DARWIN-OP HUMANOID ROBOT KINEMATICS. Robert L. Williams II, Ph.D., Mechanical Engineering, Ohio University, Athens, Ohio, USA

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1 Proceedngs of the ASME 1 Internatonal Desgn Engneerng Techncal onferences & omputers and Informaton n Engneerng onference IDET/IE 1 August 1-15, 1, hcago, IL, USA DET1-765 DARWIN-OP HUMANOID ROBOT KINEMATIS Robert L. Wllams II, Ph.D., wllar4@oho.edu Mechancal Engneerng, Oho Unversty, Athens, Oho, USA ABSTRAT Ths paper presents knematcs analyss for the DARwIn- OP (Dynamc Anthropomorphc Robot wth Intellgence Open-Platform) robot. Ths s a -dof humanod walkng robot developed by Vrgna Tech, Purdue, and the Unversty of Pennsylvana and marketed by Robots Inc. The robot verson analyzed n ths paper s 455 mm tall and has a mass of.8 kg. Oho Unversty has two of these unts for robot applcatons research and teachng. Presented are a descrpton of DARwIn-OP, the Denavt- Hartenberg Parameters for each seral chan (-dof head pan/tlt, 3-dof arms, and 6-dof legs), specfc length parameters, jont angle lmts, plus forward pose knematcs equatons and partal nverse pose knematcs solutons, wth examples. 1. INTRODUTION DARwIn-OP s a unque open-platform humanod robot sponsored by NSF, developed collaboratvely by Vrgna Tech ( Purdue, and the Unversty of Pennsylvana, and marketed by Robots Inc. ( Thanks to NSF support, Oho Unversty was one of the nsttutons who receved two DARwIn humanod robots for research and teachng. DARwIn s beng used to play robot soccer, but we are nvestgatng other humanod robot applcatons. DARwIn-OP s ntended to be open-platform, wth a communty of users contrbutng to robot developments followng the ntal collaborator s developments. The software and control archtecture s all open-source and free to the world. Multple computer platforms and varous programmng languages are sad to be allowed. Programmng enhancements should be shared wth the DARwIn communty. In addton, the desgn of the humanod robot s open-source, wth complete AD fles and parts lsts avalable for free so one could buld the robot from scratch. One could make mprovements and nnovatons to the exstng robot desgn, whch should also be shared wth the DARwIn communty. Ths paper s organzed as follows. Frst we descrbe the degrees-of-freedom (dof) DARwIn-OP robot concept and hardware. Then we present the Denavt-Hartenberg (DH) parameters for each seral chan of the humanod robot (-dof head pan/tlt, 3-dof arms, and 6-dof legs). Included n ths secton are the specfc length parameters and jont angle lmts for the 455 mm tall DARwIn-OP. Next we present forward pose knematcs equatons for each seral chan, wth examples. Last we present partal nverse pose knematcs solutons for each seral chan of the robot. All analyses n ths paper are from well-known seral robotcs technques (rag, 5). We could not fnd ths knematcs analyss from any source so ths wll be part of our contrbuton to the DARwIn communty.. DARwIn-OP DESRIPTION The -dof DARwIn Humanod Robot s shown n Fgure 1, n the standard pose wth all zero jont angles. In the hardware shown n Fgure 1, each arm has three sngle-dof revolute (R) jonts, each leg has sx sngle-dof R jonts, and the pan/tlt head has two sngle-dof R jonts. Therefore, the overall robot has dof. Each arm has a -dof (offset-u-jont) shoulder jont and a 1-dof elbow jont, for a total of 3-dof per arm. Each leg has a 3-dof (S-jont wth 3 ntersectng R jonts) hp jont, a 1-dof knee