1625:, originating from the algorithm of Lemek and Dantzig, is the most popular method. Unfortunately, however, numerical experiments show that the pivoting algorithm may fail when handling systems with a large number of unilateral contacts, even using the best optimizations. For NCPs, using a polyhedral approximation can transform the NCPs into a set of LCPs, which can then be solved by the LCP solver. Other approaches beyond these methods, such NCP-functions or cone complementarity problems (CCP) based methods are also employed to solve NCPs.
86:
1612:(N/LCP) formulation and the augmented Lagrangian formulation. With respect to the solution of contact models, the non-smooth method is more tedious, but less costly from the computational viewpoint. A more detailed comparison of solution methods using contact models and non-smooth theory was carried out by Pazouki et al.
945:
When using this method for unilateral constraints between two rigid bodies, the
Signorini condition alone is not enough to model the impact process, so impact laws, which give the information about the states before and after the impact, are also required. For example, when the Newton restitution law
93:
In this method, normal forces generated by the unilateral constraints are modelled according to the local material properties of bodies. In particular, contact force models are derived from continuum mechanics, and expressed as functions of the gap and the impact velocity of bodies. As an example, an
1603:
If the unilateral constraints are modelled by the continuum mechanics based contact models, the contact forces can be computed directly through an explicit mathematical formula, that depends on the contact model of choice. If instead the non-smooth theory based method is employed, there are two main
1620:
Following this approach, the solution of dynamics equations with unilateral constraints is transformed into the solution of N/LCPs. In particular, for frictionless unilateral constraints or unilateral constraints with planar friction, the problem is transformed into LCPs, while for frictional
1353:
1564:
1439:
218:
denotes the contact force generated by the unilateral constraints, as shown in the figure below. Moreover, in terms of the concept of proximal point of convex theory, the
Signorini condition can be equivalently expressed as:
937:
721:
1704:
292:
1935:
Alves, Janete; Peixinho, Nuno; da Silva, Miguel
Tavares; Flores, Paulo; Lankarani, Hamid M. (March 2015). "A comparative study of the viscoelastic constitutive models for frictionless contact interfaces in solids".
518:
644:
When implementing non-smooth theory based methods, the velocity
Signorini condition or the acceleration Signorini condition are actually employed in most cases. The velocity Signorini condition is expressed as:
173:
98:
is shown in the figure on the right. In such model, the contact is explained by the local deformation of bodies. More contact models can be found in some review scientific works or in the article dedicated to
1206:
810:
1017:
51:, imperfect joints, or rocket landings. In these applications, the unilateral constraints result in impacts happening, therefore requiring suitable methods to deal with such constraints.
372:
1463:
denotes the kinematic friction coefficient. Similarly to the normal contact force, the formulation above can be equivalently expressed in terms of the notion of proximal point as:
862:
1871:
Machado, Margarida; Moreira, Pedro; Flores, Paulo; Lankarani, Hamid M. (July 2012). "Compliant contact force models in multibody dynamics: Evolution of the Hertz contact theory".
2585:
Demyanov, V.F., Stavroulakis, G.E., Polyakova, L.N., Panagiotopoulos, P.D. "Quasidifferentiability and
Nonsmooth Modelling in Mechanics, Engineering and Economics" Springer 1996
1199:
1121:
1051:
758:
631:
1170:
1092:
605:
550:
1593:
576:
321:
1061:
For frictional unilateral constraints, the normal contact forces are modelled by one of the methods above, while the friction forces are commonly described by means of
1141:
216:
2631:
Potra F.A., Anitescu M., Gavrea B. and
Trinkle J. A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contacts, joints and friction.
1468:
1461:
412:
392:
196:
1362:
869:
650:
637:
2459:"A comparison of numerical methods for solving multibody dynamics problems with frictional contact modeled via differential variational inequalities"
121:
for non-penetration, and impact laws are used to define the impact process. The
Signorini condition can be expressed as the complementarity problem:
1636:
760:
denotes the relative normal velocity after impact. The velocity
Signorini condition should be understood together with the previous conditions
224:
419:
1201:
is no more than the maximum of the static friction force. This relationship can be summarised using the maximum dissipation principle, as
126:
2638:
Stewart D.E. and
Trinkle J.C. An Implicit Time-Stepping Scheme for Rigid Body Dynamics with Inelastic Collisions and Coulomb Friction.
63:, including methods using Hertz's models, penalty methods, and some regularization force models, while the second kind is based on the
1348:{\displaystyle \lambda _{\rm {T}}\in D(\mu \lambda )~~~~~~\forall S\in D(\mu \lambda )~~~~~~(S-\lambda _{\rm {T}})U_{\rm {T}}\geq 0,}
1605:
1820:
32:
763:
523:
Both the expressions above represent the dynamic behaviour of unilateral constraints: on the one hand, when the normal distance
2202:
Pazouki, Arman; Kwarta, Michał; Williams, Kyle; Likos, William; Serban, Radu; Jayakumar, Paramsothy; Negrut, Dan (2017-10-13).
