Interior singularity analysis for a 2(3HUS+S) parallel manipulator with descending matrix rank method

This research presents a new spatial Parallel Manipulator (PM) with pure translational motion in a plane or 2-DOF PM for pickand-place operation. This spatial 2-DOF PM is constructed by the 2(RRR)-2(PRRR) kinematic chain and is named Ψ2. This Ψ2 PM offers simplicity in its architecture because each limb applies a 1- DOF joint. The applicability of Ψ2 for pick-and-place operations was investigated by performing kinematic and singularity analyses at the preliminary stage. The singularity of Ψ2 was determined by relying on its Jacobian matrices, where three kinds of singularity can be evaluated. Next, the singularity and Jacobian matrix hold a key to identifying the workspace in a regular geometrical shape required for the operation of such an application. The identified operational workspace must be free of singularity and good conditioning workspace. In this study, two predetermined values of Ψ2 kinematic parameters were used to evaluate the kinematic characteristics and singularity of the manipulator as well as to identify its operational workspace. According to the results, Ψ2 has a good prospect of being applied to high-speed pick-and-place operations.

In this paper, singular configurations of a new spatial rotation 4-SPS-1-S parallel manipulator are examined. The 4-SPS-S parallel manipulator has 3 purely rotational degrees of freedom in the space, and can perform spatial roll, pitch and yaw rotation motions. It possesses high load capacity, stiffness and stability. Inverse position analysis and velocity mapping equations of the mechanism are presented, and then Jacobian matrices are built. Transformation from output velocity to input velocity is explored and computed. The square root of determinant of J <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> J matrix is adopted as an actuating performance index, serving as a criteria to judge singularities of the mechanism. A numerical example is presented to illustrate the distribution of actuating performance indexes across the workspace followed by singularity analysis. The paper provides crucial information for the design of this type of parallel manipulator.

The closed-loop structure of a parallel robot results in complex kinematic singularities in the workspace. Singularity analysis become important in design, motion, planning, and control of parallel robot. The traditional method to determine a singular configurations is to find the determinant of the Jacobian matrix. However, the Jacobian matrix of a parallel manipulator is complex in general, and thus it is not easy to find the determinant of the Jacobian matrix. In this paper, we focus on the singularity analysis of a novel 4-DOFs parallel robot H4 based on screw theory. Two types singularities, i.e., the forward and inverse singularities, have been identified.

In this paper, the kinematics, workspace, statics, dynamics of a parallel robot with three translational degrees of freedom for steel snagging are introduced. The parallel robot consists of three tracks referred to as the base platform, a movable platform, two connecting legs and a translational-motion mechanism referred to as the third leg. Its kinematics equation as well as and the Jacobian matrix is simple and explicit. Its workspace is larger. The effective workspace is a triangular camber pyramid. There is no singularity configuration and motion coupling in the whole motion space. There is no uncertainty position. The maneuverability is easy. Its maximum section and work depth are both large. The driving forces and the side direction forces of the sliders change continuously and smoothly. There is no impact. The parallel robot has good accelerate and dynamics performances within workspace. The steel-conditioning grinding experiments have also been carried out. The wheel selection, removal rate, grinding power and their influence factors are studied. Comparing with hanging grinders used in most steel mills of our country, this robotic grinder can reduce labor intensity of workers, raise efficiency and guarantee quality greatly. The parallel robot is fit for many manufacturing forms besides steel spot snagging because of its unique performance.

Parallel manipulators have a specific mechanical architecture where all the links are connected both at the base and at the gripper of the robot. By changing the lengths of these links we are able to control the position1 of the gripper. In general, for a given set of links lengths there is only one position for the gripper. However, it may be suspected that in some cases more than one solution may be found for the position of the gripper: the robot is in a singular configuration. To determine these singular configurations the classical method is to find the roots of the determinant of the jacobian matrix. In our case the jacobian matrix is complex, and it seems not to be possible to find these roots. We propose here a new method based on Grassmann line geometry. The set of lines, P3, constitutes a linear variety of rank 6. We show that a singular configuration is obtained when the variety spanned by the lines associated to the robot links has a rank less than 6. An important feature of the varieties of this geometry is that they can be described by simple geometric rules. Thus to find the singular configurations of parallel manipulators we have to find the configuration in which the robot matches these rules. Such an analysis is performed on a special parallel manip ulator. We show that we find all the well-known singular configurations, and also some new ones.

In this paper, analytic singularity analysis of a 4-DOF parallel robot H4 is addressed. Since a parallel manipulator consisting of several serial chains has complex singularities in the workspace, the determination of singular configurations is very important in design, trajectory planning, and control. The classical method to determine singular configurations is to find the determinant of the Jacobian matrix. However, the Jacobian matrix of a parallel manipulator is complex in general and thus it is not easy to find the determinant of the Jacobian matrix. Therefore, we focus on the analytic singularity analysis of a 4-DOF parallel robot H4 using Jacobian deficiencies. A subset of the whole singularities and the intuitively predictable ones are only derived using Jacobian matrix deficiency. Three type singularities, i.e., overmobility, undermobility and combined singularities, have been presented.

