10
Zeroing Neural Networks for Robot Arm Motion Generation
10.1 Introduction
The model for solving nonlinear equations bears an essential similarity with the controller for controlling the plant: their residual errors are required to decrease to an acceptable small value as soon as possible. The exploitation of this similarity provides a possibility to investigate computational methods from the perspective of control system theory. It is worth mentioning that, in the field of numerical computation and control, there is always a great demand for robustness due to the existence of various noises or disturbances, such as round‐off errors and truncation errors. From the perspective of computation, many recurrent neural network models, e.g. zeroing neural network (ZNN), are analyzed and applied to the solution of various problems [128]. To improve the robustness for solving time‐varying problems, a noise‐tolerant zeroing neural network (NTZNN) design formula is proposed in [13], which can be used to design recurrent neural networks from the viewpoint of control. Then, such a NTZNN design method is explored to design a modified NTZNN in [1] for the online solution of quadratic programming with application to the repetitive motion planning of a redundant robot arm. As discussed in [7], nonlinear activation functions can be used to accelerate the convergence speed of original ZNN models. However, to the best of the authors' knowledge, there is no systematic solution on NTZNN with the aid of nonlinear activation functions. Therefore, the extension from existing NTZNN to nonlinearly activated NTZNN (NANTZNN) remains an unsolved problem.
The redundant robot arms are robotic devices, of which the available degrees of freedom are more than those strictly required for executing the user‐specified primary end‐effector task [15]. Recent progress in this topic shows the advantages of using optimization techniques for simultaneously handling the primal task as well as other performance indices, e.g. quadratic programming. For example, Li et al. identify two limitations of the existing neural network solutions for the quadratic program (QP)‐based motion planning of a robot arm, and overcome them by proposing two modified neural network models in [84]. In comparison with an individual robot arm, multi‐arm systems are used in many schemes in order to improve performance and reliability. For example, a repetitive motion planning scheme for simultaneously controlling two robot arms is presented in [3], where the two subschemes are unified into a QP‐based scheme and then solved by a recurrent neural network. It is an important issue to tolerate noises online during the end‐effector task execution since arms often work in environments with strong electromagnetic interference.
In this chapter, via the NTZNN‐based neural‐dynamic design method presented in [13], we make progress by presenting a NANTZNN model for the distributed cooperative motion planning of multiple redundant arms.
10.2 Preliminaries, Problem Formulation, and Distributed Scheme
In this section, preliminaries, problem formulation, and distributed scheme are provided to lay a foundation for investigations.
10.2.1 Definition and Robot Arm Kinematics
We present the definition on communication topology of limited communications with 
denoting the neighbor set of the ith redundant robot arm in the communication graph as well as robot arm kinematics in the following [61].
Robot arm kinematics. Given the desired trajectory 
of the end‐effector in work space, we need to generate online the joint trajectory 
in joint space so as to command the robot arm motion. Note that the Cartesian coordinate 
in the workspace of a robot arm is uniquely determined by a nonlinear mapping:
where 
is a differentiable nonlinear function. Computing time derivations on both sides of (10.1) leads to
where 
is the Jacobian matrix of 
, and usually is abbreviated as 
. The end‐effector 
of the redundant robot arm is expected to track the desired path 
, i.e. 
.
10.2.2 Problem Formulation
The constraint for the ith robot arm can be formulated as
where 
denotes the neighbor set of the ith robot on the communication graph; 
denotes the connection weight between the ith robot and the jth one; 
with 
denoting the bias distance between the end‐effector and the reference point; 
for 
; and 
for 
. Then, the constraint can be formulated as
where 
is the Kronecker product; 
denotes a vector composed of 1; 
is an identity matrix; 
with 
being the diagonal matrix whose 
diagonal entries are the 
elements of the vector 
with the ijth element of 
being 
; 
; and 
is defined as
Then, a vector‐valued error function could be defined as
The following design formula can be used:
where design parameter 
is used to scale the displacement. Expanding (10.6), we further have
where 
with 
denoting the joint angle in the joint space of the ith robot arm and with 
denoting its time derivative; and
with 
denoting the Jacobian matrix of the ith robot arm.
10.2.3 Distributed Scheme
We consider the minimum velocity norm (MVN) performance index in this chapter:
Then, the distributed scheme can be formulated as
10.3 NANTZNN Solver and Theoretical Analyses
Here, we present a NANTZNN model to solve the MVN‐oriented distributed scheme (10.8) with guaranteed convergence and robustness.
10.3.1 NANTZNN for Real‐Time Redundancy Resolution
On the basis of [13], we propose and exploit a NANTZNN model for the online solution of the MVN‐oriented distributed scheme (10.8).
