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2345:{\displaystyle {\boldsymbol {\mathcal {F}}}={\boldsymbol {K}}_{\mathrm {x} }({\boldsymbol {x}}_{\mathrm {d} }-{\boldsymbol {x}})+{\boldsymbol {D}}_{\mathrm {x} }({\dot {\boldsymbol {x}}}_{\mathrm {d} }-{\dot {\boldsymbol {x}}})+{\hat {\boldsymbol {\Lambda }}}({\boldsymbol {q}}){\ddot {\boldsymbol {x}}}_{\mathrm {d} }+{\hat {\boldsymbol {\mu }}}({\boldsymbol {q}},{\dot {\boldsymbol {q}}})+{\hat {\boldsymbol {\gamma }}}({\boldsymbol {q}})+{\hat {\boldsymbol {\eta }}}({\boldsymbol {q}},{\dot {\boldsymbol {q}}}),}
100:. It assumes the postulate that "no controller can make the manipulator appear to the environment as anything other than a physical system." This rule of thumb can also be stated as: "in the most common case in which the environment is an admittance (e.g. a mass, possibly kinematically constrained) that relation should be an impedance, a function, possibly nonlinear, dynamic, or even discontinuous, specifying the force produced in response to a motion imposed by the environment."
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1038:{\displaystyle {\boldsymbol {\tau }}={\boldsymbol {K}}({\boldsymbol {q}}_{\mathrm {d} }-{\boldsymbol {q}})+{\boldsymbol {D}}({\dot {\boldsymbol {q}}}_{\mathrm {d} }-{\dot {\boldsymbol {q}}})+{\hat {\boldsymbol {M}}}({\boldsymbol {q}}){\ddot {\boldsymbol {q}}}_{\mathrm {d} }+{\hat {\boldsymbol {c}}}({\boldsymbol {q}},{\dot {\boldsymbol {q}}})+{\hat {\boldsymbol {g}}}({\boldsymbol {q}})+{\hat {\boldsymbol {h}}}({\boldsymbol {q}},{\dot {\boldsymbol {q}}}),}
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1852:{\displaystyle {\boldsymbol {\mathcal {F}}}={\boldsymbol {\Lambda }}({\boldsymbol {q}}){\ddot {\boldsymbol {x}}}+{\boldsymbol {\mu }}({\boldsymbol {x}},{\dot {\boldsymbol {x}}})+{\boldsymbol {\gamma }}({\boldsymbol {q}})+{\boldsymbol {\eta }}({\boldsymbol {q}},{\dot {\boldsymbol {q}}})+{\boldsymbol {\mathcal {F}}}_{\mathrm {ext} }}
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kinematics and inertias of the mechanism, we can orient those axes arbitrarily and in various coordinate systems. For example, we might cause a robotic part holder to be very stiff tangentially to a grinding wheel, while being very compliant (controlling force with little concern for position) in the radial axis of the wheel.