This research presents Pure Condition approach, which has used in analyzing simultaneously the singularity configuration and the rigidity of mechanism. The study cases analysis is implemented on variable joints orientation of 6R (Revolute) Serial Manipulators (SMs) and variable actuated joints position of 3-PRS (Prismatic-Revolute-Spherical) Parallel Manipulators (PMs) using Grassmann-Cayley Algebra (GCA). In this work we require in Projective Space both Twist System (TS) and Global Wrench System (GWS) respectively for serial and parallel manipulators which represent the Jacobian Matrix (J) in symbolic approach to Plücker coordinate vector of lines and unify framework on static and kinematics respectively. This paper, works, is designed to determine geometrically at symbolic level the vanished points of inverse form of this Jacobian Matrix (J) which called superbracket in GCA. The investigation vary to those reported early by introducing GCA approach on the singularity of serial robot, variable joints orientation and actuated positions on robot manipulators (RMs) to analyze rigidity frame work and singularity configuration which involve simultaneously Pure Condition. And the results also revealed a single singularity condition which contains all particulars cases and three general cases such as the shoulder, elbow and wrist singularity for SMs while double, single and undermined singularities respectively for 3-PRS, 3-PRS and 3-PRS PMs which contain all generals and particulars cases.

In this paper, a novel random matrix approach is utilized to characterize the uncertainty propagation in the articulated robotic systems. This method permits the probabilistic formulation to take into account the complexity and structural inter-dependencies of the kinematic manipulators. However, conventional random variable/vector based approaches are unable to address this issue. The kinematic Jacobian matrix of the manipulator is probabilistically formulated so that it adopts a random symmetric positive definite perturbation matrix. The density function of the perturbation matrix is constructed using maximum entropy principle that yields a Wishart distribution with specific parameters. The only information used in the construction of the Jacobian matrix probability model are mean and a scalar value dispersion parameter. We propose to use this framework to study the propagation of the uncertainties in both serial and parallel manipulators. Parallel architecture manipulators offer superior manipulation accuracy and structural rigidity compared to serial chain manipulators but feature more complex physical layout and limited workspace. Therefore, for the design purposes, a trade off between two different architectures is required. This necessitates the mathematical characterization of their capability with respect to the uncertainty propagation. As a case study, the developed random matrix method is implemented on a planar RRR serial and a planar 3-RRR parallel manipulator, that are performing an identical task. A Monte Carlo analysis is conducted and the statistics of the response of both systems provides appropriate measures to compare their dexterity dealing with the uncertainties. The final results verify superiority of the parallel architecture.

The emergence of surgical robots has revolutionized complex operations, improving precision, lowering operating risks, and shortening recovery periods. Given the merits, an eight degrees of freedom (DOF) hybrid surgical robot (HSR) has been proposed, which leverages the benefits of both serial and parallel manipulators. However, its performance is hindered by the constrained range of motion of its parallel platform. To address the issue, this research presents a systematic approach for designing and optimizing the proposed HSR. The first step is the design of the HSR, followed by a multi-stage design analysis of its parallel platform, concentrating on kinematic, geometrical, and singularity analysis. Higher values of the condition number indicate singular configurations in the platform’s workspace, highlighting the need for an optimized design. For optimization of the platform, performance parameters like global condition number (GCN), actuator forces, and stiffness are identified. Initially, the design is optimized by targeting GCN only through a genetic algorithm (GA). This approach compromised the other parameters and raised the need for simultaneous optimization employing a non-dominated sorting genetic algorithm (NSGA II). It offered a better trade-off between performance parameters. To further assess the working of the optimized parallel platform, workspace analysis and motion planning of a predefined trajectory have been performed.

SUMMARYWhen compared to serial manipulators, parallel manipulators have small workspaces mainly due to their closed-loop structure. As opposed to type I singularities (or inverse kinematic singularities) that are generally encountered at the workspace boundaries, type II singularities characteristically arise within the workspace, and around them, the inverse dynamic solution becomes unbounded. Hence, a desired trajectory passing through a type II singular position cannot be achieved by the manipulator, and its useful workspace becomes further and substantially reduced. It has been previously shown in the literature that if the trajectory is replanned in such a way that the dynamic equations of motion of the manipulator are consistent at a type II singularity, i.e. if the trajectory is consistent, then the manipulator passes through this singular configuration in a controllable manner, while the inverse dynamic solution remains finite. An inconsistent trajectory, on the other hand, is stated in the literature to be unrealizable. However, although seems a promising technique, trajectory replanning itself is also a deviation from the originally desired trajectory, and there might be cases in applications where, due to some task-specific reasons, the desired trajectory, although inconsistent, is not allowed to be replanned to satisfy the consistency conditions. In this paper, a method of singularity robust balancing is proposed for parallel manipulators passing through type II singular configurations while following inconsistent trajectories. By this means, an originally unrealizable inconsistent trajectory passing through a type II singularity can be followed without any deviation, while the required actuator forces remain bounded after the manipulator is balanced according to the design methodology presented in this study. The effectiveness of the introduced method is shown through numerical simulations considering a planar 3-DOF 2-PRR parallel manipulator under different balancing scenarios.

