IEEE Robotics & Automation Magazine - June 2020 - 81

θ2

E
le

l3
z

θ3
C

y
E
z

le′

DF
ld′ lf ′

C
lc′

θ1

la
B

lb′

la′

z
(a)

(b)

Figure 3. The (a) pantograph scheme and (b) positions of the centers of gravity.

depend on the masses and positions of the centers of gravity
of the links:
p 1 = (m a lla + m b llb + m c llc + m d lld + m e lle + m f llf ) g,
p 2 = (m a l 2 + m b l 2 + m c l c - m e l e) g,
p 3 = (m a l 3 + m b l b + m c l 1 - m d l d) g,
p 4 = m f l f g,
p 5 = m a l a g.

(2)

Table 1. The HomeRehab specifications.
Workspace

800 × 400 × 400 mm

Maximum force

14 N (continuous), 28 N (peak)

Actuators

Maxon DCX32L-GPX32, 128 μN · m, 4:1

Transmission

20:1 cable transmission

Encoders

Maxon ENX16, 1,024 pulses/revolution

Position resolution 6 μm
Weight

A rough estimation of the five parameters pi could be
derived using a CAD model of the mechanism. However, to
obtain reliable values for pi, an experimental fit is required.
ML-based methods do not require the resolution of analytical
equations. While HomeRehab is a mechanism whose equations may be derived with relative ease, for some parallel
mechanisms and commercial devices whose CAD models
and geometrical data are not available, the process to develop
the equations may be complex.
Among the different ML techniques and approaches, in
this article, a decision tree technique is used, since it enables
fast responses and accurate results, as many researchers have
already tested [14], [15]. A decision tree is an algorithm and
data structure oriented for supervised learning, where each
node represents an attribute or feature (in our case, 3D coordinates). For each node, the children are classified according to a
criterion until obtaining a leaf node. These leaves represent the
final decision [16].
Usually, several decision trees are used because more accurate results are obtained (each tree may give a different solution, and a vote scheme is performed to determine the final
decision). The algorithm that achieves this process is called
random forest [17]. This algorithm can be used to classify or
perform a regression prediction where each tree in the

8.2 kg

ensemble is trained on a subset of the entire training data set.
Then, each split is performed on a random subset of features
(one for each tree) [16].
The extra trees is an extension of the random forest regression
model proposed by Geurts [18]. Random forest and extra trees
are important algorithms within this class and have reported
state-of-the-art performance on many regression tasks with highdimensional inputs and outputs [19]. The differences between
extra trees and the original random forest are as follows:
1) Unlike random forest, extra trees does not use the tree-bagging step to generate the training subset for each tree.
2) Extra trees randomly selects the best feature along with the
corresponding value to split the node [20].
These two differences result in extra trees being less susceptible to overfitting and able to report better performance [18].
It is important to note that the extra trees regression is not a
classification tree. In a regression tree, as the target variable
does not have classes, we fit a regression model to the target
variable using each of the independent variables. Then, for
each independent variable, the data are split at several points.
At each split point, the error between the predicted value and
the actual values is squared to get a sum of squared errors
JUNE 2020

IEEE ROBOTICS & AUTOMATION MAGAZINE

IEEE Robotics & Automation Magazine - June 2020

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