* Professor and author of correspondence, Phone: (352) 392-9461, Fax: (352) 392-1071, Email: [email protected].
ABSTRACT
This paper introduces a reconfigurable one degree-of-freedom spatial mechanism that can be applied to repetitivemotion tasks. The concept is to incorporate five pairs of non-circular gears into a six degree-of–freedom closed-loop spatialchain. The gear pairs are designed based on the givenmechanism parameters and the user defined motion specifica-tion of a coupler link of the mechanism. It is shown in thepaper that planar gear pairs can be used if the spatial closed-loop chain is comprised of six pairs of parallel joint axes, i.e.the first joint axis is parallel to the second, the third is parallelto the fourth, …, and the eleventh is parallel to the twelfth. Thispaper presents the detailed reverse kinematic analysis of thisspecific geometry. A numerical example is presented.
The reverse kinematic analysis of spatial mechanisms hasbeen widely studied for practically any combination of revolute,cylindrical, and prismatic joints, see [1]-[16]. Many solutiontechniques were investigated to obtain the reverse kinematicanalysis. Duffy and Rooney [15] introduced a unified theoryfor the analysis of spatial mechanisms which was utilized onmany cases (Crane and Duffy [16]). Other solution techniquesinclude numerical iteration and continuation methods [17]-[21].The approach used in this paper, based on the unified theory,
has the advantage in that solutions are obtained without therequirement for any iterations or initial guess values.
The 7R mechanism is the most complicated kinematicanalysis of a one degree of freedom spatial loop and this wasreferred to as the “Mount Everest of kinematic problems” byProfessor Ferdinand Freudenstein. Duffy and Crane [22]obtained a 32nd degree input-output equation in the tan-half-angle of the output angular displacement for this mechanism.Lee and Liang [23] later obtained a 16th degree polynomialinput-output equation in the tan-half-angle of the output angulardisplacement. More recently, Husty, Pfurner and Schröcker [24]used multidimensional geometry and Segre manifolds’ theory tointerpret the nature of the general 6R-chains reverse kinematicproblem.
This paper presents the solution of a spatial 7R closed-loopmechanism with consecutive pairs of joint axes parallel. Themotivation of this work is related to future applications ofspatial motion generation incorporating non-circular gears. Fig.1(a) shows a planar motion generator where two pairs of non-circular gears have been designed in order to attain a desiredmotion of the coupler link of a six link mechanism. Fig. 1(b)shows an extension of this concept to the spatial case. Fivepairs of non-circular gears have been designed and incorporatedin a twelve link closed-loop chain to obtain a one degree-of-freedom mechanism where the coupler link (containing point P)follows a desired motion profile. Planar non-circular gears canbe utilized if adjacent joint axes are parallel as shown in Fig.1(b). The closed-loop mechanism can be separated into twosix-axis open-loop mechanisms. A reverse kinematic analysis ofthis device is required in order to design the non-circular gearsthat will position the distal link as desired for the specified
motion profile. This paper details the reverse kinematicanalysis of this particular six axis open-loop chain. The specialgeometry is that the 1st and 2nd axes are parallel, the 3rd and 4th
axes are parallel, and the 5th and 6th joint axes are parallel.
Figure 1: Incorporation of planar non-circular gears into (a)planar mechanism and (b) spatial mechanism to obtain onedegree of freedom motion generation.
2. NOMENCLATURE
aij = Link distance of the i link
Si = Offset distance of the i linkθi = Joint angle of the i linkαij = Twist angles of the i linkφ1 = Angle between the x axes of the fixed and 1st
coordinate systemsγ1 = Angle between the x axes of the hypothetical
link coordinate system and the fixed coordinatesystem
aij = Link unit vector of the i linkSi = Joint unit vector of the i linkP = Position vectorR = Rotation matrixi, j, k = Unit vectors along the x y z axessi, ci = Sine and cosine of the joint angle θi
sij, cij = Sine and cosine of the twist angle αij
si+j, ci+j = Sine and cosine of the sum of angles θi+ θj
3. REVERSE KINEMATIC ANALYSIS
3.1 Problem Statement
Figure 2 presents the joint vectors, link vectors and jointangles labeling corresponding to an open-loop six axis chainwith consecutive joint axes parallel: S1||S2, S3||S4 and S5||S6. Theproblem statement for the mechanism is summarized as
2) User defined parameters to establish the 6th coordinatesystem, i.e. the coordinate system attached to the last link:
distance S6 and direction of vector a67 relative to linkoffset S6
3). Tool point measured in the 6th coordinate system: 6Ptool
4). Desired location of tool point as measured in the fixedcoordinate system: FPtool
5). Desired orientation of the end effector link as measured in
the fixed coordinate system: FS6 and Fa67
FindThe joint angle parameters φ1, θ2, θ3, θ4, θ5, and θ6 that willposition and orient the last link as specified.
