assignment robotics

7/31/2019 Assignment Robotics

1/49

Technological Educational Institute of Piraeus

MSc ADVANCED INDUSTRIAL AND

MANUFACTURING SYSTEMS

Module: Robotics and Flexible Automation

Assignment:

Simulation of a two-dimensional

Robotic Mechanism

Module Leader: Dr Georgios Chamilothoris

Students Name: Georgios G. ROKOS

Students Signature:___________________________

Date: April 2012


2/49

Technological Professional Institute of Piraeus

1

Table of Contents

Preface ............................................................................................................................2

Section 1. Designing a 2D Robotic Arm ............................................................................3

1.1 RR Mechanism (Planar Manipulator)..................................................................6

1.2 RP Mechanism (Polar Manipulator) .................................................................19

1.3 Brief Comparison of the Mechanisms ..............................................................23

Section 2. Partitioning the Workspace-Introducing a Resolution Constraint ................. 24

Section 3. Calculating Forces and Torques.....................................................................36

References..................................................................................................................... 48


3/49


2

PPPrrreeefffaaaccceee

This paper has its origins in the Introduction to Robotics and Flexible

Automation Module of the MSc in Advanced Industrial and Management

Systems, undertaken at the Technological Educational Institute of Piraeus, in

cooperation with the Kingston University, under the aegis of Dr. Georgios

Chamilothoris.

This assignment seeks to demonstrate the kinematics of two 2- dimensional

manipulators of different geometry, both graphically and mathematically.

Deeply influenced by Craigs Introduction to Robotics methodology, the

development of mechanisms is illustrated step-by-step. In addition, the

discrepancies associated with the geometry of each mechanism are also put

under inspection.

Moreover, at a later phasis, the paper focuses on one of the manipulators,

including additional constraints (i.e. resolution) as well as target destination

points. Taking into account the updated data, the positional differences that

emerge are analyzed, while presenting the methodology followed both

manually and through Microsoft Excel, whose file copy is included in the soft

copy of the assignment.

Finally, given the measure and the orientation of a hypothesized force,

exerted on the four corners of the work envelope, the torque along one of

the mechanisms in all cases is calculated and depicted through diagrams.

Hopefully, this paper will constitute a pleasant experience for its readers.

Georgios Rokos


4/49


3

Section I: Designing a 2D robotic arm

In the first section of this paper, two robotic mechanisms, each of different geometry, shall

be designed.

Although robotic mechanisms usually consist of at least three different joints, in this case

only two joints will be included. That is because the mechanism is to operate on a two

dimensional space and a third joint would be considered a redundancy as it would provide

the robot with more Degrees of Freedom than needed (Baillieul & Martin, 1990). The gripper

shall be attached to the end of the second link, without any joint in-between.

The designs underlie certain constraints. Namely, the bases of mechanisms are expected to

be linearly offset from the closest corner of a perfect squared workspace by a distance d

equal to the length b of a side multiplied by 3. In addition, the origins are constrained tocomprise rotary articulations.

Consequently, the robotic mechanisms will unavoidably be of RR (Rotary-Rotary) and RP

(Rotary-Prismatic) geometry.

When robotic mechanisms are under development, there are two types of spatial

description for them. A Universal - Cartesian Coordinate System (Cartesian Space) provides a

general external view of the design as a whole while various Joint attached Frames are

utilized to describe positions with regard to the internal Joint Space.

Frames are situations of position and orientation pairs, consisting of four vectors which

depict the relation of frames with a coordinate system of reference (Craig, 1989), namely {U}

in this case.

To avoid unnecessary mathematical operations, the Universal Coordinate System {U} is

considered coincident with frame {A}. Yet, frame {A} shall be rotating with regard to the

Universal Coordinate System {U}, given that the first joint of the mechanism is constrained

to be rotary.

Frame {A} is that of the first articulation of the mechanism which is, concurrently, the base

the mechanism.

Given that frame {A} is of different orientation than the coordinate system {U}, as the first

joint is rotary in both mechanisms, its description equation should be deployed.

To describe a frame in terms of a coordinate system implement the following equation is

implemented:

(1.01)where

is the vector describing a point P in terms of {U} and

is the vector describing

the same point in terms of {A}.


5/49


4

We will name P the most distant point of the workspace, which is the top right corner of the

workspace, supposing that the robot origin is positioned on the left of the mechanism.

Vector

(1.02)contains the position information ofPrelative to the superscript coordinate system U.

Given that d=3b, where b is equal to 8 m (assuming that 1 unit is 1 meter), and d is equal to

24 m, then d+b=32 m is in terms {U}. This is also displayed in the figure below.

In addition, is also known, as it is equal to the length b = 8 m of a side of the squareworkspace. Since the under development mechanism is 2-dimensional, will inevitably beequal to 0.

Having gauged:

= (1.03)to arrive to the description of , i.e. point P relative to frame {A}, the transformationmatrix needs to be computed. is a 4x4 table, decomposing as follows:

(1.04)

FIGURE 1: Spatial constraints


6/49


5

where is a 3x3 rotation matrix illustrating the orientation of a frame or coordinatesystem {A} relative to a known coordinate system or frame {U}.

is a vector locating the origin of {A} relative to {U}.Explicitly:

(1.05)It is known from Geometry that the dot product of two unit vectors is equal to the cosine of

angle 1 , formed between them. Thus:

(1.06)

is a vector locating the origin of {A} relative to {U}. As aforementioned, frame {A} iscoincident with {U} and there is no linear distance between them. As a result:

(1.07)and:

(1.08)


7/49


6

1.1 RR Mechanism (Planar Manipulator)

As aforementioned, two mechanisms of different geometry need to be designed. However,

the mechanisms will have one point in common; the rotary joint in the base and, by

extension, the mapping matrix

.

At this part it is of significant to take into account the assumptions made while modeling the

mechanisms.

Basic Assumptions I

1) The Joints are of zero length

2) The Joints are of zero width

3) Frame {A} rotates by a hypothesized axis Z of {U}

4) The length is measured in meters.

To describe the RR manipulator, which is the robotic arm consisting of two rotary

articulations, a second frame {B} has to be defined, attached to the second joint of the

mechanism. Thence, a new transform mapping needs to be generated.According to Craig (1989), to arrive to the description in terms of {U} of a point known in

terms of a frame {B}, when {B} is known relative to {A} and {A} is known relative to {U}, the

following operation must take place:

(1.09)or

(1.10)where:

(1.11)

and since the rotation takes place across a hypothesized Z axis:

(1.12)

TABLE 1: Basic assumptions


8/49


7

Vectors and illustrate the positioning offset of the origins of {A} in relationto {U} and of {B} in relation to {A} respectively.

Since {U} and {A} are set to be coincident:

(1.13)and {A} and {B} are the end-sides of:

(1.14)

Subsequently:

(1.15)

and:

(1.16)

FIGURE 2: Indicative Coordinate Systems in the RR mechanism (orientations and distances)


9/49


8

From equations (1.11), (1.15) and (1.16) it arises that:

(1.17)and:

(1.18)Where:

, (1.19) , (1.20) . (1.21)

Having deployed a series of equations to define the RR mechanism, the general pattern of

articulations table remains undeveloped.

Taking into account Craigs (1989) pattern, a parameter table, also identified as DH table for

the mechanism in Figure 2 would look like the one below.

ai-1 1 0 0 0 2 0 0 : the joint between two links and

: the angle between

and

measured about

ai-1 : the distance between and measured along


10/49


9

: the distance from to measured along : the angle between and measured about

is located at joint

The coordinate system of Link is placed at the end of the link.The coordinate system placed on joint is the coordinate systemThe next step to define the RR hand shall be to find the perfect measures for and viawhich the arm covers the workspace without extending beyond the latter.

Figure 3 depicts the workspace which the arm is expected to cover.

Based on the assumption that the links have no mass, is allowed to fold backwards,towards the opposite endpoint of

(Figure 4). Thus,

may actually have smaller length

that the linear distance of closest point of the square workspace from the base/origin. The

closest point can be denoted as N, where:

FIGURE 3: Workspace


11/49


10

(1.22)

The most distant point that the arm must reach is point P, which is, with regard to the base

of the mechanism, the opposite top corner of the square. and are supposed to reachthis point when they form together the utmost length of the arm, meaning that their vectorsare collinear.

(1.23)

FIGURE 4: The mechanism reaching N

FIGURE 5: The mechanism reaching P


12/49


11

By adding (1.22) and (1.23) by elimination, the following equations emerge:

(1.24)

(1.25)When the end-effector of the mechanism reaches point P, equals to 0. The coordinatesof are known to be . Using the Pythagorean Theorem, the distance , which isequal to may now be estimated (Figure 6).

(1.26)

(1.27)From the given data (see Figure 1) and the equations (1.24) and (1.25), the optimum length

values for and may now be computed. (1.28) (1.29)

The rectilinear distance of any point in the workspace from the origin of the mechanism will

henceforth be denoted by .

FIGURE 6: Calculating the sum of

and

using the Pythagorean Theorem


13/49


12

The next step shall be to mathematically estimate the angles (the rotation of relativeto the X axis of {U}) and (the rotation of the relative to ), given the measures of and

, for any known point Q within the square workspace.

To do so, the law of cosines (Figure 7)needs to be implemented.

Since the measures of and are found earlier, remains to be calculated. At this point itis worth mentioning that unlike the measures of and , that of is not fixed and needsrecalculation for any different point of the workspace.

By drawing two auxiliary lines, one vertical from Q to the X axis of {U} and one across the of

X axis of {U}, a right triangle is formed. The auxialiary vertical sides are, in fact the

measures of

and

, while

is the hypotenuse of the triangle.

Law of Cosines:

(1.30)Law of Sins:

(1.31)

FIGURE 7: Basic Principles of Trigonometry

FIGURE 8:Graphical exemplification of equations from(1.32)to (1.44)


14/49


13

Using the Pythagorean it arises that:

(1.32)

Now, according to the law of cosines:

1) (1.33)2) , because (1.34)3) (1.35)4) (1.36)

The symbol denotes that there are two values of for the given lengths of, and .That is because the robot may follow the inverse trajectory to reach the same point, as in

the figure below, where the trajectory is mirrored about .

As illustrated in Figure 8:

(1.37)where:

(1.38)

and:

FIGURE 9: Mirroring of the mechanism about the straight line segment uniting its origin with itsend point


15/49


14

(1.39)According to the law of sins:

(1.40)

This implies that:

1) (1.41)2) arcsin (1.42)3) arcsin , because (1.43)4) (1.44)

Note: The above procedure describes the estimation of the angles, given that as inthe Pattern.

To prove the validity of the procedure, it will be implemented for the four corners of the

workspace, starting with point . (1.45)

(1.46) (1.47)

arcsin , when , and (1.48)

, when

, and (1.49)

(1.50) (1.51)Using DraftSight 2012, a Solidworks 2D CAD alternative, the above estimations are

confirmed, as depicted in the figure below.


16/49


15

FIGURE 10: The manipulator reaching point K from two paths

The two DH tables are the following

ai-1 1 0 0 0 2 0 28,4924 0 -

ai-1 1 0 0 0 2 0 28,4924 0 Similarly, for point the correct angles are the following:

(1.52) (1.53) (1.54)

arcsin , when , and (1.55) , when , and (1.56)

(1.57)


17/49


16

(1.58)This means that there is only one correct set of angles for point .

FIGURE 11: The manipulator reaching point P from the unique path

ai-1 1 0 0 0 2 0 28,4924 0 By implementing the same equations one may also find the correct angles for point . (1.59)

(1.60)

(1.61) arcsin , when , and (1.62)

, when (1.63) and (1.64)


18/49


17

(1.65)

FIGURE 12: The manipulator reaching point M from two paths

Finally, for point , the correct rotation angles are the following:

(1.66)

(1.67) (1.68)

arcsin , when , and (1.69)

ai-1 1 0 0 0 2 0 28,4924 0

ai-1

1 0 0 0 2 0 28,4924 0


19/49


18

, when (1.70)

and (1.71)

(1.72)

FIGURE 13: The manipulator reaching point N from one unique path

ai-1 1 0 0 0 2 0 28,4924 0 3,14159


20/49


19

1.2 RP Mechanism (Polar manipulator)

The second mechanism will be of Rotary-Prismatic geometry. Its first joint, which constitutes

the base of the manipulator, will be rotary while the second joint, mounted at the minimum

distance from the workspace will be prismatic.

Just like the previous mechanism, the RP mechanism will be designed in a manner to cover

the workspace but not extent beyond it linearly without reason. As a result, given that the

closest (with regard to the origin) point of the workspace is , the length of thefirst link shall be 24 m ). This way, the second prismatic joint, mounted on theendpoint of the first link will not need to expand the second link whose measure will be zero

in this occasion. The gripper, the second joint and the second link will all be positioned at the

same point, assuming that their volume is 0.

Since the first joint is identical to that of our previous example, its rotation and translation

matrices will also be identical.

Thus:

(1.08)

However, the second joint differs, given that it is prismatic. According to Craig (1989) and ,

prismatic articulations of that kind (sliding in terms of the precedent x-axis) are vertical to

the coordinate system before them.

Thus, while in the previous coordinate system the z-axis did not appear since it was pointing

outside this page, in the frame attached to the prismatic joint the x-axis is pointing outside

the page, the z-axis replaces the x-axis and the y-axis rotates about itself.

Thence the 2

nd

frame will be mapped in terms of the precedent articulation as follows:

(1.73)Where


21/49


20

(1.74)

P in terms of frame {B} will be:

(1.75)where d is the total slided distance.

(1.76)

where:

, (1.77) , (1.78)

. (1.79)d is known to be equal to 0 for point Nand 8 for point M.

For points K and P it may be easily calculated by subtracting from the corresponding values, known from the RR manipulator analysis, the new length=24 m. Thus and While is constant no matter the endpoint changes and can be gaugedeasily since the sides of the formed triangle are known.

For point P:


22/49


21

ai-1 1 0 0 0 2 / /For point K:

FIGURE 14: RP Manipulator reaching point P

FIGURE 15: RPManipulator reaching point K


23/49


22

ai-1 1 0 0 0 2

/

/

For points Mand N is know to be equal to 0.

FIGURE 16: RP Manipulator reaching point N

ai-1 1 0 0 0 2 / /

FIGURE 17: RP Manipulator reaching point M

ai-1 1 0 0 0

2 / /


24/49


23

1.3 Brief comparison of the two mechanisms

Although both manipulators can reach the target workspace, their different geometry

implies different key features as well.

To begin with, the RR mechanism is capable of covering the majority of its workspace taking

two alternative paths. As a result, it is able to avoid eventual obstacles without pausing its

operation. Of course this implies that its kinematics are more complex, and so is its

programming. Moreover, robots with uniquely rotary articulations call for more

sophisticated control approaches, which lead to higher costs.

In addition, the articulations of the RR mechanism, since both rotary, are easier to seal but

the structure of such a mechanism is not very rigid at full reach (Balafoutis).

The Rotary- Prismatic robot possesses a slightly better size-to-reach ratio, even when the

other is folded up, for instance in retracted position. Its control system is simpler and its rigid

structure enables larger payload potential and higher repeatability ratio. However, prismatic

guides are more vulnerable to dust and liquids than rotary guides.

Although both mechanisms are considered to be of zero volume in the precedent kinematic

analysis, in reality this can not be true. As a result, if the bases of the mechanisms where

positioned on the ground, the Rotary Prismatic manipulator would not be able to reach

points M and N. On the contrary, the Rotary Rotary manipulator would still be able to

reach those points by bending down.

Concerning the power supply of their motors, the RP mechanism could be equipped with

hydraulic drives which would turn even more powerful. The RR mechanism, on the other

hand suits electric motors which are cheaper and easier to implement.

Finally, it is worth noticing that although the working tool in both cases is a gripper, it may

be of different quality or material, given that its weight is added to the total loaded weight,

which might be an issue for the RR manipulator. The sensors defining whether the target

object is in grippers range may be mounted on the gripper, exchanging signals with robot

controller till the command for the grasping action is given.