jont, and a -dof (U-jont) ankle jont for a total of 6-dof per leg. The last hp jont, the knee jont, and the frst ankle jont are all about parallel Z axes. The -dof head (U-jont) enables pan and tlt for the camera va an azmuth R jont and an elevaton R jont. Each DARwIn-OP robot jont s drven by a Dynamxel MX-8 servomotor controlled by an nternal M-73 Robots servo-controller. The DARwIn-OP sensors nclude an HD dgtal vdeo camera, tr-axs gyroscope and accelerometer, and two mcrophones. The DARwIn-OP on-board computer s a 1.6 GHz Intel Atom Z53 FtP wth 4 GB SSD and the Ubuntu Lnux operatng system. DARwIn s programmed n ++ and 1 opyrght 1 by ASME Downloaded From: on /18/16 Terms of Use:

2 s able to operate both tethered and battery-powered. The LPO 3ELL 11.1v 1mAh battery provdes up to 3 operaton mnutes (Merln Robotcs, 1). Secton 3.3. Refer to the lengths defntons n Fgure 3 for these parameters. The basc torso reference frame s shown n red n Fgure 3. Ths {} dextral artesan reference frame, stands for hest. Frame {} s movng and serves as the reference frame for all fve seral chans. A fxed world {Wo} artesan reference frame (not shown) may be used for DARwIn-OP absolute knematc referencng. Fgure 1. DARwIn-OP Humanod Robot, Zero Pose The numberng conventon for the jont servomotors s shown n Fgure. The jont angle values wll be denoted by, = 1,,,, accordng to the numberng scheme n Fgure. Agan, the pose shown s for all zero jont angles. For notatonal smplcty n artesan coordnate frame defnton, frame numbers {}, {1}, {}, etc. wll be recycled for the fve seral chans. Thus, there wll be fve Z axes, four Z 3 axes, two Z 6 axes, and so on. Ths s to avod the onerous noton of Z RA, Z LL, etc. (representng rght arm and left leg). Accordng to rag (5) notaton for the Denavt- Hartenberg Parameters (1955), jont angle s supposed to rotate about R-jont axs Z. Agan for smplcty n notaton, ths rule wll be volated, n order to number the jont angles as shown n Fgure. 3. DARwIn-OP PARAMETERS The DARwIn-OP Denavt-Hartenberg (DH, Denavt and Hartenberg, 1955) parameters are presented n ths secton, for each of the seral chans (head, arm, and leg). DH parameters have been adopted for standard knematcs analyss n seralchan robots (rag, 5). The physcal length parameters are gven frst, followed by the jont angle lmts, and then the DH parameters for the DARwIn-OP humanod robot. We present DH parameters for the -dof head, the 3-dof rght arm, and the 6-dof rght leg (the 3-dof left arm and 6-dof left leg are symmetrc to ther counterparts, as seen n Fgures 1 and ). 3.1 DARwIn-OP robot lengths Ths secton presents the varous lnk lengths for the DARwIn-OP Humanod Robot. These are specfc hardware desgn values assocated wth the DH Parameters gven n Fgure. DARwIn-OP Servomotor Numberng (Robots, 11) Torso Lengths These values are not DH Parameters but wll be ntroduced later n the Forward Pose Knematcs solutons. The subscrpt h for L h s lower-case and ndcates head. Length Value (mm) L h 5.5 L 8. L TX 5. L TY 1. L TZ Head Lengths These values are not DH Parameters but wll be ntroduced later n the Forward Pose Knematcs solutons. Length Value (mm) H X 33. H Y 34.4 H Z.5 opyrght 1 by ASME Downloaded From: on /18/16 Terms of Use:

3 Length Value (mm) F X 14. F Y 15. (1. front) F Z 66. L X 5. L Z DARwIn-OP Jont Angle Lmts Ths secton presents the knematc jont angle lmts for the DARwIn-OP Humanod Robot. These are specfc hardware desgn values assocated wth the DH Parameters gven n the followng secton. Refer to the jont numberng gven n Fgure. All unts n Table I are degrees. Fgure 3. DARwIn Robot Lengths Defntons Rght and Left Arm Lengths These DH Parameters are shown n Fgure 3. L H s not a DH Parameter but wll be used later n the Forward Pose Knematcs solutons. The subscrpt H for L H s upper-case and ndcates hand. Length Value (mm) L L 6. L L H Rght and Left Leg Lengths These DH Parameters are shown n Fgure 3. L F s not a DH Parameter but wll be used later n the Forward Pose Knematcs solutons. Length Value (mm) L L L F Rght and Left Foot Lengths DARwIn s feet (see Fgure 3) are 14 mm long, 66 mm wde and 15 mm thck (taperng to 1 mm thck at the front edge). The feet are centered wth the ankle jont front-to-back (L X = 5 mm), but not sde-to-sde (L Z = 3 mm from the nner edge of each foot to the ankle jont). Table I. DARwIn-OP Jont Lmts Jont Jont name Axs MIN MAX Rght Arm 1 shoulder ptch Z shoulder roll Z elbow Z 3 16 Left Arm shoulder ptch Z shoulder roll Z elbow Z 3 16 Rght Leg 7 hp yaw Z hp roll Z 6 9 hp ptch Z knee Z ankle ptch Z ankle roll Z Left Leg 8 hp yaw Z hp roll Z 6 1 hp ptch Z knee Z ankle ptch Z ankle roll Z Head 19 head pan Z head tlt Z DARwIn-OP DH Parameters The standardzed Denavt-Hartenberg (DH) Parameters (1955) are used to descrbe the lnks/jonts geometry of a seralchan robot. In ths secton the DH Parameters are presented for each of the DARwIn-OP seral chans,.e. pan/tlt head, rght arm, and rght legs. rag (5) conventon s used for the DH Parameters n ths paper. All angular unts n the DH Parameter tables below are degrees. 3 opyrght 1 by ASME Downloaded From: on /18/16 Terms of Use:

4 The artesan reference frame defntons for three DARwIn-OP seral chans (-dof head, 3-dof rght arm, 6-dof rght leg) are gven n Fgures 4, 5, and 6. It s mportant to note that all of these DH parameters fgures are shown n the zero-angle poses. Therefore t looks lke many movng coordnate frame axes lne up, but ths wll change as jont angles rotate away from zero n all cases. The 3-dof left arm and 6-dof left leg are symmetrc to the rght arm and leg, so these cases are not presented here Two-dof Pan/Tlt Head The artesan reference frame defntons for the two-dof pan/tlt head are shown n Fgure 4. Table II gves the DH parameters for the two-dof pan/tlt head. Table III. Three-dof Rght Arm DH Parameters 1 a 1 d L 1 L L Sx-dof Rght Leg The artesan reference frame defntons for the sx-dof rght leg are shown n Fgure 6. Table IV gves the DH parameters for the sx-dof rght leg. Fgure 4. Two-dof Pan/Tlt Head oordnate Frames Table II. Two-dof Pan/Tlt Head DH Parameters 1 a 1 d Three-dof Rght Arm The artesan reference frame defntons for the three-dof rght arm are shown n Fgure 5. Table III gves the DH parameters for the three-dof rght arm. Fgure 6. Sx-dof Rght Leg oordnate Frames Fgure 5. Three-dof Rght Arm oordnate Frames (Top and Front Vews) Table IV. Sx-dof Rght Leg DH Parameters 1 a 1 d L L opyrght 1 by ASME Downloaded From: on /18/16 Terms of Use:

5 4. DARwIn-OP FORWARD POSE KINEMATIS In general, the Forward Pose Knematcs (FPK) problem for a seral-chan robot s stated: Gven the jont values, calculate the pose (poston and orentaton) of the end frame of nterest. For seral-chan robots, the FPK problem set up and soluton s straght-forward. It s based on substtutng each lne of the Denavt-Hartenberg Parameters table nto the equaton below (rag, 5), gvng the pose of frame {} wth respect to ts nearest neghbor frame { 1} back along the seral chan: c s a 1 1 sc 1 cc 1 s 1 ds 1 T ss 1 cs 1 c 1 dc 1 1 The equaton above represents pose (poston and orentaton) of frame {} wth respect to frame { 1} by usng a 4x4 homogeneous transformaton matrx. The upper left 3x3 matrx s the rotaton matrx gvng the orentaton and the upper rght 3x1 vector s the poston vector. Then transformaton equatons are used to fnd the pose of the overall end-frame of nterest wth respect to the base reference frame, to complete the FPK soluton for each seral chan. 4.1 Two-dof Pan/Tlt Head FPK Expressons The statement of the FPK problem for the two-dof pan/tlt head seral chan of the DARwIn-OP humanod robot s: Gven, calculate 19, T and X. where X s the pontng vector along the X unt vector drecton, expressed n {} coordnates. Substtute each row of the DH parameters n Table II nto 1 the above equaton for T to obtan the two neghborng homogeneous transformaton matrces as a functon of the jont angles. c19 s19 s19 c19 1T 1 1 c s 1 1 T s c 1 Where the followng abbrevatons were used: c cos, s sn, for = 19,. Now substtute these two neghborng homogeneous transformaton matrces nto the followng homogeneous transform equaton to derve the FPK result. c19c c19s s19 1 s19c s19s c19 T 1T19 T s c 1 Note that the fourth column result, the poston vector X gvng the orgn of {} from the orgn of {}, expressed n {} coordnates, s all zero components snce the pan/tlt axes are ntersectng. There are no knematc lengths or length offsets nvolved. To derve the second FPK result X, use the followng transform equaton, where X 1 1 T. c19c s19c X T X s 1 The desred pontng vector s contaned n the frst three elements of X,.e. the 1 n the fourth locaton s merely a placeholder for homogeneous transformaton matrx multplcaton. The basc pan/tlt head FPK result s T. To calculate the pose of each eye frame (RE and LE for rght and left eye, respectvely) wth respect to the DARwIn chest reference frame {}, the followng transform equatons are used. RET T T 19, RET LET T T 19, LET 1 1 H X 1 L h T 1 H Y 1 RET 1 H Z H X 1 H Y LET 1 H Z 1 The overall transform equatons above can be evaluated numercally. onstants L h, H X, H Y, and H Z were gven earler. Note that T, T RE, and T LE are not evaluated by any row n the DH parameter table, snce there s no varable assocated wth these fxed homogeneous transformaton matrces based on constant lengths and orentaton. Instead, they are determned by nspecton, usng the rotaton matrx and 5 opyrght 1 by ASME Downloaded From: on /18/16 Terms of Use:

6 poston vector components of the homogeneous transformaton matrx defnton. In the above expressons for T RE and T LE the orentaton of each eye frame {RE} and {LE} was arbtrarly assgned as dentty,.e. dentcal to the orentaton of {}. Ths can easly be changed as requred. 4. Three-dof Rght Arm FPK Expressons The statement of the FPK problem for the three-dof rght arm seral chan of the DARwIn-OP humanod robot s: Gven 1, 3, 5, calculate T 3 and T H. Where {H} s the rght-arm end-effector (hand) frame and {} s the DARwIn reference frame n the chest. As stated earler, for notatonal smplcty n artesan coordnate frame defnton, frame numbers {}, {1}, {}, etc. wll be recycled for the fve seral chans (head, rght and left arms, rght and left legs). Therefore, n ths paper many repeated frame numbers are context-dependent, whch must be sorted out n programmng the overall humanod robot. Substtute each row of the DH parameters n Table III nto the equaton for 1 T to obtan the three neghborng homogeneous transformaton matrces as a functon of the jont angles. c1 s1 s3 c3 L1 s1 c1 1T 1 1 L 3 1 T c3 s3 1 1 c5 s5 L 1 3T s5 c5 1 Where the followng abbrevatons were used: c cos, s sn, for = 1,3,5. Now substtute these three neghborng homogeneous transformaton matrces nto the followng homogeneous transform equaton to derve the FPK result. 