Dynamically consistent Jacobian inverse for mobile manipulators

By analogy to the definition of the dynamically consistent Jacobian inverse for robotic manipulators, we have designed a dynamically consistent Jacobian inverse for mobile manipulators built of a non-holonomic mobile platform and a holonomic on-board manipulator. The endogenous configuration space approach has been exploited as a source of conceptual guidelines. The new inverse guarantees a decoupling of the motion in the operational space from the forces exerted in the endogenous configuration space and annihilated by the dual Jacobian inverse. A performance study of the new Jacobian inverse as a tool for motion planning is presented.     

Iterative learning control and the singularity robust Jacobian inverse for mobile manipulators

The method of iterative learning control, to a large extent, has been inspired by robotics research, focused on the control of stationary manipulators. In this article we deal with the inverse kinematics problem for mobile manipulators, and show that a very basic singularity robust Jacobian inverse can be derived in a natural way within the framework of iterative learning control. To achieve this objective we have exploited the endogenous configuration space approach. The introduced Jacobian inverse defines the singularity robust Jacobian inverse kinematics algorithm for mobile manipulators. A Kantorovich-type estimate of the region of guaranteed convergence of the algorithm is derived. For two example kinematics, this estimate has been computed efficiently.     

Formal analysis of the kinematic Jacobian in screw theory

As robotic systems flourish, reliability has become a topic of paramount importance in the human–robot relationship. The Jacobian matrix in screw theory underpins the design and optimization of robotic manipulators. Kernel properties of robotic manipulators, including dexterity and singularity, are characterized with the Jacobian matrix. The accurate specification and the rigorous analysis of the Jacobian matrix are indispensable in guaranteeing correct evaluation of the kinematics performance of manipulators. In this paper, a formal method for analyzing the Jacobian matrix in screw theory is presented using the higher-order logic theorem prover HOL4. Formalizations of twists and the forward kinematics are performed using the product of exponentials formula and the theory of functional matrices. To the best of our knowledge, this work is the first to formally analyze the kinematic Jacobian using theorem proving. The formal modeling and analysis of the Stanford manipulator demonstrate the effectiveness and applicability of the proposed approach to the formal verification of the kinematic properties of robotic manipulators.     

Formulation of unified Jacobian for serial-parallel manipulators

This paper presents a general approach for Jacobian analysis of serial-parallel manipulators (S-PMs) formed by two lower mobility parallel manipulators (PMs) connected in serials. Based on the kinematic relation, coupling and constraint properties of each PM, the unified forward and inverse Jacobian matrices for S-PMs are derived in explicit form. It is shown that the Jacobian matrices of S-PMs have unified forms, which include the complete information of each PM. A (3-RPS)+(3-SPS/UP) S-PM is used as an example to demonstrate the proposed approach. The established model is applicable for S-PMs with various architectures.     

Extended Jacobian inverse kinematics algorithms for mobile manipulators

We consider the inverse kinematic problem for mobile manipulators consisting of a nonholonomic mobile platform and a holonomic manipulator on board the platform. The kinematics of a mobile manipulator are represented by a driftless control system with outputs together with the associated variational control system. The output reachability map of the driftless control system determines the instantaneous kinematics, while the output reachability map of the variational system plays the role of the analytic Jacobian of the mobile manipulator. Relying on a formal analogy between the kinematics of stationary and mobile manipulators we exploit the extended Jacobian construction in order to design a collection of extended Jacobian inverse kinematics algorithms for mobile manipulators. It has been proved mathematically and confirmed in computer simulations that these algorithms are capable of efficiently solving the inverse kinematic problem. Moreover, a choice of the Jacobian extension may lay down some guidelines for the platform‐manipulator motion coordination. © 2002 Wiley Periodicals, Inc.     

