A set of nonlinear equations and inequalities arising in robotics and its online solution via a primal neural network

doi:10.1016/j.neucom.2005.11.006

Neurocomputing

Volume 70, Issues 1–3, December 2006, Pages 513-524

https://doi.org/10.1016/j.neucom.2005.11.006 Get rights and content

Abstract

In this paper, for handling general minimum-effort inverse-kinematic problems, the nonuniqueness condition is investigated. A set of nonlinear equations and inequality is presented for online nonuniqueness-checking. The concept and utility of primal neural networks (NNs) are introduced in this context of dynamical inequalities and constraints. The proposed primal NN can handle well such a nonlinear online-checking problem in the form of a set of nonlinear equations and inequality. Numerical examples demonstrate the effectiveness and advantages of the primal NN approach.

Introduction

Redundant manipulators are robots having more degrees-of-freedom (DOF) than required to perform a given end-effector task [15]. Our human arm is also such a redundant system [8]. One fundamental issue in operating such systems is the redundancy-resolution problem [25], [29], [30]. By resolving the redundancy, the robots can avoid obstacles, joint physical limits, as well as to optimize various performance criteria. In addition to minimum-energy redundancy resolution [15], [25], [30], another interesting approach is the minimum-effort redundancy resolution [3], [5], [9], [16], [31]. The minimum-effort resolution may encounter discontinuities due to the nonuniqueness of the solution at some time instants [6]. The nonuniqueness of the minimum-effort solution was previously checked by a geometrical method [6]. As shown in this paper, the nonuniqueness can be checked more generally and efficiently by solving online a set of nonlinear equations and inequalities.

It is worth mentioning that in view of its fundamental role arising in numerous fields of science and engineering, the problem of solving groups of linear equations and inequalities has been investigated extensively for the past decades. For example, about the recent research based on recurrent neural networks (NNs) (specifically, the Hopfield-type NNs), see [2], [10], [11], [18], [19], [21], [26], [27] and the references therein. The NN approach is now thought to be a powerful tool for online solutions, in light of its parallel distributed computing nature and hardware implementability [1], [4], [13], [19].

In the aforementioned literature, most researches are devoted to solving linear equations and inequalities only (with the matrix-inverse problem as a special case). Among them, the mathematical description can be unified as $Ey ⩽ f$ , $Cy = g$ where y is the unknown vector to be determined, given coefficient matrices E, C and coefficient vectors f, g. In the group of linear equations and inequalities, $Cy = g$ defines the problem to be solved while $Ey ⩽ f$ defines its constraints. To our best knowledge, there have been few research results about solving nonlinear or hybrid equations/inequalities based on NN approaches. On the other hand, the nonlinear situations are encountered in the minimum-effort redundancy resolution of robot manipulators. The set of such nonlinear equations and inequalities is in the form of $Ey ⩽ f$ , $Cy = g$ and $y^{T} y = h$ with given scalar $h > 0$ . The real-time computation requirement becomes stringent, especially for sensor-based robotic systems of high DOF.

The design methods and techniques for recurrent NNs are briefly reviewed as follows. In terms of the types of decision variables involved, the NN models can be divided into two classes: the pure primal NNs and the nonprimal NNs. In detail, we have the following concepts.

Concept 1: A recurrent NN is called primal NN, if the network dynamic equation and implementation only use the original (or to say, primal) decision variables, y. For example, the neural models in [2], [19], [27] are primal NNs.

Concept 2: A recurrent NN is called nonprimal NN, if the network dynamic equation and implementation use any auxiliary decision variables, in addition to the original ones. For example, the primal–dual NNs in [29], [30], the dual NNs in [20], [25], and the Lagrange NN in [22].

The earlier primal NN models such as [19] use finite penalty parameters and generate approximate solutions only. By using auxiliary variables like dual decision variables, the nonprimal NNs usually have better convergence to exact/theoretical solutions as compared to primal NNs [20]. As the recent research shows [20], [29], however, the primal NN is preferable in the context of dynamic constraints/inequalities, provided that the solution accuracy is acceptably good. In that case, the number of constraints/inequalities is dynamically changed, and thus the hardware implementation of nonprimal NNs could become less favorable, not to mention the network complexity due to using auxiliary neurons.

In this paper, motivated by the above observations, we generalize the NN design experience to handling the nonlinear/hybrid situations in minimum-effort robotics. The concept of primal NNs is thus formally proposed in solving online a set of nonlinear equations and inequalities. The remainder of this paper is organized in four sections. The problem formulation is given in Section 2. The primal NN, together with convergence results, is presented in Section 3. Illustrative numerical examples are discussed in Section 4. Lastly, Section 5 concludes this paper with final remarks. The main contributions of the paper are as follows:

(1)
Based on linear programming and equation-solving, a more efficient criterion is proposed for checking nonuniqueness-points (as compared to geometric methods).
(2)
The concept, utility and advantages of primal NNs are illustrated by developing a primal NN for such an online-checking criterion of nonuniqueness (as compared to nonprimal NNs).
(3)
Previous robotic research results are confirmed by the conducted computer simulations and analysis. That is, nonuniqueness is nearly sufficient for the appearance of discontinuities in the manipulators with low DOF/redundancy.
(4)
New robotic research results are summarized for the manipulators with high DOF/redundancy. That is, nonuniqueness is generally a necessary condition to the appearance of discontinuities in minimum-effort inverse-kinematic solutions.

