Abstract

We propose a novel method for the robust, non-contact, and six degrees of freedom (6-DOF) motion sensing of an arbitrary rigid body using multi-view laser Doppler measurements. The proposed method reconstructs the 6-DOF motion from fragmentary velocities on the surface of the target. It is unique compared to conventional contact-less motion sensing methods since it is robust against lack-of-feature objects and environments. By discussing the formulation of motion reconstruction by fragmentary velocities, we show that at least three viewpoints are essential for 6-DOF motion reconstruction. Further, we claim that the condition number of the measurement matrix can be a measure of system accuracy, and numerical simulation is performed to find an appropriate system configuration. The proposed method was implemented using a laser Doppler velocimeter, a galvanometer scanner, and some mirrors. We introduce the methods for calibration, coordinate system selection, and the calculation pipeline, all of which contribute to the accuracy of the proposed system. For evaluation, the proposed system is compared with an off-line chessboard-tracking scheme of a 500 fps camera. Experiments of measuring six different motion patterns are demonstrated to show the robustness of the proposed method against different kinds of motion. We also conduct evaluations with different distances and velocities. The mean value error is less than 1.3 deg/s in rotation and 3.2 mm/s in translation, and is robust against changes in distance and velocity. For speed evaluation, the throughput of the proposed method is approximately 250 Hz and the latency is approximately 20 ms.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2017 (3)

2016 (1)

S. Shrestha, F. Heide, W. Heidrich, and G. Wetzstein, “Computational Imaging with Multi-camera Time-of-flight Systems,” ACM Trans. Graphic 35, 33 (2016).
[Crossref]

2015 (3)

2013 (1)

2012 (2)

M. Calonder, V. Lepetit, M. Ozuysal, T. Trzcinski, C. Strecha, and P. Fua, “BRIEF: Computing a Local Binary Descriptor Very Fast,” IEEE Trans. PAMI. 34, 1281–1298 (2012).
[Crossref]

C.-H. Cheng, C.-W. Lee, T.-W. Lin, and F.-Y. Lin, “Dual-frequency laser Doppler velocimeter for speckle noise reduction and coherence enhancement,” Opt. Express 20, 20255–20265 (2012).
[Crossref] [PubMed]

2008 (2)

V. Lepetit, F. Moreno-Noguer, and P. Fua, “EPnP: An Accurate O(n) Solution to the PnP Problem,” International Journal of Computer Vision 81, 155 (2008).
[Crossref]

J. Vass, R. Šmíd, R. Randall, P. Sovka, C. Cristalli, and B. Torcianti, “Avoidance of speckle noise in laser vibrometry by the use of kurtosis ratio: Application to mechanical fault diagnostics,” Mechanical Sys. Signal Process. 22, 647–671 (2008).
[Crossref]

2006 (1)

1994 (1)

S. Rothberg and N. A. Halliwell, “Vibration measurements on rotating machinery using laser Doppler velocimetry,” J. Vibration Acoustics 116, 326–331 (1994).
[Crossref]

1992 (1)

P. J. Besl and N. D. McKay, “A method for registration of 3-D shapes,” IEEE Trans. PAMI. 14, 239–256 (1992).
[Crossref]

1984 (1)

1981 (1)

B. K. Horn and B. G. Schunck, “Determining optical flow,” Artificial Intelligence 17, 185–203 (1981).
[Crossref]

1973 (1)

W. K. George and J. L. Lumley, “The laser-Doppler velocimeter and its application to the measurement of turbulence,” J. Fluid Mechanics 60, 321–362 (1973).
[Crossref]

1970 (1)

F. Eberhardt and F. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoustical Soc. Am. 48, 603–609 (1970).
[Crossref]

1964 (1)

Y. Yeh and H. Cummins, “Localized fluid flow measurements with an He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964).
[Crossref]

Adiv, G.

G. Adiv, “Determining three-dimensional motion and structure from optical flow generated by several moving objects,” IEEE Trans. PAMI. pp. 384–401 (1985).
[Crossref]

Andrews, F.

