Companion to derivation_7dof.md,
dynamics_modifications_7dof.md, and
posture_proof.md. Curated Jun 19, 2026. Purpose: background
references for the linear-algebra and redundancy-resolution machinery
behind our objects — the arm null direction \(\hat{\boldsymbol n}=\ker\boldsymbol
J_{\nu_e}^\oplus\), the system kernel \(\hat{\boldsymbol
k}=\ker\boldsymbol\Gamma\), the inertia-weighted covector \(\boldsymbol z_a\), the \(v_n=0\) section, the self-motion “fiber”,
and the posture lever.
Not yet a deep-research pass. These are the standard, committee-safe entry points, deliberately excluding Wikipedia. The geometric-mechanics / mechanical-connection layer (Marsden–Montgomery, Kelly–Murray, Bloch) is flagged at the end as a separate Tier D to pull later.
All papers below carry a verified DOI (one exception, flagged). To pull each:
GET /graph/v1/paper/DOI:<doi> (e.g.
DOI:10.1109/JRA.1987.1087068), request fields
title,authors,year,venue,externalIds,openAccessPdf.GET /graph/v1/paper/search?query=<exact title> then
match on author=Burdick, year=1989, venue=ICRA.Books (Tier A texts, Nakamura, Siciliano et al., Ott) are not Semantic-Scholar PDF targets — cite them by edition; ISBNs are intentionally omitted here to avoid transcription errors, fill from a library lookup. Do not source textbook PDFs from file-sharing sites.
Draft BibTeX is at the bottom — re-key against the metadata each
fetch returns before committing to the .bib; treat
page/issue fields as provisional where noted.
These ground “the kernel of a wide Jacobian is the instantaneous self-motion space,” and the SVD route by which we actually extract \(\hat{\boldsymbol n}\).
| bibkey | reference | why it’s here |
|---|---|---|
strang2016introduction |
Gilbert Strang, Introduction to Linear Algebra, 5th ed., Wellesley-Cambridge Press, 2016. Free lectures: MIT OCW 18.06. | The “four fundamental subspaces”; null space \(N(A)\) and rank–nullity — why a 13-column, rank-12 \(\boldsymbol\Gamma\) has exactly a 1-D kernel. Best non-Wikipedia intuition. |
meyer2000matrix |
Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000. | Rigorous, committee-grade statements of rank–nullity and the fundamental subspaces. |
trefethenbau1997 |
L. N. Trefethen & D. Bau, Numerical Linear Algebra, SIAM, 1997. | SVD lectures; the right singular vectors with zero singular values
are the numerical null-space basis — the theory behind
our full_matrices=True trap (economy SVD drops \(\hat{\boldsymbol n}\)). |
golubvanloan2013 |
G. H. Golub & C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins Univ. Press, 2013. | Canonical numerical reference for numerical rank and null-space-via-SVD; pairs with \(\sigma_6>0\) as distance-to-singularity. |
hornjohnson2013 |
R. A. Horn & C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge Univ. Press, 2013. | Already cited for Weyl (posture_proof Gap 2); §7.3 is the singular-value perturbation home. |
The core of the question: how the kernel becomes motion. Start with Siciliano’s tutorial.
