REVERSE KINEMATIC ANALYSIS OF THE SPATIAL SIX AXIS ROBOTIC MANIPULATOR WITH CONSECUTIVE JOINT AXES PARALLEL

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1 Proceeding of the ASME 007 International Deign Engineering Technical Conference & Computer and Information in Engineering Conference IDETC/CIE 007 September 4-7, 007 La Vega, Nevada, USA DETC REVERSE KINEMATIC ANALYSIS OF THE SPATIAL SIX AXIS ROBOTIC MANIPULATOR WITH CONSECUTIVE JOINT AXES PARALLEL Javier Roldán Mckinley Center for Intelligent Machine and Robotic Univerity of Florida, Gaineville, Florida, USA Carl Crane III * Center for Intelligent Machine and Robotic Univerity of Florida Gaineville, Florida, USA ccrane@ufl.edu David B. Dooner Univerity of Puerto Rico at Mayagüez Mayagüez, Puerto Rico d_dooner@me.uprm.edu ABSTRACT Thi paper introduce a reconfigurable one degree-offreedom patial mechanim that can be applied to repetitive motion tak. The concept i to incorporate five pair of noncircular gear into a ix degree-of freedom cloed-loop patial chain. The gear pair are deigned baed on the given mechanim parameter and the uer defined motion pecification of a coupler link of the mechanim. It i hown in the paper that planar gear pair can be ued if the patial cloedloop chain i compried of ix pair of parallel joint axe, i.e. the firt joint axi i parallel to the econd, the third i parallel to the fourth,, and the eleventh i parallel to the twelfth. Thi paper preent the detailed revere kinematic analyi of thi pecific geometry. A numerical example i preented. Keyword: Robotic manipulator; patial kinematic; pherical mechanim; revere kinematic analyi; non-circular gear.. INTRODUCTION The revere kinematic analyi of patial mechanim ha been widely tudied for practically any combination of revolute, cylindrical, and primatic joint, ee []-[]. Many olution technique were invetigated to obtain the revere kinematic analyi. Duffy and Rooney [] introduced a unified theory for the analyi of patial mechanim which wa utilized on many cae (Crane and Duffy []). Other olution technique include numerical iteration and continuation method [7]-[]. The approach ued in thi paper, baed on the unified theory, ha the advantage in that olution are obtained without the requirement for any iteration or initial gue value. The 7R mechanim i the mot complicated kinematic analyi of a one degree of freedom patial loop and thi wa referred to a the Mount Everet of kinematic problem by Profeor Ferdinand Freudentein. Duffy and Crane [] obtained a 3 nd degree input-output equation in the tan-halfangle of the output angular diplacement for thi mechanim. Lee and Liang [3] later obtained a th degree polynomial input-output equation in the tan-half-angle of the output angular diplacement. More recently, Huty, Pfurner and Schröcker [4] ued multidimenional geometry and Segre manifold theory to interpret the nature of the general R-chain revere kinematic problem. Thi paper preent the olution of a patial 7R cloed-loop mechanim with conecutive pair of joint axe parallel. The motivation of thi work i related to future application of patial motion generation incorporating non-circular gear. Fig. (a) how a planar motion generator where two pair of noncircular gear have been deigned in order to attain a deired motion of the coupler link of a ix link mechanim. Fig. (b) how an extenion of thi concept to the patial cae. Five pair of non-circular gear have been deigned and incorporated in a twelve link cloed-loop chain to obtain a one degree-offreedom mechanim where the coupler link (containing point P) follow a deired motion profile. Planar non-circular gear can be utilized if adjacent joint axe are parallel a hown in Fig. (b). The cloed-loop mechanim can be eparated into two ix-axi open-loop mechanim. A revere kinematic analyi of thi device i required in order to deign the non-circular gear that will poition the dital link a deired for the pecified * Profeor and author of correpondence, Phone: (3) 39-94, Fax: (3) 39-07, ccrane@ufl.edu. Copyright 007 by ASME

