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“Newton Euler Strategy for the Dynamic Analysis of 3-DOF Parallel UPU Manipulator”

Abstract

This projects aims to describe the dynamic analysis of the 3-DOF Parallel UPU manipulator. The 3 DOF UPU is a parallel manipulator in which the U and P notations are for universal and prismatic pairs respectively, is a well-known manipulator that provides the platform with three degrees of freedom of pure translation, pure rotation or mixed translation and rotation with respect to the base, according to the relative directions of the revolute pair axes. The pure translational parallel 3-UPU manipulators have received a great lot of attention. Over the years many studies have been reported in the literature on the kinematic analysis, workspace optimization, singularities and joint clearance influence on the platform accuracy of this manipulator. However, the evaluation of the dynamic modelling of the parallel manipulators based on the two most famous methods i.e. Newton- Euler and Lagrange’s method have not been clearly discussed so far. Therefore, this project tries to throw light on this area and covers the Newton Euler Strategy for the calculation of the dynamic model of the 3 DOF UPU Parallel Manipulator.

1. Introduction

Over the last few decades the parallel manipulators have attracted the attention of lot of researchers because of their complementary characteristic with serial manipulator. The word 'Parallel’ which is used here is to point out is not in the geometrical sense. Though these links act together, but it is not implied that they are aligned as parallel lines, here parallel means that the position of the end point of each linkage is independent of the position of the other links. High rigidity, low stiffness, high dynamic performance with limited workspace and a low dexterous manipulability are some of the many features of the parallel manipulators. Six degrees of freedom PMs generalized Stewart platforms have been widely studied but recently the study of parallel manipulators with less than 3-DOF gained a lot of attention much due to the fact that many of the tasks do not generally require 6-DOF and consequently less complex machines are worth to be studied.

[pic]
Fig .1 Gough Stewart Platform hexapods chains use prismatic joint linear actuators between any axis universal ball joints.

Overview of 3-DOF UPU Parallel manipulator

An interesting 3-DOF PM is the 3-UPU one which was presented by Lung-Wen Tsai in 1996. The prismatic pairs are normally actuated while the remaining ones are passive. This particular topology shows three serial chains which connect the base with the platform, under certain geometric conditions and provides the movable platform with 3 DOF of pure translation with respect to the base
.
[pic] Fig 2-Tsai 3-UPU Parallel manipulator

The 3-UPU manipulator moreover has been used as a sort of benchmark mechanism for the study of different type of singularities in parallel manipulators by many researchers.
The manipulator shown in the Figure 1.2 is Tsai 3-UPU manipulator. It consists of a fixed base labelled as link 0 and the moving platform labelled as link 7.The moving platform is connected to the fixed base via three identical legs by a universal joint at point Bi and another universal joint at point Ai , for i= 1, 2 and 3 respectively. Each limb is made up of an upper member and a lower member links i.e. 1 and 4 for the first limb, 2 and 5 for the second limb, and 3 and 6 for the third one. 2. Denavit- Hartenberg Algorithm

[pic]

Figure 3 Diagram showing various frames of different frames

The homogeneous matrix T is thus obtained that transforms the homogenous coordinates of a point from Sb to Sp by a successive rotations about Euler method and described as under [pic]

Where c(.) and s(.) stand for the cosine and the sine of the argument; γ1, γ2 and γ3 are the Euler angle about x, y, and z axes respectively and Δx, Δy and Δz are respectively the small translations of the platform along x, y and z axes of Sb respectively; Δγ1, Δγ2, Δγ3 are respectively the small variation of the Euler angles. Therefore Δ δ = (Δx, Δy, Δz, Δγ1, Δγ2, Δγ3) A represents a small variation of the displacement of the platform. For a platform translation of one unit along the x axis of the reference system Sb, that is for Δδ = (1,0,0,0,0,0)A, the variables θ3i and d4i, i = 1,2,3, in the ith leg can be found by solving the following system:

Ni = Σ (Δδ), where i = 1,2,3

3. Inverse Dynamic Modelling

Because the existence of a spatial kinematical structure the dynamics analysis of parallel robots is complicated, this possesses a large number of passive degrees of freedom, dominance of the inertial forces, frictional and gravitational components. Considering all gravitational effects and neglecting the frictions forces, the main objective of the inverse dynamics is to evaluate the input torques or forces, which actuators have to exert in order to produce a given trajectory of the end-effector.The two broadly adopted approaches for dynamic analysis of robot manipulators are the Euler-Lagrange formulation and Newton-Euler formulation.The complete physical description of the manipulator in the Newton Euler Approach is first incorporated in the Lagrangian function in terms of a set of generalized coordinates and velocities, and then a systematic procedure is followed to develop the Lagrangian equations of motion which is beyond the scope of this project. On the other hand, in the Newton Euler approach, Newton's law and Euler's equation for linear and angular motion are applied to individual bodies. The Newton-Euler approach is numerically more efficient so far as inverse dynamics computation is concerned.

4. Dynamic Formulation Strategy

The Newton-Euler method of dynamic formulation is based on the direct application of Newton's law and Euler's equations which are given as

[pic]

[pic]

to individual bodies. This produces six scalar equations for each body. The subject matter of this report is the set of principles required for the selection and ordering of the necessary equations for various bodies in order to relate the end-effector acceleration components to the actuator forces/torques for parallel manipulators. A parallel manipulator has its end-effector (platform) connected to the frame (base) through a number of partially actuated kinematic chains (legs) in-parallel. A parallel manipulator would have a single actuation in each leg. Even the hybrid manipulators which have more than one actuations in a leg, in fact, can be analysed by the methods pertaining to parallel manipulators. For open-chain serial manipulators, the actuated joint variable completely describes the configuration and the direct kinematics is straightforward and unique. This makes the joint-space formulation attractive for the dynamics of serial manipulators. On the other hand, the direct kinematics of parallel manipulators is found to be quite cumbersome and complicated with possibilities of branching. Therefore, a joint-space formulation of dynamics is both complicated and undesirable for parallel manipulators. For parallel manipulators, the inverse kinematics is comparatively simpler which makes task-space dynamic formulation appropriate. The dynamic formulation of parallel manipulators is considered in task-space in this project.
The salient features and steps of the strategy are described as follows

1-Decoupling of legs
For a task-space dynamic formulation, the aim is to develop the equations of motion (namely, Newton's and Euler's equations) for the platform. For writing these Newton Euler equations, the forces/torques which the legs exert on the platform are unknowns. Thus, it is important to analyse each and every leg with the intent to evaluate its contribution to the force system acting on the platform in terms of the actuation(s) in that leg. The first step in the strategy is to determine the position, velocity and acceleration of the platform-connection-point of each leg in terms of the task-space coordinates and their derivatives. After evaluating this each leg gets decoupled from the rest of the system and its kinematics and dynamics have to be analysed individually for its reaction at the platform-connection-point.

2-Kinematics and Dynamics of legs
From the position, velocity and acceleration of the platform-connection-point, the kinematics of the leg is solved up to the evaluation of the linear acceleration of the centres of gravity and the angular accelerations of all the links in the leg. In addition, from the configuration of the leg, the moment of inertia of each link is transformed to the global basis. The next step is to consider the dynamics of the leg and to select the appropriate components from the equilibrium equations. Let us consider that the number of task-space coordinates is equal to the number of degrees of freedom of the manipulator and also equal to the DOF's allowed by each leg to the platform. Suppose this to be the standard case of parallel manipulators. Then, the number of reaction components at the platform connection point will be the same as the number of freedoms available in the leg
(except those at the platform-connection), i.e. number of task-space coordinates less the degrees of freedom of the platform-connection. For example there will be three components of force (and no moment) reaction in a 6-DOF standard parallel manipulator with spherical joint at the platform-connection-point, and the leg should have joints giving a total of three degrees of freedom. Therefore we will be considering only those components of equilibrium equations where freedom is available is sufficient to determine the reaction of the leg to the platform in terms of the actuations in the leg. First, the equilibrium of the most distance (farthest from the base) link is considered and the forces or moments (or a combination, as the case may be) on it along the screw of freedom available at the preceding joint are equilibrated according to the type of joint hase been explained as follows
1. Prismatic joint- Force equilibrium along the sliding direction-one scalar equation.
2. Revolute joint- moment equilibrium about the revolute axis-one scalar equation.
3. Screw joint- appropriate linear combination of force and moment equilibrium along and about the screw axis-one scalar equation.
4. Cylindrical joint- force equilibrium along the joint axis and moment equilibrium about two independent scalar equations.
5. Universal joint: moment equilibrium about both the axes of freedom-two independent scalar equations
6. Spherical joint: moment equilibrium- vector equation with three independent scalar components.
After this, the combined equilibrium of the penultimate and the farthest links is considered and the equation(s) involving components corresponding to the joint preceding the penultimate link is/ are retained. By repeating this procedure till the final link the equilibrium of the entire leg is considered and component(s) corresponding to the freedom allowed by the base joint is/are taken. Thus, we get exactly the requisite number of equations necessary for solving the reaction components at the platform-connection in terms of the actuator force/torques. These equations are solved, either together or in some preferred order (depending upon the manipulator structure), and the reaction at the platform-connection-point is determined. It is to be noted that the above method ensures that the reactions of the joints in the leg are automatically eliminated from these equations.

