6252:
5824:
6247:{\displaystyle {\begin{aligned}\delta W&=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\right)\cdot \mathbf {d} \times {\vec {\omega }}\delta t+\left(\sum _{i=1}^{n}\mathbf {F} _{i}\right)\cdot \mathbf {v} \delta t+\left(\sum _{i=1}^{n}\mathbf {X} _{i}\times \mathbf {F} _{i}\right)\cdot {\vec {\omega }}\delta t\\&=\left(\sum _{i=1}^{n}\mathbf {F} _{i}\right)\cdot (\mathbf {v} +\mathbf {d} \times {\vec {\omega }})\delta t+\left(\sum _{i=1}^{n}\mathbf {X} _{i}\times \mathbf {F} _{i}\right)\cdot {\vec {\omega }}\delta t.\end{aligned}}}
6503:
2971:
3744:
139:
6263:
828:
691:
2120:
3485:
4056:
2811:
5677:
1666:
2822:
5094:
696:
187:
is a screw used to represent the velocity of a rigid body as an angular velocity around an axis and a linear velocity along this axis. All points in the body have the same component of the velocity along the axis, however the greater the distance from the axis the greater the velocity in the plane
191:
The points in a body undergoing a constant twist motion trace helices in the fixed frame. If this screw motion has zero pitch then the trajectories trace circles, and the movement is a pure rotation. If the screw motion has infinite pitch then the trajectories are all straight lines in the same
3531:
559:
6715:
3279:
174:
A screw is a six-dimensional vector constructed from a pair of three-dimensional vectors, such as forces and torques and linear and angular velocity, that arise in the study of spatial rigid body movement. The components of the screw define the Plücker coordinates of a line in space and the
6498:{\displaystyle {\mathsf {T}}=({\vec {\omega }},\mathbf {d} \times {\vec {\omega }}+\mathbf {v} )=(\mathbf {T} ,\mathbf {T} ^{\circ }),\quad {\mathsf {W}}=\left(\sum _{i=1}^{n}\mathbf {F} _{i},\sum _{i=1}^{n}\mathbf {X} _{i}\times \mathbf {F} _{i}\right)=(\mathbf {W} ,\mathbf {W} ^{\circ }),}
5809:
1521:
1974:
3302:
1315:
3911:
2647:
1202:
974:
552:
6593:
4858:
6979:
158:. The six parameters that define a screw motion are the four independent components of the Plücker vector that defines the screw axis, together with the rotation angle about and linear slide along this line, and form a pair of vectors called a
5542:
5488:
2209:
1548:
2966:{\displaystyle {\begin{Bmatrix}\mathbf {Q} -\mathbf {P} \\\mathbf {P} \times \mathbf {Q} \end{Bmatrix}}={\begin{bmatrix}A&0\\DA&A\end{bmatrix}}{\begin{Bmatrix}\mathbf {q} -\mathbf {p} \\\mathbf {p} \times \mathbf {q} \end{Bmatrix}}.}
4328:. In planar transformations a translation is obtained by reflection in parallel lines, and rotation is obtained by reflection in a pair of intersecting lines. To produce a screw transformation from similar concepts one must use planes in
2518:
The coordinate transformations for screws are easily understood by beginning with the coordinate transformations of the Plücker vector of line, which in turn are obtained from the transformations of the coordinate of points on the line.
2612:
5379:
4873:
3739:{\displaystyle {\textbf {V}}_{P}={\textbf {p}}={\begin{Bmatrix}{\textbf {V}}_{P}\\0\end{Bmatrix}}={\begin{bmatrix}{\dot {A}}(t)&{\dot {\textbf {d}}}(t)\\0&0\end{bmatrix}}{\begin{Bmatrix}{\textbf {p}}\\1\end{Bmatrix}}.}
7501:
Selig, J. M. (2011) "Rational
Interpolation of Rigid Body Motions," Advances in the Theory of Control, Signals and Systems with Physical Modeling, Lecture Notes in Control and Information Sciences, Volume 407/2011 213–224,
208:
of a line in space and has zero pitch. A torque, on the other hand, is a pure moment that is not bound to a line in space and is an infinite pitch screw. The ratio of these two magnitudes defines the pitch of the screw.
102:
Important theorems of screw theory include: The
Transfer Principle proves that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws.
4703:
7115:
6604:
3094:
1088:
2437:
7044:
1225:
823:{\displaystyle {\mathsf {S}}\times {\mathsf {T}}=(\mathbf {S} ,\mathbf {V} )\times (\mathbf {T} ,\mathbf {W} )=(\mathbf {S} \times \mathbf {T} ,\,\,\mathbf {S} \times \mathbf {W} +\mathbf {V} \times \mathbf {T} ).}
3900:
1943:
5688:
2299:
686:{\displaystyle {\mathsf {S}}\cdot {\mathsf {T}}=(\mathbf {S} ,\mathbf {V} )\cdot (\mathbf {T} ,\mathbf {W} )=(\mathbf {S} \cdot \mathbf {T} ,\,\,\mathbf {S} \cdot \mathbf {W} +\mathbf {V} \cdot \mathbf {T} ),}
6789:
1389:
2115:{\displaystyle \mathbf {V} _{P}(t)=\mathbf {P} +\mathbf {v} -\mathbf {d} \quad {\text{or}}\quad \mathbf {V} _{P}(t)=\mathbf {\omega } \times \mathbf {P} +\mathbf {v} +\mathbf {d} \times \mathbf {\omega } ,}
6871:
3480:{\displaystyle {\textbf {P}}(t)={\textbf {p}}={\begin{Bmatrix}{\textbf {P}}\\1\end{Bmatrix}}={\begin{bmatrix}A(t)&{\textbf {d}}(t)\\0&1\end{bmatrix}}{\begin{Bmatrix}{\textbf {p}}\\1\end{Bmatrix}}.}
1830:
1119:
848:
442:
5829:
4051:{\displaystyle ={\begin{bmatrix}\Omega &-\Omega {\textbf {d}}+{\dot {\textbf {d}}}\\0&0\end{bmatrix}}={\begin{bmatrix}\Omega &\mathbf {d} \times \omega +\mathbf {v} \\0&0\end{bmatrix}}.}
417:
271:
2806:{\displaystyle {\mathsf {Q}}=(\mathbf {Q} -\mathbf {P} ,\mathbf {P} \times \mathbf {Q} )=((\mathbf {q} -\mathbf {p} ),(\mathbf {p} \times \mathbf {q} )+\mathbf {d} \times (\mathbf {q} -\mathbf {p} ))}
2508:
6892:
In the study of robotic systems the components of the twist are often transposed to eliminate the need for the 6×6 matrix in the calculation of work. In this case the twist is defined to be
4303:
2356:
1719:
6514:
4336:, which is the line of intersection of the intersecting planes that generate the rotation of the screw. Thus four reflections in planes effect a screw transformation. The tradition of
830:
which is a screw. The dot and cross products of screws satisfy the identities of vector algebra, and allow computations that directly parallel computations in the algebra of vectors.
4718:
188:
perpendicular to this axis. Thus, the helicoidal field formed by the velocity vectors in a moving rigid body flattens out the further the points are radially from the twist axis.
6898:
6802:
If the virtual work of a wrench on a twist is zero, then the forces and torque of the wrench are constraint forces relative to the twist. The wrench and twist are said to be
5672:{\displaystyle \delta W=\mathbf {F} _{1}\cdot \mathbf {V} _{1}\delta t+\mathbf {F} _{2}\cdot \mathbf {V} _{2}\delta t+\cdots +\mathbf {F} _{n}\cdot \mathbf {V} _{n}\delta t.}
1661:{\displaystyle {\mathsf {R}}=(\mathbf {F} -\mathbf {F} ,\mathbf {A} \times \mathbf {F} -\mathbf {B} \times \mathbf {F} )=(0,(\mathbf {A} -\mathbf {B} )\times \mathbf {F} ).}
5409:
2134:
7601:
7316:
4216:
3071:
1026:
4150:
130:. Based on screw theory, an efficient approach has also been developed for the type synthesis of parallel mechanisms (parallel manipulators or parallel robots).
2552:
5089:{\displaystyle (e^{b\varepsilon }e^{-ar})q(e^{ar}e^{b\varepsilon r})=e^{b\varepsilon r}(e^{-ar}qe^{ar})e^{b\varepsilon r}=e^{2b\varepsilon r}(e^{-ar}qe^{ar}).}
5271:
7600:
Xiangke Wang, Dapeng Han, Changbin Yu, and
Zhiqiang Zheng (2012) "The geometric structure of unit dual quaternions with application in kinematic control",
6710:{\displaystyle \delta W=(\mathbf {W} \cdot \mathbf {T} ^{\circ }+\mathbf {W} ^{\circ }\cdot \mathbf {T} )\delta t={\mathsf {W}}{\mathsf {T}}\delta t,}
3274:{\displaystyle {\mathsf {S}}={\mathsf {s}},\quad (\mathbf {S} ,\mathbf {V} )=(,)(\mathbf {s} ,\mathbf {v} )=(\mathbf {s} ,\mathbf {v} +\mathbf {s} ).}
4537:
7055:
2384:
6990:
312:
7531:
7486:
7459:
7432:
7405:
7378:
5804:{\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot ({\vec {\omega }}\times (\mathbf {X} _{i}-\mathbf {d} )+\mathbf {v} )\delta t.}
3778:
1845:
4069:
of homogeneous transforms. The components of are the components of the twist screw, and for this reason is also often called a twist.
2251:
1516:{\displaystyle {\mathsf {R}}=\sum _{i=1}^{n}{\mathsf {W}}_{i}=\sum _{i=1}^{n}(\mathbf {F} _{i},\mathbf {P} _{i}\times \mathbf {F} _{i}).}
288:
are three-dimensional real vectors. The sum and difference of these ordered pairs are computed componentwise. Screws are often called
7565:
6726:
6812:
1758:
1310:{\displaystyle {\mathsf {S}}\times {\mathsf {T}}=\left|{\mathsf {S}}\right|\left|{\mathsf {T}}\right|\sin {\hat {z}}{\mathsf {N}}.}
4411:
155:
146:
A spatial displacement of a rigid body can be defined by a rotation about a line and a translation along the same line, called a
104:
34:
227:
7677:
7616:
2461:
7296:
7291:
200:
The force and torque vectors that arise in applying Newton's laws to a rigid body can be assembled into a screw called a
4357:
4243:
7285:
2314:
1677:
1197:{\displaystyle {\mathsf {S}}\cdot {\mathsf {T}}=\left|{\mathsf {S}}\right|\left|{\mathsf {T}}\right|\cos {\hat {z}};}
969:{\displaystyle \sin {\hat {z}}=(\sin \varphi ,d\cos \varphi ),\,\,\,\cos {\hat {z}}=(\cos \varphi ,-d\sin \varphi ),}
547:{\displaystyle {\hat {a}}{\mathsf {S}}=(a,b)(\mathbf {S} ,\mathbf {V} )=(a\mathbf {S} ,a\mathbf {V} +b\mathbf {S} ).}
6588:{\displaystyle \delta W=(\mathbf {W} \cdot \mathbf {T} ^{\circ }+\mathbf {W} ^{\circ }\cdot \mathbf {T} )\delta t.}
3525:
The velocity of this movement is defined by computing the velocity of the trajectories of the points in the body,
7183:
4477:
1736:
of a rigid body, we must consider its movement defined by the parameterized set of spatial displacements, D(t)=(,
4853:{\displaystyle {\frac {1}{\exp(ar-b\varepsilon r)}}=(e^{ar}e^{-br\varepsilon })^{-1}=e^{br\varepsilon }e^{-ar},}
7584:
7550:
7236:
7179:
4325:
7163:
4356:
The combination of a translation with a rotation effected by a screw displacement can be illustrated with the
7145:
4407:
4321:
6974:{\displaystyle {\check {\mathsf {T}}}=(\mathbf {d} \times {\vec {\omega }}+\mathbf {v} ,{\vec {\omega }}),}
7198:
2543:
1020:
205:
115:
88:
7174:
and metric, applied to screws, has been described by Harvey Lipkin. Other prominent contributors include
111:
proves that rotations about a rigid object's major and minor -- but not intermediate -- axes are stable.
