9
Zeroing Neural Networks for Robot Arm Motion Generation
9.1 Introduction
In recent decades, robotics has received more and more attention in scientific areas and engineering applications. Many research studies have focused on this topic, and various kinds of robots have been developed and investigated [1, 15, 63, 65–68, 98–101]. For example, a multiple collaborative manipulators system is investigated in [101], where the velocity consensus state can be achieved with the aid of the presented consensus control method. Redundant manipulators have been playing an increasingly important role and are an appealing topic in engineering fields [1, 15, 63, 98–100]. Redundant manipulators can achieve subtasks easily and dexterously such as repetitive motion and optimization of various performance criteria, since they possess more degrees of freedom than the minimum number required to execute a given primary task. One of the fundamental issues in operating such a robot system is the inverse‐kinematics problem (also termed the redundancy‐resolution problem). That is, with the desired trajectories of the manipulator's end‐effector being provided in Cartesian space, trajectories in joint space should be generated accordingly [1, 15, 63, 98–100]. Rather than individual manipulators, multi‐manipulator systems are used in many scenarios in order to improve performance and reliability. For example, in applications such as exploration, surveillance, tracking, or payload transport, in order to achieve better overall performance, a group of robots are driven to keep specific formations [101–105]. By considering the fact that manipulators often work in environments with strong electromagnetic interference, an unavoidable problem is to suppress noises during task execution. In addition to the increased complexity and computational burdens in multiple manipulator coordination, the noises involved also pose more requirements on real‐time processing for the redundancy resolution at network level. Moreover, it is prior to design distributed algorithms requiring only local neighbor‐to‐neighbor information exchange because of limited communication bandwidth and communication range in many robotic applications.
Table 9.1 Comparison of different schemes for redundancy resolution of manipulators.
| Noise tolerant | Manipulator numbers | Distributed vs. Centralized | Initial position | |
| This chapter | Yes | Multiple | Distributed | Any |
| [1, 15, 98, 99] | No | Single | NAa) | Any |
| [63, 100] | No | Two | Centralized | Any |
| [125] | No | Multiple | Distributed | Restrictiveb) |
| [126] | No | Multiple | Distributed | Restrictiveb) |
| [127] | No | Multiple | Distributed | Restrictiveb) |
a) NA means that the item does not apply to the schemes in the associated references.
b) The associated schemes require that the initial position of the end‐effector should be on the desired trajectory for tracking.
Neural network approaches, recognized as a powerful tool for real‐time processing, have long been applied widely in various control systems, e.g. multiple agents control [101, 106–111] and adaptive control [112–116]. For example, by using an adaptive neural network to approximate the nonlinear functions of the agents' dynamic, a consensus method based on the adaptive neural network is presented for the control of a class of nonlinear second‐order multi‐agent systems, thereby greatly reducing the online computation burden. Liu and Tong construct an adaptive fuzzy controller for a class of nonlinear discrete‐time systems with unknown functions and bounded disturbances in [109], which takes into account the effect of the discrete‐time dead zone with the system states not being required to be measurable. Wang et al. present a control algorithm based on neural networks and backstepping techniques for the adaptive neural control of stochastic nonlinear systems in [114]. In [115], their model is further extended to be capable of controlling the stochastic systems with strong interconnected nonlinearities. The gradient‐based neural network (GNN), being a typical class of the recurrent neural networks (RNNs), uses the norm of the error as the performance index and evolves along the gradient‐descent direction to make the error norm vanish to zero in the time‐invariant case [14]. As a special type of RNN designed for solving time‐varying problems, the zeroing neural network (also termed the Zhang neural network, ZNN) is able to perfectly track a time‐varying solution by using time derivatives of time‐varying parameters [1, 14, 117–119]. Many researchers have made progress in this direction by proposing various kinds of ZNN models for solving problems with different highlights, e.g. finite‐time convergence [120, 121] and high accuracy discretization [122, 123]. To generate repetitive motion in a closed path tracking, a drift‐free scheme based on a ZNN‐related neural‐dynamic design method is investigated at the joint‐velocity level in [124], which is further extended to schemes at joint‐acceleration level in [14, 98, 99]. Although extensive achievements have been obtained for the control of single‐arm redundant manipulators, research on the redundancy resolution of multiple manipulators is far from up‐to‐date, which severely restricts its applications in practical and academic research [125–127]. As mentioned previously, on account of their parallel computational power, RNNs are able to handle the problem of real‐time control of manipulators. For example, a ZNN‐based neural‐dynamic design method has been exploited for the repetitive motion generation of dual‐arm systems [63, 100], e.g. two arms of a humanoid robot [100].