5. Algebraic Manipulations from standard expressions

The above procedure results in complicated expressions involving the quantity[pic]representing platform acceleration, in multiple products of vectors in various terms. For inverse dynamics computations, where is [pic] known, the expressions can be evaluated numerically at each step and the final expression for the reaction of the leg to the platform is easily obtained in the form

[pic]

For the purpose of deriving the dynamic equations in closed form, however, x· is to be treated as unknown and all the terms involving x· have to be clubbed in the form Q[pic] where Q is a matrix of appropriate dimension. For this purpose, apart from the usual operations of vector algebra, the following two working rules will be used for correspondence between vector algebra and matrix notations.

Rule I. With arbitrary vectors a and b [pic]

Rule II. With arbitrary vector a, for planar systems (X[pic]R 2)

[pic]

,where al is the vector obtained by rotating vector a by a right angle in anticlockwise (positive) sense; and for spatial system (x[pic]R3 )

[pic]

[pic] is the skew symmetric matrix corresponding to the vector a.

With the help of these rules in the dynamic equations and carrying out the necessary algebraic manipulations, the reaction of the leg to the platform can be expressed in the form

[pic] where [pic]

Where

[pic]

incorporate the contributions from all the legs as well as from the platform. The force transformation matrix H transforms the actuator forces/torques to task-space while RExt is the external force system acting from the environment on the platform (end-effector).

6. Summary

The steps involved in the above methodology of dynamic formulation can be summarized as below.
1. For each leg 1. Calculate the position, velocity and acceleration of the platform-connection-point from (or in terms of) the task-space coordinates and their derivatives. 2. Solve the kinematics of the leg for position, velocity and acceleration. 3. Let m be number of links in series in the leg. Let l1, l2, to lm denote the m links and j1 j2…,jm denote the joints, starting from the base. Then, for the value of i=1-m, consider the equilibrium of links lm-i+1 to lm and take its component(s) corresponding to the freedom(s) allowed by joint jm-i+1. 4. Using the equations obtained above, solve the reaction at the connection point of the platform in terms of the actuation(s) in the leg (for inverse dynamics) or express the reaction in terms of the actuations and the platform accelerations (for deriving closed-form equation)

5. Consider equilibrium of the platform and write Newton's and Euler's equations for it. Solve the actuator forces/torques from these equations (for inverse dynamics) or simplify the equations to the standard form (resulting in the closed-form dynamic equations).Two parallel strategies for inverse dynamics and for deriving closed-form dynamic equations based on the same theme have been described above. This might seem redundant because the inverse dynamics can be solved from the dynamic equations also. However, it should be noted that, compared with deriving the equations in closed form and then using them, following the direct strategy turns out to be significantly economical for the inverse dynamics problem. In the examples presented in the paper, however, only the derivation of closed-form dynamic equations will be demonstrated for the sake of brevity and the inverse dynamics problem will not be pursued separately.

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