7687:
7449:
4237:), the homogeneous transformation to a new location and orientation can be computed with the formula,
7244:
5483:{\displaystyle \mathbf {V} _{i}={\vec {\omega }}\times (\mathbf {X} _{i}-\mathbf {d} )+\mathbf {v} ,}
4155:
and ask for the movement that has a constant twist matrix . The solution is the matrix exponential
309:. Let the addition and subtraction of these numbers be componentwise, and define multiplication as
108:
58:
2204:{\displaystyle {\mathsf {T}}=({\vec {\omega }},\mathbf {v} +\mathbf {d} \times {\vec {\omega }}),\!}
162:. For comparison, the six parameters that define a spatial displacement can also be given by three
7692:
7137:
4341:
80:
7682:
7633:
5131:
4337:
119:
7222:
5174:. These six parameters generate a subgroup of the units, the unit sphere. Of course it includes
2816:
Thus, a spatial displacement defines a transformation for Plücker coordinates of lines given by
2304:
and (ii) when = 0, that is the body does not rotate but only slides in the direction
7527:
7482:
7455:
7428:
7401:
7374:
7175:
7155:
4345:
3296:
Consider the movement of a rigid body defined by the parameterized 4x4 homogeneous transform,
2234:
is the velocity of the point in the body that corresponds with the origin of the fixed frame.
127:
91:
of a line define a unit screw, and general screws are obtained by multiplication by a pair of
7271:. Coolidge based his description simply on the tools Hamilton had used for real quaternions.
4161:
7662:
7625:
7562:
7503:
7256:
7171:
7167:
7159:
2976:
The matrix is the skew-symmetric matrix that performs the cross product operation, that is
107:
proves that any change between two rigid object poses can be performed by a single screw.
50:
38:
4061:
Recall that is the angular velocity matrix. The matrix is an element of the Lie algebra
3005:
7569:
7268:
7260:
7187:
4461:
4457:
4066:
2607:{\displaystyle {\mathsf {q}}=(\mathbf {q} -\mathbf {p} ,\mathbf {p} \times \mathbf {q} ),}
2448:
976:
which are also dual scalars. In general, the function of a dual variable is defined to be
123:
96:
42:
7162:. He also worked out elliptic geometry, and a fresh view of Euclidean geometry, with the
4072:
From the definition of the matrix , we can formulate the ordinary differential equation,
2125:
where = is the angular velocity matrix and ω is the angular velocity vector.
7348:
5374:{\displaystyle \mathbf {X} _{i}(t)=\mathbf {x} _{i}+\mathbf {d} (t)\quad i=1,\ldots ,n,}
4078:
142:
The pitch of a pure screw relates rotation about an axis to translation along that axis.
7448:
Murray, Richard M.; Li, Zexiang; Sastry, S. Shankar; Sastry, S. Shankara (1994-03-22).
5682:
Define the velocities of each point in terms of the twist of the moving body to obtain
5127:
2367:
84:
2455:
pointing define the direction of the slide, then the twist for the joint is given by,
7671:
7655:
7230:
204:. A force has a point of application and a line of action, therefore it defines the
2534:
is the translation vector. Consider the line in the body defined by the two points
7573:
7248:
4368:
2245: = 0, then the twist is a pure rotation about a line, then the twist is
163:
69:
6598:
The 6×6 matrix is used to simplify the calculation of work using screws, so that
7521:
7507:
7476:
7422:
7395:
7368:
5257:
are defined by the movement of the rigid body with rotation and the translation
7546:
7226:
7151:
5139:
4062:
304:
92:
72:
7280:
7240:
7194:
4698:{\displaystyle {\begin{pmatrix}z&0\\0&z^{*}\end{pmatrix}}=\thicksim .}
4500:
4473:
4333:
845:, then the infinite series definitions of sine and cosine yield the relations
166:
that define the rotation and the three components of the translation vector.
62:
54:
27:
Mathematical formulation of vector pairs used in physics (rigid body dynamics)
7263:
described the use of dual quaternions for screw displacements on page 261 of
7110:{\displaystyle \delta W={\mathsf {W}}\cdot {\check {\mathsf {T}}}\delta t=0,}
4221:
This formulation can be generalized such that given an initial configuration
1083:{\displaystyle |{\mathsf {S}}|={\sqrt {{\mathsf {S}}\cdot {\mathsf {S}}}}=1;}
6257:
Introduce the twist of the moving body and the wrench acting on it given by
5135:
2432:{\displaystyle \xi ={\begin{Bmatrix}\omega \\q\times \omega \end{Bmatrix}}.}
17:
7039:{\displaystyle \delta W={\mathsf {W}}\cdot {\check {\mathsf {T}}}\delta t.}
1357:) is a screw. The resultant force and moment obtained from all the forces
556:
Finally, introduce the dot and cross products of screws by the formulas:
5179:
3284:
The dual matrix = (, ) has determinant 1 and is called a
114:
Screw theory is an important tool in robot mechanics, mechanical design,
76:
7637:
7141:
138:
3895:{\displaystyle {\textbf {V}}_{P}=^{-1}{\textbf {P}}(t)={\textbf {P}},}
1938:{\displaystyle \mathbf {V} _{P}(t)=\left\mathbf {p} +\mathbf {v} (t),}
1374:, acting on a rigid body is simply the sum of the individual wrenches
1206:
Let N be the unit screw that defines the common normal to the axes of
7202:
5183:
7629:
175:
magnitudes of the vector along the line and moment about this line.
7658:, Department of Design and Innovation, the Open University, London.
2294:{\displaystyle {\mathsf {L}}=(\omega ,\mathbf {d} \times \omega ),}
7259:. However, the point of view of Sophus Lie has recurred. In 1940,
4469:
4344:
and provides a language of transformation that does not depend on
4329:
2617:
then in the fixed frame we have the transformed point coordinates
137:
122:. This is in part because of the relationship between screws and
46:
3749:
The dot denotes the derivative with respect to time, and because
1116:
is the distance between these axes along the common normal, then
6784:{\displaystyle ={\begin{bmatrix}0&I\\I&0\end{bmatrix}},}
5123: where the required rotation and translation are effected.
1341:
be the vector locating this point in a fixed frame. The wrench
7193:
The homography idea in transformation geometry was advanced by
295:
Now, introduce the ordered pair of real numbers â = (
6866:{\displaystyle \delta W={\mathsf {W}}{\mathsf {T}}\delta t=0,}
83:, where lines form the screw axes of spatial movement and the
7350:
The theory of screws: A study in the dynamics of a rigid body
1825:{\displaystyle \mathbf {P} (t)=\mathbf {p} +\mathbf {d} (t).}
7321:, Foreign Technology Division translation FTD-HT-23-1632-67
2991:
The 6×6 matrix obtained from the spatial displacement
412:{\displaystyle {\hat {a}}{\hat {c}}=(a,b)(c,d)=(ac,ad+bc).}
1748:
that is fixed in moving body coordinates to trace a curve
266:{\displaystyle {\mathsf {S}}=(\mathbf {S} ,\mathbf {V} ),}
7587:(1873), "Preliminary Sketch of Biquaternions", Paper XX,
2503:{\displaystyle \xi ={\begin{Bmatrix}0\\v\end{Bmatrix}}.}
1952:
is velocity of the origin of the moving frame, that is d
7288:
uses screws to describe rigid body motions and loading.
5814:
Expand this equation and collect coefficients of ω and
4379:, all other terms of the exponential series vanishing.
7614:
Buchheim, Arthur (1885). "A Memoir on biquaternions".
6747:
4567:
3998:
3932:
3708:
3634:
3594:
3449:
3391:
3358:
2919:
2880:
2831:
2476:
2399:
1526:
Notice that the case of two equal but opposite forces
7058:
6993:
6901:
6815:
6729:
6607:
6517:
6266:
5827:
5691:
5545:
5412:
5274:
4876:
4721:
4540:
4246:
4164:
4081:
3914:
3781:
3760:
into the velocity equation to obtain the velocity of
3534:
3305:
3097:
3008:
2825:
2650:
2555:
2464:
2387:
2317:
2254:
2137:
1977:
1848:
1761:
1680:
1551:
1392:
1329:
associated with a force acting on a rigid body. Let
1228:
1122:
1029:
851:
699:
562:
445:
315:
230:
7318:
The Screw
Calculus and Its Applications in Mechanics
7451:
7334:, William R. Spillers (ed.), Elsevier, pp. 266–281.
4332:: the parallel planes must be perpendicular to the
7109:
7038:
6973:
6865:
6783:
6709:
6587:
6497:
6246:
5803:
5671:
5482:
5373:
5088:
4852:
4697:
4297:
4210:
4144:
4050:
3894:
3738:
3479:
3273:
3065:
2965:
2805:
2606:
2502:
2431:
2370:, let the axis of rotation pass through the point
2350:
2293:
2203:
2114:
1937:
1824:
1713:
1660:
1515:
1309:
1196:
1082:
968:
822:
685:
546:
411:
265:
7602:Journal of Mathematical Analysis and Applications
4491:be half the angle of the desired turn about axis
4324:, the elemental concept of transformation is the
4312:represents the parameters of the transformation.
4298:{\displaystyle g(\theta )=\exp(\xi \theta )g(0),}
2200:
7136:The mathematical framework was developed by Sir
2237:There are two important special cases: (i) when
2351:{\displaystyle {\mathsf {T}}=(0,\mathbf {v} ).}
1714:{\displaystyle {\mathsf {M}}=(0,\mathbf {M} ),}
1015:These definitions allow the following results:
7475:Lynch, Kevin M.; Park, Frank C. (2017-05-25).
5265:) of a reference point in the body, given by
1744:is a translation vector. This causes a point
87:of forces. The pair of vectors that form the
5512:The work by the forces over the displacement
2522:Let the displacement of a body be defined by
1222:) is the dual angle between these axes, then
8:
7367:McCarthy, J. Michael; Soh, Gim Song (2010).
2378:, then the twist for the joint is given by,
3490:This notation does not distinguish between
7362:
7360:
7330:Yang, A.T. (1974) "Calculus of Screws" in
7239:initiated the use of dual quaternions for
7140:in 1876 for application in kinematics and
6984:so the calculation of work takes the form
2308:, then the twist is a pure slide given by
7520:Kong, Xianwen; Gosselin, Clément (2007).