1596:
1450:{\displaystyle {\boldsymbol {K}}({\boldsymbol {q}}_{\mathrm {d} }-{\boldsymbol {q}})+{\boldsymbol {D}}({\dot {\boldsymbol {q}}}_{\mathrm {d} }-{\dot {\boldsymbol {q}}})+{\boldsymbol {M}}({\boldsymbol {q}})({\ddot {\boldsymbol {q}}}_{\mathrm {d} }-{\ddot {\boldsymbol {q}}})={\boldsymbol {\tau }}_{\mathrm {ext} }.}
84:
Mechanical impedance is the ratio of force output to motion input. This is analogous to electrical impedance that is the ratio of voltage output to current input (e.g. resistance is voltage divided by current). A "spring constant" defines the force output for a displacement (extension or compression)
108:
Impedance control doesn't simply regulate the force or position of a mechanism. Instead it regulates the relationship between force and position on the one hand, and velocity and acceleration on the other hand, i.e. the impedance of the mechanism. It requires a position (velocity or acceleration) as
576:{\displaystyle {\boldsymbol {\tau }}={\boldsymbol {M}}({\boldsymbol {q}}){\ddot {\boldsymbol {q}}}+{\boldsymbol {c}}({\boldsymbol {q}},{\dot {\boldsymbol {q}}})+{\boldsymbol {g}}({\boldsymbol {q}})+{\boldsymbol {h}}({\boldsymbol {q}},{\dot {\boldsymbol {q}}})+{\boldsymbol {\tau }}_{\mathrm {ext} }}
400:
coordinates) and in three orientations (e.g. roll, pitch, yaw). In theory, an impedance controller can cause the mechanism to exhibit a multi-dimensional mechanical impedance. For example, the mechanism might act very stiff along one axis and very compliant along another. By compensating for the
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Mechanical admittance is the inverse of impedance - it defines the motions that result from a force input. If a mechanism applies a force to the environment, the environment will move, or not move, depending on its properties and the force applied. For example, a marble sitting on a table will
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interacts with its environment and the force position relation is of concern. Examples of such applications include humans interacting with robots, where the force produced by the human relates to how fast the robot should move/stop. Simpler control methods, such as position control or torque
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have the dimension of stiffness and damping, they are commonly referred to as stiffness and damping matrix, respectively. Clearly, the controlled robot is essentially a multi-dimensional mechanical impedance (mass-spring-damper) to the environment, which is addressed by
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2760:{\displaystyle {\boldsymbol {K}}_{\mathrm {x} }{\boldsymbol {e}}_{\mathrm {x} }+{\boldsymbol {D}}_{\mathrm {x} }{\dot {\boldsymbol {e}}}_{\mathrm {x} }+{\boldsymbol {\Lambda }}{\ddot {\boldsymbol {e}}}_{\mathrm {x} }={\boldsymbol {\mathcal {F}}}_{\mathrm {ext} }}
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input and has a resulting force as output. The inverse of impedance is admittance. It imposes position. So actually the controller imposes a spring-mass-damper behavior on the mechanism by maintaining a dynamic relationship between force
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Impedance control is used in applications such as robotics as a general strategy to send commands to a robotics arm and end effector that takes into account the non-linear kinematics and dynamics of the object being manipulated.
2823:) as well. Thus, one can choose desired impedance (mainly stiffness) in the task space. For example, one may want to make the controlled robot act very stiff along one direction while relatively compliant along others by setting
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of the spring. A "damping constant" defines the force output for a velocity input. If we control the impedance of a mechanism, we are controlling the force of resistance to external motions that are imposed by the environment.
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Albu-Schäffer, A.; Ott, C.; Hirzinger, G. (2004), "A passivity based
Cartesian impedance controller for flexible joint robots - part II: full state feedback, impedance design and experiments.",
359:) is an energy dissipating device. If we can control impedance, we are able to control energy exchange during interaction, i.e. the work being done. So impedance control is interaction control.
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are the generalized force of the
Coriolis and centrifugal term, the gravitation, further nonlinear terms, and environmental contacts. Note that this representation only applies to robots with
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1591:{\displaystyle {\boldsymbol {K}}{\boldsymbol {e}}+{\boldsymbol {D}}{\dot {\boldsymbol {e}}}+{\boldsymbol {M}}{\ddot {\boldsymbol {e}}}={\boldsymbol {\tau }}_{\mathrm {ext} }}
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2931:{\displaystyle {\boldsymbol {K}}_{\mathrm {x} }={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1000\end{pmatrix}}\mathrm {N/m} ,}
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The same principle also applies to task space. An uncontrolled robot has the following task-space representation in
Lagrangian formulation:
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1997:
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245:{\displaystyle {\boldsymbol {F}}=M{\boldsymbol {a}}+C{\boldsymbol {v}}+K{\boldsymbol {x}}+{\boldsymbol {f}}+{\boldsymbol {s}}}
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control, perform poorly when the manipulator experiences contacts. Thus impedance control is commonly used in these settings.
