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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 93

( ® , ® ¤ ) ® ( ), where ( ® , ® ¤ ) ∈ × is a regression matrix and ® ( ) ∈ ×1 is a vector of unknown
parameters regarding the robot.
Moreover, the following reasonable assumptions are made.

Assumption 1. The system can meet the alignment condition, i.e., ® (0) = ® (0), ® ¤ (0) = ® ¤ (0). The desired
joint position trajectory, namely, ® , and its th derivatives are bounded, namely, ∀ ∈ [0, ], ∀ ∈ +.
Assumption 2. The external disturbance of the robot is bounded and is subject to a positive constant:

sup k ( )k ≤ (7)
®

In view of the nonlinear time-varying robotic system with repetitive work over a finite interval time ∈ [0, ],
an open-closed loop PD-ILC law is designed. This algorithm belongs to the feedback–feedforward control law,
which can make full use of the effective information stored in the system for learning and can ensure that the
output variables converge to the bounded threshold of desired values.

The specific expression is written as follows:

® +1 ( ) = ® ( ) + ® fore + ® back (8)

where ® is the driving torque and is the number of iterations. Moreover, ® fore is the feedforward control input,
written as:

®
® fore = ® ( ) + ¤ ( ) (9)
where , are symmetric positive definite gain matrices for the feedforward control and ® = ® − ® and
® ¤ = ® ¤ − ® ¤ represent the joint errors in terms of angular displacement and angular velocity, respectively, in
the th iteration.

The feedback control ® back takes the following form:

®
® back = (1 − ) ® +1 ( ) + (1 − ) ¤ +1 ( ) (10)
where and are gain coefficients of the controller.

The scheme of the proposed controller is displayed in Figure 3. It can be seen that the information obtained
in the th iteration can be regarded as the feedforward part. The current joint errors, namely, the information
obtained in the ( + 1)th iteration, constitute the feedback part of the control law. Under the condition that
the control target and external environment remain unchanged, the target task is repeatedly executed, and the
response of the system is identical to the feedforward information. When the system deviates from the desired
trajectory, the feedback term will compensate the motion errors.

4. CONVERGENCE ANALYSIS OF THE CONTROLLER
To prove the convergence of proposed controller, the following two lemmas are introduced as the fundamen-
tals.
Lemma 1. With ∀® , ® ∈ [0, ], ∈ [0, ], assuming that the operator : [0, ] → [0, ] meets global
®
Lipschitz condition, one obtains the following two outcomes.

(1) For ∀® ∈ [0, ], there is a unique ® ∈ [0, ] that holds:

®
® ( ) + (® )( ) = ® ( ), ∀ ∈ [0, ] (11)