7081:
7079:
7078:
7069:
7068:
7057:
7016:
7014:
7013:
7004:
7003:
6992:
6954:
6953:
6945:
6931:
6930:
6922:
6905:
6903:
6902:
6900:
6842:
6841:
6826:
6825:
6814:
6742:
6728:
6692:
6691:
6676:
6675:
6658:
6649:
6644:
6634:
6629:
6620:
6606:
6568:
6559:
6554:
6544:
6539:
6530:
6516:
6483:
6478:
6469:
6452:
6447:
6437:
6432:
6425:
6414:
6401:
6396:
6389:
6378:
6360:
6359:
6346:
6341:
6332:
6318:
6304:
6303:
6295:
6281:
6280:
6268:
6267:
6265:
6220:
6219:
6205:
6200:
6190:
6185:
6178:
6167:
6135:
6134:
6126:
6118:
6101:
6096:
6089:
6078:
6042:
6041:
6027:
6022:
6012:
6007:
6000:
5989:
5966:
5952:
5947:
5940:
5929:
5900:
5899:
5891:
5877:
5872:
5865:
5854:
5828:
5826:
5784:
5773:
5764:
5759:
5741:
5740:
5728:
5723:
5716:
5705:
5690:
5654:
5649:
5639:
5634:
5612:
5607:
5597:
5592:
5576:
5571:
5561:
5556:
5544:
5472:
5461:
5452:
5447:
5429:
5428:
5419:
5414:
5411:
5332:
5323:
5318:
5281:
5276:
5273:
5071:
5052:
5030:
5011:
4995:
4976:
4957:
4935:
4922:
4897:
4884:
4875:
4835:
4819:
4803:
4784:
4771:
4722:
4720:
4662:
4652:
4630:
4591:
4562:
4539:
4245:
4190:
4163:
4086:
4085:
4080:
4020:
4006:
3993:
3959:
3957:
3956:
3947:
3946:
3927:
3913:
3883:
3882:
3855:
3854:
3845:
3803:
3802:
3790:
3784:
3783:
3780:
3712:
3711:
3703:
3663:
3661:
3660:
3638:
3637:
3629:
3605:
3599:
3598:
3589:
3580:
3579:
3556:
3555:
3543:
3537:
3536:
3533:
3453:
3452:
3444:
3409:
3408:
3386:
3362:
3361:
3353:
3344:
3343:
3307:
3306:
3304:
3260:
3240:
3223:
3200:
3192:
3148:
3140:
3127:
3126:
3112:
3111:
3099:
3098:
3096:
3013:
3012:
3007:
2947:
2939:
2930:
2922:
2914:
2875:
2859:
2851:
2842:
2834:
2826:
2824:
2792:
2784:
2764:
2753:
2745:
2722:
2714:
2688:
2680:
2672:
2664:
2652:
2651:
2649:
2593:
2585:
2577:
2569:
2557:
2556:
2554:
2471:
2463:
2394:
2386:
2337:
2319:
2318:
2316:
2274:
2256:
2255:
2253:
2183:
2182:
2174:
2166:
2152:
2151:
2139:
2138:
2136:
2104:
2096:
2088:
2080:
2072:
2054:
2049:
2042:
2036:
2019:
2011:
1984:
1979:
1976:
1918:
1910:
1877:
1855:
1850:
1847:
1805:
1797:
1762:
1760:
1700:
1682:
1681:
1679:
1647:
1636:
1628:
1605:
1597:
1589:
1581:
1573:
1565:
1553:
1552:
1550:
1501:
1496:
1486:
1481:
1471:
1466:
1456:
1445:
1432:
1426:
1425:
1418:
1407:
1394:
1393:
1391:
1333:be the point of application of the force
1298:
1297:
1286:
1285:
1269:
1268:
1254:
1253:
1240:
1239:
1230:
1229:
1227:
1180:
1179:
1163:
1162:
1148:
1147:
1134:
1133:
1124:
1123:
1121:
1063:
1062:
1053:
1052:
1050:
1042:
1036:
1035:
1030:
1028:
916:
915:
908:
907:
906:
859:
858:
850:
809:
801:
793:
785:
784:
783:
775:
767:
753:
745:
731:
723:
711:
710:
701:
700:
698:
672:
664:
656:
648:
647:
646:
638:
630:
616:
608:
594:
586:
574:
573:
564:
563:
561:
533:
522:
511:
494:
486:
459:
458:
447:
446:
444:
329:
328:
317:
316:
314:
252:
244:
232:
231:
229:
33:is the algebraic calculation of pairs of
3522:), which is hopefully clear in context.
2999:) can be assembled into the dual matrix
7308:
7197:more than a century ago. Even earlier,
4639:
7154:saw screw theory as an application of
7082:
7070:
7017:
7005:
6906:
6843:
6827:
6693:
6677:
6361:
6269:
3128:
3100:
2653:
2558:
2320:
2257:
2140:
1740:(t)), where is a rotation matrix and
1683:
1554:
1427:
1395:
1299:
1270:
1255:
1241:
1231:
1164:
1149:
1135:
1125:
1064:
1054:
1037:
712:
702:
575:
565:
460:
233:
7551:On Sir Robert Ball's Theory of Screws
7549:(1902) (D.H. Delphenich translator)
7523:Type Synthesis of Parallel Mechanisms
5240:in a rigid body. The trajectories of
5190:Work of forces acting on a rigid body
3756:Substitute the inverse transform for
2530:), where is the rotation matrix and
7:
7342:
7340:
7267:. He notes the 1885 contribution of
5393:are coordinates in the moving body.
4484:, is a screw displacement of space.
3753:is constant its derivative is zero.
1724:can be interpreted as pure moments.
431:) by the dual scalar â = (
126:which have been used to interpolate
5497:is the angular velocity vector and
3960:
3948:
3884:
3856:
3785:
3713:
3664:
3600:
3581:
3538:
3454:
3410:
3363:
3345:
3308:
3292:Twists as elements of a Lie algebra
2514:Coordinate transformation of screws
1671:This shows that screws of the form
1542:respectively, yields the resultant
1325:A common example of a screw is the
1023:of a line and satisfy the relation
833:Let the dual scalar ẑ = (
439:) is computed componentwise to be,
6835:
6733:
6685:
4460:in the eight-dimensional space of
4001:
3943:
3935:
2030:
2005:
25:
6794:and is the 3×3 identity matrix.
2374:and be directed along the vector
1752:(t) in the fixed frame given by,
1104:is the angle between the axes of
7332:Basic Questions of Design Theory
7265:A History of Geometrical Methods
7205:form of unit quaternions as exp(
6946:
6923:
6659:
6645:
6630:
6621:
6569:
6555:
6540:
6531:
6479:
6470:
6448:
6433:
6397:
6342:
6333:
6319:
6296:
6201:
6186:
6127:
6119:
6097:
6023:
6008:
5967:
5948:
5892:
5873:
5785:
5774:
5760:
5724:
5650:
5635:
5608:
5593:
5572:
5557:
5473:
5462:
5448:
5415:
5333:
5319:
5277:
4021:
4007:
3261:
3241:
3224:
3201:
3193:
3149:
3141:
2948:
2940:
2931:
2923:
2860:
2852:
2843:
2835:
2793:
2785:
2765:
2754:
2746:
2723:
2715:
2689:
2681:
2673:
2665:
2594:
2586:
2578:
2570:
2338:
2275:
2218:of the moving body. The vector
2175:
2167:
2097:
2089:
2081:
2050:
2037:
2020:
2012:
1980:
1968:) into this equation to obtain,
1919:
1911:
1851:
1806:
1798:
1763:
1701:
1648:
1637:
1629:
1606:
1598:
1590:
1582:
1574:
1566:
1497:
1482:
1467:
1112:around their common normal, and
810:
802:
794:
786:
776:
768:
754:
746:
732:
724:
673:
665:
657:
649:
639:
631:
617:
609:
595:
587:
534:
523:
512:
495:
487:
253:
245:
7617:American Journal of Mathematics
6358:
5346:
4424:and under the translation (1 +
3764:by operating on its trajectory
3136:
2047:
2041:
7663:Robotics, Geometry and Control
7481:. Cambridge University Press.
7086:
7021:
6965:
6959:
6936:
6919:
6910:
6838:
6832:
6736:
6730:
6688:
6682:
6663:
6617:
6573:
6527:
6489:
6466:
6352:
6329:
6323:
6309:
6286:
6277:
6225:
6146:
6140:
6115:
6047:
5905:
5789:
5778:
5755:
5746:
5737:
5466:
5443:
5434:
5343:
5337:
5314:
5311:
5305:
5299:
5293:
5287:
5099:Now for any quaternion vector
5080:
5045:
5004:
4969:
4947:
4915:
4909:
4877:
4800:
4764:
4755:
4734:
4689:
4659:
4645:
4642:
4636:
4608:
4559:
4541:
4289:
4283:
4277:
4268:
4256:
4250:
4197:
4191:
4180:
4177:
4171:
4165:
4136:
4133:
4127:
4121:
4118:
4112:
4106:
4103:
4097:
4082:
3921:
3915:
3879:
3873:
3867:
3861:
3842:
3838:
3832:
3826:
3823:
3820:
3814:
3799:
3680:
3674:
3655:
3649:
3576:
3573:
3567:
3552:
3421:
3415:
3403:
3397:
3340:
3337:
3331:
3325:
3319:
3313:
3265:
3257:
3248:
3237:
3231:
3220:
3214:
3211:
3205:
3189:
3186:
3183:
3174:
3168:
3162:
3159:
3153:
3137:
3123:
3117:
3108:
3057:
3054:
3045:
3039:
3033:
3030:
3024:
3018:
3009:
2800:
2797:
2781:
2778:
2772:
2758:
2742:
2739:
2733:
2727:
2711:
2708:
2702:
2699:
2693:
2661:
2598:
2566:
2342:
2328:
2285:
2265:
2194:
2188:
2157:
2148:
2066:
2060:
2033:
2027:
2008:
2002:
1996:
1990:
1929:
1923:
1892:
1886:
1867:
1861:
1816:
1810:
1794:
1791:
1785:
1779:
1773:
1767:
1705:
1691:
1652:
1641:
1625:
1616:
1610:
1562:
1507:
1462:
1291:
1185:
1043:
1031:
960:
930:
921:
900:
873:
864:
814:
764:
758:
742:
736:
720:
693:which is a dual scalar, and
677:
627:
621:
605:
599:
583:
538:
505:
499:
483:
480:
468:
452:
419:The multiplication of a screw
403:
376:
370:
358:
355:
343:
334:
322:
257:
241:
1:
7297:Twist (rational trigonometry)
7292:Twist (differential geometry)
4499:half the displacement on the
4444:) for any vector quaternions
4340:borrows some of the ideas of
79:of lines which is central to
7508:10.1007/978-3-642-16135-3_18
7370:Geometric Design of Linkages
5194:Consider the set of forces
5142:generated by the parameters
1004:) is the derivative of
7124:is reciprocal to the twist
5396:The velocity of each point
1100:) be the dual angle, where
7709:
7421:Featherstone, Roy (2008).
7394:Featherstone, Roy (1987).
5536:of each point is given by
3076:which operates on a screw
7585:Clifford, William Kingdon
7424:Robot Dynamics Algorithms
7397:Robot Dynamics Algorithms
7315:Dimentberg, F. M. (1965)
6508:then work takes the form
5134:of dual quaternions is a
4863:so, the homography sends
4531:). Now the homography is
7656:William Kingdon Clifford
7237:William Kingdon Clifford
7148:(rigid body mechanics).
4326:reflection (mathematics)
68:Screw theory provides a
7400:. Kluwer Academic Pub.
4322:transformation geometry
4211:{\displaystyle =e^{t}.}
1732:In order to define the
7678:Mechanical engineering
7661:Ravi Banavar notes on
7286:Newton–Euler equations
7221:. The idea is also in
7199:William Rowan Hamilton
7111:
7040:
6975:
6867:
6785:
6711:
6589:
6499:
6430:
6394:
6248:
6183:
6094:
6005:
5945:
5870:
5805:
5721:
5673:
5484:
5375:
5090:
4854:
4699:
4299:
4212:
4146:
4052:
3896:
3740:
3481:
3286:dual orthogonal matrix
3275:
3067:
2967:
2807:
2608:
2504:
2433:
2352:
2295:
2205:
2116:
1939:
1826:
1715:
1662:
1517:
1461:
1423:
1311:
1198:
1084:
970:
824:
687:
548:
413:
267:
143:
116:computational geometry
7561:Harvey Lipkin (1983)
7253:Geometrie der Dynamen
7112:
7041:
6976:
6868:
6786:
6712:
6590:
6500:
6410:
6374:
6249:
6163:
6074:
5985:
5925:
5850:
5806:
5701:
5674:
5501:is the derivative of
5485:
5376:
5091:
4855:
4700:
4300:
4213:
4147:
4053:
3897:
3741:
3482:
3276:
3068:
3066:{\displaystyle =(,),}
2995: = (,
2968:
2808:
2609:
2526: = (,
2505:
2434:
2353:
2296:
2241:is constant, that is
2206:
2117:
1956:/dt. Now substitute
1940:
1827:
1716:
1663:
1518:
1441:
1403:
1312:
1214:, and ẑ = (
1199:
1085:
971:
825:
688:
549:
414:
268:
141:
7347:Ball, R. S. (1876).
7245:Aleksandr Kotelnikov
7056:
6991:
6899:
6813:
6727:
6605:
6515:
6264:
5825:
5689:
5543:
5410:
5272:
4874:
4719:
4538:
4316:Screws by reflection
4244:
4162:
4079:
3912:
3779:
3532:
3303:
3095:
3006:
2823:
2648:
2553:
2462:
2385:
2315:
2252:
2135:
1975:
1846:
1759:
1678:
1549:
1390:
1226:
1120:
1027:
849:
697:
560:
443:
313:
228:
221:be an ordered pair
154:. This is known as
53:, that arise in the
7589:Mathematical Papers
7164:Cayley–Klein metric
7138:Robert Stawell Ball
5253: = 1,...,
4358:exponential mapping
4342:projective geometry
2544:Plücker coordinates
1370: = 1,...,
1092:Let ẑ = (
1021:Plücker coordinates
206:Plücker coordinates
97:addition of vectors
89:Plücker coordinates
81:rigid body dynamics
7568:2016-03-05 at the
7225:parametrizing the
7190:, J. R. Phillips.