3.2 Determination of Equivalent Closed-Loop SpatialMechanism
The solution method for this problem follows the approachdefined in Crane and Duffy [16]. First the coordinates of theorigin point of the 6th coordinate system are calculated as
FP6orig = FPtool – (6Ptool · i)Fa67
– (6Ptool · j)FS6×Fa67 – (6Ptool · k)FSF. (1)
At this point the 4x4 transformation matrix that describes theposition and orientation of the 6th coordinate system withrespect to ground, TF
6, is known since the fourth column of the
matrix is defined by FP6orig and the columns of the upper 3×3sub-matrix are defined by Fa67,
FS6×Fa67 and FS6. Thus
TF6
=
1000orig6
FF6 PR (2)
where:
RF6
= [Fa67FS6×
Fa67FS6]. (3)
Next, a hypothetical seventh joint axis is defined byarbitrarily selecting values for the parameters a67 and α67. Theselected values for these parameters are
Figure 2: Joint vectors, link vectors and joint angles of theopen-loop spatial mechanism.
A hypothetical link (see Figure 3) is then inserted betweenthe new seventh joint axis and the first joint axis. Threedistances, S7, a71, and S1 and three angles, α71, θ7, and γ1 arethen readily determined as described in [16]. The angle γ1 isshown in Figure 3 and its relation to θ1 can be written as
θ1 = φ1 + γ1. (6)
Figure 3: Hypothetical closure link
At this point an equivalent closed-loop spatial mechanism,formed by seven links and seven revolute joints, has beendefined where all the link lengths, a12 through a71, all the twistangles, α12 through α71, and all the joint offsets, S1 through S7,are known. Further the resulting closed-loop mechanism is aone degree-of-freedom device where the angle θ7 is known.
3.3 Equivalent Spherical Mechanism
Crane and Duffy [16] solve closed-loop spatial mechanismsby introducing sine, sine-cosine, and cosine laws for anequivalent spherical mechanism that is associated with theclosed-loop mechanism. These laws provide expressions thatcontain only the joint angles and twist angles of the spatialmechanism. An equivalent closed-loop spherical mechanismcan be formed from the closed-loop spatial mechanism bytranslating the directions of the joint axis vectors and the linkvectors so that they all intersect at a point. Spherical links maybe inserted on a unit sphere to maintain the angularrelationships between the joint axis vectors. The equivalentspherical mechanism for the case under consideration is shownin Figure 4. The equivalent spherical mechanism has reducedto a quadrilateral since the spatial mechanism has adjacent axesparallel.
A spherical quadrilateral is a one degree-of-freedommechanism (further information about mobility see Real et al[26]). In this case θ7 is known and it is thus possible to obtainsolutions for the remaining angles of the spherical quadrilateralwhich in this case are the three sums θ1+θ2, θ3+θ4, and θ5+θ6.
The solution starts with solving the spherical quadrilateraldepicted in Figure 4 to find the sums of the angles θ1+θ2, θ3+θ4,and θ5+θ6. A detailed solution is presented in [16] which alsopresents a derivation of the sine, sine-cosine, and cosine lawsfor a spherical quadrilateral that will be used here.
3.4.1 Solving for θ1+θ2
A spherical cosine law may be written that will relate theangle θ7 and the joint angle sum θ1+θ2. This cosine law iswritten here as
A c1+2 + B s1+2 + D = 0 (7)
where si+j and ci+j represent the sine and cosine of thesum θi+θj and where
A = -s23 (s71 c67 + c71 s67 c7)
B = s23 s67 s7
D = c23 (c71 c67 – s71 s67 c7).
The terms sij and cij represent the sine and cosine of thetwist angle αij and the terms si and ci represent the sine andcosine of the joint angle θi. Equation (7) can be solved for twosolutions for θ1+2 which will be referred to as θ(1+2)A and θ(1+2)B,where θ(i+j)= θi+θj.