2 motion profile. Thi paper detail the revere kinematic analyi of thi particular ix axi open-loop chain. The pecial geometry i that the t and nd axe are parallel, the 3 rd and 4 th axe are parallel, and the th and th joint axe are parallel. (a) planar cae 3. REVERSE KINEMATIC ANALYSIS 3. Problem Statement Figure preent the joint vector, link vector and joint angle labeling correponding to an open-loop ix axi chain with conecutive joint axe parallel: S S, S 3 S 4 and S S. The problem tatement for the mechanim i ummarized a Given ). The contant mechanim parameter: link ditance a, a 3, a 34, a 4, a twit angle α =0, α 3, α 34 =0, α 4, α =0 offet ditance S, S 3 =0, S 4, S =0. ) Uer defined parameter to etablih the th coordinate ytem, i.e. the coordinate ytem attached to the lat link: ditance S and direction of vector a 7 relative to link offet S 3). Tool point meaured in the th coordinate ytem: P tool 4). Deired location of tool point a meaured in the fixed coordinate ytem: F P tool ). Deired orientation of the end effector link a meaured in the fixed coordinate ytem: F S and F a 7 Find The joint angle parameter φ, θ, θ 3, θ 4, θ, and θ that will poition and orient the lat link a pecified. (b) patial cae Figure : Incorporation of planar non-circular gear into (a) planar mechanim and (b) patial mechanim to obtain one degree of freedom motion generation.. NOMENCLATURE a ij = Link ditance of the i link S i = Offet ditance of the i link θ i = Joint angle of the i link α ij = Twit angle of the i link φ = Angle between the x axe of the fixed and t coordinate ytem γ = Angle between the x axe of the hypothetical link coordinate ytem and the fixed coordinate ytem a ij = Link unit vector of the i link S i = Joint unit vector of the i link P = Poition vector R = Rotation matrix i, j, k = Unit vector along the x y z axe i, c i = Sine and coine of the joint angle θ i ij, c ij = Sine and coine of the twit angle α ij i+j, c i+j = Sine and coine of the um of angle θ i + θ j 3. Determination of Equivalent Cloed-Loop Spatial Mechanim The olution method for thi problem follow the approach defined in Crane and Duffy []. Firt the coordinate of the origin point of the th coordinate ytem are calculated a F P orig = F P tool ( P tool i) F a 7 ( P tool j) F S F a 7 ( P tool k) F S F. () At thi point the 4x4 tranformation matrix that decribe the poition and orientation of the th coordinate ytem with repect to ground,, i known ince the fourth column of the T F matrix i defined by F P orig and the column of the upper 3 3 ub-matrix are defined by F a 7, F S F a 7 and F S. Thu F F F T = R P orig () where: F R = [ F a F 7 S F a F 7 S ]. (3) Next, a hypothetical eventh joint axi i defined by arbitrarily electing value for the parameter a 7 and α 7. The elected value for thee parameter are a 7 =0 (4) α 7 =90. () Copyright 007 by ASME

3 A hypothetical link (ee Figure 3) i then inerted between the new eventh joint axi and the firt joint axi. Three ditance, S 7, a 7, and S and three angle, α 7, θ 7, and γ are then readily determined a decribed in []. The angle γ i hown in Figure 3 and it relation to θ can be written a θ = φ + γ. () S z F S y a 7 x γ x F y F z θ 7 S 7 S S 7 α 7 a 7 a 7 Figure 3: Hypothetical cloure link At thi point an equivalent cloed-loop patial mechanim, formed by even link and even revolute joint, ha been defined where all the link length, a through a 7, all the twit angle, α through α 7, and all the joint offet, S through S 7, are known. Further the reulting cloed-loop mechanim i a one degree-of-freedom device where the angle θ 7 i known. Figure : Joint vector, link vector and joint angle of the open-loop patial mechanim. 3.3 Equivalent Spherical Mechanim Crane and Duffy [] olve cloed-loop patial mechanim by introducing ine, ine-coine, and coine law for an equivalent pherical mechanim that i aociated with the cloed-loop mechanim. Thee law provide expreion that contain only the joint angle and twit angle of the patial mechanim. An equivalent cloed-loop pherical mechanim can be formed from the cloed-loop patial mechanim by tranlating the direction of the joint axi vector and the link vector o that they all interect at a point. Spherical link may be inerted on a unit phere to maintain the angular relationhip between the joint axi vector. The equivalent pherical mechanim for the cae under conideration i hown in Figure 4. The equivalent pherical mechanim ha reduced to a quadrilateral ince the patial mechanim ha adjacent axe parallel. A pherical quadrilateral i a one degree-of-freedom mechanim (further information about mobility ee Real et al []). In thi cae θ 7 i known and it i thu poible to obtain olution for the remaining angle of the pherical quadrilateral which in thi cae are the three um θ +θ, θ 3 +θ 4, and θ +θ. 3 Copyright 007 by ASME

