Chapter 7: Application Issues

Transcription

1 Chapter 7: Applcaton Issues hs chapter wll brefly summarze several of the ssues that arse n mplementaton of the Carpal Wrst. he specfc ssues nvolved n puttng ths wrst nto producton are both task and manpulator dependent and are therefore unque to the desred applcaton of the overall manpulator system. However general approaches to modelng and solvng task-dependent operatons are necessary. hs chapter wll focus on manpulator systems topcs that are general n nature and that can be addressed usng the knematc and dynamc wrst model developed n ths dssertaton. he followng four topcs wll be dscussed n ths chapter: ) Methods for provdng full orentatonal control on the Wrst structure. 2) Redundancy resoluton through optmal performance requrements n dexterty mechancal advantage and dynamc response. 3) Path plannng or trajectory synthess 4) rajectory synthess based on nput torque requrements or other dynamc constrants. Whle not exhaustve these topcs represent a varety of the ssues crtcal to manpulator applcaton. Performance of a manpulator through advanced control results from accurate modelng of the robot archtecture ts envronment and task. he topcs demonstrate how the Wrst model can be ncorporated wth ts ntended task to provde the performance requred n applcaton. 7. Full Orentatonal Capablty of the Carpal Wrst he Carpal Wrst s a three degree-of-freedom devce provdng two orentatonal degrees of freedom ptch and yaw and a thrd translatonal degree of freedom called plunge. hese three degrees of freedom result from the knematc constrants of ts parallel archtecture and are fully controlled through the three nput jont angles. A typcal manpulator conssts of two sectons or structures one the arm (called regonal structure) whch s used to generally poston the tool and the other the wrst (called the local structure) whch provdes the requred orentaton of the tool. In applcaton the Carpal Wrst can be mounted on the end of a robotc arm to become the local structure. he most general robotc manpulator provdes full pose control of ts tool a sx degree of freedom (dof) task. Here the arm of a sx degree-of-freedom robot should provde full three dof poston capablty whle the wrst should provde complete orentaton capablty also a three-dof task. When mounted on a typcal three dof manpulator arm the added Carpal Wrst would make the robot a sx degree of freedom devce. he Carpal Wrst however lacks full orentatonal control snce t cannot provde roll moton between ts basal and dstal plates. Dependng on the choce of arm capabltes and applcaton needs a number of possble arrangements could exst: he frst possble arrangement s to mount the Wrst on an arm that provdes two postonal and one orentatonal degrees of freedom. hs would result n a sx dof manpulator that provdes full pose control of the tool. However there are several factors that may lmt or prevent ths form of applcaton. Frst the type of arm specfed n ths system s non-standard snce current robotc systems expect roll orentaton to be performed by the wrst structure rather 69

