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2 8 Contol of Redundant Robotic Manipulatos with State Constaints Miosław Galicki Open Access Database Intoduction Recently, edundant manipulatos have attacted much attention due to thei potential abilities which ae inteesting fom both a theoetical and pactical point of view. Redundant degees of feedom make it possible to pefom some useful objectives such as collision avoidance in the wok space with both static and moving obstacles, joint limit avoidance, and/o avoidance of singula configuations when the manipulato moves. Most of pactical tasks, fo example, inseting a shaft into co-opeating elements (beaing, sleeve, o atchet-wheel), equie the knowledge of geometic paths (given in the wok space) and a pope toleance of matching that specifies the coesponding accuacy of the path following. In many othe industial tasks such as lase cutting o ac welding, the accuacy of path following is vital, and it is easonable to assume that designes and manufactues will specify pecision using an absolute toleance on tacking eo. he application of edundant manipulatos to such tasks complicates thei pefomance, since these manipulatos in geneal do not povide unique solutions. Consequently, some objective citeia should be specified to solve the obot tasks uniquely. he minimization of pefomance time is mostly consideed in the liteatue. Seveal appoaches may be distinguished in this context. Using the concept of a egula tajectoy and the extended state space, the stuctue of path-constained time-optimal contols has been studied in the woks (Galicki, 998b Galicki, 000) fo kinematically edundant manipulatos. Moeove, the efficient numeical pocedues able to find such contols wee also poposed in the woks (Galicki, 998a Galicki & Pajak, 999). Nevetheless, these algoithms equie full knowledge of manipulato Jacobian matix and obot dynamic equations, too. Although all the afoementioned algoithms poduce optimal solutions, they ae not suitable to eal-time computations due to thei computational complexity. heefoe, it is natual to attempt othe techniques in ode to contol the obot in eal-time. Using on-line tajectoy time scaling, a dynamic and computed toque laws espectively, a nealy time-optimal path tacking contol fo Souce: Industial-Robotics-heoy-Modelling-Contol, ISBN , pp. 964, ARS/plV, Gemany, Decembe 006, Edited by: Sam Cubeo 499

3 500 Industial Robotics: heoy, Modelling and Contol non-edundant obotic manipulatos with patially uncetain dynamics has been pesented in woks (Dahl, 994 Kiefe et al., 997). Howeve, these algoithms equie the solution of invese kinematic poblem along the path. A technique which avoids solving an invese of obot kinematic equations and uses the exact Jacobian matix, has been offeed in (Galicki, 00 Galicki 004) fo detemining a collision-fee tajectoy of edundant manipulatos opeating in both a static envionment (Galicki, 00) and in a dynamic one (Galicki, 004). Recently, a genealized tanspose Jacobian contolle with gavity compensation and a non-linea (satuating) deivative tem has been intoduced in (Galicki, 006a) to geneate obot contols subject to geometic path and actuato constaints. As is known, many obotic contolles have been poposed to solve both a set point contol poblem (a egulation task) (akegaki & Aimoto, 98 Aimoto, 996 Canudas de Wit et al., 996 Sciavicco & Siciliano, 996 Aimoto, 990 Kelly, 999 Galicki, 00 Galicki, 005) and the tajectoy tacking (Slotine & Li, 987 Slotine &Li, 99 Feng & Palaniswami, 99 Beghuis et al., 993 Lewis et al., 993 omei, 000), espectively. Howeve, most of these contolles have assumed full knowledge of manipulato kinematic equations. Recently, seveal appoximate Jacobian setpoint contolles have been poposed (Cheach et al., 999 Yazael & Cheach, 00 Cheach et al., 003) to tackle uncetainties in both obot kinematics and dynamics. he contolles poposed do not equie the exact knowledge of Jacobian matix and dynamic equations. Howeve, the esults in (Cheach et al., 999 Yazael & Cheach, 00 Cheach et al., 003) ae applicable only to a setpoint contol of a obot. his pape, which is based on ou ecent wok (Galicki, 006b), intoduces a new class of adaptive path following contolles not equiing the full knowledge of both kinematic and dynamic equations in the contol laws. Consequently, they ae suitable fo contolling uncetain obotic manipulatos. Motivated in pat by the dissipativity and adaptivity methodology (Slotine & Li, 987), we develop path following contolles whose stuctue is composed of tanspose adaptive Jacobian contolle plus a non-linea tem including an estimated contol. Unde the assumption of the full ank adaptive Jacobian matix, the poposed contol scheme has been deived based on the Lyapunov stability theoy. By using sensoy feedback of the end-effecto position, it is also shown that the end-effecto is able to follow a pescibed geometic path fo obots with both uncetain kinematics and dynamics. Futhemoe, new adaptive laws extending the adaptive algoithm fom (Slotine & Li, 987) to tackle kinemetic uncetainties too, ae poposed. It is to notice that appoximate Jacobian setpoint contolles fom (Cheach et al., 999 Yazael & Cheach, 00 Cheach et al., 003) can not be diectly applicable to ou task. he eason is that appoximate Jacobian matix in (Cheach et al., 999 Yazael & Cheach, 00 Cheach et al., 003) does not include kinematic paametes to be adapted and the eo of appoximation is a'pioi bounded. On the othe

