ResearchArticle On the Direct Kinematics Problem …downloads.hindawi.com/journals/jr/2018/2412608.pdfJournalofRobotics .. - Mechanisms. Consider an - mechanism where the linear actuators

Research ArticleOn the Direct Kinematics Problem of Parallel Mechanisms

Arthur Seibel Stefan Schulz and Josef Schlattmann

Workgroup on System Technologies and Engineering Design Methodology Hamburg University of Technology21073 Hamburg Germany

Correspondence should be addressed to Arthur Seibel arthurseibeltuhhde

Received 26 October 2017 Revised 1 January 2018 Accepted 9 January 2018 Published 12 March 2018

Academic Editor Gordon R Pennock

Copyright copy 2018 Arthur Seibel et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The direct kinematics problem of parallel mechanisms that is determining the pose of the manipulator platform from the linearactuatorsrsquo lengths is in general uniquely not solvable For this reason instead of measuring the lengths of the linear actuators wepropose measuring their orientations and in most cases also the orientation of the manipulator platformThis allows the design ofa low-cost sensor system for parallel mechanisms that completely renounces length measurements and provides a unique solutionof their direct kinematics

1 Introduction

A typical six-degrees-of-freedom parallel mechanism con-sists of a (fixed) base platform and a (movable) manipulatorplatform The position and orientation (also known as pose)of the manipulator platform are commanded by fixing thedistances between 119899 points on the base platform and119898 pointson the manipulator platform where 119899119898 isin 3 6 Theremay be different ways for realizing such a mechanism Themost commonone is to use six linear actuators for connectingthe platforms together

Determining the pose of the manipulator platform fromthe linear actuatorsrsquo lengths (also known as direct kinematicsproblem) generally leads to a system of algebraic equationsthat has at most 40 different solutions [1ndash8] This number ofsolutions can be further reduced by introducing additionalconstraints for example combinatorial or planarity con-straints [9] Nonetheless a closed-form solution cannot berealized by onlymeasuring the lengths of the linear actuators

Current sensor concepts for solving the direct kinematicsproblem can be basically classified into two groups [10] Thefirst group consists of using the minimal number of sensorsin our case six length sensors and then including addi-tional numerical procedures to uniquely identify the parallelmechanismrsquos pose [11ndash20] These procedures however aregenerally not real-time capable require an initial estimate ofthe solution and may exhibit convergence problems or even

converge to a wrong solution The requirement of an initialsolution estimate is especially then problematic when startingthe mechanism at an arbitrary pose

In contrast the second approach consists of addingextra sensors for obtaining additional information aboutthe parallel mechanismrsquos state [21ndash28] These can be forexample angular sensors that are placed on the base orthe manipulator platform joints or linear andor angularsensors that are placed on supplementary passive legs Herethe number and location of the sensors must be carefullychosen because otherwise this may cause specific problemssuch as workspace limitations due to the passive leg or jointarrangement Furthermore using different sensor types leadsto a higher complexity and may even negatively affect theperformance due to possible time delays andor conflictingmeasurement values

For this reason in order to provide a unique solutionof the direct kinematics problem without using additionalnumerical procedures or sensors instead of measuring thelengths of the linear actuators we propose measuring theirorientations and if necessary also the orientation of themanipulator platformTheorientations of the linear actuatorsand the roll-pitch orientation of the manipulator platformcan be measured for example by acceleration sensors withthree axes and themeasurement of the yaw orientation of themanipulator platform can be realized for example by usinga magnetic sensor [29]

HindawiJournal of RoboticsVolume 2018 Article ID 2412608 9 pageshttpsdoiorg10115520182412608

2 Journal of Robotics

y2

z2x2p2J2

p12

Jk2

pJ1J2

Jk1p1J1 y1

z1 x1

1

2

Figure 1 Nomenclature for the description of parallel mechanisms

The remainder of this paper is organized as follows InSection 2 a classification of six-degrees-of-freedom parallelmechanisms based on the number of base and manipulatorplatform joints as well as combinatorial classes is introducedIn Section 3 we investigate if a closed-form solution forthe direct kinematics problem of the mechanism typespresented in Section 2 is possible by only measuring theorientations of the linear actuators In Section 4 for themechanism types where a closed-form solution of the directkinematics problem is not possible by only measuring thelinear actuatorsrsquo orientations we also include the informationabout the roll-pitch orientation of the manipulator platformIn Section 5 we discuss the last remaining case wherealso the information about the manipulator platformrsquos yaworientation is included In order to complete our systematicinvestigation in Section 6 we extend our results to three-degrees-of-freedom planar mechanisms Section 7 discussessome practical considerations regarding the sensor selectionand implementation of the proposed algorithms in a real-time control Finally in Section 8 our results are summarizedand discussed

Throughout the paper we use the following notationreferring to Figure 1 The body-fixed frame of the baseplatform is denoted as 1 and the body-fixed frame of themanipulator platform as 2 The position vector of the 119896thjoint 119869119896119894 of platform 119894 is denoted as p119894119869119896119894 and the connectionvector between the joints 1198691198961 and 1198691198962 of platforms 1 and 2as p11986911989611198691198962 with 119896 isin 1 6 Using inverse kinematics thisvector can be determined from

1p11986911989611198691198962 = 1p12 + 1R2 sdot 2p21198691198962 minus 1p11198691198961 (1)

with respect to platform 1 Here 1R2 denotes the rotationmatrix from frame 2 into frame 1 and p12 is the vectorconnecting the origins of platforms 1 and 2The roll pitchand yaw angles of the manipulator platform shall be denotedas 120572 120573 and 120574 and the direction or orientation of p11986911989611198691198962 isreferred to as r11986911989611198691198962 which has unit length

(a) 3-3 ( ) (b) 6-3 ( ) (c) 6-6 ( )6

Figure 2 Examples of combinatorial classes of 119899-119898mechanisms

2 Classification of Six-Degrees-of-FreedomParallel Mechanisms

Typically parallel mechanisms are classified by the numberof joints on the base and the manipulator platform Thistype of classification however is not sufficient for catchingall descriptive parameters of a parallel mechanism For thisreason Faugere and Lazard [9] introduced the notion of acombinatorial class which is represented by a graph wherethe edges are the linear actuators and the vertices are thejoints (see Figure 2) Here we use both approaches togetherto classify parallel mechanisms

21 119899-3 Mechanisms This group of mechanisms contains119899 base platform joints with 119899 isin 3 6 and three ma-nipulator platform joints Each manipulator platform jointis connected to one two or three linear actuators We canclassify this group into two types In the first type which shallbe referred to as 119899-3-I mechanisms all three manipulatorplatform joints are connected to exactly two linear actuatorsAccording to [9] this type of mechanisms corresponds toseven combinatorial classes

3 (2)

In the second type which shall be referred to as 119899-3-II mechanisms the first manipulator platform joint isconnected to three linear actuators the second manipulatorplatform joint to two linear actuators and the third manip-ulator platform joint to one linear actuator This type ofmechanisms corresponds to ten combinatorial classes [9]