2 Stephen L. Canfeld Chapter 7: Applcaton Issues 70 than the arm. Also ths arrangement requres the Carpal Wrst to provde one of the postonng degrees of freedom for the manpulator. he plunge or translaton moton of the Carpal Wrst s lmted n range of travel however and would not satsfy the postonng requrements of a typcal manpulator. A second arrangement could consst of the Wrst mounted on a standard three degree of freedom postonng arm. hs arrangement results n a sx dof manpulator that s not capable of provdng tool roll but wth one redundant degree of postonal freedom.. hs arrangement has many advantages that would encourage ts use n producton. It s smlar to the typcal manpulator systems that use a three dof postonng arm. Also ths arrangement does not alter the Carpal Wrst structure. Fnally the redundant degree of freedom can be used to ncrease manpulator performance through a refned control algorthm. he drawback of ths arrangement s obvously ts lack of ablty to provde tool roll. However many applcatons do not requre orentaton of the tool about ts pontng drecton. hese applcatons or tasks are called axssymmetrc and nclude processes such as arc weldng spot weldng deburrng applyng adhesves polshng grndng drllng spray-pantng scannng many pck-and-place jobs and mountng and fxturng tasks. In applcatons that do not requre orentaton about the tool axs (or orentaton about any axs fxed relatve to the tool) ths arrangement provdes the best combnaton of arm and Wrst capabltes. he thrd arrangement proposed s to mount the Wrst on a standard three dof postonng arm and provde an extra roll degree of freedom to the manpulator. hs results n a system wth full pose control (6 dof) plus a redundant degree of freedom (for a total of 7 dof). However complcatons arse n provdng the addtonal dof n such a way that t does not lmt the advantages provded by the Carpal Wrst. Addng the roll degree of freedom can be accomplshed n a number of ways. he strongest and least complcated approach s to provde a rotatonal degree of freedom located between the manpulator arm and the Wrst. hs would rotate the basal plate normal as: θ roll B Z (7.) where θ roll s the nput roll angle and B Z s the z axs of the basal frame the frame of the Wrst base. In order to provde pure roll about the dstal plate extra orentaton control must be provded by the Wrst such that only the component of θ roll about the D Z axs s seen by the tool and the D X and D Y components are removed. Gven an nput θroll at the base the tool would be rotated by: D D B θ = R 3 θ 3 (7.2) roll B j roll where: D R 3 s the rotaton between the basal and dstal plates expressed n tensor notaton and B j D roll θ are the components of orentaton at the dstal plate for = -3. θ he negatve = 2 components of D roll are ncluded n the path plannng requrements and are sent through the nverse knematcs of the Wrst. he advantage of ths form of roll control s

3 Stephen L. Canfeld Chapter 7: Applcaton Issues 7 prmarly the locaton of the roll axs. Locatng t at the base of the Wrst mnmzes the sze and weght constrants that are placed on the Wrst components resultng n a stronger jont and lghter Wrst. he prmary dsadvantage s that the arm archtecture must be modfed an opton that s not attractve to companes who would adapt the Carpal Wrst to current robotc arms. o avod modfyng the arm requrements the roll axs can be ntegrated wth the Carpal Wrst archtecture. hs approach seems to be the most acceptable to robot manufacturers for applcatons requrng the full sx dof pose control snce t provdes a wrst wth full orentatonal control. A conceptual desgn of ths four-degree-of-freedom Wrst has been prepared n response to requests from several robot manufacturers. he conceptual desgn s shown on the Wrst n Fg. 7. and n an exploded vew n Fg hs arrangement provdes a compact hghstrength hgh-resoluton roll axs that mantans the hollow tunnel characterstcs of the Carpal Wrst. Knematcally the conceptual desgn conssts of addng a roll axs on the dstal plate drected along the dstal plate normal ( D Z). he forward and nverse knematcs of ths four degree-of-freedom arrangement have been presented n the postonal knematc analyss and demonstrate no addtonal dffculty n control. he conceptual desgn for the added roll dof conssts of four basc elements: a motor a drve-shaft a gear reducer and a bearng support structure. A hollow-shaft pancake-type DC servo motor s mounted on the bottom of the dstal plate. A pancake-type 000: harmonc-drve gear reducer s mounted on the opposng sde of the dstal plate along wth a flange-type bearng retaner that carres all axal and radal loads. A hollow drve shaft runnng through the dstal plate connects the motor to gear reducer. Fgure 7.: Conceptual Roll Axs Desgn

4 Stephen L. Canfeld Chapter 7: Applcaton Issues 72 Bearng Flange Dstal Plate Harmonc Drve Reducer Hollow Drve- Shaft Hollow-Shaft DC Servo Fgure 7.2: Exploded Vew of Roll dof Addng a roll axs to the Wrst creates a unque knematc devce n that the structure s no longer parallel but a hybrd parallel-seral devce. Hybrd manpulator devces have been dscussed n the lterature prmarly n devces that employ n-parallel actuaton. Snce the advantages of the Carpal Wrst stem from ts parallel archtecture addng an axs n seres would create a potental lmtng factor. However ths hybrd arrangement avods the major dsadvantage of seral manpulators; accumulatng moment loads on jont axes due to lnks farther along the chan. In the proposed Wrst hybrd arrangement the roll axs s placed at the end of the manpulator and thus does not see moment loads from any other lnk. A fnal approach to provdng full pose control wth the Wrst s to requre the tool to provde the roll moton. hs would be logcal n many applcatons for example a drllng tool or a rotary drvng tool. In effect ths approach s smlar to addng the roll on the dstal plate. 7.2 Redundancy Resoluton When matng the Carpal Wrst wth a typcal three degree-of-freedom postonng arm one redundant degree of freedom wll result due to the Wrst s plunge capablty. In ths case the redundancy s apparent from physcal consderatons of the manpulator system dof s whch can be thought of as postonng and orentng components. Mathematcally redundances are demonstrated n the system Jacoban matrx J where the rows of J represent output degrees of freedom (a maxmum of sx) of the manpulator and the columns represent system nputs or controlled jonts. hus a square n x n Jacoban of full rank represents a non-redundant system wth n degrees of freedom whle a non-square n x m wth m>n ( short ) Jacoban represents a