4 Contol of Redundant Robotic Manipulatos with State Constaints 50 hand, the contolle poposed in ou wok adaptively vaies both kinematic paametes of the Jacobian matix and the dynamic ones in such a way as to stably follow by the end-effecto a geometic path. he pape is oganized as follows. Section fomulates the obotic task to be accomplished in tems of a contol poblem. Section 3 descibes how to employ the Lyapunov stability theoy to detemine contols (if they exist). Section 4 povides us with a compute example of geneating the obot contols in a two dimensional task space fo a plana edundant manipulato compising thee evolute kinematic pais. Finally, some conclusions ae dawn.. Fomulation of the adaptive contol poblem he contol scheme designed in the next section is applicable to holonomic mechanical systems compising both non-edundant and edundant manipulatos consideed hee which ae descibed, in geneal, by the following dynamic equations, n expessed in genealized co-odinates (joint co-odinates) q = (q,...,q n ) R ( q) + C( q, ) + G( q) u M = () whee M (q) denotes the n n inetia matix C( q, ) is the n -dimensional vecto epesenting centifugal and Coiolis foces G (q) stands fo the n - dimensional vecto of gavity foces u ( u,..., ) = is the vecto of contols (toques/foces) n denotes the numbe of kinematic pais of the V-th class. In most applications of obotic manipulatos, a desied path fo the end-effecto is specified in task space such as visual space o Catesian space. he aim is to follow by the end-effecto a pescibed geometic path (given in the m - dimensional task space) descibed by the following equations p ( q) Θ( s) = 0 () whee p n m : R R denotes an m -dimensional, non-linea (with espect to vecto q ) mapping constucted fom the kinematic equations of the manipulato = ( p ( q),..., p ( q ) m ( s ) = ( Θ ( s),..., ( ) ) Θ p q) ) ( m s u n Θ stands fo a given geometic path s is the cuent paamete of the path (e.g. its length) s [ 0, smax ] s max is the maximal path length. he mapping Θ is assumed to be bounded togethe dθ with the fist and second deivatives and not degeneated, i.e. > 0. ds

5 50 Industial Robotics: heoy, Modelling and Contol It should be: he kinematic equations of a manipulato ae independent p p ank = m. In geneal (i.e. when ank m, we should equie that q q p p dθ ank = ank in ode to guaantee consistency of the obotic q q ds task (). he poblem is to detemine contol u which geneates manipulato tajectoy q = q(t) and path paameteization s = s(t) satisfying the equation () fo each t [ 0, ], whee denotes an (unknown) time hoizon of task pefomance. It is natual to assume that at the initial moment t = 0, fo which s ( 0) = 0, a given (by definition) initial configuation q ( 0) = q0 satisfies (), i.e. p ( q 0 ) Θ(0) = 0 (3) Final path paameteization fulfils the equality s ( ) smax = 0 (4) Futhemoe, at the initial and the final time moment, the manipulato and path velocities equal zeo, i.e. ( 0) = ( ) = 0 (5) and s ( 0) = s ( ) = 0 (6) As is known, task space velocity p is elated to joint space velocity as follows p = J ( q, Y ) (7) whee p J(q,Y) = q is the mn Jacobian matix Y stands fo an odeed set Y = Y,..., Y k such as link lengths, joint offsets k denotes the numbe of kinematic paametes. Seveal impotant popeties of dynamic equations () may be deived (Spong & Vidyasaga, 989) of kinematic paametes ( )

