Direct Position Analysis of a Particular Translational 3-URU Manipulator

2021 ◽  
pp. 1-22
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Problem Formulation ◽  
Parallel Manipulators ◽  
Equation System ◽  
Point Of View ◽  
Solution Technique ◽  
Algebraic Equations ◽  
Similar System ◽  
Position Analysis ◽  
Problem Formalization ◽  
Actuated Joints

Abstract Direct position analysis (DPA) of parallel manipulators (PMs) is in general difficult to solve. Over on PMs' topology, DPA complexity depends on the choice of the actuated joints. From an analytic point of view, the system of algebraic equations that one must solve to implement PMs' DPA is usually expressible in an apparently simple form, but such a form does not allow an analytic solution and even the problem formalization is relevant in PMs' DPAs. The ample literature on the DPA of Stewart platforms well document this point. This paper addresses the DPA of a particular translational PM of 3-URU type, which has the actuators on the frame while the actuated joints are not adjacent to the frame. The problem formulation brings to a closure-equation system consisting of three irrational equations in three unknowns. Such a system is transformed into an algebraic system of four quadratic equations in four unknowns that yields a univariate irrational equation in one of the four unknowns and three explicit expressions of the remaining three unknowns. Then, an algorithm is proposed which is able to find only the real solutions of the DPA. The proposed solution technique can be applied to other DPAs reducible to a similar system of irrational equations and, as far as this author is aware, is novel. Keywords: Kinematics, Position Analysis, Parallel Manipulators, Lower Mobility, Translational Manipulators

Position Analysis of a New Family of 3-DOF Translational Parallel Manipulators*

2003 ◽  
Vol 125 (2) ◽  
pp. 316-322 ◽  
Author(s):  
Marco Carricato ◽  
Vincenzo Parenti-Castelli
Keyword(s):  
Inverse Analysis ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Numerical Examples ◽  
Position Analysis ◽  
New Family ◽  
Revolute Joints ◽  
Univariate Polynomial ◽  
Position Problem ◽  
Actuated Joints

This article presents the position analysis of a new family of 3-limbed 3-dof parallel manipulators, whose output links exhibit a motion of pure translation with respect to the base. Each limb contains four passive revolute joints and an active prismatic pair, which can be mounted anywhere along the kinematic chain or replaced by a fifth revolute one. A univariate polynomial has been found that solves the direct position problem wherever the actuators are placed. The inverse analysis has been carried out in closed-form for all possible locations of the actuated joints. Finally, numerical examples are provided.

Analytic Form Solution of the Direct Position Analysis of the SP-2RS Architectures

2004 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Polynomial Equation ◽  
Analytic Form ◽  
Parallel Manipulators ◽  
Equation System ◽  
Rigid Bodies ◽  
Form Solution ◽  
Position Analysis ◽  
Non Linear ◽  
Non Linear System

When the actuators are locked, parallel manipulators (PMs) become parallel structures, that are structures constituted by two rigid bodies (platform and base) connected by a number of kinematic chains (limbs) with only passive kinematic pairs. A set of PMs is the one collecting the manipulators (SP-2RS architectures) which become structures with one limb of type SP and two limbs of type RS (P, R and S stand for prismatic pair, revolute pair and spherical pair respectively). The analytic determination of the assembly modes of the SP-2RS structures (i.e. the solution in analytic form of the direct position analysis of the SP-2RS architectures) has not been presented in the literature yet. This paper presents the solution in analytic form of the DPA of the SP-2RS architectures. In particular, the closure equation system of a generic SP-2RS structure is written in the form of three non-linear equations in three unknowns. The solution of the non-linear system is reduced to the determination of the roots of a sixteenth-degree univariate polynomial equation plus a simple back substitution procedure. The proposed solution algorithm is applied to a real case. The result of this study is that the solutions of the direct position analysis of all the SP-2RS architectures are at most sixteen and can be analytically determined through the proposed algorithm.

Position Analysis of a New Family of 3-DOF Translational Parallel Manipulators

2001 ◽  
Author(s):  
Marco Carricato ◽  
Vincenzo Parenti-Castelli
Keyword(s):  
Inverse Analysis ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Numerical Examples ◽  
Position Analysis ◽  
New Family ◽  
Revolute Joints ◽  
Univariate Polynomial ◽  
Position Problem ◽  
Actuated Joints

Abstract This article presents the position analysis of a new family of 3-limbed 3-dof parallel manipulators, whose output links exhibit a motion of pure translation with respect to the base. Each limb contains four revolute joints and a prismatic pair, which can be mounted anywhere along the kinematic chain or replaced by a fifth revolute one. A univariate polynomial has been found that solves the direct position problem wherever the actuators are placed. The inverse analysis has been carried out in closed-form for all possible locations of the actuated joints. Finally, numerical examples are provided.