5 Stephen L. Canfeld Chapter 7: Applcaton Issues 73 redundant manpulator system wth (m-n) redundances. For a manpulator that does not requre roll the Carpal Wrst can be attached to a three dof arm and the resultng Jacoban would be a 5 x 6 matrx demonstratng 6-5 = redundant degree of freedom. he redundancy resultng n ths system s of nterest snce t provdes a free degree of freedom (.e. one that s not rgdly defned from the knematc path specfcatons) that can be used to mprove the knematc and dynamc performance of the system. he general approach for takng advantage of ths free dof s to specfy some performance crtera that provdes an addtonal constrant on the system at the knematc level thus resolvng the redundancy. hs wll be demonstrated on three levels for a Wrst-ntegrated manpulator system: One approach wll optmze on dexterty characterstcs at the nstantaneous knematcs level another wll provde a desred mechancal advantage or force control output also at the velocty level. he fnal approach wll optmze on dynamc performance at the acceleraton level Redundancy Resoluton Lterature Revew Usng manpulator system redundances to enhance knematc and dynamc performance has been studed by many researchers. One of the most common uses for redundances n robotc systems s obstacle-avodance. hs area has been studed by a number of researchers ncludng Macejewsk and Klen 985 Krcansk and Vukobratovc 986 Nearchou and Aspragathos 996 and McLean and Cameron 996. Whtney 969 demonstrated mprovements n rate resoluton based on the pseudo-nverse of the Jacoban matrx. hs approach has been refned and mproved by several other researchers ncludng Hollerbach and Suh 985 Nakamuraa and Hanafusa 987 Klen et al. 995 and Roberts and Macejewsk 996. Usng the system redundancy to mnmze the requred actuator nput energy was demonstrated by Vukobratovc and Krcansk 984 whle usng the system redundancy to mnmze nstantaneous jont torques was studed by Hollerbach and Suh 985. Usng redundancy to optmze the mechancal advantage of the manpulator was demonstrated by Dubrey and Luh 986 and Park et al Fnally Km and Rastegar 997 used redundancy n avodng actuatng torques that may excte the natural modes of vbratons of the manpulator system Optmzng on Dexterty In ths secton an approach for redundancy resoluton of the Carpal Wrst based on dexterty crtera s demonstrated. he dexterty crtera used n ths technque s formulated from the dexterty defntons presented by Soper et al. 997 Canfeld et al. 997 and the nstantaneous knematc analyss developed for the Carpal Wrst n Canfeld et al. 996 and the materal presented n Chap. 4. he dexterty defnton presented n ths work s based on the dea of the relatve stretchng between the nput and output velocty vectors defned by the Jacoban matrx: v D = = (7.3) ω µ