F. Eberhardt and F. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoustical Soc. Am. 48, 603–609 (1970).
[Crossref]

Besl, P. J.

P. J. Besl and N. D. McKay, “A method for registration of 3-D shapes,” IEEE Trans. PAMI. 14, 239–256 (1992).
[Crossref]

Bin, Z.

Bradski, G.

E. Rublee, V. Rabaud, K. Konolige, and G. Bradski, “ORB: An efficient alternative to SIFT or SURF,” in 2011 International Conference on Computer Vision (IEEE, 2011), pp. 2564–2571.
[Crossref]

Calonder, M.

M. Calonder, V. Lepetit, M. Ozuysal, T. Trzcinski, C. Strecha, and P. Fua, “BRIEF: Computing a Local Binary Descriptor Very Fast,” IEEE Trans. PAMI. 34, 1281–1298 (2012).
[Crossref]

Chen, B.

Cheng, C.-H.

Chetverikov, D.

D. Chetverikov, D. Svirko, D. Stepanov, and P. Krsek, “The Trimmed Iterative Closest Point algorithm,” in Object Recognition Supported by User Interaction for Service Robots, vol. 3 (IEEE, 2002), vol. 3, pp. 545–548.
[Crossref]

Cremers, D.

J. Engel, T. Schöps, and D. Cremers, “LSD-SLAM: Large-Scale Direct Monocular SLAM,” in Computer Vision – ECCV 2014 (Springer, 2014), pp. 834–849.

Cristalli, C.

J. Vass, R. Šmíd, R. Randall, P. Sovka, C. Cristalli, and B. Torcianti, “Avoidance of speckle noise in laser vibrometry by the use of kurtosis ratio: Application to mechanical fault diagnostics,” Mechanical Sys. Signal Process. 22, 647–671 (2008).
[Crossref]

Cuifang, K.

Cummins, H.

Y. Yeh and H. Cummins, “Localized fluid flow measurements with an He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964).
[Crossref]

Cunxing, C.

Demarest, F. C.

Droeschel, D.

S. May, D. Droeschel, S. Fuchs, D. Holz, and A. Nüchter, “Robust 3D-mapping with time-of-flight cameras,” in 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2009), pp. 1673–1678.
[Crossref]

Eberhardt, F.

F. Eberhardt and F. Andrews, “Laser heterodyne system for measurement and analysis of vibration,” J. Acoustical Soc. Am. 48, 603–609 (1970).
[Crossref]

Engel, J.

J. Engel, T. Schöps, and D. Cremers, “LSD-SLAM: Large-Scale Direct Monocular SLAM,” in Computer Vision – ECCV 2014 (Springer, 2014), pp. 834–849.

Fenglin, Y.

Fitzgibbon, A. W.

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle Adjustment — A Modern Synthesis,” in Proceedings of International Workshop on Vision Algorithms (Springer, 1999), pp. 298–372.

Forster, C.

C. Forster, M. Pizzoli, and D. Scaramuzza, “SVO: Fast semi-direct monocular visual odometry,” in 2014 IEEE International Conference on Robotics and Automation (IEEE, 2014), pp. 15–22.
[Crossref]

Forsyth, D.

D. Forsyth and J. Ponce, Computer Vision: A Modern Approach (Prentice Hall, 2011).

Fua, P.

M. Calonder, V. Lepetit, M. Ozuysal, T. Trzcinski, C. Strecha, and P. Fua, “BRIEF: Computing a Local Binary Descriptor Very Fast,” IEEE Trans. PAMI. 34, 1281–1298 (2012).
[Crossref]

V. Lepetit, F. Moreno-Noguer, and P. Fua, “EPnP: An Accurate O(n) Solution to the PnP Problem,” International Journal of Computer Vision 81, 155 (2008).
[Crossref]

Fuchs, S.

S. May, D. Droeschel, S. Fuchs, D. Holz, and A. Nüchter, “Robust 3D-mapping with time-of-flight cameras,” in 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2009), pp. 1673–1678.
[Crossref]

George, W. K.

W. K. George and J. L. Lumley, “The laser-Doppler velocimeter and its application to the measurement of turbulence,” J. Fluid Mechanics 60, 321–362 (1973).
[Crossref]

Gill, P. E.

P. E. Gill, W. Murray, M. H. Wright, and et al., Numerical Linear Algebra and Optimization, vol. 1 (Addison-Wesley, 1991).

Halliwell, N. A.

S. Rothberg and N. A. Halliwell, “Vibration measurements on rotating machinery using laser Doppler velocimetry,” J. Vibration Acoustics 116, 326–331 (1994).
[Crossref]

Han, S.

Hartley, R.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University, 2003).

Hartley, R. I.

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle Adjustment — A Modern Synthesis,” in Proceedings of International Workshop on Vision Algorithms (Springer, 1999), pp. 298–372.

Heide, F.

S. Shrestha, F. Heide, W. Heidrich, and G. Wetzstein, “Computational Imaging with Multi-camera Time-of-flight Systems,” ACM Trans. Graphic 35, 33 (2016).
[Crossref]

F. Heide, G. Wetzstein, M. Hullin, and W. Heidrich, “Doppler Time-of-flight Imaging,” in Proceedings of ACM SIGGRAPH 2015 Emerging Technologies (ACM, 2015), pp. 9.

Heidrich, W.

S. Shrestha, F. Heide, W. Heidrich, and G. Wetzstein, “Computational Imaging with Multi-camera Time-of-flight Systems,” ACM Trans. Graphic 35, 33 (2016).
[Crossref]

F. Heide, G. Wetzstein, M. Hullin, and W. Heidrich, “Doppler Time-of-flight Imaging,” in Proceedings of ACM SIGGRAPH 2015 Emerging Technologies (ACM, 2015), pp. 9.

Higham, N. J.

N. J. Higham, Accuracy and Stability of Numerical Algorithms (SIAM, 2002).
[Crossref]

Holz, D.

S. May, D. Droeschel, S. Fuchs, D. Holz, and A. Nüchter, “Robust 3D-mapping with time-of-flight cameras,” in 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2009), pp. 1673–1678.
[Crossref]

Horn, B. K.

B. K. Horn and B. G. Schunck, “Determining optical flow,” Artificial Intelligence 17, 185–203 (1981).
[Crossref]

Hsieh, H.-L.

Huang, Q.

A. H. Khawaja, Q. Huang, J. Li, and Z. Zhang, “Estimation of current and sag in overhead power transmission lines with optimized magnetic field sensor array placement,” IEEE Trans. Magnetics 53, 1–10 (2017).
[Crossref]

Hullin, M.

F. Heide, G. Wetzstein, M. Hullin, and W. Heidrich, “Doppler Time-of-flight Imaging,” in Proceedings of ACM SIGGRAPH 2015 Emerging Technologies (ACM, 2015), pp. 9.

Ishikawa, M.

L. Miyashita, R. Yonezawa, Y. Watanabe, and M. Ishikawa, “3D Motion Sensing of Any Object Without Prior Knowledge,” ACM Trans. Graphic. 34, 218 (2015).
[Crossref]

Khawaja, A. H.

A. H. Khawaja, Q. Huang, J. Li, and Z. Zhang, “Estimation of current and sag in overhead power transmission lines with optimized magnetic field sensor array placement,” IEEE Trans. Magnetics 53, 1–10 (2017).
[Crossref]

Kim, S.-W.

Kim, Y.-J.

Konolige, K.

E. Rublee, V. Rabaud, K. Konolige, and G. Bradski, “ORB: An efficient alternative to SIFT or SURF,” in 2011 International Conference on Computer Vision (IEEE, 2011), pp. 2564–2571.
[Crossref]

Krsek, P.

D. Chetverikov, D. Svirko, D. Stepanov, and P. Krsek, “The Trimmed Iterative Closest Point algorithm,” in Object Recognition Supported by User Interaction for Service Robots, vol. 3 (IEEE, 2002), vol. 3, pp. 545–548.
[Crossref]

Lee, C.-W.

Lepetit, V.

M. Calonder, V. Lepetit, M. Ozuysal, T. Trzcinski, C. Strecha, and P. Fua, “BRIEF: Computing a Local Binary Descriptor Very Fast,” IEEE Trans. PAMI. 34, 1281–1298 (2012).
[Crossref]

V. Lepetit, F. Moreno-Noguer, and P. Fua, “EPnP: An Accurate O(n) Solution to the PnP Problem,” International Journal of Computer Vision 81, 155 (2008).
[Crossref]

Levoy, M.

S. Rusinkiewicz and M. Levoy, “Efficient variants of the ICP algorithm,” in Proceedings of Third International Conference on 3-D Digital Imaging and Modeling (IEEE, 2001), pp. 145–152.
[Crossref]

Li, J.

A. H. Khawaja, Q. Huang, J. Li, and Z. Zhang, “Estimation of current and sag in overhead power transmission lines with optimized magnetic field sensor array placement,” IEEE Trans. Magnetics 53, 1–10 (2017).
[Crossref]

Lin, F.-Y.

Lin, T.-W.

Lou, Y.

Lumley, J. L.

W. K. George and J. L. Lumley, “The laser-Doppler velocimeter and its application to the measurement of turbulence,” J. Fluid Mechanics 60, 321–362 (1973).
[Crossref]

May, S.

S. May, D. Droeschel, S. Fuchs, D. Holz, and A. Nüchter, “Robust 3D-mapping with time-of-flight cameras,” in 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2009), pp. 1673–1678.
[Crossref]

McKay, N. D.

P. J. Besl and N. D. McKay, “A method for registration of 3-D shapes,” IEEE Trans. PAMI. 14, 239–256 (1992).
[Crossref]

McLauchlan, P. F.

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle Adjustment — A Modern Synthesis,” in Proceedings of International Workshop on Vision Algorithms (Springer, 1999), pp. 298–372.

Miyashita, L.

L. Miyashita, R. Yonezawa, Y. Watanabe, and M. Ishikawa, “3D Motion Sensing of Any Object Without Prior Knowledge,” ACM Trans. Graphic. 34, 218 (2015).
[Crossref]

Moreno-Noguer, F.

V. Lepetit, F. Moreno-Noguer, and P. Fua, “EPnP: An Accurate O(n) Solution to the PnP Problem,” International Journal of Computer Vision 81, 155 (2008).
[Crossref]

Murray, W.

P. E. Gill, W. Murray, M. H. Wright, and et al., Numerical Linear Algebra and Optimization, vol. 1 (Addison-Wesley, 1991).

Nüchter, A.

S. May, D. Droeschel, S. Fuchs, D. Holz, and A. Nüchter, “Robust 3D-mapping with time-of-flight cameras,” in 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2009), pp. 1673–1678.
[Crossref]

Ozuysal, M.

M. Calonder, V. Lepetit, M. Ozuysal, T. Trzcinski, C. Strecha, and P. Fua, “BRIEF: Computing a Local Binary Descriptor Very Fast,” IEEE Trans. PAMI. 34, 1281–1298 (2012).
[Crossref]

Pan, S.-W.

Pizzoli, M.

C. Forster, M. Pizzoli, and D. Scaramuzza, “SVO: Fast semi-direct monocular visual odometry,” in 2014 IEEE International Conference on Robotics and Automation (IEEE, 2014), pp. 15–22.
[Crossref]

Pollefeys, M.

M. Ye, X. Wang, R. Yang, L. Ren, and M. Pollefeys, “Accurate 3D pose estimation from a single depth image,” in IEEE International Conference on Computer Vision (IEEE, 2011), pp. 731–738.

Ponce, J.

D. Forsyth and J. Ponce, Computer Vision: A Modern Approach (Prentice Hall, 2011).

Qibo, F.

Rabaud, V.

E. Rublee, V. Rabaud, K. Konolige, and G. Bradski, “ORB: An efficient alternative to SIFT or SURF,” in 2011 International Conference on Computer Vision (IEEE, 2011), pp. 2564–2571.
[Crossref]

Randall, R.

J. Vass, R. Šmíd, R. Randall, P. Sovka, C. Cristalli, and B. Torcianti, “Avoidance of speckle noise in laser vibrometry by the use of kurtosis ratio: Application to mechanical fault diagnostics,” Mechanical Sys. Signal Process. 22, 647–671 (2008).
[Crossref]

Ren, L.

M. Ye, X. Wang, R. Yang, L. Ren, and M. Pollefeys, “Accurate 3D pose estimation from a single depth image,” in IEEE International Conference on Computer Vision (IEEE, 2011), pp. 731–738.

Rothberg, S.

S. Rothberg, “Numerical simulation of speckle noise in laser vibrometry,” Appl. Opt. 45, 4523–4533 (2006).
[Crossref] [PubMed]

S. Rothberg and N. A. Halliwell, “Vibration measurements on rotating machinery using laser Doppler velocimetry,” J. Vibration Acoustics 116, 326–331 (1994).
[Crossref]

Rublee, E.

E. Rublee, V. Rabaud, K. Konolige, and G. Bradski, “ORB: An efficient alternative to SIFT or SURF,” in 2011 International Conference on Computer Vision (IEEE, 2011), pp. 2564–2571.
[Crossref]

Rusinkiewicz, S.

S. Rusinkiewicz and M. Levoy, “Efficient variants of the ICP algorithm,” in Proceedings of Third International Conference on 3-D Digital Imaging and Modeling (IEEE, 2001), pp. 145–152.
[Crossref]

Scaramuzza, D.

C. Forster, M. Pizzoli, and D. Scaramuzza, “SVO: Fast semi-direct monocular visual odometry,” in 2014 IEEE International Conference on Robotics and Automation (IEEE, 2014), pp. 15–22.
[Crossref]

Schöps, T.

J. Engel, T. Schöps, and D. Cremers, “LSD-SLAM: Large-Scale Direct Monocular SLAM,” in Computer Vision – ECCV 2014 (Springer, 2014), pp. 834–849.

Schunck, B. G.

B. K. Horn and B. G. Schunck, “Determining optical flow,” Artificial Intelligence 17, 185–203 (1981).
[Crossref]

Shrestha, S.

S. Shrestha, F. Heide, W. Heidrich, and G. Wetzstein, “Computational Imaging with Multi-camera Time-of-flight Systems,” ACM Trans. Graphic 35, 33 (2016).
[Crossref]

Šmíd, R.

J. Vass, R. Šmíd, R. Randall, P. Sovka, C. Cristalli, and B. Torcianti, “Avoidance of speckle noise in laser vibrometry by the use of kurtosis ratio: Application to mechanical fault diagnostics,” Mechanical Sys. Signal Process. 22, 647–671 (2008).
[Crossref]

Sommargren, G. E.

Sovka, P.

J. Vass, R. Šmíd, R. Randall, P. Sovka, C. Cristalli, and B. Torcianti, “Avoidance of speckle noise in laser vibrometry by the use of kurtosis ratio: Application to mechanical fault diagnostics,” Mechanical Sys. Signal Process. 22, 647–671 (2008).
[Crossref]

Stepanov, D.

D. Chetverikov, D. Svirko, D. Stepanov, and P. Krsek, “The Trimmed Iterative Closest Point algorithm,” in Object Recognition Supported by User Interaction for Service Robots, vol. 3 (IEEE, 2002), vol. 3, pp. 545–548.
[Crossref]

Strecha, C.

M. Calonder, V. Lepetit, M. Ozuysal, T. Trzcinski, C. Strecha, and P. Fua, “BRIEF: Computing a Local Binary Descriptor Very Fast,” IEEE Trans. PAMI. 34, 1281–1298 (2012).
[Crossref]

Sun, J.

Svirko, D.

D. Chetverikov, D. Svirko, D. Stepanov, and P. Krsek, “The Trimmed Iterative Closest Point algorithm,” in Object Recognition Supported by User Interaction for Service Robots, vol. 3 (IEEE, 2002), vol. 3, pp. 545–548.
[Crossref]

Torcianti, B.

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Supplementary Material (1)

NameDescription
» Visualization 1       An intuitive demonstration of the experiment in Sec. 4.2.

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Figures (11)

Fig. 1
Fig. 1 The 6-DOF motion of a rigid body measured by a laser from the LDV. The rigid body is moving with rotational velocity ω and translational velocity v, and is hit by a laser beam Li from the LDV at point pi. Points on Li can be denoted by oi + sli, where oi is any point on Li, li is the direction vector, and s is a scalar. For ease of description, we refer to oi as “viewpoint” and li as “direction” for each laser beam Li. The velocity of pi, namely vi, consists of two parts introduced by the rotation velocity and translation velocity of the object. The measured velocity by LDV is the projection of vi in the laser beam’s direction li, namely vi = vi · li.
Fig. 2
Fig. 2 Numerical simulation of different system configurations. (a) Illustration of constraints of system configuration during simulation. The three viewpoints o1, o2, and o3 are on the gray plane αz=0. In the simulation, where the laser beams hit the target is assumed to be fixed at certain points on the surface of the target. The color of a laser beam represents the viewpoint that it passes through. (b) Illustration of κA when d changes.
Fig. 3
Fig. 3 Schematic of the proposed 6-DOF motion sensing system with a single LDV.
Fig. 4
Fig. 4 System implementation. The corresponding distances between viewpoints, i.e. ‖o2o1‖ and ‖o3o1‖ are 685 mm and 651 mm, respectively. The angle between the two laser beams from one viewpoint is 4.31 deg and κA = 520.
Fig. 5
Fig. 5 Simulation of κAa(c) when the origin of the coordinate system is changed. The origins are randomly generated in a cube with size 1000 × 1000 × 3000. The corresponding condition number is illustrated by the color.
Fig. 6
Fig. 6 An example of the calculation pipeline.
Fig. 7
Fig. 7 Experiment with six different motion patterns. See Visualization 1 for details. Note that the target is a chessboard in the practical experiment for the camera, but the proposed method does not need the chessboard; hence, a white board is used in the video to demonstrate this property.
Fig. 8
Fig. 8 Results for six motion patterns.
Fig. 9
Fig. 9 Experimental setup for providing constant rotational and translational velocities.
Fig. 10
Fig. 10 Experiment results over different velocities.
Fig. 11
Fig. 11 Experiment over different distances when the target was rotating at constant velocity. The rotational velocity is represented by the norm of rotational speed ‖ω‖. The experiment was performed under two different system configurations, namely config1 (κA = 520) and config2 (κA = 1100). The direction of MR2 and MR4 were changed between these two configurations in order to enable all laser beams to hit the same target when it was farther from the system.

Tables (1)

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Table 1 Results of 10-s motion measurements

Equations (15)

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v i = ω × p i + v
v i = ( ω × p i + v ) l i = p i × l i ω + l i v
v i = ( ( p i o i ) × l i + o i × l i ) ω + l i v
v i = ( o i × l i ) ω + l i v
( ( o 1 × l 1 ) T l 1 T ( o 2 × l 2 ) T l 2 T ( o N × l N ) T l N T ) ( ω v ) = ( v 1 v 2 v N )
X = A + b
r j ( 1 ) = [ ( o 1 × l j ) T l j T ]
dim ( span { r j ( 1 ) } ) 3
r k ( 2 ) = [ ( ( o 1 + q 1 ) × l k ) T l k T ] = [ ( o 1 × l k ) T l k T ] + [ ( q 1 × l k ) T 0 ]
dim ( span { r j ( 1 ) , r k ( 2 ) } ) 5
r m ( 3 ) = [ ( ( o 1 + q 2 ) × l m ) T l m T ] = [ ( o 1 × l m ) T l m T ] + [ ( q 2 × l m ) T 0 ]
X ˜ X 2 X 2 κ A * b ˜ b 2 b 2
A ( a ) ( c ) = ( ( ( o 1 c ) × l 1 ) T l 1 T ( ( o 2 c ) × l 2 ) T l 2 T ( ( o 6 c ) × l 6 ) T l 6 T )
c * = arg min c ( κ A ( a ) ( c ) )
p w = n T n ( K c 1 p i ) K c 1 p i

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