IEEE Robotics & Automation Magazine - September 2017 - 80

(a)

(b)

input sequence is iteratively found
such that the output of the system is as
close as possible to a desired output.
Iterative Learning Control [2] (ILC)
permits embedding this rule in a general theory. ILC exploits the error evolution of the whole interval [t 0, t f ) of a
previous iteration to update a feedforward command, according to the law

(c)

i

(e)

(d)

(f)

Figure 8. The evolution resulting from the application of high gain feedback with low
stiffness: robot positions (without an obstacle) at (a) t = 0 s, (b) t = 1 s, and (c) t = 2 s,
and (with an obstacle) at (d) t = 0 s, (e) t = 1 s, and (f) t = 2 s. With an obstacle
present, the robot drops the bar, as in the stiff case.

(b)

(c)

(d)

(e)

(f)

Figure 9. The evolution resulting from the application of the ILC algorithm with low
stiffness: robot positions (without an obstacle) at (a) t = 0 s, (b) t = 1 s, and (c) t = 2 s,
and (with an obstacle) at (d) t = 0 s, (e) t = 1 s, and (f) t = 2 s. With an obstacle present,
the robot adapts to the external environment (i.e., the mechanical stiffness is preserved by
the proposed anticipatory control).

i

i +1

= fi i + ae i,

(7)
i

m

where f, a are two constants, and i : [t 0, t f ) " R and
e i: [t 0, t f ) " R m are the whole control action and error
evolution, respectively, at the ith iteration. In this way, an
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SEPTEMBER 2017

(8)

i +1

= Q (i i) + R (e i, e i +1),

(9)

where the presence of e i +1 permits
incorporating the feedback action in
the same framework. In this manner,
ILC can be used to design an appropriate algorithm that permits learning
the feedforward action in a humanlike manner.
To illustrate the application of the
ILC framework to an SR, we use in the
following a combination of currentstate ILC and LQR feedback. The control law [of type (9)] is
i

of the same action, derived from a statistical model of error
evolution over iterations:

= Q (i i) + R (e i),

where the function R (e i) identifies the
ILC algorithm, and Q (i i) is a function
that maps the old control in the new one
(typically a smoothing function). It is
interesting to note that there is evidence
(e.g., reported in [20]) that in humans
feedback motor correction plays a crucial role in motor learning. Hence, a
more general algorithm able to merge all
these contributions should be considered. Leveraging this observation, we
can take advantage of the ILC literature
rewriting the control law (8), as in the
so-called current-iteration ILC [2],
i

(a)

i +1

i +1

= Q (i i) + K off e i + K on e i +1,
(10)

where i i, e i are the control action and
the error at the i-th iteration. Q (·) is a
suitable average mean filter, and K off ,
K on are two linear gains. Figure 4 shows the block diagram of
the algorithm. For this control law, (4) becomes
K on # d 2T (0, v)
2q

-1

.

(11)

Hence, it is always possible to choose K on such that (4) is satisfied. Here K on is the result of an LQR and K off is such that
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