6 Contol of Redundant Robotic Manipulatos with State Constaints 503. he inetia matix M ( q ) is symmetic and positive definite fo all q R n. Matix M(q) C(q,q) is skew-symmetic so that n M(q) v,q, R v, C(q, v = 0 (8) whee, is the scala poduct of vectos. 3. he dynamic equations () ae linea with espect to an odeed set of physical paametes X = (X,...,X ), i.e. d M ( q) + C( q, ) + G( q) = D( q,,, )X (9) whee whee D( q,,, ) is called the ( d ) n dynamic egesso matix d stands fo the numbe of the physical paametes such as link masses and inetias. Diffeential equation (7) has the following popety. 4. he ight hand side of (7) is linea with espect to e Y. Consequently, equation (7) can be expessed as follows p = J ( q, Y ) = K( q, )Y (0) whee K( q,q) is called the (m k)kinematic egesso matix. In ode to simplify futhe computations s max is assumed to be equal to, i.e. s max =. Let us define eos e and em+ of path following (task eos) as e = e ( e,..., e ) = p( q) Θ ( s) = ( p Θ,..., p Θ ) = m + s m. m m () Many commecial sensos ae available fo measuement of end-effecto position p, such as vision systems, electomagnetic measuement systems, position sensitive detectos o lase tacking systems. Hence, path following eo e in () is also assumed to be available (fo a given s ) fom measuement. Fo evolute kinematic pais, consideed hee, mapping p(.) is bounded. Consequently, we have the following popety.

7 504 Industial Robotics: heoy, Modelling and Contol 5. Boundedness of mappings p( ),Θ( ), implies that task eo e is bounded. Expessions ()-(6) fomulate the obot task as a contol poblem. he fact that thee exist state equality constaints makes the solution of this poblem difficult. he next section will pesent an appoach that endes it possible to solve the contol poblem ()-(6) making use of the Lyapunov stability theoy. 3. Adaptive path contol of the manipulato Ou aim is to contol the manipulato such that the end-effecto fulfils ()-(6). heefoe, we popose adaptive Jacobian path following contolles fo obots with both uncetain kinematics and dynamics. In ou appoach, the exact knowledge of both obot dynamic equations and Jacobian matix is not equied in updating the uncetain paametes. In the pesence of kinematic uncetainty, the paametes of the Jacobian matix ae uncetain and hence equality (0) can be expessed as follows Jˆ = K( q, )Yˆ () whee Jˆ = J(q,Y) ˆ R mn is an adaptive Jacobian matix and Ŷ stands fo the vecto of estimated kinematic paametes. In ode to show the stability of the path following contol system in the pesence of both kinematic and dynamic uncetainties, we define an adaptive joint-space sliding vectoz as z = λjˆ e + (3) whee λ is a positive scala coefficient. he adaptivity of vecto z is undestood in the sense that the paametes of the adaptive Jacobian matix will be updated by a paamete update law, defined late. Diffeentiating equation (3) with espect to time yields z = Jˆ e + λjˆ e + λ (4) Based on equations () and (3)-(4), we can easily expess obot dynamic e- quations by means of vecto z and its time deivative, as ( q, ) z + M ( q) + C( q, ) q + G( q u M ( q) z + C ) = (5)

8 Contol of Redundant Robotic Manipulatos with State Constaints 505 ˆ Whee = ( ) = λj e J e and q ( q q ) = J e, n λ ˆ, n λ =,...,,,...,, Moeove, we know fom popety 3., that the last thee tems of equation (5) become linea with espect to vecto X and hence they can be witten as follows ˆ ( q, ) + G( q) D( q,,, ). M ( q) + C = X (6) Inseting the ight hand side of (6) into (5), obot dynamic equations (5) take the following fom ( q, ) z + D( q,, ) X u M ( q) z + C, = (7) Based on (3), (4) and (7), lets popose the following adaptive Jacobian contolle u = kz k P Jˆ e + D( q,,, )Xˆ (8) whee ˆX stands fo the estimated physical paamete vecto defined below k and k p ae some positive scalas which could be eplaced by diagonal matices of positive constants without affecting the stability esults obtained futhe on (this should lead to impoved pefomance). Hee, scala constants ae chosen fo simplicity of pesentation. In ode to measue eo e, path paameteization s= s(t) is equied which is computed by solving the following scala diffeential equation s = k s s k P dθ( s) dγ e, + γ ( s ) + s ds ds ( ) (9) whee k s denotes a positive coefficient γ is assumed to be a stictly positive ( ) 0 function { } inf γ > of s with bounded fist and second deivatives fo any γ0 s +, whee γ0 γ( 0) { } inf γ = (as will be seen futhe on, γ 0 detemines the uppe bound on the accuacy of the path following and may be specified by the use). he choice of function γ is cucial fo computational effectiveness of contol dγ scheme (8), (9). One possibility is γ = γ(s) with 0 ds >. An altenative choi-

