- Email: [email protected]
Identification of Human Mass Properties From Motion
Proceedings of the 15th IFAC Symposium on System Identification Saint-Malo, France, July 6-8, 2009
Identification of Human Mass Properties From Motion ⋆ Gentiane Venture ∗ Ko Ayusawa ∗ Yoshihiko Nakamura ∗ ∗
The University of Tokyo, 7-3-1 Hongo Bunkyo-ku 113-8656 Tokyo, Japan, (e-mail: [email protected])
Abstract: We experimentally identify the human body dynamics from motion capture data and contact force measurements. We make use of the 6 base-link equations to write the identification model. We have shown that the obtained system is sufficient to estimate the whole base inertial parameters. The method is computationally efficient; and allows to identify the pure inertial parameters without transmission, friction considerations. However finding persistent exciting trajectories is a drawback of the method. In this paper we put emphasis in describing persistent exciting trajectories that are suitable to obtain accurate results. We propose a method to automatically select the set of motions used for the identification in order to optimize the computation time and the accuracy of the identified parameters. For the experiments we make use of gymnastic motions. Each of the 3 subjects -of different body conformations- perform 29 types of motions. The identification method and the automatic motions selection method are both applied to obtain the inertial parameters of one subject. Keywords: Parameter Identification, Excitation, Least-squares Identification, Dynamic Modelling, Human-body, Biomedical 1. INTRODUCTION
Dynamics Identification
The mass properties of the human body are crucial in biomechanics studies, in gait studies and for some medical applications: orthopedics, neurology, musculoskeletal disorders. Gait stability is often examined using the trajectory of the wholebody center of mass (COM). To monitor the variation of muscle mass during hospitalization, rehabilitation, the measurement of the inertia and the position of the COM of each link of the body is a key-data to refine the diagnosis and to provide personalized health-care. The recent portable technologies allow to develop systems to estimate in-vivo the position of the global COM Betker et al. [2006], Han et al. [2006]. However the estimation of the moment of inertia is usually computed by an interpolation of literature data Young et al. [1983], Durkin and Dowling [2003]. Which are obtained by inaccurate techniques such as photogrammetry Jensen [1978] or by expensive and time-consuming techniques based on 3D imaging (CT-scan or MRI) and 3D modeling interpolations Pearsall and Reid [1994], Cheng [2000]. Consequently, there is a pressing need to develop reliable and robust method to estimate the mass properties of the human body. In this paper we present a methodology to estimate the inertial parameters of humans from motion and contact force measurements. In section II we present the identification method that is applied to the human body. In section III we propose a method to optimize the selection of exciting trajectories, in order to guaranty reliable and fast computations. In section IV we apply the proposed scheme experimentally to identify the mass properties of young woman. We compare the identification results obtained with different sets of motions. The results obtained with the proposed method are validate and compared with literature data. ⋆ This research is supported by the Special Coordination Funds for Promoting Science and Technology: "IRT Foundation to Support Man and Aging Society".
978-3-902661-47-0/09/$20.00 © 2009 IFAC
988
Fk
Force Sensing
φB
?
YB . .. q, q,q. q
Motion otion Sensing S
Fig. 1. Principle of the identification of bipeds from base-link dynamics 2. IDENTIFICATION OF HUMAN DYNAMICS FROM BASE-LINK EQUATIONS 2.1 Robotics background In robotics, the identification of the inertial parameters of a system is a well-known problem Sujan and Dubowsky [2003], Liu et al. [1998] and general formalisms and methodologies that apply to a wide range of dynamic systems are available Khalil and Dombre [2002], Swevers et al. [1997]. Generally methods implies the measurements of joint torques, however a few method using the contact forces information have been developed West et al. [1989]. We have proposed to identify the mass properties of mobile legged systems using only the dynamics of the base link Venture et al. [2008a]. In this paper we use this method that applies to humanoid robots to identify the human inertial parameters. It allows to identify the wholebody inertial parameters making use of a motion capture studio and force-plates as it is shown in Fig. 1. 2.2 Formulation The inverse dynamics of a legged system can be written in a linear form with respect to the inertial parameters Venture et al.
