Inverse Position Problems Research Articles

Direct control of the endpoint of the manipulator under non-smooth uncertainty and reference trajectories

The problem of tracking the reference trajectory for the manipulator endpoint is considered under the following assumptions: the matrices of the mechanical system have uncertain parameters; external non-smooth disturbances act on the system; and only the vector of generalized coordinates of the manipulator is measured. This vector is uniquely recalculated into the vector of positions of the endpoint. A direct method for the synthesis of a tracking system is proposed, which does not require solving the inverse position problem. The manipulator model is represented in the canonical input-output form considering the vector of the endpoint. The input is a vector of generalized moments developed by the actuators. The complexity of regulation consists of the uncertainty of the input matrix. To stabilize tracking errors under these conditions, a method of linearization by dynamic feedback has been developed. For estimating mixed variables and disturbances, an observer of the minimum possible dynamic order is proposed. A cascade procedure for tuning S-shaped smooth and nonlinear (sigmoid) corrective actions of the observer is presented. This procedure provides an estimation of non-smooth disturbances in a given time with a given accuracy. To smooth primitive trajectories that define, as a first approximation, the desired movement of the endpoint in the workspace, tracking differentiators with sigmoid local feedbacks are used. To tune a tracking differentiator of arbitrary dimension, a decomposition procedure is developed. This procedure considers physical constraints on the velocity and higher derivatives of the manipulator endpoint. The variables of such a differentiator generate smoothed trajectories and their derivatives of any desired order. These trajectories become achievable for the robot. They are used in the tracking system of the control plant as new reference trajectories. The simulation results for a 3-link manipulator demonstrate the excellent performance of the proposed algorithms.

Analysis of rotation capacity of a novel 2R1T mechanism based on origami of thick panels

A foldable and symmetrical lower-mobility parallel mechanism was proposed based on Waterbomb origami of thick panels. It consists of a moving platform, a base plate and three deployable foldable legs between moving platform and base plate. Firstly, constraint wrenches of each leg were formulated based on screw theory and the results illustrated that the moving platform is in possession of two degrees of orientation freedom and one translational degree of freedom. Secondly, it was approved that base and moving platform are always symmetrical about a middle plane and the moving platform can rotate continuously about any axis chosen freely on this plane. Solving models including forward and inverse position problems were established to determine the maximum rotational angle and workspace. Finally, performance indexe of maximum rotational angle of the PM was analyzed, and effects of two structural variables to the performance were summarized. Conclusions obtained can provide a theoretical basis for the structural design and engineering application of this 2T1R parallel mechanism.

Design and Optimization of a 6-DOF Singularity-Free Parallel Manipulator

A novel six degrees-of-freedom (DOF) singularity-free parallel manipulator (PM) is presented. The PM consists of five peripheral limbs and a centre limb, which makes the mechanism have high stiffness and large tilting capability . Due to the special architecture, the doubly actuated centre limb of the manipulator could have infinite inverse solutions. In every configuration of the end-effector, the manipulator can adapt its centre limb to the position and orientation with best dexterities. Targeting for further analysis of the manipulator, its detailed kinematic analysis is developed, including the solutions of the inverse position problems, the singularity, the dexterity and the workspace. The analysis results show that the proposed manipulator has good performance and thus may be suitable candidate for complicated surface machining.

Analysis and Synthesis of Dynamic Characteristics of Machine-Tool Equipment Based on Rotary Linear Modules

The use of mechatronic rotary linear modules in manipulators of machine-tool equipment is considered. A solution to the inverse position problem is presented, and the kinematic capabilities of a device based on mechatronic rotary linear modules for machine-tool equipment are evaluated. Dynamic equations have been derived used to evaluate the dynamic characteristics and the precision of the finishing of complex surfaces by the example of the machining of a gas-turbine engine blade feather.

