full Paper - Nguyen Dang Binh

Table 4: Master and Slave position vectors
Symbol Meaning
P M P S
R M S
O M S
position of spherical joint centre
respect to local frame of master
and slave arm
rotation matrix from master’s
local frame
to slave’s one
position vector of slave’s local
frame respect to master’s one
Since the control procedure will provide to use the right
arm as motion driver, in the following it will be indicated as
master arm and the other one as slave.
As mentioned before, the control procedure will provide
to store the positions of the path points expressed respect to
local frame of two arms (Table 4).
By using any free set of three among the stored points (P M 0
P M 1 P M 2 respect to master’s frame and P S 0 P S 1 P S 2 respect to
slave’s one), it is possible identify a mobile reference frame
which is uniquely identi£ed in both reference systems (Σ0
and Σ1).
Such a system can be used to £nd the rotation matrix between
master’s and slave’s local frames by the following procedure:
1. using stored point it is possible to determine two intermediate
frames with versors
i =(P1 − P0)/|P1 − P0|
Massimo Bergamasco / Haptic Rendering: Control Strategy
j =(P2 − P0) − ((P2 − P0) ·
i)i k =i ×j
2. the rotation matrixes between the master’s local frame
and the intermediate frame and between the slave’s local
frame and the intermediate frame simply result
R M I =[i M ,j M ,k M ] R S I =[i S ,j S ,k S ]
3. the rotation matrix between the master’s local frame and
the slave’s local frame simply results
R M S = R M I (R S I )T
4. Once rotation matrix is known, by using one of three
stored points P0P1P2 (for example P0) it is possible to
£nd the origin position vector of slave’s local frame respect
to master’s one:
O M S = P M 0 −R M S P S 0
It worths to mention that the procedure is independent from
the speci£c trajectory used to £nd the six points set (P M 0
P M 1 P M 2 P S 0 P S 1 P S 2 ). Therefore even in presence of unaccurate
position control the calibration procedure can provide
exact calibration provided that a synchronized and accurate
measure of these points is available.
64
Errors in evaluating stored vectors position can occur
when the procedure is carried on. The reasons could be
within the following ones:
mechanical device could be different from simpli£ed
kinematic model:
– dimensions of mechanical parts may slightly differs respect
to design speci£cations;
– the mechanical device is not absolutely rigid therefore
elasticity of transmissions and of structures may alter
the effective measurement;
– backlash of coupled mechanical parts;
error in angular offsets evaluation, that will led to imprecise
calibration procedure and digital resolution of incremental
position sensors can affect computational procedures.
In order to reduce calibration errors, a recursive procedure
has been implemented by storing a large number of points
and by calculating £nal relative position and orientation between
two frames using average values.
The procedure adopted is the follow:
1. master arm is controlled in order to move the centre of
Cardano’s joint along a circumference as shown in Fig.
6;
2. a number of circumference points are stored as indicated
during the movement;
3. a group of three points (P0 P1 P2)i are considered for determining
the correspondent matrix rotation (R M S )i;
4. from any rotation matrix it is necessary to associate the
Roll-Pitch-Yaw angles ( RPY )i;
5. the £nal RPY are calculated as the average values;
6. from the £nal RPY is calculated the £nal rotation matrix.
8. Results
Once the calibration has been achieved the control software
provides to store it in speci£c OS registry entries and reuse
them whenever the system is restarted.
In this way the user can operate on the systems as the
arms belong to unique OS without caring too much (nor in
use, nor in SW development) with the organization of mechanics
and system control. If the user will notice that collocation
of objects within VE appears to be corrupted he can
manage a recalibration of the device at any time. This can
be achieved by issuing the relative command from the HI
control panel. Some extensive tests have been carried out in
order to match the effectiveness of the calibration all over the
workspace. The procedure works as follows: different “virtual”
axes-aligned square-boxes have been placed on several
positions of the arms workspace. HIs have then been moved
onto the boxes in order to arrange thimbles is a parallel fashion
on the same surface. Relative position has been measured
according to the normal to the considered surface. Whenever
c○ The Eurographics Association 2005.