2118:"A comparative study between the augmented Lagrangian method and the complementarity approach for modeling the contact problem"
1710:. A comparative study between LCP formulations and the augmented Lagrangian formulation was carried out by Mashayekhi et al.
1609:
2303:"Numerical method for dynamics of multi-body systems with two-dimensional Coulomb dry friction and nonholonomic constraints"
953:
1633:
Different from the N/LCP formulations, the augmented
Lagrangian formulation uses the proximal functions described above,
2572:
Acary V., Brogliato B. Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics.
947:
2599:
Glocker Ch. and Studer C. Formulation and preparation for Numerical Evaluation of Linear Complementarity Systems.
1757:
326:
68:
2654:
Numerics of Unilateral Contacts and Friction -- Modeling and Numerical Time Integration in Non-Smooth Dynamics
1740:
1707:
815:
35:
which prevents penetration between two rigid/flexible bodies. Constraints of this kind are omnipresent in
1175:
1097:
1821:"A parametric study on the dynamic response of planar multibody systems with multiple clearance joints"
59:
There are mainly two kinds of methods to model the unilateral constraints. The first kind is based on
2656:, Lecture Notes in Applied and Computational Mechanics, Volume 47, Springer, Berlin, Heidelberg, 2009
2470:
2215:
2176:
2165:"A matrix-free cone complementarity approach for solving large-scale, nonsmooth, rigid body dynamics"
2087:
2000:
1746:
1022:
729:
36:
610:
60:
1743: – Numerical methods for computing the motion and effect of a large number of small particles
2669:
2330:
2283:
2164:
2145:
2075:
2051:
2016:
1853:
1801:
1752:
1719:
1146:
1068:
581:
526:
2350:"Nonlinear equation approach for inequality elastostatics: a two-dimensional BEM implementation"
1572:
1559:{\displaystyle \lambda _{\rm {T}}={\rm {proj}}_{D(\mu \lambda )}(\lambda _{T}-\rho U_{\rm {T}})}
555:
300:
2500:"Posing Multibody Dynamics With Friction and Contact as a Differential Complementarity Problem"
2419:
2260:
1778:
552:
is above zero, the contact is open, which means that there is no contact force between bodies,
117:
In non-smooth method, unilateral interactions between bodies are fundamentally modelled by the
2521:
2439:
2400:
2322:
2241:
2233:
2137:
2036:
Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics
1734:
1622:
1126:
636:
201:
118:
100:
95:
80:
20:
2511:
2478:
2431:
2392:
2361:
2314:
2275:
2223:
2184:
2129:
2095:
2043:
2008:
1985:
1953:
1945:
1917:
1888:
1880:
1843:
1835:
1793:
1728:
112:
64:
48:
44:
640:
Figure 2: a) unilateral contact, b) the Signorini graph, c) continuum mechanics based model
2204:"Compliant contact versus rigid contact: A comparison in the context of granular dynamics"
1908:
Gilardi, G.; Sharf, I. (October 2002). "Literature survey of contact dynamics modelling".
1446:
2624:
Pfeiffer F., Foerg M. and Ulbrich H. Numerical aspects of non-smooth multibody dynamics.
1949:
1884:
2474:
2219:
2180:
2091:
2004:
1172:
is equal to zero, namely when the two bodies are relatively steady, the friction force
397:
377:
181:
85:
2365:
2012:
1921:
1434:{\displaystyle D(\mu \lambda )=\{\forall x|-\mu \lambda \leq \|x\|\leq \mu \lambda \}}
2663:
2334:
2302:
2149:
2117:
2055:
2020:
1857:
2287:
1805:
1065:. Coulomb's friction law can be expressed as follows: when the tangential velocity
40:
932:{\displaystyle {\ddot {g}}\geq 0,\quad \lambda \geq 0,\quad {\ddot {g}}\lambda =0}
2553: – Open source scientific software for modeling non-smooth dynamical systems
2546:
Open-source codes and non-commercial packages using the non-smooth based method:
1722: – Tool to study dynamic behavior of interconnected rigid or flexible bodies
1621:
unilateral constraints, the problem is transformed into NCPs. To solve LCPs, the
1094:
is not equal to zero, namely when the two bodies are sliding, the friction force
2261:"An iterative approach for cone complementarity problems for nonsmooth dynamics"
1779:"An iterative approach for cone complementarity problems for nonsmooth dynamics"
716:{\displaystyle U_{\rm {N}}^{+}\geq 0,\quad \lambda \geq 0,\quad U^{+}\lambda =0}
2381:"Equivalence of the Complementarity Problem to a System of Nonlinear Equations"
2228:
2203:
2483:
2458:
2457:
Melanz, Daniel; Fang, Luning; Jayakumar, Paramsothy; Negrut, Dan (June 2017).
2435:
2318:
2279:
2188:
2133:
2099:
1839:
1797:
2525:
2443:
2404:
2326:
2237:
2141:
2047:
2380:
2349:
942:
where the overdots denote the second-order derivative with respect to time.
812:. The acceleration Signorini condition is considered under closed contact (
2556:
2245:
1706:. Together with dynamics equations, this formulation is solved by means of
2034:
Pfeiffer, Friedrich (14 March 2012). "On non-smooth multibody dynamics".