4 A 34 (7 7 ) (7c7 c7 7c7 ) c. S,S α 4 θ 3 +θ 4 S 3,S 4 The correponding value for θ (3+4)A and θ (3+4)B for each value of θ + are obtained from (8) and (9). θ +θ α 7 a 3 Figure 4: Equivalent pherical quadrilateral. 3.4 Solution of Spherical Quadrilateral θ 7 The olution tart with olving the pherical quadrilateral depicted in Figure 4 to find the um of the angle θ +θ, θ 3 +θ 4, and θ +θ. A detailed olution i preented in [] which alo preent a derivation of the ine, ine-coine, and coine law for a pherical quadrilateral that will be ued here. a 7 a 4 α 7 a 7 α 3 S 7 S,S θ +θ Solving for θ +θ Correponding value for the um θ +θ can be determined from the following ine and ine-coine law: ( 3 )c 7 ( 7c 3 c 7 3c ) 7 (0) 4 c c 7A 7 (c7c 7 7 7c7 ) () 4 where: A ( ) ( c c c ) c7 The correponding value for θ + are obtained from (0) and () for each value of θ +. At thi point of the analyi, two value for θ + and unique correponding value for the um θ 3+4 and θ + have been determined. A olution tree of the um of angle olution i preented in Fig Solving for θ +θ A pherical coine law may be written that will relate the angle θ 7 and the joint angle um θ +θ. Thi coine law i written here a A c + + B + + D = 0 (7) where i+j and c i+j repreent the ine and coine of the um θ i +θ j and where A = - 3 ( 7 c 7 + c 7 7 c 7 ) B = D = c 3 (c 7 c c 7 ). The term ij and c ij repreent the ine and coine of the twit angle α ij and the term i and c i repreent the ine and coine of the joint angle θ i. Equation (7) can be olved for two olution for θ + which will be referred to a θ (+)A and θ (+)B, where θ (i+j) = θ i +θ j Solving for θ 3 +θ 4 Correponding value for the um θ 3 +θ 4 can be determined from a ine and ine-coine law for the equivalent pherical quadrilateral. Thee equation are written a 3 4 ( 77 )c ( 7c7 c77c7 ) (8) 4 c3 4 c 3 A 34 3 (c 7c 7 7 7c7 ) (9) 4 where: Figure : Solution tree of the pherical quadrilateral to find the um of conecutive angle joint 3. Spatial Mechanim Joint Angle In thi ection, the individual joint angle θ through θ are obtained from projection of the vector loop equation of the patial mechanim and from knowledge of the um of angle that have been previouly determined. 3.. Vector Loop Equation The vector loop equation for the equivalent cloed-loop mechanim i written a S S + a a + S S + a 3 a 3 + a 34 a 34 + S 4 S 4 + a 4 a 4 + a a + S S + S 7 S 7 + a 7 a 7 = 0. () The vector term will all be evaluated in term of a righthanded coordinate ytem whoe Z axi i along S 3 and whoe X axi i along a 3. Thi yield the following three equation: A c +A c 3 +A 3 c +A 4 +A = 0, (3) B +B 3 +B 3 c +B 4 +B = 0, (4) D +D +D 3 = 0 () where the coefficient A through D 3 can be expreed in term of known quantitie a A = a, A = a 34, A 3 = a c 3+4, A 4 = -a c 4 3+4, A = S S (c 7 c c + ) + a 3 + a 7 c + + a 4 c 3+4, 4 Copyright 007 by ASME