In implementations of a neural network model for the cooperative motion generation of a network of redundant robot manipulators, we usually assume that it is free of all kinds of noises or external errors. However, there always exist some realization errors in hardware implementations or disturbances in the real‐time control, which can be deemed as noises. Sometimes these noises have significant impacts on the accuracy of the neural network for solving theses time‐varying problems, and in some cases, they may cause failure of the solving task [4]. In addition, for time critical tasks, it is preferable to integrate denoising with problem solving for real‐time processing. That is, time is precious for time‐varying problem solving (or the plant with time‐varying demands) in practice, and preprocessing for noise reduction may consume extra time, possibly violating the requirement of real‐time computation. A noise‐tolerant ZNN (NTZNN) model is presented in [118] from the control perspective for solving time‐varying problems, which can be employed for the cooperative motion generation of a network of redundant robot manipulators with inherent tolerance to noises. Comparisons between existing solutions for manipulator redundancy resolution and the proposed solution are summarized in Table 9.1. To the best of our knowledge, there is no systematic solution of noise‐tolerant neural network design for the cooperative motion generation in a distributed network of redundant robot manipulators.
9.2 Preliminaries
In this section, definitions, assumptions, and modeling of the kinematics used in this chapter are provided to lay a foundation for investigations.
9.2.1 Problem Definition and Assumption
Definitions on the communication graph and communication topology presented in [105] are provided in the following.
A communication graph is a graph with the nodes being redundant manipulators and the edges being communication links. Moreover, 
is used to denote a set of redundant manipulators with communication links to the ith redundant manipulator, which represents the neighbor set of the ith redundant manipulator on the communication graph.
Moveover, the definition of communication topology of limited communications is presented in the following [61, 105].
9.2.1.1 Assumption
Limited communication, with each redundant manipulator as a node and the communication link between one‐hop neighboring robots as edges, is the communication topology of a connected undirected graph. We use 
to denote the neighbor set of the ith redundant manipulator in the communication graph.
9.2.2 Manipulator Kinematics
Given the desired trajectory 
of the end‐effector, we want to generate online the joint trajectory 
so as to command the manipulator motion. Note that the Cartesian coordinate 
in the workspace of a manipulator is uniquely determined by a nonlinear mapping:
where 
is a differentiable nonlinear function with a known structure and parameters for a given manipulator. Computing time derivations on both sides of (9.1) leads to
where 
is the Jacobian matrix of 
, and usually is abbreviated as 
. The end‐effector 
of the redundant manipulator is expected to track the desired path 
, i.e. 
.
9.3 Problem Formulation and Distributed Scheme
In this section, the problem formulation is constructed and then the associated distributed scheme is presented.
9.3.1 Problem Formulation and Neural‐Dynamic Design
During the executing process of the cooperative motion generation of redundant robot manipulators, the desired formation of all end‐effectors should be maintained to avoid stretching or squeezing of the payload generated by the relative movement between them. In this chapter, the commanding signal is only accessible to its neighboring manipulators, instead of all manipulators. Therefore, the constraint for the ith manipulator can be formulated as
where 
denotes the neighbor set of the ith manipulator on the communication graph; 
denotes the connection weight between the ith manipulator and the jth one, with its value being 1 for 
and 0 for 
; 
with 
denoting the constant distance vector between the end‐effector and the reference point; and 
for 
and 
for 
with 
denoting the neighbor set of the command center. By combining the constraints of all manipulators in the group, the constraint for cooperative control of redundant robot manipulators can be formulated as
where 
is the Kronecker product; 
denotes a vector composed of 1; 
is an identity matrix; Laplacian matrix 
with 
being the diagonal matrix whose 
diagonal entries are the 
elements of the vector 
with the ijth element of matrix 
being 
; 
; and 
is a diagonal matrix defined as
A vector‐valued error function could be defined to construct the Jacobian equality constraint:
By exploiting the neural‐dynamic method presented in [115], we have
where 
. Expanding (9.6) leads to
where 
with 
denoting its time derivative; and
9.3.2 Distributed Scheme
We consider the following minimum velocity norm (MVN) performance index in this chapter:
Then, the distributed scheme is formulated as
So far, the MVN‐oriented distributed scheme has been formulated as a quadratic programming problem with limited communications. In the following section, the NTZNN model will be constructed to solve such a problem online. Note that the differences of different types of QP‐based redundancy‐resolution schemes (e.g. MVN‐type scheme, RMP‐type scheme, or even their weighted combinations) lie in the performance index. Therefore, the proposed distributed scheme is capable of other performance indices and can be extended by following similar steps to those presented in this chapter.
