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.

Direct position analysis of parallel manipulators which generate SP-2PS structures

Robotica ◽  
2005 ◽  
Vol 23 (4) ◽  
pp. 521-526 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Polynomial Equation ◽  
Analytic Form ◽  
Parallel Manipulators ◽  
Real Case ◽  
End Effector ◽  
Position Analysis ◽  
Parallel Structures ◽  
Univariate Polynomial

The determination of the assembly modes of the parallel structures with three legs of type PS or SP (P and S stand for prismatic pair and spherical pair, respectively) consists of solving the direct position analysis of all the three-legged parallel manipulators which have, in each leg, one not actuated prismatic pair, one not actuated spherical pair and one or two one-dof actuated pairs of any type, placed along the leg in any order. There are two types of such structures: (i) 3PS structures and (ii) SP-2PS structures. The procedure to determine the assembly modes of the SP-2PS structures has not been presented yet, in the literature. This paper presents the analytic form determination of the assembly modes of the SP-2PS structures. In particular, the closure equations of a generic SP-2PS structure will be written and their solution will be reduced to the solution of an eight-degree univariate polynomial equation with real coefficients. Finally, the proposed algorithm will be applied to a real case. The result of this study is that the assembly modes of any SP-2PS structure are at most eight, and the end-effector poses, which solve the direct position analysis of the parallel manipulators that generate those structures, are also eight.

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.

Closed-Form Determination of the Location of a Rigid Body by Seven In-Parallel Linear Transducers

1996 ◽  
Author(s):  
Carlo Innocenti
Keyword(s):  
Rigid Body ◽  
Linear Equations ◽  
Parallel Manipulators ◽  
Rigid Bodies ◽  
Body Location ◽  
Position And Orientation ◽  
General Geometry ◽  
Two Bodies

Abstract The paper presents an original analytic procedure for unambiguously determining the relative position and orientation (location) of two rigid bodies based on the readings from seven linear transducers. Each transducer connects two points arbitrarily chosen on the two bodies. The sought-for rigid-body location simply results by solving linear equations. The proposed procedure is suitable for implementation in control of fully-parallel manipulators with general geometry. A numerical example shows application of the reported results to a case study.

An L1 adaptive closed-loop guidance law for an orbital injection problem

2009 ◽  
Vol 223 (6) ◽  
pp. 753-761 ◽  
Author(s):  
J Roshanian ◽  
M Zareh ◽  
H H Afshari ◽  
M Rezaei
Keyword(s):  
Closed Loop ◽  
Adaptive Strategy ◽  
Analytical Form ◽  
Open Loop ◽  
Control Input ◽  
Guidance Law ◽  
Non Linear ◽  
Non Linear System ◽  
Orbital Injection

The current paper presents the determination of a closed-loop guidance law for an orbital injection problem using two different approaches and, considering the existing time-optimal open-loop trajectory as the nominal solution, compares the advantages of the two proposed strategies. In the first method, named neighbouring optimal control (NOC), the perturbation feedback method is utilized to determine the closed-loop trajectory in an analytical form for the non-linear system. This law, which produces feedback gains, is in general a function of small perturbations appearing in the states and constraints separately. The second method uses an L1 adaptive strategy in determination of the non-linear closed-loop guidance law. The main advantages of this method include characteristics such as improvement of asymptotic tracking, guaranteed time-delay margin, and smooth control input. The accuracy of the two methods is compared by introducing a high-frequency sinusoidal noise. The simulation results indicate that the L1 adaptive strategy has a better performance than the NOC method to track the nominal trajectory when the noise amplitude is increased. On the other hand, the main advantage of the NOC method is its ability to solve a non-linear, two-point, boundary-value problem in the minimum time.

Inverse position analysis, workspace determination and position synthesis of parallel manipulators with 3-RSR topology

Robotica ◽  
2003 ◽  
Vol 21 (6) ◽  
pp. 627-632 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Parallel Manipulators ◽  
End Effector ◽  
New Approach ◽  
Position Analysis ◽  
Starting Point ◽  
Inverse Position Analysis ◽  
The Common ◽  
Three Degree Of Freedom ◽  
Position Synthesis

Manipulators with 3-RSR topology are three-degree-of-freedom parallel manipulators that may be either spherical or mixed-motion manipulators. The inverse position analysis (IPA) and the workspace determination of 3-RSR manipulators are addressed by means of a new approach. The new approach is centered on a particular form of the closure equations called compatibility equations. The compatibility equations contain only the six coordinates (end-effector coordinates) which locates the end-effector pose (position and orientation) with respect to the frame, and the geometric constants of the manipulator. When the manipulator geometry is assigned, the common solutions of the compatibility equations are the end-effector coordinates which identify the end-effector poses belonging to the manipulator workspace. Moreover, they can be the starting point to easily solve the IPA. The presented compatibility equations can be also used to solve the position synthesis of the 3-RSR manipulator. This way of solving the position synthesis will demonstrate that only approximated solutions exist when more than eight end-effector poses are given.

A Non-Linear Stiffness Model for Serial and Parallel Manipulators

2017 ◽  
Vol 5 (1) ◽  
pp. 34-62
Author(s):  
Manoj Kumar
Keyword(s):  
Numerical Technique ◽  
Sudden Change ◽  
Parallel Manipulators ◽  
Rigid Bodies ◽  
Stiffness Analysis ◽  
Cartesian Stiffness Matrix ◽  
Stiffness Model ◽  
Non Linear ◽  
Linear Behaviour ◽  
Cartesian Stiffness

The paper presents a methodology to enhance the stiffness analysis of serial and parallel manipulators with passive joints. It directly takes into account the loading influence on the manipulator configuration and, consequently, on its Jacobians and Hessians. The main contributions of this paper are the introduction of a non-linear stiffness model for the manipulators with passive joints, a relevant numerical technique for its linearization and computing of the Cartesian stiffness matrix which allows rank-deficiency. Within the developed technique, the manipulator elements are presented as pseudo-rigid bodies separated by multidimensional virtual springs and perfect passive joints. Simulation examples are presented that deal with parallel manipulators of the Ortholide family and demonstrate the ability of the developed methodology to describe non-linear behaviour of the manipulator structure such as a sudden change of the elastic instability properties (buckling).

Forward Position Analysis of Nearly General Stewart Platforms

1992 ◽  
Author(s):  
Change-de Zhang ◽  
Shin-Min Song
Keyword(s):  
Closed Form Solution ◽  
Polynomial Equation ◽  
Stewart Platform ◽  
Form Solution ◽  
Constraint Equations ◽  
Position Analysis ◽  
Order Polynomial ◽  
Moving Platform ◽  
Order Polynomial Equation ◽  
Simultaneous Elimination

Abstract This paper presents the closed-form solution of forward position analysis of the nearly general stewart platform, which consists of a base and a moving planar platforms connected by six extensible limbs through spherical joints in the two planar platforms. It becomes a general stewart platform if the centers are not constrained to those two planes. In this study, transformation matrix is used to represent the position of the moving platform. Based on the six dependency equations of the rotation matrix and the six constraint equations related to the six link lengths, a set of six 4-th degree equations in three unknowns are derived. Further derivations produce twenty-one dependent constraint equations. By simultaneous elimination of two unknowns a 20-th order polynomial equation in one unknown is obtained. Due to dual solutions of other unknowns, this indicates a maximum of forty possible solutions. The roots of this polynomial are solved numerically and the realistic solutions are constructed using computer graphics.

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

Sign in / Sign up

Export Citation Format

Share Document