1 3T 1T 1 T 3 3T 5 s1s5csc sc 1 5css cc 1 3 Lc 1 1Lcs 1 3Ls 3 1 cs 1 5ssc cc 1 5sss sc 1 3 Ls 1 1Lss 1 3Lc 3 1 cc 3 5 cs 3 5 s3 Lc 3 1 T H The basc rght-arm FPK result s 3 T. To calculate, the pose of the rght-arm end-effector frame {H} wth respect to the DARwIn chest reference frame {}, the followng transform equaton s used. 3 HT T 3T 1, 3, 5 HT 1 1 LH 1 T 3 1 L 3 1 L HT The overall transform equaton above can be evaluated numercally. onstants L, L 3, and L H were gven earler. Note that T and 3 T H are not evaluated by any row n the DH parameter table, snce there s no varable assocated wth these fxed homogeneous transformaton matrces based on constant lengths and orentaton. Instead, they are determned by nspecton, usng the rotaton matrx and poston vector components of the homogeneous transformaton matrx defnton. For use n nverse pose knematcs, here s the resultng poston vector of the hand tp wth respect to the {} frame: X 3 H 3T 1, 3, 5 XH Lc 1 1Lcs 1 3L3( s1sc 1 5cs 1 3s5) LH ( ss 1 5csc 1 3 5) Ls 1 1 Lss 1 3 L3( c1 cc 1 5 sss 1 3 5) LH ( cs 1 5 ssc 1 3 5) Lc 3Lcs 3 3 5LH cc Sx-dof Rght Leg FPK Expressons The statement of the FPK problem for the sx-dof rght leg seral chan of the DARwIn-OP humanod robot s: Gven 7, 11, 9, 13, 17, 15, calculate T 6 and T F. Where {F} s the rght-leg end-effector (foot) frame and {} s the DARwIn chest reference frame. As stated earler, for notatonal smplcty n artesan coordnate frame defnton, frame numbers {}, {1}, {}, etc. wll be recycled for the fve seral chans (head, rght and left arms, rght and left legs). Therefore, n ths paper many repeated frame numbers are context-dependent, whch must be sorted out n programmng the overall humanod robot. Substtute each row of the DH parameters n Table IV nto 1 the equaton for T to obtan the sx neghborng homogeneous transformaton matrces as a functon of the jont angles. c7 s7 s11 c11 s7 c7 1T T c11 s opyrght 1 by ASME Downloaded From: on /18/16 Terms of Use:

7 c9 s9 c13 s13 L4 1 3T 3 s13 c13 4 s9 c9 T c17 s17 L5 c15 s15 4 s17 c17 5T T s15 c Where the followng abbrevatons were used: c cos, s sn, for = 7,11,9,13,17,15. Now substtute these three neghborng homogeneous transformaton matrces nto the followng homogeneous transform equaton to derve the FPK result T 1T 7 T 11 3T 9 4T 13 5T 17 6T15 Snce there are three parallel Z axes (3,4,5) n the rght leg, we can group the above matrx multplcatons as follows for a sgnfcant smplfcaton T 1T 7 T 11 5T 9, 13, 17 6T15 where: cabc sabc L4c9L5cab 1 5T sabc cabc L4s9 L5s ab 1, abc abc were used. The rght leg FPK result s: r11 r1 r13 px r1 r r3 p y 6T r31 r3 r33 p z 1 where the abbrevatons cab cos9 13, sab sn 9 13 c cos, s sn where: r11 s7sabc c7s11cabc c15 c7c11s15 r c s s s c c s c s r s s c c c 1 7 abc 7 11 abc abc r1 s7sabc c7s11cabc s15 c7c11c15 r c s s s c s s c c r s c c s c 7 abc 7 11 abc abc r s c c s s r c c s s s r c s 13 7 abc 7 11 abc 3 7 abc 7 11 abc abc px L4s9L5sab s7 L4c9L5cab c7s11 p L s L s c L c L c s s p L c L c c y ab ab 7 11 z ab 11 The basc rght-leg