Relation between Press Intensity and Angular Velocity at a RPPP Mechanism

Abstract

We study some properties of RPPP. RPPP is discussed by rising with constant velocity along a given axis. The constant pressure which it stresses on a constant axis is defined by the increasing PPP. Some relations between the increase at PPP and angular velocity at R are analyzed and relations of correlation are investigated at Matlab.

1. Introduction

Mechanisms may have different types of joints, such as linear, rotary, sliding, or spherical. Although spherical joints are common in many systems, since they posses multiple degrees of freedom, and thus, are difficult to control, spherical joints are not common in robotics, except in research. Most robots have either a linear (prismatic) joint or a rotary (revolute) joint [1]. To describe the translational and rotational relationships between adjacent links, Denavit and Hartenberg proposed a matrix method of systematically establishing a coordinate system to each link of an articulated chain. The Denavit-Hartenberg (D-H) representation results in a 4 × 4 homogeneous transformation matrix representing each link's coordinate system at the joint with respect to the previous link's coordinate system. Thus, through sequential transformations, the end effector expressed in the hand coordinates can be transformed and expressed in the base coordinates which make up the inertial frame of this dynamic system [2].

Calculating the position and orientation of the hand of robot is called forward kinematic analysis. In other words, if all robot joint variables are known, using forward kinematic equations, one can calculate where the robot is at any instant [1].

A robot manipulator is often designed as spatial open chain in which each joint is actuated. The kinematics equations of the open chain define the position of the end effector for a given set of values for the joint parameters. It is also necessary to be able to compute the joint parameter values that provide a desired position for end effector. This is known as the inverse kinematics problem in robotics [3].

Three serial chains with 3-dof each connect the fixed link to the end effector link [4, 5].

Mechanism of RPPP robot is studied in this paper. This mechanism is used for rising with constant velocity along fixed axis at screw plane. Design and equations of matrix of mechanisms are given in Section 2.

2. Presentation of RPPP

RPPP mechanism and its special tasks are dealt with in this paper. The first main function is to rise with a constant velocity of the end effector at a constant direction. Its other functions are to come back and to displace the constant axis (Figures 1 and 2).

Figure 1 

Settle of frames and end effector according to steps.

Figure 2 

Skid motion.

At the rising and returning time, a constant pressure is done to the constant axis. The pressure is fixed if pistons have an expressed pressure at any time. Firstly let us write kinematic equations for RPPP.

The frames of mechanism which we investigated can be seen in Figure 1, where 𝐿=𝑎₀+𝐿₁+𝐿₂+𝐿₃. (2.1)   𝑎₁, 𝑎₂, and 𝑎₃ are latest lengths of 𝐿₁, 𝐿₂, 𝐿₃.𝑎₁, 𝑎₂, and 𝑎₃ are known as 𝑎₁=𝐿1+tan²𝜃₁,𝑎−1₂=𝐿1+tan²𝜃₁+𝜃₂−1−𝑎₁,𝑎₃=𝐿1+tan²𝜃₁+𝜃₂+𝜃₃−1−𝑎₁+𝑎₂. (2.2)

𝑎_𝑖 is a function of 𝜃_𝑖 and 𝜃_𝑖 is a function of 𝑎_𝑖 too, in Figure 3. Point values at the constant axis for 𝜃_𝑖 are as follows.𝑀_𝑖=𝑂^′𝑃𝑦_𝑖=𝐿tan^𝑖_𝑗=1𝜃_𝑗𝑀,1≤𝑖≤3,₁=𝑂^′𝑃𝑦₀=𝐿tan𝜃₁,𝑀₂=𝐿tan𝜃₁+𝜃₂,𝑀₃=𝐿tan𝜃₁+𝜃₂+𝜃₃. (2.3)

Figure 3 

Relation between 𝜃𝑖 and 𝑎𝑖.

The end effector rises with fixed speed along vector 𝑎=(0,𝑦,𝑧₀) when the base frame is at point (0,0,0).

The turning back motion is an inverse of rising. The skid motion is a displacement of fixed frame from (𝑥₀,0,𝑧₀) to (𝑥₀,0,𝑧_𝑜+𝑧).

Now we can define the general procedure, based on Denavit-Hartenberg representation, to assign frames to each joint.

D-H parameters belonging to mechanism RPPP are calculated as in Table 1.

Table 1 

D-H parameters.

Table 2 

Correlation relation among 𝐿_𝑖, 𝜔_𝑖, and 𝜀_𝑖.