(3)

22 119899-4 Mechanisms Similar to 119899-3 mechanisms this groupof mechanisms can be also classified into two types Thefirst type which shall be referred to as 119899-4-I mechanisms ischaracterized by two manipulator platform joints with eachof them connected to two linear actuators and two furthermanipulator platform joints with each of them connected to

Journal of Robotics 3

one linear actuator According to [9] this type ofmechanismscorresponds to sixteen combinatorial classes

22222

2 (4)

The second type which shall be referred to as 119899-4-IImechanisms is characterized by one manipulator platformjoint connected to three linear actuators and three furthermanipulator platform joints with each of them connected toone linear actuatorThis type of 119899-4mechanisms correspondsto nine combinatorial classes [9]

3 2

(5)

23 119899-5Mechanisms This group of mechanisms is describedby twelve combinatorial classes [9]

2

2 2 234

(6)

24 119899-6Mechanisms This group ofmechanisms is associatedwith six combinatorial classes [9]

3 3226 4 (7)

3 Closed-Form Solution by Only Measuringthe Linear Actuatorsrsquo Orientations

In this section wewill investigate if a closed-form solution forthe direct kinematics problem of the mechanisms introducedin the previous section is possible by only measuring theorientations of the linear actuators

31 119899-3 Mechanisms

311 Type I Consider an 119899-3-I mechanism where the linearactuators 119896 = 1 and 119896 = 2 are connected to the manipulatorplatform joint 11986912 the linear actuators 119896 = 3 and 119896 = 4 to themanipulator platform joint 11986922 and the linear actuators 119896 = 5and 119896 = 6 to themanipulator platform joint 11986932The positionsof these joints 1p111986912

1p111986922 and1p111986932 are defined by the

intersection points between the straight lines 119892119896 through thelinear actuators 119896 = 1 6 with

119892119896 1p119896 = 1p11198691198961 + 120582119896 sdot 1r11986911989611198691198962 120582119896 isin R (8)

where 1p119896 denote the coordinates of 119892119896 and 1r11986911989611198691198962 themeasured orientations of the linear actuators In particularthe intersection point between 1198921 and 1198922 defines 1p111986912 theintersection point between 1198923 and 1198924 defines 1p111986922 and theintersection point between 1198925 and 1198926 defines 1p111986932 These

three joint positions on the other hand define a plane withthe normal vector

1n = 1p1198691211986922 times 1p1198691211986932 (9)

for example where

1p1198691211986922 = 1p111986922 minus 1p111986912 1p1198691211986932 = 1p111986932 minus 1p111986912 (10)

Theorientation angles120572120573 and 120574 of themanipulator platformcan be then determined from

120572 = cosminus1( 1e1199111 sdot 1n1199101119911110038161003816100381610038161n11991011199111 1003816100381610038161003816 ) 120573 = cosminus1( 1e1199111 sdot 1n1199091119911110038161003816100381610038161n11990911199111 1003816100381610038161003816 ) 120574 = cosminus1( 1e1199091 sdot 1p1199091119910111986912119869221003816100381610038161003816100381610038161p119909111991011198691211986922 100381610038161003816100381610038161003816 )

= cosminus1( 1e1199101 sdot 1p1199091119910111986912119869221003816100381610038161003816100381610038161p119909111991011198691211986922 100381610038161003816100381610038161003816 )

(11)

where 1e1199091 1e1199101 and

1e1199111 are the unit vectors of the baseplatform in 1199091 1199101 and 1199111 direction 1n11991011199111 is the projectionof 1n on the 1199101-1199111 plane 1n11990911199111 is the projection of 1n on the1199091-1199111 plane and 1p119909111991011198691211986922 is the projection of 1p1198691211986922 on the1199091-1199101 plane with1n11991011199111 = (1n sdot 1e1199101) sdot 1e1199101 + (1n sdot 1e1199111) sdot 1e1199111 1n11990911199111 = (1n sdot 1e1199091) sdot 1e1199091 + (1n sdot 1e1199111) sdot 1e1199111 1p119909111991011198691211986922 = (1p1198691211986922 sdot 1e1199091) sdot 1e1199091 + (1p1198691211986922 sdot 1e1199101)

sdot 1e1199101 (12)

The manipulator platformrsquos position 1p12 on the other handcan be obtained for example from

1p12 = 1p111986912 minus 1R2 sdot 2p211986912 (13)

where

1R2=1R2120572 sdot 1R2120573 sdot 1R2120574 (14)

with

1R2120572 = [[[1 0 00 cos120572 sin1205720 minus sin120572 cos120572

]]]


1R2120573 = [[[cos120573 0 minus sin1205730 1 0sin120573 0 cos120573

]]]

1R2120574 = [[[cos 120574 sin 120574 0minus sin 120574 cos 120574 00 0 1

]]]

(15)

We can see that by only measuring the orientations of thelinear actuators a unique solution for the direct kinematicsproblem of 119899-3-I mechanisms can be found

312 Type II Now consider an 119899-3-II mechanism where thelinear actuators 119896 = 1 119896 = 2 and 119896 = 3 are connectedto the manipulator platform joint 11986912 the linear actuators119896 = 4 and 119896 = 5 are connected to the manipulator platformjoint 11986922 and the linear actuator 119896 = 6 is connected to themanipulator platform joint 11986932 The position 1p111986912 of thejoint 11986912 is defined by the intersection point between two ofthe straight lines 1198921 1198922 and 1198923 through the linear actuators119896 = 1 119896 = 2 and 119896 = 3 Hence only two orientations ofthese linear actuators are necessary for defining this positionThe position 1p111986922 of the joint 11986922 on the other hand isdefined by the intersection point between the straight lines1198924 and 1198925 through the linear actuators 119896 = 4 and 119896 = 5 Bothpositions define a straight line around which themanipulatorplatform can virtually rotate We can now define a sphere forexample with the center 1p111986922 and the radius |2p1198692211986932 | Theintersection points between the sphere and the straight line1198926 through the linear actuator 119896 = 6 define the two possiblepositions 1p111986932 of the manipulator platform joint 11986932 So byonly measuring the orientations of the linear actuators it isnot possible to find a unique solution for the direct kinematicsproblem of 119899-3-II mechanisms by only measuring the linearactuatorsrsquo orientations


321 Type I Consider an 119899-4-I mechanism where the linearactuators 119896 = 1 and 119896 = 2 are connected to the manipulatorplatform joint 11986912 the linear actuators 119896 = 3 and 119896 = 4 areconnected to the manipulator platform joint 11986922 the linearactuator 119896 = 5 is connected to the manipulator platformjoint 11986932 and the linear actuator 119896 = 6 is connected to themanipulator platform joint 11986942 The position 1p111986912 of themanipulator platform joint 11986912 is defined by the intersectionpoint between the straight lines 1198921 and 1198922 through the linearactuators 119896 = 1 and 119896 = 2 and the position 1p111986922 of themanipulator platform joint 11986922 is defined by the intersectionpoint between the straight lines 1198923 and 1198924 through the linearactuators 119896 = 3 and 119896 = 4 Both positions define a straightline around which the manipulator platform can virtuallyrotate We can now define a sphere for example with thecenter 1p111986922 and the radius |2p1198692211986932 | The intersection pointsbetween the sphere and the straight line 1198925 through the

linear actuator 119896 = 5 define the two possible positions 1p111986932of the manipulator platform joint 11986932 So it is not possibleto find a unique solution for the direct kinematics problemof 119899-4-I mechanisms by only measuring the orientations ofthe linear actuators

