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1 Documnt ownloa from: This papr must b cit as: Zotovic Stanisic, R.; Valra Frnánz, Á. (1. Ajusting th paramtrs of th mchanical impancfor vlocity, impact an forc control. Robotica. 3(4:1-5. oi:1.117/s Th final publication is availabl at Copyright Cambrig Univrsity Prss (CUP

2 Ranko Zotovic Stanisic Ángl Valra Frnánz Dpartmnt of Systms Enginring an Control Univrsia Politécnica Valncia P.O. Bo. 1, E-4671, Valncia, Spain. ADJUSTING THE PARAETERS OF THE ECHANICAL IPEDANCE FOR VELOCITY, IPACT AND FORCE CONTROL Abstract: This work is icat to th analysis of th application of activ impanc control for th ralisation of thr objctivs simultanously: vlocity rgulation in fr motion, impact attnuation an finally forc tracking. At first, a brif analysis of activ impanc control is ma, ucing th valu of ach paramtr in orr to achiv th thr objctivs. It is monstrat that th systm may b ma ovramp with th aquat slction of th paramtrs if th charactristics of th nvironmnt ar known, avoiing high ovrshoots of forc uring th impact. Th scon an most important contribution of this work is an aitional masur for impact control in th cas whn th charactristics of th nvironmnt ar unknown. It consists in switching among iffrnt valus of th paramtrs of th impanc in orr to issipat fastr th nrgy of th systm, limiting th paks of forc an avoiing losss of contact. Th optimal switching critria ar uc for vry paramtr in orr to issipat th nrgy of th systm as fast as possibl. Th rsults ar vrifi in simulation. Kywors: robot control, impact, forc control, impanc control, switching. 1. INTRODUCTION Possibly th most charactristic problm of robot forc control is th abrupt chang from fr to constrain motion. Th importanc of this iscontinuity is mphasiz by th fact that in th typical inustrial applications th nvironmnt is vry stiff an th ynamics of th systm is much fastr uring th constrain motion. Th systm is highly unramp. In th transint phas (impact th forc may rach angrous paks. Bfor achiving th contact, th magnitu to b controll is th vlocity, an aftr, th forc. Sinc iffrnt magnitus ar to b controll an th charactristics of th systm hav an important chang, it sms logical to us on controllr for ach phas. In orr to mak th transition from fr to constrain motion as smooth as possibl a thir controllr may b introuc. It is call impact control. Th impact is th most important phas bcaus high paks of forc may occur an caus irrvrsibl amag to th robot, th nvironmnt or th tool. Evn if that osn t happn, smallr paks of forc triorat graually th mchanics of th robot. Anothr potntial problm of th impact phas is th possibility of bouncing. All ths rawbacks coul b asily avoi signing an ovramp controllr if th charactristics of th nvironmnt wr known. Unfortunatly it is oftn not th cas. For this rason, any a priori slct paramtrs of th rgulator may not b aquat, an aitional actions coul b ncssary if th systm appars to b unramp whn th contact is achiv. An aitional inconvnint of th impact control is th fact that this phas is trmly brif an may last just a fw sampling prios. This implis that, for ampl, an aaptiv controllr may b too slow to protct th systm. Th impact control has bn tnsivly rsarch an vry ivrs solutions hav bn propos. Th most compilation from iffrnt sourcs as wll as a vry hausting analysis of th impact control has bn publish by Brogliato 1 in It shoul b mphasiz that thr ar two ways to trat th impact 1, : th rigi an th flibl mol. Th formr osn t consir what is happning uring th contact phas, just bfor an aftr it. It is assum that th uration of th impact is infinitly short. Th rlation btwn th vlocitis in th momnt of th contact an aftr th rboun is givn

3 by th cofficint of rstitution. In th flibl mol th impact is trat analytically consiring th robot an/ or th nvironmnt as lastic bois. This mol will b us in this articl. Following will b numrat som mthos for impact control. With th rigi mol, Brogliato t al. 3 propos two mthos for limiting th numbr of rbouns an assuring in this way th stability of th systm. Thr ar mor works bas on th flibl mol. Volp an Khosla 4 propos thr mthos for impact control. All thr ar orint to avoi contact loss rathr than th protction against paks of forc. Hy an Cutkosky 5 propos in 1994 th moulation with pulss of th fforwar. Ths pulss ar comput to supprss th transitory harmonics. Xu, Hollrbach an a 6 implmnt a PD forc control with fforwar is us for forc rfrnc tracking as wll as for th impact control. Th paramtrs of th rgulators vary accoring a non linar law. Thy cras if th robot is approaching th rfrnc in orr to ruc th rsiual nrgy. In th contrary cas, thy incras. Frrtti, agnani an Zavala Río 7 propos to apply uring th impact a fforwar trmin mpirically combin with th forc rgulator, in orr to avoi contact losss. Th switching of paramtrs was introuc in forc control by B. Armstrong t al. 8,9, in a stuy similar to th prsnt work. Th authors switch th gain matri accoring to th stat of th systm. Th ssntial ia is th sam, but a iffrnt mathmatical mthoology was us. In th work of B. Armstrong t al. LI was us to monstrat th valiity of th mtho. From th mntion sourcs, it can b uc that th tchniqus for impact ar control ar vry htrognous. Thy on t us th sam mol. Som ar icat to avoi contact losss rgarlss of th possibl paks of forc. Othrs ar limit to guarant th convrgnc of th systm aftr a finit numbr of rbouns. Som consir th charactristics of th nvironmnt ar compltly known. As stat arlir, in aition to th impact controllr, a position/ vlocity an forc rgulators ar also ncssary. A forc control task may consist of thr controllrs an two procsss with vry iffrnt ynamics. Switching among thm is a potntial sourc of bouncing, sliing rgim an vn stability loss. In orr to avoi th switching among controllrs, impanc control may b us. This is a control stratgy thortically aquat for th thr phass, propos by Nvill Hogan 1. Its objctiv is to impos a sir ynamics to th robot rathr than tracking th forc rfrnc. Its main avantag is that th sam controllr can b us for both fr an rstrict motion. Th main isavantag of impanc control, as it has bn stat in many sourcs, for ampl by D Schuttr t al. 11, is that it is ncssary to hav an act mol of th nvironmnt in orr to rach th forc rfrnc. This assumption may b impossibl for som ral applications. Th first contribution of this articl is th uction how to slct th impanc paramtrs in orr to ovrcom this limitation, assuming th charactristics of th nvironmnt ar known. It is monstrat that th aquat combination of paramtrs allows raching th rfrnc valu of th forc rgarlss of th nvironmnt, achiving vlocity control uring fr motion an attnuating th impact. Nvrthlss, this tchniqu os not guarant th bhaviour of th systm uring th impact if th paramtrs of th nvironmnt ar unknown. Th scon an most important contribution of this papr is an impact controllr bas on th switching of th paramtrs of th impanc pning on th issipat an th gnrat forms of nrgy in th givn instant. Th basic ias of this work hav bn publish by th authors in 5 1. Also, th authors of this papr publish a mtho for simultanous vlocity, forc, an impact control 13. Th approach is similar to this articl, but it was appli to plicit insta of implicit forc control. Th paramtrs ar iffrnt, as wll as th way to ajust thm. Th propos mtho guarants an improvmnt of th amping of th systm, rgarlss of th charactristics of th nvironmnt. It ns just a fw sampling prios to b ffctiv.. SOE CONSIDERATIONS REGARDING IPEDANCE CONTROL Th ynamics quation of a robot arm subjct to an trnal forc is wll known 14, 15, 16 : τ J T ( q F = D( q & + H ( q, + G( q 1 T = D ( q( τ J ( q F H( q, G( q X & = J ( q X & = J ( q & + J& ( q q & = J ( q J ( q J& ( q (1 Whr τ is th vctor of motor torqus, J(q th Jacobian matri of th robot, F th vctor of trnal forcs acting on th robot s n ffctor, D(q th inrtia matri of th robot, H ( q, th matri of cntrifugal an Coriolis torqus G(q th vctor of gravity torqus on th motors, an q, an & ar th vctor of joint positions, vlocitis an acclrations, rspctivly. Solving th quation (1 for th acclration: ( On th othr han, th rlation btwn th Cartsian an joint vlocitis is 14, 15, 16 : An th acclrations: Solving (4 for & : Rplacing ( in (5 an solving for X & : (3 (4 (5