5 B = -a c 3, B = a 34, B 3 = a 3+4, B 4 = a c 4 c 3+4, B = S 3 + S 3 + S 7 ( 3 c 7 +c 3 7 c + ) + S [c 3 ( c 7 c 7 c + ) 3 7 c 7 ] a 7 c a 4 3+4, D = a 3, D = a 4, D 3 = S c 3 + S c 3 + S 7 (c 3 c c + ) + S [ 3 ( 7 + -c 7 c 7 c + ) c 3 7 c 7 ] + a S 4. () The known um of angle θ +, θ 3+4, and θ + have been introduced in the term c i+j and i+j in the coefficient A through D 3. In the next ection, equation (3) through () are olved for the three joint angle θ, θ 3 and θ. 3.. Solution for θ and θ In thi ection equation (3), (4), and () are manipulated in order to obtain one equation in term of the unknown ine and coine of θ. Equation (3) can be olved for c 3 and (4) for 3 to yield Ac A 3c A 4 A c3 (7) A B B3c B4 B 3. (8) B Squaring (7) and (8) and adding the two yield the following equation in term of the two unknown θ and θ (Ac A 3c A 4 A ) A (B B3c B4 B ) 0. (9) B Multiplying Eq. (9) by (A B ) and expanding give where E = A B E c +E c +E 3 +E 4 +E = 0 (0) E = E A c +E B +E c, E 3 = A B E 4 = E 4A c +E 4B +E 4C E = E A c +E B +E C c +E D c +E E + E F, () and where E A = A B A 3, E B = A B A 4, E C = A B A, E 4A = B A B 3, E 4B = B A B 4, E 4C = B A B, E A = A B 3 +B A 3, E B = A 4 B +B 4 A, E C = (A 3 A 4 B + B 3 B 4 A ), E D = (A 3 A B + B 3 B A ), E E = (A 4 A B + B 4 B A ), E F = B (A -A ) + A B. () Solving Eq. () for give D D3. (3) D Squaring Eq. (3), ubtituting =-c and olving for c give D D 3. (4) D c Subtituting (3) and (4) into (0) and regrouping give D D3 D D3 E E 3 D D D D3 E 4 E E c D. () Squaring () and ubtituting (4) for c yield an equation in the ine and coine of θ which i written a F 4 + F 3 + F 3 3 c + F 4 c + F c + F c + F 7 c + F 8 c + F 9 c + F 0 + F + F = 0 () where F =D (D E 4B +D E B D E 3 D E 4B +D E 3 + D E D E 4B +D E D E E 3 ) F =D (D D 3 E 4D D 3 E 3 E +D D 3 E 3 D E 3 D E 4C + D E D E 4C +D D E B E C 3D E 3 D E 4B D 3 + D D E 4B E 4C +3D E D E 4B D 3 +D E B D 3 +D E 4B D 3 F 3 = D D (-D E 4A E 3 +D E 4A E +D E A E B +D E 4B E 4A ) F 4 = D D (E 4A +E A ) F = D D D 3 (E 4A +E A ) F = D (D E A +E 4A D 3 +E A D 3 ) F 7 =D D (D D E A E C +D D E 4A E 4C 3D E 3 E 4A D 3 + 3D E E 4A D 3 +D E A E B D 3 +D E 4A E 4B D 3 ), F 8 = D (-E A E B D 3 D E 4A D E D D E 4A E + D E A E B D 3 +D E 4A D 3 E 4B +D E A E C D D 3 + D E 4A D E 4C D 3 +3E D D 3 E 4A 3E 3 D D 3 E 4A ) F 9 =D (-E A E C D 3 D E E 4A D 3 D E 4A D 3 E + D E 4A D 3 E 4C +D E A E C D 3 +E D 3 3 E 4A E 3 D 3 3 E 4A ) F 0 = D [(E 3 +E -E E 3 )D 3 +(E D E 4C - E 3 D E 4C )D 3 E E D +D E C + D E 4C E D + E D E 3 + E 3 E D ] +D [(E D E 4B -E 3 D E 4B )D 3 + (4D E B E C +4D E 4B E 4C )D 3 D 3 E 4B E E D 3 E 4B ]+(D E 4B +D E B )D 3 D 4 E B F = D [(4E +4E 3-8E E 3 )D 3 3 +(E D E 4C -E 3 D E 4C )D 3 +(4E D E 3-4E E D +D E 4C +D E C - 4E D +4E 3 E D )D 3 E D 3 E 4C D 3 E 4C E ]+ (E D E 4B -E 3 D E 4B )D D E B E C +D E 4B E 4C )D 3 (D 3 E 4B E +E D 3 E 4B )D 3 D 4 E B E C F = D 3 4 (E +E 3 -E E 3 )+D 3 3 (E D E 4C -E 3 D E 4C )+ D 3 (D E C +D E 4C -E E D - E D +E D E 3 +E 3 E D ) D 3 (E D 3 E 4C +D 3 E 4C E )+E D 4 +E D 4 E + E D 4 D 4 E C. (7) Copyright 007 by ASME

