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arxiv: 2605.06498 · v1 · pith:L7UN5C3Hnew · submitted 2026-05-07 · 💻 cs.RO · cs.SY· eess.SY

Lie Group Formulation of Recursive Dynamics Algorithms of Higher Order for Floating-Base Robots

Ahmed Ali , Chiara Gabellieri , Antonio Franchi This is my paper

Pith reviewed 2026-05-08 08:42 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords lie group dynamicsfloating base robotshigher order derivativesrecursive algorithmsnewton-eulerarticulated body inertiaaerial manipulatorquadratic complexity
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The pith

Recursive Lie-group algorithms enable higher-order dynamics computation for floating-base robots at quadratic cost.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops procedures for computing higher-order time derivatives of the Lie-group Newton-Euler, articulated-body inertia, and hybrid dynamics algorithms for floating-base tree robots. These recursions are collected into closed-form equations of motion that preserve an admissible Coriolis matrix satisfying passivity and keep the articulated inertia tensor unchanged across derivatives. The methods are demonstrated on a 12-DoF aerial manipulator with analytical expressions for the first derivative and numerical evaluation up to the fifth order. A sympathetic reader would care because this provides an efficient way to obtain the higher derivatives required for advanced control, optimization, and simulation in robotics, avoiding the exponential cost of automatic differentiation.

Core claim

We describe recursive procedures to compute the higher-order time derivatives of the Lie-group Newton-Euler, Articulated-Body Inertia, and hybrid dynamics algorithms for floating-base trees. The resulting recursions are collected into closed-form equations of motion, identifying an admissible Coriolis matrix satisfying the passivity property and showing that the articulated inertia tensor remains unchanged across all time derivatives. Application to a 12-DoF aerial manipulator yields analytical geometric forward and inverse dynamics along with their first time derivatives, while numerical simulations evaluate these dynamics up to fifth order. Benchmarks demonstrate quadratic computational成本

What carries the argument

Recursive higher-order extensions of the Lie-group Newton-Euler and Articulated-Body Inertia algorithms using spatial twists for SE(3) base and tree joints.

If this is right

  • The dynamics algorithms can be extended to arbitrary derivative orders via recursion.
  • The articulated inertia tensor is invariant under time differentiation of any order.
  • An admissible Coriolis matrix satisfying passivity exists for the closed-form equations at each order.
  • Analytical expressions for first-order derivatives are derivable for systems like the 12-DoF aerial manipulator.
  • Computation up to fifth order is practical and scales quadratically in cost.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This could support derivative-based methods in real-time robot control and motion planning that were previously computationally prohibitive.
  • The invariance of the inertia tensor across orders may simplify the analysis of energy properties or stability in higher-order dynamic systems.
  • Similar recursive structures could be adapted for robots with different topologies or additional constraints.
  • The quadratic scaling suggests these algorithms are suitable for integration into optimization frameworks requiring multiple derivatives.

Load-bearing premise

The higher-order recursive procedures can be collected into closed-form equations of motion while preserving an admissible Coriolis matrix that satisfies passivity and leaving the articulated inertia tensor unchanged across derivatives.

What would settle it

Computing the fifth-order dynamics for the 12-DoF aerial manipulator and verifying that the articulated inertia tensor matches the base case while the Coriolis matrix satisfies passivity, alongside confirming quadratic computation time scaling.

read the original abstract

In this paper, we describe procedures for computing higher-order time derivatives of the Lie-group Newton-Euler, Articulated-Body Inertia, and hybrid dynamics algorithms for floating-base trees, where the base configuration evolves on SE(3) and the attached mechanism is an open kinematic tree with configuration on the (n1+n2)-dimensional manifold T^{n1} \times R^{n2}, using spatial representation of twists. After presenting the algorithms, we collect the resulting recursions into closed-form equations of motion, identifying an admissible Coriolis matrix satisfying the passivity property, and showing that the articulated inertia tensor remains unchanged across all time derivatives. We then apply the developed methods to a 12-DoF aerial manipulator to derive analytical expressions for its geometric forward and inverse dynamics along with their first time derivatives whereas the numerical simulations successfully evaluate these dynamics up to fifth order. Finally, to demonstrate their practical utility, we benchmark the proposed extensions and show that, in the considered tests, their computational cost scales quadratically with the derivative order, whereas the automatic-differentiation baseline exhibits exponential scaling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper describes recursive procedures for higher-order time derivatives of Lie-group Newton-Euler, ABA, and hybrid dynamics algorithms for floating-base trees on SE(3) with tree mechanisms. The recursions are collected into closed-form equations, with an admissible Coriolis matrix for passivity and invariance of the articulated inertia tensor. Analytical expressions up to first derivatives are given for a 12-DoF aerial manipulator, numerical evaluations to fifth order, and benchmarks show quadratic cost scaling with derivative order versus exponential for automatic differentiation.

