Determining the rigid-body inertia properties of a knee prosthesis by FRF measurements

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1 Proceedings of the IMAC-XXVII February 9-12, 2009 Orlando, Florida USA 2009 Society for Experimental Mechanics Inc. Determining the rigid-body inertia properties of a knee prosthesis by FRF measurements Emiliano Mucchi (*), iuliamarta Bottoni, Raffaele Di regorio (*) corresponding author EnDIF Engineering Department in Ferrara Università Degli Studi di Ferrara Via Saragat,1 I Ferrara ITALY Tel: Fax: emiliano.mucchi@unife.it ABSTRACT The precise evaluation of the rigid-body (RB) inertia proprieties of a prosthesis is important for its implantation in the patient. Nowadays, in the motion analysis of humans carrying prostheses, the RB inertia properties of the healthy limbs are used instead of the prosthesis ones due to their difficult evaluation. This forced approximation may yield significant errors in the motion-analysis results. In fact, the prosthesis is, in general, heavier and has a different center of gravity (CO) with respect to the healthy limb. Moreover, the computation of the forces applied by the muscles during motion (i.e., the inverse dynamics) cannot be correctly performed without the RB inertia properties of the prosthesis. In this paper, a prosthesis for transfemoral amputees is studied, namely C-Leg 3C-100. In particular, an experimental technique working in the frequency domain has been used for estimating prosthesis inertia properties. Such a methodology evaluates the RB inertia properties from Frequency Response Function (FRF) measurements. First, a dedicated specimen has been designed and manufactured specifically to validate the methodology. Then, the assessed methodology has been applied to the real prosthesis. 1. INTRODUCTION Rigid-body (RB) inertia properties of simple obects can be easily estimated by using analytical approaches, but, when the obect s mass distribution is not completely known (i.e., for complex structures), experimental approaches become necessary for evaluating the RB inertia properties. In the literature, researchers have developed several alternative methods for the identification of the RB inertia properties. These methods can be gathered into two main categories: time-domain and frequency-domain methods. The classical time-domain methods (TDM) are based on the measurements of the small-oscillation period of a pendulum [5] [6]. It is an experimental test conceptually simple, which hides some traps for unskilled operators, due to its cumbersome set-up; moreover, it is time-consuming for large and heavy obects [18]. Recently, a new time-domain technique, named InTenso method, has been proposed in [2] [3]. The InTenso method, which is mainly used for vehicles or part of them, employs a cage where the vehicle under test is embedded. Furthermore, some authors have developed a few variants for the TDM; for instance, in [4] the estimate of the RB inertia properties comes through a selection of rigid and flexible body modes. The frequency-domain methods (FDM) extract the RB inertia properties from the Frequency Response Functions (FRFs) measured on a structure under free-free boundary conditions. Three main types of FDM can be distinguished: - Inertia Restrain Methods (IRM). - Direct System Identification Methods (DSIM). - Modal Model Methods (MMM). The Inertia Restrain Method is based on the principle that, in the low-frequency range, located between the last rigid mode and the first flexible mode, the dynamic response of a freely-supported structure is characterized by a constant plateau named mass line [1]. In the IRM, the evaluation of the mass line has to be as accurate as possible, since it is the input datum and input-data errors greatly affect the computed RB inertia properties [14]. When the separation between the last rigid mode and the first flexible mode is not clear, a modal analysis of the structure has to be performed in order to extract the first flexible mode and to remove its influence from the mass line [9]. Moreover, for noisy FRFs in the mass line zone, in [9] [8] is

2 proposed to evaluate the lower residual of the first flexible mode, since it is representative of the RB behavior of the structure. The Direct System Identification Method allows the evaluation of the mass, stiffness and damping parameters through the use of measured FRFs. In particular, this method deduces the analytical expressions of the FRFs from the equation of motion in the frequency domain, where the entries of the system matrices (mass, stiffness and damping matrices) are considered unknowns. Then, it uses them to fit the measured FRFs so that an over-determined set of equations is written, and solved with the least-squares method, for determining the system matrices [5] [10] [11]. In the Modal Model Method [13] the RB inertia properties are calculated from a system of equations based upon the orthogonality relationship between the RB modes which contains the mass matrix. MMM requires as many excitation points as necessary to well excite the six rigid-body modes of the structure under test. The precise evaluation of the RB inertia proprieties (mass, center of gravity (CO) and inertia tensor) of knee prostheses is important for their implantation in the patient. Nowadays, the RB inertia properties of healthy limbs are used in the motion analysis of humans with prostheses, instead of the actual properties of the prosthesis, which are difficult to evaluate. This forced approximation may yield significant errors in the motion-analysis results. In fact, the prosthesis is, in general, heavier and has a different center of gravity with respect to the healthy limb. Moreover, the computation of the forces applied by the muscles during motion (i.e., the inverse dynamics) cannot be correctly performed without the RB inertia properties of the prosthesis. This scenario motivated the authors work. In this paper, a prosthesis for trans-femoral amputees is studied, namely the above-knee prosthesis C-Leg 3C100 (see Figure 1 left). It is an electronically-controlled knee with hydraulic adustment in the static as well as in the dynamic stage of the motion. It consists of an upper part and a lower part. The upper part of the C- Leg contains hydraulic valves and electronic components for the control, while the lower part is a moving pipe to connect to an artificial foot. Hydraulic part Moving pipe Figure 1: C-Leg 3C100 (left) and the prosthesis suspended for FRF measurements (right). The estimate of the RB inertia properties of the C-Leg 3C100 has been performed through the IRM. Such an experimental technique has been already applied to complex mechanical systems, i.e. engines, automotive components and simple structures [3] [16] [17], but, to authors knowledge, it has not been applied to medical devices, yet. The method is validated by applying it to a specimen that was designed and manufactured specifically for this aim. Such a specimen is an obect with oil in the inside and with oints; moreover, its mass distribution is similar to the one of the C-Leg prosthesis. All the measurements carried out in this work and the farther analyses aimed at calculating the RB inertia property have been performed in LMS Test.Lab environment [19]. 2. THE INERTIA RESTRAIN METHOD The IRM uses three different algorithms, depending on the shape of the measured FRFs [15] [16] [17]: Unchanged FRFs (to be used when the rigid-body modes and the flexible ones are well separated).

3 Corrected FRFs (to be used when insufficient bandwidth exists between the RB modes and the flexible ones; such an algorithm corrects the measured FRFs, by subtracting the synthesized FRFs of the flexible modes, before fitting them). Lower residual (to be used when accurately measured FRFs are not available in the low frequency range of the mass line; such an algorithm takes into account the lower residual of the first flexible mode in the FRFs analytical expressions that fit the measured FRFs). In the IRM, an accurate geometrical wire-frame, collecting the precise spatial locations of the excitation/response points and the weight of the component is an essential requirement. FRFs are acquired by using either a hammer or a shaker as excitation of the specimen, which has to be suspended under freefree boundary conditions. Two excitations and six response points, in three directions, are theoretically sufficient to implement the algorithms. Nevertheless, practical tests [15] [16] showed that the best results are obtained with at least six excitation degrees of freedom (DOFs) and 24 response DOFs. The RB inertia properties are calculated from the measured FRFs by solving an over-determined equation system with the least-squares method. Such an equation system is written by selecting all the FRF spectral lines belonging to a specified frequency band between the last rigid-body mode and the first flexible mode (i.e., where the mass line takes place) (see Figure 2). Log e Phase Sum FRF SUM Sum FRF SUM Figure 2: Example of FRF (amplitude and phase) in the low frequency band. In general, the dynamic response of a mechanical system is a superposition of several mode shapes. However, if, in a given frequency band, only one mode dominates, say the r th one, the dynamic response can be approximated as follows [20][21]: [ π ] T r { ψ} { ψ} ( i2π f λ ) Q k r r Hi (2 f) = (2 i π f) + HF + LF where H : FRFs matrix; Q r : modal scaling factor r ; λ r : system pole r ; ψ : modal vector r ; r Rigid Body Modes Mass Line [ HF ] : high-frequency-residual matrix; r [ ] [ ] Flexible Mode [ LF ] : low-frequency-residual matrix; f : frequency; i : imaginary unit; k: integer exponent depending on the form of FRF (i.e. K=0,1,2 for receptance, mobility, inertance, respectively). The residual matrices take the influence of the modes in the neighborhood of the frequency band into account. If the FRF matrix is the inertance matrix (i.e. the output variable is the acceleration) and the r th mode is the last rigid mode, the high frequency residual does not depend on the frequency, its behavior is controlled by the RB inertia properties of the system, and is a flat line ( mass line ), whereas the lowfrequency residual depends on the suspension stiffness and on the frequency square. Thus, above the RB modes the trend of FRF does not depend on the frequency as far as the contribute of the flexible modes is negligible. The frequency of the first flexible mode limits how high in frequencies above RB modes the FRF is a constant mass line. In the following part of this section, the kernel of the IRM algorithms will be illustrated. (1)

4 The introduction of a Cartesian reference system, O-xyz, fixed to the RB to be analyzed and with origin O, allows the acceleration of an RB excitation/response point, P i, to be written as follows (all the vector are proected in O-xyz): (2) P 2 i = O + ω + ω P i where ω and ω are the RB angular velocity and acceleration, respectively. P i is the position vector of P i in O- xyz. And the accent put on a vector symbol denotes the skew-symmetric matrix associated to that vector. After having neglected the squares of the angular velocity components in (2), the division of (2) by the magnitude, F, of the excitation force exerted on an excitation point P, yields, after some rearrangements: P i F I O F P = 3 i ω F (3) where I 3 is the identity 3 3 matrix. Relationship (3) relates the measured FRFs (i.e., the P i F for i = 1,2, ) to six characteristic FRFs (i.e., O F and ω F ) that uniquely identify the RB acceleration field in response to a unit excitation at P. By writing (3) for a sufficient number of response points (more than three non-aligned points, between eight and twelve response points in three directions are suggested), an over-determined equation system is obtained which can be solved with the least-squares method to determine the six characteristic FRFs. This solution technique reduces the effect of the measurement errors in the measured accelerations. The dynamic model of a free RB excited by a force F n (n is a unit vector which identifies the direction of the force) applied to P is: Fn = m F( Pn ) = Fn + ω + ω ω (4) where m and are the mass and the center of gravity of the RB, respectively, and is the RB inertia tensor referred to a Cartesian reference that has the origin at and the same orientation as O-xyz: xx xy = xy yy yz (5) yz zz The division of both the vector equations (4) by F, after the terms containing the squares of the angularvelocity components have been neglected and F has been expressed through relationship (3), yields: ( ) m ω F 03 6 n m O F = Pn n N (6) where is the 3 6 null matrix, 6 1 is equal to ( xx, yy, zz, xy, yz, ) T, and N ω x 0 0 ω y 0 ω z 1 = 0 ω 0 ω ω y x z F 0 0 ω z 0 ω y ω x (7) The introduction of m and of the previously determined characteristic FRFs into (6) makes the RB inertia properties and 6 1 the only unknowns. System (6) can be written for all the excitation points. By writing (6) for a sufficient number of excitation points (at least two, but six are suggested), an overdetermined equation system is obtained which can be solved with the least-squares method to determine the RB inertia properties.

5 3. METHOD ASSESSMENT The IRM method is tested on a specimen specifically designed and manufactured for this purpose. Such a specimen is an obect composed of two aluminium pipes, mounted one inside the other, with the following dimensions: external pipe: length 480 mm, external diameter 65 mm, internal diameter 55 mm. Internal pipe: length 90 mm, external diameter mm, internal diameter mm. The internal pipe is filled up with oil and the lateral ends are sealed with two threaded plugs. This pipe is fixed to the other by using six M6 screws. The specimen is suspended by using bungee cords (Figure 3), so that the whole system can be considered as a free body. Twelve input DOFs were excited by using an impact hammer (PCB 086D05) and the acceleration responses were measured in thirty output DOFs by a PCB miniaturized accelerometer (frequency range ). The signals were acquired by using sampling frequency of 2048 and frequency resolution of The measurement points were taken regularly spread over the structure in order to improve the spatial observability. The excitation was given so that all the RB modes were excited. It is worth noting that the excitation of the rotational RB mode around the external-pipe axis (y axis) required the introduction of two extra excitation points located on two screws (S1 an S2 in Figure 3) to be hammered along the z and x directions. A plastic tip for the hammer is used to allow the low-frequency range to be excited at the best. The FRF measurement results (see Figure 6, left) reveal that the mass line is clear and well-defined for all the FRFs and for the FRF-sum too (i.e. the complex sum of the FRFs of all the measured structural points). So, the Unchanged FRF algorithm can be used for the RB inertia property calculation. A frequency range between 70 and 120 is selected for the analysis (see Figure 4). Table 1 collects the 3D-CAD nominal values (2 nd column) of the CO coordinates and of the inertia-tensor entries, and their evaluation (3 rd column) by means of the IRM. The values obtained from the 3D-CAD model are considered the reference ones. The analysis of Table 1 reveals that the IRM yields highly accurate results for the CO coordinates, for the diagonal inertia-tensor entries (i.e., the moments of inertia about the coordinate axes), and for the product of inertia yz. On the contrary, the IRM values for the xy and products of inertia exhibit high relative errors when compared to the nominal values. These mismatches could be ustified by considering that the products of inertia xy and have a absolute value much lower than the other entries of the inertia tensor. Therefore, it may be concluded that, for this type of specimens, the sensitivity of the IRM is about kg m 2, when used to measure moment/product of inertia, and about 3 mm, when used to measure the CO coordinates. Bungee cords S2 y S1 y x z Figure 3 : aluminium specimen under test and reference frame xyz. S1 A few tests were performed to verify the influence of the suspension stiffness. Three different suspension system configurations have been set. First, in the a-configuration of Figure 5, the system was suspended with flexible bungee cords fixed at a single point of the supporting structure. Then, in the b- and c- configurations of Figure 5, the bungee cords were attached in nine points to the supporting structure but the number of cords in the c-configuration was increased yielding in a stiffer suspension system. Figure 6 shows a measured FRF on the low-frequency range where the RB modes occur for the three kind of suspension configurations. In all the three cases, the RB modes are well recognizable but increasing the number of

6 cords, and therefore increasing the stiffness of the suspension system, they rise in frequency becoming closer to the first flexible mode and reducing the mass line zone. As above described, two extra excitation points have been introduced on the screws for exciting a particular rotational RB mode, due to the cylindrical shape of the specimen. In order to understand the importance of exciting all the RB modes, a further measurement campaign was carried out without those two excitation points and a new FRF set was obtained. The analysis with this FRF set gave large error for the only yy moment of inertia (about 41.90%), on the contrary the other entries of the inertia tensor and the CO were accurately calculated. This large error is due to the incorrect excitation of the RB rotational mode around the y axis. Therefore, it may be concluded that the excitation energy must be about-equally distributed among all the RB modes in order to get FRFs that evaluate correctly all the inertia-tensor entries. Particular attention must be paid when the specimen has a shape that does not guide the choice of the excitation points like a cylindrical one. RB Inertia Nominal values IRM Property (3D-CAD) x y z xx yy zz xy yz Table 1: CO coordinates in mm, evaluated in the assigned Cartesian reference, and entries of the inertia tensor in kg m 2, referred to a Cartesian reference with origin at the CO for the aluminium specimen weighting kg. The values refer to the 3D-CAD technique and to the IRM Rigid body modes First flexible mode Log e-3 FRF pipe2:n1:-y/pipe2:n1:+y selected band 3.00e e Figure 4: FRF-sum with selected frequency band for the aluminium specimen

7 Supporting structure a-configuration) specimen b- configuration) c-configuration) Figure 5: the three suspension system configurations for the aluminium specimen. Log e-3 300e-3 200e-3 100e e e e e e e e e-3 500e-6 300e-6 200e-6 100e Amplitude a-configuration b-configuration c-configuration Linear 1.00 Amplitude 0.00 Figure 6: (left) mass lines for the measured FRFs and (right) zoom around the RB modes of a measured FRF in the three different suspension configurations. 4. TEST CASE: THE ABOVE-KNEE PROSTHESIS The C-Leg 3C100 was suspended with bungee cords to a supporting structure as shown in Figure 7,left. These suspension cords were chosen as soft as possible checking that they were not completely stretched. Nineteen input DOFs were excited by using an impact hammer with steel tip and the acceleration responses were measured in 24 response DOFs by a PCB miniaturized accelerometer. The signals were acquired by using a sampling frequency of 4096 and a resolution of 0.5, which allowed to catch the low-frequency behavior. Furthermore, the FRFs were calculated by using the H1 estimator [20] [21]. The coherence function was evaluated and the reciprocity of the FRFs were checked in order to have a reliable FRF set. Figure 7 shows the obtained FRF-sum. The mass line of this FRF-sum seems clear enough and not influenced by the first flexible mode. So the Unchanged FRF algorithm can be used to calculate the RB inertia properties. A frequency range between 95 and 105 was chosen for the analysis where the mass line appears straight. The results are shown in the 2 nd column of Table 2. In order to evaluate the repeatability of the IRM, the RB inertia properties both of the hydraulic part (upper part) and of the modular pipe (lower part) of the prosthesis were separately estimated through the IRM. Then, they were used to calculate the RB inertia properties of the whole prosthesis by using the composition laws of the inertia properties. The 3 rd column of Table 2 collects the CO coordinates and the entries of the inertia tensor obtained from this alternative procedure. These results are near to those obtained by directly measuring the FRFs on the whole prosthesis, and the differences fall within the estimated sensitivity of the IRM. Therefore, the repeatability of the IRM is proven and its sensitivity is confirmed for this type of specimens.

8 Rigid body modes Log Sum FRF SUM Selected frequency band e Figure 7: Amplitude of the FRF-sum for the C-Leg 3C100 and selected frequency band. RB Inertia Hydraulic part+ Prosthesis Property modular pipe x y ,3020 z xx yy zz xy yz Table 2: CO coordinates in cm, evaluated in an assigned Cartesian reference, and entries of the inertia tensor in kg m2, referred to a Cartesian reference with origin at the CO, for the prosthesis weighting 1,414 kg. the 2nd column refers to the analysis on the entire prosthesis, the 3rd column refers to the values analytically calculated from the results on the separated parts (hydraulic part and modular pipe) of the prosthesis. 5. SYNOPSYS AND CONCLUSIONS The paper has presented an experimental frequency-domain method for the evaluation of the RB inertia properties of a prosthesis, namely C-Leg 3C100. To authors knowledge, this is the first time the IRM is applied on medical devices. In particular, the method has been assessed on an aluminium specimen specifically designed and manufactured for this purpose. The results of this analysis have highlighted that the IRM sensitivity is about kg m 2, when evaluating the inertia-tensor entries, and about 3 mm, when evaluating the CO coordinates, which makes it accurate for the prostheses. The effects of the stiffness, of the suspension system, and of the selection of the excitation/response points have been also analyzed and discussed. The assessed method is applied to a C-Leg 3C100 prosthesis for the evaluation of its RB inertia properties. Successively, the IRM repeatability has been tested by separately applying it to the determination of the RB inertia properties both of the upper part and of the lower part of the prosthesis, and, then, by using the soobtained data for calculating again the RB inertia properties of the whole prosthesis. This alternative procedure has proved the IRM repeatability and has confirmed the IRM sensitivity for this type of specimens.

