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Wang et al. Intell Robot 2023;3(3):479-94 I http://dx.doi.org/10.20517/ir.2023.26 Page 7 of 16

4. HYBRID MOTION/ FORCE CONTROL BASED ON RNN
Considering the uncertainty of the upper-bound error in the controller, a primary RNN is utilized. By design-
ing adaptive laws of network weights based on the Lyapunov stability theory, the parameters of learning factors
in neural networks are adjusted. In addition, the boundary value estimation algorithm is utilized to compen-
sate for the estimation error. In order to analyze the system stability, the Poincare return map is utilized, in
which the manifold of a continuous subsystem can pass through the impact cross section.

4.1. Principle and structure of RNN
A primary RNN is adopted to deal with the complicated optimization problem. It will remember previous
information and use it to affect the output of subsequent nodes. The mapping equation of an RNN is expressed
as follows:

Õ
( ) =
( ) − , , , ( − 1) (14)
¤

=1
where represent input variables, are output variables, is denoted by connective weights between the
hidden layer and output layer, and and are the center and the width of the Gaussian function - th block
membership function - in the th input. is the internal feedback gain. The base function of the th block
acceptance field corresponding to the th input can be written as

"
Õ ( ) + ( − 1) −
( ) = exp − , (15)
2
=1

The th multidimensional acceptance domain function and the corresponding multidimensional domain
space are respectively denoted by

h i
 Î Í 2
  Φ =
 ( ) = exp ( − ) , = 1, 2 · · · ,
=1 =1 2 (16)


 Φ = Φ 1 Φ 2 · · · Φ

where is the number of receptive fields.

Define the weight memory space matrix = 1 2 · · · · · · , where = 1 2 · · · .
Then, the coverage of multidimensional function space Φ to the input state will also change. The output
of the system is the product of the weight matrix and the vector of the receptive field Φ, which can be

described in the form of = 1 2 · · · = Φ.
4.2. Design of a hybrid force/motion controller
The control structure of a hybrid motion/force controller is shown in Figure 3. In order to derive optimized
contact force and motion, hybrid motion/force control is proposed based on RNNs to approximate dynamic
functions. Several assumptions are made in advance.

Assumption 4.1. According to the approximation principle of neural networks, suppose that there exists the
desired weight , Gauss function Φ , desired center value , width , and recurrent gain coefficient ,
∗
∗
∗
∗
∗
which makes an RNN approach any smooth nonlinear function .