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Note that mechanical systems are inherently multi-dimensional - a typical robot arm can place an object in three dimensions (
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3060:"Force, compliance, impedance and interaction control, Summer school dynamic walking and running with robots"
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as the closed-loop system, which is essentially a multi-dimensional mechanical impedance to the environment (
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is an approach to dynamic control relating force and position. It is often used in applications where a
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2617:{\displaystyle {\boldsymbol {e}}_{\mathrm {x} }={\boldsymbol {x}}_{\mathrm {d} }-{\boldsymbol {x}}}
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1499:{\displaystyle {\boldsymbol {e}}={\boldsymbol {q}}_{\mathrm {d} }-{\boldsymbol {q}}}
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In
Proceedings of the 2004 IEEE International Conference on Robotics and Automation
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assuming the task space is a three-dimensional
Euclidean space. The damping matrix
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summarizes all the external forces from the environment. The actuation torque
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includes further torques from, e.g., inherent stiffness, friction, etc., and
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react very differently to a given force than will a log floating in a lake.
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the symmetric and positive-definite task-space inertia matrix. The terms
174:{\displaystyle ({\boldsymbol {x}},{\boldsymbol {v}},{\boldsymbol {a}})}
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An uncontrolled robot can be expressed in
Lagrangian formulation as
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The key theory behind the method is to treat the environment as an
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on the left side corresponds to the input torque of the robot.
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are the internal model of the corresponding mechanical terms.
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are the internal model of the corresponding mechanical terms.
339:) are energy storing elements, whereas a damper (with damping
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2816:{\displaystyle {\boldsymbol {\mathcal {F}}}_{\mathrm {ext} }}
2026:{\displaystyle {\boldsymbol {\mathcal {F}}}_{\mathrm {ext} }}
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Analogously, one may propose the following control law:
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Whole-Body
Impedance Control of Wheeled Humanoid Robots
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are the task-space stiffness and damping matrices, and
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is the symmetric and positive-definite inertia matrix,
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1675:{\displaystyle {\boldsymbol {\tau }}_{\mathrm {ext} }}
741:{\displaystyle {\boldsymbol {\tau }}_{\mathrm {ext} }}
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One may propose a control law of the following form:
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on the left side is the input variable to the robot.
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3009:"Impedance Control: An Approach to Manipulation"
2964:{\displaystyle {\boldsymbol {D}}_{\mathrm {x} }}
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2409:{\displaystyle {\boldsymbol {K}}_{\mathrm {x} }}
2378:{\displaystyle {\boldsymbol {x}}_{\mathrm {d} }}
1086:{\displaystyle {\boldsymbol {q}}_{\mathrm {d} }}
2469:{\displaystyle {\hat {\boldsymbol {\Lambda }}}}
2527:{\displaystyle {\hat {\boldsymbol {\gamma }}}}
8:
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2054:{\displaystyle {\boldsymbol {\mathcal {F}}}}
1093:denotes the desired joint angular position,
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61:Learn how and when to remove this message
45:, without removing the technical details.
1921:{\displaystyle {\boldsymbol {\Lambda }}}
1246:{\displaystyle {\hat {\boldsymbol {h}}}}
1217:{\displaystyle {\hat {\boldsymbol {g}}}}
1188:{\displaystyle {\hat {\boldsymbol {c}}}}
1159:{\displaystyle {\hat {\boldsymbol {M}}}}
137:and position, velocity and acceleration
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3118:from the original on September 7, 2017
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3025:from the original on December 21, 2021
3104:. Springer International Publishing.
1987:{\displaystyle {\boldsymbol {\eta }}}
763:{\displaystyle {\boldsymbol {\tau }}}
667:the Coriolis and centrifugal torque,
43:make it understandable to non-experts
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1943:{\displaystyle {\boldsymbol {\mu }}}
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130:{\displaystyle ({\boldsymbol {F}})}
3065:. pp. 212–243. Archived from
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1899:{\displaystyle {\boldsymbol {x}}}
1877:{\displaystyle {\boldsymbol {q}}}
1637:{\displaystyle {\boldsymbol {D}}}
1615:{\displaystyle {\boldsymbol {K}}}
1130:{\displaystyle {\boldsymbol {D}}}
1108:{\displaystyle {\boldsymbol {K}}}
704:{\displaystyle {\boldsymbol {h}}}
682:{\displaystyle {\boldsymbol {g}}}
660:{\displaystyle {\boldsymbol {c}}}
638:{\displaystyle {\boldsymbol {M}}}
616:{\displaystyle {\boldsymbol {q}}}
289:{\displaystyle {\boldsymbol {s}}}
267:{\displaystyle {\boldsymbol {f}}}
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3043:: CS1 maint: date and year (
3058:Buchli, J. (July 12, 2011).