25/49


24

Section II: Partitioning the Workspace

Introducing a resolution constraint

In this section, the workspace shall be partitioned into points on a 8x8 planar grid (Figure

18). Working with the RR mechanism, 81 endpoints, offset by one meter (on the y and x axis)

from each other, will come along and once again the corresponding rotation angles of the

links for the mechanism to reach these points will have to be calculated. But this preliminary

analysis will correspond to theoretical target values on the workspace.

Having estimated the angular values, a new constraint shall be introduced during the

analysis of the geometry of the mechanism; resolution.

According to Sisiliano and Khatib (2007, pp. 83) resolution represents the smallest

incremental motion that can be produced by the manipulator. In the case of the RR

mechanism, what they describe as motion constitutes the values of the rotation angles and

which are restricted to take values strictly multiple of/1024 rad.

Following the revised and values, the real coordinates of the endpoints, i.e. thosecorresponding to the revised angles, will be gauged.

FIGURE 18.: 8x8 planar grid


26/49


25

In addition, the position error, namely the absolute value of the distance between the

theoretical and the actual coordinates of the end points, will be computed both in terms of

the x- and in terms of the y- axis. So will the position error in the radial sense.

Finally, the average position error and its standard deviation will be estimated separately in

terms of the x-axis, in terms of y-axis and in radial sense.

Of course, these computations are too time-consuming without the assistance of electronic

means. During the analysis, the computations for one random point will be performed

manually and then the configurations needed for Microsoft Excel 2007 to automatically

generate the required values for the remaining angles and endpoints will be presented.

The theoretical coordinates of point , as shown in figure 18, are .From equations (1.32),(1.36), (1.44) it may be estimated that:

(2.01) (2.02) (2.03)

To implement the resolution constraint the following procedure is put forward:

(2.04)

(2.05) (2.06)

(2.07) (2.08)

(2.09)Moving back to equations (1.19) and (1.20), the way to estimate the closest to theoretical

endpoint coordinates with the revised values of and is disclosed:

24,986776= x-, where (2.10)

25,008241= x+, where (2.11)


27/49


26

7,038702=y-, where (2.12)

6,962052 =y+, where (2.13)The position errors in terms of x will therefore be:

, where (2.14) , where (2.15)In terms of y it will be:

, where (2.16) , where (2.17)The position error in radial sense will be:

(2.18) (2.19)

In the next page the excel formulations are presented for each of the columns of the

following pages where the revised and values, the revised coordinated, the positionerrors, the averages and standard deviations of the position errors are computed.


28/49


27

Data Table

Column Excel Formulation Equation ID Value

l1 (8*SQRT(17)+24)/2 (1.24) 28,492423

l2 (8*SQRT(17)-24)/2 (1.25) 4,492423

l3 SQRT(xi^2+yi^2) (1.33) Variable

2>0 ACOS(((l3)-(l1^2)-(l2^2))/(2*l1*l2)) (1.36) Variable

20)*-1 (1.36) Variable

ATAN(yi/xi) (1.39) Variable

+ ASIN(l2/l3*SIN(2>0)) Variable

- ASIN(l2/l3*SIN(20 SIGN(2>0)*MROUND(ABS(2>0);PI()/1024) Variable

Res 2


29/49


28


30/49


29


31/49


30


32/49


31


33/49


32


34/49


33


35/49


34


36/49


35


37/49


36

Section III: Calculating forces and torques

In the last section of this paper, we will consider that forces are exerted on the four corners

of the workspace. These forces are oriented towards the center of the workspace.

The goal is to find the torque on the two links of robot, given that the measure of the forces

is equal to unit.

It is known that when an object exerts some force to a second object, then the second

object exerts the same measure of force to the first object at an inverse direction, assuming

that both remain static in the end and that gravity is excluded from the procedure (Principle

of Action and Reaction).

As pre-mentioned, the robotic mechanism developed is believed to be of zero volume andmass. Thus, if the force is exerted from point P towards the center of the mechanism, then

its reaction would look like the one in the figure below.