7107:
7036:
6971:
6888:Twists in robotics
6863:
6781:
6772:
6707:
6585:
6495:
6244:
6242:
5801:
5669:
5480:
5371:
5217:act on the points
5086:
4850:
4695:
4599:
4338:inversive geometry
4295:
4208:
4145:{\displaystyle =,}
4142:
4048:
4039:
3984:
3892:
3736:
3727:
3697:
3620:
3477:
3468:
3438:
3377:
3271:
3063:
2963:
2954:
2908:
2866:
2803:
2604:
2500:
2491:
2429:
2420:
2348:
2291:
2201:
2112:
1935:
1822:
1711:
1658:
1513:
1307:
1194:
1080:
966:
820:
683:
544:
409:
263:
144:
128:rigid-body motions
120:multibody dynamics
7604:389(2):1352 to 64
7563:Metrical Geometry
7533:978-3-540-71990-8
7488:978-1-107-15630-2
7461:978-0-8493-7981-9
7434:978-0-387-74315-8
7407:978-0-89838-230-3
7380:978-1-4419-7892-9
7353:. Hodges, Foster.
7156:elliptic geometry
7089:
7049:In this case, if
7024:
6962:
6939:
6913:
6798:Reciprocal screws
6312:
6289:
6228:
6143:
6050:
5908:
5749:
5437:
5138:. A subgroup has
4759:
4685:
4679:
4625:
4619:
4555:
4549:
4346:analytic geometry
4094:
4065:of the Lie group
3967:
3962:
3950:
3886:
3858:
3811:
3787:
3715:
3671:
3666:
3646:
3602:
3583:
3564:
3540:
3456:
3412:
3365:
3347:
3310:
3120:
3021:
2451:, let the vector
2191:
2160:
2045:
1904:
1534:acting at points
1294:
1188:
1069:
980:(ẑ) = (
924:
867:
455:
337:
325:
213:Algebra of screws
109:Poinsot's theorem
16:(Redirected from
7700:
7642:
7641:
7611:
7605:
7598:
7592:
7582:
7576:
7559:
7553:
7544:
7538:
7537:
7517:
7511:
7499:
7493:
7492:
7472:
7466:
7465:
7445:
7439:
7438:
7418:
7412:
7411:
7391:
7385:
7384:
7364:
7355:
7354:
7344:
7335:
7328:
7322:
7313:
7257:Wilhelm Blaschke
7184:F. M. Dimentberg
7172:von Staudt conic
7168:symmetric matrix
7160:Erlangen Program
7120:then the wrench
7116:
7114:
7113:
7108:
7091:
7090:
7085:
7080:
7074:
7073:
7045:
7043:
7042:
7037:
7026:
7025:
7020:
7015:
7009:
7008:
6980:
6978:
6977:
6972:
6964:
6963:
6955:
6949:
6941:
6940:
6932:
6926:
6915:
6914:
6909:
6904:
6884:are reciprocal.
6876:then the screws
6872:
6870:
6869:
6864:
6847:
6846:
6831:
6830:
6790:
6788:
6787:
6782:
6777:
6776:
6716:
6714:
6713:
6708:
6697:
6696:
6681:
6680:
6662:
6654:
6653:
6648:
6639:
6638:
6633:
6624:
6594:
6592:
6591:
6586:
6572:
6564:
6563:
6558:
6549:
6548:
6543:
6534:
6504:
6502:
6501:
6496:
6488:
6487:
6482:
6473:
6462:
6458:
6457:
6456:
6451:
6442:
6441:
6436:
6429:
6424:
6406:
6405:
6400:
6393:
6388:
6365:
6364:
6351:
6350:
6345:
6336:
6322:
6314:
6313:
6305:
6299:
6291:
6290:
6282:
6273:
6272:
6253:
6251:
6250:
6245:
6243:
6230:
6229:
6221:
6215:
6211:
6210:
6209:
6204:
6195:
6194:
6189:
6182:
6177:
6145:
6144:
6136:
6130:
6122:
6111:
6107:
6106:
6105:
6100:
6093:
6088:
6062:
6052:
6051:
6043:
6037:
6033:
6032:
6031:
6026:
6017:
6016:
6011:
6004:
5999:
5970:
5962:
5958:
5957:
5956:
5951:
5944:
5939:
5910:
5909:
5901:
5895:
5887:
5883:
5882:
5881:
5876:
5869:
5864:
5810:
5808:
5807:
5802:
5788:
5777:
5769:
5768:
5763:
5751:
5750:
5742:
5733:
5732:
5727:
5720:
5715:
5678:
5676:
5675:
5670:
5659:
5658:
5653:
5644:
5643:
5638:
5617:
5616:
5611:
5602:
5601:
5596:
5581:
5580:
5575:
5566:
5565:
5560:
5489:
5487:
5486:
5481:
5476:
5465:
5457:
5456:
5451:
5439:
5438:
5430:
5424:
5423:
5418:
5380:
5378:
5377:
5372:
5336:
5328:
5327:
5322:
5286:
5285:
5280:
5095:
5093:
5092:
5087:
5079:
5078:
5063:
5062:
5044:
5043:
5022:
5021:
5003:
5002:
4987:
4986:
4968:
4967:
4946:
4945:
4930:
4929:
4908:
4907:
4892:
4891:
4859:
4857:
4856:
4851:
4846:
4845:
4830:
4829:
4811:
4810:
4798:
4797:
4779:
4778:
4760:
4758:
4723:
4708:The inverse for
4704:
4702:
4701:
4696:
4683:
4677:
4670:
4669:
4657:
4656:
4635:
4634:
4623:
4617:
4604:
4603:
4596:
4595:
4553:
4547:
4519:) and z* = exp((
4462:dual quaternions
4304:
4302:
4301:
4296:
4217:
4215:
4214:
4209:
4204:
4203:
4151:
4149:
4148:
4143:
4096:
4095:
4087:
4057:
4055:
4054:
4049:
4044:
4043:
4024:
4010:
3989:
3988:
3969:
3968:
3963:
3958:
3952:
3951:
3901:
3899:
3898:
3893:
3888:
3887:
3860:
3859:
3853:
3852:
3813:
3812:
3804:
3795:
3794:
3789:
3788:
3745:
3743:
3742:
3737:
3732:
3731:
3717:
3716:
3702:
3701:
3673:
3672:
3667:
3662:
3648:
3647:
3639:
3625:
3624:
3610:
3609:
3604:
3603:
3585:
3584:
3566:
3565:
3557:
3548:
3547:
3542:
3541:
3486:
3484:
3483:
3478:
3473:
3472:
3458:
3457:
3443:
3442:
3414:
3413:
3382:
3381:
3367:
3366:
3349:
3348:
3312:
3311:
3280:
3278:
3277:
3272:
3264:
3244:
3227:
3204:
3196:
3152:
3144:
3132:
3131:
3122:
3121:
3113:
3104:
3103:
3072:
3070:
3069:
3064:
3023:
3022:
3014:
2972:
2970:
2969:
2964:
2959:
2958:
2951:
2943:
2934:
2926:
2913:
2912:
2871:
2870:
2863:
2855:
2846:
2838:
2812:
2810:
2809:
2804:
2796:
2788:
2768:
2757:
2749:
2726:
2718:
2692:
2684:
2676:
2668:
2657:
2656:
2613:
2611:
2610:
2605:
2597:
2589:
2581:
2573:
2562:
2561:
2542:, which has the
2509:
2507:
2506:
2501:
2496:
2495:
2443:Prismatic joints
2438:
2436:
2435:
2430:
2425:
2424:
2357:
2355:
2354:
2349:
2341:
2324:
2323:
2300:
2298:
2297:
2292:
2278:
2261:
2260:
2210:
2208:
2207:
2202:
2193:
2192:
2184:
2178:
2170:
2162:
2161:
2153:
2144:
2143:
2121:
2119:
2118:
2113:
2108:
2100:
2092:
2084:
2076:
2059:
2058:
2053:
2046:
2043:
2040:
2023:
2015:
1989:
1988:
1983:
1944:
1942:
1941:
1936:
1922:
1914:
1909:
1905:
1903:
1895:
1878:
1860:
1859:
1854:
1835:The velocity of
1831:
1829:
1828:
1823:
1809:
1801:
1766:
1720:
1718:
1717:
1712:
1704:
1687:
1686:
1667:
1665:
1664:
1659:
1651:
1640:
1632:
1609:
1601:
1593:
1585:
1577:
1569:
1558:
1557:
1522:
1520:
1519:
1514:
1506:
1505:
1500:
1491:
1490:
1485:
1476:
1475:
1470:
1460:
1455:
1437:
1436:
1431:
1430:
1422:
1417:
1399:
1398:
1316:
1314:
1313:
1308:
1303:
1302:
1296:
1295:
1287:
1278:
1274:
1273:
1263:
1259:
1258:
1245:
1244:
1235:
1234:
1203:
1201:
1200:
1195:
1190:
1189:
1181:
1172:
1168:
1167:
1157:
1153:
1152:
1139:
1138:
1129:
1128:
1089:
1087:
1086:
1081:
1070:
1068:
1067:
1058:
1057:
1051:
1046:
1041:
1040:
1034:
1019:Unit screws are
975:
973:
972:
967:
926:
925:
917:
869:
868:
860:
829:
827:
826:
821:
813:
805:
797:
789:
779:
771:
757:
749:
735:
727:
716:
715:
706:
705:
692:
690:
689:
684:
676:
668:
660:
652:
642:
634:
620:
612:
598:
590:
579:
578:
569:
568:
553:
551:
550:
545:
537:
526:
515:
498:
490:
464:
463:
457:
456:
448:
418:
416:
415:
410:
339:
338:
330:
327:
326:
318:
287:
281:
272:
270:
269:
264:
256:
248:
237:
236:
156:Chasles' theorem
152:
151:
124:dual quaternions
105:Chasles' theorem
21:
7708:
7707:
7703:
7702:
7701:
7699:
7698:
7697:
7668:
7667:
7651:
7646:
7645:
7630:10.2307/2369176
7613:
7612:
7608:
7599:
7595:
7583:
7579:
7570:Wayback Machine
7560:
7556:
7545:
7541:
7534:
7519:
7518:
7514:
7500:
7496:
7489:
7478:Modern Robotics
7474:
7473:
7469:
7462:
7447:
7446:
7442:
7435:
7420:
7419:
7415:
7408:
7393:
7392:
7388:
7381:
7366:
7365:
7358:
7346:
7345:
7338:
7329:
7325:
7314:
7310:
7305:
7277:
7269:Arthur Buchheim
7261:Julian Coolidge
7223:Euler's formula
7188:Kenneth H. Hunt
7166:. The use of a
7134:
7054:
7053:
6989:
6988:
6897:
6896:
6890:
6811:
6810:
6800:
6771:
6770:
6765:
6759:
6758:
6753:
6743:
6725:
6724:
6643:
6628:
6603:
6602:
6553:
6538:
6513:
6512:
6477:
6446:
6431:
6395:
6373:
6369:
6340:
6262:
6261:
6241:
6240:
6199:
6184:
6162:
6158:
6095:
6073:
6069:
6060:
6059:
6021:
6006:
5984:
5980:
5946:
5924:
5920:
5871:
5849:
5845:
5838:
5823:
5822:
5758:
5722:
5687:
5686:
5648:
5633:
5606:
5591:
5570:
5555:
5541:
5540:
5532:
5523:
5446:
5413:
5408:
5407:
5402:
5392:
5317:
5275:
5270:
5269:
5248:
5239:
5230:
5223:
5216:
5207:
5200:
5192:
5067:
5048:
5026:
5007:
4991:
4972:
4953:
4931:
4918:
4893:
4880:
4872:
4871:
4831:
4815:
4799:
4780:
4767:
4727:
4717:
4716:
4658:
4648:
4626:
4598:
4597:
4587:
4585:
4579:
4578:
4573:
4563:
4536:
4535:
4402:= 0. Note that
4354:
4318:
4242:
4241:
4229:), and a twist
4186:
4160:
4159:
4077:
4076:
4038:
4037:
4032:
4026:
4025:
4004:
3994:
3983:
3982:
3977:
3971:
3970:
3938:
3928:
3910:
3909:
3841:
3782:
3777:
3776:
3726:
3725:
3719:
3718:
3704:
3696:
3695:
3690:
3684:
3683:
3658:
3630:
3619:
3618:
3612:
3611:
3597:
3590:
3535:
3530:
3529:
3467:
3466:
3460:
3459:
3445:
3437:
3436:
3431:
3425:
3424:
3406:
3387:
3376:
3375:
3369:
3368:
3354:
3301:
3300:
3294:
3093:
3092:
3004:
3003:
2953:
2952:
2936:
2935:
2915:
2907:
2906:
2901:
2892:
2891:
2886:
2876:
2865:
2864:
2848:
2847:
2827:
2821:
2820:
2646:
2645:
2641:, which yield.