Significance. This contribution is significant for robotics as it enables efficient computation of higher-order dynamics derivatives using recursive methods rather than costly automatic differentiation. The quadratic scaling demonstrated in benchmarks is particularly valuable for real-time applications. The work credits the extension of established algorithms with Lie-group structure, provides concrete analytical results, and includes numerical validation, making the claims falsifiable and the methods reproducible in principle. The proof of inertia invariance and passivity-preserving Coriolis matrix add theoretical strength without introducing free parameters.

major comments (1)
  1. [Numerical evaluation] The manuscript reports successful numerical evaluation of the dynamics up to fifth order but does not provide error bounds or full verification details against independent calculations. This is important for substantiating the higher-order results and the quadratic scaling claim in the benchmarks.
minor comments (2)
  1. [Configuration description] The manifold is specified as T^{n1} × R^{n2} without defining n1 and n2 in the aerial manipulator example, which could be clarified for readers.
  2. [Benchmark results] Including the specific automatic differentiation library and hardware specifications used in the timing comparisons would enhance reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. We address the major comment below and will incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Numerical evaluation] The manuscript reports successful numerical evaluation of the dynamics up to fifth order but does not provide error bounds or full verification details against independent calculations. This is important for substantiating the higher-order results and the quadratic scaling claim in the benchmarks.

    Authors: We agree that the original manuscript would benefit from explicit error bounds and more detailed verification against independent calculations. While the numerical simulations in the paper successfully compute the dynamics up to fifth order, we did not include quantitative error analysis or comparisons in the submitted version. In the revised manuscript, we will add a dedicated verification subsection that compares the recursive higher-order derivatives to finite-difference approximations (with multiple step sizes) and reports the resulting error norms and observed convergence orders. This will directly substantiate the accuracy of the fifth-order results and support the quadratic scaling benchmarks. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends established recursive Lie-group dynamics independently

full rationale

The paper presents explicit recursive procedures for higher-order derivatives of the Lie-group Newton-Euler, ABA, and hybrid algorithms on SE(3) floating-base trees, then collects them into closed-form equations while preserving an admissible Coriolis matrix and proving invariance of the articulated inertia tensor. These steps follow directly from differentiating the base recursions (which are taken as given from prior literature) without any reduction to fitted parameters, self-definitions, or unverified self-citations. The analytic expressions for the 12-DoF aerial manipulator and the quadratic-vs-exponential benchmark timings are independent numerical/analytical outputs, not forced by construction from the inputs. No enumerated circularity pattern applies.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard Lie-group properties of SE(3) and the product manifold for joints, plus the existence of recursive factorizations that extend to arbitrary order without introducing new fitted quantities.

axioms (1)
  • domain assumption Base configuration evolves on SE(3) and joints on T^{n1} × R^{n2} with spatial twist representation
    Explicitly stated as the configuration space for floating-base trees.

pith-pipeline@v0.9.0 · 5499 in / 1115 out tokens · 64416 ms · 2026-05-08T08:42:03.863570+00:00 · methodology

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Reference graph

Works this paper leans on

71 extracted references · 71 canonical work pages

  1. [1]

    and Roth, B., 1990,Theoretical kinematics, Vol

    Bottema, O. and Roth, B., 1990,Theoretical kinematics, Vol. 24, Courier Cor- poration

    work page 1990

  2. [2]

    Siciliano, B., Sciavicco, L., Villani, L., and Oriolo, G., 2010,Robotics: Mod- elling, Planning and Control, Advanced Textbooks in Control and Signal Pro- cessing, Springer London

    work page 2010

  3. [3]

    Angeles, J., 2003,Fundamentals of robotic mechanical systems: theory, meth- ods, and algorithms, Springer

    work page 2003

  4. [4]