We wrote the motion matrix with a different method, instead of using the D-H para- meters.

A translation vector according to Figure 4 is 𝐴𝐵=𝐵−𝐴=(𝐿−𝐿2cos𝜃,𝐿cos𝜃−𝐿tan𝜃). (2.4) Consequently we obtained the motion matrix as ()𝑇=cos𝜃−sin𝜃0𝐿−𝐿cos𝜃sin𝜃cos𝜃0𝐿cos2𝜃−tan𝜃00100001. (2.5)

Figure 4 

Relation between 𝐿 and 𝜀 at any 𝑡 time.

There is not any pressure to vertical axis in this machine as it is. If the machine is used with different purposes, such as painting, sandpaper, it must make a press to vertical axis.

While the output lengths of pistons are 𝐿_𝑖, if the output of pistons are 𝐿_𝑖+𝜀_𝑖, the pistons make a pressure to a barrier. The fixed pressure belongs to the initial value which we determined. So 𝐿_𝑖−1𝐿_𝑖−1+𝜀_𝑖−1=𝐿_𝑖𝐿_𝑖+𝜀_𝑖 (2.6) is obtained with Thales theorem and from Figure 4.

At the same time, the following equalities are obtained with information from Figure 4.𝐿_𝑖=𝐿₀cos𝜃_𝑖,𝑂𝑃_𝑦𝑖=𝐿tan𝜃_𝑖,𝑤_𝑖=𝑣𝐿_𝑖,𝜀_𝑖=𝐿_𝑖𝐿_𝑖−1𝜀_𝑖𝐿−1,_𝑖𝑤_𝑖𝜀=𝑣∶constant,_𝑖=𝑤_𝑖−1𝑤_𝑖𝜀_𝑖−1. (2.7) These equalities are used at correlation between 𝐿,𝑤 and 𝜀.

cor(𝜀,𝑤) approaches to −1 from 𝜀_𝑖𝑤_𝑖=𝑤_𝑖−1𝜀_𝑖−1. (2.8)cor(𝜀,𝐿) approaches to 1 because 𝜀_𝑖𝜀_𝑖−1=𝐿_𝑖𝐿_𝑖−1. (2.9)cor(𝐿,𝑤) approaches to −1 because 𝐿_𝑖𝑤_𝑖∶constant. (2.10)

3. Experimental Result and Discussions

For 𝐿=100, 𝜀₀=5, 0≤𝜃≤65⁰, we calculate 𝐿, 𝜀_𝑖, and 𝑤_𝑖 at Matlab. The programme at Matlab and its figure (Figure 5) are as Algorithm 1.

clear all, close all, clc  axis([-5 110 -5 300])  hold on  L=100  a=square((L-L*cosd(50))̂‌2+((L*cosd(50)-tand(50)))̂‌2)  for t=0:5:65  for u=50:5:65  t  R=[cosd(t) -sind(t) L-L*cosd(t)  sind(t) cosd(t) L*(tand(t)-sind(t))  0 0 1]  A=[L;0;1]  M=R*A  line([0, M(1)], [0, M(2)], 'Marker', '.', 'LineStyle', '-')  plot(L*cosd(t), L*sind(t), 'Marker', '.', 'LineStyle', '-')  pause(0.1)  cla  end  line([0, M(1)], [0, M(2)], 'Marker', '.', 'LineStyle', '-')  plot(L*cosd(t), L*sind(t), 'Marker', '.', 'LineStyle', '-')  end

Algorithm 1 

Figure 5 

If correlation is calculated between 𝜀,𝑤, and 𝐿 values,()()()cor𝜀,𝜔=−0,976350874,cor𝜀,𝐿=0,999677409,cor𝐿,𝜔=−0,975049457 (3.1) are obtained. This is a desired result.

4. Conclusions

In this paper, RPPP mechanism, in which we used kinematical equations and we gave a design and a numeric sampling, is a prototype for a pressing. The machine is apt for being used especially in the specialized aims.

The pistons conducting translation are of hydraulic structure and should be considered to be controlled with double sensors. And finally, if we add 𝜀 to 𝐿, then we can make a little press to fixed axis. Results of significance tests between 𝐿, 𝜀 and 𝑤 are appropriate for the expectations. While 𝜀 and 𝑤 have positive correlation, both (𝜀,𝐿) and (𝐿,𝑤) have negative correlation but significant.