The Inverse 6 D Robot Kinematics Mapping

Figure 8.5: The same 27 training data vectors (cmp. Fig. 8.4) for the bi-directional PSOM mapping: (left) in the Cartesian space

~r

, (middle) the corresponding joint angle space

~

^. (Right:) The corresponding node locations^a2Ain the param-eter manifold

S

. Neighboring nodes are connected by lines, which reveals now the “banana” structure on the left.

Here,

¹

:::

denote the joint angles,

~r

is the Cartesian position of the end effector of length

l

z in world coordinates.

~a

^and

~n

denote the normalized approach vector and the vector normal to the hand plane. The last nine components vectors are part of the homogeneous coordinate transforma-tion matrix

(The missing second matrix column

~o

is the cross product of the normal-ized orientation vectors

~a

^and

~n

and therefore bears no further informa-tion, see Fig. 8.6 and e.g. (Fu et al. 1987; Paul 1981).)

In this space, we must construct the

m

^{= 6} dimensional embedding manifold

M

that represents the configuration manifold of the robot. With three nodes per axis direction we require^{36 =726}reference vectors^{wa 2}

X

. The distribution of these vectors might have been found with a SOM, however, for the present demonstration we generated the values for the

w

aby augmenting 729 joint angle vectors on a rectangular 333333 grid in joint angle space

~

with the missing

r

x

r

y

r

z

a

x

a

y

a

z

n

x

n

y

n

z–

θ

θ

θ θ θ

θ

l_z

o n r a

Figure 8.6: The 15 com-ponents of the training data vectors for the PSOM net-works: The six Puma axes and the position

~r

^and

orien-tation vectors

~n

^,

~o

^{, and}

~a

^of

the tool frame.

components, using the forward kinematics transform equations (Paul 1981) (

^{12[-135,-45],}

^{22[-180,-100],}

^32[-35,55],

^42[-45,45],

^52[-90,0],

^62[45,135], and tool length

l

z={0,200} mm in

z

direction of the

T

^6frame,

see Fig. 8.6.

Similar to the previous example, we then test the PSOM based on the³⁶ points in the inverse mapping direction. To this end, we specify Cartesian goal positions

~r

and orientation values

~a~n

at 200 randomly chosen inter-mediate test points and use the PSOM to obtain the missing joint angles

~

^.

Thus, nine dimensions of the embedding space

X

are selected as in-put sub-space. The three components^f

r

x

r

y

r

z^gare given in length units ([mm] or [m]) and span intervals of range {1.5, 1.2, 1.6} meters for the given training set, in contrast to the other six dimensionless orientation compo-nents, which vary in the interval [-1,+1]. Here the question arises what to do with these incommensurable components of different unit and magni-tude? The answer is to account for this in the distance metric

dist

^{(). The}

best solution is to weight each component

k

in Eq. 4.7 reciprocally to the measurement variance

p

k ⁼var⁽

w

k^)];1

:

^(8.2)

If the number of measurements is small, as it is usual for small data sets,

8.2 The Inverse 6 D Robot Kinematics Mapping 115 PSOM³⁶ Cartesian position approach vector normal vector

deviation

~r

^deviation

~a

^deviation

~n

l

z^{[mm] Sampling Mean NRMS Mean NRMS Mean NRMS}

0 bounded 19 mm 0.055 0.035 0.055 0.034 0.057

200 bounded 23 mm 0.053 0.035 0.055 0.034 0.057

0 Chebyshev 12 mm 0.033 0.022 0.035 0.020 0.035

200 Chebyshev 14 mm 0.034 0.022 0.035 0.021 0.035 Table 8.2: Full 6 DOF inverse kinematics accuracy using a 333333 PSOM for a Puma robot with two different tool lengths

l

z. The training set was sampled in a rectangular grid in

¹:::⁶, in each axis centered at the working range midpoint. The bordering samples were taken at the range borders (bounded), or according to the zeros of the Chebyshev polynomial

T

^{3(Eq. 6.3).}

we may roughly approximate the variance by the following computational shortcut. In Eq. 8.2 the non-zero diagonal elements

p

k of the projection matrix ^Pare set according to the interval spanned by the set of reference vectors^wa:

p

k ⁼⁽

w

^maxk ^;

w

^mink ^);2

:

^(8.3)

With

w

^maxk ⁼ max

8a2A

w

k^a and

w

k^min= min

8a2A

w

k^a (8.4)

the distance metric becomes invariant to a rescaling of any component of the embedding space

X

. This method can be generally recommended when input components are of uneven scale, but considered equally sig-nificant. As seen in the next section, the differential scaling of the compo-nents can by employed to serve further needs.

