A least-squares algorithm for the equiform transformation from spatial marker co-ordinates

Veldpaus, F. E.<sup>1</sup>; Woltring, H. J.<sup>2,3</sup>; Dortmans, J. M. G.<sup>1</sup>

Affiliations

¹Faculty of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, NL 5600 MB Eindhoven, The Netherlands

²University of Nijmegen, The Netherlands

³Philips Medical Systems, Building QA-2, P.O. Box 10.000, NL 5680 DA Best, The Netherlands

Abstract

The present paper describes an algorithm for estimating the translation vector and the rotation matrix of a moving body from noisy measurements on the spatial co-ordinates of at least three non-collinear markers. A sensitivity analysis of the estimated parameters and of the helical axis is presented in terms of characteristics of the marker distribution. The implementation of the proposed algorithm in a FORTRAN-subroutine is appended.

Links

DOI: 10.1016/0021-9290(88)90190-X

PubMed: 3339026

WoS: A1988L199100008

History

Received: 1986-05

Revised: 1986-12

Cited Works (1)

Year	Entry
1980	Spoor CW, Veldpaus FE. Rigid body motion calculated from spatial co-ordinates of markers. J Biomech. 1980;13(4):391-393.

Cited By (39)

Year	Entry
2007	Siegmund GP, Davis MB, Quinn KP, Hines E, Myers BS, Ejima S, Ono K, Kamiji K, Yasuki T, Winkelstein BA. Cervical facet capsule response to whiplash loading with a rotated head posture. In: Proceedings of the 35th International Workshop on Human Subjects for Biomechanical Research. 2007.123-136.
2009	Lessley DJ, Purtsezov SV, Shaw CG, Parent DP, Riley PO, Kent R, Crandall JR. Assessment and validation of a methodology for measuring anatomical kinematics during impact loading. In: Proceedings of the 37th International Workshop on Human Subjects for Biomechanical Research. 2009.
2013	Meijer R, Broos J, Elrofai H, de Bruijn E, Forbes P, Happee R. Modelling of bracing in a multi‐body active human model. In: Proceedings of the 2013 International IRCOBI Conference on the Biomechanics of Injury. September 11-13, 2013; Gothenburg, Sweden.576-587.
1999	Winkelstein BA, Nightingale RW, Richardson WJ, Myers BS. Cervical facet joint mechanics: its application to whiplash injury. In: Proceedings of the 43rd Stapp Car Crash Conference. October 25-27, 1999; San Diego, CA. Warrendale, PA: Society of Automotive Engineers:243-252. SAE 99SC15.
2000	Siegmund G, Myers BS, Davis MB, Bohnet HF, Winkelstein BA. Human cervical motion segment flexibility and facet capsular ligament strain under combined posterior shear, extension and axial compression. Stapp Car Crash J. 2000;44:159-170. SAE 2000-01-SC12.
2000	Winkelstein B, Myers BS. Experimental and computational characterization of three-dimensional cervical spine flexibility. Stapp Car Crash J. 2000;44:139-158. SAE 2000-01-SC11.
1995	Cappozzo A, Catani F, Della Croce U, Leardini A. Position and orientation in space of bones during movement: anatomical frame definition and determination. Clin Biomech (Bristol, Avon). June 1995;10(4):171-178.
2009	Andersen MS, Damsgaard M, Rasmussen J. Kinematic analysis of over-determinate biomechanical systems. Comput Methods Biomech Biomed Eng. 2009;12(4):371-384.
1993	Söderkvist I, Wedin P-Å. Determining the movements of the skeleton using well-configured markers. J Biomech. December 1993;26(12):1473-1477.
1994	Ateshian GA, Kwak SD, Soslowsky LJ, Mow VC. A stereophotogrammetric method for determining in situ contact areas in diarthrodial joints, and a comparison with other methods. J Biomech. January 1994;27(1):111-124.
1994	van den Bogert AJ, Smith GD, Nigg BM. In vivo determination of the anatomical axes of the ankle joint complex: an optimization approach. J Biomech. December 1994;27(12):1477-1488.
1999	Lu T-W, O’Connor JJ. Bone position estimation from skin marker co-ordinates using global optimisation with joint constraints. J Biomech. February 1999;32(2):129-134.
2010	Andersen MS, Benoit DL, Damsgaard M, Ramsey DK, Rasmussen J. Do kinematic models reduce the effects of soft tissue artefacts in skin marker-based motion analysis? an in vivo study of knee kinematics. J Biomech. 2010;43(2):268-273.
1992	Soslowsky LJ, Flatow EL, Bigliani LU, Pawluk RJ, Ateshian GA, Mow VC. Quantitation of in situ contact areas at the glenohumeral joint: a biomechanical study. J Orthop Res. July 1992;10(4):524-534.
2004	Gupta S, van der Helm FCT, Sterk JC, van Keulen F, Kaptein BL. Development and experimental validation of a three-dimensional finite element model of the human scapula. Proc Inst Mech Eng Part H-J Eng Med. March 2004;218(3):127-142.
2000	Winkelstein BA, Nightingale RW, Richardson WJ, Myers BS. The cervical facet capsule and its role in whiplash injury: a biomechanical investigation. Spine. May 15, 2000;25(10):1238-1246.
2001	Siegmund GP, Myers BS, Davis MB, Bohnet HF, Winkelstein BA. Mechanical evidence of cervical facet capsule injury during whiplash: a cadaveric study using combined shear, compression, and extension loading. Spine. October 1, 2001;26(9):2095-2101.
1996	Kelkar R. Normal and Abnormal Mechanics of the Shoulder: Studies of Articular Geometry, Contact and Kinematics [PhD thesis]. Columbia University; 1996.
1997	Kwak SKD. Experimental and Mathematical Investigation of the Human Knee: Anatomy, Kinematics and Contact [PhD thesis]. Columbia University; 1997.
1999	Xu L. Biomechanics of the Thumb Carpometacarpal Joint: Topography, Ligamentous Structure, Articular Contact, and Kinematics [PhD thesis]. Columbia University; 1999.
2001	Wang VM-W. Biomechanics of the Normal and Surgically Reconstructed Shoulder [PhD thesis]. Columbia University; 2001.
1999	Winkelstein BA. A Mechanical Basis for Whiplash Injury: The Cervical Facet Joint, Spinal Motion Segment, and Combined Loading [PhD thesis]. Duke University; 1999.
1994	Simmons CA. Optimal Surface Coating Distribution on a Femoral Endoprosthesis [Master's thesis]. Cambridge, MA: Massachusetts Institute of Technology; February 1994.
1997	Murray WM. The Functional Capacity of the Elbow Muscles: Anatomical Measurements, Computer Modeling, and Anthropometric Scaling [PhD thesis]. Evanston, IL: Northwestern University; June 1997.
1999	Arnold AS. Quantitative Descriptions of Musculoskeletal Geometry in Persons With Cerebral Palsy: Guidelines for the Evaluation and Treatment of Crouch Gait [PhD thesis]. Evanston, IL: Northwestern University; December 1999.
1990	McClay IS. A Comparison of Tibiofemoral and Patellofemoral Joint Motion in Runners With and Without Patellofemoral Pain [PhD thesis]. State College, PA: Pennsylvania State University; December 1990.
2014	De Bruijn E. Isometric and Dynamic Control of Neck Muscles: Reflexive Contributions and Muscle Synergies [PhD thesis]. Delft, The Netherlands: Technische Universiteit Delft; 2014.
2018	Synek A. Predicting Habitual Activities from Bone Architecture Using a Biomechanical Approach [PhD thesis]. Vienna University of Technology; 2018.
2004	McWalter EJ. Links Between Three-Dimensional Patellar Kinematics, Cartilage Morphology and Varus/valgus Alignment in Early Knee Osteoarthritis [Master's thesis]. Vancouver, BC: University of British Columbia; December 2004.
2010	McWalter EJ. In Vivo Assessments of Patellofemoral Kinematics and Contact Areas With Applications to Osteoarthritis [PhD thesis]. Vancouver, BC: University of British Columbia; September 2010.
1997	Boyd SK. A 3D in-Situ Model for Patellofemoral Joint Contact Analysis in the Normal and Anterior Cruciate Ligament Deficient Knee [Master's thesis]. Calgary, AB: University of Calgary; April 1997.
2002	Couillard S. Cartilage Deformation of the Feline Patellofemoral Joint Obtained from Laser Scanning [Master's thesis]. Calgary, AB: University of Calgary; April 2002.
2006	Monteleone BJ. Effects of Functional Ankle Instability on Ankle Joint Complex Dynamics and Motor Control [PhD thesis]. Calgary, AB: University of Calgary; August 2006.
2008	Küpper J. A Novel Measure of in-Vivo Knee Joint Laxity [Master's thesis]. Calgary, AB: University of Calgary; January 2008.
2011	Westover LM. Quantifying in Vivo Laxity in the ACL and Individual Knee Joint Structures: A Numerical Modeling Approach [Master's thesis]. Calgary, AB: University of Calgary; December 2011.
2009	Metzger MF. Evaluation of Spinal Reconstructive Surgeries Using Multi-Body System Dynamics [PhD thesis]. San Francisco, University of California; 2009.
2009	Gates DH. The Role of Muscle Fatigue on Movement Timing and Stability During Repetitive Tasks [PhD thesis]. University of Texas at Austin; May 2009.
1994	Gilchrist LA. A Computer Simulation of Human Gait [PhD thesis]. University of Waterloo; 1994.
2007	Reeves NP. Tuning-Out Instability: The Importance of Feedback Control in the Spine [PhD thesis]. Yale University; December 2007.