322 Type II Now consider an 119899-4-II mechanism where thelinear actuators 119896 = 1 119896 = 2 and 119896 = 3 are connected to themanipulator platform joint 11986912 the linear actuator 119896 = 4 isconnected to the manipulator platform joint 11986922 the linearactuator 119896 = 5 is connected to the manipulator platformjoint 11986932 and the linear actuator 119896 = 6 is connected to themanipulator platform joint 11986942 The position 1p111986912 of thejoint 11986912 is defined by the intersection point between two ofthe straight lines 1198921 1198922 and 1198923 through the linear actuators119896 = 1 119896 = 2 and 119896 = 3 Hence only two orientations ofthese linear actuators are necessary for defining this positionWe can now define two spheres for example the first spherewith the center 1p111986912 and the radius |2p1198691211986922 | and the secondsphere with the same center but with the radius |2p1198691211986932 |Theintersection points between the first sphere and the straightline 1198924 through the linear actuator 119896 = 4 define the twopossible positions 1p111986922 of the manipulator platform joint11986922 and the intersection points between the second sphereand the straight line 1198925 through the linear actuator 119896 = 5define the two possible positions 1p111986932 of the manipulatorplatform joint 11986932 So in total we obtain four differentpossible orientations of the manipulator platform and it ishence not possible to find a unique solution for the directkinematics problem of 119899-4-II mechanisms by onlymeasuringthe linear actuatorsrsquo orientations

33 119899-5 Mechanisms Consider an 119899-5 mechanism wherethe linear actuators 119896 = 1 and 119896 = 2 are connectedto the manipulator platform joint 11986912 the linear actuator119896 = 3 is connected to the manipulator platform joint 11986922the linear actuator 119896 = 4 is connected to the manipulatorplatform joint 11986932 the linear actuator 119896 = 5 is connected tothe manipulator platform joint 11986942 and the linear actuator119896 = 6 is connected to the manipulator platform joint 11986952The position 1p111986912 of the manipulator platform joint 11986912 isdefined by the intersection point between the straight lines1198921 and 1198922 through the linear actuators 119896 = 1 and 119896 =2 We can now define two spheres for example the firstsphere with the center 1p111986912 and the radius |2p1198691211986922 | andthe second sphere with the same center but with the radius|2p1198691211986932 | The intersection points between the first sphereand the straight line 1198923 through the linear actuator 119896 = 3define the two possible positions 1p111986922 of the manipulatorplatform joint 11986922 and the intersection points between thesecond sphere and the straight line 1198924 through the linearactuator 119896 = 4 define the two possible positions 1p111986932 of themanipulator platform joint 11986932 So in total we obtain fourdifferent possible orientations of the manipulator platformand it is hence not possible to find a unique solution forthe direct kinematics problem of 119899-5 mechanisms by onlymeasuring the orientations of the linear actuators


34 119899-6 Mechanisms Consider an 119899-6 mechanism wherethe linear actuators 119896 isin 1 6 are connected to themanipulator platform joints 1198691198962 We can now choose threearbitrary linear actuators 119897 119901 and 119902 with 119897 119901 119902 isin 1 6and 119897 = 119901 = 119902 and define three straight lines through theselinear actuators

119892119897 1p119897 = 1p11198691198971 + 120582119897 sdot 1r11986911989711198691198972 120582119897 isin R119892119901 1p119901 = 1p11198691199011 + 120582119901 sdot 1r11986911990111198691199012 120582119901 isin R119892119902 1p119902 = 1p11198691199021 + 120582119902 sdot 1r11986911990211198691199022 120582119902 isin R

(16)

Here 1p119897 1p119901 and 1p119902 denote the coordinates of 119892119897 119892119901and 119892119902 and 1r11986911989711198691198972 1r11986911990111198691199012 and 1r11986911990211198691199022 denote the measuredorientations of the linear actuators 119896 = 119897 119896 = 119901 and 119896 = 119902Next we can define two spheres the first sphere with thecenter 1p119897 and the radius |2p11986911989721198691199012 | and the second sphere withthe same center but with the radius |2p11986911989721198691199022 | In this contextthe first sphere has to intersect 119892119901 and the second sphere 119892119902so that we can write the following two equations

(1p119901 minus 1p119897)2 = 1003816100381610038161003816100381610038162p11986911989721198691199012 1003816100381610038161003816100381610038162 (17)

(1p119902 minus 1p119897)2 = 1003816100381610038161003816100381610038162p11986911989721198691199022 1003816100381610038161003816100381610038162 (18)

Since we know the angle between 2p11986911989721198691199012 and2p11986911989721198691199022 we can

also write

(1p119901 minus 1p119897) sdot (1p119902 minus 1p119897) = 2p11986911989721198691199012 sdot 2p11986911989721198691199022 (19)

We now have a system of three nonlinear equations (17) (18)and (19) in three variables (120582119897 120582119901 and 120582119902) which in generalis uniquely not solvable So it is not possible to find a uniquesolution for the direct kinematics problemof 119899-6mechanismsby only measuring the linear actuatorsrsquo orientations

4 Closed-Form Solution by Measuringthe Linear Actuatorsrsquo Orientations andthe Roll-Pitch Orientation ofthe Manipulator Platform

In Section 3 we have shown that by only measuring theorientations of the linear actuators it is only possible to find aunique solution for 119899-3-I mechanisms In this section we willinvestigate if a closed-form solution for the direct kinematicsproblem is possible by also including the information aboutthe roll-pitch orientation of the manipulator platform

Consider an 119899-119898 mechanism with 119899 isin 3 6 and119898 isin 3 4 5 Each of these mechanisms contains at leasttwo linear actuators that are connected to one manipulatorplatform joint 1198691198972 We assume that the measured roll-pitchorientation of the manipulator platform is given by the unitnormal vector 1n The position 1p11198691198972 of the manipulatorplatform joint 1198691198972 and the unit normal vector 1n define aplane and the desired positions 1p11198691199012 and

1p11198691199022 of two other

y2

z2 x2

2Jp2 p2J2

rJ2J1gp1

gp2dp

rJ1J2Jp1

p1J1 y1

z1

p12

p2J2Jq2

gq1

rJ1J2

p1J1

rJ2J1dq

gq2

Jq1

1

x1

Figure 3 Schematic diagram for the algorithm from Section 5

linear actuators 119896 = 119901 and 119896 = 119902 where 119901 = 119902 aredefined by the intersection points between these two linearactuators and the plane So by measuring the orientations offour linear actuators as well as the roll-pitch orientation ofthe manipulator platform it is possible to obtain a closed-form solution for the direct kinematics problem of 119899-119898mechanisms with 119898 = 6 For 119898 = 6 however our solutionstrategy fails since 119899-6 mechanisms do not have a commonconnection of at least two linear actuators at the manipulatorplatform In this case the information about the manipulatorplatformrsquos orientation leads to the following plane equation