4 = J ( q D + J& ( q Z( s = = s + B + ( q( τ J f ( s s X ( s + BsX ( s + KX ( s = = v( s sx ( s (9 K s T ( q F H ( q, G( q + T = J( q D ( q J ( q F T F = ( J( q D ( q J ( q = T = J ( q D( q J ( q = ( q Whr: ( q = J ( q D( q J T 1 ( q (6 This systm is highly non-linar. Th acclration pns not only on th motor torqu an th trnal forc, but also on th inrtia matri, cntrifugal, Corilois an gravity forcs an th robot Jacobian. Ths magnitus vary pning th joint positions an vlocitis. Applying a constant forc on th n ffctor, th acclration vctor woul vary both its intnsity an its irction uring motion. Th bhaviour of th robot whn in contact with th nvironmnt woul b complicat to prict for th robot oprator. Impanc control allows an intuitiv raction of th robot to trnal forcs. It may b obtain manipulating th input variabl τ. Som ways to achiv it ar numrat an brifly scrib in subsction.1. Assuming that th torqu vctor τ is st to th act valu that compnsat th cntrifugal, coriolis an gravity forcs as wll as th trm J ( q, thn quation 6 bcoms: (7 (8 Rprsnts th ral Cartsian inrtia matri of th systm, i.. th rlation btwn th forc an th acclration. It shoul b not that it varis with th configuration of th robot. Also, it is non iagonal (in th gnral cas. For ths rasons, th inrtia of a robot changs uring motion. Th mchanical impanc is th rlation btwn th forc an th vlocity of th systm: Whr Z rprsnts th impanc, f th trnal forc, v th vlocity, th position, th mass, B th amping an K th stiffnss of th systm. Th impanc control consists in imposing to th systm th sir mass, amping an stiffnss (, B an K rspctivly insta of th ral ons. Svral formulations can b foun for th mchanical impanc. Som of thm ar: F = F = F = F = B X& + K ( X X F = B ( X& X& F = + B X& Th first on is most common 14, 15, 16. Th scon on has bn us in th original papr of Hogan 1. Th first thr can b foun in th work of Sraji an Colbaugh 17. Th fourth formulation is also call stiffnss or complianc control 16. It has th avantag that it ns nithr acclration nor forc snsor to b implmnt. Nvrthlss, th inrtia is not controll an thus th bhaviour of th robot varis in iffrnt configurations. Th sam formulation has bn aopt by Christian Ott l al. 18, 19 for th control of lastic robots. It shoul b notic that th ral robot is a scon orr systm, whil th fourth formulation is first orr. That is u to th fact that that th formulation rprsnts th stationary bhaviour of th systm whil th inrtia, prsnt only in th transint phas, is not inclu in th quation. Th fifth formulation has bn us by Lu t al. In this cas th orr of th systm is ruc by using a sliing mo controllr. A first orr systm is obtain an thus oscillations ar avoi, an nithr paks of forc nor contact losss may occur. Finally th sith formulation may b us for human frinly robots that cut tasks in coopration with humans 1 or for mulation of human muscls. Sinc th stiffnss is zro, if th robot is isplac by an trnal forc, it will not rturn to its initial position. F = K X ( + B ( X& X& + B X& + K ( X X rf rf rf rf F = + B ( X& X& + K ( X X + B X& B X& rf + K X (11 + B X& + K X FF rf rf + K ( X X rf rf rf + K ( X X rf rf (1 Th first formulation is th most gnral on an all th othrs may b consir spcial cass of it. It may b writtn in th form: Whr FF is th fforwar trm. Thrfor, th impanc is fin by four paramtrs: th mass, th amping, th stiffnss an th fforwar. Following, th ffct of all of thm will b analys in constrain an fr motion in orr to obtain an aquat prformanc in all th phass of th task. Th following figur rprsnts a schma of th ral, physical, ynamics of th robot as wll as th ynamics achiv by mans of impanc control.