2551:
2550:
2516:
2490:
2489:
2483:
2482:
2472:
2460:
2459:
2449:prismatic joint
2445:
2419:
2418:
2406:
2405:
2395:
2383:
2382:
2364:
2362:Revolute joints
2313:
2312:
2250:
2249:
2133:
2132:
2048:
1978:
1973:
1972:
1960: = (
1896:
1879:
1873:
1849:
1844:
1843:
1757:
1756:
1730:
1676:
1675:
1547:
1546:
1495:
1480:
1465:
1424:
1388:
1387:
1382:
1365:
1323:
1264:
1249:
1224:
1223:
1158:
1143:
1118:
1117:
1025:
1024:
847:
846:
695:
694:
558:
557:
441:
440:
311:
310:
283:
277:
226:
225:
215:
198:
181:
172:
149:
148:
136:
85:lines of action
43:linear velocity
28:
23:
22:
15:
12:
11:
5:
7706:
7704:
7696:
7695:
7690:
7685:
7680:
7670:
7669:
7666:
7665:
7659:
7650:
7649:External links
7647:
7644:
7643:
7624:(4): 293–326.
7606:
7593:
7591:, p. 381.
7577:
7554:
7539:
7532:
7512:
7494:
7487:
7467:
7460:
7440:
7433:
7413:
7406:
7386:
7379:
7356:
7336:
7323:
7307:
7306:
7304:
7301:
7300:
7299:
7294:
7289:
7283:
7276:
7273:
7243:, followed by
7201:displayed the
7180:W. K. Clifford
7176:Julius Plücker
7133:
7130:
7118:
7117:
7106:
7103:
7100:
7097:
7094:
7088:
7084:
7077:
7072:
7067:
7064:
7061:
7047:
7046:
7035:
7032:
7029:
7023:
7019:
7012:
7007:
7002:
6999:
6996:
6982:
6981:
6970:
6967:
6961:
6958:
6952:
6948:
6944:
6938:
6935:
6929:
6925:
6921:
6918:
6912:
6908:
6889:
6886:
6874:
6873:
6862:
6859:
6856:
6853:
6850:
6845:
6840:
6837:
6834:
6829:
6824:
6821:
6818:
6799:
6796:
6792:
6791:
6780:
6775:
6769:
6766:
6764:
6761:
6760:
6757:
6754:
6752:
6749:
6748:
6746:
6741:
6738:
6735:
6732:
6718:
6717:
6706:
6703:
6700:
6695:
6690:
6687:
6684:
6679:
6674:
6671:
6668:
6665:
6661:
6657:
6652:
6647:
6642:
6637:
6632:
6627:
6623:
6619:
6616:
6613:
6610:
6596:
6595:
6584:
6581:
6578:
6575:
6571:
6567:
6562:
6557:
6552:
6547:
6542:
6537:
6533:
6529:
6526:
6523:
6520:
6506:
6505:
6494:
6491:
6486:
6481:
6476:
6472:
6468:
6465:
6461:
6455:
6450:
6445:
6440:
6435:
6428:
6423:
6420:
6417:
6413:
6409:
6404:
6399:
6392:
6387:
6384:
6381:
6377:
6372:
6368:
6363:
6357:
6354:
6349:
6344:
6339:
6335:
6331:
6328:
6325:
6321:
6317:
6311:
6308:
6302:
6298:
6294:
6288:
6285:
6279:
6276:
6271:
6255:
6254:
6239:
6236:
6233:
6227:
6224:
6218:
6214:
6208:
6203:
6198:
6193:
6188:
6181:
6176:
6173:
6170:
6166:
6161:
6157:
6154:
6151:
6148:
6142:
6139:
6133:
6129:
6125:
6121:
6117:
6114:
6110:
6104:
6099:
6092:
6087:
6084:
6081:
6077:
6072:
6068:
6065:
6063:
6061:
6058:
6055:
6049:
6046:
6040:
6036:
6030:
6025:
6020:
6015:
6010:
6003:
5998:
5995:
5992:
5988:
5983:
5979:
5976:
5973:
5969:
5965:
5961:
5955:
5950:
5943:
5938:
5935:
5932:
5928:
5923:
5919:
5916:
5913:
5907:
5904:
5898:
5894:
5890:
5886:
5880:
5875:
5868:
5863:
5860:
5857:
5853:
5848:
5844:
5841:
5839:
5837:
5834:
5831:
5830:
5812:
5811:
5800:
5797:
5794:
5791:
5787:
5783:
5780:
5776:
5772:
5767:
5762:
5757:
5754:
5748:
5745:
5739:
5736:
5731:
5726:
5719:
5714:
5711:
5708:
5704:
5700:
5697:
5694:
5680:
5679:
5668:
5665:
5662:
5657:
5652:
5647:
5642:
5637:
5632:
5629:
5626:
5623:
5620:
5615:
5610:
5605:
5600:
5595:
5590:
5587:
5584:
5579:
5574:
5569:
5564:
5559:
5554:
5551:
5548:
5528:
5519:
5491:
5490:
5479:
5475:
5471:
5468:
5464:
5460:
5455:
5450:
5445:
5442:
5436:
5433:
5427:
5422:
5417:
5400:
5388:
5382:
5381:
5370:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5345:
5342:
5339:
5335:
5331:
5326:
5321:
5316:
5313:
5310:
5307:
5304:
5301:
5298:
5295:
5292:
5289:
5284:
5279:
5244:
5235:
5228:
5221:
5212:
5205:
5198:
5191:
5188:
5128:group of units
5126:Evidently the
5097:
5096:
5085:
5082:
5077:
5074:
5070:
5066:
5061:
5058:
5055:
5051:
5047:
5042:
5039:
5036:
5033:
5029:
5025:
5020:
5017:
5014:
5010:
5006:
5001:
4998:
4994:
4990:
4985:
4982:
4979:
4975:
4971:
4966:
4963:
4960:
4956:
4952:
4949:
4944:
4941:
4938:
4934:
4928:
4925:
4921:
4917:
4914:
4911:
4906:
4903:
4900:
4896:
4890:
4887:
4883:
4879:
4861:
4860:
4849:
4844:
4841:
4838:
4834:
4828:
4825:
4822:
4818:
4814:
4809:
4806:
4802:
4796:
4793:
4790:
4787:
4783:
4777:
4774:
4770:
4766:
4763:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4733:
4730:
4726:
4706:
4705:
4694:
4691:
4688:
4682:
4676:
4673:
4668:
4665:
4661:
4655:
4651:
4647:
4644:
4641:
4638:
4633:
4629:
4622:
4616:
4613:
4610:
4607:
4602:
4594:
4590:
4586:
4584:
4581:
4580:
4577:
4574:
4572:
4569:
4568:
4566:
4561:
4558:
4552:
4546:
4543:
4464:. This 3-flat
4353:
4350:
4317:
4314:
4306:
4305:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4252:
4249:
4219:
4218:
4207:
4202:
4199:
4196:
4193:
4189:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4153:
4152:
4141:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4108:
4105:
4102:
4099:
4093:
4090:
4084:
4059:
4058:
4047:
4042:
4036:
4033:
4031:
4028:
4027:
4023:
4019:
4016:
4013:
4009:
4005:
4003:
4000:
3999:
3997:
3992:
3987:
3981:
3978:
3976:
3973:
3972:
3966:
3955:
3945:
3942:
3939:
3937:
3934:
3933:
3931:
3926:
3923:
3920:
3917:
3903:
3902:
3891:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3851:
3848:
3844:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3810:
3807:
3801:
3798:
3793:
3747:
3746:
3735:
3730:
3724:
3721:
3720:
3710:
3709:
3707:
3700:
3694:
3691:
3689:
3686:
3685:
3682:
3679:
3676:
3670:
3659:
3657:
3654:
3651:
3645:
3642:
3636:
3635:
3633:
3628:
3623:
3617:
3614:
3613:
3608:
3596:
3595:
3593:
3588:
3578:
3575:
3572:
3569:
3563:
3560:
3554:
3551:
3546:
3488:
3487:
3476:
3471:
3465:
3462:
3461:
3451:
3450:
3448:
3441:
3435:
3432:
3430:
3427:
3426:
3423:
3420:
3417:
3407:
3405:
3402:
3399:
3396:
3393:
3392:
3390:
3385:
3380:
3374:
3371:
3370:
3360:
3359:
3357:
3352:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3293:
3290:
3282:
3281:
3270:
3267:
3263:
3259:
3256:
3253:
3250:
3247:
3243:
3239:
3236:
3233:
3230:
3226:
3222:
3219:
3216:
3213:
3210:
3207:
3203:
3199:
3195:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3151:
3147:
3143:
3139:
3135:
3130:
3125:
3119:
3116:
3110:
3107:
3102:
3080: = (
3074:
3073:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3020:
3017:
3011:
2974:
2973:
2962:
2957:
2950:
2946:
2942:
2938:
2937:
2933:
2929:
2925:
2921:
2920:
2918:
2911:
2905:
2902:
2900:
2897:
2894:
2893:
2890:
2887:
2885:
2882:
2881:
2879:
2874:
2869:
2862:
2858:
2854:
2850:
2849:
2845:
2841:
2837:
2833:
2832:
2830:
2814:
2813:
2802:
2799:
2795:
2791:
2787:
2783:
2780:
2777:
2774:
2771:
2767:
2763:
2760:
2756:
2752:
2748:
2744:
2741:
2738:
2735:
2732:
2729:
2725:
2721:
2717:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2691:
2687:
2683:
2679:
2675:
2671:
2667:
2663:
2660:
2655:
2615:
2614:
2603:
2600:
2596:
2592:
2588:
2584:
2580:
2576:
2572:
2568:
2565:
2560:
2515:
2512:
2511:
2510:
2499:
2494:
2488:
2485:
2484:
2481:
2478:
2477:
2475:
2470:
2467:
2444:
2441:
2440:
2439:
2428:
2423:
2417:
2414:
2411:
2408:
2407:
2404:
2401:
2400:
2398:
2393:
2390:
2368:revolute joint
2363:
2360:
2359:
2358:
2347:
2344:
2340:
2336:
2333:
2330:
2327:
2322:
2302:
2301:
2290:
2287:
2284:
2281:
2277:
2273:
2270:
2267:
2264:
2259:
2212:
2211:
2199:
2196:
2190:
2187:
2181:
2177:
2173:
2169:
2165:
2159:
2156:
2150:
2147:
2142:
2123:
2122:
2111:
2107:
2103:
2099:
2095:
2091:
2087:
2083:
2079:
2075:
2071:
2068:
2065:
2062:
2057:
2052:
2039:
2035:
2032:
2029:
2026:
2022:
2018:
2014:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1987:
1982:
1946:
1945:
1934:
1931:
1928:
1925:
1921:
1917:
1913:
1908:
1902:
1899:
1894:
1891:
1888:
1885:
1882:
1876:
1872:
1869:
1866:
1863:
1858:
1853:
1833:
1832:
1821:
1818:
1815:
1812:
1808:
1804:
1800:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1765:
1729:
1726:
1722:
1721:
1710:
1707:
1703:
1699:
1696:
1693:
1690:
1685:
1669:
1668:
1657:
1654:
1650:
1646:
1643:
1639:
1635:
1631:
1627:
1624:
1621:
1618:
1615:
1612:
1608:
1604:
1600:
1596:
1592:
1588:
1584:
1580:
1576:
1572:
1568:
1564:
1561:
1556:
1524:
1523:
1512:
1509:
1504:
1499:
1494:
1489:
1484:
1479:
1474:
1469:
1464:
1459:
1454:
1451:
1448:
1444:
1440:
1435:
1429:
1421:
1416:
1413:
1410:
1406:
1402:
1397:
1378:
1361:
1322:
1319:
1318:
1317:
1306:
1301:
1293:
1290:
1284:
1281:
1277:
1272:
1267:
1262:
1257:
1252:
1248:
1243:
1238:
1233:
1204:
1193:
1187:
1184:
1178:
1175:
1171:
1166:
1161:
1156:
1151:
1146:
1142:
1137:
1132:
1127:
1090:
1079:
1076:
1073:
1066:
1061:
1056:
1049:
1045:
1039:
1033:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
923:
920:
914:
911:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
866:
863:
857:
854:
819:
816:
812:
808:
804:
800:
796:
792:
788:
782:
778:
774:
770:
766:
763:
760:
756:
752:
748:
744:
741:
738:
734:
730:
726:
722:
719:
714:
709:
704:
682:
679:
675:
671:
667:
663:
659:
655:
651:
645:
641:
637:
633:
629:
626:
623:
619:
615:
611:
607:
604:
601:
597:
593:
589:
585:
582:
577:
572:
567:
543:
540:
536:
532:
529:
525:
521:
518:
514:
510:
507:
504:
501:
497:
493:
489:
485:
482:
479:
476:
473:
470:
467:
462:
454:
451:
423: = (
408:
405:
402:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
345:
342:
336:
333:
324:
321:
274:
273:
262:
259:
255:
251:
247:
243:
240:
235:
214:
211:
197:
194:
180:
177:
171:
168:
135:
134:Basic concepts
132:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7705:
7694:
7691:
7689:
7686:
7684:
7681:
7679:
7676:
7675:
7673:
7664:
7660:
7657:
7653:
7652:
7648:
7639:
7635:
7631:
7627:
7623:
7619:
7618:
7610:
7607:
7603:
7597:
7594:
7590:
7586:
7581:
7578:
7575:
7571:
7567:
7564:
7558:
7555:
7552:
7548:
7543:
7540:
7535:
7529:
7525:
7524:
7516:
7513:
7509:
7505:
7498:
7495:
7490:
7484:
7480:
7479:
7471:
7468:
7463:
7457:
7454:. CRC Press.