3.4.2 Solving for θ3+θ4
Corresponding values for the sum θ3+θ4 can be determinedfrom a sine and sine-cosine law for the equivalent sphericalquadrilateral. These equations are written as
2176771677121767
45
43 s)csccs(c)ss(s
1s (8)
)cssc(csAcs
1c 767716771233423
45
43 (9)
where:
217677167712176734 c)cscc(ss)s(sA .
The corresponding values for θ(3+4)A and θ(3+4)B for each value ofθ1+2 are obtained from (8) and (9).
3.4.3 Solving for θ5+θ6
Corresponding values for the sum θ5+θ6 can be determinedfrom the following sine and sine-cosine laws:
7212371237172123
45
65 s)cscc(sc)s(ss
1s (10)
)cssc(csAcs
1c 767716771675667
45
65 (11)
where:
721237123717212356 c)cscc(ss)s(sA .
The corresponding values for θ5+6 are obtained from (10) and(11) for each value of θ1+2.
At this point of the analysis, two values for θ1+2 and uniquecorresponding values for the sums θ3+4 and θ5+6 have beendetermined. A solution tree of the sum of angles solution ispresented in Fig 5.
Figure 5: Solution tree of the spherical quadrilateral to findthe sums of consecutive angle joints
3.5 Spatial Mechanism Joint Angles
In this section, the individual joint angles θ1 through θ6 areobtained from projections of the vector loop equation of thespatial mechanism and from knowledge of the sums of anglesthat have been previously determined.
3.5.1 Vector Loop Equation
The vector loop equation for the equivalent closed-loopmechanism is written as
The vector terms will all be evaluated in terms of a right-handed coordinate system whose Z axis is along S3 and whoseX axis is along a23. This yields the following three equations:
The known sums of angles θ1+2, θ3+4, and θ5+6 have beenintroduced in the terms ci+j and si+j in the coefficients A1 throughD3. In the next sections, equations (13) through (15) are solvedfor the three joint angles θ2, θ3 and θ5.
3.5.2 Solution for θ5 and θ6
In this section equations (13), (14), and (15) are manipu-lated in order to obtain one equation in terms of the unknownsine and cosine of θ5. Equation (13) can be solved for c3 and(14) for s3 to yield
2
55453213
A
AsAcAcAc
(17)
2
55453213
B
BsBcBsBs
. (18)
Squaring (17) and (18) and adding the two yields the followingequation in terms of the two unknowns θ2 and θ5
2
2
25545321
A
)AsAcAc(A
01B
)BsBcBs(B2
2
25545321
. (19)
Multiplying Eq. (19) by (A2B2)2 and expanding gives
E1c22+E2c2+E3s2
2+E4s2+E5 = 0 (20)
where
E1 = A12 B2
2
E2 = E2Ac5+E2Bs5+E2c, E3 = A22 B1
2
E4 = E4Ac5+E4Bs5+E4C
E5 = E5Ac52+E5Bs5
2+E5Cs5c5+E5Dc5+E5E s5+ E5F, (21)
and where
E2A = 2A1B22 A3, E2B = 2A1B2
2 A4, E2C = 2A1B22 A5,
E4A = 2 B1A22 B3, E4B = 2 B1A2
2 B4, E4C = 2 B1A22 B5,
E5A = A22 B3
2+B22A3
2, E5B = A42 B2
2+B42 A2
2,
E5C = 2(A3A4B22 + B3B4A2
2),
E5D = 2(A3A5B22 + B3B5A2
2), E5E = 2(A4A5B22 + B4B5A2
2),
E5F = B22(A5
2-A22) + A2
2B52. (22)
Solving Eq. (15) for s2 gives
1
3522
D
DsDs
. (23)
Squaring Eq. (23), substituting s22=1-c2
2 and solving for c22
gives
2
1
3522
2D
DsD1c
. (24)
Substituting (23) and (24) into (20) and regrouping gives
2
1
3523
2
1
3521
D
DsDE
D
DsD1E
225
1
3524 cEE
D
DsDE
. (25)
Squaring (25) and substituting (24) for c22 yields an equation in
Thus an eight degree polynomial in the variable x5, i.e.equation (31), can be obtained for each of the two sets ofsolutions for the sums θ1+2, θ3+4, and θ5+6. Values of θ5 areobtained from (31) for each of the eight solutions of thepolynomial as
θ5 = 2tan-1(x5). (33)
Corresponding values of θ6 are obtained fromθ6 = (θ5+θ6) – θ5. (34)
3.5.3 Solution for θ2, θ1, and φ1
Values for the angle θ2 are now to be determined whichcorrespond to each solution set of θ5. The terms A6, B6, and D6
The corresponding value for sinθ2 can be obtained directly from(40) as
1
62
D
Ds
. (41)
The value for cosθ2 will be obtained by writing (38) and (39) as-c3 = (A1c2 + A6)/A2, (42)-s3 = (B1s2 + B6)/B2. (43)
Squaring and adding these equations yields
01B
BsB
A
AcA2
2
621