| bibkey | reference | DOI | why it’s here |
|---|---|---|---|
siciliano1990tutorial |
B. Siciliano, “Kinematic control of redundant robot manipulators: A tutorial,” J. Intelligent & Robotic Systems 3(3):201–212, 1990. | 10.1007/BF00126069 |
Best single entry point. Null-space optimization, task augmentation, extended Jacobian — our \(\boldsymbol z_a=\nabla g\) “algorithmic singularity” warning (deriv. §4) is straight from here. |
liegeois1977supervisory |
A. Liégeois, “Automatic supervisory control of the configuration and behavior of multibody mechanisms,” IEEE Trans. SMC 7(12):868–871, 1977. | 10.1109/TSMC.1977.4309644 |
Origin of gradient projection / the null-space projector \((\boldsymbol E-\boldsymbol J^{+}\boldsymbol
J)\) — ancestor of our \(\boldsymbol
J^{+}\boldsymbol J_{\nu_e}^\oplus=\boldsymbol E_7-\hat{\boldsymbol
n}\hat{\boldsymbol n}^T\) (dyn. mods Prop. 2). S2:
abee84595bae9de9b3a89d11760e95acf2482c15. |
kleinhuang1983pseudoinverse |
C. A. Klein & C.-H. Huang, “Review of pseudoinverse control for use with kinematically redundant manipulators,” IEEE Trans. SMC SMC-13:245–250, 1983. | 10.1109/TSMC.1983.6313123 |
Establishes non-cyclicity of the pseudoinverse — exactly posture_proof §2: “where you end up on each fiber depends on where you were on the last one.” (Issue no. listed as 2 or 3 across indexes; DOI is authoritative.) |
chiaverini2008redundant |
S. Chiaverini, G. Oriolo & I. D. Walker, “Kinematically Redundant Manipulators,” in Springer Handbook of Robotics (Siciliano & Khatib, eds.), Springer, 2008. | 10.1007/978-3-540-30301-5_12 |
Modern survey of redundancy-resolution schemes. (Retitled “Redundant
Robots” in 2nd ed., 2016, DOI
10.1007/978-3-319-32552-1_10.) |
nakamura1991advanced |
Y. Nakamura, Advanced Robotics: Redundancy and Optimization, Addison-Wesley, 1991. | — (book) | Classic monograph tying SVD, singularities, and redundancy. |
siciliano2009robotics |
B. Siciliano, L. Sciavicco, L. Villani & G. Oriolo, Robotics: Modelling, Planning and Control, Springer, 2009. | — (book) | Textbook §3.5 null-space projection if you want a “standard reference” cite. |
| bibkey | reference | DOI | maps to |
|---|---|---|---|
burdick1989selfmotion |
J. W. Burdick, “On the inverse kinematics of redundant manipulators: characterization of the self-motion manifolds,” Proc. IEEE ICRA, Scottsdale AZ, pp. 264–270, 1989. | TODO — confirm via S2 title search | The global structure of \(\ker\) — our fiber / self-motion circle and the \(\varphi\)-coordinate of Gap 3 (posture_proof §3–4). |
khatib1987operational |
O. Khatib, “A unified approach for motion and force control of robot manipulators: The operational space formulation,” IEEE J. Robotics & Automation 3(1):43–53, 1987. | 10.1109/JRA.1987.1087068 |
The dynamically-consistent generalized inverse =
our \(\boldsymbol z_a=\hat{\boldsymbol
k}^T\boldsymbol M/(\hat{\boldsymbol k}^T\boldsymbol M\hat{\boldsymbol
k})\) and the \(\boldsymbol
M\)-orthogonality of Thm. 1. S2:
33576c0fc316c45c3672523114b20a5bb996e1f4. |
dubowsky1993spacerobots |
S. Dubowsky & E. Papadopoulos, “The kinematics, dynamics, and control of free-flying and free-floating space robotic systems,” IEEE Trans. Robotics & Automation 9(5):531–543, 1993. | 10.1109/70.258046 |
Canonical free-flying vs free-floating reference; grounds deriv. §3 (“don’t call ours an RNS in front of the committee”). |
ott2008cartesian |
C. Ott, Cartesian Impedance Control of Redundant and Flexible-Joint Robots, Springer Tracts in Advanced Robotics, 2008. | — (book) | Already cited in deriv. §4c; the inertia-weighted / dynamically-consistent null space in monograph form. |
Not fetched yet. When we formalize the \(\boldsymbol z_a\) = horizontal-space-of-the-mechanical-connection framing (our Marsden–Montgomery / Kelly–Murray asides, dyn. mods §5.2 remark): Bloch, Nonholonomic Mechanics and Control; Marsden & Ratiu, Introduction to Mechanics and Symmetry; Kelly & Murray, “Geometric phases and robotic locomotion,” J. Robotic Systems, 1995. Say the word and I’ll verify + add these to a Tier D block.