3007:Hogan, N. (June 6–8, 1984).
3016:American Control Conference
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689:the gravitational torque,
96:and the manipulator as an
2037:. The generalized force
393:{\displaystyle (x,y,z)}
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3098:Dietrich, A. (2016).
3018:. pp. 304, 313.
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3072:on October 16, 2017.
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2035:redundant kinematics
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3144:Control engineering
2566:Similarly, one has
1600:Since the matrices
405:Mathematical basics
274:being friction and
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977:
959:
936:
913:
889:
871:
848:
599:
598:
544:
491:
460:
352:{\displaystyle C}
332:{\displaystyle K}
312:{\displaystyle M}
74:Impedance control
71:
70:
63:
3151:
3128:
3127:
3125:
3123:
3095:
3089:
3088:
3080:
3074:
3073:
3071:
3064:
3055:
3049:
3048:
3042:
3034:
3032:
3030:
3024:
3013:
3004:
2970:
2968:
2967:
2962:
2960:
2959:
2958:
2952:
2937:
2935:
2934:
2929:
2924:
2920:
2911:
2910:
2845:
2844:
2843:
2837:
2822:
2820:
2819:
2814:
2812:
2811:
2810:
2798:
2797:
2776:
2766:
2764:
2763:
2758:
2756:
2755:
2754:
2742:
2741:
2731:
2730:
2729:
2723:
2722:
2714:
2710:
2702:
2701:
2700:
2694:
2693:
2685:
2681:
2680:
2679:
2673:
2664:
2663:
2662:
2656:
2650:
2649:
2648:
2642:
2627:
2623:
2621:
2620:
2615:
2613:
2605:
2604:
2603:
2597:
2588:
2587:
2586:
2580:
2562:
2560:
2559:
2554:
2552:
2551:
2543:
2533:
2531:
2530:
2525:
2523:
2522:
2514:
2504:
2502:
2501:
2496:
2494:
2493:
2485:
2475:
2473:
2472:
2467:
2465:
2464:
2456:
2446:
2444:
2443:
2438:
2436:
2435:
2434:
2428:
2415:
2413:
2412:
2407:
2405:
2404:
2403:
2397:
2384:
2382:
2381:
2376:
2374:
2373:
2372:
2366:
2351:
2349:
2348:
2343:
2335:
2334:
2326:
2320:
2312:
2311:
2303:
2294:
2286:
2285:
2277:
2268:
2267:
2259:
2253:
2245:
2244:
2236:
2230:
2229:
2228:
2222:
2221:
2213:
2206:
2198:
2197:
2189:
2180:
2179:
2171:
2165:
2164:
2163:
2157:
2156:
2148:
2141:
2140:
2139:
2133:
2121:
2113:
2112:
2111:
2105:
2096:
2095:
2094:
2088:
2079:
2078:
2060:
2058:
2057:
2052:
2050:
2049:
2032:
2030:
2029:
2024:
2022:
2021:
2020:
2008:
2007:
1993:
1991:
1990:
1985:
1983:
1971:
1969:
1968:
1963:
1961:
1949:
1947:
1946:
1941:
1939:
1927:
1925:
1924:
1919:
1917:
1905:
1903:
1902:
1897:
1895:
1883:
1881:
1880:
1875:
1873:
1858:
1856:
1855:
1850:
1848:
1847:
1846:
1834:
1833:
1820:
1819:
1811:
1805:
1797:
1786:
1778:
1767:
1766:
1758:
1752:
1744:
1736:
1735:
1727:
1721:
1713:
1705:
1704:
1681:
1679:
1678:
1673:
1671:
1670:
1669:
1657:
1643:
1641:
1640:
1635:
1633:
1621:
1619:
1618:
1613:
1611:
1597:
1595:
1594:
1589:
1587:
1586:
1585:
1573:
1564:
1563:
1555:
1552:
1544:
1543:
1535:
1532:
1524:
1519:
1505:
1503:
1502:
1497:
1495:
1487:
1486:
1485:
1479:
1470:
1456:
1454:
1453:
1448:
1443:
1442:
1441:
1429:
1417:
1416:
1408:
1402:
1401:
1400:
1394:
1393:
1385:
1375:
1367:
1356:
1355:
1347:
1341:
1340:
1339:
1333:
1332:
1324:
1317:
1306:
1298:
1297:
1296:
1290:
1281:
1252:
1250:
1249:
1244:
1242:
1241:
1233:
1223:
1221:
1220:
1215:
1213:
1212:
1204:
1194:
1192:
1191:
1186:
1184:
1183:
1175:
1165:
1163:
1162:
1157:
1155:
1154:
1146:
1136:
1134:
1133:
1128:
1126:
1114:
1112:
1111:
1106:
1104:
1092:
1090:
1089:
1084:
1082:
1081:
1080:
1074:
1054:
1044:
1042:
1041:
1036:
1028:
1027:
1019:
1013:
1005:
1004:
996:
987:
979:
978:
970:
961:
960:
952:
946:
938:
937:
929:
923:
922:
921:
915:
914:
906:
899:
891:
890:
882:
873:
872:
864:
858:
857:
856:
850:
849:
841:
834:
823:
815:
814:
813:
807:
798:
790:
776:
769:
767:
766:
761:
759:
747:
745:
744:
739:
737:
736:
735:
723:
710:
708:
707:
702:
700:
688:
686:
685:
680:
678:
666:
664:
663:
658:
656:
644:
642:
641:
636:
634:
622:
620:
619:
614:
612:
593:
582:
580:
579:
574:
572:
571:
570:
558:
546:
545:
537:
531:
523:
512:
504:
493:
492:
484:
478:
470:
462:
461:
453:
447:
439:
431:
417:
399:
397:
396:
391:
358:
356:
355:
350:
338:
336:
335:
330:
318:
316:
315:
310:
295:
293:
292:
287:
285:
273:
271:
270:
265:
263:
251:
249:
248:
243:
241:
233:
225:
214:
203:
192:
180:
178:
177:
172:
167:
159:
151:
136:
134:
133:
128:
123:
66:
59:
55:
52:
46:
26:
25:
18:
3159:
3158:
3154:
3153:
3152:
3150:
3149:
3148:
3134:
3133:
3132:
3131:
3121:
3119:
3112:
3097:
3096:
3092:
3082:
3081:
3077:
3069:
3062:
3057:
3056:
3052:
3035:
3028:
3026:
3022:
3011:
3006:
3005:
3001:
2996:
2987:
2947:
2942:
2941:
2905:
2904:
2899:
2894:
2888:
2887:
2882:
2877:
2871:
2870:
2865:
2860:
2850:
2832:
2827:
2826:
2791:
2786:
2785:
2774:
2735:
2711:
2682:
2668:
2651:
2637:
2632:
2631:
2592:
2575:
2570:
2569:
2536:
2535:
2507:
2506:
2478:
2477:
2449:
2448:
2423:
2418:
2417:
2392:
2387:
2386:
2361:
2356:
2355:
2210:
2145:
2128:
2100:
2083:
2068:
2067:
2039:
2038:
2001:
1996:
1995:
1974:
1973:
1952:
1951:
1930:
1929:
1908:
1907:
1886:
1885:
1864:
1863:
1827:
1694:
1693:
1688:
1652:
1647:
1646:
1624:
1623:
1602:
1601:
1568:
1510:
1509:
1474:
1461:
1460:
1424:
1382:
1321:
1285:
1272:
1271:
1226:
1225:
1197:
1196:
1168:
1167:
1139:
1138:
1117:
1116:
1095:
1094:
1069:
1064:
1063:
1052:
903:
838:
802:
781:
780:
750:
749:
718:
713:
712:
691:
690:
669:
668:
647:
646:
625:
624:
603:
602:
591:
553:
422:
421:
412:
407:
364:
363:
341:
340:
321:
320:
301:
300:
276:
275:
254:
253:
183:
182:
139:
138:
111:
110:
106:
67:
56:
50:
47:
39:help improve it
36:
27:
23:
12:
11:
5:
3157:
3155:
3147:
3146:
3136:
3135:
3130:
3129:
3110:
3090:
3075:
3050:
2998:
2997:
2995:
2992:
2986:
2983:
2957:
2951:
2927:
2923:
2919:
2915:
2909:
2903:
2900:
2898:
2895:
2893:
2890:
2889:
2886:
2883:
2881:
2878:
2876:
2873:
2872:
2869:
2866:
2864:
2861:
2859:
2856:
2855:
2853:
2848:
2842:
2836:
2809:
2806:
2803:
2796:
2780:
2779:
2770:
2768:
2753:
2750:
2747:
2740:
2734:
2728:
2720:
2717:
2709:
2705:
2699:
2691:
2688:
2678:
2672:
2667:
2661:
2655:
2647:
2641:
2612:
2608:
2602:
2596:
2591:
2585:
2579:
2549:
2546:
2520:
2517:
2491:
2488:
2462:
2459:
2433:
2427:
2402:
2396:
2371:
2365:
2341:
2338:
2332:
2329:
2323:
2319:
2315:
2309:
2306:
2300:
2297:
2293:
2289:
2283:
2280:
2274:
2271:
2265:
2262:
2256:
2252:
2248:
2242:
2239:
2233:
2227:
2219:
2216:
2209:
2205:
2201:
2195:
2192:
2186:
2183:
2177:
2174:
2168:
2162:
2154:
2151:
2144:
2138:
2132:
2127:
2124:
2120:
2116:
2110:
2104:
2099:
2093:
2087:
2082:
2077:
2048:
2019:
2016:
2013:
2006:
1982:
1960:
1938:
1916:
1894:
1872:
1845:
1842:
1839:
1832:
1826:
1823:
1817:
1814:
1808:
1804:
1800:
1796:
1792:
1789:
1785:
1781:
1777:
1773:
1770:
1764:
1761:
1755:
1751:
1747:
1743:
1739:
1733:
1730:
1724:
1720:
1716:
1712:
1708:
1703:
1687:
1684:
1668:
1665:
1662:
1656:
1632:
1610:
1584:
1581:
1578:
1572:
1567:
1561:
1558:
1551:
1547:
1541:
1538:
1531:
1527:
1523:
1518:
1506:, one obtains
1494:
1490:
1484:
1478:
1473:
1469:
1446:
1440:
1437:
1434:
1428:
1423:
1420:
1414:
1411:
1405:
1399:
1391:
1388:
1381:
1378:
1374:
1370:
1366:
1362:
1359:
1353:
1350:
1344:
1338:
1330:
1327:
1320:
1316:
1312:
1309:
1305:
1301:
1295:
1289:
1284:
1280:
1239:
1236:
1210:
1207:
1181:
1178:
1152:
1149:
1125:
1103:
1079:
1073:
1058:
1057:
1048:
1046:
1034:
1031:
1025:
1022:
1016:
1012:
1008:
1002:
999:
993:
990:
986:
982:
976:
973:
967:
964:
958:
955:
949:
945:
941:
935:
932:
926:
920:
912:
909:
902:
898:
894:
888:
885:
879:
876:
870:
867:
861:
855:
847:
844:
837:
833:
829:
826:
822:
818:
812:
806:
801:
797:
793:
789:
758:
734:
731:
728:
722:
699:
677:
655:
633:
611:
597:
596:
587:
585:
569:
566:
563:
557:
552:
549:
543:
540:
534:
530:
526:
522:
518:
515:
511:
507:
503:
499:
496:
490:
487:
481:
477:
473:
469:
465:
459:
456:
450:
446:
442:
438:
434:
430:
411:
408:
406:
403:
389:
386:
383:
380:
377:
374:
371:
348:
328:
308:
284:
262:
240:
236:
232:
228:
224:
220:
217:
213:
209:
206:
202:
198:
195:
191:
170:
166:
162:
158:
154:
150:
146:
126:
122:
118:
105:
102:
69:
68:
30:
28:
21:
13:
10:
9:
6:
4:
3:
2:
3156:
3145:
3142:
3141:
3139:
3117:
3113:
3107:
3103:
3102:
3094:
3091:
3086:
3079:
3076:
3068:
3061:
3054:
3051:
3046:
3040:
3029:September 19,
3021:
3017:
3010:
3003:
3000:
2993:
2991:
2984:
2982:
2980:
2976:
2975:
2938:
2925:
2917:
2907:
2901:
2896:
2891:
2884:
2879:
2874:
2867:
2862:
2857:
2851:
2846:
2824:
2778:
2771:
2769:
2767:
2732:
2718:
2703:
2689:
2665:
2629:
2628:
2625:
2606:
2589:
2567:
2564:
2352:
2339:
2330:
2321:
2298:
2272:
2263:
2254:
2231:
2217:
2184:
2175:
2166:
2152:
2125:
2114:
2080:
2065:
2062:
2036:
1860:
1824:
1815:
1806:
1790:
1771:
1762:
1753:
1737:
1731:
1706:
1691:
1685:
1683:
1598:
1565:
1559:
1545:
1539:
1525:
1507:
1488:
1471:
1457:
1444:
1421:
1412:
1403:
1389:
1360:
1351:
1342:
1328:
1310:
1299:
1269:
1267:
1266:
1261:
1260:
1254:
1056:
1049:
1047:
1045:
1032:
1023:
1014:
991:
965:
956:
947:
924:
910:
877:
868:
859:
845:
827:
816:
791:
778:
777:
774:
771:
595:
588:
586:
584:
550:
541:
532:
516:
497:
488:
479:
463:
457:
432:
419:
418:
415:
409:
404:
402:
384:
381:
378:
375:
372:
360:
346:
326:
306:
297:
234:
226:
218:
215:
207:
204:
196:
193:
160:
152:
103:
101:
99:
95:
90:
86:
82:
79:
75:
65:
62:
54:
51:February 2014
44:
40:
34:
31:This article
29:
20:
19:
16:
3122:September 1,
3120:. Retrieved
3100:
3093:
3084:
3078:
3067:the original
3053:
3027:. Retrieved
3015:
3002:
2988:
2985:Applications
2972:
2939:
2825:
2783:
2772:
2630:
2568:
2565:
2353:
2066:
2063:
1861:
1692:
1689:
1599:
1508:
1458:
1270:
1263:
1257:
1255:
1061:
1050:
779:
772:
600:
589:
420:
413:
361:
298:
107:
91:
87:
83:
73:
72:
57:
48:
32:
15:
1256:Inserting (
410:Joint space
78:manipulator
2994:References
1686:Task space
94:admittance
2719:¨
2708:Λ
2690:˙
2607:−
2548:^
2545:η
2519:^
2516:γ
2490:^
2487:μ
2461:^
2458:Λ
2331:˙
2308:^
2305:η
2282:^
2279:γ
2264:˙
2241:^
2238:μ
2218:¨
2194:^
2191:Λ
2176:˙
2167:−
2153:˙
2115:−
1981:η
1959:γ
1937:μ
1915:Λ
1816:˙
1795:η
1776:γ
1763:˙
1742:μ
1732:¨
1711:Λ
1655:τ
1571:τ
1560:¨
1540:˙
1489:−
1427:τ
1413:¨
1404:−
1390:¨
1352:˙
1343:−
1329:˙
1300:−
1238:^
1209:^
1180:^
1151:^
1024:˙
1001:^
975:^
957:˙
934:^
911:¨
887:^
869:˙
860:−
846:˙
817:−
788:τ
757:τ
721:τ
556:τ
542:˙
489:˙
458:¨
429:τ
104:Principle
98:impedance
3138:Category
3116:Archived
3039:cite web
3020:Archived
1262:) into (
299:Masses (
252:, with
37:Please
3108:
2979:stable
2534:, and
2354:where
1994:, and
1862:where
1224:, and
1062:where
601:where
3070:(PDF)
3063:(PDF)
3023:(PDF)
3012:(PDF)
2977:) is
3124:2017
3106:ISBN
3045:link
3031:2013
2902:1000
2416:and
1622:and
1459:Let
1115:and
41:to
3140::
3114:.