The parallel manipulators with n&#60;6 spherical joint-prismatic joint-spherical joint (SPS) type active legs and one passive constrained leg have attracted more attention and been used in many practical applications. This kind of parallel manipulator has a larger capability of load bearing and a simple structure of active leg. The paper used 4-SPS/PS four degree-of-freedom (DOF) parallel manipulator for the simulation of ship's heave and swing motion. Mobility property of the 4-SPS/PS parallel manipulator is analyzed by screw theory. The inverse position kinematics model is established and the analytical solution is obtained. It uses extra sensor and Sylvester resultant elimination to derive analytical solutions of forward position kinematics including two real solutions and two imaginary solutions, and uses vector cross product to derive the Jacobian matrix of 4-SPS/PS parallel manipulator. Furthermore, it introduces three performance indicators to analyze the singularity. In order to obtain accurate analysis results of singularity, it calculates the minimum singular value, condition number and manipulability degree of Jacobian matrix for different combination of pose, respectively.

This paper presents the Jacobian analysis of a parallel manipulator that has a fully decoupled 4-DOF remote center-of-motion for application in minimally invasive surgery. Owing to the special structure of the manipulator, the Jacobian matrix of the manipulator is expressed as a combination of three special Jacobian matrices, namely the Jacobian of motion space, Jacobian of constraints, and Jacobian of actuations. Based on these Jacobian matrices, the singular configurations of the manipulator are then identified. It shows that the configuration singularity only exists at the central point and the boundary of the reachable workspace of the manipulator.

The establishment of an elastostatic stiffness model for over constrained parallel manipulators (PMs), particularly those with over constrained subclosed loops, poses a challenge while ensuring numerical stability. This study addresses this issue by proposing a systematic elastostatic stiffness model based on matrix structural analysis (MSA) and independent displacement coordinates (IDCs) extraction techniques. To begin, the closed-loop PM is transformed into an open-loop PM by eliminating constraints. A subassembly element is then introduced, which considers the flexibility of both rods and joints. This approach helps circumvent the numerical instability typically encountered with traditional constraint equations. The IDCs and analytical constraint equations of nodes constrained by various joints are summarized in the appendix, utilizing multipoint constraint theory and singularity analysis, all unified within a single coordinate frame. Subsequently, the open-loop mechanism is efficiently closed by referencing the constraint equations presented in the appendix, alongside its elastostatic model. The proposed method proves to be both modeling and computationally efficient due to the comprehensive summary of the constraint equations in the Appendix, eliminating the need for additional equations. An example utilizing an over constrained subclosed loops demonstrate the application of the proposed method. In conclusion, the model proposed in this study enriches the theory of elastostatic stiffness modeling of PMs and provides an effective solution for stiffness modeling challenges they present.

A methodology is proposed for unified solving active wrench of the limited-degree of freedom (DOF) parallel manipulators (PMs). First, the geometric constraints and the inverse displacement kinematics are analysed. Second, the formulae for unified solving the inverse/forward velocity and the translational/rotational Jacobian matrices and inverse/forward Jacobian matrices are derived. Third, the analytic formulae for unified solving the active wrench of limited-DOF PMs are derived based on the principle of virtual work. Finally, a 3-DOF PM with linear/rotational active legs is presented to illustrate the use of the methodology.

A new 3T1R parallel manipulator (PM) 2PPaRR-2PPaRU with rotational pitch motion is proposed in this paper. In such a mechanism, there are four limbs connected to the moving platform (MP) through several revolute joints and universal joints, and all actuated joints are prismatic joints with an axis along a vertical direction. Mobility and motion characteristics are verified by constraint screw theory. The inverse kinematic solutions and the singularity configurations are derived through kinematic analysis. Next, the workspace analysis and the connection between [Formula: see text] and the volume of the reachable workspace are performed. The local motion/force transmission index is selected to evaluate the local transmission performance of 2PPaRR-2PPaRU PM. Finally, the global motion/force transmission index and the volume of reachable workspace are considered simultaneously to optimize the structural parameters. The 2PPaRR-2PPaRU PM with improved local transmission performance and reachable workspace has been achieved after optimization.