9.4 NTZNN Solver and Theoretical Analyses
Here, we present an NTZNN model to solve the MVN‐oriented distributed scheme (9.8) for the motion generation of multiple redundant robot manipulators with guaranteed convergence and robustness.
9.4.1 ZNN for Real‐Time Redundancy Resolution
As a novel type of RNN specifically designed from the perspective of control for solving time‐varying problems, NTZNN is able to accurately track a time‐varying solution with the guaranteed capability of noise tolerance and suppression. Therefore, we exploit an NTZNN model for the online solution of the MVN‐oriented distributed scheme (9.8).
Define a Lagrange function as follows:
where 
is the Lagrange‐multiplier vector. According to the Lagrange‐multiplier method [58], the solution has to satisfy the following
Then, solving (9.8) can be done by solving the following equation:
where
with 
. We define the following vector‐valued indefinite error function as
From a control perspective, the error 
can be interpreted as a distance measure between 
and 
. If the error 
approaches zero with time, then the variable 
is correspondingly driven to 
of (9.11). To force 
to be zero, the following NTZNN design formula is adopted:
where 
and 
. We further obtain the following distributed NTZNN model:
In addition, the architecture for an electronic implementation of the distributed NTZNN model (9.13) is depicted in Figure 9.1.
Figure 9.1 Circuit schematic of the distributed NTZNN model (9.13).
To lay a basis for further investigation on the robustness of the MVN‐oriented distributed scheme (9.8) aided by distributed NTZNN model (9.13) under the pollution of unknown noises, we have the following equation:
where 
denotes the vector‐form noises.
Consider the following dynamical system:
Such a problem can be described in control terminology as follows: a controller 
should be designed to drive 
to zero, and, consequently, 
is driven to converge to the desired solution 
. Therefore, such a computation problem can thus be formulated as a regulation problem in control fields, in which the output 
needs to be regulated to zero. In this sense, from NTZNN design formula (9.12), the controller is designed as follows:
From the above expression, 
(or the distributed NTZNN model) can be viewed as a generalized PID controller. Specifically, 
is the control gain, 
is the proportional term, 
is the integral term, and 
is the derivative term.
9.4.2 Theoretical Analyses and Results
The analyses on the stability, convergence and robustness of the proposed distributed NTZNN model (9.13) are conducted via the following theorems.
There always exist communication noises, computational errors, perturbations or even their superpositions for the cooperative motion generation of redundant robot manipulators, which can be deemed as noises. Sometimes those noises have significant impacts on the accuracy of the computational model for the problem solving, and in some cases, they may cause failure of the solving task. Therefore, it is worth investigating the robustness of the distributed NTZNN model (9.13). The following theorem is presented to discuss the performance of the distributed NTZNN model (9.13) with constant noise 
.
In addition, regarding the random noise, we have the following theorem.
9.5 Illustrative Examples
There are two procedures for a distributed network of redundant manipulators cooperatively transporting a payload. The first procedure: each end‐effector of all manipulators should reach a desired configuration (the consensus of redundant robot manipulators) for holding a different place or a handle on the payload. The second procedure: each end‐effector of all manipulators should keep a desired format during the execution to fulfill certain tasks (the cooperative control of redundant robot manipulators). That is to say, a reference point on the payload, e.g. the center of mass, is expected to stay at the reference position or track the reference trajectory. In this section, computer simulations on the motion generation (consensus and control) of redundant robot manipulators are conducted based on a group of PUMA 560 manipulators to illustrate the effectiveness of the proposed MVN‐oriented distributed scheme (9.8) as well as the distributed NTZNN model (9.13). The PUMA 560 robot manipulator has six revolutionary joints, which connect seven links in series with the last one being an end‐effector. In general, when we only consider the position of the end‐effector, a PUMA 560 robot manipulator can be deemed as a functionally redundant robot. For the modeling details on the kinematics of a PUMA 560 robot manipulator, please refer to [125].
9.5.1 Consensus to a Fixed Configuration
In this consensus example, we choose 
, 
, 
, 
, constant noise 
, and task duration is 5 s; the initial joint state of each manipulator is randomly generated. In addition, 
is set as
and 
with 
for 
. Besides, the initial values of distributed NTZNN model (9.13) are set as zero (e.g. 