6 Stephen L. Canfeld Chapter 7: Applcaton Issues 74 where D s the dexterty µ s the mechancal advantage and the nput and output veloctes are related through the Jacoban as: v = Jω. (7.4) Expandng D: and then solvng for ω from Eq. 7.4 Substtutng ths result n Eq. 7.5 gves: D= D= ( v v) 2( ) ω = J 2 ω ω (7.5) v (7.6) ( ( ) ) v v v J J v. (7.7) hs equaton gves a general equaton for the dexterty of a manpulator as a functon of workspace poston n a desred drecton. Note that t assumes a square full rank (nonredundant non-sngular) Jacoban. When m>n a non-sngular Jacoban requres the rank of J to equal n. In order for the output velocty to be consstent n Eq. 7.4 t must be n the column space of J. When ths system of equatons s consstent the nput velocty s gven by the pseudo-nverse of J as: ω =( J J) J v he dexterty D s then found by substtutng Eq. 7.8 nto Eq. 7.5: D= [( ) ] ( ) v v v J J J J J J v (7.8). (7.9) Lookng at Eq. 7.4 a polar decomposton can be used to decompose the Jacoban nto two matrx operators: a matrx R whch stretches and a matrx Q whch rotates the nput velocty vector. hs s demonstrated for the m x n Jacoban for two cases (Soper et al. 997): m n: J = QR (7.0) where R s postve defnte R = ( J J) m < n: where R s postve defnte R = ( JJ ) 2 2 and Q s orthonormal Q= JR. J = RQ (7.) and Q s orthonormal Q= R J. he matrx R s of prmary nterest for the dexterty defnton because t contans the relatve stretchng nformaton between the nput and output veloctes. Consder the egenvalue decomposton of R: R = L L (7.2)

7 Stephen L. Canfeld Chapter 7: Applcaton Issues 75 where Σ s the dagonal matrx of the egenvalues of R. he elements of Σ represent the extremes n stretchng v relatve to ω and wll be called the prncpal dextertes. he assocated egenvectors wll be called the prncpal drectons. for m n: QRω = v (7.3) defnng y as y = Q v gves: Rω = y (7.4) and Eq. 7.3 becomes: D = y ω (7.5) whch s the rato of maxmum to mnmum prncpal dextertes. he drectons assocated wth these prncpal dextertes y prncpal are the egenvectors of R (u )rotated by Q: y prncpal = Qu (7.6) Fnally for the case where m<n the same approach s appled: RQω = v (7.7) and defnng y = Qω gves: Ry = v (7.8) and Eq. 7.3 becomes D = v y (7.9) whch s agan the rato of maxmum to mnmum prncpal dextertes. Now the drectons of the prncpal dextertes are the drectons of the egenvectors u. he polar decomposton for the case m<n allows the dexterty to be calculated as gven n Eqs. 7.5 and 7.9: D= ( ( ) ) v v v R R v. (7.20) Wth the dexterty equatons developed dexterty can be optmzed to resolve the redundant plunge degree of freedom n the Wrst. he approach wll proceed as follows. Frst the optmal dexterty wll be quantfed for example as the largest stretchng possble between the output and nput veloctes whch s gven n the maxmum prncpal dexterty. he above dexterty defntons wll then be used to fnd the maxmum prncpal dexterty and ts drecton. Wth ths Eq. 7.8 determnes the plunge velocty component of the vector v whch s then mposed on the knematcs to provde the addtonal constrant needed for redundancy resoluton.