(1p119901 minus 1p119897) sdot 1n = 0 (20)

We now have a system of two nonlinear equations (17)and (20) in two variables (120582119897 and 120582119901) which in general isuniquely not solvable So by including the information aboutthe roll-pitch orientation of themanipulator platform it is notpossible to find a unique solution for the direct kinematicsproblem of 119899-6 mechanisms

5 Closed-Form Solution by Measuringthe Linear Actuatorsrsquo Orientations andthe Roll-Pitch-Yaw Orientation ofthe Manipulator Platform

We have seen in Section 4 that the information about theorientations of the linear actuators and the roll-pitch orien-tation of the manipulator platform is not enough to obtaina closed-form solution for the direct kinematics problem of119899-6 mechanisms However we have shown in [29] that byalso including the information about the yaw orientation ofthe manipulator platform the direct kinematics problem of119899-6 mechanisms can be uniquely solved In the following wewill review our solution concept from [29] in the context of ageneral 119899-119898mechanism with 119899119898 isin 3 6

Consider the orientations 1r11986911990111198691199012 and1r11986911990211198691199022 of two arbi-

trarily chosen linear actuators 119896 = 119901 and 119896 = 119902 where 119901 =119902 These orientations define two pairs of straight lines 1198921199011and 1198921199012 as well as 1198921199021 and 1198921199022 with the base vectors 1p11198691199011and 1p21198691199012 as well as

1p11198691199021 and1p21198691199022 (see Figure 3) We can


now define two distance vectors between these two pairs ofstraight lines

d119901 = 1r11986911990111198691199012 times (1p12 + 1R2 sdot 2p21198691199012 minus 1p11198691199011) d119902 = 1r11986911990211198691199022 times (1p12 + 1R2 sdot 2p21198691199022 minus 1p11198691199021) (21)

where the measured roll pitch and yaw orientation of themanipulator platform is summarized in the rotation matrix1R2 Using the identity

a times b equiv a sdot b (22)

where

a = [[[[0 minus119886119911 119886119910119886119911 0 minus119886119909minus119886119910 119886119909 0

]]]] (23)

we can rewrite (21) as

d119901 = 1r11986911990111198691199012 sdot (1p12 + 1R2 sdot 2p21198691199012 minus 1p11198691199011)= 1r11986911990111198691199012⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

šA119901

sdot 1p12⏟⏟⏟⏟⏟⏟⏟šx

+ 1r11986911990111198691199012 sdot (1R2 sdot 2p21198691199012 minus 1p11198691199011)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟šminusc119901

d119902 = 1r11986911990211198691199022 sdot (1p12 + 1R2 sdot 2p21198691199022 minus 1p11198691199021)

= 1r11986911990211198691199022⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟šA119902

sdot 1p12⏟⏟⏟⏟⏟⏟⏟šx

+ 1r11986911990211198691199022 sdot (1R2 sdot 2p21198691199022 minus 1p11198691199021)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟šminusc119902

(24)

Now in order to find the unknown position 1p12 š x wehave to solve the linear least-squares problem

Ax minus c2 = min (25)

where

A = [A119901A119902

] isin R6times3

c = [c119901c119902] isin R

6(26)

This linear least-squares problem can be reduced to the set oflinear equations

(A⊤A⏟⏟⏟⏟⏟⏟⏟šA

) x = A⊤c⏟⏟⏟⏟⏟⏟⏟šc

(27)

with the unique solution

x = Aminus1c = (A⊤A)minus1 A⊤c (28)

The measurement of the orientations of the other four linearactuators is not necessary here since the positions of thecorrespondingmanipulator platform joints are defined by themanipulator platformrsquos geometry Note that a robust way ofcomputing x in this case is by using for example the QRdecomposition [30]

6 Extension to Planar Mechanisms

In this section we extend our previous findings for spa-tial mechanisms to three-degrees-of-freedom planar mecha-nisms Note that here only the roll orientation of the manip-ulator platform is measured due to the planarity constraint

61 119899-2 Mechanisms This group of mechanisms contains 119899base platform joints with 119899 isin 2 3 and two manipulatorplatform joints It can be described by two combinatorialclasses

(29)

Now assume that the linear actuators 119896 = 1 and 119896 = 2are connected to the manipulator platform joint 11986912 andthe linear actuator 119896 = 3 is connected to the manipulatorplatform joint 11986922 The position 1p111986912 of the manipulatorplatform joint 11986912 is defined by the intersection point betweenthe straight lines 1198921 and 1198922 through the linear actuators 119896 = 1and 119896 = 2 We can now define a circle with the center 1p111986912and the radius |2p1198691211986922 | The intersection points between thecircle and the straight line 1198923 through the linear actuator 119896 =3 define the two possible positions 1p111986922 of the manipulatorplatform joint 11986922 So by only measuring the orientations ofthe linear actuators it is not possible to find a unique solutionfor the direct kinematics problem of 119899-2 mechanisms

In the next step we assume that the roll orientation of themanipulator platform ismeasured in terms of the unit normalvector 1n Then the angle 120574 between the manipulator and thebase platform can be determined as follows

120574 = cosminus1 (1e1199101 sdot 1n) (30)

where 1e1199101 denotes the unit vector of the base platform in 1199101direction The manipulator platformrsquos position 1p12 on theother hand can be obtained for example from

1p12 = 1p111986912 minus 1R2 sdot 2p211986912 (31)

where

1R2 = [ cos 120574 sin 120574minus sin 120574 cos 120574] (32)

The measurement of the orientation of the third linearactuator is not necessary here since the position of thecorresponding manipulator platform joint is defined by themanipulator platformrsquos geometry


62 119899-3 Mechanisms In contrast to 119899-2 mechanisms thisgroup of mechanisms contains three manipulator platformjoints It can be also described by two combinatorial classes

3 (33)

The measured orientations 1r11986911989611198691198962 define the straightlines 119892119896 through the linear actuators 119896 = 1 3 with

119892119896 1p119896 = 1p11198691198961 + 120582119896 sdot 1r11986911989611198691198962 120582119896 isin R (34)

where 1p119896 denote the coordinates of 119892119896 We can now definetwo circles for example the first circle with the center 1p1and the radius |2p1198691211986922 | and the second circle with the samecenter but with the radius |2p1198691211986932 | In this context the firstcircle has to intersect 1198922 and the second circle 1198923 so that wecan write the following two equations

(1p2 minus 1p1)2 = 100381610038161003816100381610038162p1198691211986922 100381610038161003816100381610038162 (1p3 minus 1p1)2 = 100381610038161003816100381610038162p1198691211986932 100381610038161003816100381610038162

(35)