6 The tan-half angle of θ i now introduced a x tan, (8) and the ine and coine of θ may be written in term of x via the trig identitie x (9) x c x. (30) x Subtituting (9) and (30) into (), dividing throughout by (+x ) 4, and regrouping yield C 8 x 8 + C 7 x 7 + C x + C x + C 4 x 4 + C 3 x 3 + C x + C x + C 0 = 0 (3) where C 8 = F + F F 9 C 7 = (F F 8 + F ) C = (F 4 F 7 F 9 + F 0 + F ) C = (4F 4 F 3 F F 8 + 3F ) C 4 = (8F 4F 4 F + 4F 0 + 3F ) C 3 = (4F + 4 F 3 F + F 8 + 3F ) C = (F 4 + F 7 + F 9 + F 0 + F ) C = (F + F 8 + F ) C 0 = F 9 + F + F. (3) Thu an eight degree polynomial in the variable x, i.e. equation (3), can be obtained for each of the two et of olution for the um θ +, θ 3+4, and θ +. Value of θ are obtained from (3) for each of the eight olution of the polynomial a θ = tan - (x ). (33) Correponding value of θ are obtained from θ = (θ +θ ) θ. (34) 3..3 Solution for θ, θ, and φ Value for the angle θ are now to be determined which correpond to each olution et of θ. The term A, B, and D are now defined in term of known parameter a A = A 3 c + A 4 + A (3) B = B 3 c + B 4 + B (3) D = D + D 3. (37) Subtituting thee term into (3), (4), and () give A c + A c 3 + A = 0 (38) B + B 3 + B = 0 (39) D + D = 0. (40) The correponding value for inθ can be obtained directly from (40) a D. (4) D The value for coθ will be obtained by writing (38) and (39) a -c 3 = (A c + A )/A, (4) - 3 = (B + B )/B. (43) Squaring and adding thee equation yield Ac A B B 0 A B. (44) Expanding thi equation yield A c A A c A A B BB B 0. (4) B Equation (4) may be ued to directly ubtitute for in term of known parameter. The term co θ may be replaced by quaring (4), ubtituting c =-, and olving for c a c D. (4) D Subtituting (4) into (4) and olving for c give where: c A = B D B D A A A B A B D (47) D B A A B = D (B B ) + B B D D B D. Knowledge of the ine and coine of θ from (4) and (47) give a unique value for the angle θ. The correponding value of the angle θ i obtained from θ = (θ +θ ) θ. (48) Latly, value for the firt joint parameter, φ, are obtained from () a φ = θ γ. (49) 3..4 Solution for θ 3 and θ 4 The angle θ 3 and θ 4 are the lat two remaining parameter to be obtained. The ine and coine of θ 3 that correpond to each olution et of θ, θ, θ, and θ can be obtained directly from (4) and (43) yielding a unique correponding value for θ 3. The correponding value for θ 4 i then obtained from θ 4 = (θ 3 + θ 4 ) θ 3. (0) At thi point all the joint angle parameter have been obtained. Figure illutrate the ixteen olution. 4. NUMERICAL EXAMPLE The contant mechanim parameter of an example cae are preented in Table. The free choice value for the offet S, the hypothetical link length a 7 and the hypothetical twit angle α 7 are preented in Table. The poition and orientation requirement are ummarized in Table. The cloed-the-loop parameter obtained for thi cae are ummarized in Table 3. The olution for the equivalent pherical quadrilateral for the um θ +θ, θ 3 +θ 4, and θ +θ i preented in Table 4. Fourteen real olution and two complex olution were found for thi numerical example and are ummarized in Table. Copyright 007 by ASME