9 506 Industial Robotics: heoy, Modelling and Contol ce could be γ γ( e ) =, whee γ attains its maximum fo e = 0 and smoothly deceases as e inceases. Fo simplicity of futhe consideations, we take the fist fom of γ. Assumption. Function γ is equied not to satisfy diffeential equation dγ γ + ( s ) = 0 ds As will be seen futhe on, Assumption. esults in an asymptotic convegence of s to. Let us note, that the fist two tems fom (8) pesent an adaptive tanspose Jacobian contolle with adaptively vaying kinematic paamete vecto Ŷ. he last tem in dependence (8) is an estimated contol based on equation (6). Es- Ĵ = J q,y ˆ timated kinematic paametes Ŷ of the adaptive Jacobian matix ( ) ae updated accoding to the following law Yˆ = w k k P K ( q, )e (0) and estimated physical paametes ˆX of the dynamic equations ae updated by Xˆ = w d D ( q,,, )z () whee wk, w d, ae, similaly as befoe, positive gains (scalas) which could be eplaced by diagonal matices of positive constants without affecting the stability of contolle (8). Although some kinematic paametes appea in ˆX, we should adapt on them sepaately inŷ to peseve lineaity. Estimated kinematic paametes Ŷ (updated accoding to ule (0)) ae then used to compute adaptive Jacobian Ĵ and its time deivative Ĵ using fo this pupose the ight hand side of (0) which is only a mapping of q,e, and. Having obtained the adaptive Jacobian matix and its time deivative, we detemine quantities and. It is woth noticing, that thei computation does not equie any pseudoinvese of matix Ĵ which esults in numeical stability of contolle (8). Finally, based on q,q,q,, and, we may detemine dy- D q,q,q,q, which is then used to update estimated namic egesso matix ( ) physical paamete vecto ˆX.

10 Contol of Redundant Robotic Manipulatos with State Constaints 507 Let us note, that setpoint contolles poposed in woks (Cheach et al., 999 Yazael & Cheach, 00 Cheach et al., 003), which ae computationally somewhat simple, can not be applicable to ou task. he eason is that the eo of appoximation in (Cheach et al., 999 Yazael & Cheach, 00 Cheach et al., 003) is bounded by a constant and appoximate Jacobian matix does not include paametes to be adapted. Due to adaptation law (0), one can not guaantee in ou task to satisfy assumption egading the appoximation eo made in woks (Cheach et al., 999 Yazael & Cheach, 00 Cheach et al., 003). he closed-loop eo dynamics is obtained by inseting the ight hand side of equation (8) into equation (7) M ( q) z = C ( q, ) z + D( q,,, ) dθ( s) e = J ( q, Y ) e m+ ds dθ( s) e m+ = kse m+ kp e, ds ~ Y = wkkpk ( q, ) e ~ X = w D ( q,,, )z d ~ X kz k P Jˆ e dγ + γ em+ + em+ () ds ~ = ˆ whee Y Y Y X X X deive the following esult. ~ = ˆ. Applying the Lyapunov stability theoy, we heoem. If thee exists a solution to the poblem ()-(6) and adaptive Jacobian matix Ĵ is non-singula along end-effecto path () and function γ fulfils Assumption., then contol scheme (8) geneates manipulato tajectoy whose limit point ( ( ), e m+ ( ), e( ), em+ ( ) ) = (0,0,0,0),i.e. satisfying state constaints ()-(6), is asymptotically stable. Poof. Conside a Lyapunov function candidate V ~ ~,. Mz z + X + Y + e m+ + kpγem + kpe wd w (3) k = + he time deivative of V is given by