1699:{\displaystyle \lambda ={\rm {proj}}_{\mathbb {R} ^{+}}(\lambda -\rho g)}
1062:
287:{\displaystyle \lambda ={\rm {proj}}_{\mathbb {R} ^{+}}(\lambda -\rho g)}
513:{\displaystyle {\rm {proj}}_{\bf {C}}(x)={\rm {argmin}}_{y\in C}\|y-x\|}
2550:
2516:
2499:
1958:
1893:
1848:
2396:
2647:
Augmented time-stepping integration of non-smooth dynamical systems
2559:, an open source multi-physics simulation engine, see also project
2074:
Pfeiffer, Friedrich; Foerg, Martin; Ulbrich, Heinz (October 2006).
16:
Mechanical constraint which prevents penetration between two bodies
635:
168:{\displaystyle g\geq 0,\quad \lambda \geq 0,\quad \lambda \perp g}
84:
2619:
Non-smooth Mechanics and Applications, CISM Courses and Lectures
1737: – Study of the deformation of solids that touch each other
2615:
Unilateral Contact and Dry Friction in Finite Freedom Dynamics,
1604:
formulations for the solution of the Signorini conditions: the
2116:
Jalali Mashayekhi, Mohammad; Kövecses, József (August 2017).
2498:
Negrut, Dan; Serban, Radu; Tasora, Alessandro (2018-01-01).
2560:
805:{\displaystyle g\geq 0,\;\lambda \geq 0,\;\lambda \perp g}
2649:, PhD Thesis ETH Zurich, ETH E-Collection, to appear 2008
2590:
Dynamik von Starrkoerpersystemen mit Reibung und Stoessen
2259:
Anitescu, Mihai; Tasora, Alessandro (26 November 2008).
1777:
Anitescu, Mihai; Tasora, Alessandro (26 November 2008).
1724:
Pages displaying short descriptions of redirect targets
1012:{\displaystyle e=-{U_{\rm {N}}^{+}}/{U_{\rm {N}}^{-}}}
607:
is equal to zero, the contact is closed, resulting in
67:, which models the system with unilateral contacts as
2608:
Computer Methods in Applied mechanics and Engineering
2463:
Computer Methods in Applied Mechanics and Engineering
2169:
Computer Methods in Applied Mechanics and Engineering
2080:
Computer Methods in Applied Mechanics and Engineering
1993:
Computer Methods in Applied Mechanics and Engineering
1639:
1575:
1471:
1449:
1365:
1209:
1178:
1149:
1129:
1100:
1071:
1025:
956:
872:
818:
766:
732:
653:
613:
584:
558:
529:
422:
400:
380:
329:
303:
227:
204:
184:
129:
2301:
Xu, Ziyao; Wang, Qi; Wang, Qingyun (December 2017).
2076:"Numerical aspects of non-smooth multibody dynamics"
1053:
denotes the relative normal velocity before impact.
2111:
2109:
1698:
1587:
1558:
1455:
1433:
1347:
1193:
1164:
1135:
1115:
1086:
1045:
1011:
931:
856:
804:
752:
715:
625:
599:
570:
544:
512:
406:
386:
366:
315:
286:
210:
190:
167:
2606:Jean M. The non-smooth contact dynamics method.
2594:VDI Fortschrittsberichte Mechanik/Bruchmechanik.
2504:Journal of Computational and Nonlinear Dynamics
578:; on the other hand, when the normal distance
2580:Communications and Control Engineering Series
8:
2574:Springer Verlag, LNACM 35, Heidelberg, 2008.
1428:
1416:
1410:
1384:
1123:is proportional to the normal contact force
507:
495:
198:denotes the distance between two bodies and
2420:"A special newton-type optimization method"
2268:Computational Optimization and Applications
1979:
1977:
1975:
1973:
1971:
1969:
1786:Computational Optimization and Applications
113:Contact dynamics § Non-smooth approach
792:
779:
2515:
2482:
2227:
2163:Tasora, A.; Anitescu, M. (January 2011).
1957:
1892:
1847:
1670:
1666:
1665:
1663:
1648:
1647:
1638:
1574:
1546:
1545:
1529:
1504:
1489:
1488:
1477:
1476:
1470:
1448:
1393:
1364:
1329:
1328:
1314:
1313:
1215:
1214:
1208:
1184:
1183:
1177:
1155:
1154:
1148:
1128:
1106:
1105:
1099:
1077:
1076:
1070:
1037:
1031:
1030:
1024:
1002:
996:
995:
990:
985:
978:
972:
971:
966:
955:
909:
908:
874:
873:
871:
842:
836:
835:
817:
765:
744:
738:
737:
731:
698:
665:
659:
658:
652:
612:
590:
589:
583:
557:
535:
534:
528:
483:
462:
461:
441:
440:
425:
424:
421:
399:
379:
374:represents the proximal point in the set
367:{\displaystyle {\rm {proj}}_{\bf {C}}(x)}
348:
347:
332:
331:
328:
302:
258:
254:
253:
251:
236:
235:
226:
203:
183:
128:
1986:"The non-smooth contact dynamics method"
1143:; when instead the tangential velocity
2582:Springer-Verlag, London, 1999 (2dn Ed.)