    A Recursive Lagrangian Formulation of Maniputator Dynamics and a Comparative Study of Dynamics Formulation Complexity,

    Hollerbach, J. M., 1980, “A Recursive Lagrangian Formulation of Maniputator Dynamics and a Comparative Study of Dynamics Formulation Complexity,” IEEE Transactions on Systems, Man, and Cybernetics,10(11), pp. 730–736

    work page 1980

  5. [5]

    Dynamics of articulated open- chain active mechanisms,

    Stepanenko, Y. and Vukobratović, M., 1976, “Dynamics of articulated open- chain active mechanisms,” Mathematical Biosciences,28(1), pp. 137–170

    work page 1976

  6. [6]

    Kinematic and kinetic analysis of open-chain linkages utilizing Newton-Euler methods,

    Orin, D., McGhee, R., Vukobratović, M., and Hartoch, G., 1979, “Kinematic and kinetic analysis of open-chain linkages utilizing Newton-Euler methods,” Mathematical Biosciences,43(1), pp. 107–130

    work page 1979

  7. [7]

    Featherstone, R., 2014,Rigid body dynamics algorithms, Springer

    work page 2014

  8. [8]

    Jain, A., 2010,Robot and multibody dynamics: analysis and algorithms, Springer Science & Business Media

    work page 2010

  9. [9]

    Efficient O (n) recursive computation of the operational space inertia matrix,

    Lilly, K. W. and Orin, D. E., 2002, “Efficient O (n) recursive computation of the operational space inertia matrix,” IEEE Transactions on Systems, Man, and Cybernetics,23(5), pp. 1384–1391

    work page 2002

  10. [10]

    Kalman filtering, smoothing, and recursive robot arm forward and inverse dynamics,

    Rodriguez, G., 1987, “Kalman filtering, smoothing, and recursive robot arm forward and inverse dynamics,” IEEE Journal on Robotics and Automation, 3(6), pp. 624–639

    work page 1987

  11. [11]

    Spatial operator factorization and inversion of the manipulator mass matrix,

    Rodriguez, G. and Kreutz-Delgado, K., 1992, “Spatial operator factorization and inversion of the manipulator mass matrix,” IEEE Transactions on Robotics and Automation,8(1), pp. 65–76

    work page 1992

  12. [12]

    Analytical dynamics of mechanisms—a computer oriented overview,

    Paul, B., 1975, “Analytical dynamics of mechanisms—a computer oriented overview,” Mechanism and Machine Theory,10(6), pp. 481–507

    work page 1975

  13. [13]

    A Sparsity-Oriented Approach to the Dynamic Analysis and Design of Mechanical Systems—Part 1,

    Orlandea, N., Chace, M. A., and Calahan, D. A., 1977, “A Sparsity-Oriented Approach to the Dynamic Analysis and Design of Mechanical Systems—Part 1,” Journal of Engineering for Industry,99(3), pp. 773–779

    work page 1977

  14. [14]

    A new perspective on O (n) mass-matrix inversion for serial revolute manipulators,

    Lee, K. and Chirikjian, G. S., 2005, “A new perspective on O (n) mass-matrix inversion for serial revolute manipulators,”Proceedings of the 2005 IEEE Inter- national Conference on Robotics and Automation, IEEE, pp. 4722–4726

    work page 2005

  15. [15]

    General Dynamic Algorithm for Floating Base Tree Structure Robots With Flexible Joints and Links,

    Khalil, W., Boyer, F., and Morsli, F., 2017, “General Dynamic Algorithm for Floating Base Tree Structure Robots With Flexible Joints and Links,” Journal of Mechanisms and Robotics,9(3), p. 031003

    work page 2017

  16. [16]

    Modeling and Mechanical Analysis of Snake Robots on Cylinders,

    Tang, C., Li, P., Zhou, G., Meng, D., Shu, X., Guo, S., and Li, Z., 2019, “Modeling and Mechanical Analysis of Snake Robots on Cylinders,” Journal of Mechanisms and Robotics,11(4), p. 041013

    work page 2019

  17. [17]

    Vectorized Formula- tion of Newton-Euler Dynamics for Efficiently Computing Three-Dimensional Folding Chains,

