# Relation between Press Intensity and Angular Velocity at a RPPP Mechanism

Authors: Şenay Baydaş; Bülent Karakaş
Journal: Mathematical Problems in Engineering
 (2011)

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## 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.


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## Body

## 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 Section2.


## 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 (Figures1 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 Figure1, where
(2.1)L=a0+L1+L2+L3.
a1, a2, and a3 are latest lengths of L1, L2, L3.a1, a2, and a3 are known as (2.2)a1=L(1+tan2θ1-1),a2=L(1+tan2(θ1+θ2)-1)-a1,a3=L(1+tan2(θ1+θ2+θ3)-1)-(a1+a2).ai is a function of θi and θi is a function of ai too, in Figure 3. Point values at the constant axis for θi are as follows.(2.3)Mi=‖O′Pyi‖=Ltan(∑j=1iθj),1≤i≤3,M1=‖O′Py0‖=Ltan(θ1),M2=Ltan(θ1+θ2),M3=Ltan(θ1+θ2+θ3).Figure 3
Relation betweenθi and ai.The end effector rises with fixed speed along vectora⃗=(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,zo+ẑ).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 Table1.Table 1
D-H parameters.
θidiaiαiθ10a10θ20a20θ30a30Table 2
Correlation relation amongLi, ωi, and ɛi.
Liωiɛiɛ0=511000, 15, 000762100, 01520, 15, 003053100, 0610, 09995, 006864100, 13720, 09995, 012215100, 24420, 09985, 01916100, 3820, 09965, 027547100, 55080, 09955, 037558100, 7510, 09935, 049149100, 98280, 0995, 06232510101, 24650, 09885, 07713511101, 54270, 09855, 09358512101, 87170, 09825, 11170513102, 23410, 09785, 1315214102, 63040, 09745, 1530715103, 06140, 0975, 1763816103, 52760, 09665, 20149517104, 02990, 09615, 2284618104, 56920, 09565, 2573119105, 14620, 09515, 28810520105, 76210, 09465, 3208921106, 41780, 0945, 35572522107, 11450, 09345, 39267523107, 85350, 09275, 431824108, 6360, 09215, 4731825109, 46360, 09145, 5168926110, 33780, 09065, 5630127111, 26020, 08995, 6116328112, 23260, 08915, 6628529113, 2570, 08835, 7167730114, 33540, 08755, 77350531115, 47010, 08665, 83316532116, 66330, 08575, 8958933117, 91780, 08485, 96181534119, 23630, 08396, 0310935120, 62180, 08296, 10387536122, 07750, 08196, 1803437123, 60680, 08096, 2606838125, 21360, 07996, 3450939126, 90180, 07886, 433840128, 6760, 07776, 52703541130, 54070, 07666, 62506542132, 50130, 07556, 72816543134, 56330, 07436, 83663544136, 73270, 07316, 9508245139, 01640, 07197, 0710746141, 42140, 07077, 19778547143, 95570, 06957, 33139548146, 62790, 06827, 47238549149, 44770, 06697, 62126550152, 42530, 06567, 7786251155, 57240, 06437, 9450852158, 90160, 06298, 12134553162, 42690, 06168, 308254166, 1640, 06028, 5065155170, 13020, 05888, 71723556174, 34470, 05748, 9414657178, 82920, 05599, 1803958183, 60780, 05459, 435459188, 7080, 0539, 7080260194, 16040, 051510612000, 0510, 31332562206, 26650, 048510, 6502763213, 00540, 046911, 01344564220, 26890, 045411, 4058665228, 11720, 043811, 8310166236, 62020, 042311, 83101We wrote the motion matrix with a different method, instead of  using the D-H para- meters.A translation vector according to Figure4 is (2.4)AB⃗=B-A=(L-L2cos⁡⁡θ,Lcos⁡⁡θ-Ltan⁡θ).
Consequently we obtained the motion matrix as
(2.5)T=[cos⁡⁡θ-sin⁡θ0L-Lcos⁡⁡θsin⁡θcos⁡⁡θ0L(cos⁡⁡2θ-tan⁡θ)00100001].Figure 4
Relation betweenL and ɛ 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 areLi, if the output of pistons are Li+ɛi, the pistons make a pressure to a barrier. The fixed pressure belongs to the initial value which we determined. So
(2.6)Li-1Li-1+ɛi-1=LiLi+ɛi
is obtained with Thales theorem and from Figure 4.At the same time, the following equalities are obtained with information from Figure4.(2.7)Li=L0cos⁡⁡θi,‖OPyi⃗‖=Ltan⁡θi,wi=vLi,ɛi=LiLi-1ɛi-1,Liwi=v:constant,ɛi=wi-1wiɛi-1.
These equalities are used at correlation between L,w and ɛ.cor(ɛ,w) approaches to −1 from
(2.8)ɛiwi=wi-1ɛi-1.cor(ɛ,L) approaches to 1 because (2.9)ɛiɛi-1=LiLi-1.cor(L,w) approaches to −1 because (2.10)Liwi:constant.


## 3. Experimental Result and Discussions

ForL=100, ɛ0=5, 0≤θ≤650, we calculate L, ɛi, and wi at Matlab. The programme at Matlab and its figure (Figure 5) are as Algorithm 1.Algorithm 1
clear all, close all, clcaxis([-5 110 -5 300])hold onL=100a=square((L-L*cosd(50))̂2+((L*cosd(50)-tand(50)))̂2)for t=0:5:65for u=50:5:65tR=[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*Aline([0, M(1)], [0, M(2)], 'Marker', '.', 'LineStyle', '-')plot(L*cosd(t), L*sind(t), 'Marker', '.', 'LineStyle', '-')pause(0.1)claendline([0, M(1)], [0, M(2)], 'Marker', '.', 'LineStyle', '-')plot(L*cosd(t), L*sind(t), 'Marker', '.', 'LineStyle', '-')endFigure 5If correlation is calculated betweenɛ,w, and L values,(3.1)cor(ɛ,ω)=-0,976350874,cor(ɛ,L)=0,999677409,cor(L,ω)=-0,975049457
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 addɛ to L, then we can make a little press to fixed axis. Results of significance tests between L, ɛ and w are appropriate for the expectations. While ɛ and w have positive correlation, both (ɛ,L) and (L,w) have negative correlation but significant.



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*Source: 638565-2011-03-30.xml*