Analytic Form Solution of the Direct Position Analysis of a Wide Family of Three-Legged Parallel Manipulators

2005 ◽  
Vol 128 (1) ◽  
pp. 264-271 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Analytic Form ◽  
Parallel Manipulators ◽  
Analytical Form ◽  
Form Solution ◽  
Kinematic Chains ◽  
Position Analysis ◽  
In Series ◽  
The One ◽  
Generic Structures ◽  
Wide Family

A wide family of parallel manipulators (PMs) is the one that groups all PMs with three legs where the legs become kinematic chains constituted of a passive spherical pair (S) in series with either a passive prismatic pair (P) or a passive revolute pair (R) when the actuators are locked. The topologies of the structures generated by these manipulators, when the actuators are locked, are ten. Two out of these topologies are the SR-2PS topology (one SR leg and two PS legs) and the SP-2RS topology (one SP leg and two RS legs). This paper presents two algorithms. The first one determines all the assembly modes of the SR-2PS structures. The second one determines all the assembly modes of the SP-2RS structures. The presented algorithms can be applied without changes to solve, in analytical form, the direct position analysis (DPA) of all the parallel manipulators that generate a SR-2PS structure or a SP-2RS structure when the actuators are locked. In particular, the closure equations of two generic structures, one of type SR-2PS and the other of type SP-2RS, are written. The eliminants of the two systems of equations are determined and the solution procedures are presented. Finally, the proposed procedures are applied to real cases. This work demonstrates that (i) the DPA solutions of any PM that becomes a SR-2PS structure are at most eight, and (ii) the DPA solutions of any PM that becomes a SP-2RS structure are at most sixteen.

A Statically Balanced Gough/Stewart-Type Platform: Conception, Design, and Simulation

2009 ◽  
Vol 1 (3) ◽  
Author(s):  
Marco Carricato ◽  
Clément Gosselin
Keyword(s):  
Sufficient Conditions ◽  
Parallel Manipulators ◽  
Point Of View ◽  
Theoretical Point ◽  
Constant Force ◽  
Parallel Mechanisms ◽  
End Effector ◽  
Neutral Equilibrium ◽  
The Difference ◽  
Elastic Elements

Gravity compensation of spatial parallel manipulators is a relatively recent topic of investigation. Perfect balancing has been accomplished, so far, only for parallel mechanisms in which the weight of the moving platform is sustained by legs comprising purely rotational joints. Indeed, balancing of parallel mechanisms with translational actuators, which are among the most common ones, has been traditionally thought possible only by resorting to additional legs containing no prismatic joints between the base and the end-effector. This paper presents the conceptual and mechanical designs of a balanced Gough/Stewart-type manipulator, in which the weight of the platform is entirely sustained by the legs comprising the extensible jacks. By the integrated action of both elastic elements and counterweights, each leg is statically balanced and it generates, at its tip, a constant force contributing to maintaining the end-effector in equilibrium in any admissible configuration. If no elastic elements are used, the resulting manipulator is balanced with respect to the shaking force too. The performance of a study prototype is simulated via a model in both static and dynamic conditions, in order to prove the feasibility of the proposed design. The effects of imperfect balancing, due to the difference between the payload inertial characteristics and the theoretical/nominal ones, are investigated. Under a theoretical point of view, formal and novel derivations are provided of the necessary and sufficient conditions allowing (i) a body arbitrarily rotating in space to rest in neutral equilibrium under the action of general constant-force generators, (ii) a body pivoting about a universal joint and acted upon by a number of zero-free-length springs to exhibit constant potential energy, and (iii) a leg of a Gough/Stewart-type manipulator to operate as a constant-force generator.

FORCE-UNCONSTRAINED POSES FOR PARALLEL MANIPULATORS WITH REDUNDANT ACTUATED BRANCHES

2005 ◽  
Vol 29 (3) ◽  
pp. 343-356 ◽  
Author(s):  
Flavio Firmani ◽  
Ron P. Podhorodeski
Keyword(s):  
Dimensional Space ◽  
Parallel Manipulators ◽  
End Effector ◽  
Input And Output ◽  
Planar Manipulators ◽  
Order Polynomial ◽  
Prismatic Joints ◽  
Redundantly Actuated ◽  
Actuated Joints ◽  
Planar Parallel Manipulators

A study of the effect of including a redundant actuated branch on the existence of force-unconstrained configurations for a planar parallel layout of joints is presented1. Two methodologies for finding the force-unconstrained poses are described and discussed. The first method involves the differentiation of the nonlinear kinematic constraints of the input and output variables with respect to time. The second method makes use of the reciprocal screws associated with the actuated joints. The force-unconstrained poses of non-redundantly actuated planar parallel manipulators can be mathematically expressed by means of a polynomial in terms of the three variables that define the dimensional space of the planar manipulator, i.e., the location and orientation of the end-effector. The inclusion of redundant actuated branches leads to a system of polynomials, i.e., one additional polynomial for each redundant branch added. Elimination methods are employed to reduce the number of variables by one for every additional polynomial. This leads to a higher order polynomial with fewer variables. The roots of the resulting polynomial describe the force-unconstrained poses of the manipulator. For planar manipulators it is shown that one order of infinity of force-unconstrained configurations is eliminated for every actuated branch, beyond three, added. As an example, the four-branch revolute-prismatic-revolute mechanism (4-RPR), where the prismatic joints are actuated, is presented.