10.3182/20090706-3-FR-2004.0163
15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 [2008b], Khalil and Dombre [2002] as shown in Eq.1, where the inertial parameters are grouped in the vector φ and the joint information is grouped in the regressor Y . The upper part of the equation describes the 6 DOF motion of the base-link in the 3D space. The lower part of the equation describes the motion of the N bodies of the various kinematic chains constituting the whole system. Usually the base-link is chosen as the lower torso, the chains are the limbs, the upper-torso, the head. where:
X Nc Y1 0 K k1 φ= + Fk Y2 K k2 τ
(1)
k=1
Y1 is the regressor, which is function of the Y2 ¨ system joint angles θ, velocity θ˙ and acceleration θ, and of the vector of generalized coordinates q0 and its derivatives; • φ is the vector of inertial parameters to estimate such that: T (2) φ = φT0 φT1 ·· φTN
• Y =
- φj is the vector of standard parameters for each link Bj (j = 0 to N ), such that: φj = [ mi msi,x msi,y msi,z Ii,xx (3) Ii,yy Ii,zz Ii,yz Ii,zx Ii,xy ]T - mj is the mass, - Ij,xx , Ij,yy , Ij,zz , Ij,yz , Ij,zx , Ij,xy are the 6 independent components of the inertia matrix I j , - msj,x , msj,y , msj,z are the first moments components of the vector msj
Not all of the 10N parameters are used to describe the dynamics. The necessary parameters are called base parameters φB They are the only identifiable parameters Khalil and Dombre [2002]. They are computed from the standard parameters φ Gautier [1991], Kawasaki et al. [1991], Khalil and Bennis [1995] to obtain the minimal identification model Eq.4. Y B φB =
Y B1 Y B2
X Nc 0 K k1 φB = + Fk K k2 τ
(4)
k=1
Considering the upper part of Eq. 4 we obtain the reduced system given by Eq. 5 that is no longer function of the joint torque τ . Eq. 5 is as a classical identification problem . The sampled system is solved by the least squares method. Y B1 φB =
Nc X
K k1 F k
(5)
k=1
In the following, to simplify notations we will use E to denote the contribution of the external forces. E=
Nc X
K k1 F k
k=1
2.3 Advantages of the method We have mathematically demonstrated in Venture et al. [under review], Ayusawa et al. [2008], that the structural identifiability of the base parameters is maintained by showing that dim(col(Y B1 )) = dim(col(Y B )). Using the identification
989
Table 1. Human body model name of joint waist neck left shoulder left elbow left wrist right shoulder right elbow right wrist left hip left knee left ankle right hip right knee right ankle
type of joint free spherical spherical spherical revolute spherical spherical revolute spherical spherical revolute spherical spherical revolute spherical
DOF 6 3 3 3 1 3 3 1 3 3 1 3 3 1 3
link lower torso upper torso head left upper arm left arm left hand right upper arm right arm right hand left thigh left shank left foot right thigh right shank right foot
index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
model given by Eq. 4 or the identification model given by Eq. 5 leads to similarly identify the base parameters. However when using Eq.4 it is necessary to have a measurement of the base-link position and orientation, the joint coordinates for each joint, the contact forces and the joint torque for each joint. It is also necessary to take into account the visco-elastic properties of the joints if there are some Khalil and Dombre [2002], Venture et al. [2006]. And eventually to use a model of the actuation to obtain the joint torques: friction model for humanoids, muscle model for humans, which are still open problems. In conventional methods the identified inertial parameters are thus contaminated by the inaccuracies in the respective models. These issues are eliminated when using the base-link approach given in Eq. 5, as it is neither a function of the joint torque τ nor the visco-elasticities as shown in Fig. 1. The inertial parameters are then estimated straightforwardly. Nevertheless, as there is no local measurements of the joint torque a drawback of the base-link approach resides in the design of exciting motions. 2.4 Application to the human body Definition of a 34 DOF human model We consider a model of the human body made of 34 DOF and 15 rigid links Nakamura and Yamane [2000], as described in Table 1. It represents the most important DOF that are used in daily activities such as locomotion, grasping. In addition to the 34 DOF of its kinematics 6 DOF are used to define the generalized coordinates. Identification model and notations This kinematic leads to a 128 × 1 vector of base parameters φB given by Eq. 6. h iT (6) φB = φTB0 φTB1 ... φTB14
such that: [Mi M Si,x M Si,y M Si,z Ji,xx (i = 0) Ji,yy Ji,zz Ji,yz Ji,zx Ji,xy ]T [M Si,x M Si,y Ji,xx − Ji,yy Ji,zz revolute φBi = Ji,yz Ji,zx Ji,xy ]T [M Si,x M Si,y M Si,z Ji,xx Ji,yy spherical Ji,zz Ji,yz Ji,zx Ji,xy ]T where:
- Mi is the base parameter of link i representing the sum of the masses of links that are lower in the chain: Mi−1 = mi−1 + Mi ,