). The corresponding simulation results based on the MVN‐oriented distributed scheme (9.8) for consensus of redundant robot manipulators solved by distributed NTZNN model (9.13) are illustrated in Figure 9.3.
Figure 9.3 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 and R
denotes the ith redundant robot manipulator. (a) Motion trajectories; (b) profiles of end‐effector position errors; (c) profiles of joint angle; and (d) profiles of joint velocity.
Specifically, Figure 9.3b plots that trajectories of end‐effectors of eight PUMA 560 manipulators are all driven to the desired positions. Figure 9.3b shows that, after the transient state, the position errors of the end‐effector converge to zero rapidly in spite of noises. In addition, Figure 9.3c and d illustrate the smooth profiles of joint angle and joint velocity, respectively, which further demonstrates the effectiveness of the proposed MVN‐oriented distributed scheme (9.8) for the consensus of multiple redundant manipulators in the presence of noises.
9.5.2 Cooperative Motion Generation Perturbed by Noises
In this section, we consider eight PUMA 560 manipulators for cooperative payload transport perturbed with noise 
. The desired motion issued by the command center is to track a circular path with radius 0.3 m at an angular speed of 0.2 rad/s around its center. In addition, only manipulators 1 and 5 are able to access the desired motion information provided by the command center. The parameters are chosen as 
, 
, 
, and task duration is 
s. In addition, the rest parameters are set the same as those in the previous example. The simulations run from a random initialization are generated as shown in Figures 9.4 and 9.5.
Figure 9.4 Three‐dimensional view of motion trajectories of eight PUMA 560 robot manipulators synthesized by MVN‐oriented distributed scheme (9.8) as well as distributed NTZNN model (9.13) perturbed with noise 
for cooperative payload transport along a time‐varying circular reference.
Figure 9.5 Computer simulations synthesized by MVN‐oriented distributed scheme (9.8) as well as distributed NTZNN model (9.13) perturbed with noise 
for cooperative payload transport of eight PUMA 560 robot manipulators along a time‐varying circular reference with limited communications. Profiles of (a) position errors, (b) joint angle, and (c) joint velocity.
To show the detailed task execution process, the three‐dimensional view of motion trajectories is illustrated in Figure 9.4. It can be observed from this figure that, in spite of different poses for all manipulators as well as noises, they all execute the task denoted by the circular trajectory cooperatively and successfully. In addition, all the end‐effector position tracking errors shown in Figure 9.5a are of order 
m, which remain at a very tiny value during the task execution. The corresponding profiles of joint angle and joint velocity are shown in Figure 9.5c and d, respectively. It is observable in Figure 9.5c that the resulting 
move smoothly. These simulation results substantiate the effectiveness of MVN‐oriented distributed scheme (9.8) and distributed NTZNN model (9.13).
9.5.3 ZNN‐Based Solution Perturbed by Noises
In this section, we show the cooperative motion generation with the aid of the original ZNN model perturbed by noises. A ZNN‐based solution is produced when 
with the rest of the parameters set the same as those in the previous example. It can be observed from Figure 9.6a that, perturbed with noise 
, the end‐effector position errors are larger than those illustrated in Figure 9.5a. In addition, it can be found in Figure 9.6b and c that the corresponding joint‐angle and joint‐velocity profiles oscillate with the time, which further substantiates the superiority and effectiveness of MVN‐oriented distributed scheme (9.8) with the aid of distributed NTZNN model (9.13).
Figure 9.6 Computer simulations synthesized by MVN‐oriented distributed scheme (9.8) as well as distributed ZNN model perturbed with noise 
for cooperative payload transport of eight PUMA 560 robot manipulators along a time‐varying circular reference with limited communications. Profiles of (a) end‐effector position errors, (b) joint angle, and (c) joint velocity.
9.6 Summary
In this chapter, a distributed scheme has been proposed for the cooperative motion generation of multiple redundant manipulators. The proposed scheme can simultaneously achieve the specified primary task to reach global cooperation under limited communications among manipulators and optimality in terms of a specified optimization index of redundant robot manipulators. The proposed distributed scheme has been reformulated as a quadratic program. To inherently suppress noises, a NTZNN has been constructed to solve the quadratic program problem online. Then, theoretical analyses have shown that, without noise, the proposed distributed scheme is able to execute a given task with exponentially convergent position errors. Moreover, in the presence of noise, the proposed distributed scheme with the aid of a NTZNN model has a satisfactory performance. Furthermore, simulations and comparisons based on PUMA 560 redundant robot manipulators have substantiated the effectiveness and accuracy of the proposed distributed scheme with the aid of a NTZNN model.