Design and kinematic analysis of a new 3-DOF spherical parallel manipulator for a prosthetic wrist

This paper presents a new three-degrees-of-freedom spherical parallel manipulator, which is designed to be used as a prosthetic wrist. The inverse position problem is solved and a closed form equation is obtained. Afterward, using the screw theory approach, the mobility analysis is performed to demonstrate that the proposed mechanism performs spherical motion. The velocity analysis is carried out by means of screw theory, and an input–output velocity equation is obtained. Furthermore, a preliminary virtual design of the mechanism is presented and the workspace and analytical static analysis is performed. Finally, a set of dynamic simulations are carried out to demonstrate the three movements (i.e., pronation–supination, flexion–extension, and ulnar–radial deviation) of the wrist, and the simulation results show that the proposed mechanism is capable of performing the full range of motion required for daily living and the required actuation torques are obtained for the future development.

Spherical manipulator with parallel structure

A spherical manipulator with parallel structure is considered. The advantages and disadvantages of its various configurations are assessed. The inverse position problem is solved in general form.

An integrated study of the workspace and singularity for a Schönflies parallel manipulator

This paper presents a simple and systematic approach to formulate the inverse position problem of a Schonflies parallel manipulator. As aresult, the inverse position problem is solved in closed form and leads directly to the automatic generation of the workspace of the manipulator.Additionally, a systematic velocity analysis is also presented, which allows to detect and characterize all the singularities related to this manipulator.

Position Analysis of a Hybrid Serial-Parallel Manipulator in Immersion Lithography

This paper proposes a novel hybrid serial-parallel mechanism with 6 degrees of freedom. The new mechanism combines two different parallel modules in a serial form. 3-P̲(PH) parallel module is architecture of 3 degrees of freedom based on higher joints and specializes in describing two planes’ relative pose. 3-P̲SP parallel module is typical architecture which has been widely investigated in recent researches. In this paper, the direct-inverse position problems of the 3-P̲SP parallel module in the couple mixed-type mode are analyzed in detail, and the solutions are obtained in an analytical form. Furthermore, the solutions for the direct and inverse position problems of the novel hybrid serial-parallel mechanism are also derived and obtained in the analytical form. The proposed hybrid serial-parallel mechanism is applied to regulate the immersion hood’s pose in an immersion lithography system. Through measuring and regulating the pose of the immersion hood with respect to the wafer surface simultaneously, the immersion hood can track the wafer surface’s pose in real-time and the gap status is stabilized. This is another exploration to hybrid serial-parallel mechanism’s application.

Combined Inverse Kinematic and Static Analysis and Optimal Design of a Cable-Driven Mechanism with a Spring Spine

A special humanoid neck with low motion noise requirements yields a cable-driven parallel mechanism to imitate the rotational motion of a human neck. The fixed base and moving platform of the mechanism are connected by four cables and a column compression spring. The four cables are actuated separately, while the spring can support weight on the moving platform. Although similar mechanisms exist in the literature, the analysis of them is scarce because a flexible spring instead of a rigid kinematic chain is used as the spine. With the spring’s lateral buckling motion, a new approach must be adopted to solve the kinematics. In this paper, we propose a method that combines the kinematics with the statics to solve them simultaneously. The configuration of the moving platform is parameterized with four parameters, one of which is considered as parasitic motion. Using the spring’s lateral buckling equation, we can obtain the parasitic motion and solve the inverse position problem. The optimal design for cable placements is then performed to minimize the actuation force. The method in this paper provides a novel way to analyze parallel mechanisms with a spring spine and it can be applied to other mechanisms with flexible spines.

Kinematics of a star-triangle planar parallel manipulator

This paper addresses kinematics of a 3 degree-of-freedom (DOF) planar parallel manipulator called Star-Triangle manipulator. The manipulator has good accuracy and a relatively large singularity-free workspace. First, position analysis of the manipulator is implemented indicating that both the inverse and forward position problems of the manipulator have only one solution. Then, velocity and singularity analyses of the manipulator are carried out and its isotropic configurations are identified using two Jacobian matrices. Based on the isotropic conditions, the workspace of manipulator is determined geometrically and it is shown that all points in the workspace are free of singularities. At last, kinematic accuracy of the manipulator is investigated on the workspace through a kinematic conditioning index (KCI).