Solving (10.8) can be done by solving the following equation:
where
with 
. We define the error function as
To force 
to be zero, the following NANTZNN design formula is adopted:
where 
and 
. Besides, 
denotes a vector array of activation function and the ith element of 
is denoted by 
. Generally speaking, any monotonically increasing odd activation function 
can be used for constructing the NANTZNN model. In this chapter, the power‐sigmoid and hyperbolic sine activation functions are applied in constructing the NANTZNN model:
- The power‐sigmoid (ps) activation function (with
and 
):
- The hyperbolic sine (hs) activation function (with
):
We further obtain the following distributed NANTZNN model:
To lay a basis for further investigation on the robustness of MVN‐oriented distributed scheme (10.8) aided by distributed NTZNN model (10.12) under the pollution of unknown noises, we have the following equation:
where 
denotes the vector‐form noises originated from communication noises, computational errors, perturbations, or even their superposition.
10.3.2 Theoretical Analyses and Results
The analyses on the stability, convergence, and robustness of the proposed distributed NANTZNN model (10.12) are conducted in this section using the following theorems.
It is worth investigating the robustness of distributed NANTZNN model (10.12). The following theorem is presented to discuss the performance of distributed NANTZNN model (10.12) with constant noise 
.
10.4 Illustrative Examples
In this section, we conduct computer simulations in the situation of zero noise and constant noise.
10.4.1 Cooperative Motion Planning without Noises
In this section, we consider six PUMA 560 robot arms for cooperative motion planning perturbed without noise. In addition, only robot arm 1 is able to access the desired motion information provided by the command center. The initial joint state of each robot arm is chosen on the desired trajectory. In addition, 
is set as
The parameters are chosen as 
, 
, 
, and task duration is 
s. In addition, the remaining parameters are set as zero (e.g. 
). A typical simulation is generated in Figures 10.1 and 10.2.
Figure 10.1 Computer simulations synthesized by MVN‐oriented distributed scheme (9.8) and distributed NTZNN model (9.13) for consensus of eight redundant PUMA 560 robot manipulators with limited communications and perturbed with constant noise 
, where initial joint states of manipulators are randomly generated. (a) Motion trajectories; (b) profiles of end‐effector position errors; (c) profiles of joint angle; and (d) profiles of 
.
Figure 10.2 Computer simulations synthesized by MVN‐oriented distributed scheme (9.8) and distributed NTZNN model (9.13) for consensus of eight redundant PUMA 560 robot manipulators with limited communications and perturbed with constant noise 
, where initial joint states of manipulators are randomly generated. Profiles of (a) joint angle and (b) joint velocity.
It can be observed from Figure 10.3a that all robot arms execute the task denoted by the circular trajectory cooperatively and successfully. In addition, Figure 10.3b shows that the end‐effector position tracking errors all remain with a very tiny value during the task execution. The profiles of end‐effector velocities, Lagrange‐multiplier vector 
, joint angle, and joint velocity are shown in Figures 10.3c, 10.3d, 10.4a, and 10.4b, respectively. It can be observed from these figures that the resulting profiles move smoothly. These simulation results substantiate the effectiveness of MVN‐oriented distributed scheme (10.8) and distributed hyperbolic‐sine activation function activated NANTZNN model (10.12).
Figure 10.3 Computer simulations synthesized by MVN‐oriented distributed scheme (10.8) and hyperbolic‐sine activation function activated NANTZNN model (10.12) perturbed with noise 
for motion planning of 6 redundant PUMA 560 robot arms with limited communications. (a) Motion trajectories; (b) profiles of end‐effector position errors; (c) profiles of joint angle; and (d) profiles of 
.
Figure 10.4 Computer simulations synthesized by MVN‐oriented distributed scheme (10.8) and hyperbolic‐sine activation function activated NANTZNN model (10.12) perturbed with noise 
for motion planning of six redundant PUMA 560 robot arms with limited communications. Profiles of (a) joint angle and (b) joint velocity.
10.4.2 Cooperative Motion Planning with Noises
In this section, we consider cooperative motion planning perturbed with noise 
. The parameters and communication topology are set the same as those in the previous example with simulation results shown in Figure 10.2. In addition, simulation results based on power‐sigmoid activation function activated NANTZNN model (10.12) are presented in Figure 10.5. Note that the detailed descriptions in Figures 10.2 and 10.5 are omitted due to space limitations. In summary, these simulation results substantiate the effectiveness of MVN‐oriented distributed scheme (10.8) and distributed NANTZNN model (10.12).
Figure 10.5 Computer simulations synthesized by MVN‐oriented distributed scheme (10.8) and power‐sigmoid activation function activated NANTZNN model (10.12) perturbed with noise 
for motion planning of six redundant PUMA 560 robot arms with limited communications. Profiles of (a) end‐effector position errors and (b) joint angle.
10.5 Summary
In this chapter, we have proposed a NANTZNN for solving the distributed motion planning of multiple robot arms in the presence of noises. Theoretical analyses have been presented to show the superiorities of the proposed NANTZNN model compared with the existing linearly activated NTZNN model. Furthermore, simulation results have shown the effectiveness and accuracy of the presented distributed scheme with the aid of the NANTZNN model.