1769:
55:Modelling of the unilateral constraints
2348:Stavroulakis, G.E.; Antes, H. (2000).
1749: – Modeling approach in mechanics
39:applications, such as granular flows,
857:{\displaystyle g=0,U_{\rm {N}}^{+}=0}
7:
2069:
2067:
2065:
1950:10.1016/j.mechmachtheory.2014.11.020
1885:10.1016/j.mechmachtheory.2012.02.010
323:denotes an auxiliary parameter, and
2385:SIAM Journal on Applied Mathematics
1731: – Motion of multibody systems
2640:Int. J. Numer. Methods Engineering
2578:Brogliato B. Nonsmooth Mechanics.
1658:
1655:
1652:
1649:
1547:
1499:
1496:
1493:
1490:
1478:
1443:represents the friction cone, and
1387:
1330:
1315:
1258:
1216:
1194:{\displaystyle \lambda _{\rm {T}}}
1185:
1156:
1116:{\displaystyle \lambda _{\rm {T}}}
1107:
1078:
1032:
997:
973:
837:
739:
660:
591:
536:
478:
475:
472:
469:
466:
463:
435:
432:
429:
426:
342:
339:
336:
333:
246:
243:
240:
237:
14:
2626:Comput. Methods Appl. Mech. Engrg
2307:Applied Mathematics and Mechanics
1057:Frictional unilateral constraints
2379:Mangasarian, O. L. (July 1976).
1629:Augmented Lagrangian formulation
1595:denotes an auxiliary parameter.
442:
349:
1046:{\displaystyle U_{\rm {N}}^{-}}
907:
894:
753:{\displaystyle U_{\rm {N}}^{+}}
693:
680:
155:
142:
1819:Flores, Paulo (7 March 2010).
1693:
1678:
1610:linear complementarity problem
1553:
1522:
1517:
1508:
1394:
1378:
1369:
1321:
1300:
1279:
1270:
1237:
1228:
626:{\displaystyle \lambda \geq 0}
454:
448:
361:
355:
281:
266:
1:
2366:10.1016/S0045-7949(99)00111-X
2013:10.1016/S0045-7825(98)00383-1
1922:10.1016/S0094-114X(02)00045-9
37:non-smooth multibody dynamics
2596:VDI Verlag, Düsseldorf, 1995
2418:Fischer, A. (January 1992).
1938:Mechanism and Machine Theory
1910:Mechanism and Machine Theory
1873:Mechanism and Machine Theory
94:illustration of the classic
1165:{\displaystyle U_{\rm {T}}}
1087:{\displaystyle U_{\rm {T}}}
600:{\displaystyle g_{\rm {N}}}
545:{\displaystyle g_{\rm {N}}}
107:Non-smooth contact dynamics
65:non-smooth contact dynamics
2686:
2633:Int. J. Numer. Meth. Engng
2628:195(50-51):6891-6908, 2006
2229:10.1103/PhysRevE.96.042905
1588:{\displaystyle \rho >0}
948:coefficient of restitution
571:{\displaystyle \lambda =0}
316:{\displaystyle \rho >0}
110:
78:
2601:Multibody System Dynamics
2484:10.1016/j.cma.2017.03.010
2436:10.1080/02331939208843795
2319:10.1007/s10483-017-2285-8
2280:10.1007/s10589-008-9223-4
2189:10.1016/j.cma.2010.06.030
2134:10.1007/s11044-016-9510-2
2122:Multibody System Dynamics
2100:10.1016/j.cma.2005.08.012
1840:10.1007/s11071-010-9676-8
1798:10.1007/s10589-008-9223-4
2354:Computers and Structures
2048:10.1177/1464419312438487
1758:Variational inequalities
1136:{\displaystyle \lambda }
211:{\displaystyle \lambda }
69:variational inequalities
1741:discrete element method
1708:root-finding algorithms
75:Smooth contact dynamics
61:smooth contact dynamics
31:, denotes a mechanical
2642:39(15):2673-2691, 1996
2621:. Springer, Wien, 1988
2610:177(3-4):235-257, 1999
1984:Jean, M. (July 1999).
1700:
1589:
1560:
1457:
1435:
1349:
1195:
1166:
1137:
1117:
1088:
1063:Coulomb's friction law
1047:
1013:
933:
858:
806:
754:
717:
641:
627:
601:
572:
546:
514:
408:
388:
368:
317:
288:
212:
192:
169:
90:
2635:66(7):1079-1124, 2006
1701:
1590:
1561:
1458:
1436:
1350:
1196:
1167:
1138:
1118:
1089:
1048:
1014:
934:
859:
807:
755:
718:
639:
628:
602:
573:
547:
515:
409:
389:
369:
318:
289:
213:
193:
170:
88:
29:unilateral constraint
2542:Open-source software
2086:(50–51): 6891–6908.