9 ACKNOWLEDMENTS The authors wish to thank Centro Protesi INAIL (Vigorso di Budrio, Bologna, Italy) and its technicians for cooperation and assistance in the collection of experimental data. This work has been developed within the laboratory of research and technology transfer InterMech (LAV - Acoustics and Vibrations) realized through the contribution of Regione Emilia Romagna - Assessorato Attivita' Produttive, Sviluppo Economico, Piano telematico, PRRIITT misura 3.4 azione A. REFERENCES [1] Lamontia M., On the Determination and Use of Residual Flexibilities, Inertia Restraints and Rigid Body Modes, Proceedings of IMAC I, pp , [2] Previati., Mastinu., obbi M., Piccardi C., Bolzoni L., Rinaldi S., A new test rig for measuring the inertia properties of vehicles and their subsystems, Proceedings of IMECE 2004, Anaheim, California, USA, November 13-20, [3] Previati., obbi M., Pennati M., Mastinu., Accurate measurement of mass properties of round vehicles and their subsystems, Proceedings of the 7th International Symposium on Advanced Vehicle Control AVEC 2004, Arnhem, the Netherlands, August, [4] Pandit S., Hu Z.Q., Yao Y.X., Experimental technique for accurate determination of rigid body characteristics, Proceedings of IMAC X, Los Angeles, February 1992, pp [5] Schedlinski C., Link M., A survey of current inertia parameter identification methods, Mechanical Systems and Signal Processing Vol. 15(1), pp , [6] enta., Delprete C., Some considerations on the experimental determination of moments of inertia, Meccanica Vol. 29, pp , [7] Bretl., Conti P., Rigid Body Mass Properties from Test Data, ournal of Vibration Acoustics Stress and Reliability in Design, Vol. 111, pp , [8] Weis Y.S., Reis., Experimental Determination of Rigid Body Inertia Properties, Proceedings of IMAC VII, pp , [9] Toivola., Nuutila O., Comparison of three methods for determining rigid body inertia properties from Frequency Response Function, Proceedings of IMAC XI, [10] Huang S.., Cogan S., Lallement., Experimental identification of the characteristics of a rigid structure on an elastic suspension, Proceedings of IMAC XIII, Nashville, TN, pp , [11] Huang S.., Lallement., Direct estimation of rigid body properties from harmonic forced responses, Proceedings of IMAC XV, Orlando, FL, pp , [12] Okuma M., Heylen W., Sas P., Identification of the rigid body properties of 3-d frame structure by MCK identification method, Proceedings of ISMA25, Leuven, Belgium, September 13-15, [13] Almeida R.A.B., Urgueira A.P.V., Maia N.M.M., Identification of Rigid Body Properties from vibration measurements, ournal of Sound and Vibration Vol. 299, pp , [14] Fregolent A., Saginario., Un modello per la stima degli errori introdotti nel campo delle basse frequenze in misure FRF di sistemi liberi, Proceedings of AIMETA, Trento, Italy, September 28, [15] Leurs W., ielen L., Brughmans M., Dierckx B, Calculation of rigid body properties from FRF data: practical implementation and test case, Proceedings of IMAC XV, Tokyo, apan, [16] LMS International, Advanced FRF based determination of structural inertia properties, LMS technical report. [17] Pauwels, Measuring Rigid Body Properties of an Exhaust System, LMS technical report, [18] E. Mucchi, S. Fiorati, R. Di regorio,. Dalpiaz, Determining the rigid-body inertia properties of cumbersome systems: comparison of techniques in time and frequency domain, Proceedings of the IMAC XXVII, Orlando 9-12 February [19] LMS Test.Lab 8A, LMS International, [20] D.. Ewins, Modal testing: Theory and Practice, [21] Heylen, Lammens, Sas. Modal analysis theory and testing, Katholieke Universitait Leuven.