To measure the accuracy of the inverse kinematics approximation, we determine the deviation between the goal pose and the actually attained position after back-transforming (true map) the resulting angles computed by the PSOM. Two further question are studied in this case:

1. What is the influence of using tools with different length

l

z mounted on the last robot segment?

2. What is the influence of standard and Chebyshev-spaced sampling of training points inside their working interval? When the data val-ues (here 3 per axis) are sampled proportional to the Chebyshev ze-ros in the unit interval (Eq. 6.3), the border samples are moved by a constant fraction (here 16 %) towards the center.

Tab. 8.2 summarizes the resulting mean deviation of the desired Carte-sian positions and orientations. While the tool length

l

zhas only marginal influence on the performance, the Chebyshev-spaced PSOM exhibits a sig-nifcant advantage. As argued in Sect. 6.4, Chebyshev polynomials have ar-guably better approximation capabilities. However, in the case

n

^=3both

sampling schemes have equidistant node-spacing, but the Chebyshev-spacing approach contracts the marginal sampling points inside the working inter-val. Since the vicinity of each reference vector is principally approximated with high-accuracy, this advantage is better exploited if the reference train-ing vector is located within the given workspace, instead of located at the border.

Figure 8.7: Spatial dis-tribution of positioning errors of the PUMA robot arm using the 6 D inverse kinematics transform computed with a 333333 C-PSOM. The six-dimensional man-ifold is embedded in a 15-dimensional

~r~a~n~

^-space.

The spatial distribution of the resulting

~r

deviations is displayed in Fig. 8.7 (of the third case in Tab. 8.2). The local deviations are indicated

8.2 The Inverse 6 D Robot Kinematics Mapping 117 by little (double sized) cross-marks in the perspective view of the Puma's

workspace.

Cartesian position

~r

PSOM Type Average NRMS

333 PSOM 17 mm 0.041

333 C-PSOM 11 mm 0.027

444 PSOM 2.4 mm 0.0061

444 C-PSOM 1.7 mm 0.0042

555 PSOM 0.11 mm 0.00027

555 C-PSOM 0.091 mm 0.00023 333 L-PSOM of 444 6.7 mm 0.041 333 L-PSOM of 555 2.4 mm 0.0059 333 L-PSOM of 777 1.3 mm 0.018

Table 8.3: 3 DOF inverse Puma robot kinematics accuracy using several PSOM architectures including the equidistantly (“PSOM”), Chebyshev spaced (“C-PSOM”), and the local PSOM (“L-PSOM”).

The full 6-dimensional kinematics problem is already a rather demand-ing task. Most neural network applications in this problem domain have considered lower dimensional transforms, for instance (Kuperstein 1988) (

m

^{= 5}), (Walter, Ritter, and Schulten 1990) (

m

^{= 3}), (Ritter et al. 1992) (

m

^{= 3and}

m

⁼⁵), and (Yeung and Bekey 1993) (

m

^{= 3}); all of them use several thousand training samples.

To set the present approach into perspective with these results, we in-vestigate the same Puma robot problem, but with the three wrist joints fixed. Then, we may reduce the embedding space

X

to the essential vari-ables⁽

¹

²

³

p

x

p

y

p

z⁾. Again using only three nodes per axis we require only 27 reference vectors ^wa to specify the PSOM. Using the same joint ranges as in the previous case we obtain the results of Tab. 8.3 for several PSOM network architectures and training set sizes.

0 20 40 60 80 100 120 140 160

0 100 200 300 400 500 600 700 800

Number of Training Examples Mean Cartesian Deviation [mm]

Mean Joint Angle Deviation [deg]

Figure 8.8: The positioning capabilities of the 333 PSOM network over the course of learning. The graph shows the mean Cartesian ^hj

~r

^ji and angular

hj

~

^jideviation versus the number of already experienced learning examples.

After 400 training steps the last arm segment was suddenly elongated by 150 mm (10 % of the linear work-space dimensions.)

8.3 Puma Kinematics: Noisy Data and

Dans le document Jörg Walter (Page 127-132)