarity to the constraint of holding on to an object with all fingers in the case of doing fingertip manupulations of a grasped object.)
Consider case 2 above where rank(J) < m. Why would we want to observe more measures than needed to get a nonsingular control problem? The argument for doing so is robustness. The argument against is computational cost. There are strong indications that human control takes advantage of any sensory input correlated with the task. Evidence supporting this include the method we believe is used for human learning, and that humans, when impairing one sensory system, most of the time without much performance degradation can do the task using the remaining sensory systems. In the perception vector we can take advantage of redundant measures for noise reduction and sensor error suppression (identifying outliers). Another advantage particular to vision based control is that with multiple redundant features we can deal with occlusion by simply dropping a feature while it is occluded and adding it back in when it reappears.
A somewhat different reason relates to how we choose our sensory signals, and the perception vector. If we want to control a n DOF system with n = m features, then we have to be very careful to choose linearly independent features. If on the other hand we randomly choose very many visual features (m >> n) we are likely to among those have several groups of n which are linearly independent.