Since 2p1198691211986922 and2p1198691211986932 are linearly dependent we can also

write

(1p2 minus 1p1) sdot (1p3 minus 1p1) = 1 (36)

We now have a system of three nonlinear equations (35) and(36) in three variables (1205821 1205822 and 1205823) which in generalis uniquely not solvable So it is not possible to find a uniquesolution for the direct kinematics problemof 119899-3mechanismsby only measuring the orientations of the linear actuatorsHowever by also including the roll orientation in terms ofthe rotation matrix (32) we can always apply our generalalgorithm from Section 5 which always provides a closed-form solution of the direct kinematics problem Note that inthis case only the measurement of the orientations of twolinear actuators is necessary since the position of the thirdmanipulator platform joint is defined by the manipulatorplatformrsquos geometry

7 Practical Considerations

Currently there are many possible sensors available on themarket spreading from very expensive precalibrated high-precision sensors to uncalibrated low-cost sensors In [29]we used the InvenSense MPU-9150 inertial measurementunits (IMUs) consisting of an acceleration sensor with threeaxes a gyroscope and a magnetic sensor for determiningthe closed-form solution for the direct kinematics problemof a general 119899-119898 mechanism We obtained the correctsolution for selected static poses but the results showedrelatively high mean errors and standard deviations espe-cially for the yaw orientation of the manipulator platformThis was primarily caused by the noisy and uncalibratedIMUs The calibration problem however can be solvedwithout any additional external equipment by using the

approach from [31] For example the acceleration sensor hasto be calibratedcorrected in terms of sensor bias scalingerror and nonorthogonality In this context the calibratedmeasurement data acal can be obtained by the followingtransformation

acal = TS (au + b) (37)

where au is the uncalibrated acceleration vector and ba constant bias term The matrix S is a diagonal matrixcomprising of scaling factors in each axis and T is an uppertriangular matrix to correct nonorthogonality

Another problem related to calibration is sensor place-ment In [29] we mounted the IMUs on the gearboxes ofthe linear actuators and on top of the manipulator platformIn this context the alignment of the IMUs regarding thebase platformrsquos coordinate system has to be determined verycarefully One possibility is to use very precise measurementequipment for example by using optical or angular sensorsto obtain the location of the IMUs on the linear actuatorsAn alternative way is to determine the sensor alignmentby comparing the target orientations with the measuredorientations for several predefined poses

In order to achieve a closed-form solution of the directkinematics problem in hard real-time the introduced algo-rithms where the linear actuatorsrsquo orientations and if neces-sary the roll-pitch orientation of the manipulator platformare measured are preferable due to the precision of theacceleration sensors However the algorithm where alsothe yaw orientation of the manipulator platform is neededrequires the usage of a magnetic sensor which in general isvery imprecise Several information filters such as Kalmanfilter or complementary filters were proposed to obtain theoptimal measurement data fusion (see eg [32 33]) Thesefilters can perform very quickly (between 13 and 7 120583s [32])but they do not calculate the correct yaw orientation withthe first measurement value Instead the calculated yaworientation only converges towards the correct value withinseveral measurements

As already mentioned above we tested our approach forthe general 119899-119898 mechanism on several static poses [29]In the static case the acceleration sensors measure theconstant gravity vector of the earth without any disturbancesUnder dynamic conditions however in addition to theearthrsquos gravity field the acceleration sensors also measurethe acceleration of the mechanism itself Since we can alsomeasure the angular velocities by the available gyroscopeswe are able to compensate these erroneous measurementsIn particular by implementing an information filter forfusing the measurement data of the acceleration sensorsthe gyroscopes and the magnetic sensors of the IMUs theorientations of the linear actuators can be robustly obtained(see eg [34])

Figure 4 shows the concept for the pose control of parallelmechanisms associated with the introduced algorithms forsolving the direct kinematics problem Here a target poseptarget is defined and compared with the actual pose pisleading to the pose deviationΔp By using inverse kinematicswe can convert Δp into the required length deviation Δl of


Inverse kinematics Controller System

MeasurementsSensor fusion filterDirect kinematics

minuspＮＬＡＮ Δp

pＣＭ

Δl

z

u

m

y

Figure 4 Pose control concept for parallel mechanisms using the algorithms from Sections 3ndash6

the linear actuators and give it to the controller for examplea PID controller as input The controller then generates thecontrol input u for the system that in turn produces thesystem output y The measurement vector m that includesthe raw data of the acceleration sensors the gyroscopesand the magnetic sensors is sent to the sensor fusion filterfor example a Kalman filter Here the orientations of thelinear actuators and if necessary also the orientation of themanipulator platform are calculated The filter output z isthen used to calculate the actual pose of the manipulatorplatform pis by using the algorithms introduced in Sections3ndash6

In conclusion the pose of an 119899-119898 parallel mechanism canbe determined by only measuring the linear actuatorsrsquo orien-tations and if necessary the orientation of the manipulatorplatform The accuracy mainly depends on three things (1)the precision of the used sensors (2) their calibration andaccurate alignment on the linear actuators and the manipula-tor platform and (3) whether we have to measure the yaworientation or not For a dynamic pose determination wehave to estimate the linear actuatorsrsquo orientations by a suitablesensor fusion

8 Conclusions

We showed that for 119899-3-I mechanisms it is possible to finda unique solution for the direct kinematics problem by onlymeasuring the orientations of the linear actuators By alsoincluding the information about the roll-pitch orientationof the manipulator platform it is also possible to uniquelysolve the direct kinematics problem for 119899-3-II 119899-4 and 119899-5mechanisms Finally we demonstrated that the most generalcase of 119899-6 mechanisms also requires the information aboutthe yaw orientation of the manipulator platform

We then extended our approach to planar mechanismsand showed that the direct kinematics problem can beuniquely solved by measuring the linear actuatorsrsquo orienta-tions and the roll orientation of the manipulator platform

The results suggest that inmost cases it is not even neces-sary to measure the orientations of all six linear actuators Inparticular for 119899-3-II 119899-4 and 119899-5 mechanisms additionallyto the roll-pitch orientation of themanipulator platform onlythe orientations of four linear actuators are needed By alsomeasuring the yaw orientation of the manipulator platformthe number of required linear actuatorsrsquo orientations can beeven reduced to two

The case where only the linear actuatorsrsquo orientationsare measured is advantageous because then the sensorscan be placed close to the base platform thus reducing

the wiring effort Furthermore only measuring the roll-pitch orientation of the manipulator platform provides betterresults compared to an additional measurement of the yaworientation [29]This is especially advantageous for exampleformillingmachines where the yawdegree-of-freedom is notused

Our results enable the design of a low-cost sensor systemfor parallel mechanisms that provides a unique solution oftheir direct kinematics problem This concept is particularlyimportant if no information about the previous states of theparallel mechanism is available for example if it is switchedon in a certain pose Furthermore acceleration or magneticsensors are significantly smaller than the usual sensors formeasuring the linear actuatorsrsquo lengths thus allowing fora reduction of moving equipment as well as extending theworkspace

The real-time performance of the proposed sensor con-cept and the associated closed-form solutions for the directkinematics problem of parallel mechanisms can be improvedby sensor fusion including the information of additionallinear actuatorsrsquo orientations or sensors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the German Research Founda-tion (DFG) (Grant SCHL 27515-1) The publication of thiswork was also supported by the DFG and Hamburg Uni-versity of Technology (TUHH) in the funding programmeldquoOpen Access Publishingrdquo

References

[1] F Ronga and T Vust ldquoStewart platforms without computerrdquo inProceedings of the International Conference on Real Analytic andAlgebraic Geometry pp 197ndash212 Trento Italy 1992

[2] K H Hunt and E J F Primrose ldquoAssembly configurationsof some in-parallel-actuated manipulatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 31ndash42 1993

[3] D Lazard ldquoOn the representation of rigid-body motionsand its application to generalized platform manipulatorsrdquo inComputational Kinematics G M L Gladwell J Angeles GHommel and P Kovacs Eds vol 28 of Solid Mechanics and ItsApplications pp 175ndash181 Springer DordrechtTheNetherlands1993


[4] B Mourrain ldquoThe 40 ldquogenericrdquo positions of a parallel robotrdquoin Proceedings of the International Symposium on Symbolic andAlgebraic Computation pp 173ndash182 Kiev Ukraine July 1993

[5] M Raghavan ldquoStewart platform of general geometry has 40configurationsrdquo Journal of Mechanical Design vol 115 no 2 pp277ndash280 1993

[6] M Raghavan and B Roth ldquoSolving polynomial systems forthe kinematic analysis and synthesis of mechanisms and robotmanipulatorsrdquo Journal of Mechanical Design vol 117 pp 71ndash791995

[7] M L Husty ldquoAn algorithm for solving the direct kinematicsof general Stewart-Gough platformsrdquoMechanism and MachineTheory vol 31 no 4 pp 365ndash379 1996

[8] PDietmaier ldquoThe Stewart-Gough platformof general geometrycan have 40 real posturesrdquo in Proceedings of the InternationalSymposium on Advances in Robot Kinematics pp 7ndash16 StroblAustria 1998

[9] J C Faugere and D Lazard ldquoCombinatorial classes of parallelmanipulatorsrdquo Mechanism and Machine Theory vol 30 no 6pp 765ndash776 1995

[10] J-P Merlet Parallel Robots Springer Dordrecht The Nether-lands 2nd edition 2006

[11] Z Geng and L Haynes ldquoNeural network solution for theforward kinematics problem of a Stewart platformrdquo in Pro-ceedings of the IEEE International Conference on Robotics andAutomation pp 2650ndash2655 Sacramento CA USA April 1991

[12] N Mimura and Y Funahashi ldquoNew analytical system apply-ing 6 DOF parallel link manipulator for evaluating motionsensationrdquo in Proceedings of the IEEE International Conferenceon Robotics and Automation pp 227ndash233 Nagoya Japan May1995

[13] R Boudreau and N Turkkan ldquoSolving the forward kinematicsof parallel manipulators with a genetic algorithmrdquo Journal ofRobotic Systems vol 13 no 2 pp 111ndash125 1996

[14] B Dasgupta and T S Mruthyunjaya ldquoA constructive predictor-corrector algorithm for the direct position kinematics problemfor a general 6-6 Stewart platformrdquo Mechanism and MachineTheory vol 31 no 6 pp 799ndash811 1996

[15] C M Gosselin ldquoParallel computational algorithms for thekinematics and dynamics of planar and spatial parallel manipu-latorsrdquo Journal of Dynamic Systems Measurement and Controlvol 118 no 1 pp 22ndash28 1996

[16] P R McAree and R W Daniel ldquoA fast robust solution tothe Stewart platform forward kinematicsrdquo Journal of RoboticSystems vol 13 no 7 pp 407ndash427 1996

[17] K Der-Ming ldquoDirect displacement analysis of a Stewart plat-formmechanismrdquoMechanism and MachineTheory vol 34 no3 pp 453ndash465 1999

[18] A K Dhingra A N Almadi and D Kohli ldquoA Grobner-Sylvester hybrid method for closed-form displacement analysisof mechanismsrdquo Journal of Mechanical Design vol 122 no 4pp 431ndash438 2000

[19] K H Hunt and P R McAree ldquoThe octahedral manipulatorgeometry and mobilityrdquo The International Journal of RoboticsResearch vol 17 no 8 pp 868ndash885 1998

[20] Z Sika V Kocandrle and V Stejskal ldquoAn investigation of prop-erties of the forward displacement analysis of the generalizedStewart platform by means of general optimization methodsrdquoMechanism and Machine Theory vol 33 no 3 pp 245ndash2531998

[21] R Stoughton and T Arai ldquoOptimal sensor placement forforward kinematics evaluation of a 6-DOFparallel linkmanipu-latorrdquo in Proceedings of the IEEERSJ International Workshop onIntelligent Robots and Systems pp 785ndash790 Osaka Japan 1991

[22] K C Cheok J L Overholt and R R Beck ldquoExact methodsfor determining the kinematics of a Stewart platform usingadditional displacement sensorsrdquo Journal of Robotic Systemsvol 10 no 5 pp 689ndash707 1993

[23] J-P Merlet ldquoClosed-form resolution of the direct kinematics ofparallelmanipulators using extra sensors datardquo in Proceedings ofthe IEEE International Conference on Robotics and Automationpp 200ndash204 Singapore Singapore May 1993

[24] R Nair and J H Maddocks ldquoOn the forward kinematics ofparallel manipulatorsrdquo The International Journal of RoboticsResearch vol 13 no 2 pp 171ndash188 1994

[25] L Baron and J Angeles ldquoThe direct kinematics of parallelmanipulators under joint-sensor redundancyrdquo IEEE Transac-tions on Robotics and Automation vol 16 no 1 pp 12ndash19 2000

[26] Y-J Chiu and M-H Perng ldquoForward kinematics of a generalfully parallel manipulator with auxiliary sensorsrdquo The Interna-tional Journal of Robotics Research vol 20 no 5 pp 401ndash4142001

[27] J Hesselbach C Bier I Pietsch et al ldquoPassive joint-sensorapplications for parallel robotsrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems pp3507ndash3512 Sendai Japan October 2004

[28] R Vertechy and V Parenti Castelli ldquoAccurate and fast bodypose estimation by three point position datardquo Mechanism andMachine Theory vol 42 no 9 pp 1170ndash1183 2007

[29] S Schulz A Seibel D Schreiber and J Schlattmann ldquoSensorconcept for solving the direct kinematics problem of theStewart-Gough platformrdquo in Proceedings of the IEEERSJ Inter-national Conference on Intelligent Robots and Systems pp 1959ndash1964 Vancouver Canada September 2017

[30] G H Golub and C F Van Loan Matrix Computations vol3 of Johns Hopkins Series in the Mathematical Sciences JohnsHopkins Baltimore MD USA 1983

[31] U Qureshi and F Golnaraghi ldquoAn algorithm for the in-fieldcalibration of a MEMS IMUrdquo IEEE Sensors Journal 2017

[32] R G Valenti I Dryanovski and J Xiao ldquoKeeping a goodattitude a quaternion-based orientation filter for IMUs andMARGsrdquo Sensors vol 15 no 8 pp 19302ndash19330 2015

[33] S O H Madgwick A J L Harrison and R VaidyanathanldquoEstimation of IMU and MARG orientation using a gradientdescent algorithmrdquo in Proceedings of the IEEE InternationalConference on Rehabilitation Robotics pp 179ndash185 ZurichSwitzerland July 2011

[34] X Yun M Lizarraga E R Bachmann and R B McGhee ldquoAnimproved quaternion-basedKalmanfilter for real-time trackingof rigid body orientationrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems pp1074ndash1079 Las Vegas NV USA October 2003

International Journal of

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Hindawiwwwhindawicom Volume 2018


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Submit your manuscripts atwwwhindawicom


y2

z2x2p2J2

p12

Jk2

pJ1J2

Jk1p1J1 y1

z1 x1

1

2

Figure 1 Nomenclature for the description of parallel mechanisms

The remainder of this paper is organized as follows InSection 2 a classification of six-degrees-of-freedom parallelmechanisms based on the number of base and manipulatorplatform joints as well as combinatorial classes is introducedIn Section 3 we investigate if a closed-form solution forthe direct kinematics problem of the mechanism typespresented in Section 2 is possible by only measuring theorientations of the linear actuators In Section 4 for themechanism types where a closed-form solution of the directkinematics problem is not possible by only measuring thelinear actuatorsrsquo orientations we also include the informationabout the roll-pitch orientation of the manipulator platformIn Section 5 we discuss the last remaining case wherealso the information about the manipulator platformrsquos yaworientation is included In order to complete our systematicinvestigation in Section 6 we extend our results to three-degrees-of-freedom planar mechanisms Section 7 discussessome practical considerations regarding the sensor selectionand implementation of the proposed algorithms in a real-time control Finally in Section 8 our results are summarizedand discussed

Throughout the paper we use the following notationreferring to Figure 1 The body-fixed frame of the baseplatform is denoted as 1 and the body-fixed frame of themanipulator platform as 2 The position vector of the 119896thjoint 119869119896119894 of platform 119894 is denoted as p119894119869119896119894 and the connectionvector between the joints 1198691198961 and 1198691198962 of platforms 1 and 2as p11986911989611198691198962 with 119896 isin 1 6 Using inverse kinematics thisvector can be determined from

1p11986911989611198691198962 = 1p12 + 1R2 sdot 2p21198691198962 minus 1p11198691198961 (1)

with respect to platform 1 Here 1R2 denotes the rotationmatrix from frame 2 into frame 1 and p12 is the vectorconnecting the origins of platforms 1 and 2The roll pitchand yaw angles of the manipulator platform shall be denotedas 120572 120573 and 120574 and the direction or orientation of p11986911989611198691198962 isreferred to as r11986911989611198691198962 which has unit length

(a) 3-3 ( ) (b) 6-3 ( ) (c) 6-6 ( )6

Figure 2 Examples of combinatorial classes of 119899-119898mechanisms

2 Classification of Six-Degrees-of-FreedomParallel Mechanisms

Typically parallel mechanisms are classified by the numberof joints on the base and the manipulator platform Thistype of classification however is not sufficient for catchingall descriptive parameters of a parallel mechanism For thisreason Faugere and Lazard [9] introduced the notion of acombinatorial class which is represented by a graph wherethe edges are the linear actuators and the vertices are thejoints (see Figure 2) Here we use both approaches togetherto classify parallel mechanisms

21 119899-3 Mechanisms This group of mechanisms contains119899 base platform joints with 119899 isin 3 6 and three ma-nipulator platform joints Each manipulator platform jointis connected to one two or three linear actuators We canclassify this group into two types In the first type which shallbe referred to as 119899-3-I mechanisms all three manipulatorplatform joints are connected to exactly two linear actuatorsAccording to [9] this type of mechanisms corresponds toseven combinatorial classes

3 (2)

In the second type which shall be referred to as 119899-3-II mechanisms the first manipulator platform joint isconnected to three linear actuators the second manipulatorplatform joint to two linear actuators and the third manip-ulator platform joint to one linear actuator This type ofmechanisms corresponds to ten combinatorial classes [9]

(3)

22 119899-4 Mechanisms Similar to 119899-3 mechanisms this groupof mechanisms can be also classified into two types Thefirst type which shall be referred to as 119899-4-I mechanisms ischaracterized by two manipulator platform joints with eachof them connected to two linear actuators and two furthermanipulator platform joints with each of them connected to



22222

2 (4)


3 2

(5)


2

2 2 234

(6)


3 3226 4 (7)







119892119896 1p119896 = 1p11198691198961 + 120582119896 sdot 1r11986911989611198691198962 120582119896 isin R (8)



1n = 1p1198691211986922 times 1p1198691211986932 (9)

for example where





(11)

where 1e1199091 1e1199101 and


sdot 1e1199101 (12)


1p12 = 1p111986912 minus 1R2 sdot 2p211986912 (13)

where

1R2=1R2120572 sdot 1R2120573 sdot 1R2120574 (14)

with


]]]



]]]


]]]

(15)











(16)


(1p119901 minus 1p119897)2 = 1003816100381610038161003816100381610038162p11986911989721198691199012 1003816100381610038161003816100381610038162 (17)

(1p119902 minus 1p119897)2 = 1003816100381610038161003816100381610038162p11986911989721198691199022 1003816100381610038161003816100381610038162 (18)


also write






1p11198691199022 of two other

y2

z2 x2

2Jp2 p2J2

rJ2J1gp1

gp2dp

rJ1J2Jp1

p1J1 y1

z1

p12

p2J2Jq2

gq1

rJ1J2

p1J1

rJ2J1dq

gq2

Jq1

1

x1



(1p119901 minus 1p119897) sdot 1n = 0 (20)












where


]]]] (23)



šA119901

sdot 1p12⏟⏟⏟⏟⏟⏟⏟šx



= 1r11986911990211198691199022⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟šA119902

sdot 1p12⏟⏟⏟⏟⏟⏟⏟šx


(24)



where

A = [A119901A119902

] isin R6times3

c = [c119901c119902] isin R

6(26)


(A⊤A⏟⏟⏟⏟⏟⏟⏟šA

) x = A⊤c⏟⏟⏟⏟⏟⏟⏟šc

(27)







(29)



120574 = cosminus1 (1e1199101 sdot 1n) (30)


1p12 = 1p111986912 minus 1R2 sdot 2p211986912 (31)

where





3 (33)


119892119896 1p119896 = 1p11198691198961 + 120582119896 sdot 1r11986911989611198691198962 120582119896 isin R (34)



(35)


write






acal = TS (au + b) (37)










pＣＭ

Δl

z

u

m

y




8 Conclusions










Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




22222

2 (4)