一种新的PUMA类型机器人奇异回避算法

传统的奇异回避方法运算量大, 本文提出了一种新的 PUMA 类型机器人奇异回避方法—奇异分离加阻尼倒数法. 首先, 分析产生奇异的条件, 将导致 Jacobian 奇异的参数分离出来, 然后用阻尼倒数代替其普通倒数, 以回避运动学奇异的影响. 该方法无需对 Jacobian 进行 SVD 分解, 也无需估计其最小奇异值, 因而运算量小, 实时性好, 仅牺牲末端部分方向的精度, 适合于预定轨迹和实时轨迹的跟踪. 仿真和实验结果证明了算法的有效性.     

Scaled Jacobian transpose based control for robotic manipulators

This paper presents a scaled Jacobian transpose based control method for robotic manipulators as a modification of a conventional Jacobian transpose based method. The proposed method has several advantages such as it shows faster convergence and better tracking performance than the conventional method, furthermore, it does not have any singularity problem similar to the conventional method. The scaled Jacobian transpose is obtained by collecting each pseudoinverse of the column vector of the Jacobian matrix. The proposed method performs a given task well under singular configurations while minimizing the task error. Finally, a few comparative studies with the conventional method are provided to show the effectiveness of the proposed method through simulations.     

Inverse statics analysis of planar parallel manipulators via Grassmann-Cayley algebra

The wrench Jacobian matrix plays an important role in the statics and singularity analysis of planar parallel manipulators (PPMs). The Jacobian matrix can be calculated based on the conventional Plücker coordinate method. However, this method cannot be applied when two links are in parallel. A new approach is proposed for the analysis of the forward and inverse wrench Jacobian matrix using Grassmann-Cayley algebra (GCA). A symbolic formula for the inverse statics analysis is obtained based on the Jacobian. The proposed method can be applied when two links are in parallel. The approach is explained in detail based on a planar 3-RPR PPM example, and the analysis procedure for nine other PPMs is also presented. This novel approach to deriving the statics can be applied to spatial parallel manipulators and redundant cases of PPMs.

     

Comparison of Extended Jacobian and Lagrange Multiplier Based Methods for Resolving Kinematic Redundancy

Several methods have been proposed in the past for resolving the control of kinematically redundant manipulators by optimizing a secondary criterion. The extended Jacobian method constrains the gradient of this criterion to be in the null space of the Jacobian matrix, while the Lagrange multiplier method represents the gradient as being in the row space. In this paper, a numerically efficient form of the Lagrange multiplier method is presented and is compared analytically, computationally, and operationally to the extended Jacobian method. This paper also presents an improved method for tracking algorithmic singularities over previous work.     

Singularity Analysis of Lower Mobility Parallel Manipulators Using Grassmann–Cayley Algebra

This paper introduces a methodology to analyze geometrically the singularities of manipulators, of which legs apply both actuation forces and constraint moments to their moving platform. Lower mobility parallel manipulators and parallel manipulators, of which some legs have no spherical joint, are such manipulators. The geometric conditions associated with the dependency of six PlUumlcker vectors of finite lines or lines at infinity constituting the rows of the inverse Jacobian matrix are formulated using Grassmann-Cayley algebra (GCA). Accordingly, the singularity conditions are obtained in vector form. This study is illustrated with the singularity analysis of four manipulators.     

Extended Jacobian inverse kinematics algorithm for nonholonomic mobile robots

By a generalization of the well-known extended Jacobian method for stationary manipulators, we derive the extended Jacobian inverse kinematics algorithm for nonholonomic mobile robots. Key points of the derivation consist in defining the kinematics of a mobile robot as the end-point map of a driftless control system, decomposing the space of control functions of this system into a finite and an infinite dimensional subspaces, and introducing an augmenting kinematics map subordinated to this decomposition. The original kinematics and the augmenting kinematics constitute the extended kinematics. The inverse Jacobian of the extended kinematics defines the extended Jacobian inverse kinematics algorithm. By design, the algorithm is repeatable. As an example, we derive a specific extended Jacobian inverse kinematics algorithm and illustrate its performance with the computer simulations.     