    Fass, T. H., Hao, G., and Cantillon-Murphy, P., 2022, “Vectorized Formula- tion of Newton-Euler Dynamics for Efficiently Computing Three-Dimensional Folding Chains,” Journal of Mechanisms and Robotics,14(6), p. 060911

    work page 2022

  18. [18]

    Symbolic Differentiation Algorithm for InverseDynamicsofSerialRobotsWithFlexibleJoints,

    Do, T.-T., Vu, V.-H., and Liu, Z., 2021, “Symbolic Differentiation Algorithm for InverseDynamicsofSerialRobotsWithFlexibleJoints,”JournalofMechanisms and Robotics,13(6), p. 064501. Journal of Mechanisms and Robotics / 9 0 10 20 30 -1 0 1 p [m] x y z 0 10 20 30 -20 0 20[deg] 0 10 20 30 -0.1 0 0.1[deg/s] x y z 0 10 20 30 -0.02 0 0.02(1) [deg/s2] 0 10 20 30 ...

    work page 2021

  19. [19]

    Robotic manipulators and the product of exponen- tials formula,

    Brockett, R. W., 2005, “Robotic manipulators and the product of exponen- tials formula,”Mathematical Theory of Networks and Systems: Proceedings of the MTNS-83 International Symposium Beer Sheva, Israel, June 20–24, 1983, Springer, pp. 120–129

    work page 2005

  20. [20]

    Robots and screw theory: applicationsofkinematicsandstaticstorobotics,

    Davidson, J. K., Hunt, K. H., and Pennock, G. R., 2004, “Robots and screw theory: applicationsofkinematicsandstaticstorobotics,”J.Mech.Des.,126(4), pp. 763–764

    work page 2004

  21. [21]

    M., Li, Z., and Sastry, S

    Murray, R. M., Li, Z., and Sastry, S. S., 2017,A mathematical introduction to robotic manipulation, CRC press

    work page 2017

  22. [22]

    Lynch, K., 2017,Modern Robotics, Cambridge University Press

    work page 2017

  23. [23]

    TheLiegroupof rigidbodydisplacements, afundamentaltool for mechanism design,

    Hervé, J., 1999, “TheLiegroupof rigidbodydisplacements, afundamentaltool for mechanism design,” Mechanism and Machine Theory,34(5), pp. 719–730

    work page 1999

  24. [24]

    An Analytic and Computational Condition for the Finite Degree-of-Freedom of Linkages, and Its Relation to Lie Group Methods,

    Müller, A., 2022, “An Analytic and Computational Condition for the Finite Degree-of-Freedom of Linkages, and Its Relation to Lie Group Methods,” Jour- nal of Mechanisms and Robotics,14(4), p. 041011

    work page 2022

  25. [25]

    Singularity-Free Dynamic Equations of Open-Chain Mechanisms With General Holonomic and Nonholonomic Joints,

    Duindam, V. and Stramigioli, S., 2008, “Singularity-Free Dynamic Equations of Open-Chain Mechanisms With General Holonomic and Nonholonomic Joints,” IEEE Transactions on Robotics,24(3), pp. 517–526

    work page 2008

  26. [26]

    On the inertially decoupled structure of the floating base robot dynamics,

    Garofalo, G., Henze, B., Englsberger, J., and Ott, C., 2015, “On the inertially decoupled structure of the floating base robot dynamics,” IFAC-PapersOnLine, 48(1), pp.322–327, 8thViennaInternationalConferenceonMathematicalMod- elling

    work page 2015

  27. [27]

    Centroidal dynamics of a humanoid robot,

    Orin, D. E., Goswami, A., and Lee, S.-H., 2013, “Centroidal dynamics of a humanoid robot,” Autonomous robots,35, pp. 161–176

    work page 2013

  28. [28]

    A Lie group formulation of robot dynamics,

    Park, F. C., Bobrow, J. E., and Ploen, S. R., 1995, “A Lie group formulation of robot dynamics,” The International journal of robotics research,14(6), pp. 609–618

    work page 1995

  29. [29]

    and Cardona, A., 2001,Flexible multibody dynamics: A finite element approach, John Wiley

    Geradin, M. and Cardona, A., 2001,Flexible multibody dynamics: A finite element approach, John Wiley

    work page 2001

  30. [30]