Generally, the forces on the four corners of the workspace are depicted in the figure below.

In yellow color are the forces exerted from the arm to the workspace and in green color are

the reaction forces of the workspace to the arm when the final result is inertia.

FIGURE 19: Force and reaction force on point P


38/49


37

Taking into account:

a) the measure of F, which is always equal to the unit,

b) the angles formed from the x-axis to each of the reaction forces of the workspace (or their

extensions till their intersection with the x axis).

c) the principle in Figure .

the vectors of the forces are the following:

(3.01)

(3.02)

(3.03)

(3.04)

FIGURE 20: Actions and reactions on the four corners of the workspace


39/49


38

Next, a point A needs to be selected on the second link of our mechanism.

Starting from corner P, the distance between A and P shall be

denoted by .Taking into account the principle depicted in Figure 21, it arises that:

(3.05)To calculate the torque on a point of the second link of the RR mechanism and haveto be multiplied (taking the external product of the two vectors).

(3.06) and (3.07)

(3.08)

FIGURE 21: Calculating the coordinates of


40/49


39

For the actual point A on the RR mechanism, when the latter reaches point P of the

workspace:

(3.09)

(3.10)It is known from Section 1 of this paper that for point P the sum of and equals to0,244979r.

Assuming that point A is mounted 1 meter (along the mechanism) from point P, then the

measure of is 1 and: (3.11)

For a point B, placed on the same link, whose meters the measure of torque would be: (3.12)

To examine the measure of torque on the first link of the mechanism, a point C needs to be

selected.

Although this is not the case when reaching point P, and will be considered as if theywere not collinear so as to come up with a generally applicable equation.

FIGURE 22: Vectors and lengths supposing thatand are not collinear


41/49


40

As illustraded in the Figure above:

and (3.13)

(3.14)

In the meantime, it is known by definition (Figure..) that:

(3.15)Where :

(3.16)And :

(3.17)

Thus:

(3.18)

Figure 23: Addition of vectors:


42/49


41

According to equation (3.16), the general description of a vector starting from a point on the

first link of the mechanism and ending on another point on the workspace is the following:

(3.19)

The general equation for , where C is a point on the first link of the mechanism would bethe following:

(3.20)

and (3.21) (3.22)

Were point C selected at a distance from point P in the actual RR mechanism,then:

(3.23) (3.24)

(3.25)

Similarly, for a point D whose (3.26)At this point, it will be attempted to verify that there are no flaws in the general equations.

At a point J, where the joint between the two links is mounted, the torque should be

standard no matter the aspect from which it is examined, i.e. both the first equation for a

point on the second link and the second equation for a point on the first link should

generate the same result. It is known that for pointJ

a) As a point on the second link:


43/49


42

(3.27)b) As a point on the first link:

(3.28)

Note that the second equation is applicable even when examining the torque on a point on

the second link but for verification reasons two distinct methodologies where put forward.

To complete the calculation of torque along the mechanism, its measures on the endsides of

the arm have to be gauged. When estimating the torque on the end-point P of the arm

( , it is mathematically proven that: (3.29)

As for the origin of the mechanism where : (3.30)

The diagram of the torque distribution along the mechanism, when the latter reaches point

P is the following:

It appears that thougout the mechanism, length and torque increase pro rata. This is an

expected outcome since both links share the same rotation measure and act as if they were

one. The highest rate of torque is placed on the origin of the mechanism,where

reaches its

greatest value.

0,00

5,00

10,00

15,00

20,00

0,00 5,00 10,00 15,00 20,00 25,00 30,00 35,00

Torque

Distance l from endpoint

Torque along mechanism when

reaching for any 2

DIAGRAM 1: Torque along mechanism when reaching pointP.


44/49


43

Next, the torque distribution will be estimated when the mechanism reaches point K by

applying the formerly gererated equations on the same points (same values) upon themechanism. Note that the mechanism may reach point K through two trajectories. Starting

from

, where

is upward and

dawnward:

(3.31) (3.32) (3.33)

5,492423sin(0,439480) (3.34) 9,492423sin(0,439480)=0,525683 (3.35)

32,984845sin(0,439480)=15,559683 (3.36)

(3.37)

In this case, the measure of torque is negative along the second link, until the joint between

the two links, and then starts increasing along the first link, getting positive eventually.

