IEEE Robotics & Automation Magazine - June 2020 - 82
(SSE). The split-point errors across the variables are compared,
and the variable/point yielding the lowest SSE is chosen as the
root node/split point. This process is recursively continued.
Both methods, analytical and ML-based, need experimental data to fit the model or train the algorithm. The inputs to
the methods are the angles measured at the three active joints
of the device in any Cartesian position of its workspace, while
the outputs are the three torques that must be applied to each
joint to hold the device still in each Cartesian position. For
other devices, the number of inputs and outputs should be
equal to the number of active DoF of the device. To generate
such a force and position database, the workspace of the
device is divided into a finite number of points, and an experiment is designed to compute the torque values necessary to
hold the system still at each point.
Experiments
This section first describes how the force database is collected
and the procedure to fit the analytical equations for gravity
compensation and to train the ML-based method. A final
experiment is carried out on a set of points of HomeRehab's
workspace, different from the points used for the training, to
compare the performance of each method.
Experimental Setup
The HomeRehab workspace is divided into 4,920 points covering 79.4% of the workspace used for rehabilitation applications. The tested positions (Figure 4) form a rectangular
Figure 4. The tested positions.
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JUNE 2020
parallelepiped grid in the Cartesian axes, 0.8 m × 0.22 m ×
0.18 m. The distance between points is 2 cm.
A proportional-integral-derivative (PID) controller forces
the system to move automatically from one position to another. When the mechanism reaches the steady state at each tested position (with a position error lower than 0.1 mm), the
torques provided by the controller are precisely the torques
that compensate for the gravity. These torques τ1, τ2, and τ3
are recorded in a database together with the angles θ1, θ2, and
θ3 measured at each active joint of the device. The generation
of this database takes 2.5 h.
Analytical Model
A least-squares optimization method is carried out to obtain
the values of pi that best fit model (1) with the training database. The following values are obtained:
p 1 = 3.33 N $ m,
p 2 = 3.97 N $ m,
p 3 = 3.98 N $ m,
p 4 = 2.05 N $ m,
p 5 = 0.77 N $ m.
(3)
The experimental torques and those that are obtained from
the gravity model (1) with the proposed parameters (3) are
shown in Figure 5 for 2,000 consecutive points of the training
database. It is worth noting that the experimentally fitted gravity model is robust to some assumptions of the scheme
IEEE Robotics & Automation Magazine - June 2020
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