A 6-DOF adaptive parallel manipulator with large tilting capacity

In this paper, a novel 6 degrees of freedom (DOFs) adaptive parallel manipulator with large tilting capacity is presented. The manipulator consists of four identical peripheral limbs and one center limb connecting the base and the moving platform. Due to the special architecture, the doubly actuated center limb of the manipulator could have infinite inverse solutions. In every configuration of the end-effector, the manipulator can adapt its center limb to the position and orientation with best dexterity. An optimization equation for obtaining the optimized dexterity of the manipulator is introduced to solve this nonholonomic problem, which also makes the manipulator capable of large tilting capacity. Targeting for the application of five-face machining, the detailed kinematic analysis of the manipulator is developed, which includes the closed-form solutions of inverse position problems, the singularity, dexterity, workspace and tilting capability. The analysis developed in this paper shows that the proposed manipulator has large tilting capacity and thus a suitable candidate for five-face machining.

A symmetric parallel Schönflies-motion manipulator for pick-and-place operations

SUMMARYThis paper presents a new symmetric parallel Schönflies-motion generator. The design is an evolution of a previous robot with linear inputs. The complete kinematic analysis of the 4-degree-of-freedom (dof) parallel manipulator is presented. The degrees of freedom are obtained from the Group Theory, the direct and inverse position problems are solved obtaining the manipulator's workspace, and the Jacobian analysis is presented. Then the isotropic configurations of the manipulator are discussed and the local dexterity map within the workspace is produced. Finally, two alternatives of a rotational mechanical device, which will increase the angular end-effector range, are proposed.

Solving the Inverse Position Problem of a 7R Robot

In this paper a new computational technique for the inverse position problem of a 7R robot is presented. Instead of reducing the problem to one highly complicated input-output equation, we work with a system of 10 very simple polynomial equations. We show the total degree of the system is 16, in agreement with previous works. Moreover we present a numerical example confirms the technique. The whole process is simple and easy to program.

Workspace Decomposition Based Dimensional Synthesis of a Novel Hybrid Reconfigurable Robot

A novel 5-axis hybrid reconfigurable robot named Tricept-IV, including one 4-degree-of-freedom (4DOF) hybrid module and one 2DOF end effector, is investigated. Compared with extensive research that has been pursued for the parallel kinematic machines such as Tricept, Sprint Z3 Head, and so on, the hybrid kinematic machines have not adequately been studied. This paper demonstrates a method of workspace decomposition applied in the dimensional synthesis of a 4DOF hybrid module, which is the underlying architecture of the newly invented robot. This paper starts with an introduction of the Tricept-IV robot. After dividing the 4DOF hybrid module into one position-orientation coupling (POC) subsystem and one position-feed (PF) subsystem, its workspace is decomposed into POC and PF subspaces accordingly, and then the inverse position problem may be solved by means of one prejudgment method. Furthermore, the Jacobian matrix of the POC subsystem is obtained by the screw theory so as to formulate its kinematic performance index. Finally, the dimensional synthesis based on workspace decomposition of the hybrid module is carried out by taking a global view of the dimensional synthesis of these two subsystems.

Synthesis and Design of a Novel 3T1R Fully-Parallel Manipulator

In this paper a new topology of four degrees-of-freedom 3T1R fully-parallel manipulator is presented, which is defined only using lower kinematic pairs. The paper starts with a complete type synthesis of different topologies of fully-parallel manipulators that can perform the so-called Schönflies motion, based on the Theory of Groups of Displacements. After imposing some practical requirements, the different possibilities of manipulators are reduced to only one topology of fully-parallel and fully-symmetrical parallel manipulator. Then, the kinematic analysis of the manipulator is shown, including the closed-form resolution of both forward and inverse position problems, the velocity and the singularity analysis. Finally, a prototype of the manipulator is presented, which is intended to be used in pick and place applications.

A Parallelogram-Based Parallel Manipulator for Schönflies Motion

A parallelogram-based 4 degrees-of-freedom parallel manipulator is presented in this paper. The manipulator can generate the so-called Schönflies motion that allows the end effector to translate in all directions and rotate around an axis parallel to a fixed direction. The theory of group of displacements is applied in the synthesis of this manipulator, which employs parallelograms in every limb. The planar parallelogram kinematic chain provides a high rotational capability and an improved stiffness to the manipulator. This paper shows the kinematic analysis of the manipulator, including the closed-form resolution of the forward and inverse position problems, the velocity, and the singularity analysis. Finally, a prototype of the manipulator, adding some considerations about its singularity-free design, and some technical applications in which the manipulator can be used are presented.

