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Li et al. Intell Robot 2022;2(1):89–104 I http://dx.doi.org/10.20517/ir.2022.02 Page 91

are deployed in parallel robots [15] . For instance, an open-loop PD-type ILC algorithm was proposed for a
class of nonlinear time-varying systems with control delay and arbitrary initial value [16] . In this manner, the
learning convergence curve is not smooth, although it solves the problem of initial shift. The robustness of
the controller can be ensured by designing a robust term, aiming at the control of a 3-dof permanent magnet
spherical actuator [17] . Open-loop PD-type ILCs have also been applied in the Delta robot; however, the test on
the controller showed that convergence requires a number of iterations and plenty of computation time, i.e., an
unacceptable computational burden in real applications [18] . To speed up the convergence of the controller, the
constant gain of the PD control can be changed to a time-varying one [19] , but this introduces glitches during
the convergence procedure. Alternatively, an adaptive controller can be integrated, where the controller gain
is defined as a function of the number of iterations [20] ; sequentially, both the position and velocity tracking
errors can be monotonically and rapidly reduced. In addition, to realize the automatic tuning of a controller, a
method with generalization capabilities is proposed in [21] that can effectively tune the parameters to improve
the trajectory tracking accuracy for robots. Besides, ILC can also be applied in repetitive rehabilitation train-
ing [22] , in which a high-order ILC can improve the transient performance and decrease the steady-state error,
compared to traditional PID controllers. Since ILC is equivalent to an integrator along with the iterations, it is
sensitive to external disturbances [23] . The focus of this work is the design of an ILC considering disturbances
for high-speed parallel robots for a pick-and-place application.

In the practical application of industrial robots, classical PD control is still the mainstream algorithm, and
studies on the iterative learning theory applied to control of parallel robots have not been extensively reported.
Consequently, the present work is to illustrate the effectiveness and feasibility of such algorithms for parallel
robots. In this paper, an open-closed-loop PD-type ILC method is proposed and illustrated with a parallel
robot producing Schönflies motion. The proposed ILC law consists of classical PD control and ILC. The iter-
ative learning term can be regarded as feedforward compensation, which can use the information stored in
the last movement. The PD control part belongs to the feedback item and performs real-time compensation.
The controller convergence is proved based on Q operator theory, and the tracking performance is tested by
tracking a pick-and-place trajectory and compared with the classical D-ILC controller. Moreover, different
trajectories and working frequencies are selected to verify the effectiveness of the controller.

2. ROBOT STRUCTURE AND DYNAMIC MODEL
Figure 2 depicts the detailed CAD model of the robot shown in Figure 1, which is composed of a mounting
frame, a screw-pair-based moving platform, and four identical limbs. Each limb consists of a big (inner) arm
and a small (outer) arm. A drive motor and a reducer are installed on the rotating shaft of the big arm. The
outer arm is composed of two carbon fiber rods in a -shape. The inner and outer arms are connected by ball
joints, as well as the connection between the outer arm and the mobile platform. The mobile platform can be
split into two subparts, i.e., upper and lower sub-platforms. Through the helix joint, the rotation in the vertical
direction of the end-effector can be generated by the differential motion of the two sub-platforms.

The kinematics and dynamics of the robot have been well documented in the previous work [24] , which is
revisited by skipping the details. When ignoring un-modeled errors and external disturbance, the dynamic
model of the robot can be expressed as:

® (1)
® = ( ® ) ® ¥ + ( ® , ® ¤ ) ® ¤ + ( ® )
with
( ® ) = T T (2)
up
low p,low low + p,up up + b
T
( ® , ® ¤ ) = T p,low low + p,up up (3)
¤
¤
low up
T
T
( ® ) = − low p,low ® − p,up ® − b ® (4)
®
up