IEEE Robotics & Automation Magazine - June 2020 - 83

Validation
A validation experiment is carried out with HomeRehab to
test the performance of both methods. Analytical gravity
4

τ1

Torque (N·m)

3
2

τ3

1
0
-1

τ2

0
80
0
3,
00
0
3,
20
0
3,
40
0
3,
60
0
3,
80
0
4,
00
0
2,

60
2,

40
2,

20
2,

-2

ML-Based Method Development
The input data for the ML algorithm is the same as for the analytical method (4,920 training points). These data are arranged
in six columns (x, y, z, τ1, τ2, and τ3). No editing, cleaning, or
any other technique was used on the data set. The ML method
development consisted of evaluating several scenarios for the
extra trees algorithm, changing the percentage of the data for
training and testing. The ML algorithm is implemented in
Python 3.6 using the Anaconda 5.0.1 64-b environment and
scikit-learn 0.19 library. Table 2 shows the results of applying
the extra trees algorithm to different scenarios.
The tests were performed using a fixed seed for NumPy
12345 to be reproducible. The accuracy was evaluated using
other, different seeds, but there was no significant difference
in the results. The base function used was ExtraTreesRegressor
with 50 estimators. It is relevant to consider that increasing
this parameter adds complexity to the calculation by adding
trees to the forest. We tried to increase it, but found no significant improvement. However, we noticed a high penalty to the
point calculation. The maximum features are three, since the
data set has only three inputs. Finally, the random state is set
to zero.
Table 2 shows that, by using 80% of the data, the best results
are obtained. Other approaches, such as the random forest
regressor, decision tree regressor, and others, were also tested
with those data sets to check our initial hypothesis of using the

extra trees regression method. The results were not improved
in terms of accuracy (the coefficient of determination R2 in
Table 3). However, the MultiO/P GBR method achieves good
performance, with a lower prediction time and memory usage.
Thus, if real-time implementation specifications are very relevant (less than 1 ms), the final method would be the preferred choice.

depicted in Figure 3 because the estimated parameters pi contain the contribution of several distributed masses. It is not
especially relevant for the validity of the model to find the
exact value of each length and mass of the model.
Taking into account that the dc motors of the mechanism
can exert up to 10.24 N . m after the transmission, a nonnegligible amount of torque is used to compensate for the gravity of
the device. In the case of torque τ1, which is the worst case of
the three motors (Figure 5), the mean value within all the positions of the workspace is 2.27 N . m, which represents 22% of
the maximum continuous torque. The maximum gravity
torque τ1 is 3.91 N . m, 38% of the available torque.

Experiment
Experimental

Fitted Model

Figure 5. The experimental and theoretical torques of the fitted
model.

Table 2. The results for different scenarios of the
extra trees algorithm.
Scenario

Training

80%

40%

20%

Testing

20%

60%

80%

95%

0.958

0.943

0.931

0.899

Table 3. The results for different ML algorithms.
Name

Mean Square
Error

Training Time
(Mean)

Prediction Time
(Mean)

Memory
(MB)

Extra trees

0.0156

0.9489

1.05 s

2.64 ms

278.61

k-nearest neighbors

0.0213

0.928

1.87 ms

353 μs

278.81

Linear regression

0.0655

0.7516

465 μs

45.3 μs

279.15

Ridge regression

0.0655

0.7509

1.37 ms

43.9 μs

279.57

Lasso

0.3305

598 μs

50.6 μs

279.61

Random forest regression

0.0176

0.9419

667 ms

5.32 ms

279.61

Decision trees

0.0363

0.8879

9.85 ms

50.1 μs

271.8

MultiO/P GBR

0.0199

0.9342

1.39 s

747 μs

271.92

MultiO/P GBR: multioutput gradient-boosting regressor.

JUNE 2020

IEEE ROBOTICS & AUTOMATION MAGAZINE

IEEE Robotics & Automation Magazine - June 2020

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