11 508 Industial Robotics: heoy, Modelling and Contol V = z, Mz + e M z, z + k P e, + w ~ ~ X, X dθ + e m+ + kp e, e m+ + kpγem+ e m+ + k ds J + w dγ e ds ~ ~ Y, Y + m P m+ em +. d k (4) Substituting Mz,.e,.e m+,y and X fom V fo the ight-hand sides of closedloop eo dynamics () and using the skew-symmetic popety of matix M C [popety. eqn. (8)], we obtain afte simple calculations, that V ˆˆ = k z, z kp JJ e, e kse m +. Since V 0, function V is bounded. heefoe, z,x, and Y ae bounded vectos. his implies that ˆX and Y ae bounded, too. Consequently, it follows fom (3), that is also bounded. Moeove,s and s ae bounded, too. As can be seen, V is negative fo all ( z,e,e m + ) ( z,e,e m + ) = 0 (Kstic et al., 995) that ( z,e,e m + ) z( ) 0,e( ) 0, and e 0,as. and is zeo only when 0, which implies (using LaSalle-Yoshizawa invaiant theoem tends asymptotically to zeo, i.e. m+, as. By diffeentiating e m + in () 3 de with espect to time, it is also easy to see, that m + is bounded function by 3 dt assumptions egading Θ and γ. his means, that e m + is unifomly continuous. Hence, em+ ( ) 0,, as,, too. he convegence of path velocity and acceleation yields the following equation γe dγ ( ) + e + ( ) = 0. ds m+ m (5) dγ m 0 + m+ = 0. On account of Assumption ds, the second equality is not fulfilled. hus, e m + ( ) = 0 (o equivalently s = ). On account of (3), q( ) 0, as, too. Consequently, Consequently, e + ( ) = o γ e ( ) ( ) bounday conditions (4)-(6) ae (asymptotically) fulfilled and limit point q,e,e,e = 0000,,, is asymptotically stable. Finally, it ( ( ) m+ ( ) ( ) m+ ( )) ( ) should be emphasized, that the chosen Lyapunov function does not guaantee convegence of paamete estimations ˆX and Ŷ to thei tue values.

12 Contol of Redundant Robotic Manipulatos with State Constaints 509 X t 0 Y k 0 pγ = t= On account of (3)-(6), we have Vt = 0 = + + w w Fo sufficiently lage w d and w k, the fist two tems in this equality may be kpγ0 omitted. Hence, we obtain V t = 0 SinceV kpγ0 is not positive, function V fulfils the inequality V. Consequently, the following bound on e may easily be obtained, based on (3) and the last dependence e γ 0 (6) d k 0. An impotant emak may be deived fom the poof caied out. Namely, Inequality (6) pesents an uppe bound (path independent) on the accuacy of path following by the end-effecto accoding to the contol law (8). Let us note that estimation of the uppe bound on path following eo (6) is vey consevative. Consequently, contol gains w and w do not equie lage values to achieve a good path following accuacy, as the numeical simulations (given in the next section) show. Moeove, seveal obsevations can be made egading the contol stategy (8). Fist note, that the poposed contol law equies, in fact no infomation concening the obot kinematic and dynamic equations. Second, the choice of contolle paametes k,k p,k s, w d and w k accoding to dependencies (8)-() guaantees asymptotic stability of the closed-loop eo dynamics () duing the manipulato movement. Moeove, the tanspose of Ĵ (instead of a pseudoinvese) in contol scheme (8) does not esult in numeical instabilities due to (possible) kinematic singulaities met on the obot tajectoy. Nevetheless, (8) has been deived unde the assumption of full-ank adaptive Jacobian matix along the path. Futhemoe, contolle (8) does not equie the knowledge of task space velocity. Due to consevative estimation of the path following accuacy, contol algoithm (8) esults in a bette accuacy of the path following as compaed to uppe bound given fom (6), as the numeical computations caied out in the next section show. In ode to pevent contol (toque) oscillations at the vey beginning of time histoies (caused by e.g. the non-zeo initial path following eo) k s fom (9) should be a bounded, quickly deceasing time dependent function as t (see the next section). d k

13 50 Industial Robotics: heoy, Modelling and Contol Due to eal-time natue of obot contolle (8), we shall ty to estimate the numbe of aithmetic opeations equied to implement the algoithm pesented in this section. he dimension of the obot task space is assumed in estimation to be constant. Opeations equied fo computation of sin,cos, and Θ() functions ae not taken into account. Futhemoe, matices Ĵ, K and D ae assumed in estimation to be given. Moeove, estimations ae caied out at any time instant of the obot task accomplishment. It follows fom (3) and (8) that tems kz,kpˆ J e, equie On ( ) opeations. Computation of the ight hand sides (9) and (0) involves O () and On ( ) opeations, espectively assuming that k = O( n). Computational complexity fo the ight hand side of () equals O(n ) by assumption that d = O( n). Computation of estimated contol ˆ D(q,q,q,q )X equies also the same ode of complexity, i.e. O(n ) opeations. Finally, computational complexity of the whole obot contolle (8) is of the ode O(n ). 4. A numeical example he aim of this section is to illustate the pefomance of the poposed adaptive contol algoithm using a dynamic thee-joint diect-dive am (n = 3) of SCARA-type obotic manipulato opeating in a two-dimensional (m = ) task space. Kinematic scheme of this manipulato and the task to be accomplished is shown in Fig.. In the simulations, SI units ae used. he components of dynamic equations of this manipulato ae as follows (Spong & Vidyasaga, 989): M = [ M ij ] i, j 3 = + + ci=cos( q ), ( ) whee M X X4c3 X6c3 i i s ij = sin( q i + q j ) c ijk = cos( q ) i + qj + qk M = X + X 4c + X 5c3 M = X + X 6 c3 M 33 = X 3 M = M M 3 = X 3 M 3 = M 3 M 3 = M 3 si=sin q, cij=cos(q i+q j ) s ijk = sin qi + qj + qk M 3 = X 3 + X 6c3 ( ).