1747:Non-smooth mechanics
1637:
1573:
1469:
1456:{\displaystyle \mu }
1447:
1363:
1207:
1176:
1147:
1127:
1098:
1069:
1023:
954:
950:will be defined as:
870:
816:
764:
730:
651:
611:
582:
556:
527:
420:
398:
378:
327:
301:
225:
202:
182:
127:
2603:13(4):447-463, 2005
2592:, volume 18/182 of
2475:2017CMAME.320..668M
2220:2017PhRvE..96d2905P
2181:2011CMAME.200..439T
2092:2006CMAME.195.6891P
2005:1999CMAME.177..235J
1599:Solution techniques
1042:
1007:
983:
847:
749:
670:
119:Signorini condition
96:Hertz contact model
89:Hertz contact model
2567:Books and articles
1828:Nonlinear Dynamics
1753:Collision response
1720:Multibody dynamics
1696:
1623:pivoting algorithm
1616:N/LCP formulations
1585:
1556:
1453:
1431:
1345:
1191:
1162:
1133:
1113:
1084:
1043:
1026:
1009:
991:
967:
929:
854:
831:
802:
750:
733:
713:
654:
642:
623:
597:
568:
542:
510:
404:
384:
364:
313:
284:
208:
188:
165:
91:
25:unilateral contact
2517:10.1115/1.4037415
2313:(12): 1733–1752.
2208:Physical Review E
1916:(10): 1213–1239.
1735:contact mechanics
1299:
1296:
1293:
1290:
1287:
1284:
1257:
1254:
1251:
1248:
1245:
1242:
917:
882:
407:{\displaystyle x}
387:{\displaystyle C}
191:{\displaystyle g}
101:contact mechanics
81:Contact mechanics
21:contact mechanics
2677:
2530:
2529:
2519:
2495:
2489:
2488:
2486:
2454:
2448:
2447:
2430:(3–4): 269–284.
2415:
2409:
2408:
2376:
2370:
2369:
2345:
2339:
2338:
2298:
2292:
2291:
2265:
2256:
2250:
2249:
2231:
2199:
2193:
2192:
2175:(5–8): 439–453.
2160:
2154:
2153:
2113:
2104:
2103:
2071:
2060:
2059:
2031:
2025:
2024:
1999:(3–4): 235–257.
1990:
1981:
1964:
1963:
1961:
1932:
1926:
1925:
1905:
1899:
1898:
1896:
1868:
1862:
1861:
1851:
1825:
1816:
1810:
1809:
1783:
1774:
1729:contact dynamics
1725:
1705:
1703:
1702:
1697:
1677:
1676:
1675:
1674:
1669:
1662:
1661:
1594:
1592:
1591:
1586:
1565:
1563:
1562:
1557:
1552:
1551:
1550:
1534:
1533:
1521:
1520:
1503:
1502:
1483:
1482:
1481:
1462:
1460:
1459:
1454:
1440:
1438:
1437:
1432:
1397:
1354:
1352:
1351:
1346:
1335:
1334:
1333:
1320:
1319:
1318:
1297:
1294:
1291:
1288:
1285:
1282:
1255:
1252:
1249:
1246:
1243:
1240:
1221:
1220:
1219:
1200:
1198:
1197:
1192:
1190:
1189:
1188:
1171:
1169:
1168:
1163:
1161:
1160:
1159:
1142:
1140:
1139:
1134:
1122:
1120:
1119:
1114:
1112:
1111:
1110:
1093:
1091:
1090:
1085:
1083:
1082:
1081:
1052:
1050:
1049:
1044:
1041:
1036:
1035:
1018:
1016:
1015:
1010:
1008:
1006:
1001:
1000:
989:
984:
982:
977:
976:
938:
936:
935:
930:
919:
918:
910:
884:
883:
875:
863:
861:
860:
855:
846:
841:
840:
811:
809:
808:
803:
759:
757:
756:
751:
748:
743:
742:
722:
720:
719:
714:
703:
702:
669:
664:
663:
632:
630:
629:
624:
606:
604:
603:
598:
596:
595:
594:
577:
575:
574:
569:
551:
549:
548:
543:
541:
540:
539:
519:
517:
516:
511:
494:
493:
482:
481:
447:
446:
445:
439:
438:
413:
411:
410:
405:
394:to the variable
393:
391:
390:
385:
373:
371:
370:
365:
354:
353:
352:
346:
345:
322:
320:
319:
314:
293:
291:
290:
285:
265:
264:
263:
262:
257:
250:
249:
217:
215:
214:
209:
197:
195:
194:
189:
174:
172:
171:
166:
49:particle damping
45:vehicle dynamics
2685:
2684:
2680:
2679:
2678:
2676:
2675:
2674:
2660:
2659:
2569:
2544:
2539:
2537:Further reading
2534:
2533:
2497:
2496:
2492:
2456:
2455:
2451:
2417:
2416:
2412:
2397:10.1137/0131009
2378:
2377:
2373:
2347:
2346:
2342:
2300:
2299:
2295:
2263:
2258:
2257:
2253:
2201:
2200:
2196:
2162:
2161:
2157:
2115:
2114:
2107:
2073:
2072:
2063:
2033:
2032:
2028:
1988:
1983:
1982:
1967:
1934:
1933:
1929:
1907:
1906:
1902:
1870:
1869:
1865:
1823:
1818:
1817:
1813:
1781:
1776:
1775:
1771:
1766:
1723:
1716:
1664:
1646:
1635:
1634:
1631:
1618:
1601:
1571:
1570:
1541:
1525:
1487:
1472:
1467:
1466:
1445:
1444:
1361:
1360:
1324:
1309:
1210:
1205:
1204:
1179:
1174:
1173:
1150:
1145:
1144:
1125:
1124:
1101:
1096:
1095:
1072:
1067:
1066:
1059:
1021:
1020:
952:
951:
946:is employed, a
868:
867:
814:
813:
762:
761:
728:
727:
694:
649:
648:
609:
608:
585:
580:
579:
554:
553:
530:
525:
524:
460:
423:
418:
417:
396:
395:
376:
375:
330:
325:
324:
299:
298:
252:
234:
223:
222:
200:
199:
180:
179:
125:
124:
115:
109:
83:
77:
57:
17:
12:
11:
5:
2683:
2681:
2673:
2672:
2662:
2661:
2658:
2657:
2650:
2643:
2636:
2629:
2622:
2617:volume 302 of
2611:
2604:
2597:
2586:
2583:
2576:
2568:
2565:
2564:
2563:
2554:
2543:
2540:
2538:
2535:
2532:
2531:
2490:
2449:
2410:
2371:
2360:(6): 631–646.
2340:
2293:
2274:(2): 207–235.
2251:
2194:
2155:
2128:(4): 327–345.
2105:
2061:
2042:(2): 147–177.
2026:
1965:
1927:
1900:
1863:
1834:(4): 633–653.
1811:
1792:(2): 207–235.
1768:
1767:
1765:
1762:
1761:
1760:
1755:
1750:
1744:
1738:
1732:
1726:
1715:
1712:
1695:
1692:
1689:
1686:
1683:
1680:
1673:
1668:
1660:
1657:
1654:
1651:
1645:
1642:
1630:
1627:
1617:
1614:
1600:
1597:
1584:
1581:
1578:
1555:
1549:
1544:
1540:
1537:
1532:
1528:
1524:
1519:
1516:
1513:
1510:
1507:
1501:
1498:
1495:
1492:
1486:
1480:
1475:
1452:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1396:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1344:
1341:
1338:
1332:
1327:
1323:
1317:
1312:
1308:
1305:
1302:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1239:
1236:
1233:
1230:
1227:
1224:
1218:
1213:
1187:
1182:
1158:
1153:
1132:
1109:
1104:
1080:
1075:
1058:
1055:
1040:
1034:
1029:
1005:
999:
994:
988:
981:
975:
970:
965:
962:
959:
928:
925:
922:
916:
913:
906:
903:
900:
897:
893:
890:
887:
881:
878:
853:
850:
845:
839:
834:
830:
827:
824:
821:
801:
798:
795:
791:
788:
785:
782:
778:
775:
772:
769:
747:
741:
736:
712:
709:
706:
701:
697:
692:
689:
686:
683:
679:
676:
673:
668:
662:
657:
622:
619:
616:
593:
588:
567:
564:
561:
538:
533:
509:
506:
503:
500:
497:
492:
489:
486:
480:
477:
474:
471:
468:
465:
459:
456:
453:
450:
444:
437:
434:
431:
428:
414:, defined as:
403:
383:
363:
360:
357:
351:
344:
341:
338:
335:
312:
309:
306:
283:
280:
277:
274:
271:
268:
261:
256:
248:
245:
242:
239:
233:
230:
207:
187:
164:
161:
158:
154:
151:
148:
145:
141:
138:
135:
132:
108:
105:
76:
73:
56:
53:
27:, also called
15:
13:
10:
9:
6:
4:
3:
2:
2682:
2671:
2668:
2667:
2665:
2655:
2651:
2648:
2644:
2641:
2637:
2634:
2630:
2627:
2623:
2620:
2616:
2612:
2609:
2605:
2602:
2598:
2595:
2591:
2588:Glocker, Ch.
2587:
2584:
2581:
2577:
2575:
2571:
2570:
2566:
2562:
2558:
2555:
2552:
2549:
2548:
2547:
2541:
2536:
2527:
2523:
2518:
2513:
2510:(1): 014503.
2509:
2505:
2501:
2494:
2491:
2485:
2480:
2476:
2472:
2468:
2464:
2460:
2453:
2450:
2445:
2441:
2437:
2433:
2429:
2425:
2421:
2414:
2411:
2406:
2402:
2398:
2394:
2390:
2386:
2382:
2375:
2372:
2367:
2363:
2359:
2355:
2351:
2344:
2341:
2336:
2332:
2328:
2324:
2320:
2316:
2312:
2308:
2304:
2297:
2294:
2289:
2285:
2281:
2277:
2273:
2269:
2262:
2255:
2252:
2247:
2243:
2239:
2235:
2230:
2225:
2221:
2217:
2214:(4): 042905.
2213:
2209:
2205:
2198:
2195:
2190:
2186:
2182:
2178:
2174:
2170:
2166:
2159:
2156:
2151:
2147:
2143:
2139:
2135:
2131:
2127:
2123:
2119:
2112:
2110:
2106:
2101:
2097:
2093:
2089:
2085:
2081:
2077:
2070:
2068:
2066:
2062:
2057:
2053:
2049:
2045:
2041:
2037:
2030:
2027:
2022:
2018:
2014:
2010:
2006:
2002:
1998:
1994:
1987:
1980:
1978:
1976:
1974:
1972:
1970:
1966:
1960:
1955:
1951:
1947:
1943:
1939:
1931:
1928:
1923:
1919:
1915:
1911:
1904:
1901:
1895:
1890:
1886:
1882:
1878:
1874:
1867:
1864:
1859:
1855:
1850:
1845:
1841:
1837:
1833:
1829:
1822:
1815:
1812:
1807:
1803:
1799:
1795:
1791:
1787:
1780:
1773:
1770:
1763:
1759:
1756:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1721:
1718:
1717:
1713:
1711:
1709:
1690:
1687:
1684:
1681:
1671:
1643:
1640:
1628:
1626:
1624:
1615:
1613:
1611:
1607:
1598:
1596:
1582:
1579:
1576:
1567:
1542:
1538:
1535:
1530:
1526:
1514:
1511:
1505:
1484:
1473:
1464:
1450:
1441:
1425:
1422:
1419:
1413:
1407:
1404:
1401:
1398:
1390:
1381:
1375:
1372:
1366:
1358:
1355:
1342:
1339:
1336:
1325:
1310:
1306:
1303:
1276:
1273:
1267:
1264:
1261:
1234:
1231:
1225:
1222:
1211:
1202:
1180:
1151:
1130:
1102:
1073:
1064:
1056:
1054:
1038:
1027:
1003:
992:
986:
979:
968:
963:
960:
957:
949:
943:
940:
926:
923:
920:
914:
911:
904:
901:
898:
895:
891:
888:
885:
879:
876:
865:
851:
848:
843:
832:
828:
825:
822:
819:
799:
796:
793:
789:
786:
783:
780:
776:
773:
770:
767:
745:
734:
724:
710:
707:
704:
699:
695:
690:
687:
684:
681:
677:
674:
671:
666:
655:
646:
638:
634:
620:
617:
614:
586:
565:
562:
559:
531:
521:
504:
501:
498:
490:
487:
484:
457:
451:
415:
401:
381:
358:
310:
307:
304:
295:
278:
275:
272:
269:
259:
231:
228:
220:
205:
185:
176:
162:
159:
156:
152:
149:
146:
143:
139:
136:
133:
130:
122:
120:
114:
106:
104:
102:
97:
87:
82:
74:
72:
70:
66:
62:
54:
52:
50:
46:
42:
38:
34:
30:
26:
22:
2653:
2646:
2639:
2632:
2625:
2618:
2614:
2613:Moreau J.J.
2607:
2600:
2593:
2589:
2579:
2573:
2545:
2507:
2503:
2493:
2466:
2462:
2452:
2427:
2424:Optimization
2423:
2413:
2391:(1): 89–92.
2388:
2384:
2374:
2357:
2353:
2343:
2310:
2306:
2296:
2271:
2267:
2254:
2211:
2207:
2197:
2172:
2168:
2158:
2125:
2121:
2083:
2079:
2039:
2035:
2029:
1996:
1992:
1941:
1937:
1930:
1913:
1909:
1903:
1876:
1872:
1866:
1831:
1827:
1814:
1789:
1785:
1772:
1632:
1619:
1602:
1568:
1465:
1442:
1359:
1356:
1203:
1060:
944:
941:
866:
725:
647:
643:
522:
416:
296:
221:
177:
123:
116:
92:
58:
41:legged robot
28:
24:
18:
2469:: 668–693.
1944:: 172–188.
23:, the term
2652:Studer C.
2645:Studer C.
1959:1822/31823
1894:1822/19623
1879:: 99–121.
1849:1822/23520
1764:References
111:See also:
79:See also:
33:constraint
2670:Mechanics
2526:1555-1415
2444:0233-1934
2405:0036-1399
2335:125402414
2327:0253-4827
2238:2470-0045
2150:123789094
2142:1384-5640
2056:123605632
2021:120827881
1688:ρ
1685:−
1682:λ
1641:λ
1606:nonlinear
1577:ρ
1539:ρ
1536:−
1527:λ
1515:λ
1512:μ
1474:λ
1451:μ
1426:λ
1423:μ
1420:≤
1417:‖
1411:‖
1408:≤
1405:λ
1402:μ
1399:−
1388:∀
1376:λ
1373:μ
1337:≥
1311:λ
1307:−
1277:λ
1274:μ
1265:∈
1259:∀
1235:λ
1232:μ
1223:∈
1212:λ
1181:λ
1131:λ
1103:λ
1039:−
1019:, where
1004:−
964:−
921:λ
915:¨
899:≥
896:λ
886:≥
880:¨
797:⊥
794:λ
784:≥
781:λ
771:≥
705:λ
685:≥
682:λ
672:≥
618:≥
615:λ
560:λ
508:‖
502:−
496:‖
488:∈
305:ρ
276:ρ
273:−
270:λ
229:λ
206:λ
160:⊥
157:λ
147:≥
144:λ
134:≥
2664:Category
2246:29347540
1858:92980088
1714:See also
2561:website
2551:Siconos
2471:Bibcode
2288:1107494
2216:Bibcode
2177:Bibcode
2088:Bibcode
2001:Bibcode
1806:1107494
864:), as:
2557:Chrono
2524:
2442:
2403:
2333:
2325:
2286:
2244:
2236:
2148:
2140:
2054:
2019:
1856:
1804:
1569:where
1357:where
1298:
1295:
1292:
1289:
1286:
1283:
1256:
1253:
1250:
1247:
1244:
1241:
726:where
297:where
178:where
2331:S2CID
2284:S2CID
2264:(PDF)
2146:S2CID
2052:S2CID
2017:S2CID
1989:(PDF)
1854:S2CID
1824:(PDF)
1802:S2CID
1782:(PDF)
2522:ISSN
2440:ISSN
2401:ISSN
2323:ISSN
2242:PMID
2234:ISSN
2138:ISSN
1580:>
308:>
2512:doi
2479:doi
2467:320
2432:doi
2393:doi
2362:doi
2315:doi
2276:doi
2224:doi
2185:doi
2173:200
2130:doi
2096:doi
2084:195
2044:doi
2040:226
2009:doi
1997:177
1954:hdl
1946:doi
1918:doi
1889:hdl
1881:doi
1844:hdl
1836:doi
1794:doi
19:In
2666::
2520:.
2508:13
2506:.
2502:.
2477:.
2465:.
2461:.
2438:.
2428:24
2426:.
2422:.
2399:.
2389:31
2387:.
2383:.
2358:75
2356:.
2352:.
2329:.
2321:.
2311:38
2309:.
2305:.
2282:.
2272:47
2270:.
2266:.
2240:.
2232:.
2222:.
2212:96
2210:.
2206:.
2183:.
2171:.
2167:.
2144:.
2136:.
2126:40
2124:.
2120:.
2108:^
2094:.
2082:.
2078:.
2064:^
2050:.
2038:.
2015:.
2007:.
1995:.
1991:.
1968:^
1952:.
1942:85
1940:.
1914:37
1912:.
1887:.
1877:53
1875:.
1852:.
1842:.
1832:61
1830:.
1826:.
1800:.
1790:47
1788:.
1784:.
1566:,
939:,
723:,
633:.
520:.
294:,
175:,
103:.
71:.
47:,
43:,
2528:.
2514::
2487:.
2481::
2473::
2446:.
2434::
2407:.
2395::
2368:.
2364::
2337:.
2317::
2290:.
2278::
2248:.
2226::
2218::
2191:.
2187::
2179::
2152:.
2132::
2102:.
2098::
2090::
2058:.
2046::
2023:.
2011::
2003::
1962:.
1956::
1948::
1924:.
1920::
1897:.
1891::
1883::
1860:.
1846::
1838::
1808:.
1796::
1694:)
1691:g
1679:(
1672:+
1667:R
1659:j
1656:o
1653:r
1650:p
1644:=
1608:/
1583:0
1554:)
1548:T
1543:U
1531:T
1523:(
1518:)
1509:(
1506:D
1500:j
1497:o
1494:r
1491:p
1485:=
1479:T
1429:}
1414:x
1395:|
1391:x
1385:{
1382:=
1379:)
1370:(
1367:D
1343:,
1340:0
1331:T
1326:U
1322:)
1316:T
1304:S
1301:(
1280:)
1271:(
1268:D
1262:S
1238:)
1229:(
1226:D
1217:T
1186:T
1157:T
1152:U
1108:T
1079:T
1074:U
1033:N
1028:U
998:N
993:U
987:/
980:+
974:N
969:U
961:=
958:e
927:0
924:=
912:g
905:,
902:0
892:,
889:0
877:g
852:0
849:=
844:+
838:N
833:U
829:,
826:0
823:=
820:g
800:g
790:,
787:0
777:,
774:0
768:g
746:+
740:N
735:U
711:0
708:=
700:+
696:U
691:,
688:0
678:,
675:0
667:+
661:N
656:U
621:0
592:N
587:g
566:0
563:=
537:N
532:g
505:x
499:y
491:C
485:y
479:n
476:i
473:m
470:g
467:r
464:a
458:=
455:)
452:x
449:(
443:C
436:j
433:o
430:r
427:p
402:x
382:C
362:)
359:x
356:(
350:C
343:j
340:o
337:r
334:p
311:0
282:)
279:g
267:(
260:+
255:R
247:j
244:o
241:r
238:p
232:=
186:g
163:g
153:,
150:0
140:,
137:0
131:g
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
