Analytic singularity analysis of a 4-DOF parallel robot based on Jacobian deficiencies

Singularity analysis is one of the basic problems for parallel manipulators. When a manipulator moves in a singular configuration, the motion and transmission performance are poor. In certain serious cases, the normal operation could be damaged. Based on the topology structure and kinematics analysis of a 2(3HUS+S) parallel manipulator, the Jacobian matrices were established. Then, the singular locus surface was obtained by numerical simulation. In addition, the relationship between the motion path curve and the singular locus surface was analyzed. In this study, α, β, and γ are the attitude angles that describe the motion of moving platforms. There is a nonsingular attitude space in singular locus surfaces, and the singular locus surface is a single surface in a small attitude angle range. The nonsingular attitude space increases as the absolute value of γ increases, and singularity could be avoided when γ is large. Furthermore, the motion path curve passes through the singular locus surface two times, and the two intersection points are consistent with the positions where the motion dexterity is equal to zero. This study provides new insights on the singularity analysis of parallel manipulators, particularly for the structure parameter optimization of the nonsingular attitude space.

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, 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.

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.

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.

² This paper deals with a spherical parallel manipulator developed for a master device, used in a teleoperation system for surgical application. The aim is to improve the calculation of the direct kinematic model (DKM) even when the parallel manipulator is i n a singular configuration by using an extra sensor for a passive joint. A new simple approach for the DKM calculation of the haptic device, in order to meet requirement of real time applications, is developed. The measure of a passive joint's angle reduc es the calculation time and improves the accuracy of the direct model particularly in the vicinity of the singular positions. Experiments were conducted on tele -operation system despite of the presence of singularities in the master workspace. The obtaine d results demonstrate the feasibility of the approach. I. I NTRODUCTION The use of parallel structure as haptic device in tele -operation systems has substantially increased in past years, because parallel mechanisms have the advantage of being accurate and rig id. Tobergte et al. [1] proposed a delta parallel structure as haptic device called sigma.7 to control the DLR lightweight robot MIRO [2]. Ma et al. [3] have experiment a 4 -DOF haptic device in a virtual environment. The spherical parallel manipulator ( SPM ) is class of the parallel robot that has a fixed center of rotation. It was used by Gosselin et al . to control the orientation of a camera [4]. A new SPM is presented in this paper used as a haptic interface in a tele -operation system for a surgical application. The parallel robots suffer from the parallel singularity. This type of singularity, for the case of SPM, appears inside the workspace and generates large actuator torques and a loss of stiffness. The forces (and/or) torques applied to the end -effector, when it is located in parallel singularity, cannot be balanced by the tor ques produced by the actuators . Therefore, uncontrollable movements appear instantly . In the literature, several studies have attempted to remove this type of singularity of the workspace by proposing an optimized structure [5]-[6]. However by improving the kinematic behavior, the parallel robot structure often becomes larger and a problem of interference

Robotic manipulators inevitably encounter singular configurations in the process of movement, which seriously affects their performance. Therefore, the identification of singular configurations is extremely important. However, serial manipulators that do not meet the Pieper criterion cannot obtain singular configurations through analytical methods. A joint angle parameterization method, used to obtain singular configurations, is here creatively proposed. First, an analytical method based on the Jacobian determinant and the proposed method were utilized to obtain their respective singular configurations of the Stanford manipulator. The singular configurations obtained through the two methods were consistent, which suggests that the proposed method can obtain singular configurations correctly. Then, the proposed method was applied to a seven-degree-of-freedom (7-DOF) serial manipulator and a planar 5R parallel manipulator. Finally, the correctness of the singular configurations of the 7-DOF serial manipulator was verified through the shape of the end-effector velocity ellipsoid, the value of the determinant, the value of the condition number, and the value of the manipulability measure. The correctness of singular configurations of the planar 5R parallel manipulator was verified through the value of the determinant, the value of the condition number, and the value of the manipulability measure.

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.

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.

The kinematic and dynamic analysis of the high-degree of freedom (dof) hybrid systems that using serial and parallel robots together on fixed or mobile platform such as legged robots and service robots, is a rather difficult and complex, time consuming problem and a topic of great interest for researchers. There are many educational and commercial toolboxes which analyze serial and parallel robots separately. With the toolbox developed here, high-dof serial, parallel and hybrid robotics systems used on both fixed and mobile platforms can be designed and studied easily. Forward and inverse kinematics, Jacobian matrix, singularities, trajectory planning, torques and forces in joint and operational space can be examined with different robot designs. Using the toolbox that proposed in this paper, robot designer, researchers, students and educators can easily develop their own robotics systems and visualize them in the virtual reality environment provided by MATLAB. Also, it helps understanding fundamentals of high-dof multi-body hybrid robotics systems through user friendly interactive simulation. The most of existing toolboxes in the literature allow only user to analyze the predefined robots in their library with limited interactive capabilities. In some of them, the user can define a new robot with limited dof. But, the user must already have a prior knowledge about designing process of robots. In the proposed toolbox, designing and defining any new robotics systems become very easy. What makes this toolbox unique compared with the existing toolboxes is that any type of robot structures with high dof can be developed, analyzed and studied in detail.

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, an algorithm for the determination of the singularity loci of spherical three-degree-of-freedom parallel manipulators with prismatic atuators is presented. These singularity loci, which are obtained as curves or surfaces in the Cartesian space, are of great interest in the context of kinematic design. Indeed, it has been shown elsewhere that parallel manipulators lead to a special type of singularity which is located inside the Cartesian workspace and for which the end-effector becomes uncontrollable. It is therfore important to be able to identify the configurations associated with theses singularities. The algorithm presented is based on analytical expressions of the determinant of a Jacobian matrix, a quantity that is known to vanish in the singular configurations. A general spherical three-degree-of-freedom parallel manipulator with prismatic actuators is first studied. Then, several particular designs are investigated. For each case, an analytical expression of the singularity locus is derived. A graphical representation in the Cartesian space is then obtained.

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.