Backlash elimination in parallel manipulators using actuation redundancy

Robotica ◽  
2011 ◽  
Vol 30 (3) ◽  
pp. 379-388 ◽  
Author(s):  
Roger Boudreau ◽  
Xu Mao ◽  
Ron Podhorodeski
Keyword(s):  
Nonlinear Optimization ◽  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Redundant Actuation ◽  
Joint Torques ◽  
Accuracy Enhancement ◽  
Actuation Redundancy ◽  
Actuated Joints

SUMMARYIn this work, accuracy enhancement through backlash elimination is considered. When a nonredundantly actuated parallel manipulator is subjected to a wrench while following a trajectory, required actuator torque switching (going from positive to negative or vice versa) may occur. If backlash is present in the actuation hardware for a manipulator, torque switching compromises accuracy. When in-branch redundant actuation is added, a pseudoinverse torque solution requires smaller joint torques, but torque switching may still occur. A method is presented where concepts of exploiting a nullspace basis of the joint torques are used to ensure that single sense joint torques can be achieved for the actuated joints. The same sense torque solutions are obtained using nonlinear optimization. The methodology is applied to several examples simulating parallel manipulators in machining applications.

Direct Position Analysis in Analytical Form of Parallel Manipulators Generating Structures With Topology SR-2PS

2004 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Analytical Form ◽  
Solution Procedure ◽  
Generic Structure ◽  
Kinematic Chains ◽  
Position Analysis ◽  
In Series ◽  
The One ◽  
Wide Family

A wide family of parallel manipulators (PMs) is the one that groups all the PMs with three legs where the legs become kinematic chains constituted of a passive spherical pair (S) in series with either a passive prismatic pair (P) or a passive revolute pair (R) when the actuators are locked. The topologies of the structures generated by these manipulators, when the actuators are locked, are ten. One out of these topologies is the SR-2PS topology (one SR leg and two PS legs). This paper presents an algorithm that determines all the assembly modes of the structures with topology SR-2PS in analytical form. The presented algorithm can be applied without changes to solve, in analytical form, the direct position analysis of any parallel manipulator which generates a SR-2PS structure when the actuators are locked. In particular, the closure equations of a generic structure with topology SR-2PS are written. The eliminant of this system of equations is determined and the solution procedure is presented. Finally, the proposed procedure is applied to a real case. This work demonstrates that the solutions of the direct position analysis of any parallel manipulator which generates a SR-2PS structure when the actuators are locked are at most eight.

scholarly journals The Pantelides algorithm for delay differential-algebraic equations

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Ines Ahrens ◽  
Benjamin Unger
Keyword(s):  
Differential Algebraic Equations ◽  
Equivalence Classes ◽  
Point Of View ◽  
Algebraic Equations ◽  
Sufficient Criterion ◽  
Differential Algebraic Equation ◽  
Graph Theoretical Approach ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Structural Point

Abstract We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.

Evaluation of Topological Properties of Parallel Manipulators Based on the Topological Characteristic Indexes

Robotica ◽  
2019 ◽  
Vol 38 (8) ◽  
pp. 1381-1399 ◽  
Author(s):  
Huiping Shen ◽  
Ting-Li Yang ◽  
Ju Li ◽  
Dan Zhang ◽  
Jiaming Deng ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Dynamic Properties ◽  
Kinematic Chain ◽  
Parallel Manipulators ◽  
Design Guidelines ◽  
Design Knowledge ◽  
Topological Properties ◽  
Topological Characteristic ◽  
Actuated Joints ◽  
Selection Of

SUMMARYThe topological structure of a parallel manipulator (PM) determines its intrinsic topological properties (TPs). The TPs further determine essential kinematic and dynamic properties of the mechanism. TPs can be expressed through topological characteristics indexes (TCI). Therefore, defining a set of TCIs is an important issue to evaluate the TPs of PMs. This article addresses the evaluation of topological properties (ETP) of PMs based on TCI. A general and effective ETP method for PMs is proposed. Firstly, 12 TCIs are proposed, including 8 quantitative TCIs, that is, position and orientation characteristics sets (POC), dimension of the POC set, degrees of freedom (DOF), number of independent displacement equations, types and number of an Assur kinematic chain (AKC), coupling degrees of the AKCs, degrees of redundancy and the number of overs; as well as 4 qualitative TCIs, that is, selection of actuated joints, identification of inactive joints, DOF type and Input–Output motion decoupling. Secondly, the ETP method is illustrated by evaluating some well-known PMs including the Delta, Tricept, Exechon, Z3, H4 and the Gough–Stewart platform manipulators, as well as 28 other typical PMs. Via the ETP analysis of these mechanisms also some valuable design knowledge is derived and guidelines for the design of PMs are established. Finally, a 5-DOF decoupled hybrid spraying robot is developed by applying the design knowledge and the design guidelines derived from the ETP analysis.

Sign in / Sign up

Export Citation Format

Share Document