FPK result s 6 T. To calculate T F, the pose of the rght-leg end-effector frame {F} wth respect to the DARwIn chest reference frame {}, the followng transform equaton s used. 6 FT T 6T 7, 11, 9, 13, 17, 15 FT 1 LTX 1 L TY T 1 L TZ 1 1 LF 6 1 FT 1 1 The overall transform equaton above can be evaluated numercally. onstants L TX, L TY, L TZ, and L F were gven earler. Note that T and 6 T F are not evaluated by any row n the DH parameter table, snce there s no varable assocated wth these fxed homogeneous transformaton matrces based on constant lengths and orentaton. Instead, they are determned by nspecton, usng the rotaton matrx and poston vector components of the basc homogeneous transformaton matrx defnton. 4.4 FPK Examples Two-dof Pan/Tlt Head, 1, : Gven T Three-dof Rght Arm,, 1,,3 Gven T H : X opyrght 1 by ASME Downloaded From: on /18/16 Terms of Use:

8 4.4.3 Sx-dof Rght Leg,,,,, 1, 3,, 35, 45,6 Gven T F These FPK results were valdated by comparng the symbolc formulae derved n ths paper wth a numercal approach n each case. Fgure 7 shows the MATLAB graphcal results for the zero confguraton (see Fgures 1 and ). Fgure 8 shows the MATLAB graphcal results for the FPK examples confguraton (symmetry s lost snce we are usng dentcal, rather than symmetrc, jont angles for the rght and left arms and legs). Fgure 8. DARwIn-OP MATLAB Smulaton, examples confguraton 5. DARwIn-OP INVERSE POSE KINEMATIS In general, the Inverse Pose Knematcs (IPK) problem for a seral-chan robot s stated: Gven the pose (poston and orentaton) of the end frame of nterest, calculate the jont values. For seral-chan robots, the IPK soluton starts wth the FPK equatons. The soluton of coupled nonlnear algebrac equatons s requred and multple soluton sets generally result. The head IPK soluton s complete but the rght arm and rght leg IPK solutons are partal below. The FPK examples can also serve as IPK examples when the nput s reversed. Fgure 7. DARwIn-OP MATLAB Smulaton, zero confguraton 5.1 Two-dof Pan/Tlt Head IPK Soluton The statement of the IPK problem for the two-dof pan/tlt head seral chan of the DARwIn-OP humanod robot s: Gven X, calculate 19,. X s the pontng vector along the X unt vector drecton, expressed n {} coordnates; t must be entered as a unt vector. Its FPK expresson s recalled below: c19c x X s19c y s z where x, y, z are the gven components of the desred unt vector X. 8 opyrght 1 by ASME Downloaded From: on /18/16 Terms of Use:

9 The IPK soluton for the pan/tlt head s obtaned n the followng order. Frst use a rato of the y to x equatons to elmnate c : 19 atan( yx, ) where atan s the quadrant-specfc nverse tangent functon. Next use a rato of the z to x equatons: atan( zx, / c ) 19 or, equvalently, use a rato of the z to y equatons: atan( zy, / s ) 19 If 19 9 use the y equaton soluton, and f 19,18 use the x equaton soluton. If nether of these artfcal sngularty condtons exst, both solutons wll yeld the same result for. There s a sngle soluton set,.e. no multple solutons n ths case. 5. Three-dof Rght Arm IPK Soluton (partal) The statement of the IPK problem for the three-dof rght arm seral chan of the DARwIn-OP humanod robot s: Gven,,. T H, calculate Where {H} s the rght-arm end-effector hand frame and {} s the DARwIn chest reference frame. The gven pose T H represents 6-dof, whle the unknown jont space n ths case s only 3-dof. Therefore, the entre T H cannot be gven, but only a 3-dof subspace. Let us specfy the poston vector