Mecademic Meca500 R3, Programming Manual

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4 Programming manual for Meca500 (for firmware 9.2.x)
BASIC THEORy ANd dEfINITIONS
1.1.6 Joint set and robot posture
There are several possible solutions for joint angle values, for a desired location of the robot end-
effector with respect to the robot base (i.e., several possible sets { θ
1
, θ
2
, θ
3
, θ
4
, θ
5
, θ
6
}). The simplest way
to describe how the robot is postured, is by giving its set of joint angles. This set will be referred to as
the joint set , and occasionally as joint position .
For example, in Figure 3 , the joint set is {0°, 0°, 0°, 0°, 0°, 0°}, although, it could have been {0°, 0°, 0°, 0°,
0°, 360°}, and you wouldn't be able to tell the difference.
A joint set completely defines the relative poses (i.e., the "arrangement," of the seven robot links,
starting with the base and ending with the end-effector). This arrangement is called the robot posture .
Thus, the joint sets { θ
1
, θ
2
, θ
3
, θ
4
, θ
5
, θ
6
} and { θ
1
, θ
2
, θ
3
, θ
4
, θ
5
, θ
6
+ c
t
360°}, where −180° < θ
6
≤ 180° and
c
t
is the axis 6 turn configuration, correspond to the same robot posture. Therefore, a joint set has the
same information as a robot posture AND an axis 6 turn configuration.
1.2. Configurations, singularities and workspace
1.2.1 Inverse kinematic solutions and configuration parameters
Meca500's inverse kinematics generally provide up to eight feasible robot postures for a desired pose
of the TRF with respect to the WRF ( Figure 4 ), and many more joint sets (since if θ
6
is a solution, then
θ
6
± n 360°, where n is an integer, is also a solution). Each of these solutions is associated with one of
eight robot posture configurations , defined by three parameters: c
s
, c
e
and c
w
. Each of these parameters
corresponds to a specific geometric condition on the robot posture (see Figure 5 ):
• c
s
(shoulder configuration parameter):
– c
s
= 1, if the wrist center (where the axes of joints 4, 5 and 6 intersect) is on the "front" side of
the plane passing through the axes of joints 1 and 2 (see Figure 5 a ). The condition c
s
= 1 is often
referred to as "front".
– c
s
= −1, if the wrist center is on the "back" side of this plane (see Figure 5 c ) .
• c
e
(elbow configuration parameter):
– c
e
= 1, if θ
3
> −arctan(60/19) ≈ −72.43° ("elbow up" condition, see Figure 5 d ) ;
– c
e
= −1, if θ
3
< −arctan(60/19) ≈ −72.43° ("elbow down" condition, see Figure 5 f ).
• c
w
(wrist configuration parameter):
– c
w
= 1, if θ
5
> 0° ("no flip" condition, see Figure 5 g );
– c
w
= −1, if θ
5
< 0° ("flip" condition, see Figure 5 i ).
Figure 4 shows an example all eight possible robot postures, described by the posture configuration
parameters { c
s
, c
e
, c
w
}, for the pose {77 mm, 210 mm, 300 mm, −103°, 36°, 175°} of the FRF with respect
to the BRF.
Figure 5 shows an example of each robot posture configuration parameter, and limit conditions, which
are called singularities . Note that the popular terms front/back and elbow-up/elbow-down are misleading
as they are not relative to the robot base but to specific planes that move when some of the robot joints
rotate.