7453:
7452:
7444:
7441:
7436:
7430:
7426:
7425:
7417:
7414:
7409:
7403:
7399:
7398:
7390:
7387:
7382:
7376:
7372:
7371:
7363:
7361:
7357:
7352:
7351:
7343:
7341:
7337:
7333:
7327:
7324:
7320:
7319:
7312:
7309:
7302:
7298:
7295:
7293:
7290:
7287:
7284:
7282:
7279:
7278:
7274:
7272:
7270:
7266:
7262:
7258:
7254:
7250:
7246:
7242:
7238:
7234:
7232:
7231:complex plane
7228:
7224:
7220:
7216:
7212:
7208:
7204:
7200:
7196:
7191:
7189:
7185:
7181:
7177:
7173:
7169:
7165:
7161:
7157:
7153:
7149:
7147:
7143:
7139:
7131:
7129:
7127:
7123:
7104:
7101:
7098:
7095:
7092:
7075:
7065:
7062:
7059:
7052:
7051:
7050:
7033:
7030:
7027:
7010:
7000:
6997:
6994:
6987:
6986:
6985:
6968:
6956:
6950:
6942:
6933:
6927:
6916:
6895:
6894:
6893:
6887:
6885:
6883:
6879:
6860:
6857:
6854:
6851:
6848:
6822:
6819:
6816:
6809:
6808:
6807:
6805:
6797:
6795:
6778:
6773:
6767:
6762:
6755:
6750:
6744:
6739:
6723:
6722:
6721:
6704:
6701:
6698:
6672:
6669:
6666:
6655:
6650:
6640:
6635:
6625:
6614:
6611:
6608:
6601:
6600:
6599:
6582:
6579:
6576:
6565:
6560:
6550:
6545:
6535:
6524:
6521:
6518:
6511:
6510:
6509:
6492:
6484:
6474:
6463:
6459:
6453:
6443:
6438:
6426:
6421:
6418:
6415:
6411:
6407:
6402:
6390:
6385:
6382:
6379:
6375:
6370:
6366:
6355:
6347:
6337:
6326:
6315:
6306:
6300:
6292:
6283:
6274:
6260:
6259:
6258:
6237:
6234:
6231:
6222:
6216:
6212:
6206:
6196:
6191:
6179:
6174:
6171:
6168:
6164:
6159:
6155:
6152:
6149:
6137:
6131:
6123:
6112:
6108:
6102:
6090:
6085:
6082:
6079:
6075:
6070:
6066:
6064:
6056:
6053:
6044:
6038:
6034:
6028:
6018:
6013:
6001:
5996:
5993:
5990:
5986:
5981:
5977:
5974:
5971:
5963:
5959:
5953:
5941:
5936:
5933:
5930:
5926:
5921:
5917:
5914:
5911:
5902:
5896:
5888:
5884:
5878:
5866:
5861:
5858:
5855:
5851:
5846:
5842:
5840:
5835:
5832:
5821:
5820:
5819:
5817:
5798:
5795:
5792:
5781:
5770:
5765:
5752:
5743:
5734:
5729:
5717:
5712:
5709:
5706:
5702:
5698:
5695:
5692:
5685:
5684:
5683:
5666:
5663:
5660:
5655:
5645:
5640:
5630:
5627:
5624:
5621:
5618:
5613:
5603:
5598:
5588:
5585:
5582:
5577:
5567:
5562:
5552:
5549:
5546:
5539:
5538:
5537:
5535:
5531:
5527:
5522:
5518:
5515:
5510:
5508:
5504:
5500:
5496:
5477:
5469:
5458:
5453:
5440:
5431:
5425:
5420:
5406:
5405:
5404:
5399:
5394:
5391:
5387:
5368:
5365:
5362:
5359:
5356:
5353:
5350:
5347:
5340:
5329:
5324:
5308:
5302:
5296:
5290:
5282:
5268:
5267:
5266:
5264:
5260:
5256:
5252:
5247:
5243:
5238:
5234:
5227:
5220:
5215:
5211:
5204:
5197:
5189:
5187:
5185:
5181:
5177:
5173:
5169:
5165:
5161:
5157:
5153:
5149:
5145:
5141:
5137:
5133:
5129:
5124:
5122:
5118:
5114:
5110:
5106:
5102:
5083:
5075:
5072:
5068:
5064:
5059:
5056:
5053:
5049:
5040:
5037:
5034:
5031:
5027:
5023:
5018:
5015:
5012:
5008:
4999:
4996:
4992:
4988:
4983:
4980:
4977:
4973:
4964:
4961:
4958:
4954:
4950:
4942:
4939:
4936:
4932:
4926:
4923:
4919:
4912:
4904:
4901:
4898:
4894:
4888:
4885:
4881:
4870:
4869:
4868:
4866:
4847:
4842:
4839:
4836:
4832:
4826:
4823:
4820:
4816:
4812:
4807:
4804:
4794:
4791:
4788:
4785:
4781:
4775:
4772:
4768:
4761:
4752:
4749:
4746:
4743:
4740:
4737:
4731:
4728:
4724:
4715:
4714:
4713:
4711:
4692:
4686:
4680:
4674:
4671:
4666:
4663:
4653:
4649:
4631:
4627:
4620:
4614:
4611:
4605:
4600:
4592:
4588:
4582:
4575:
4570:
4564:
4556:
4550:
4544:
4534:
4533:
4532:
4530:
4526:
4522:
4518:
4514:
4510:
4506:
4502:
4498:
4494:
4490:
4485:
4483:
4479:
4476:constructed,
4475:
4471:
4467:
4463:
4459:
4455:
4451:
4447:
4443:
4439:
4435:
4431:
4427:
4423:
4420:
4416:
4413:
4409:
4405:
4401:
4397:
4393:
4389:
4385:
4380:
4378:
4374:
4370:
4366:
4361:
4359:
4351:
4349:
4347:
4343:
4339:
4335:
4331:
4327:
4323:
4315:
4313:
4311:
4292:
4286:
4280:
4274:
4271:
4265:
4262:
4259:
4253:
4247:
4240:
4239:
4238:
4236:
4232:
4228:
4224:
4205:
4200:
4194:
4187:
4183:
4174:
4168:
4158:
4157:
4156:
4139:
4130:
4124:
4115:
4109:
4100:
4091:
4088:
4075:
4074:
4073:
4070:
4068:
4064:
4045:
4040:
4034:
4029:
4017:
4014:
4011:
3995:
3990:
3985:
3979:
3974:
3964:
3953:
3940:
3929:
3924:
3918:
3908:
3907:
3906:
3889:
3876:
3870:
3864:
3849:
3846:
3835:
3829:
3817:
3808:
3805:
3796:
3791:
3775:
3774:
3773:
3771:
3767:
3763:
3759:
3754:
3752:
3733:
3728:
3722:
3705:
3698:
3692:
3687:
3677:
3668:
3652:
3643:
3640:
3631:
3626:
3621:
3615:
3606:
3591:
3586:
3570:
3561:
3558:
3549:
3544:
3528:
3527:
3526:
3523:
3521:
3517:
3513:
3509:
3505:
3501:
3497:
3493:
3474:
3469:
3463:
3446:
3439:
3433:
3428:
3418:
3400:
3394:
3388:
3383:
3378:
3372:
3355:
3350:
3334:
3328:
3322:
3316:
3299:
3298:
3297:
3291:
3289:
3287:
3268:
3254:
3251:
3245:
3234:
3228:
3217:
3208:
3197:
3180:
3177:
3171:
3165:
3156:
3145:
3133:
3114:
3105:
3091:
3090:
3089:
3088:) to obtain,
3087:
3083:
3079:
3060:
3051:
3048:
3042:
3036:
3027:
3015:
3002:
3001:
3000:
2998:
2994:
2989:
2987:
2984: ×
2983:
2980: =
2979:
2960:
2955:
2944:
2927:
2916:
2909:
2903:
2898:
2895:
2888:
2883:
2877:
2872:
2867:
2856:
2839:
2828:
2819:
2818:
2817:
2789:
2775:
2769:
2761:
2750:
2736:
2730:
2719:
2705:
2696:
2685:
2677:
2669:
2658:
2644:
2643:
2642:
2640:
2637: +
2636:
2633: =
2632:
2628:
2625: +
2624:
2621: =
2620:
2601:
2590:
2582:
2574:
2563:
2549:
2548:
2547:
2545:
2541:
2537:
2533:
2529:
2525:
2520:
2513:
2497:
2492:
2486:
2479:
2473:
2468:
2465:
2458:
2457:
2456:
2454:
2450:
2442:
2426:
2421:
2415:
2412:
2409:
2402:
2396:
2391:
2388:
2381:
2380:
2379:
2377:
2373:
2369:
2361:
2345:
2334:
2331:
2325:
2311:
2310:
2309:
2307:
2288:
2282:
2279:
2271:
2268:
2262:
2248:
2247:
2246:
2244:
2240:
2235:
2233:
2230: ×
2229:
2226: +
2225:
2222: =
2221:
2217:
2197:
2185:
2179:
2171:
2163:
2154:
2145:
2131:
2130:
2129:
2126:
2109:
2105:
2101:
2093:
2085:
2077:
2073:
2069:
2063:
2055:
2024:
2016:
1999:
1993:
1985:
1971:
1970:
1969:
1967:
1964: −
1963:
1959:
1955:
1951:
1932:
1926:
1915:
1906:
1900:
1897:
1889:
1883:
1880:
1874:
1870:
1864:
1856:
1842:
1841:
1840:
1838:
1819:
1813:
1802:
1788:
1782:
1776:
1770:
1755:
1754:
1753:
1751:
1747:
1743:
1739:
1735:
1727:
1725:
1708:
1697:
1694:
1688:
1674:
1673:
1672:
1655:
1644:
1633:
1622:
1619:
1613:
1602:
1594:
1586:
1578:
1570:
1559:
1545:
1544:
1543:
1541:
1537:
1533:
1529:
1510:
1502:
1492:
1487:
1477:
1472:
1457:
1452:
1449:
1446:
1442:
1438:
1433:
1419:
1414:
1411:
1408:
1404:
1400:
1386:
1385:
1384:
1381:
1377:
1373:
1369:
1364:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1320:
1304:
1288:
1282:
1279:
1275:
1265:
1260:
1250:
1246:
1236:
1221:
1217:
1213:
1209:
1205:
1191:
1182:
1176:
1173:
1169:
1159:
1154:
1144:
1140:
1130:
1115:
1111:
1107:
1103:
1099:
1095:
1091:
1077:
1074:
1071:
1059:
1047:
1022:
1018:
1017:
1016:
1013:
1011:
1007:
1003:
999:
995:
991:
987:
983:
979:
963:
957:
954:
951:
948:
945:
942:
939:
936:
933:
927:
918:
912:
909:
903:
897:
894:
891:
888:
885:
882:
879:
876:
870:
861:
855:
852:
844:
840:
836:
831:
817:
806:
798:
790:
780:
772:
761:
750:
739:
728:
717:
707:
680:
669:
661:
653:
643:
635:
624:
613:
602:
591:
580:
570:
554:
541:
530:
527:
519:
516:
508:
502:
491:
477:
474:
471:
465:
449:
438:
434:
430:
426:
422:
406:
400:
397:
394:
391:
388:
385:
382:
379:
373:
367:
364:
361:
352:
349:
346:
340:
331:
319:
308:
307:
302:
298:
293:
291:
286:
280:
260:
249:
238:
224:
223:
222:
220:
212:
210:
207:
203:
195:
193:
189:
186:
178:
176:
169:
167:
165:
161:
157:
153:
140:
133:
131:
129:
125:
121:
117:
112:
110:
106:
100:
98:
94:
90:
86:
82:
78:
74:
71:
66:
64:
60:
56:
52:
48:
44:
40:
36:
32:
19:
7688:Rigid bodies
7621:
7615:
7609:
7596:
7588:
7580:
7574:Georgia Tech
7557:
7542:
7526:. Springer.