-10,00

-5,00

0,00

5,00

10,0015,00

20,00

25,00

0,00 5,00 10,00 15,00 20,00 25,00 30,00 35,00

Torque



reaching for 2


45/49


44

Now, when , i.e. the first link is downward while the second link is upward:

(3.38)

(3.39) (3.40)

5,492423sin(0,204022) (3.41)

9,492423sin(0,204022)=2,994624 (3.42)

32,984845sin(0,204022)=19,261745 (3.43)

(3.44)

The behavioiur of torque does not change significantly despite the sign alteration of ,however it is less intensive when moving negatively (while on the second link) and catches

up when moving positevely (while on the first link), when

is positive.

In both cases, the measure of torque is equal upon the basis of the mechanism.

-5,00

0,00

5,00

10,00

15,00

20,00

25,00

0,00 5,00 10,00 15,00 20,00 25,00 30,00 35,00

Torque



reaching for 2>0

DIAGRAM 3: Torque along mechanism when reachingK


46/49


45

When the mechanism reaches point M and :

(3.45)

(3.46) (3.47)

5,492423sin(0,092992) (3.48)

9,492423sin(0,092992) (3.49)

32,984845sin(0,092992) (3.50)

(3.51)

When reaching point M, unlike point K, the torque does not meet any sign change and

remains negative throughout the mechanism. Yet, a mentionable difference in its distibution

(between the two links) is that it gets more intensive after the second joint.

-25,00

-20,00

-15,00

-10,00

-5,00

0,00

0,00 5,00 10,00 15,00 20,00 25,00 30,00 35,00

Torque



reaching for 2


47/49


46

When the mechanism reaches point M and :

(3.52)

(3.53) (3.54)

5,492423sin (3.55)

9,492423sin (3.56) 32,984845sin (3.57)

(3.58)

Compared to the previous occasion, when the measure of torque is more intensivenegatively upon the second link and catches up on the proportion of the first link to end up

with a same measure upon the origin of the mechanism.

-25,00

-20,00

-15,00

-10,00

-5,00

0,00

0,00 5,00 10,00 15,00 20,00 25,00 30,00 35,00

Torque


Torque along mechanism whenreaching for 2>0

DIAGRAM 5: Torque along mechanism when reachingM


48/49


47

When the mechanism reaches point N, there is only one trajectory available, just like when

it reaches point P.

(3.59) (3.60) (3.61) (3.62)

(3.63)

(3.64) (3.65)

Once again, the measure of torque is positively proportional to upon the second link andinverses on the joint between the two links, getting negative.

-20,00

-15,00

-10,00

-5,00

0,00

5,00

0,00 5,00 10,00 15,00 20,00 25,00 30,00 35,00

Torque

Distancel

from endpoint


reaching for any 2


49/49


References

Baillieul, J., & Martin, D.P., (1990), Proceedings of Symposia in Applied Mathematics, Vol. 41,

Robotics, American Mathematical society, pp 49-59

Balafoutis, G., (unknown date), Lecture notes, Introduction to Robotics and Flexible

Automation, Piraeus: Technological Professional Institute or Piraeus, Lectures 3, 4, 5

Craig, J.J., (1989), Introduction to Robotics, Mechanics and Control, 2nd

Edition, Addison

Wesley Longman, pp 19-103

Siciliano, B., & Khatib, O. (2007), Springer Handbook of Robotics, Springer-Verlag Berlin

Heidelberg, p.p. 67-84

Van den Berg, J., (2011), Lecture notes, Robotics,

http://www.eng.utah.edu/~cs5310/chapters.html, accessed in 15/03, 2012
http://www.eng.utah.edu/~cs5310/chapters.htmlhttp://www.eng.utah.edu/~cs5310/chapters.htmlhttp://www.eng.utah.edu/~cs5310/chapters.html

assignment robotics

Documents