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Page 92 Li et al. Intell Robot 2022;2(1):89–104 I http://dx.doi.org/10.20517/ir.2022.02

Figure 2. CAD model of the 4-dof robot with a revolute-spherical-spherical limb and a screw pair-based mobile platform.

Table 1. Geometric and dynamic parameters of the robot
Parameters value
Length of inner arm 0.296 m
Length of outer arm 0.600 m
Mass of upper platform 0.855 kg
Mass of lower platform 1.080 kg
Mass of inner arm 0.842 kg
Mass of outer arm 0.073 kg

where ® ∈ is the driving torque and ® ¤ , ® ¥ ∈ represent the joint angular velocity and acceleration, respec-
4
4
tively. Moreover, ( ® ) ∈ 4×4 is the inertia matrix, ( ® , ® ¤ ) ∈ 4×4 is a vector resulting from Coriolis and
4
®
centrifugal forces, ( ® ) ∈ represents gravity, and b is the moment of inertia of inner arms. Jacobians up
and down relate the motion of the upper and lower sub-platforms to the actuated joints, while up and down ,
¤
¤
respectively, represent their derivatives with respect to time. In addition, b, p,up, and p,down are the mass
matrices of the inner arm and the upper and lower sub-platforms. The detailed modeling procedure can be
found in Ref [24] . The main geometric and dynamic parameters of the parallel robot are listed in Table 1.

3. ITERATIVE LEARNING CONTROLLER DESIGN
Prior to the ILC design for the robot, the following properties generalized to the robotic manipulators are
considered.

Property 1. The inertia matrix is bounded and positive definite, thus ∃ > 0, > 0 satisfies the following
inequalities:

0 < < k ( ® , )k < (5)

Property 2. The inertia matrix satisfies the global Lipschitz condition; therefore, a positive constant exists
that meets:

k ( ® , ) − ( ® −1 , )k ≤ k ® ( ) − ® −1 ( )k (6)
where represents the number of iterations and ® is the angular displacement of the joint.

®
Property 3. Coriolis, centrifugal, and gravitational force matrices meet the equation ( ® , ® ¤ ) ® ¤ + ( ® ) =