7522:
7515:
7497:
7477:
7470:
7450:
7443:
7427:. Springer.
7423:
7416:
7396:
7389:
7373:. Springer.
7369:
7349:
7331:
7326:
7317:
7311:
7264:
7252:
7249:Eduard Study
7235:
7218:
7214:
7210:
7206:
7192:
7150:
7135:
7125:
7121:
7119:
7048:
6983:
6891:
6881:
6877:
6875:
6803:
6801:
6793:
6719:
6597:
6507:
6256:
5815:
5813:
5681:
5533:
5529:
5525:
5520:
5516:
5513:
5511:
5506:
5502:
5498:
5494:
5492:
5397:
5395:
5389:
5385:
5383:
5262:
5258:
5254:
5250:
5245:
5241:
5236:
5232:
5225:
5218:
5213:
5209:
5202:
5195:
5193:
5175:
5171:
5167:
5163:
5159:
5155:
5151:
5147:
5143:
5125:
5120:
5116:
5112:
5108:
5104:
5100:
5098:
4864:
4862:
4709:
4707:
4528:
4524:
4520:
4516:
4512:
4508:
4504:
4503:. Then form
4496:
4492:
4488:
4486:
4481:
4465:
4453:
4449:
4445:
4441:
4437:
4433:
4429:
4425:
4421:
4418:
4414:
4403:
4399:
4395:
4391:
4387:
4383:
4381:
4376:
4372:
4369:dual numbers
4364:
4362:
4355:
4319:
4309:
4307:
4234:
4230:
4226:
4222:
4220:
4154:
4071:
4060:
3904:
3769:
3765:
3761:
3757:
3755:
3750:
3748:
3524:
3519:
3515:
3511:
3507:
3503:
3499:
3495:
3491:
3489:
3295:
3285:
3283:
3085:
3081:
3077:
3075:
2996:
2992:
2990:
2985:
2981:
2977:
2975:
2815:
2638:
2634:
2630:
2626:
2622:
2618:
2616:
2539:
2535:
2531:
2527:
2523:
2521:
2517:
2452:
2446:
2375:
2371:
2365:
2305:
2303:
2242:
2238:
2236:
2231:
2227:
2223:
2219:
2215:
2213:
2127:
2124:
1965:
1961:
1957:
1953:
1949:
1947:
1836:
1834:
1749:
1745:
1741:
1737:
1733:
1731:
1723:
1670:
1539:
1535:
1531:
1527:
1525:
1379:
1375:
1371:
1367:
1362:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1324:
1219:
1215:
1211:
1207:
1113:
1109:
1105:
1101:
1097:
1093:
1014:
1009:
1005:
1001:
997:
993:
989:
985:
981:
977:
842:
838:
834:
832:
555:
436:
432:
428:
424:
420:
305:
300:
296:
294:
290:dual vectors
289:
284:
278:
275:
218:
216:
201:
199:
190:
184:
182:
173:
164:Euler angles
159:
150:screw motion
147:
145:
113:
101:
93:real numbers
70:mathematical
67:
63:rigid bodies
31:Screw theory
30:
29:
18:Screw motion
7654:Joe Rooney
7547:Felix Klein
7227:unit circle
7152:Felix Klein
6806:that is if
6804:reciprocal,
5140:Lie algebra
4468:represents
3772:), that is
2128:The screw
841:) define a
306:dual scalar
303:) called a
192:direction.
73:formulation
7693:Kinematics
7672:Categories
7303:References
7281:Screw axis
7241:kinematics
7195:Sophus Lie
7146:mechanisms
5818:to obtain
4501:screw axis
4478:restricted
4474:homography
4472:, and the
4410:under the
4352:Homography
4334:screw axis
4225:(0) in SE(
3506:, 1), and
1383:, that is
996:)), where
843:dual angle
55:kinematics
37:, such as
7683:Mechanics
7510:Springer.
7093:δ
7087:ˇ
7076:⋅
7060:δ
7028:δ
7022:ˇ
7011:⋅
6995:δ
6960:→
6957:ω
6937:→
6934:ω
6928:×
6911:ˇ
6849:δ
6836:Π
6817:δ
6734:Π
6699:δ
6686:Π
6667:δ
6656:⋅
6651:∘
6636:∘
6626:⋅
6609:δ
6577:δ
6566:⋅
6561:∘
6546:∘
6536:⋅
6519:δ
6485:∘
6444:×
6412:∑
6376:∑
6348:∘
6310:→
6307:ω
6301:×
6287:→
6284:ω
6232:δ
6226:→
6223:ω
6217:⋅
6197:×
6165:∑
6150:δ
6141:→
6138:ω
6132:×
6113:⋅
6076:∑
6054:δ
6048:→
6045:ω
6039:⋅
6019:×
5987:∑
5972:δ
5964:⋅
5927:∑
5912:δ
5906:→
5903:ω
5897:×
5889:⋅
5852:∑
5833:δ
5793:δ
5771:−
5753:×
5747:→
5744:ω
5735:⋅
5703:∑
5693:δ
5661:δ
5646:⋅
5628:⋯
5619:δ
5604:⋅
5583:δ
5568:⋅
5547:δ
5459:−
5441:×
5435:→
5432:ω
5360:…
5136:Lie group
5054:−
5038:ε
5016:ε
4978:−
4962:ε
4940:ε
4899:−
4889:ε
4837:−
4827:ε
4805:−
4795:ε
4786:−
4750:ε
4744:−
4732:
4664:−
4654:∗
4640:∼
4632:∗
4593:∗
4275:θ
4272:ξ
4266:
4254:θ
4092:˙
4015:ω
4012:×
4002:Ω
3965:˙
3944:Ω
3941:−
3936:Ω
3847:−
3809:˙
3669:˙
3644:˙
3562:˙
3118:^
3019:^
2945:×
2928:−
2857:×
2840:−
2790:−
2770:×
2751:×
2720:−
2686:×
2670:−
2591:×
2575:−
2466:ξ
2416:ω
2413:×
2403:ω
2389:ξ
2283:ω
2280:×
2269:ω
2189:→
2186:ω
2180:×
2158:→
2155:ω
2106:ω
2102:×
2078:×
2074:ω
2031:Ω
2025:−
2006:Ω
1645:×
1634:−
1603:×
1595:−
1587:×
1571:−
1493:×
1443:∑
1405:∑
1292:^
1283:
1237:×
1186:^
1177:
1131:⋅
1060:⋅
958:φ
955:
946:−
940:φ
937:
922:^
913:
898:φ
895:
883:φ
880:
865:^
856:
807:×
791:×
773:×
740:×
708:×
670:⋅
654:⋅
636:⋅
603:⋅
571:⋅
453:^
335:^
323:^
7566:Archived
7275:See also
7158:and his
5180:3-sphere
5178:and the
5150:, where
4432:) = 1 +
4412:rotation
4390: :
4375:) = 1 +
4367:= 0 for
1337:and let
988:),
77:geometry
75:for the
59:dynamics
7638:2369176
7255:), and
7229:in the
7209:)= cos
7142:statics
7132:History
5184:versors
5130:of the
5117:pε
4525:bε
4513:bε
4507:= exp((
4430:εs
4426:εr
4388:εr
4386:= {1 +
4377:aε
4373:aε
2214:is the
1218:,
1096:,
837:,
435:,
427:,
299:,
51:moments
39:angular
35:vectors
7636:
7530:
7485:
7458:
7431:
7404:
7377:
7203:versor
7170:for a
6720:where
5493:where
5384:where
5162:, and
5115:= 1 +
5111:, let
4684:
4678:
4624:
4618:
4554:
4548:
4495:, and
4458:3-flat
4434:ε
4428:)(1 +
4408:stable
4400:ε
4371:, exp(
4365:ε
4363:Since
4308:where
4233:in se(
3905:where
2447:For a
2366:For a
1948:where
1327:wrench
1321:Wrench
276:where
217:Let a
202:wrench
196:Wrench
47:forces
7634:JSTOR
7572:from
5107:* = −
4712:* is
4470:space
4456:is a
4330:space
4067:SE(3)
4063:se(3)
2216:twist
1734:twist
1728:Twist
1530:and −
219:screw
185:twist
179:Twist
170:Screw
160:screw
45:, or
7528:ISBN
7483:ISBN
7456:ISBN
7429:ISBN
7402:ISBN
7375:ISBN
7217:sin
6880:and
5231:...
5208:...
5146:and
5132:ring
4487:Let
4448:and
4382:Let
2629:and
2538:and
1538:and
1210:and
1108:and
282:and
118:and
95:and
65:.
57:and
49:and
41:and
7626:doi
7504:doi
7207:a r
7144:of
5509:).
5403:is
5182:of
5148:b s
5144:a r
4867:to
4729:exp
4480:to
4406:is
4398:},
4320:In
4263:exp
3510:= (
3494:= (
1839:is
1345:= (
1280:sin
1174:cos
1012:).
952:sin
934:cos
910:cos
892:cos
877:sin
853:sin
61:of
7674::
7632:.
7620:.
7359:^
7339:^
7247:,
7233:.
7213:+
7186:,
7182:,
7178:,
7128:.
5534:δt
5249:,
5224:,
5201:,
5186:.
5170:∈
5166:,
5158:∈
5154:,
5119:∈
5103:,
4523:−
4511:+
4497:br
4452:.
4440:+
4422:qp
4417:→
4394:∈
4360:.
4348:.
3518:,
3514:,
3502:,
3498:,
3288:.
2988:.
2546:,
2044:or
1366:,
1349:,
1000:′(
998:df
992:′(
990:df
292:.
183:A
99:.
7640:.
7628::
7622:7
7536:.
7506::
7491:.
7464:.
7437:.
7410:.
7383:.
7251:(
7219:a
7215:r
7211:a
7126:T
7122:W
7105:,
7102:0
7099:=
7096:t
7083:T
7071:W
7066:=
7063:W
7034:.
7031:t
7018:T
7006:W
7001:=
6998:W
6969:,
6966:)
6951:,
6947:v
6943:+
6924:d
6920:(
6917:=
6907:T
6882:T
6878:W
6861:,
6858:0
6855:=
6852:t
6844:T
6839:]
6833:[
6828:W
6823:=
6820:W
6779:,
6774:]
6768:0
6763:I
6756:I
6751:0
6745:[
6740:=
6737:]
6731:[
6705:,
6702:t
6694:T
6689:]
6683:[
6678:W
6673:=
6670:t
6664:)
6660:T
6646:W
6641:+
6631:T
6622:W
6618:(
6615:=
6612:W
6583:.
6580:t
6574:)
6570:T
6556:W
6551:+
6541:T
6532:W
6528:(
6525:=
6522:W
6493:,
6490:)
6480:W
6475:,
6471:W
6467:(
6464:=
6460:)
6454:i
6449:F
6439:i
6434:X
6427:n
6422:1
6419:=
6416:i
6408:,
6403:i
6398:F
6391:n
6386:1
6383:=
6380:i
6371:(
6367:=
6362:W
6356:,
6353:)
6343:T
6338:,
6334:T
6330:(
6327:=
6324:)
6320:v
6316:+
6297:d
6293:,
6278:(
6275:=
6270:T
6238:.