5 F J T + + τ = G(q H(q,sq s q -1 1 sq 1 q D (q s s 1 ( F t a J(q + J(q,sq + FF B X& K D.K. J(q X X sx s X 1 X F + + s + B s+ K ff b Figur 1. Schma of th ral (a ynamics of th robot an th ynamics obtain by impanc control (b. s rprsnts th Laplac oprator an th block D.K. th irct kintics. As it may b obsrv, in th ral ynamics of th robot (Figur 1, schma a, th rlation btwn th trnal forc F an th Cartsian motion is compl an highly non- linar. It pns on th actual configuration of th robot, th join vlocitis an th motor torqus. In th impanc control (Figur 1, schma b th bhaviour of th systm is linar. Physically, th ynamics is th sam as in th schma a in figur 1, but th motor torqus τ ar manipulat to obtain th ynamics rprsnt in schma b, which is also givn by quation (11. Nt subsction scribs th way it may b achiv..1 Th implmntation of th impanc control Th impanc control may b implmnt in svral ways. Canuas Witt t al. 3 scrib two mthos: via linar stat fback an by invrs ynamics. Th formr is aquat for a on gr of from robot. Th lattr is bttr for svral grs of from. Anothr mtho for th implmntation of th impanc control is th sliing mo control. In this articl th invrs ynamics mtho will b us bcaus it achivs th linalization an th coupling of th systm. Following will b prsnt a brif scription of this mtho. In orr to obtain th ynamics of th systm scrib in quation (11, th acclration must b: (1 Accoring to (5, th acclration in joint spac shoul b: (13 & = J ( q( J& ( q = = J ( q( ( F t + FF B & K J& ( q Th final prssion for motor torqus is obtain substituting (13 in (1: τ = D( q J ( q( ( Ft + FF B X& K T J& ( q + H ( q, + G( q + J ( q F X (14 In this way, not only th systm is ma linar, but also it is coupl. Th bhaviour of th systm in irction of any Cartsian ais is inpnnt of th othr irctions. Th torqu from quation (14 cannot b comput whn th robot gos through a singularity, i.. a configuration whr th Jacobian matri is noninvrtibl. Th singularitis ar a major problm in robotics an its solution is byon th scop of this articl. Nvrthlss, following will b mntion two solutions propos by othr rsarchrs. In th first on 4 th controllr is split in two parts. Th first on controls th istanc from th singularity. Th scon on controls th motion in th irction orthogonal to th singular irction. Whil this mtho is aquat in fr motion, it may hav problms uring th contact. In th scon solution 19, it is prfrr simply to avoi th singularitis. Th introuction of a scon controllr which forcs th robot to mov away from singularitis is propos. It is activat only in th proimity of singularitis. Th final control action is obtain as a sum of th outputs of th two controllrs. Anothr cas whn th Jacobian matri is noninvrtibl is whn it is not squar. That happns if th numbr of grs of from of th robot is iffrnt than si. If th robot has fiv or lss grs of from, it will b unabl to control its motion/ forc along all th irctions of spac. This may b angrous in contact tasks sinc th trajctory in th non- controll irctions is unprictabl an th robot may pntrat ply in th nvironmnt causing high paks of forc. It may b unavisabl to us a robot with lss than si grs of from in contact tasks. If th robot has mor than si grs of from it may achiv th sam n ffctor trajctory with iffrnt combinations of joint motions. It is call a runant robot. Although th invrs jacobian os not ist, svral solutions for th psuo invrs may b foun in th litratur. On of thm is th right psuo-invrs 15. A mor gnral cas is a psuoinvrs that minimizs th quaratic cost function of joint vlocitis 16. It shoul b notic that th runancy may hlp in many cass to solv th problm of th singularitis by tracting all th linarly inpnnt quations 16.. Th bhaviour of th systm in constrain motion Sinc it has bn monstrat in sction.1. that th systm may b coupl, in th rst of th articl th cas of a singl gr of from will b trat without loss of gnrality. Lowr cas lttrs will b us insta of capitals for on imnsional variabls lik forc, position, vlocity, tc.

6 Assuming that th formation of th nvironmnt is lastic, th raction forc of th nvironmnt will b: f f = K ( B & ff + K = && + ( B + B & + ( K + K s 1, ( B = + B ± ( B + B 4( K + K ff + ( K + K = if K K + K ff + K = if K = K Th final valu of th forc main b obtain as: = K f = K ff + K K + K = ff ff = && + B & + K f ( B + B ( if if 4( K K K + K = (15 Whr is th coorinat of th nvironmnt s surfac. Th ynamics of th position in constrain motion may b obtain from quations (3 an (1: Th roots of th charactristic polynomial ar: an th iscriminant: (16 (17 (18 In orr to mak th systm as amp as possibl, it is convnint to assign a high valu to B an low valus to an K. Th final valu of th position will b: Thus: (19 ( (1 This mans that choosing K = not only amps th systm, but also allows us to rach th rfrnc valu of forc rgarlssly of th characrristics of th nvironmnt..3 Th bhaviour of th systm in fr motion In this cas F t= an th ynamics of th systm is fin by th following quation: Th final valus of th position an vlocity will b: ff = K K ff & = K = B B ( (3 Thrfor, a stiffnss iffrnt of zro will mak th robot rach th istanc givn by quation (3, whr it will stop. This corrpons to position control. Assigning a stiffnss qual to zro will mak th systm go to a constant sp. This corrspons to vlocity control. Th lattr is mor practical for impact control sinc it os not rquir prvious knowlg about th position of th nvironmnt. On th othr han, th amping shoul b chosn th way to assur th sir final vlocity v rf: B ff = v rf ( B + B < 4( K + K (4.4 Conclusions about th slction of th impanc paramtrs This sction will contain a rcapitulaton of th prvious conclusions for th slction of th paramtrs in orr to achiv th sir prformanc both in fr an constrain motion. Th stiffnss K shoul b st to zro for two rasons. At first, in fr motion, vlocity control is obtain insta of position control, what is bttr suit for impact achiving a softr impact in th cas whn th act position of th nvironmnt is unknown. Scon, th final valu of forc os not pn on th charactristics of th nvironmnt an th forc rfrnc may b always rach. Th valu of ff shoul b slct qual to th rfrnc forc in orr to achiv tracking rror zro accoring to (1. Th amping B is us to assur th systm will hav a vlocity quivalnt to th rfrnc valu uring th fr motion accoring to (4. Rgaring th mass, it is th only paramtr that practically has no importanc uring fr motion.th valu assign to th mass shoul b low in orr to amp th systm uring th impact. If th stiffnss of th nvironmnt is known, th valu of th mass that maks th systm unramp may b obtain from (18: Brifly th valus to b assign ar th following: - ff=f rf, whr F rf is th forc rfrnc. - K =. (5 - B : accoring to quation (4 in orr to attain th rfrnc sp. - : if th charachtristics of th systm ar known, in orr to mak th systm ovramp accoring to (5. Othrwis as small as possibl. Usually in th applications of forc control, th activ amping is ajust for smoothing th impact. In this