and calculate the unknown jont angles: Gven X H, calculate 1, 3, 5. From FPK, here are the equatons to solve: L1c1 Lcs 1 3L3( s1sc 1 5css 1 3 5) LH ( ss 1 5csc 1 3 5) x L1s1 Lss 1 3 L3( c1 cc 1 5 sss 1 3 5) LH ( cs 1 5 ssc 1 3 5) XH y Lc 3 Lcs 3 3 5LHcc 3 5 z 1 where x, y, z are the gven components of the desred tp poston vector X H. We must frst smplfy the gven X H to X H usng the followng transform equaton: 1 X H T XH Where constant transform matrx T was gven n the FPK secton for the rght arm. The above set of three equatons fully coupled n the three unknowns,, s nonlnear (transcendental). In order to smplfy ths set of equatons, use the followng transform equaton: x 1 3 X H y 1T1 T3 3T5 XH z x T T y T X z H xc s yss zc Ls L Ls L c H 5 xc c ys c zs L c xs yc L L L c L s H 5 Ths soluton has not yet been completed. 5.3 Sx-dof Rght Leg IPK Soluton (partal) The statement of the IPK problem for the sx-dof rght leg seral chan of the DARwIn-OP humanod robot s: Gven,,,,,. T F, calculate Where {F} s the rght-leg end-effector foot frame and {} s the DARwIn chest reference frame. We can frst smplfy the gven T F to T 6 usng the followng transform equaton: T T FT FT Where constant transform matrces T and 6 T F were gven n the FPK secton for the rght leg. Let the followng symbols represent the numercal values for T 6, derved from the orgnally-gven T F. r11 r1 r13 x r1 r r3 y 6T r31 r3 r33 z 1 These numercal values are equated to the FPK soluton as a functon of the sx unknown jont angles: T 1T 7 T 11 3T 9 4T 13 5T 17 6T15 The above represents 16 equatons (4 trval) n the sx unknown jont angles. The three poston vector component equatons are all ndependent, but there are only three ndependent equatons among the 9 equatons of the rotaton matrx. The equatons are coupled and nonlnear (transcendental). 9 opyrght 1 by ASME Downloaded From: on /18/16 Terms of Use:

10 Lookng at the FPK terms gven earler, we need some smplfcaton pror to solvng these equatons. There are three possbltes: 1. Notce that the hp s sphercal (.e. frames {}, {1}, {}, {3} share a common orgn pont);. FPK already took advantage of the three parallel Z axes (last hp Z 9, knee Z 13, and frst ankle Z 17 ); 3. Snce the hp s sphercal and the - dof ankle also rotates about a common pont, the dstance from the hp center to the ankle center s only a functon of one jont angle, 13. The thrd smplfcaton wll be exploted frst. 6 5 P P x y z 3 3 P L L c L s P L L L L c x y z cos 13 1 P L L LL Note ths s equvalent to the Law of osnes, lookng at the plane contanng the hp, knee, ankle, and both leg lengths L 4 and L 5. Ths soluton has not yet been completed. 6. ONLUSION Ths paper has presented knematc analyss for the -dof DARwIn-OP humanod walkng robot, ncludng the Denavt- Hartenberg Parameters for each seral chan (-dof head pan/tlt, 3-dof arm, and 6-dof leg), specfc length parameters, jont angle lmts, plus forward pose knematcs equatons and partal nverse pose knematcs solutons, wth examples. Future work plans nclude fnshng the IPK solutons. REFERENES J.J. rag, 5, Introducton to Robotcs: Mechancs and ontrol, Thrd Edton, Pearson Prentce Hall, Upper Saddle Rver, NJ. J. Denavt and R.S. Hartenberg, 1955, A Knematc Notaton for Lower-Par Mechansms Based on Matrces, Journal of Appled Mechancs: Merln Robotcs, 1, ndex.php/bpedal-robots/darwn-op-1.html Robots, 11, DARwIn-OP Quck Start Operaton and Programmng Gude, 1 opyrght 1 by ASME Downloaded From: on /18/16 Terms of Use: