Skip to main content
SpringerLink
Log in

A bone-wise approach for modeling the human hind-midfoot. Part I: kinematics

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

Multi-line segment foot models propose a triple-line segmentation into hindfoot, midfoot and forefoot; oftentimes, the midfoot is regarded as a rigid line segment. We here begin our presentation of a new model of the hind-midfoot bone complex, regarded as a system of rigid blocks, the bones, with perfect mutual constraints, the bone joints. Firstly, we give a qualitative description of the deformations of the foot per se, as if it were severed from the body or made to move passively in vivo, the talocrural joint being kept locked; the descriptive capabilities of our model are illustrated by progressively locking more and more bone joints. Secondly, we derive two interdependent systems of nonlinear algebraic equations which render the geometrical and kinematic complexity of the hind-midfoot complex: one system locates individual bones in space, the other selects all location sets classifiable as unbroken kinematic chains, that is to say, all admissible hind-midfoot configurations. We define the relative multi-parameter configuration manifold, indicate how to inspect it hierarchically, and furnish a representative example amenable to geometrical description. The accompanying dataset document features graphical information and configurational data helping to motivate our modelling choices, as well as analytical information concerning certain mathematical developments we here omit.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

Notes

  1. A sketch by Leonardo da Vinci (Windsor, RL 19008r) tells us that he too considered this possibility, and quickly discarded it, when thinking about building a model of the human foot (see Fig. 1 in DSS2 and [1]).

  2. It is important to realise that in our study of HMF kinematics the values of certain anatomical parameters are kept fixed while others do change when the HMF chain changes its shape.The chain parameters keeping their reference values are the radii of hinges 01 and 02 and of cylinder 26; the parameters in the need of adjournment according to shape changes are the axes of those three joints, the normal to plane 56 and the centres of spheres 13, 34, 35, and 56.

  3. Surface corrugation is meant to suggest that translations along the axes of these tubes are forbidden.

  4. Here is a relevant quotations from [5]: “The talo-navicular articulation being of ball-and-socket shape ...”.

  5. Alternatively, we might have chosen to make use here of \(R^{(13)}\), the rotation of the navicular with respect to the talus, which satisfies the composition condition \(\,R^{(13)}_{\circ } = R^{(01)} R^{(13)}\).

  6. Here \(m^{(36)}_{*}\) is an arbitrarily chosen unit vector orthogonal to \(n^{(36)}_{*}\).

  7. Cf. the proposition proved in DSS4.

  8. Here \(m^{(56)}_{*}\) is an arbitrarily chosen unit vector orthogonal to \(n^{(56)}_{*}\).

  9. A linearization of problem (25–52) about a chosen admissible configuration is available in DSS5. A kinematically admissible motion of the HMF chain is a one-parameter path over \({\mathcal {S}}\); granted differentiability, and on interpreting the evolution parameter as time, the motion-velocity multivector is a solution of the linear problem just mentioned.

  10. Since \(\vartheta _3\in [0,\pi /2)\) and \(|N^{(13)}_{\circ }|^2=2\), (31) implies that

    $$\begin{aligned} \sin \vartheta _3=\frac{1}{\sqrt{2}}\,|\mathrm {skw}(R^{(02)}R^{(26)}R^{(36)T})| \end{aligned}$$

    and

    $$N_{^\circ }^{{(13)}} = \sqrt 2 {\mkern 1mu} \frac{{skw(R^{{(02)}} R^{{(26)}} R^{{(36)T}} )}}{{|skw(R^{{(02)}} R^{{(26)}} R^{{(36)T}} )|}}{\text{ }} $$

    (composition of the rotations on the right side can be performed with perusal of the Gibbs relations to be found in DSS4).

  11. We reiterate that this sublist locates body 5 in space with respect to body 3 (the location of the latter being determined as a part of CLC1), while the sublist \(\ell _3^{(\beta )}=(\vartheta _{8},\xi _{4},\xi _{5})\) locates body 5 with respect to body 6, the proximal cuneiform, another part of CLC1.

  12. See e.g. https://www.youmath.it/lezioni/algebra-elementare/equazioni/569-equazioni-goniometriche-lineari-in-seno-coseno.html. Visited on 15/05/2021.

  13. Relation (40) yields

    $$\begin{aligned}&\vartheta _5=2\arctan \sqrt{\frac{r_2\circ r_1+r_3+r_3\times (r_2\circ r_1)}{1-(r_2\circ r_1)\cdot r_3}},\\&n^{(35)}_{\circ }=\big (\tan \frac{\vartheta _5}{2}\big )^{-1}\frac{r_2\circ r_1+r_3+r_3\times (r_2\circ r_1)}{1-(r_2\circ r_1)\cdot r_3}\,. \end{aligned}$$

  14. Either sublist of \(\ell _2\) is equivalent to 3 scalar variables providing enough information to locate in space body 4, the proximal cuneiform; these triplets are not assignable freely, though: in fact, as we have just shown, compatibility with CLC1 and CLC3 reduces to 1 the number of DFs of CLC2. The geometrical reason for this is that the referential and current centres of the spheres associated with ball joints 34 and 35 coincide. Consequently, when body 5 entrains the centre of ball joint 45 in any motion allowed by joint 35, the new position of joint 45 is always compatible with that of joint 34: the only possible motions of body 4 are rotations about the axis passing through the current centres of the spheres associated with joints 34 and 45.

Abbreviations

CLC:

Closed-loop chain

DF:

Degree of freedom

DSS:

DataSet section

HMF:

Hind-MidFoot

MFM:

Multi-line segment foot model

OLC:

Open-loop chain

References

  1. Brocato M, Podio-Guidugli P (2019) Un modello cinematico del piede umano e i disegni anatomici di Leonardo. In: Leonardo. Il Corpo dell’Uomo, Atti dei Convegni Lincei, vol 330. Bardi Edizioni, Roma

  2. Brocato M, Podio-Guidugli P, Sancisi N, Conconi M, Belvedere C, Parenti-Castelli V, Leardini A (2021) Subject-specific geometric definition and validation of a novel kinematic model of human hind+midfoot. In: Int. foot & ankle biomechanics meeting 2021, Book of abstracts: kinematics methodological aspects II

  3. Gosselin C, Angeles J (1990) Singularity analysis of closed-loop kinematic chains. IEEE Trans Robot Autom 6(3):281–290. https://doi.org/10.1109/70.56660

    Article  Google Scholar 

  4. Gray: http://www.bartleby.com/107/63.html

  5. Hicks CJH (1953) The mechanics of the foot 1. Joints J Anat 87:345–357

    Google Scholar 

  6. Huson A (1984) Mechanics of joints. Int J Sports Med 5:83–87

    Article  Google Scholar 

  7. Huson A (1985) Biomechanics of the subtalar joint: a re-appraisal. Soc Belga Med Chir Piede 9:389–398

    Google Scholar 

  8. Huson A (1987) Joints and movements of the foot: terminology and concepts. Acta Morphol Neerl Scand 25:117–130

    Google Scholar 

  9. Huson A (1991) Disorders of the foot and ankle : medical and surgical treatment. In: Janss MH (Eed), vol 1, chap 15. Functional anatomy of the foot. Philadephia: Saunders, pp 408–431

  10. Leardini A, Benedetti M, Berti L, Bettinelli D, Nativo R, Giannini S (2007) Rear-foot, mid-foot and fore-foot motion during the stance phase of gait. Gait Posture 25(3):453–462. https://doi.org/10.1016/j.gaitpost.2006.05.017

    Article  Google Scholar 

  11. Leardini A, Caravaggi P, Theologis T, Stebbins J (2019) Multi-segment foot models and their use in clinical populations. Gait Posture 69:50–59 (2019). https://doi.org/10.1016/j.gaitpost.2019.01.022..http://www.sciencedirect.com/science/article/pii/S0966636218317053

  12. Logan BM (2012) McMinn’s color atlas of foot and ankle anatomy, 4th edn. Elsevier Saunders

  13. MacWilliams B, Cowley M, Nicholson D (2003) Foot kinematics and kinetics during adolescent gait. Gait Posture 17:214–224

    Article  Google Scholar 

  14. Portinaro N, Leardini A, Panou A, Monzani V, Caravaggi P (2014) Modifying the rizzoli foot model to improve the diagnosis of pes-planus: application to kinematics of feet in teenagers. J Foot Ankle Res 7(1):57. https://doi.org/10.1186/s13047-014-0057-2

    Article  Google Scholar 

  15. Ridola CG, Cappello F, Marcianò V, Montalbano A, Farina-Lipari E, Palma A (2007) The synovial joints of the human foot. Ital J Anat Embryol 112(2):61–80

    Google Scholar 

  16. Wolfram: https://reference.wolfram.com/language/ref/AnatomyData.html

Download references

Acknowledgements

The early stages of this work took place at the Okinawa Institute of Science and Technology (OIST) Graduate University, Japan, whose hospitality we both gratefully acknowledge, where we profited from many discussions with Prof. Eliot Fried, head of the Mathematics, Mechanics, and Materials Unit, and with Mr. Michael Grunwald, a member of the same Unit. PPG also benefited from a couple of short stays at and supported by the GSA Laboratory – ENSA Paris-Malaquais, Paris, France. We are indebted to Alberto Leardini, DPhil, head of Movement Analysis Laboratory, IRCCS Istituto Ortopedico Rizzoli, Bologna, Italy, for many illuminating conversations about current status and future trends of foot-mechanics related research; in particular, Dr. Leardini suggested the all-important connection between our approach to foot modelling and Huson’s [6,7,8,9]. Finally, it is a pleasure to thank Prof. Vincenzo Parenti Castelli, Prof. Nicola Sancisi and Dr. Michele Conconi, Department of Industrial Engineering, University of Bologna, Italy, for their many competent comments about our work.

Funding

None.

Author information

Authors and Affiliations

  1. GSA Laboratory, ENSA Paris-Malaquais, PSL University, 14, Rue Bonaparte, 75006, Paris, France

    Maurizio Brocato

  2. Accademia Nazionale dei Lincei, Palazzo Corsini, via della Lungara, 10, 00165, Roma, Italy

    Paolo Podio-Guidugli

Authors
  1. Maurizio Brocato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paolo Podio-Guidugli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maurizio Brocato.

Ethics declarations

Conflict of interest

The authors declares that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brocato, M., Podio-Guidugli, P. A bone-wise approach for modeling the human hind-midfoot. Part I: kinematics. Meccanica 57, 977–998 (2022). https://doi.org/10.1007/s11012-021-01460-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-021-01460-x

Keywords

Mathematics Subject Classification

Search

Navigation