Approximate Jacobian control with task-space damping for robot manipulators

In this note, we propose two new approximate Jacobian control laws with task-space damping for setpoint control of robot manipulators. The proposed controllers do not require exact knowledge of the Jacobian matrix and dynamics of the robots. We will show that the end-effector's position converges to a desired position in a finite task space even when the kinematics and Jacobian matrix are uncertain. Experimental results are presented to illustrate the performance of the proposed controllers.     

Inverse Jacobian Regulator With Gravity Compensation: Stability and Experiment

Task-space regulation of robot manipulators can be classified into two fundamental approaches, namely, transpose Jacobian regulation and inverse Jacobian regulation. In this paper, two inverse Jacobian regulators with gravity compensations are presented, and the stability problems are formulated and solved. It is shown that the inverse Jacobian systems can be stabilized, and there exists a region of attraction such that the system remains stable. Our results show that the two fundamental approaches are two dual controllers, in the sense that the transpose Jacobian matrix can be replaced by the inverse Jacobian matrix and vice versa. The theoretical results are verified experimentally by implementing the inverse Jacobian regulators on an industrial robot, PUMA560.     

Inverse jerk analysis of symmetric zero-torsion parallel manipulators

In this work the inverse velocity, acceleration and jerk analyses of a class of three-degrees-of-freedom parallel manipulators are approached by means of the theory of screws. The concept of reciprocal screws allows to obtain simple and decoupled expressions to compute the joint rates, up to the third time derivative, of the class of manipulators under study given the instantaneous kinematic properties of the center of the moving platform. The interdependency between the angular and linear kinematic properties of the moving platform is also derived. A case study is included.     

Performance investigations on optimum mechanical design aspects of planar parallel manipulators

This paper presents a comparative analysis of three degrees of freedom planar parallel robotic manipulators (x, y and θ_z motion platforms) namely 2PRP-PPR, 2PRR-PPR, 3PPR (Hybrid), 3PRP (Hephaist) and 3PPR U-base in terms of optimal kinematic design performance, static structural stiffness and dynamic performance (energy and power consumption). Kinematic and dynamic performance analyses of these platforms have been done using multibody dynamics software (namely ADAMS/View). Static stiffness of the above-mentioned manipulators have been analysed, compared using the conventional joint space Jacobian stiffness matrix method, and this method has been verified through a standard finite-element software (namely NASTRAN) as well. The size of the fixed base or aspect ratio (width/height) can be varied for various working conditions to understand its design parameters and optimal design aspects which are depending on the fixed base structure. Different aspect ratios (fixed base size) are considered for the comparative analyses of isotropy, manipulability and stiffness for the above-mentioned planar parallel manipulators. From the numerical simulation results, it is observed that the 2PRP-PPR manipulator is associated with a few favourable optimum design aspects such as singularity-free workspace, better manipulability, isotropy, higher stiffness and better dynamic performance in terms of power and energy requirement as compared to other planar parallel manipulators.     

Global positioning of robot manipulators with mixed revolute and prismatic joints

The existing controllers for robot manipulators with uncertain gravitational force can globally stabilize only robot manipulators with revolute joints. The main obstacles to the global stabilization of robot manipulators with mixed revolute and prismatic joints are unboundedness of the inertia matrix and the Jacobian of the gravity vector. In this note, a class of globally stable controllers for robot manipulators with mixed revolute and prismatic joints is proposed. The global asymptotic stabilization is achieved by adding a nonlinear proportional and derivative term to the linear proportional-integral-derivative (PID) controller. By using Lyapunov's direct method, the explicit conditions on the controller parameters to ensure global asymptotic stability are obtained.     

Task-Space Modular Dynamics for Dual-Arms Expressed through a Relative Jacobian

This paper presents modular dynamics for dual-arms, expressed in terms of the kinematics and dynamics of each of the stand-alone manipulators. The two arms are controlled as a single manipulator in the task space that is relative to the two end-effectors of the dual-arm robot. A modular relative Jacobian, derived from a previous work, is used which is expressed in terms of the stand-alone manipulator Jacobians. The task space inertia is expressed in terms of the Jacobians and dynamics of each of the stand-alone manipulators. When manipulators are combined and controlled as a single manipulator, as in the case of dual-arms, our proposed approach will not require an entirely new dynamics model for the resulting combined manipulator. But one will use the existing Jacobians and dynamics model for each of the stand-alone manipulators to come up with the dynamics model of the combined manipulator. A dual-arm KUKA is used in the experimental implementation.     

