Analytic minimum-norm solution for rate coordination in redundant manipulators

In this paper, we study minimal zero norm solutions of the linear complementarity problems, defined as the solutions with smallest cardinality. Minimal zero norm solutions are often desired in some real applications such as bimatrix game and portfolio selection. We first show the uniqueness of the minimal zero norm solution for Z-matrix linear complementarity problems. To find minimal zero norm solutions is equivalent to solve a difficult zero norm minimization problem with linear complementarity constraints. We then propose a p norm regularized minimization model with p in the open interval from zero to one, and show that it can approximate minimal zero norm solutions very well by sequentially decreasing the regularization parameter. We establish a threshold lower bound for any nonzero entry in its local minimizers, that can be used to identify zero entries precisely in computed solutions. We also consider the choice of regularization parameter to get desired sparsity. Based on the theoretical results, we design a sequential smoothing gradient method to solve the model. Numerical results demonstrate that the sequential smoothing gradient method can effectively solve the regularized model and get minimal zero norm solutions of linear complementarity problems.

The redundant manipulator as a ubiquitous and essential component of robots, acts as a momentous role in mechanized production. The motion control of a redundant manipulator is an essential problem that must be handled. Consequently, for the repetitive motion control of redundant manipulators, a recurrent neural network (RNN) model for the double-index (DI) scheme is researched in this paper. The DI scheme sets the minimum kinetic energy (MKE) and minimum joint-angle offset (MJAO) as optimization indexes, for which the time-dependent weights are designed, and takes the dynamic equations of redundant manipulators and joint limit constraints into consideration as well. Then, the DI scheme is reformulated as a quadratic programming (QP) problem. Besides, the RNN model based on the iteration of related parameters is derived to solve the QP problem, through which joint data (i.e., joint angles, joint velocities, and joint accelerations) are obtained to drive the motion of the redundant manipulator. To verify the effectiveness of the DI scheme for motion control of redundant manipulator, simulation results of the DI scheme solved by the RNN model, the minimum acceleration norm (MAN) scheme and MKE scheme are compared.

The minimum torque norm control scheme for redundant manipulators has long been known to exhibit instabilities. In the paper an analysis is made of a class of acceleration-level redundancy resolution schemes that includes this and other well-known schemes. It is proved that divergence of the joint velocity norm to infinity in finite time is possible, and in fact does occur for self-motions of mechanisms with one degree of redundancy under almost all such schemes in the class, i.e., generically. The schemes that do not diverge in finite time are associated with conserved quantities and include minimum acceleration norm and dynamically-consistent redundancy resolution schemes. It is also proved that exponential dissipation of nullspace velocity cannot eliminate the divergence.

In view of the advantages of the solution of the reduced biquaternion matrix equation, which was originally employed to solve the color image deblurred problems, we study the minimal norm Hermitian solution, pure imaginary Hermitian solution and pure real Hermitian solution of the reduced biquaternion matrix equation by presenting a new real representation method and establish the necessary and sufficient conditions for the existence of the above solutions. It is demonstrated that our real representation method can also obtain the minimal norm least squares Hermitian solution, the minimal norm least squares pure imaginary Hermitian solution and the minimal norm least squares pure real Hermitian solution of the studied reduced biquaternion matrix equation when it is inconsistent. Finally, we provide some numerical examples to illustrate the feasibility and validity of our method in comparison with the complex representation method, especially in terms of computing error and CPU time. Experimental results demonstrate that the proposed method is feasible and much more effective for color image restoration than the complex representation method.

A redundant manipulator is defined when more degrees of freedom (DOF) are available than the minimum number of DOF required to execute a given end-effector primary task. Our human arm, elephant trunk, and snake are also such redundant systems. Compared to non-redundant manipulators, the redundant manipulator naturally has wider operational space and extra degrees to meet more functional constraints, such as the online avoidance of joint physical limits and environmental obstacles. One of the most fundamental issues in operating the redundant manipulators is the redundancy-resolution problem. That is, given the Cartesian velocity/acceleration trajectories of the end-effector, we are required to generate the corresponding joint velocity, acceleration, and/or torque trajectories in real time.

Closed-loop inverse kinematics (CLIK) algorithm mostly resolves the redundancy at the velocity level. In this paper we extend the CLIK algorithm to the acceleration level to meet some applications that require the joint accelerations. The redundancy resolutions at the velocities and acceleration levels via pseudoinverse method are analyzed respectively. The objective function of joint limits avoidance(JLA) is combined into the redundancy resolution as an optimization approach of the null space motion. A seven-DOF redundant manipulator is designed to do the computer simulations and the real experiments are carried out on a Powercube modular manipulator. Their results demonstrated the effectiveness of the proposed algorithm.

Fault-tolerant motion planning and control of redundant manipulator

A redundant manipulator can achieve multiple tasks using the degree of redundancy.In this paper, a redundancy resolution problem is formulated with multiple criteria into a local equality and inequality constrained optimization problem.To achieve multiple tasks, the task with the highest priority can be performed by optimizing a corresponding objective function.The forward kinematic relation and the other tasks with higher priority are performed by satisfying an equality constraint and a set of inequality constraints, respectively.The solution of the path planning is then obtained using an optimization procedure.Different to the conventional approach, the inverse of the Jacobian matrix is not used in the proposed method.The proposed method also provides a unified approach to solve motion planning problems with multiple tasks.