14 Contol of Redundant Robotic Manipulatos with State Constaints 5 p [m] l q l q q 3 l 3 the geometic path p [m] Figue. A kinematic scheme of the manipulato and the task to be accomplished C = [ C ij ] i, j 3 whee C = ( X 4s + X 5s ) ( X 5s3 + X 6s3 ) 3 C = ( X 4 s + X 5 s)( + ) ( X 5 s + X 6 s) 3 C3 = ( X 5s + X 6s)( + + 3) C = ( X 5s + X 5s) + X 6s3 3 C = ( X 5 s + X 6 s) 3 C3 = X 6s3( ) C3 = ( X 4s + X 5s) X 6s3 C = X s ( + ) q C 33 = 0. = G, ( G G ) G, 3 whee G = X 7c + X 8c + X 9c3 Paametes X, i : 9 i G = X c + X 9c3 8 = take the following nominal values: G = 3 X 9c 3.

15 5 Industial Robotics: heoy, Modelling and Contol X =.956, X = , X 3 = 0.05, X 4 = 0.475, X 5 = 0.80, X 6 = 0.5, X 7 = ( ml c + ml + m3l )g = ( m l m l )g X 8 c + 3 X 9 = m3lc3g,,, whee g stands fo the gavity acceleation m,l, i i and l ci, i = : 3denote link mass, length and location of the mass cente which is assumed to be equal to l ci = l i / l = 0.4 l = 0.36 l 3 = 0.3 m = 3.6 m =.6 m 3 =. Jacobian matix J(q,Y) equals J Ys Ys Y3s3 Ys Y3s3 Y = Yc + Yc + Y3c3 Yc + Y3c3 Y3c3 3 s3 Whee Y = l i = : 3 i i and the kinematic egesso matix takes the following fom K sq = cq c s( + ) s3( ) ( + ) c3( + + ) 3 he end-effecto position p = (pp ) (see Fig. ) epesents in the simulations the task space coodinates (m = ). he uppe bound on the accuacy of the path following in all the compute simulations, is assumed to be equal to 7 γ 0 = 006., whee γ(s) = 000, + 000, e, s. Let us intoduce path following eos

16 Contol of Redundant Robotic Manipulatos with State Constaints 53 e e e 3 p Θ = p Θ s to evaluate the pefomance of the obot contolle (8). In ode to examine the effects of both kinematic and dynamic uncetainties, initial values fo vectos ˆX and Ŷ wee set in the simulations as ˆX( 0) = ( ), Ŷ( 0) = ( ). he task of the obot is to tansfe the end-effecto a- long the geometic path (the dotted line in Fig. ), expessed by the following equations Θ ( s) = s Θ ( s) = s. he initial configuation q 0 equals q 0 = ( π / π / π / ) 0t Paametes k 0,.k 3000,.λ k = 6, e + 30 have been whee, s [ 0, ] = p = = and s ( ) chosen expeimentally to achieve pactically easonable time hoizon of task w = diag 000 and pefomance and elatively small contols with k ( ) w diag ( ) d =. he esults of compute simulation ae pesented in Figs x 0 3 e [m] t [sec] Figue. Path following eo e vs. time

17 54 Industial Robotics: heoy, Modelling and Contol 5 x e [m] t [sec] Figue 3. Path following eo e vs. time e t [sec] Figue 4. Path following eo e3 vs. time 30 5 u [Nm] t [sec] Figue 5. Input toque u vs. time

18 Contol of Redundant Robotic Manipulatos with State Constaints u [Nm] t [sec] Figue 6. Fig. 6 Input toque u vs. time 4 3 u 3 [Nm] t [sec] Figue 7. Input toque u3 vs. time Figue 8. ime couse of adaptive estimate Ŷ