3041:}}
3037:{{
3014:.
2981:.
2624:,
2505:,
2476:,
1972:,
1950:,
1859:,
1682:.
1195:,
1166:,
583:,
181::
3126:.
3047:)
3033:.
2974:3
2956:x
2950:D
2926:,
2922:m
2918:/
2914:N
2908:)
2897:0
2892:0
2885:0
2880:1
2875:0
2868:0
2863:0
2858:1
2852:(
2847:=
2841:x
2835:K
2808:t
2805:x
2802:e
2795:F
2777:)
2775:3
2773:(
2752:t
2749:x
2746:e
2739:F
2733:=
2727:x
2716:e
2704:+
2698:x
2687:e
2677:x
2671:D
2666:+
2660:x
2654:e
2646:x
2640:K
2611:x
2601:d
2595:x
2590:=
2584:x
2578:e
2432:x
2426:D
2401:x
2395:K
2370:d
2364:x
2340:,
2337:)
2328:q
2322:,
2318:q
2314:(
2299:+
2296:)
2292:q
2288:(
2273:+
2270:)
2261:q
2255:,
2251:q
2247:(
2232:+
2226:d
2215:x
2208:)
2204:q
2200:(
2185:+
2182:)
2173:x
2161:d
2150:x
2143:(
2137:x
2131:D
2126:+
2123:)
2119:x
2109:d
2103:x
2098:(
2092:x
2086:K
2081:=
2076:F
2047:F
2018:t
2015:x
2012:e
2005:F
1893:x
1871:q
1844:t
1841:x
1838:e
1831:F
1825:+
1822:)
1813:q
1807:,
1803:q
1799:(
1791:+
1788:)
1784:q
1780:(
1772:+
1769:)
1760:x
1754:,
1750:x
1746:(
1738:+
1729:x
1723:)
1719:q
1715:(
1707:=
1702:F
1667:t
1664:x
1661:e
1631:D
1609:K
1583:t
1580:x
1577:e
1566:=
1557:e
1550:M
1546:+
1537:e
1530:D
1526:+
1522:e
1517:K
1493:q
1483:d
1477:q
1472:=
1468:e
1445:.
1439:t
1436:x
1433:e
1422:=
1419:)
1410:q
1398:d
1387:q
1380:(
1377:)
1373:q
1369:(
1365:M
1361:+
1358:)
1349:q
1337:d
1326:q
1319:(
1315:D
1311:+
1308:)
1304:q
1294:d
1288:q
1283:(
1279:K
1265:1
1259:2
1235:h
1206:g
1177:c
1148:M
1124:D
1102:K
1078:d
1072:q
1055:)
1053:2
1051:(
1033:,
1030:)
1021:q
1015:,
1011:q
1007:(
998:h
992:+
989:)
985:q
981:(
972:g
966:+
963:)
954:q
948:,
944:q
940:(
931:c
925:+
919:d
908:q
901:)
897:q
893:(
884:M
878:+
875:)
866:q
854:d
843:q
836:(
832:D
828:+
825:)
821:q
811:d
805:q
800:(
796:K
792:=
733:t
730:x
727:e
698:h
676:g
654:c
632:M
610:q
594:)
592:1
590:(
568:t
565:x
562:e
551:+
548:)
539:q
533:,
529:q
525:(
521:h
517:+
514:)
510:q
506:(
502:g
498:+
495:)
486:q
480:,
476:q
472:(
468:c
464:+
455:q
449:)
445:q
441:(
437:M
433:=
388:)
385:z
382:,
379:y
376:,
373:x
370:(
347:C
327:K
307:M
283:s
261:f
239:s
235:+
231:f
227:+
223:x
219:K
216:+
212:v
208:C
205:+
201:a
197:M
194:=
190:F
169:)
165:a
161:,
157:v
153:,
149:x
145:(
125:)
121:F
117:(
64:)
58:(
53:)
49:(
35:.
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