Human-machine Cooperative Manipulation with Vision-based Motion Constraints - Visual Servoing via Advanced Numerical Methods - page 53

robot equipped with a force sensing handle at the endpoint. Tools are mounted at
the endpoint, and “manipulated” by an operator holding the force handle. The robot
responds to the applied force, thus implementing a means of direct control for the
operator. The robot has been designed to provide micron-scale accuracy, and to be
ergonomically appropriate for minimally invasive microsurgical tasks [30].
In this section, we introduce the basic admittance control model used for the
SHR, extend this control to anisotropic compliances, and finally relate anisotropic
compliances to an underlying task geometry.
In the remainder of this chapter, transpose is denoted by , scalars are written low-
ercase in normal face, vectors are lowercase and boldface, and matrices are normal
face uppercase.

4.2.1 Virtual Fixtures as a Control Law

In what follows, we model the robot as a purely kinematic Cartesian device with
tool tip position x ∈ SE (3) and a control input that is endpoint velocity v = x ∈

6 , all expressed in the robot base frame. The robot is guided by applying forces
and torques f

ℜ

6

∈ ℜ on the manipulator handle, likewise expressed in robot base

coordinates.
In the steady-hand paradigm, the relationship between velocity and motion is
derived by considering a “virtual contact” between the robot tool tip and the en-
vironment. In most cases, this contact is modeled by a linear viscous friction law

k v = f , (4.1)

or equivalently

v = 1

k f

, (4.2)

where k 0 controls the stiffness of the contact. In what follows, it will be more

convenient to talk in terms of a compliance c

>

≡ 1 / k .

the effect is that the manipulator is equally compliant in all
directions. Suppose we now replace the single constant c with a diagonal matrix
C .Makinguseof C in (4.2) gives us the freedom to change the compliance of the
manipulator in the coordinate directions. For example, setting all but the first two
diagonal entries to zero would create a system that permitted motion only in the x - y
plane. It is this type of anisotropic compliance that we term a virtual fixture. In the
case above, the fixture is “hard ” meaning it permits motion in a subspace of the
workspace. If we instead set the first two entries to a large value, and the remaining
entries to a small one, the fixture becomes “soft.” Now, motion in all directions
is allowed, but some directions are easier to move in than others. We refer to the
motions with high compliance as preferred directions, and the remaining directions
as nonpreferred directions.

When using (4 . 2) ,