19 56 Industial Robotics: heoy, Modelling and Contol Figue 9. ime couse of adaptive estimate Ŷ Figue 0. ime couse of adaptive estimate Ŷ 3 Figue.. ime couse of adaptive estimate ˆX

20 Contol of Redundant Robotic Manipulatos with State Constaints 57 Figue. ime couse of adaptive estimate ˆX Figue 3. ime couse of adaptive estimate ˆX 3 Figue 4. ime couse of adaptive estimate ˆX 4

21 58 Industial Robotics: heoy, Modelling and Contol Figue 5. ime couse of adaptive estimate ˆX 5 Figue 6. ime couse of adaptive estimate ˆX 6 Figue 7. ime couse of adaptive estimate ˆX 7

22 Contol of Redundant Robotic Manipulatos with State Constaints 59 Figue 8. ime couse of adaptive estimate ˆX 8 Figue 9. ime couse of adaptive estimate ˆX p [m] p [m] Figue 0. Manipulato motion along the geometic path

23 50 Industial Robotics: heoy, Modelling and Contol As might be expected, the path following eos fom Figs -3 ae much smalle than those obtained fom the consevative dependence (6). Moeove, as one can obseve fom Figs -7, the time dependent damping function k s deceases (eliminates) eos and toques oscillations at the vey beginning of time histoies. Futhemoe, as seen fom Figs 8-9, estimations Xˆ, Yˆ do not convege to thei eal (nominal) values. 5. Conclusion his study has pesented an adaptive obot contolle fo the path following by the end-effecto. he contol geneation scheme has been deived using the Lyapunov stability theoy. An advantage of the poposed contol law (8) is that it equies, in fact no infomation egading the paametes of the obot dynamic equations. he contol stategy (8) is shown to be asymptotically stable (by fulfilment of pactically easonable assumptions). he poposed obot contolle has been applied to a plana edundant manipulato of thee evolute kinematic pais opeating in a two dimensional task space. Numeical simulations have shown that the esults obtained ae in accodance with the theoetical analysis. he novelty of the stategy poposed lies in its elative simplicity in design, pogam code and eal-time implementation. he appoach pesented hee will also be in futue diectly applicable to coopeating kinematically edundant manipulatos. Acknowledgement. his wok was suppoted by the DFG Ga 65/--,. 6. Refeences Aimoto, S. (990). Design of obot contol systems, Adv. Robot., vol. 4, no., pp Aimoto, S. (996). Contol theoy of non-linea mechanical systems, Oxfod, U. K., Claendon. Beghuis, H. R. Otega, & H. Nijmeije (993). A obust adaptive obot contolle, IEEE ans. on Robotics and Automat., vol. 9, no. 6, pp Canudas, de Wit, C. B. Siciliano & G. Bastin (996). heoy of obot contol, New Yok: Spinge-Velag. Cheah, C. C. S. Kawamoa & S. Aimoto (999). Feedback contol fo obotic manipulatos with an uncetain Jacobian matix, J. Robot. Syst., vol. 6, no., pp

24 Contol of Redundant Robotic Manipulatos with State Constaints 5 Cheach, C. C. M. Hiano S. Kawamua & S. Aimoto (003). Appoximate jacobian contol fo obots with uncetain kinematics and dynamics, IEEE ans. on Robotics and Automation, vol. 9, no. 4, pp Dahl, O. (994). Path-constained obot contol with limited toques - Expeimental evaluation, IEEE J. Robot. Automat., vol. 0, pp Feng, G. & M. Palaniswami (99). Adaptive contol of obot manipulatos in task space, IEEE ans. on Automatic Contol, vol. 38, no., pp Galicki, M. (998a). he planning of obotic optimal motions in the pesence of obstacles, Int. J. Robotics Res., vol. 7, no. 3, pp Galicki M. (998). he stuctue of time optimal contols fo kinematically edundant manipulatos with end-effecto path constaints, Poc. IEEE Conf. Robotics Automat., pp.0-06, Leuven, Belgium. Galicki, M. & I. Pajak (999). Optimal motion of edundant manipulatos with state equality constaints, Poc. IEEE Int. Symp. on Assembly and ask Planning, pp. 8-85, Poto, Potugal. Galicki, M. (000). ime-optimal contols of kinematically edundant manipulatos with geometic constaints, IEEE ans. Robotics Automat., vol. 6, no., pp Galicki, M. (00). Real-time tajectoy geneation fo edundant manipulatos with path constaints, Int. J. Robotics Res., vol. 0, no. 7, pp Galicki, M. (00). Motion contol of obotic manipulatos in task space, Poc. IEEE/RSJ Int. Conf. Intell. Robots and Systems, pp , Lausanne. Galicki, M. (004). Path following by the end-effecto of a edundant manipulato opeating in a dynamic envionment, IEEE ans. Robotics, vol. 0, no 6, pp Galicki, M. (005). Collision-fee contol of obotic manipulatos in the task space, J. Robot., Syst., vol., no 8, pp Galicki, M. (006a). Path-constained contol of a edundant manipulato in a task space, Robotics and Autonomous Systems, vol. 54, pp Galicki, M. (006b). Adaptive path-constained contol of a obotic manipulato in a task space, Robotica (to appea). Kelly R. (999). Regulation of manipulatos in geneic task space: An enegy shaping plus damping injection appoach, IEEE ans. Rob. Automat., vol. 5, no., pp Kiefe, J. A. J. Cahill & M. R. (997). James, Robust and accuate time-optimal path-tacking contol fo obot manipulatos, IEEE ans. Automat. Robot., vol. 3, pp Kstic,M. I. Kanellakopoulos & P. Kokotovic (995). Nonlinea and adaptive contol design, J. Wiley and Sons, New Yok. Lewis, F. L. C.. Abdallach & D. M. Dawson (993). Contol of obot manipulatos, New Yok: Macmillan Publishing Company.

25 5 Industial Robotics: heoy, Modelling and Contol Sciavicco, L. & B. Siciliano (996). Modeling and contol of obot manipulatos, New Yok: McGaw-Hill. Slotine, J. J. E. & W. Li (987). On adaptive contol of obot manipulatos, Int. J. Robotics Res., no. 6, pp Slotine, J. J. E. & W. Li (99). Applied nonlinea contol, Englewood Cliffs, New Jesey: Pentice Hall. Spong, M. & M. Vidyasaga (989). Robot dynamics and contol, Wiley, New Yok. akegaki, M. & S. Aimoto (98). A new feedback method fo dynamic contol of manipulatos, ans. ASME: J. Dyn. Syst., Meas., Cont., vol. 0, pp omei, P. (000). Robust adaptive fiction compensation fo tacking contol of obot manipulatos, IEEE tans. on Automatic Contol, vol. 45, no., pp , 000. Yazael, H. & C. C. Cheach (00). ask-space adaptive contol of obotic manipulatos with uncetainties in gavity egesso matix and kinematics, IEEE ans. on Automatic Contol, vol. 47, no. 9, pp

26 Industial Robotics: heoy, Modelling and Contol Edited by Sam Cubeo ISBN Had cove, 964 pages Publishe Po Liteatu Velag, Gemany / ARS, Austia Published online 0, Decembe, 006 Published in pint edition Decembe, 006 his book coves a wide ange of topics elating to advanced industial obotics, sensos and automation technologies. Although being highly technical and complex in natue, the papes pesented in this book epesent some of the latest cutting edge technologies and advancements in industial obotics technology. his book coves topics such as netwoking, popeties of manipulatos, fowad and invese obot am kinematics, motion path-planning, machine vision and many othe pactical topics too numeous to list hee. he authos and edito of this book wish to inspie people, especially young ones, to get involved with obotic and mechatonic engineeing technology and to develop new and exciting pactical applications, pehaps using the ideas and concepts pesented heein. How to efeence In ode to coectly efeence this scholaly wok, feel fee to copy and paste the following: Mioslaw Galicki (006). Contol of Redundant Robotic Manipulatos with State Constaints, Industial Robotics: heoy, Modelling and Contol, Sam Cubeo (Ed.), ISBN: , Inech, Available fom: botic_manipulatos_with_state_constaints Inech Euope Univesity Campus SeP Ri Slavka Kautzeka 83/A 5000 Rijeka, Coatia Phone: +385 (5) Fax: +385 (5) Inech China Unit 405, Office Block, Hotel Equatoial Shanghai No.65, Yan An Road (West), Shanghai, 00040, China Phone: Fax:

27 006 he Autho(s). Licensee IntechOpen. his chapte is distibuted unde the tems of the Ceative Commons Attibution-NonCommecial-ShaeAlike-3.0 License, which pemits use, distibution and epoduction fo non-commecial puposes, povided the oiginal is popely cited and deivative woks building on this content ae distibuted unde the same license.