2
2
621
. (44)
Expanding this equation yields
2
2
2
6261
2
2
2
1
A
AcAA2cA
01B
BsBB2sB2
2
2
6261
2
2
2
1 . (45)
Equation (41) may be used to directly substitute for s2 in termsof known parameters. The term cos2θ2 may be replaced bysquaring (41), substituting c2
2=1-s22, and solving for c2
2 as2
1
62
2D
D1c
. (46)
Substituting (46) into (45) and solving for c2 gives
61
2
2
2
1
2
1
2
2
2
6126
2
2126
2
12
AABD2
DBABAAAc
(47)
where:
A126 = B22D6
2 – B22D1
2
B126 = D12(B2
2 – B62) + 2B1B6D1D6 – B1
2D62.
Knowledge of the sine and cosine of θ2 from (41) and (47)gives a unique value for the angle θ2. The corresponding valueof the angle θ1 is obtained from
θ1 = (θ1+θ2) – θ2. (48)Lastly, values for the first joint parameter, φ1, are obtained from(6) as
φ1 = θ1 – γ1. (49)
3.5.4 Solution for θ3 and θ4
The angles θ3 and θ4 are the last two remaining parametersto be obtained. The sine and cosine of θ3 that correspond toeach solution set of θ1, θ2, θ5, and θ6 can be obtained directlyfrom (42) and (43) yielding a unique corresponding value forθ3. The corresponding value for θ4 is then obtained from
θ4 = (θ3 + θ4) – θ3. (50)
At this point all the joint angle parameters have beenobtained. Figure 6 illustrates the sixteen solutions.
4. NUMERICAL EXAMPLE
The constant mechanism parameters of an example case arepresented in Table 1. The free choice values for the offset S6,the hypothetical link length a67 and the hypothetical twist angleα67 are presented in Table 1. The position and orientationrequirements are summarized in Table 2. The closed-the-loopparameters obtained for this case are summarized in Table 3.
The solution for the equivalent spherical quadrilateral forthe sums θ1+θ2, θ3+θ4, and θ5+θ6 is presented in Table 4.Fourteen real solutions and two complex solutions were foundfor this numerical example and are summarized in Table 5.
Figure 7 illustrates the fourteen real configurations forthe open-loop manipulator in this numerical example. Aforward analysis was performed as a check and all sixteensolutions, including the complex solutions, position andorient the end effector as desired.
`Efforts to identify a numerical example that would yieldsixteen real solutions have not yet been successful. However,the fact that the two complex solutions in this example dosatisfy the problem statement indicates that sixteen is the correctdegree of the solution for this geometry.
The reverse kinematic solution of a spatial open-loopmechanism consisting of six revolute joints and six seriallinks, with consecutive pairs of joint axes parallel waspresented in this paper. The solution technique incorporateda hypothetical closure-link to form a one degree-of-freedomclosed-loop spatial mechanism where one of the joint angles,θ7, is known. An analysis of the equivalent closed-loopspherical mechanism resulted in the solution of two sets ofsolutions for the joint angle sums (θ1+θ2), (θ3+θ4), and(θ5+θ6). Projection of the vector loop equation of the spatialmechanisms on three independent directions then resulted ina set of equations from which a total of sixteen solution setsfor the joint angles could be obtained. A numerical examplewas presented which verifies the degree of the solution.
The motivation of this work is related to futureapplications of spatial motion generation incorporating non-circular gears. Using the geometry discussed here, five pairsof planar non-circular gears can be incorporated into a spatialtwelve axis closed-loop mechanism to result in a one degree-of-freedom device where the coupler link is positioned andoriented along a desired path. This problem is currentlyunder investigation.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support provided bythe Department of Energy via the University Research Programin Robotics (URPR), grant number DE-FG04-86NE37967.
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