7 Figure : Solution tree for the patial ix axi manipulator with conecutive axe parallel. Table : Contant mechanim parameter for numerical example Offet Ditance [in] Link Length [in] Twit Angle [deg] S = a = 4.38 α = 0 S 3 = 0 a 3 = 0.74 α 3 = 9.99 S 4 =.34 a 34 =.9009 α 34 = 0 S = 0 a 4 =.349 α 4 = S * =.0 a = α = 0 a 7 * = 0 α 7 * = 90 *: Free choice. Table : Deired poition and orientation F a 7 [ ] F S [ ] F P tool [in] [ ] P tool [in] [ 7 8] 7 Copyright 007 by ASME

8 Table 3: Calculated cloe-the-loop parameter Ditance [in] Angle [degree] a 7 = 4.74 α 7 = S 7 = 0.83 θ 7 = -9.7 S = -9.9 γ = Table 4: Sum of conecutive joint angle obtained from the equivalent pherical quadrilateral olution Sum of Angle Solution A [degree] Solution B [degree] θ θ θ Sol φ [deg] θ [deg] θ 3 [deg] θ 4 [deg] θ [deg] θ [deg] A B C D E Table : Joint angle correponding to the ixteen olution of the numerical example i i i i i i F G H I J K L M N O P Copyright 007 by ASME

9 Figure 7 illutrate the fourteen real configuration for the open-loop manipulator in thi numerical example. A forward analyi wa performed a a check and all ixteen olution, including the complex olution, poition and orient the end effector a deired. `Effort to identify a numerical example that would yield ixteen real olution have not yet been ucceful. However, the fact that the two complex olution in thi example do atify the problem tatement indicate that ixteen i the correct degree of the olution for thi geometry. Figure 7: Fourteen real olution in Table 9 Copyright 007 by ASME

10 Figure 7: Fourteen real olution in Table CONCLUSION The revere kinematic olution of a patial open-loop mechanim coniting of ix revolute joint and ix erial link, with conecutive pair of joint axe parallel wa preented in thi paper. The olution technique incorporated a hypothetical cloure-link to form a one degree-of-freedom cloed-loop patial mechanim where one of the joint angle, θ7, i known. An analyi of the equivalent cloed-loop pherical mechanim reulted in the olution of two et of olution for the joint angle um (θ+θ), (θ3+θ4), and (θ+θ). Projection of the vector loop equation of the patial mechanim on three independent direction then reulted in a et of equation from which a total of ixteen olution et for the joint angle could be obtained. A numerical example wa preented which verifie the degree of the olution. The motivation of thi work i related to future application of patial motion generation incorporating noncircular gear. Uing the geometry dicued here, five pair of planar non-circular gear can be incorporated into a patial twelve axi cloed-loop mechanim to reult in a one degreeof-freedom device where the coupler link i poitioned and oriented along a deired path. Thi problem i currently under invetigation. ACKNOWLEDGMENTS The author gratefully acknowledge the upport provided by the Department of Energy via the Univerity Reearch Program in Robotic (URPR), grant number DE-FG04-8NE Copyright 007 by ASME

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