% --- Tier A: linear algebra (books; cite by edition) ---
@book{strang2016introduction,
author = {Strang, Gilbert},
title = {Introduction to Linear Algebra},
edition = {5th}, publisher = {Wellesley-Cambridge Press}, year = {2016}}
@book{meyer2000matrix,
author = {Meyer, Carl D.},
title = {Matrix Analysis and Applied Linear Algebra},
publisher = {SIAM}, year = {2000}}
@book{trefethenbau1997,
author = {Trefethen, Lloyd N. and Bau, David},
title = {Numerical Linear Algebra},
publisher = {SIAM}, year = {1997}}
@book{golubvanloan2013,
author = {Golub, Gene H. and Van Loan, Charles F.},
title = {Matrix Computations},
edition = {4th}, publisher = {Johns Hopkins University Press}, year = {2013}}
@book{hornjohnson2013,
author = {Horn, Roger A. and Johnson, Charles R.},
title = {Matrix Analysis},
edition = {2nd}, publisher = {Cambridge University Press}, year = {2013}}
% --- Tier B: redundancy resolution ---
@article{siciliano1990tutorial,
author = {Siciliano, Bruno},
title = {Kinematic Control of Redundant Robot Manipulators: A Tutorial},
journal = {Journal of Intelligent and Robotic Systems},
volume = {3}, number = {3}, pages = {201--212}, year = {1990},
doi = {10.1007/BF00126069}}
@article{liegeois1977supervisory,
author = {Li{\'e}geois, Alain},
title = {Automatic Supervisory Control of the Configuration and Behavior of Multibody Mechanisms},
journal = {IEEE Transactions on Systems, Man, and Cybernetics},
volume = {7}, number = {12}, pages = {868--871}, year = {1977},
doi = {10.1109/TSMC.1977.4309644}}
@article{kleinhuang1983pseudoinverse,
author = {Klein, Charles A. and Huang, Ching-Hsiang},
title = {Review of Pseudoinverse Control for Use with Kinematically Redundant Manipulators},
journal = {IEEE Transactions on Systems, Man, and Cybernetics},
volume = {SMC-13}, pages = {245--250}, year = {1983},
doi = {10.1109/TSMC.1983.6313123}} % verify issue no. on fetch
@incollection{chiaverini2008redundant,
author = {Chiaverini, Stefano and Oriolo, Giuseppe and Walker, Ian D.},
title = {Kinematically Redundant Manipulators},
booktitle = {Springer Handbook of Robotics},
editor = {Siciliano, Bruno and Khatib, Oussama},
publisher = {Springer}, address = {Berlin, Heidelberg}, year = {2008},
doi = {10.1007/978-3-540-30301-5_12}} % pages from fetch
@book{nakamura1991advanced,
author = {Nakamura, Yoshihiko},
title = {Advanced Robotics: Redundancy and Optimization},
publisher = {Addison-Wesley}, year = {1991}}
@book{siciliano2009robotics,
author = {Siciliano, Bruno and Sciavicco, Lorenzo and Villani, Luigi and Oriolo, Giuseppe},
title = {Robotics: Modelling, Planning and Control},
publisher = {Springer}, year = {2009}}
% --- Tier C: our specific objects ---
@inproceedings{burdick1989selfmotion,
author = {Burdick, Joel W.},
title = {On the Inverse Kinematics of Redundant Manipulators: Characterization of the Self-Motion Manifolds},
booktitle = {Proc. IEEE International Conference on Robotics and Automation (ICRA)},
address = {Scottsdale, AZ, USA}, pages = {264--270}, year = {1989}}
% doi = {TODO: confirm via Semantic Scholar title search}
@article{khatib1987operational,
author = {Khatib, Oussama},
title = {A Unified Approach for Motion and Force Control of Robot Manipulators: The Operational Space Formulation},
journal = {IEEE Journal on Robotics and Automation},
volume = {3}, number = {1}, pages = {43--53}, year = {1987},
doi = {10.1109/JRA.1987.1087068}}
@article{dubowsky1993spacerobots,
author = {Dubowsky, Steven and Papadopoulos, Evangelos},
title = {The Kinematics, Dynamics, and Control of Free-Flying and Free-Floating Space Robotic Systems},
journal = {IEEE Transactions on Robotics and Automation},
volume = {9}, number = {5}, pages = {531--543}, year = {1993},
doi = {10.1109/70.258046}}
@book{ott2008cartesian,
author = {Ott, Christian},
title = {Cartesian Impedance Control of Redundant and Flexible-Joint Robots},
series = {Springer Tracts in Advanced Robotics}, publisher = {Springer}, year = {2008}}DOIs/metadata web-verified Jun 19, 2026. One unconfirmed: Burdick 1989 DOI (fetch by title). Page/issue fields flagged inline are the only soft spots; everything else (author, title, year, venue, DOI) is verified.