Section snippets

Problem formulation

The minimum-effort redundancy resolution of robot manipulators is the following infinity-norm minimization problem [3], [9], [16]: $minimize ∥ \dot{θ} ∥_{\infty}$ $subject to J \dot{θ} = \dot{r},$ $θ^{-} ⩽ θ ⩽ θ^{+},$ ${\dot{θ}}^{-} ⩽ \dot{θ} ⩽ {\dot{θ}}^{+},$ where $θ \in R^{n}$ is the joint vector with physical limits $[θ^{-}, θ^{+}]$ , $\dot{θ} \in R^{n}$ is the joint-velocity vector with physical limits $[{\dot{θ}}^{-}, {\dot{θ}}^{+}]$ , and $\dot{r} \in R^{m}$ is the commanded Cartesian velocity vector at the manipulator's end-effector that can be planned off-line or given in real time. The manipulator Jacobian matrix is an $m \times n$ rectangular

Primal neural network

Locating the nonuniqueness points of minimum-effort inverse kinematics has now become the solution to a set of nonlinear/hybrid equations and inequality. The problem of interest can be generalized to the following form: $Ey ⩽ f, Cy = g, y^{T} y = h,$ where, in our situation, $f = 0$ , $g = 0$ , $h = 1$ . The aforementioned linear situation is a special case of (11). Here, we concentrate on the recurrent NN approach.

As shown in Concepts 1 and 2, there could exist two kinds of recurrent NNs: primal NNs and nonprimal NNs.

Computer simulations

Theoretical results about nonuniqueness-checking (9) and primal NN (13) depicted in the previous section are substantiated by the following computer simulations. The first two examples are the static ones of solving (11) with constant coefficients E and C. The remaining three examples are the robotic application of primal NN (13) to a four-link planar robot [6], a 6DOF PUMA560 robot arm [25], [28], and a 7DOF PA10 industrial manipulator [31].

Example 1

Consider the simplified model of LP (5)–(8) with $n = 2$

Concluding remarks

Minimum-effort solutions could complement the research on redundancy resolution in terms of low individual magnitude, even distribution of workload, and analyzing motion diversity. In this paper, for investigating the problem of discontinuities/nonuniqueness appearing in minimum-effort solutions, a set of nonlinear equations and inequalities has been formulated as a nonuniqueness criterion. Due to the real-time computation requirement, a primal NN has been developed to solve online this

Acknowledgements

The author would like to thank the editors and the anonymous reviewers for their time and effort in providing many constructive comments. Here, the gratitude is expressed to them, especially considering that I was making revisions around Thanksgiving day.

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Yunong Zhang was born in Xinyang, Henan, PR China in October 1973. He received the B.E. degree from the Huazhong University of Science and Technology (HUST) in 1996, the M.E. degree from the South China University of Technology (SCUT) in 1999. He completed his Ph.D. study in the Chinese University of Hong Kong (CUHK) in November 2002 with the degree received in 2003. Then, as a research fellow he had been with the National University of Singapore (NUS), and the University of Strathclyde, United Kingdom, respectively, in 2003 and 2004. After that, he has been with the National University of Ireland, Maynooth (NUIM) as a research scientist. His current research interests are redundant robot manipulators and the related biomechanics research, recurrent neural networks and their hardware/circuits implementation, and scientific computing and optimization such as Gaussian process regression.

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A set of nonlinear equations and inequalities arising in robotics and its online solution via a primal neural network

Abstract

Introduction

Section snippets

Problem formulation

Primal neural network

Computer simulations

Concluding remarks

Acknowledgements

J. Biomechanics

Linear Algebra Appl.

Appl. Math. Comput.

Hardware implementation of an artificial neural network using field programmable gate arrays (FPGAs)

IEEE Trans. Ind. Electron.

Neural networks for solving systems of linear equation and related problems

IEEE Trans. Circuits Syst. I

Minimum effort inverse kinematics for redundant manipulators

IEEE Trans. Robotics Autom.

Neural circuits in silicon

Nature

On the structure of minimum effort solutions with application to kinematic redundancy resolution

IEEE Trans. Robotics Autom.

Control of Human Movement

A structured algorithm for minimum l∞-norm solutions and its application to a robot velocity workspace analysis

Robotica

An improved upper bound on step-size parameters of discrete-time recurrent neural networks for linear inequality and equation system

IEEE Trans. Circuits Syst.

Neural network approach to computing matrix inversion

Appl. Math. Comput.

Analog VLSI and Neural Systems

Linear Programming: a Modern Integrated Analysis

A structured algorithm for minimum $l_{\infty}$ -norm solutions and its application to a robot velocity workspace analysis