3 2

(5)


2

2 2 234

(6)


3 3226 4 (7)







119892119896 1p119896 = 1p11198691198961 + 120582119896 sdot 1r11986911989611198691198962 120582119896 isin R (8)



1n = 1p1198691211986922 times 1p1198691211986932 (9)

for example where





(11)

where 1e1199091 1e1199101 and


sdot 1e1199101 (12)


1p12 = 1p111986912 minus 1R2 sdot 2p211986912 (13)

where

1R2=1R2120572 sdot 1R2120573 sdot 1R2120574 (14)

with


]]]



]]]


]]]

(15)











(16)


(1p119901 minus 1p119897)2 = 1003816100381610038161003816100381610038162p11986911989721198691199012 1003816100381610038161003816100381610038162 (17)

(1p119902 minus 1p119897)2 = 1003816100381610038161003816100381610038162p11986911989721198691199022 1003816100381610038161003816100381610038162 (18)


also write






1p11198691199022 of two other

y2

z2 x2

2Jp2 p2J2

rJ2J1gp1

gp2dp

rJ1J2Jp1

p1J1 y1

z1

p12

p2J2Jq2

gq1

rJ1J2

p1J1

rJ2J1dq

gq2

Jq1

1

x1



(1p119901 minus 1p119897) sdot 1n = 0 (20)












where


]]]] (23)



šA119901

sdot 1p12⏟⏟⏟⏟⏟⏟⏟šx



= 1r11986911990211198691199022⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟šA119902

sdot 1p12⏟⏟⏟⏟⏟⏟⏟šx


(24)



where

A = [A119901A119902

] isin R6times3

c = [c119901c119902] isin R

6(26)


(A⊤A⏟⏟⏟⏟⏟⏟⏟šA

) x = A⊤c⏟⏟⏟⏟⏟⏟⏟šc

(27)







(29)



120574 = cosminus1 (1e1199101 sdot 1n) (30)


1p12 = 1p111986912 minus 1R2 sdot 2p211986912 (31)

where





3 (33)


119892119896 1p119896 = 1p11198691198961 + 120582119896 sdot 1r11986911989611198691198962 120582119896 isin R (34)



(35)


write






acal = TS (au + b) (37)










pＣＭ

Δl

z

u

m

y




8 Conclusions










Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




]]]


]]]

(15)











(16)


(1p119901 minus 1p119897)2 = 1003816100381610038161003816100381610038162p11986911989721198691199012 1003816100381610038161003816100381610038162 (17)

(1p119902 minus 1p119897)2 = 1003816100381610038161003816100381610038162p11986911989721198691199022 1003816100381610038161003816100381610038162 (18)


also write






1p11198691199022 of two other

y2

z2 x2

2Jp2 p2J2

rJ2J1gp1

gp2dp

rJ1J2Jp1

p1J1 y1

z1

p12

p2J2Jq2

gq1

rJ1J2

p1J1

rJ2J1dq

gq2

Jq1

1

x1



(1p119901 minus 1p119897) sdot 1n = 0 (20)












where


]]]] (23)



šA119901

sdot 1p12⏟⏟⏟⏟⏟⏟⏟šx



= 1r11986911990211198691199022⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟šA119902

sdot 1p12⏟⏟⏟⏟⏟⏟⏟šx


(24)



where

A = [A119901A119902

] isin R6times3

c = [c119901c119902] isin R

6(26)


(A⊤A⏟⏟⏟⏟⏟⏟⏟šA

) x = A⊤c⏟⏟⏟⏟⏟⏟⏟šc

(27)







(29)



120574 = cosminus1 (1e1199101 sdot 1n) (30)


1p12 = 1p111986912 minus 1R2 sdot 2p211986912 (31)

where





3 (33)


119892119896 1p119896 = 1p11198691198961 + 120582119896 sdot 1r11986911989611198691198962 120582119896 isin R (34)



(35)


write






acal = TS (au + b) (37)










pＣＭ

Δl

z

u

m

y




8 Conclusions










Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





(16)


(1p119901 minus 1p119897)2 = 1003816100381610038161003816100381610038162p11986911989721198691199012 1003816100381610038161003816100381610038162 (17)

(1p119902 minus 1p119897)2 = 1003816100381610038161003816100381610038162p11986911989721198691199022 1003816100381610038161003816100381610038162 (18)


also write






1p11198691199022 of two other

y2

z2 x2

2Jp2 p2J2

rJ2J1gp1

gp2dp

rJ1J2Jp1

p1J1 y1

z1

p12

p2J2Jq2

gq1

rJ1J2

p1J1

rJ2J1dq

gq2

Jq1

1

x1



(1p119901 minus 1p119897) sdot 1n = 0 (20)












where


]]]] (23)



šA119901

sdot 1p12⏟⏟⏟⏟⏟⏟⏟šx



= 1r11986911990211198691199022⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟šA119902

sdot 1p12⏟⏟⏟⏟⏟⏟⏟šx


(24)



where

A = [A119901A119902

] isin R6times3

c = [c119901c119902] isin R

6(26)


(A⊤A⏟⏟⏟⏟⏟⏟⏟šA

) x = A⊤c⏟⏟⏟⏟⏟⏟⏟šc

(27)







(29)



120574 = cosminus1 (1e1199101 sdot 1n) (30)


1p12 = 1p111986912 minus 1R2 sdot 2p211986912 (31)

where





3 (33)


119892119896 1p119896 = 1p11198691198961 + 120582119896 sdot 1r11986911989611198691198962 120582119896 isin R (34)



(35)


write






acal = TS (au + b) (37)










pＣＭ

Δl

z

u

m

y




8 Conclusions










Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






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Advances in

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where


]]]] (23)



šA119901

sdot 1p12⏟⏟⏟⏟⏟⏟⏟šx



= 1r11986911990211198691199022⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟šA119902

sdot 1p12⏟⏟⏟⏟⏟⏟⏟šx


(24)



where

A = [A119901A119902

] isin R6times3

c = [c119901c119902] isin R

6(26)


(A⊤A⏟⏟⏟⏟⏟⏟⏟šA

) x = A⊤c⏟⏟⏟⏟⏟⏟⏟šc

(27)







(29)



120574 = cosminus1 (1e1199101 sdot 1n) (30)


1p12 = 1p111986912 minus 1R2 sdot 2p211986912 (31)

where





3 (33)


119892119896 1p119896 = 1p11198691198961 + 120582119896 sdot 1r11986911989611198691198962 120582119896 isin R (34)



(35)


write






acal = TS (au + b) (37)










pＣＭ

Δl

z

u

m

y




8 Conclusions










Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




3 (33)


119892119896 1p119896 = 1p11198691198961 + 120582119896 sdot 1r11986911989611198691198962 120582119896 isin R (34)



(35)


write






acal = TS (au + b) (37)










pＣＭ

Δl

z

u

m

y




8 Conclusions










Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






pＣＭ

Δl

z

u

m

y




8 Conclusions










Acknowledgments


References






































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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