8 Stephen L. Canfeld Chapter 7: Applcaton Issues Optmzng for Mechancal Advantage Usng mechancal advantage as the crtera n elmnatng redundancy follows the procedure for maxmzng dexterty. From the dexterty defnton proposed by Soper et al. 997 and repeated n Eq. 7.3 above the mechancal advantage s nversely related to dexterty. herefore the greatest mechancal advantage wll occur n the drecton of mnmum prncple dexterty. Equaton 7.8 s used agan to determne the plunge velocty component of v whch s then added as a constrant equaton n the knematcs provdng fully defned jont parameter specfcatons Optmzng for Dynamc Performance Optmzng the manpulator dynamc performance provdes a thrd method of redundancy resoluton. hs approach conssts of expressng the jont space coordnate motons as trajectores. hese trajectores are a functon of tme and coeffcents known as trajectory parameters. he trajectory parameters are determned from the knematc and dynamc constrants of the user-specfed tool-task and path. For jont space trajectores contanng k trajectory parameters one degree of redundancy adds an addtonal k parameters that can be used to satsfy path/task requrements. o optmze on dynamc performance there must be a suffcent number of total parameters to exceed the knematc moton requrements. he remanng parameters are avalable to satsfy dynamc requrements. hs s demonstrated n full detal n the followng sectons. 7.3 Path Plannng In operaton a manpulator typcally moves through a specfed path or trajectory. he trajectory defnes the tme hstory of poston velocty and acceleraton of the manpulator. he trajectory exsts n both tool space and jont space (whch are related through the manpulator knematcs) but s generally specfed n tool space. he tool space specfcatons are commonly user specfed n terms of goal poses wth some added path crtera. For example a smple pckand-place task has an ntal pck pose specfed as a goal a fnal place pose specfed as a goal and path crtera whch may nclude the speed of travel general obstacle avodance and tmng events. A general approach to path plannng appled to the Carpal Wrst s presented that can be extended to nclude the arm knematcs and thus the entre manpulator. hs approach s descrbed n the followng eght steps: ) he desred output path s gven n tool-space coordnates. hs ncludes the path crtera event tmng etc. 2) he output path s descrbed wth a fnte set of precson postons called va ponts (Crag 989). 3) he precson postons or va ponts are descrbed n jont space as: ( ϑ θ θ ) ( α β ) 2 3 = f p (7.2)

9 Stephen L. Canfeld Chapter 7: Applcaton Issues 77 where the jont angles θ -θ 3 are determned from α β and p (the tool-space Wrst coordnates) through the functon f whch represents the nverse knematcs presented n Chap. 2. 4) Wth the precson ponts specfed n jont space trajectores for the jonts between the precson postons can be descrbed usng trajectory curves to meet the specfc trajectory crtera. 5) he trajectory curves are wrtten as a functon of ther trajectory parameters and tme: n θ = k j f j j= () t (7.22) where n represents the degree order or number of harmoncs dependng on the type of path chosen. 6) Moton crtera are then gven as parameters on the trajectory posed as equatons. he type of moton and degree of crtera determne the type and degree of the trajectory path and ts parameters. 7) he jont trajectory equatons (Eq. 7.22) are substtuted nto the moton crtera equatons and the resultng system of equatons s solved for the trajectory parameters. 8) Wth the trajectory parameters the jont trajectores are known and are ready to be sent to the control system to result n the desred tool path. For example a specfed path and set of precson postons can be gven requrng contnuous poston velocty and acceleraton values between all va ponts wth specfc end condtons. hs crtera can be expressed as: θ( t fk ) = θ θ k ( t fk ) = θ θ k ( t fk ) = θ k (7.23) θ( t fk ) = θk θ ( t fk ) = θ k + θ( t fk + ) = θ k + he jont space angles can be suffcently descrbed usng a ffth-order polynomal: θ () t = a5 t + a4 t + a3 t + a2 t + a t + a0 (7.24) Solvng for the k th nterval gves:

10 Stephen L. Canfeld Chapter 7: Applcaton Issues 78 a a a a a a = θ = θ k k = θ k 2 ( ) t ( ) 20θ 20θ 8θ + 2θ 3θ θ = 2t 2 k+ k o k+ k k+ k k+ k+ 3 k+ ( ) t ( ) 30θ 30θ + 4θ + 6θ 3θ 2θ = 4 2t 2 k o k+ k+ k k+ k k+ k+ k+ ( ) t ( ) 2θ 2θ 6θ + 6θ θ θ = 5 2t 2 k+ k o k+ k k+ k k+ k+ k+ t t t. (7.25) 7.4 Path Plannng Based on Input orque Requrements / Constrants rajectory synthess has been demonstrated n meetng the knematc requrements of the desred tool path. he general procedure conssted of descrbng the jont space varables as a functons of tme and trajectory parameters. hen the knematc requrements were cast as constrant equatons and used to solve the trajectory parameters thus descrbng the jont space trajectores. In ths dscusson ths general approach wll be extended to nclude dynamc / nput actuaton requrements as well as the knematc requrements placed on the jont space trajectores. Summarzng the results from Chap. 5 the equatons of moton of the Carpal Wrst manpulator can be wrtten as: = A θθ + B θθθ + Cθ (7.26) where: M s the th nput motor moment () ( ) () M A(θ) s the mass matrx representng nertal components of the wrst and tool B( θθθ ) represents the Corols and centrfugal nerta terms and C(θ) represents the gravtatonal terms. Let the jont space trajectores be descrbed as: n general or for example as a polynomal: ( ) r k θ = f k a k t k= 0 r k θ = a k t k= 0 (7.27) (7.28)

