In this work we present the construction and evaluation of recursive Bayesian estimation algorithms for human motion tracking using fractional order models. The presented schemes include both fixed fractional order as well as variable order models with implicit and explicit model order estimation schemes. The performance of the designed techniques is compared to a set of conventional integral order methods using several measurements of human hand and head motions.
BaleanuDDiethelmKScalasETrujilloJ (2012a) Fractional Calculus: Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos), Singapore: World Scientific.
2.
BaleanuDPetrasIAsadJHVelascoMP (2012b) Fractional Pais–Uhlenbeck oscillator. International Journal of Theoretical Physics51: 1253–1258.
3.
Bar-ShalomYLiXRKirubarajanT (2001) Estimation with Applications to Tracking and Engineering: Theory, Algorithms and Software, New York: John Wiley & Sons, Inc.
4.
BarbosaRSMachadoJATFerreiraIM (2003) A fractional calculus perspective of PID tuning. Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C. ASME: New York, pp. 651–659.
5.
BlairWKazakosD (1993) Second order interacting multiple model algorithm for systems with Markovian switching coefficients. American Control Conference. 484–488.
6.
BlomHBar-ShalomY (1988) The interacting multiple model algorithm for systems with Markovian switching coefficients. IEEE Transactions on Automatic Control33: 780–783.
7.
ChenW (2006) A speculative study of 2/3-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures. Chaos16: 023126–023126.
8.
ChenYRuiY (2004) Real-time speaker tracking using particle filter sensor fusion. Proceedings of the IEEE92: 485–494. DOI: 10.1109/JPROC.2003.823146.
9.
ChenYSunRZhouA (2007) An overview of fractional order signal processing (FOSP) techniques. Proceedings ASME 2007 International Design Engineering Technical Conferences, Computers and Information in Engineering Conference. ASME: New York, pp. 1205–1222.
10.
ChenYXueDDouH (2004) Fractional calculus and biomimetic control. IEEE International Conference on Robotics and Biomimetics, 2004 (ROBIO 2004). 901–906.
11.
CoimbraC (2003) Mechanics with variable-order differential operators. Annalen der Physik12: 692–703.
12.
DebnathL (2003) Recent applications of fractional calculus to science and engineering. International Journal Mathematical Sciences54: 3413–3442.
13.
DorcakLPetrasITerpakJZborovjanM (2003) Comparison of the methods for discrete approximation of the fractional-order operator4. Proceedings of the ICCC'2003 Conference. 236–239.
14.
DzielinskiASierociukD (2005) Adaptive feedback control of fractional order discrete state-space systems1. International Conference on Computational Intelligence for Modelling, Control and Automation, 2005 and International Conference on Intelligent Agents, Web Technologies and Internet Commerce. 804–809. DOI: 10.1109/CIMCA.2005.1631363.
15.
FaieghiMRDelavariHBaleanuD (2012) Control of an uncertain fractional-order Liu system via fuzzy fractional-order sliding mode control. Journal of Vibration and Control18: 1366–1374.
16.
GrewalMSAndrewsAP (2001) Kalman Filtering. Theory and Practice using MATLAB, 2nd ed, New York: John Wiley & Sons, Inc.
17.
JulierS (2003) The spherical simplex unscented transformation3. Proceedings of the 2003 American Control Conference. 2430–2434. DOI: 10.1109/ACC.2003.1243439.
18.
JulierSUhlmannJ (2004) Unscented filtering and nonlinear estimation. Proceedings of the IEEE92: 401–422. DOI: 10.1109/JPROC.2003.823141.
19.
KilbasAASrivastavaHMTrujilloJJ (2006) Theory and Applications of Fractional Differential Equations (North Holland Mathematics Studies) Vol. 204, San-Diego, CA: Elsevier.
20.
LiangYChenW (2013) A survey on computing levy stable distributions and a new matlab toolbox. Signal Processing93: 242–251. DOI: 10.1016/j.sigpro.2012.07.035.
21.
LorenzoCFHartleyTT (2002) Variable order and distributed order fractional operators. Nonlinear Dynamics29: 57–98.
22.
MachadoJAT (2003) A probabilistic interpretation of the fractional-order differentiation. Journal of Fractional Calculus and Applied Analysis6: 73–80.
23.
MachadoJATSilvaMFBarbosaRS (2010) Some applications of fractional calculus in engineering. Mathematical Problems in Engineering2010: 639801–639801.
24.
OldhamKBSpanierJ (1974) The Fractional Calculus. (Mathematics in Science and Engineering) Vol. 111, New York: Academic Press.
25.
OrtigueiraMD (2000a) Introduction to fractional linear systems. Part 1: continuous-time case. IEE Proceedings of Vision, Image and Signal Processing147: 62–70.
26.
OrtigueiraMD (2000b) Introduction to fractional linear systems. Part 2: discrete-time case. IEE Proceedings of Vision, Image and Signal Processing147: 71–78.
27.
OstalczykPRybickiT (2008) Variable-fractional-order dead-beat control of an electromagnetic servo. Journal of Vibration and Control14: 1457–1471.
28.
PodlubnyI (1999) Fractional Differential Equations (Mathematics in Science and Engineering) Vol. 198, San Diego, CA: Academic Press.
29.
PodlubnyI (2002) Geometric and physical interpretation of fractional integration and fractional differentiation. ractional Calculus and Applied Analysis5: 367–386.
30.
PodlubnyIDorcakLKostialI (1997) On fractional derivatives, fractional-order dynamic systems and pi lambda d mu-controllers5. Proceedings of the 36th IEEE Conference on Decision and Control, 1997. 4985–4990. DOI: 10.1109/CDC.1997.649841.
31.
PuYWangWZhouJWangYJiaH (2008) Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation. Science in China Series F: Information Sciences51: 1319–1339.
32.
RomanovasMKlingbeilLTraechtlerMManoliY (2010) Fractional model based Kalman filters for angular rate estimation in vestibular systems. 2010 IEEE 10th International Conference on Signal Processing (ICSP). 1–4. DOI: 10.1109/ICOSP.2010.5656463.
33.
RomanovasMKlingbeilLTraechtlerMManoliY (2011a) Explicit fractional model order estimation using unscented and ensemble Kalman filters. The Fifth Symposium on Fractional Derivatives and Their Applications (FDTA'11) at The 7th ASME/IEEE International Conference on Mechatronics and Embedded Systems and Applications (ASME/IEEE MESA 2011).
34.
RomanovasMKlingbeilLTraechtlerMManoliYAl-JawadA (2011b) Implicit fractional model order estimation using interacting multiple model Kalman filters. In: IFAC 18th World Congress, Milano, Italy: IFAC.
35.
SierociukDDzielinskiA (2006) Fractional Kalman filter algorithms for the states, parameters and order of fractional system estimation. International Journal of Applied Mathematics and Computer Science16: 129–140.
36.
ThrunSBurgardWFoxD (2005) Probabilistic Robotics (Intelligent Robotics and Autonomous Agents), Cambridge, MA: The MIT Press.
37.
WanEVan Der MerweR (2000) The Unscented Kalman filter for nonlinear estimation. In: The IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (AS-SPCC), pp.153–158. DOI: 10.1109/ASSPCC.2000.882463.