7 cas this is achiv by mans of th mass. Th amping is us for vlocity control in fr motion. It shoul b mphasiz that th slction of th paramtrs in th way it is scrib abov assurs that both vlocity rfrnc an forc rfrnc will b rach uring fr an constrain motion rspctivly, vn if th stiffnss of th nvironmnt is unknown. Nvrthlss, it is not possibl to guarant that th systm will b ovramp, nithr its bhaviour uring th transition phas accoring to (17 an ( SWITCHING THE VALUES OF THE PARAETERS Th prvious rsults show how to slct th impanc paramtrs if th charactristics of th nvironmnt ar known. Nvrthlss, in som cass this is not tru, an th bhaviour of th systm in th transition cannot b controll. Givn th potntial angr of th impact phas it is convnint to introuc an aitional masur in orr to mak th transition as soft as possibl. Th propos mtho is bas on th transformation of th nrgy. For ampl, whn th robot pntrats into th nvironmnt, th kintic nrgy it ha in fr motion is transform in lastic potntial nrgy of th nvironmnt. It is convnint to assign a low valu of mass in orr to ruc th kintic nrgy an a high stiffnss to avoi a p pntration into th nvironmnt. Whn th robot starts rbouning, th invrs nrgy transformation occurs. Thn th mass shoul b high to limit th acquir vlocity an th stiffnss low to ruc th lastic forc. Bfor starting a pr analysis of th propos mtho, som assumptions ma in th articl will b mntion in this paragraph. It will b consir for simplicity that all th magnitus ar normalis, nonimnsional quantitis. This assumption os not hav any influnc on th gnrality of th conclusions. It will b also consir that th position of th nvironmnt is on a positiv coorinat, i.. that positiv vlocity mans th robot is laning towars th nvironmnt an ngativ vlocity mans it is moving away. Th contrary cas is compltly symmtric an th conclusions obtain for on cas ar also vali for th othr on Physical principl of th nrgy issipation by mans of th switching of th paramtrs During constrain motion, th systm is typically unramp. Whil it oscillats aroun th quilibrium point, th kintic nrgy is transform to potntial an vic vrsa. Consiring an ial cas, a systm without issipation th total nrgy in vry momnt shoul b th sum of th potntial an th kintic nrgy: 1 1 E E E k m = k + p = + & (6 Whr k is th stiffnss an m th mass of th systm. In th instant whn vlocity is zro, th position rachs its trm point an th systm has only potntial nrgy (it will b assum for simplicity that th origin of th coorinat systm is in th quilibrium point: 1 E = Ep = k (7 ma Th trm valus of th position: E trm = ± k (8 In th instant whn th position is zro, th sp has an trm point an thr is only kintic nrgy: 1 E = Ek = m& (9 ma Th trm valu of th vlocity: E & trm = ± (3 m In th phas plan th systm is rprsnt by an llips. It is obvious from (8 that crasing k will mak llips highr an from (3 that crasing th mass will mak it wir, as it may b apprciat in th following figurs. Vlocity Position Figur. Phas iagrams of th systm whn th stiffnss is highr than th mass for thr valus of th nrgy. Vlocity Position Figur 3. Phas iagrams of th systm whn th mass is highr than th stiffnss for thr valus of th nrgy. Assigning a high mass an low stiffnss in th scon an fourth quarant, an oing th opposit in th first

8 an th thir on, th systm will b closr to th quilibrium point vry tim it intrscts any of th as. This is rprsnt in th following figur Th rivativ of V: V & = &&& + & ( Assuming th ynamics of th systm: && + ( B + B & + ( K + K = ff + K (3 (33 Vlocity Position 1-5 Figur 4. Phas iagram of th systm whn th mass is highr than th stiffnss in th vn quarants (ash lin an th opposit in th o ons (full lin. Th initial stat is (,.1. Switching th paramtrs in th instants of changs of quarant a consrvativ systm is ma issipativ. It is a form of nrgy issipation, an thus it may b us for impact control. It shoul b mphasiz that th prvious rasoning has bn ma for an ializ systm rathr than a ral on. Th main iffrnc is that th quilibrium point changs whn th stiffnss is switch, accoring to (19. Also, th amping has not bn takn into account. Th analysis ma in this sction is mor scriptiv than prcis. Th act on is lft for th nt sction. 3.. Switching critria an sliing rgims In orr to uc th optimal switching critria th following nrgy Lyapunov-lik function is us: V 1 & 1 = + ( (31 It rprsnts th Eucliian istanc from th quilibrium point in th phas plan. It is vint that fastr convrgnc mans fastr nrgy issipation. On th othr han, th trm 1 ( is quivalnt to th lastic potntial nrgy of th nvironmnt, scal by a factor that may pn on th units. In th sam way, th trm 1 & is proportional to th kintic nrgy. Thrfor, it may b stat that V quivalnt to th total nrgy of th systm. Any quaratic function of vlocity an istanc of th origin woul hav th sam ffct. It shoul b mphasiz that in th typical applications of Lyapunov functions, th origin of th coorinat systm is locat at th quilibrium point, an hnc it is not takn into account. Nvrthlss, whn a paramtr is switch, th quilibrium point may also chang, which may influnc th stability an gnrally th bhaviour of th systm. For this rason, is inclu in th consirations. && An ff + K B + B K K & + = ff K B B V& + + & & = K + K & + &( Introucing th prssion for from (19: (34 Th substitution of th prssion (34 in th prssion (3 givs: (35 ff K B B V& + + = & & (36 K + K ff + K & + & ( K + K This formula will us to uc th switching critria for th paramtrs. Thy will b analys on by on in th following subsctions Th mass This subsction is icat to th uction of th switching critria of th mass in function of th stat of th systm in orr to issipat th nrgy an thus to softn th impact. Also, th possibility of sliing rgims provok by th switching will b analyz. Finally, a stuy of th ffct of th nois of th acclration snsor will b ma. It will b assum that th mass is switch btwn two valus: th minimal an th maimal on. It will also b assum that only non- ngativ valu may b assign to th mass although th contrary woul b possibl in impanc control. For th achivmnt of a soft impact it is ncssary to issipat th nrgy of th systm vry fast. Th rivativ of th nrgy shoul b always as small as possibl. In orr to apprciat th ffct of th mass on th bhaviour of th systm, th partial rivativ of V & with rspct to th mass is uc: δv& δ ff + K = & + B + B K + K + & + Substituting (34 in (37: δv& δ &&& = & (37 (38