    2011,Two Lie Group Formulations for Dynamic Multibody Systems With Large Rotations, Vol. Volume 4: 8th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A and B of International Design Engi- neering Technical Conferences and Computers and Information in Engineering Conference

    work page 2011

  31. [31]

    A Formulation on the Special Euclidean Group for Dynamic Analysis of Multibody Systems,

    Sonneville, V. and Brüls, O., 2014, “A Formulation on the Special Euclidean Group for Dynamic Analysis of Multibody Systems,” Journal of Computational and Nonlinear Dynamics,9(4), p. 041002

    work page 2014

  32. [32]

    Coordinate-invariant algorithms for robot dynam- ics,

    Ploen, S. and Park, F., 1999, “Coordinate-invariant algorithms for robot dynam- ics,” IEEE Transactions on Robotics and Automation,15(6), pp. 1130–1135

    work page 1999

  33. [33]

    Geometric algorithms for robot dynamics: A tutorial review,

    Park, F. C., Kim, B., Jang, C., and Hong, J., 2018, “Geometric algorithms for robot dynamics: A tutorial review,” Applied Mechanics Reviews,70(1), p. 010803

    work page 2018

  34. [34]

    Bobrow, J., Park, F., and Sideris, A., 2006,Recent Advances on the Algorithmic Optimization of Robot Motion, Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 21–41

    work page 2006

  35. [35]

    ScrewandLiegrouptheoryinmultibodykinematics: Motion representation and recursive kinematics of tree-topology systems,

    Müller,A.,2018,“ScrewandLiegrouptheoryinmultibodykinematics: Motion representation and recursive kinematics of tree-topology systems,” Multibody System Dynamics,43(1), pp. 37–70

    work page 2018

  36. [36]

    Screw and Lie group theory in multibody dynamics: recur- sive algorithms and equations of motion of tree-topology systems,

    Müller, A., 2018, “Screw and Lie group theory in multibody dynamics: recur- sive algorithms and equations of motion of tree-topology systems,” Multibody System Dynamics,42(2), pp. 219–248

    work page 2018

  37. [37]

    An Ana- lytical and Modular Software Workbench for Solving Kinematics and Dynamics of Series-Parallel Hybrid Robots,

    Kumar, S., Szadkowski, K. A. v., Mueller, A., and Kirchner, F., 2020, “An Ana- lytical and Modular Software Workbench for Solving Kinematics and Dynamics of Series-Parallel Hybrid Robots,” Journal of Mechanisms and Robotics,12(2), p. 021114

    work page 2020

  38. [38]

    Reduced Euler-Lagrange Equations of Floating-Base Robots: Com- putation, Properties, & Applications,

    Mishra, H., Garofalo, G., Giordano, A. M., De Stefano, M., Ott, C., and Kugi, A., 2023, “Reduced Euler-Lagrange Equations of Floating-Base Robots: Com- putation, Properties, & Applications,” IEEE Transactions on Robotics,39(2), pp. 1439–1457

    work page 2023

  39. [39]

    Hamel, G., 2013,Theoretische Mechanik: Eine einheitliche Einführung in die gesamte Mechanik, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg

    work page 2013

  40. [40]

    Dynamics of Mobile Manipulators Using Dual Quaternion Algebra,

    Afonso Silva, F. F., José Quiroz-Omaña, J., and Vilhena Adorno, B., 2022, “Dynamics of Mobile Manipulators Using Dual Quaternion Algebra,” Journal of Mechanisms and Robotics,14(6), p. 061005

    work page 2022

  41. [41]

    Dual Quaternions Representation ofLagrange’sDynamicEquations,

    Cohen, A., Taub, B., and Shoham, M., 2023, “Dual Quaternions Representation ofLagrange’sDynamicEquations,” JournalofMechanismsandRobotics,16(4), p. 041004

    work page 2023

  42. [42]

    The PenduMAV: A Six-Input Omnidirectional MAV without Internal Forces – Design, Dynamics, and SE(3) Control,

    Ali, A., Gabellieri, C., and Franchi, A., 2026, “The PenduMAV: A Six-Input Omnidirectional MAV without Internal Forces – Design, Dynamics, and SE(3) Control,” 2510.15071, https://arxiv.org/abs/2510.15071

    work page arXiv 2026

  43. [43]

    OntheFeedbackLinearization ofRobotswithVariableJointStiffness,

    Palli,G.,Melchiorri,C.,andDeLuca,A.,2008,“OntheFeedbackLinearization ofRobotswithVariableJointStiffness,”2008IEEEInternationalConferenceon Robotics and Automation, pp. 1753–1759, doi: 10.1109/ROBOT.2008.4543454

    work page doi:10.1109/robot.2008.4543454 2008

  44. [44]

    Modeling and Control of Elastic Joint Robots,

    Spong, M. W., 1987, “Modeling and Control of Elastic Joint Robots,” Journal of Dynamic Systems, Measurement, and Control,109(4), pp. 310–318

    work page 1987

  45. [45]

    A recursive Newton-Euler algorithm for robots with elastic joints and its application to control,

    Buondonno, G. and De Luca, A., 2015, “A recursive Newton-Euler algorithm for robots with elastic joints and its application to control,”2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 5526– 5532, doi: 10.1109/IROS.2015.7354160

    work page doi:10.1109/iros.2015.7354160 2015

  46. [46]

    Smooth and time-optimal trajectory planning for industrial manipulators along specified paths,

    Constantinescu, D. and Croft, E. A., 2000, “Smooth and time-optimal trajectory planning for industrial manipulators along specified paths,” Journal of Robotic Systems,17(5), pp. 233–249

    work page 2000

  47. [47]

    Gasparetto, A., Boscariol, P., Lanzutti, A., and Vidoni, R., 2015,Path Planning and Trajectory Planning Algorithms: A General Overview, Springer Interna- tional Publishing, Cham, pp. 3–27

    work page 2015

  48. [48]

    Crocod- dyl: An efficient and versatile framework for multi-contact optimal control,

    Mastalli, C., Budhiraja, R., Merkt, W., Saurel, G., Hammoud, B., Naveau, M., Carpentier, J., Righetti, L., Vijayakumar, S., and Mansard, N., 2020, “Crocod- dyl: An efficient and versatile framework for multi-contact optimal control,” 2020IEEEInternationalConferenceonRoboticsandAutomation(ICRA),IEEE, pp. 2536–2542

    work page 2020

  49. [49]

    Newton-type al- gorithms for dynamics-based robot movement optimization,

    Lee, S.-H., Kim, J., Park, F., Kim, M., and Bobrow, J., 2005, “Newton-type al- gorithms for dynamics-based robot movement optimization,” IEEE Transactions on Robotics,21(4), pp. 657–667

    work page 2005

  50. [50]

    Optimalrobotmotionsforphysicalcriteria,

    Bobrow, J. E., Martin, B., Sohl, G., Wang, E. C., Park, F. C., and Kim, J., 2001, “Optimalrobotmotionsforphysicalcriteria,” JournalofRoboticSystems, 18(12), pp. 785–795

    work page 2001

  51. [51]

    A geometric formulation of multirotor aerial vehicle dynamics,

    Hong, Y., Rashad, R., Noh, S., Lee, T., Stramigioli, S., and Park, F. C., 2022, “A geometric formulation of multirotor aerial vehicle dynamics,” Nonlinear Dynamics,107(1), pp. 495–513

    work page 2022

  52. [52]

    Closed-form time derivatives of the equations of motion of rigid body systems,

    Müller, A. and Kumar, S., 2021, “Closed-form time derivatives of the equations of motion of rigid body systems,” Multibody System Dynamics,53(3), pp. 257–273

    work page 2021

  53. [53]

    Nth Order Analytical Time Derivatives of Inverse Dynamics in Recursive and Closed Forms,

    Kumar, S. and Müller, A., 2021, “Nth Order Analytical Time Derivatives of Inverse Dynamics in Recursive and Closed Forms,”2021 IEEE Inter- national Conference on Robotics and Automation (ICRA), pp. 1918–1924, doi: 10.1109/ICRA48506.2021.9561773. 12 / Transactions of the ASME

    work page doi:10.1109/icra48506.2021.9561773 2021

  54. [54]

    An O(n)-Algorithm for the Higher-Order Kinematics and In- verse Dynamics of Serial Manipulators Using Spatial Representation of Twists,

    Müller, A., 2021, “An O(n)-Algorithm for the Higher-Order Kinematics and In- verse Dynamics of Serial Manipulators Using Spatial Representation of Twists,” IEEE Robotics and Automation Letters,6(2), pp. 397–404

    work page 2021

  55. [55]

    On the closed form com- putation of the dynamic matrices and their differentiations,

    Garofalo, G., Ott, C., and Albu-Schäffer, A., 2013, “On the closed form com- putation of the dynamic matrices and their differentiations,”2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, IEEE, pp. 2364– 2359

    work page 2013

  56. [56]

    CasADi – A software framework for nonlinear optimization and optimal con- trol,

    Andersson, J. A. E., Gillis, J., Horn, G., Rawlings, J. B., and Diehl, M., 2018, “CasADi – A software framework for nonlinear optimization and optimal con- trol,” Mathematical Programming Computation

    work page 2018

  57. [57]

    A geometrical formulation of the dynamical equations describing kinematic chains,

    Brockett, R., Stokes, A., and Park, F., 1993, “A geometrical formulation of the dynamical equations describing kinematic chains,”[1993] Proceedings IEEE International Conference on Robotics and Automation, pp. 637–641 vol.2, doi: 10.1109/ROBOT.1993.291887

    work page doi:10.1109/robot.1993.291887 1993

  58. [58]

    A micro lie theory for state estimation in robotics,

    Sola, J., Deray, J., and Atchuthan, D., 2018, “A micro Lie theory for state estimation in robotics,” arXiv preprint arXiv:1812.01537

    work page arXiv 2018

  59. [59]

    Higher derivatives of the kinematic mapping and some ap- plications,

    Müller, A., 2014, “Higher derivatives of the kinematic mapping and some ap- plications,” Mechanism and Machine Theory,76, pp. 70–85

    work page 2014

  60. [60]

    Modeling,control and design optimization for a fully-actuated hexarotor aerial vehicle with tilted propellers,

    Rajappa,S.,Ryll, M.,Bülthoff,H.H.,andFranchi,A., 2015,“Modeling,control and design optimization for a fully-actuated hexarotor aerial vehicle with tilted propellers,”2015 IEEE International Conference on Robotics and Automation (ICRA), pp. 4006–4013, doi: 10.1109/ICRA.2015.7139759

    work page doi:10.1109/icra.2015.7139759 2015

  61. [61]

    Past, present, and future of aerial robotic manipulators,

    Ollero, A., Tognon, M., Suarez, A., Lee, D., and Franchi, A., 2021, “Past, present, and future of aerial robotic manipulators,” IEEE Transactions on Robotics,38(1), pp. 626–645

    work page 2021

  62. [62]

    Design and Control of an Aerial Manipulator With Invariant Center of Gravity for Physical Interaction,

    Rong, Y. and Chou, W., 2023, “Design and Control of an Aerial Manipulator With Invariant Center of Gravity for Physical Interaction,” Journal of Mecha- nisms and Robotics,16(7), p. 071001

    work page 2023

  63. [63]

    The Pinocchio C++ library – A fast and flexible imple- mentation of rigid body dynamics algorithms and their analytical derivatives,

    Carpentier, J., Saurel, G., Buondonno, G., Mirabel, J., Lamiraux, F., Stasse, O., and Mansard, N., 2019, “The Pinocchio C++ library – A fast and flexible imple- mentation of rigid body dynamics algorithms and their analytical derivatives,” IEEE International Symposium on System Integrations (SII)

    work page 2019

  64. [64]

    A Recursive Lie-Group For- mulation for the Second-Order Time Derivatives of the Inverse Dynamics of Parallel Kinematic Manipulators,

    Müller, A., Kumar, S., and Kordik, T., 2023, “A Recursive Lie-Group For- mulation for the Second-Order Time Derivatives of the Inverse Dynamics of Parallel Kinematic Manipulators,” IEEE Robotics and Automation Letters,8(6), pp. 3804–3811

    work page 2023

  65. [65]

    M., 2007,Geometric fundamentals of robotics, Springer Science & Business Media

    Selig, J. M., 2007,Geometric fundamentals of robotics, Springer Science & Business Media

    work page 2007

  66. [66]

    and Ratiu, T., 2013,Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathemat- ics, Springer New York

    Marsden, J. and Ratiu, T., 2013,Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathemat- ics, Springer New York. Appendix A: Proof of Lemma 1 The proof follows from direct computation of the expression 1 2 ̇¯M−Cwith ¯Mdefined in (5). 1 2 ̇¯M−C= 1 2 ( ̇S𝑇 G𝑐MG𝑝S+S 𝑇 G𝑐 ̇MG𝑝S+S 𝑇 G𝑐MG𝑝 ̇S) −S 𝑇 G...

    work page 2013

  67. [67]

    Hence, the new value of the𝑾1, denoted by ¯𝑾1, has to account for the wrench𝑾2 coming from the child body𝐵2 as follows ¯𝑾1 =𝑾 1 +𝑾 2 = (︁(M0 1 +M 0

    Note that the product (︁𝑺𝑇 2 M0 2𝑺2 )︁is a non-zero scalar when multi-DoF joints are treated as an arrangement of 1-DoF joints. Hence, the new value of the𝑾1, denoted by ¯𝑾1, has to account for the wrench𝑾2 coming from the child body𝐵2 as follows ¯𝑾1 =𝑾 1 +𝑾 2 = (︁(M0 1 +M 0

    work page

  68. [68]

    (C2) where the articulated inertiaM𝐴 1 and bias𝑾 𝐴 1 change to these new values ¯M𝐴 1 and ¯𝑾 𝐴 1 , respectively ¯M𝐴 1 =(︁(M0 1 +M 0

    −M 0 2𝑺2 (𝑺 𝑇 2 M0 2𝑺2) −1 𝑺𝑇 2 M0 2 )︁̇𝑽0 1 +M 0 2 (𝑺 2 ̈˜𝑞2 + ̇𝑺2 ̇𝑞2) −ad 𝑇 𝑽 0 1 M0 1𝑽0 1 −ad 𝑇 𝑽 0 2 M0 2𝑽0 2 −𝑾 0 app,1 −𝑾 0 grav,1 −𝑾 0 app,2 −𝑾 0 grav,2 = ¯M𝐴 1 ̇𝑽0 1 + ¯𝑾 𝐴 1 . (C2) where the articulated inertiaM𝐴 1 and bias𝑾 𝐴 1 change to these new values ¯M𝐴 1 and ¯𝑾 𝐴 1 , respectively ¯M𝐴 1 =(︁(M0 1 +M 0

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  69. [69]

    The articulated body𝐴 1 now has a handle𝐵 1 with updated equations of motion (C2)

    −M 0 2𝑺2 (𝑺 𝑇 2 M0 2𝑺2) −1 𝑺𝑇 2 M0 2 )︁, ¯𝑾 𝐴 1 =M0 2 (𝑺 2 ̈˜𝑞2 + ̇𝑺2 ̇𝑞2) −ad 𝑇 𝑽 0 1 M0 1𝑽0 1 −ad 𝑇 𝑽 0 2 M0 2𝑽0 2 −𝑾 0 app,1 −𝑾 0 grav,1 −𝑾 0 app,2 −𝑾 0 grav,2 . The articulated body𝐴 1 now has a handle𝐵 1 with updated equations of motion (C2). Body𝐴1 can be connected to another body as its child or its parent through a joint between the handle 𝐵1 and ...

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  70. [70]

    −M 0 2𝑺2 (𝑺 𝑇 2 M0 2𝑺2) −1 𝑺𝑇 2 M0 2 )︁̈𝑽0 1 +M 0 2 ̈˜𝑽0 2 +M 0 2𝑺2 ⃛˜𝑞2 + ̈˜𝚷0 1 + ̈˜𝚷0 2 − ̇𝑾0 app,1 − ̇𝑾0 grav,1 − ̇𝑾0 app,2 − ̇𝑾0 grav,2 = ¯M𝐴 1 ̈𝑽0 1 + ̇¯𝑾 𝐴 1 , where ̈˜𝚷0 1 := ̈𝚷0 1 −M 0 1 ̈𝑽0

    work page

  71. [71]

    Hence, it can be calculated once and reused for any order

    with the property ¯M𝐴 1 being the same as the acceleration level (𝑟=0). Hence, it can be calculated once and reused for any order. The procedure continues by connecting𝐵1 as a child to its pre- decessor in the chain until the base body is reached. Then, the following system of six linear equations is solved for̈𝑽0 base using 𝐿𝐷 𝐿𝑇 decomposition: ̇𝑾0 1,pro...

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