Identification and resolution of structural and algorithmic singularity in redundancy control of serial manipulations

A major difficulty that has haunted most researchers in the process of optimal redundancy resolution of robotic manipulators is the instability observed in even very simple numerical simulations. This numerical instability is not related to the structurally singular configurations of the manipulators, and in the literature has been referred to as “algorithmic singularity,” “artificial singularity,” or “unavoidable singularity.” In this work, conditions on both structural and algorithmic singularities are studied based on the Singular Value Decomposition of the Jacobian matrix, and, hence, a singularity-free control algorithm for redundant manipulators is developed and resolved as the Lagrange problem of optimal control. It is shown that many well-known methods for optimal redundant manipulation in the literature, including the Extended Jacobian Technique, most of constraint function-based methods, and most of the previously reported methods on global optimization techniques, are all special cases of the formulation provided here. Further, the necessary conditions of the global optimality for this general formulation are derived in explicit form and the source of “algorithmic singularity” is rigorously identified and resolved. © 2995 John Wiley & Sons, Inc.     

Approximation of Jacobian Inverse Kinematics Algorithms: Differential Geometric vs. Variational Approach

This paper addresses the approximation problem of Jacobian inverse kinematics algorithms for redundant robotic manipulators. Specifically, we focus on the approximation of the Jacobian pseudo inverse by the extended Jacobian algorithm. The algorithms are defined as certain dynamic systems driven by the task space error, and identified with vector field distributions. The distribution corresponding to the Jacobian pseudo inverse is non-integrable, while that associated with the extended Jacobian is integrable. Two methods of devising the approximating extended Jacobian algorithm are examined. The first method is referred to as differential geometric, and relies on the approximation of a non-integrable distribution (in fact: a codistribution) by an integrable one. As an alternative, the approximation problem has been formulated as the minimization of an approximation error functional, and solved using the methods of the calculus of variations. Performance of the obtained extended Jacobian inverse kinematics algorithms has been compared by means of computer simulations involving the kinematics model of the 7 dof industrial manipulator POLYCRANK. It is concluded that the differential geometric method offers a rapid, while the variational method a systematic tool for solving inverse kinematic problems.     

Modified Transpose Effective Jacobian Law for Control of Underactuated Manipulators

Underactuated manipulators consist of active and passive joints, and developing a control technique that can manage such systems is an attractive, challenging problem. Most works in this area present model-based control laws that require a full dynamics model, and are consequently affected from uncertainties and time delays due to massive computations. Non-model-based control approaches provide an efficient alternative for practical implementation. The Modified Transpose Jacobian (MTJ) algorithm is one of these controllers that has been recently proposed for fully actuated manipulators with a square matrix Jacobian. Based on an approximated feedback linearization approach, the MTJ does not need a priori knowledge of the plant dynamics. In this paper, this scheme is extended to the complicated control problem of underactuated robots in Cartesian space. To this end, the notion of the Transpose Effective Jacobian (TEJ) is presented and so the proposed algorithm is called the Modified TEJ (MTEJ) algorithm. The MTEJ control law employs stored data of the control command in the previous time step, as a learning tool to yield an improved performance. Therefore, the proposed law needs just to a portion of mass matrix that corresponds to passive joint(s), and it is much less affected by inaccuracies in system properties. The gains of the proposed MTEJ can be selected more systematically and do not need to be large; hence, the noise rejection characteristics of the algorithm are improved. Also, no need for the pseudo-inversion of the Jacobian matrix in the proposed controller makes further convenience in the underactuated cases. In addition, the relationship between kinematic and dynamic manipulability measures is discussed for underactuated manipulators. Obtained results show its superior performance even compared to that of the model-based algorithms that need full dynamics models, while the proposed MTEJ requires much lower computation effort.