9 Whn this prssion is positiv th nrgy is issipat slowr as th mass incrass. Whn it is ngativ th nrgy is issipat fastr as th mass incrass. It sms logical to assign a high valu to th mass whn (38 is ngativ an a low on in th contrary cas. For this rason th following switching law is propos: (39 If &&& is positiv, th trm associat to th mass is absorbing nrgy from th systm. In th contrary cas it is livring nrgy. It shoul b notic that &&& corrspons to th rivativ of th squar of th vlocity an thus of th kintic nrgy. Thrfor, &&& < mans that th kintic nrgy is crasing, i.. bing transform into lastic potntial nrgy. Assigning a small valu of th mass will man rucing th amount th kintic nrgy to b issipat. In th contrary cas, whn &&& >, th kintic nrgy is incrasing, i.. potntial nrgy is transform in kintics. Th switching critria (39 hav bn vrifi by mans of simulations. Ths wr ma at first assigning smallr an smallr valus to m min, whil kping m ma constant. Nt, th contrary was ma: m min was kpt constant, whil th valus of m ma wr incras in svral succssiv primnts. Tsting th two cass sparatly, th ffctivnss of th switching critria is vrifi. Othrwis, th positiv rsults in on cas coul compnsat th ngativ ons of th othr, giving a fals apparanc of th valiity of th mtho. Th aopt valus for th simulations ar th following: =1 kg (if not switch, B =1 Ns/m, K =, ff=1 N (if not switch K =1 N/m, B=1 Ns/m, = (for simplicity. It is assum that that th robot impacts with th nvironmnt in th instant t=. Th aopt valu for th stiffnss of th nvironmnt is vry high an th systm is highly unramp, what corrspons to th rality in th forc control applications. Th rsults ar rprsnt on th following graphics. Forc [N mmin if &&& < = mma if &&& > It can b apprciat that whn th valu of m min is cras, th paks of forc ar ruc. Also, th convrgnc of th systm is fastr. Forc [N Tim [miliscons Figur 6. Forc in function of th tim whn switching m ma whil m min=1 in all th cass. a m ma =1 (full lin b m ma = (circls c m ma =5 (crosss. It can b apprciat that incrasing m ma also rucs th paks of forc. Nvrthlss, it slows th convrgnc of th systm. It may b conclu that th simulations rsults confirm th valiity of th switching critria (39 rgaring th protction of th systm, as both crasing m min an incrasing m ma ruc th paks Sliing rgims whn switching th mass Th analysis of sliing rgims an sliing mos is a vry tnsiv fil an thus byon th scop of this work. Nvrthlss a brif planation will b givn in orr to improv th clarity of th articl. Furthr information can b foun in iffrnt sourcs 5. A paramtr switchs whn th stat variabls satisfy a givn conition that can b prss as S= (4 Whn this conition is tru it sai that th systm is on th switching surfac. If th systm is not on th surfac, thr ar four possibl cass: S < an S& > (41 S > an S& < (4 S > an S& (43 > S < an S& (44 < Tim [miliscons Figur 5. Forc in function of th tim whn switching m min an kping th valu m ma=1 in all th cass. a m min=1 (full lin b m min=5 (circls c m min=1 (crosss.

10 S = S < S & > S & > (a (c S & < S & < S > Fig. 7. Th four possibl cass. Th thick lin rprsnts th switching surfac S=. Abov th surfac S>, an blow it S<. Th following cass ar possibl: a S< an S & >, th systm tns towars th surfac. b S< an S & <, th systm is moving away from th surfac, c S> an S & >, th systm is moving away from th surfac, S> an S & <, th systm has for th surfac. If any of th conitions (43 or (44 is tru, th systm is moving away from th switching surfac. Th mathmatical monstration of this statmnt will b omitt, but its maning is rathr logical. In th first cas, both th valu of S an its rivativ ar positiv. In th scon cas S an its rivativ ar ngativ. Thus, in both cass th istanc from th surfac is incrasing. In th contrary cas, if any of th conitions (41 or (4 is tru, th systm is tning towars th switching surfac. In summary, if th valus of S an its rivativ hav th sam sign, th istanc of th switching surfac is incrasing. If thir signs ar opposit, th istanc is incrasing. If a systm satisfis both conitions (41 an (4, it will rmain on th surfac onc it has rach it. This is call a sliing rgim. It is a harmful phnomnon, bcaus th systm is stuck on th surfac insta of tracking th rfrnc valus. In impanc control, th systm is scon orr, typically unramp. Its bhaviour is oscillatory. It crosss th switching surfac in vry prio. Whn this happns, it is obvious that th sign of S changs. In orr to cross th surfac, th systm must ha for it (conitions (41 or (4, passs through th surfac (S= for an instant, an mov away from it (conitions (43 or (44. Thrfor, in an oscillatory systm, if no switching is prform, th sign of S changs whn crossing th surfac, whil th sign of rmains S & th sam. Nvrthlss, th switching of a paramtr may provok a chang of th sign of S &. In this cas, th systm is push back to th surfac whichvr is th sign of S. Th systm bcoms unabl to lav th surfac an rmains on it. In summary, a sliing rgim may occur if thr is a possibility th switching of a paramtr to provok th chang of th sign of S & whn crossing th surfac. ( (b In orr to analyz th conitions of th sliing rgim for th concrt cas of th switching of th mass, it is important to mphasiz that, accoring to (39 th mass switchs four tims in vry prio as both th sp an th acclration chang thir sign twic. Thrfor, thr ar two switching surfacs, whn vlocity an acclration go through zro: S = & = S 1 = && = (45 Givn that th acclration is th rivativ of th vlocity, whn th formr passs through zro, th lattr has an trm point. Whn th acclration passs from positiv to ngativ, it is vint that th vlocity has a maimum an thrfor it is positiv. Th prouct & && gos from positiv to ngativ. In th scon cas of chang of sign acclration, whn it bcoms positiv, it corrspons to a minimum of th vlocity, which is thrfor ngativ. Th prouct & && gos from positiv to ngativ lik in th prvious cas. Thus, taking into account th conition (39 in both cass of chang of sign of th acclration th mass switchs to its minimal valu from th maimal valu. Ths conclusions ar rprsnt in th following tabls. Tabl 1. Signs of rlvant magnitus an valu of th mass whn acclration changs its sign from positiv to ngativ. agnitu Bfor Aftr crossing crossing th th surfac surfac Acclration > < Vlocity > > & && > < mass m ma m min Tabl. Signs of rlvant magnitus an valu of th mass whn acclration changs its sign from ngativ to positiv. agnitu Bfor Aftr crossing crossing th th surfac surfac Acclration < > Vlocity < < & && > < mass m ma m min A similar analysis can b ma for th changs of th sign of th vlocity. In orr to chang its sign from positiv to ngativ th acclration must b ngativ, an thrfor th prouct & && bcoms positiv. To chang from ngativ to positiv, th acclration must b positiv, an thus & && gos positiv again. This is summariz in th following tabls:

11 Tabl 3. Signs of rlvant magnitus an valu of th mass whn vlocity changs its sign from positiv to ngativ. agnitu Bfor Aftr crossing crossing th th surfac surfac Vlocity > < Acclration < < & && < > mass m min m ma Tabl 4 Signs of rlvant magnitus an valu of th mass whn vlocity changs its sign from ngativ to positiv. agnitu Bfor Aftr crossing crossing th th surfac surfac Vlocity < > Acclration > > & && < > mass m min m ma Summarizing, th mass switchs to: - m min whn th acclration changs its sign. - m ma whn th vlocity changs its sign. Following will b ralis th analysis of th possibility of occurrnc of a sliing rgim in both surfacs. Th rivativ of S is: 1 & && ff + K B + B K + K & S1 = = (46 Givn that both position an vlocity ar continuous an thrfor on t chang whn th mass is switch, an that ff, K, B, K, B an ar constant, th chang of th valu of th mass (assuming it is always positiv osn t influnc irctly th sign of S & 1. As consqunc, th switching of th mass cannot chang th sign of th surfac nor provok th apparanc of a sliing rgim. Th rivativ of S is: & B + B K + K &&& && & S = = (47 Assuming that in th proimity of th switching surfac th valu of th acclration is narly zro, th following can b assum: && B + B K + K && << & (48 K K S& + & Thrfor, th switching of th mass woul not chang th sign of th rivativ of th surfac S, an thus thr cannot b sliing rgim in this surfac Th ffct of th nois of th acclration masurmnt Th switching critria of th mass (39 rquir th knowlg of th valu of th acclration. Nvrthlss, th acclration masurmnt is subjct to nois. Th analysis of th ffct of this nois will b prsnt in this sction. In orr to hav ralistic valus, th ata for th analysis will b takn from th atasht of th Analog Dvics ADXL33. It is a low cost, 3-ais, on-chip acclromtr. It is tn in th rsarch as wll as in th commrcial applications. Th banwith in ach ais is slct by th usr by mans of capacitors connct to th masur outputs. It is basically a low pass filtr. With a lowr banwith th nois filtring is improv but th rsolution of th acclromtr is triorat. A tra-off shoul b foun for ach application. Th usr shoul limit th banwith to th lowst frquncy n by th application to maimiz th rsolution an ynamic rang of th acclromtr. Accoring to th atasht, th root man squar nois shoul b calculat by th formula: N = N 1. 6B (49 Whr N is th r.m.s. nois, N th nois magnitu an B th banwith. Th nois magnitu N for th ADXL33 is 8 in an y, an 35 µ g / Hz in th z a. On th othr han, th typical sampling prios us for robot control ar btwn on an tn kilohrtz. A highr banwith for th snsor os not mak sns. For a banwith of 1 KHz an a nois magnitu of 35 µ g / Hz (z a, th worst cas, th obtain r.m.s. nois will b: N = 35X X1 =.47g (5 Simulating th systm with this valu of nois practically no iffrnc has bn obsrv with rspct to th ial cas (no nois at all. For this rason, th cas of th filtr ajust to 1 KHz has not bn rprsnt on th figur. Simulations hav bn ma with th filtr ajust to cut-off frquncis of 1 KHz an 1Hz. Both of ths valus ar unralistically high. Although such high sampling rats coul b implmnt, many problms woul appar, lik th lay of th lctronics (A/D convrsion, tc or th frquncy of puls with moulation of th powr stag of th motor (slom highr than 1 KHz. Nvrthlss som simulations hav bn ma in ths unralistically unfavourabl conitions in orr to stimat th ffct of th nois.

12 Accoring to formula (49 a valu of r.m.s. nois of.14g has bn obtain for 1 KHz, an.447g for 1Hz. Th following figur rprsnts th rsults of th simulations of th systm without nois, with a cut-off frquncy of 1HZ as wll as for th cas without switching. Forc [Nwtons] Tim [miliscons] Figur 8. Th ffct of nois on th switching. Th full lin rprsnts th forc in th ial cas (no nois at all an th lin with crosss th cas whn th filtr is ajust to 1 Hz. Th maimal mass is kg an th minimal on 1 kg. Th ott lin rprsnts th cas whn no switching at all is ma (th mass is 1 kg all th tim. It may b apprciat that th nois triorats th ffct of switching. Th iffrnc btwn th cas with nois an th ial on incrass with ach switching, its ffct is accumulativ. This may b plain by th fact th switching critria (39 ar chosn for th optimal issipation of nrgy. Any othr switching law worsns th nrgy issipation. Th nois affcts th switchings whn acclration gos through zro. Whn thy occur, th switching will not b prform actly accoring to (39 u to th nois. Thus lss than optimal nrgy will b issipat in vry switching whn th acclration changs its sign. So, thr will b mor an mor nrgy accumulat rspcting th cas without th cas nois. Nvrthlss, th rsults ar much bttr than in th cas without switching, vn with th unralistically unfavorabl lvl of nois. On th othr, th nois maks almost no iffrnc for th initial paks that ar th most angrous ons. In summary, th nois affcts vry slightly th nrgy issipation, an almost not at all in th first, most important, paks Th stiffnss Th partial rivativ of (36 rspcting th stiffnss is: δv& 1 ff + K = & + & δ K K K ( + (51 Accoring to (51 th following switching law is propos in orr to maimiz th nrgy issipation: K 1 ff + K kma if & + & < ( K + K = 1 ff + K kmin if & + & > ( K + K δv& & & = δ ff K + K << K + K δv& δ ff & (5 It may b apprciat that th critria pn on th nvironmnt stiffnss, which was assum to b unknown initially. As consqunc, th switching conitions cannot b tct. Thus, th switching of th stiffnss accoring to (5 cannot b implmnt. It coul b aquat to assign th valu K = to th stiffnss accoring to th conclusions of th scon sction, in orr to rach th forc rfrnc. Nvrthlss, if th task rquirs stiffnss iffrnt thn zro (typical pg-in-a-hol problm, its valu may b assign in th momnt th contact is achiv Th amping Th partial rivativ of (36 rspcting th amping is: δv& δ B 1 = (53 Sinc it is always ngativ, this trm is always issipativ. As consqunc, th switching of th amping os not improv th prformanc of th impact control Th fforwar Th partial rivativ of (36 rspcting th fforwar is: Sinc in th practical robotic applications: It can b assum: & (54 (55 Thus th switching critria for th fforwar shoul b: ffma if & < ff = ffmin if & > (56 Ths critria ar rathr logical. Whn th vlocity is positiv, th robot is pntrating into th nvironmnt. A low fforwar will hav as consqunc a lowr pntrating pth. Whn th vlocity is ngativ, th robot is rtiring from th nvironmnt an th fforwar shoul b high to push him back insi an prvnt th rbouning. In both cass th valu of (37 th fforwar is slct to b oppos to th motion of th robot an is acting as a sort of brak. Hnc, it is a form of nrgy issipation.

13 Sliing rgims whn switching th fforwar Th only switching surfac is: S = & = (57 An its rivativ: S& = && = (58 Th systm crosss th switching surfac twic in vry prio: whn vlocity gos from positiv to ngativ an vic vrsa. Following both cass will b analys on by on. Th switching instant will b call t sw, th instant immiatly bfor will b call t sw- an th on immiatly aftr t sw+. Whn vlocity passs from positiv to ngativ th following statmnts can b ma: 1. & ( t > & ( t < (59 sw sw+. && <, othrwis th first statmnt coul not b satisfi. 3. Accoring to (56: ff ( t = ff ff ( t = ff (6 sw min sw+ ma 4. Th sliing rgim occurs if S & changs its sign whn th fforwar is switch. Its valu just bfor switching: ff + K B + B K + K && t & min ( sw = < 5. S & aftr switching: (61 ff + K B + B K + K && t & ma ( sw+ = > (6 6. Assuming th vlocity is zro nar th surfac, an that K is also zro, this prssion bcoms: && t ff + K K ff f ma ma ( sw+ = > > (63 Thrfor, a sliing rgim happns if th assign fforwar is lowr than th actual forc. It can b avoi simply by not prforming th switching if th forc is too low. This can prict bcaus ff ma is known an th forc can b masur. Anothr altrnativ is assigning ff ma a valu lowr than th actual forc. A similar rasoning may b ma for th cas whn th vlocity gos from ngativ to positiv: 1. & ( tsw < & ( tsw+ > (64. && >, othrwis th first statmnt coul not b satisfi. 3. Accoring to (56: ff ( t = ff ff ( t = ff (65 sw ma sw+ 4. Th sliing rgim occurs if S & changs its sign whn th fforwar is switch. S & bfor switching: min ff + K B + B K + K && t & ma ( sw = > (66 5. In orr a sliing rgim to occur S & must bcom ngativ aftr th switching: ff + K B + B K + K && t & min ( sw+ = < (67 6. Assuming th vlocity is zro on th surfac, an also K is zro accoring this prssion bcoms: && t ff + K K ff f min min ( sw+ = < < (68 Thrfor, th sliing rgim can b avoi by assigning to ff min a valu highr than th actual forc. Following will b rprsnt th simulation rsults. Th simulations will b ralis first incrasing th valus of ff ma. Thn thy will b prform crasing ff min. Forc [Nwtons] Tim [iliscons] Figur 9. Diagram of th forc in function of th tim whn ff ma is switch. a ff min =ff ma=1 (Full lin b ff min =1 an ff ma =1 (Circls c ff min = 1 an ff ma = (Crosss. It can b obsrv that th valu of ff ma os not hav any influnc on th first pak, bcaus it is not activat until aftr th pak is rach. Nvrthlss, th incras of ff ma rucs th subsqunt paks. Th switching happns in all th cass nar th trm valus of th forc what corrspons approimatly to th valu zro of th vlocity. Whn ff ma is augmnt to 1, th systm is amp an th trm valus ar ruc until th thir pak, aftr approimatly 15 milliscons, whn th systm ntrs into a sliing rgim. It happns bcaus th forc in th instant of switching is lowr thn ff ma (1N. For th cas ff ma is N, th sliing rgim happns at th first switching, bcaus th pak in this instant is lowr than ff ma (N. Th sliing rgim appars accoring th thortical rsults rflct in quation (63. Th sliing rgims may b avoi by omitting th switching or by moifying ff ma. Th nt figurs rprsnt th thr possibl cass. In th first on, th switching of th fforwar is ma whn any of th switching critria (56 is tru. As consqunc, th systm ntrs a sliing rgim aftr th thir pak (Figur 1. In th nt cas (Figur 11, th

14 possibility of sliing rgim is prict accoring to conition (63, an th switching is not prform. Th sliing rgim is avoi but th amping of th systm is poor. At last, th sliing rgims ar prict accoring th quation (63 an th valu of ff ma is moifi in orr to avoi thm (Figur 1. In this ampl th law for comput ff ma has bn: f ffmin ffma = (69 In this way it is always smallr than th actual forc, an conition (63 is not vrifi. Th sliing rgims ar avoi an th amping is improv. Forc [Nwtons] Tim [miliscons] Figur 1. Diagram of th forc (ash lin an fforwar (full lin for ff min =1 an ff ma =1. A sliing rgim appars at th thir pak of forc. Forc [Nwtons] Tim [miliscons] Figur 11. Diagram of th forc (ash lin an fforwar (full lin for ff min =1 an ff ma =1 whn switching of th fforwar is not prform at th thir pak avoiing th sliing rgim. Forc [Nwtons] Tim [miliscons] Figur 1. Diagram of th forc (ash lin an fforwar (full lin for ff min =1 an ff ma =1 whn ff ma is rajust accoring to (69. Th figur 13 rprsnts th forc in function of th tim whn for iffrnt valus of ff min. It may b apprciat that as th paks of forc ar smallr as ff min crass, but th also th systm ntrs soonr a sliing rgim. Forc [N] Tim [miliscons] Figur 13. Diagram of th forc in function of th tim whn ff min is switch. a ff min = ff ma =1 (full lin b ff min =8 (circls. c ff min = 6 (crosss. It may b apprciat that crasing ff min rucs th paks of forc until a sliing rgim happns. In this cas, a solution coul b assigning a highr valu to ff min accoring to (6. In summary, in both cass th switching of ff improvs th impact control until th systm ntrs a sliing rgim, what can b asily avoi. 4. THE CONTACT LOSS All th prvious analysis trat th cas whn th robot an th nvironmnt ar in contact. It has bn monstrat that th switching of th paramtrs accor to laws (39 an (56 incrass th issipation of th nrgy an thus softns th impact. Givn that it rucs th amplitus of th oscillations, it also rucs th probability of contact loss. Nvrthlss, th contact loss may happn. This sction stuis th ffct of th switching of th mass an th fforwar in th cas whn contact is lost. In th cas of contact loss, th systm is in fr motion an its bhaviour is givn by quation (. Accoring to prvious conclusions (subsctions.4 an 3.., it will b assum that th activ stiffnss is zro (K =.Th ynamics of th systm is givn by th following quation: ff = & + B & (7 In th nt subsctions will b stui th ffcts of th switching of th mass an th fforwar rspctivly for th ynamics of th systm givn in (7. Th main critrion for th valuation will b th vlocity of th robot on a nw impact, aftr a contact loss. It will b assum that th switching of a paramtr improvs th impact control if it rucs th vlocity on th instant th contact is achiv again. On th othr han, it is mpirically known that a boy that rbouns on th groun, without any human

15 intrvntion, is a stabl systm an th numbr of rbouns is finit. So, it will b consir that, if th switching rucs th impact vlocity in ach subsqunt contact, th nrgy issipation is improv compar with th cas whn no action at all has bn appli. For this rason, th fact that th impact vlocity is ruc in ach impact will b consir as a sufficint conition to monstrat th stability of th systm, as wll as a countabl numbr of impacts Th ffct of th switching of th mass In th instant of th loss of contact, th robot is moving away from th nvironmnt. Th fforwar trm is pushing it back to th surfac an it slows until it stops an rvrss th sp. Thrfor, it may b stat that a rboun consists of two phass. In th first on it is going in th opposit irction of th nvironmnt an thus its vlocity is ngativ. It gos slowr an slowr, thus th acclration is positiv. In th scon phas, th robot is moving towars th nvironmnt so its vlocity is positiv. Sinc it is moving fastr an fastr, th acclration is also positiv. If th switching critria ar ma accoring to (39, th valu of th mass is givn in th following tabl. Tabl 5 Signs of rlvant magnitus an valu of th mass for th two phass of a rboun. agnitu First phas Scon phas Acclration > > Vlocity < > & && < > mass m min m ma Thrfor th mass is m min whil th robot is rbouning an m ma whil it is rturning back to th surfac. Th fact th mass is low whil th robot is bouncing back mans that, for a givn initial vlocity, it will stop fastr sinc it has lss inrtia. Th istanc it will covr will b shortr. Whn th robot has for th surfac again, its mass will b highr. Thus, th acclration will b lovr an it will rach lss vlocity whil travling th sam istanc. Thrfor, th vlocity of th impact will b smallr than th on th robot ha in th instant of contact loss. Th following simulations confirm th prvious conclusion. It will b assum that th robot lav th nvironmnt with th vlocity of -.1 m/s. It will also b assum that th travll istanc in both irctions is qual. Vlocity [m/s] Tim [miliscons] Figur 14. Th chang of th vlocity whn switching m ma (m min =1 kg in all th cass. Th full lin rprsnts th cas whn m ma =1 kg (no switching, th lin mark with m ma = kg, th on with iamons m ma =5 kg an th on with circls m ma =1 kg. Th two phass may b apprciat in th iagram: th first on lasts until th vlocity rachs zro. Th scon phas ns whn th contact is achiv again. Sinc th amping is rlativly small compar to th mass an th fforwar, th acclration is practically constant uring ach phas. For this rason th chang of vlocity sms linar an thus looks lik a straight lin in th figur. It shoul b mphasiz that uring th first phas th mass has th valu of m min accoring to tabl 5 (In this simulation it corrspons to 1 milliscons. In th figur this lin is th sam for all four valus an is lft as a full lin, without any aitional lmnts. Finally, it may b apprciat in th figur that incrasing m ma th impact vlocity is cras. This sms a logical rsult, bcaus th acclration will b smallr for a gratr inrtia. Thus, th vlocity rach for th sam travll istanc will also b smallr. In th nt figur ar rprsnt th simulations whn switching m min. Vlocity [m/s] Tim [miliscons] Figur 15. Th chang of th vlocity whn switching m min (m ma =1 kg in all th cass. Th full lin rprsnts th cas whn m min =1 kg (no switching, th lin mark with m min =5 kg, th on with iamons m min = kg an th on with circls m min =1 kg. It may b conclu that crasing m min rucs impact vlocity. It also maks shortr th tim from contact loss to th nw impact.

16 4.. Th ffct of th switching of th fforwar Switching th fforwar accoring to (56 mans two things: 1. It will b high whil th robot is bouncing back. Thus th robot will b stopp fastr limiting th travl istanc.. Th fforwar will b low whn th robot is haing for th surfac. Thus th robot will acclrat slowr an rach a lowr vlocity in th instant of th impact. For this rasons th switching th fforwar softns th impact: at first th travl istanc is lowr, an scon, th acclration whn it rturns back to th nvironmnt is cras. Both facts ar favorabl for rucing th impact vlocity an thus for softning th impact. Following will b rprsnt som simulation rsults. Vlocity [m/s] Tim [miliscons] Figur 16. Th chang of th vlocity whn switching ffmin(ff ma =1 N in all th cass. Th full lin rprsnts th cas whn ff min =1 N (no switching, th lin mark with ff min =5 N, th on with iamons ff min = N an th on with circls ff min =1 N. Vlocity [m/s] Tim [miliscons] Figur 17. Th chang of th vlocity whn switching ff ma (ff min =1 N in all th cass. Th full lin rprsnts th cas whn ff ma =1 N (no switching, th lin mark with ff ma = N, th on with iamons ff ma =5 N an th on with circls ff ma =1 N. It may b apprciat in th figur that crasing ff min, rucs both th impact vlocity an th uration th tim btwn th contact loss an th nt impact. Concluing, th switching th mass an fforwar improvs th impact control in th cas of contact loss. On th othr han, in th cas of loss of contact, stratgis 3 iffrnt than switching of paramtrs may b us. 5. CONCLUSIONS This articl is icat to th analysis of th application of impanc control in orr to obtain vlocity control in fr motion, impact attnuation an forc rfrnc tracking. Th mphasis is put on th impact bcaus it is th most angrous part of th task an possibly th most intrsting from th point of viw of control. At first, a gnral analysis of th impanc control was ma. Th conclusion has bn ma that an impanc controllr is fin by four paramtrs: th fforwar, th mass, th amping an th stiffnss. Also, it has bn monstrat that all thr phass of th task can b controll with th sam impanc controllr. Th fforwar must b st to th valu of th forc rfrnc in orr to rach it. Th valu of th stiffnss must b zro. Contrary to th usual trn in impanc control, th amping is us for vlocity control in th fr motion. If th stiffnss of th nvironmnt is known, th valu of th mass may b comput to mak th systm unramp, avoiing forc ovrshoots an th possibility of bouncing an thus achiving a prfct impact control. If th stiffnss of th nvironmnt is unknown, an improvmnt of th impact control may b ma by mans of switching btwn paramtrs of th mchanical impanc. This is th most important contribution of this articl. Th four paramtrs hav bn analys an it has bn monstrat that th ffct of switching is iffrnt for ach of thm. Th switching critria for th stiffnss cannot b tct unlss th charactristics of th nvironmnt ar known. Thrfor thr is no sns to switch this paramtr. Th switching of th amping os not provi any avantag, sinc th trm associat to it always issipats nrgy. Th switching of th fforwar improvs th prformanc of th systm initially, but aftr a fw prios th systm may go into a sliing rgim. Nvrthlss, th first prios ar th most critical an thus th switching of this paramtr is usful whn it is most important. Th sliing rgims may b prict an avoi. Finally, th mass is possibly th bst-suit paramtr for switching. Th switching critria o not pn on