Kinematics of a Family of Translational Parallel Mechanisms with Three 4‐DOF Legs and Rotary Actuators

AbstractThis paper focuses on the kinematics of a family of translational parallel mechanisms equipped with three 4‐DOF legs and rotary actuators. The direct and the inverse position problems are solved in analytical form, the velocity analysis is carried out, the workspace is determined and the loci of both kinematic singularities and isotropic configurations are derived. Furthermore, the problem of singularity avoidance by means of actuator redundancy is addressed and some solutions are proposed. Two special architectures are finally considered as case studies: in the first, the three actuation axes are mutually orthogonal; in the second, two actuation axes are parallel to each other and the third is perpendicular to them. The comparison of the two architectures on the basis of kinematic considerations allows for the selection of the second one as a preferable solution. © 2003 Wiley Periodicals, Inc.

Inverse position problem in highly redundant multibody systems in environments with obstacles

This paper looks at a method for the analysis of highly redundant multibody systems (e.g. in the case of cellular adaptive structures of variable geometry) in environments with obstacles. It is sought to solve the inverse problem in successive positions of multibody systems, avoiding the obstacles in its work environment; i.e. the computation of the increment that has to be assigned to the actuators throughout the movement of the multibody system so that it does not collide with obstacles, as one or more nodes perform a pre-established function (e.g. a certain path). The multibody systems are modelled via rod-type finite elements, both deformable and rigid, and the coordinates of their nodes are chosen as variables. The obstacles are modelled via a mesh of points that exert repulsive forces on the nodes of the model of the multibody, so that interference between the two is avoided. Such forces have been chosen inversely proportional to the Nth power of the distance between the corresponding points of the obstacle and of the multibody system. The method is based on a potential function and on its minimization using the Lagrange Multiplier Method. The solution of the resulting equations is undertaken iteratively with the Newton–Raphson method. The 2D and 3D examples provided attest to the good performance of the algorithms and procedure here set forth.

Singularity-Free Fully-Isotropic Translational Parallel Mechanisms

Parallel mechanisms show desirable characteristics such as a large payload to robot weight ratio, considerable stiffness, low inertia and high dynamic performances. In particular, parallel manipulators with fewer than six degrees of freedom have recently attracted researchers' attention, as their employ may prove valuable in those applications in which a higher mobility is uncalled-for. The attention of this paper is focused on translational parallel mechanisms (TPMs), that is on parallel mechanisms whose output link (platform) is provided with a pure translational motion with respect to the frame. It deals with the general problem of the topological synthesis and classification of TPMs and it investigates both their constraint and direct singularities. In particular, it identifies for the first time special families of fully-isotropic mechanisms. Such manipulators exhibit outstanding properties, as they are free from singularities and show a constant orthogonal Jacobian matrix throughout their workspace. As a consequence, both the direct and the inverse position problems are linear and the kinematic analysis proves straightforward.

A φ-model solution for the inverse position problem of calibrated robots using virtual elementary motions

It is central to the control of manipulators to calculate the set/sets of joint-displacements which correspond to a given spatial pose (position and orientation) of the end-effector. This problem, which is referred to as the inverse position problem, represents one of the most difficult mathematical challenges in the field of robotics, particularly when performed for calibrated robots (or robots with general structures). In such cases, closed form solutions are too impractical to implement and iterative solutions suffer from numerical singularities. In the present work a procedure is introduced to obtain multiple inverse position solutions for serial robotic structures. For calibrated robots, the procedure involves a simple iterative technique designed to ensure fast convergence and eliminate the occurrence of singularity. However, inverse position solutions for spherical-wrist manipulators will be obtained in a straight-forward non-iterative fashion. A published kinematic notation, referred to as the ϕ-model, was used to develop the system equations.