Information on how each of the experimental conditions functions is
critical for correctly interpreting the experimental results.
A formal and concise method, based on a combination of mathematical
models and the state-transition diagram of input devices (Buxton,
Hill, and Rowley, 1985), will be applied in this section to describe
the mechanism of each of the four input techniques in Experiment
1. The basis of the mathematical formulation used here can be
found in numerous books in mathematics (e.g. Altmann 1986) (mathematically
most rigorous), in aerospace engineering (e.g. Hughes 1986) (comprehensive
and practical) and in computer graphics (e.g. Foley, van Dam,
Feiner, and Hughes 1990) (introductory).
A1.1 Isotonic Position Control
While using the isotonic position control technique, the user operated in one of two interaction states (Figure A1.1). When the user's hand was open (State 1), the hand movement did not have any effect on the cursor. The user could re-position or re-orient the hand location in this state without affecting the cursor's position. When the user's hand was closed (by pressing the clutch down), the controller became engaged and the manipulated object would be slaved to the hand motion (State 2). In State 2, when the hand reached its limit (rotational or translational), the user had to release the clutch to enter state 1 and reset the hand gesture. This process might have to be repeated a few times when the object needed to be rotated, for example, 180 degrees around the vertical axis.
More formally, state 1 is a zero order hold. The cursor object remains unchanged, regardless of the hand gesture change, i.e.,
where X(k) is a 4 X 4 homogenous transformation matrix representing the location and orientation of the cursor; k is the sampling step
t is the current time; S = 0.0667 (second) is the sampling period; k1 is the step of entering state 1.
In State 2, the cursor movement will 'copy' the hand's movement relative to the initial hand gesture at moment t = k2S, i.e,
Where k2 is the beginning step of state 2. T is the transition matrix from the hand state at time k2S, represented by a 4 X 4 homogenous matrix H(k2), to hand state at the current moment represented by H(k), i.e.,
Matrix H(k) is formulated according to the following equation* :
(A1.5)
where 
are 6 DOF data (scaled by control
gains) collected from the hand tracker at step k, representing
positional and angular displacements of the tracker from the transmitter
(signal source). 
is the translation along
the horizontal axis, 
is along the vertical
axis and 
is along the depth axis. 
are rotations about x, y and z axes.
Similar to the isotonic position control, the user operated in one of two interaction states with the isotonic rate control scheme (Figure A 1.2). When the user's hand was open (State 1), the hand movement did not have any effect on the cursor. When the user's hand was closed, the controller became engaged but the cursor velocity, rather than the cursor position, was proportional to the hand displacement (State 2).
More formally, in State 1:
Where k1 is the beginning step of state 1 and k is the current step. Again, state 1 is a zero order hold.
State 2:
Where k2 is the beginning step of state 2. T(k) is the transition from hand status H(k2) to current hand status H(k), i.e.,
H(k) is formulated in the same way as in equation (A1.5).
Note that equation (A1.8) is different from equation (A1.3). In
equation (A1.3), hand movement is mapped to the amount of cursor
movement. In equation (A1.8), hand movement is mapped to the velocity
of cursor movement.
A1.3 Isometric Position Control
The 6 DOF isometric sensor used in the experiment was a SpaceballTM.
When used for isometric position control, the button on the Spaceball
was employed to switch between two states of operation (Figure
A1.3). When the button was pressed, the controller became engaged
and the cursor's movement will be proportional to the force/torque
that the user applied to the ball. Once the button was released,
the cursor remained where it was. The user might have to switch
between the two states several times in order to move the cursor
over large distance without excessive force/torque.
More formally, in State 1:
Where k is the current step and k1 is the beginning step of state 1.
In State 2:
Where k2 is the beginning of state 2. T is the transformation
matrix based on the force vector 
and torque
vector 
of the Spaceball at time t = kS,
scaled by control gains.
A1.4 Isometric Rate Control
The isometric rate control was a single state scheme (Figure A1.4). The object's velocity was proportional to the force/torque applied on the Spaceball.
More formally, in State 1
Where T(k) is defined in equation (A1.11).
(A1.11)