The morphological design of kinematically redundant robotic manipulators is a non-trivial, task-specific process. The availability of redundant degrees of freedom (DOF) affords a redundant manipulator an infinite number of motion solutions for a given task, thereby allowing the achievement secondary manipulation goals. Motion control methods play an integral role in the quality of redundancy resolution, but the degree of resolution attainable by these methods is ultimately determined by morphological design. In this study, a weighted isotropy measure is developed for use in optimising kinematically redundant manipulator design quality. This measure is designed to quantify manipulator performance on multiple secondary motion goals. The measure is employed as a morphological fitness metric for a constrained non-linear morphology optimisation problem. Optimisation results show that using the weighted isotropy measure significantly increased the performance levels of a kinematically redundant manipulator, with respect to secondary manipulation goals, for the intended set of tasks.

To solve the problem of designing a one fault tolerant, one degree of redundancy, space manipulator; a new multi-mode function optimization algorithm is presented. Kinematics redundancy is an important method to improve the reliability of a space manipulator. A kinematics redundant manipulator design bears multi-parameters, non-linearity and multi-mode. This problem can’t be solved by a conventional optimization method. By simulating antibody search mechanism, combined with immune system theory, the antibody diversity is maintained, and full-scale searching is carried out- realizing the multi-parameters non-linear multi-mode manipulator design. The design example shows that the presented algorithm is effective.

Compressive sensing relies on the sparse prior imposed on the signal of interest to solve the ill-posed recovery problem in an under-determined linear system. The objective function used to enforce the sparse prior information should be both effective and easily optimizable. Motivated by the entropy concept from information theory, in this paper we propose the generalized Shannon entropy function and R\'{e}nyi entropy function of the signal as the sparsity promoting regularizers. Both entropy functions are nonconvex, non-separable. Their local minimums only occur on the boundaries of the orthants in the Euclidean space. Compared to other popular objective functions, minimizing the generalized entropy functions adaptively promotes multiple high-energy coefficients while suppressing the rest low-energy coefficients. The corresponding optimization problems can be recasted into a series of reweighted $l_1$-norm minimization problems and then solved efficiently by adapting the FISTA. Sparse signal recovery experiments on both the simulated and real data show the proposed entropy functions minimization approaches perform better than other popular approaches and achieve state-of-the-art performances.

The paper presents a knowledge based approach to redundancy resolution of redundant manipulators using self motion topology knowledge. The approach can work under joint limits. End effector positions with negligible error can be attained by increasing the resolution of the configuration space maps being processed. Using the presented approach, performance of different path planning alternatives can also be evaluated for redundant manipulators.

In this paper, we present a tutorial report of the literature on the damped-least squares method which has been used for computing velocity inverse kinematics of robotic manipulators. This is a local optimization method that can prevent infeasible joint velocities near singular configurations by using a damping factor to control the norm of the joint velocity vector. However, the exactness of the inverse kinematic solution has to be sacrificed in order to achieve feasibility. The damping factor is an important parameter in this technique since it determines the trade-off between the accuracy and feasibility of the inverse kinematic solution. Various methods that have been proposed to compute an appropriate damping factor are described. Redundant manipulators, possessing extra degrees of freedom, afford more choice of inverse kinematic solutions than do non-redundant ones. The damped least-squares method has been used in conjunction with redundancy resolution schemes to compute feasible joint velocities for redundant arms while performing an additional subtask. We outline the different techniques that have been proposed to achieve this objective. In addition, we introduce an iterative method to compute the optimal damping factor for one of the redundancy resolution techniques.

l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimization was utilized in the imputation of microarray missing values, which is an important procedure in bioinformatics experiments. Two l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> approaches, based on the framework of local least squares (LLS) and iterative bicluster-based least squares (bicluster-iLLS) respectively, were employed. Imputed datasets of the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> approaches were compared with those of traditional l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> methods. The imputation error rates showed that the assumption of sparsity is not supported by the microarray datasets. Singular value decompositions in biclusters and in the neighborhoods of target gens were computed to show the structure of a microarray dataset. The coefficients of l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> minimization solutions were also analyzed to reveal possible reasons for the performance of l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> approaches.

A kinematically-(joint)-redundant spatial manipulator will not be velocity degenerate (singular) if any combination of six of its joints span the 6-system of general velocity. Therefore, all 6x6 sub-matrices of the matrix of joint screw coordinates J (commonly referred to as the Jacobian) must be rank deficient for a spatial joint-redundant manipulator to be velocity degenerate. Concurrency of zero 6x6 sub-matrix determinants of J allows sets of joint-displacement conditions leading to velocity-degenerate configurations to be identified. For actual manipulators, each set of degeneracy conditions must be checked to see if the conditions are feasible for existing device dimensions and occur within existing joint displacement limits. Within this work, the concept of concurrency of zero 6x6-sub-matrix determinants of J is considered and applied to the identification of velocity degeneracies for the Canadian Space Agency (CSA)/International Submarine Engineering (ISE) STEAR Testbed Manipulator (STM). The STM consists of two 7-jointed arms that are kinematically similar to the arms of the Special Purpose Dextrous Manipulator (SPDM) proposed for use on the International Space Station (ISS). Five unique 2-condition families of velocity degenerate configurations are found to cause velocity degeneracies for each of the 7-jointed arms. It is shown that only four of these five potential degeneracies correspond to reachable families of joint-displacement configurations for the CSA/ISE STM arms.