11 Stephen L. Canfeld Chapter 7: Applcaton Issues 79 where: θ are the jont space coordnates cast n the form of a polynomal (n ths example) a k k=0-r are the r+ trajectory parameters and t s tme. Let s be the number of knematc constrants or requrements gven by the specfed tool path. hus s trajectory parameters are requred to satsfy the s specfed end condtons. Snce a total of r+ trajectory parameters are avalable n Eq or 7.28 above the extra (r+ - s) trajectory parameters can be used to satsfy dynamc / motor torque requrements. he torque requrements can be represented as: m ( ) ( ) M = f p t ; l = r+ k (7.29) l j j j= where M l s the h torque constrant for l = -> (r+-k) of the th motor ( j ) f j p t represents the m functons makng up the th order torque constrant equaton and are gven along wth the tool-path specfcatons. All that remans s solvng the r+ trajectory parameters a k. he soluton s relatvely straghtforward. It conssts of takng the frst and second dervatves of Eqs (or 7.28): k k ( ka k t ) f k ( a k t ) θ = r k= 0 k 2 k k k [( kk ( ) ak t ) f k ( ak t ) ( kak t ) f k ( ak t )] θ = + r k= 0 n general or the dervatves of the polynomal example: θ = r k= 0 k k θ = kk a t r k= 0 ka t ( ) k k 2 (7.30) (7.3) and then substtutng these nto Eq he moments n Eq are also replaced wth the motor torque requrements Eq Fnally the resultng system of equatons s solved for the (r+-k) trajectory parameters gvng the jont space trajectores. 7.5 Optmal Path Plannng Based on Input orque Requrements he procedure demonstrated above can be altered to generate an optmzaton algorthm. he value of ths les n meetng general motor torque requrements that are not easly satsfed wth a fnte number of constrant equatons. hus the procedure above s altered at Eq to wrte the motor requrements as a functon to be optmzed followed by a gven set of constrant

12 Stephen L. Canfeld Chapter 7: Applcaton Issues 80 equatons. Optmzaton follows yeldng a set of trajectory parameters to best meet the desred motor torque profle whle satsfyng knematc moton constrants. 7.6 Summary of Applcaton Issues hs chapter has addressed several of the ssues that wll arse n applcaton of the Carpal Wrst. hs chapter s not ntended to cover all mplementaton ssues Rather ts purpose s to demonstrate the applcaton of the Carpal Wrst model knematcs and dynamcs n resolvng applcaton conflcts and n mprovng performance. A fourth degree of freedom can be added to the Carpal Wrst to provde full orentatonal capabltes he solutons presented however wll ncrease costs and are thus ntended only n applcatons requrng the addtonal degree of freedom. Redundancy resoluton technques were demonstrated at both the nstantaneous knematc and the dynamc level. Results from the applcaton of one of these specfc technques can be seen n Fg. 4.8 a mnmum dexterty plot of the Carpal Wrst. Secton and able 4.3 demonstrate the sgnfcant mprovement n Wrst dexterty due n part to the redundant degree of freedom. hs mprovement n dexterty over conventonal robotc wrsts (or the Carpal Wrst operated n fxed-plunge mode) s provded at the cost of one addtonal actuator. rajectory Synthess approaches for the Carpal Wrst and assocated arm were presented. Path plannng s requred n all manpulator systems. he technques gven n ths secton demonstrate how the knematcs and dynamc equatons of moton can be derved to optmze Wrst and manpulator performance under varous operatng crtera.