6235:t
6213:)
6207:i
6202:F
6192:i
6187:X
6180:n
6175:1
6172:=
6169:i
6160:(
6156:+
6153:t
6147:)
6128:d
6124:+
6120:v
6116:(
6109:)
6103:i
6098:F
6091:n
6086:1
6083:=
6080:i
6071:(
6067:=
6057:t
6035:)
6029:i
6024:F
6014:i
6009:X
6002:n
5997:1
5994:=
5991:i
5982:(
5978:+
5975:t
5968:v
5960:)
5954:i
5949:F
5942:n
5937:1
5934:=
5931:i
5922:(
5918:+
5915:t
5893:d
5885:)
5879:i
5874:F
5867:n
5862:1
5859:=
5856:i
5847:(
5843:=
5836:W
5816:v
5799:.
5796:t
5790:)
5786:v
5782:+
5779:)
5775:d
5766:i
5761:X
5756:(
5738:(
5730:i
5725:F
5718:n
5713:1
5710:=
5707:i
5699:=
5696:W
5667:.
5664:t
5656:n
5651:V
5641:n
5636:F
5631:+
5625:+
5622:t
5614:2
5609:V
5599:2
5594:F
5589:+
5586:t
5578:1
5573:V
5563:1
5558:F
5553:=
5550:W
5530:i
5526:v
5524:=
5521:i
5517:r
5514:δ
5507:t
5505:(
5503:d
5499:v
5495:ω
5478:,
5474:v
5470:+
5467:)
5463:d
5454:i
5449:X
5444:(
5426:=
5421:i
5416:V
5401:i
5398:X
5390:i
5386:x
5369:,
5366:n
5363:,
5357:,
5354:1
5351:=
5348:i
5344:)
5341:t
5338:(
5334:d
5330:+
5325:i
5320:x
5315:]
5312:)
5309:t
5306:(
5303:A
5300:[
5297:=
5294:)
5291:t
5288:(
5283:i
5278:X
5263:t
5261:(
5259:d
5255:n
5251:i
5246:i
5242:X
5237:n
5233:X
5229:2
5226:X
5222:1
5219:X
5214:n
5210:F
5206:2
5203:F
5199:1
5196:F
5176:F
5172:H
5168:s
5164:r
5160:R
5156:b
5152:a
5121:F
5113:q
5109:p
5105:p
5101:p
5084:.
5081:)
5076:r
5073:a
5069:e
5065:q
5060:r
5057:a
5050:e
5046:(
5041:r
5035:b
5032:2
5028:e
5024:=
5019:r
5013:b
5009:e
5005:)
5000:r
4997:a
4993:e
4989:q
4984:r
4981:a
4974:e
4970:(
4965:r
4959:b
4955:e
4951:=
4948:)
4943:r
4937:b
4933:e
4927:r
4924:a
4920:e
4916:(
4913:q
4910:)
4905:r
4902:a
4895:e
4886:b
4882:e
4878:(
4865:q
4848:,
4843:r
4840:a
4833:e
4824:r
4821:b
4817:e
4813:=
4808:1
4801:)
4792:r
4789:b
4782:e
4776:r
4773:a
4769:e
4765:(
4762:=
4756:)
4753:r
4747:b
4741:r
4738:a
4735:(
4725:1
4710:z
4693:.
4690:]
4687:1
4681::
4675:z
4672:q
4667:1
4660:)
4650:z
4646:(
4643:[
4637:]
4628:z
4621::
4615:z
4612:q
4609:[
4606:=
4601:)
4589:z
4583:0
4576:0
4571:z
4565:(
4560:]
4557:1
4551::
4545:q
4542:[
4529:r
4527:)
4521:a
4517:r
4515:)
4509:a
4505:z
4493:r
4489:a
4482:F
4466:F
4454:F
4450:s
4446:r
4442:s
4438:r
4436:(
4419:p
4415:q
4404:F
4396:H
4392:r
4384:F
4310:θ
4293:,
4290:)
4287:0
4284:(
4281:g
4278:)
4269:(
4260:=
4257:)
4251:(
4248:g
4235:n
4231:ξ
4227:n
4223:g
4206:.
4201:t
4198:]
4195:S
4192:[
4188:e
4184:=
4181:]
4178:)
4175:t
4172:(
4169:T
4166:[
4140:,
4137:]
4134:)
4131:t
4128:(
4125:T
4122:[
4119:]
4116:S
4113:[
4110:=
4107:]
4104:)
4101:t
4098:(
4089:T
4083:[
4046:.
4041:]
4035:0
4030:0
4022:v
4018:+
4008:d
3996:[
3991:=
3986:]
3980:0
3975:0
3961:d
3954:+
3949:d
3930:[
3925:=
3922:]
3919:S
3916:[
3890:,
3885:P
3880:]
3877:S
3874:[
3871:=
3868:)
3865:t
3862:(
3857:P
3850:1
3843:]
3839:)
3836:t
3833:(
3830:T
3827:[
3824:]
3821:)
3818:t
3815:(
3806:T
3800:[
3797:=
3792:P
3786:V
3770:t
3768:(
3766:P
3762:P
3758:p
3751:p
3734:.
3729:}
3723:1
3714:p
3706:{
3699:]
3693:0
3688:0
3681:)
3678:t
3675:(
3665:d
3656:)
3653:t
3650:(
3641:A
3632:[
3627:=
3622:}
3616:0
3607:P
3601:V
3592:{
3587:=
3582:p
3577:]
3574:)
3571:t
3568:(
3559:T
3553:[
3550:=
3545:P
3539:V
3520:Z
3516:Y
3512:X
3508:P
3504:Z
3500:Y
3496:X
3492:P
3475:.
3470:}
3464:1
3455:p
3447:{
3440:]
3434:1
3429:0
3422:)
3419:t
3416:(
3411:d
3404:)
3401:t
3398:(
3395:A
3389:[
3384:=
3379:}
3373:1
3364:P
3356:{
3351:=
3346:p
3341:]
3338:)
3335:t
3332:(
3329:T
3326:[
3323:=
3320:)
3317:t
3314:(
3309:P
3269:.
3266:)
3262:s
3258:]
3255:A
3252:D
3249:[
3246:+
3242:v
3238:]
3235:A
3232:[
3229:,
3225:s
3221:]
3218:A
3215:[
3212:(
3209:=
3206:)
3202:v
3198:,
3194:s
3190:(
3187:)
3184:]
3181:A
3178:D
3175:[
3172:,
3169:]
3166:A
3163:[
3160:(
3157:=
3154:)
3150:V
3146:,
3142:S
3138:(
3134:,
3129:s
3124:]
3115:A
3109:[
3106:=
3101:S
3086:v
3084:.
3082:s
3078:s
3061:,
3058:)
3055:]
3052:A
3049:D
3046:[
3043:,
3040:]
3037:A
3034:[
3031:(
3028:=
3025:]
3016:A
3010:[
2997:d
2993:D
2986:y
2982:d
2978:y
2961:.
2956:}
2949:q
2941:p
2932:p
2924:q
2917:{
2910:]
2904:A
2899:A
2896:D
2889:0
2884:A
2878:[
2873:=
2868:}
2861:Q
2853:P
2844:P
2836:Q
2829:{
2801:)
2798:)
2794:p
2786:q
2782:(
2779:]
2776:A
2773:[
2766:d
2762:+
2759:)
2755:q
2747:p
2743:(
2740:]
2737:A
2734:[
2731:,
2728:)
2724:p
2716:q
2712:(
2709:]
2706:A
2703:[
2700:(
2697:=
2694:)
2690:Q
2682:P
2678:,
2674:P
2666:Q
2662:(
2659:=
2654:Q
2639:d
2635:q
2631:Q
2627:d
2623:p
2619:P
2602:,
2599:)
2595:q
2587:p
2583:,
2579:p
2571:q
2567:(
2564:=
2559:q
2540:q
2536:p
2532:d
2528:d
2524:D
2498:.
2493:}
2487:v
2480:0
2474:{
2469:=
2453:v
2427:.
2422:}
2410:q
2397:{
2392:=
2376:ω
2372:q
2346:.
2343:)
2339:v
2335:,
2332:0
2329:(
2326:=
2321:T
2306:v
2289:,
2286:)
2276:d
2272:,
2266:(
2263:=
2258:L
2243:v
2239:d
2232:ω
2228:d
2224:v
2220:V
2198:,
2195:)
2176:d
2172:+
2168:v
2164:,
2149:(
2146:=
2141:T
2110:,
2098:d
2094:+
2090:v
2086:+
2082:P
2070:=
2067:)
2064:t
2061:(
2056:P
2051:V
2038:d
2034:]
2028:[
2021:v
2017:+
2013:P
2009:]
2003:[
2000:=
1997:)
1994:t
1991:(
1986:P
1981:V
1966:d
1962:P
1958:p
1954:d
1950:v
1933:,
1930:)
1927:t
1924:(
1920:v
1916:+
1912:p
1907:]
1901:t
1898:d
1893:)
1890:t
1887:(
1884:A
1881:d
1875:[
1871:=
1868:)
1865:t
1862:(
1857:P
1852:V
1837:P
1820:.
1817:)
1814:t
1811:(
1807:d
1803:+
1799:p
1795:]
1792:)
1789:t
1786:(
1783:A
1780:[
1777:=
1774:)
1771:t
1768:(
1764:P
1750:P
1746:p
1742:d
1738:d
1709:,
1706:)
1702:M
1698:,
1695:0
1692:(
1689:=
1684:M
1656:.
1653:)
1649:F
1642:)
1638:B
1630:A
1626:(
1623:,
1620:0
1617:(
1614:=
1611:)
1607:F
1599:B
1591:F
1583:A
1579:,
1575:F
1567:F
1563:(
1560:=
1555:R
1540:B
1536:A
1532:F
1528:F
1511:.
1508:)
1503:i
1498:F
1488:i
1483:P
1478:,
1473:i
1468:F
1463:(
1458:n
1453:1
1450:=
1447:i
1439:=
1434:i
1428:W
1420:n
1415:1
1412:=
1409:i
1401:=
1396:R
1380:i
1376:W
1372:n
1368:i
1363:i
1359:F
1355:F
1353:×
1351:P
1347:F
1343:W
1339:P
1335:F
1331:P
1305:.
1300:N
1289:z
1276:|
1271:T
1266:|
1261:|
1256:S
1251:|
1247:=
1242:T
1232:S
1220:d
1216:φ
1212:T
1208:S
1192:;
1183:z
1170:|
1165:T
1160:|
1155:|
1150:S
1145:|
1141:=
1136:T
1126:S
1114:d
1110:T
1106:S
1102:φ
1098:d
1094:φ
1078:;
1075:1
1072:=
1065:S
1055:S
1048:=
1044:|
1038:S
1032:|
1010:φ
1008:(
1006:f
1002:φ
994:φ
986:φ
984:(
982:f
978:f
964:,
961:)
949:d
943:,
931:(
928:=
919:z
904:,
901:)
889:d
886:,
874:(
871:=
862:z
839:d
835:φ
818:.
815:)
811:T
803:V
799:+
795:W
787:S
781:,
777:T
769:S
765:(
762:=
759:)
755:W
751:,
747:T
743:(
737:)
733:V
729:,
725:S
721:(
718:=
713:T
703:S
681:,
678:)
674:T
666:V
662:+
658:W
650:S
644:,
640:T
632:S
628:(
625:=
622:)
618:W
614:,
610:T
606:(
600:)
596:V
592:,
588:S
584:(
581:=
576:T
566:S
542:.
539:)
535:S
531:b
528:+
524:V
520:a
517:,
513:S
509:a
506:(
503:=
500:)
496:V
492:,
488:S
484:(
481:)
478:b
475:,
472:a
469:(
466:=
461:S
450:a
437:b
433:a
429:V
425:S
421:S
407:.
404:)
401:c
398:b
395:+
392:d
389:a
386:,
383:c
380:a
377:(
374:=
371:)
368:d
365:,
362:c
359:(
356:)
353:b
350:,
347:a
344:(
341:=
332:c
320:a
301:b
297:a
285:V
279:S
261:,
258:)
254:V
250:,
246:S
242:(
239:=
234:S
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