15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 - M Si is the base parameter of link i representing the sum of the first moment of inertia, for i from 30 to 1: M S i−1 = msi−1 + Mi i−1 pi + i−1 r i , - Ji is the base parameter of link i representing the inertia, for i from 30 to 1: J i−1 = I i−1 + Mi [i−1 pi ×]T [i−1 pi ×] + [i−1 pi ×]T [i−1 r i ×] + [i−1 r i ×]T [i−1 pi ×] + Ji,yy i−1 Ri U i−1 RTi i−1 Ri is the rotation matrix from the frame attached to link i − 1 to the frame attached to link i, - i−1 pi is the translational vector from the frame attached to link i − 1 to the frame attached to link i, - i−1 r i and U defined as follow: " # " # 0 1 0 0 i−1 i−1 0 ri = Ri ,U = 0 1 0 M S i−1,z 0 0 0
-
It corresponds to a regressor with 128 columns; it is decomposed in 15 sub-regressors (one for each link) as shown in Eq. 7, where each sub-regressor has as many columns as its attached link has base parameters. [ Y Blink0 Y Blink1 . . . Y Blink14 ] (7) As it is sometimes valuable to look also at the excitation properties at the limb level we also define sub-regressors for each limb as given in Eq 8. Y BLarm Y BRarm Y BLleg Y BRleg
= = = =
[ Y Blink3 Y Blink4 Y Blink5 ] [ Y Blink6 Y Blink7 Y Blink8 ] [ Y Blink9 Y Blink10 Y Blink11 ] [ Y Blink12 Y Blink13 Y Blink14 ]
(8)
a-posteriori computation of global excitation criteria Venture et al. [2006]. To avoid massive, time-consuming computations it often requires to closely looking at each motion to extract the excitation properties, it thus prevents to use systematic selection of motions and automatic identification. Hence, identification of the dynamics of complex systems is often regarded as a cumbersome task and is consequently seldom utilized. Usually exciting the whole dynamics with only one motion is impossible. Using the linearity in the parameters to estimate of the identification model, it is common to use a vertical concatenation of M different prescribed motions that each excites specific dynamics to create the regressor Mayeda et al. [1984], Venture et al. [2006], as shown in Eq. 9. Y Bmotion1 E motion1 Y Bmotion2 E motion2 φB = .. .. . . Y BmotionM E motionM
(9)
The actual excitation properties of the motion are verified aposteriori by looking at the condition number of each individual regressor and at the resulting regressor Venture et al. [2006]. However with complex systems it is difficult to interpret directly the obtained values of the condition number of the regressor and to understand the actual excitation properties in detail as two regressors, obtained with two different motions, can have the same condition number as it will be detailed in section IV. In that case, to limit the computation time it is necessary to closely look at the executed motions to define their excitation properties and chose the appropriate motions to concatenate. Thus systematic identification is not possible and post-processing is a manual, time-consuming task. 3.2 Proposed method outlines
3. EXCITING TRAJECTORIES FOR OPTIMAL IDENTIFICATION RESULTS 3.1 State of the art Once modeling and measuring issues are solved, it is widely recognized that the remaining issue resides in appropriately choosing the motions along which the identification model is sampled Gautier and Khalil [1992], Pfeiffer and Holzl [1995], Swevers et al. [1997]. To accurately solve the identification problem adequately chosen motions that excite the dynamics must be used. These motions are generally named persistent exciting trajectories Gautier and Khalil [1992] or optimal excitation trajectories Swevers et al. [1997]. For a robot they are prescribed according the kinematics and the joint limits using statistical information Swevers et al. [1997], linear algebra computations Pressé and Gautier [1993] or Fourier decomposition Park [2006]. Then, the designed motions are executed by the robot, the joint and the torque information are measured to feed the identification model to compute the inertial parameters. Nevertheless, these methods developed for manipulators are difficult to utilize when: (1) the number of degrees of freedom (DOF) of the system is large, as methods require large-scale non-linear optimizations, which complexity increases with the number of DOF; (2) the generation of precise prescribed motions not feasible; as it is the case for human identification. The exciting motions used for identification are thus chosen according to a detailed knowledge of the system, common sense, and
990
We propose an approach to select systematically the persistent exciting trajectories from a set of recorded data. The method is based on the computation of the regressor from the base-link dynamics and the linearity in the inertial parameters that allows to concatenate several motions into one regressor. Usually the motions are chosen from considerations on the condition number of the regressor and from a-priori excitation properties. This can not stand when the system is complex and when exciting motions can not be properly designed. To optimize the calculation procedure the motions to be concatenated are automatically selected without looking at the motion itself. It consists in looking not only at the condition number of the regressor, but also at the condition number of sub-matrices, called sub-regressors, obtained from the columns of the regressor and given by Eq. 7 and Eq. 8. 3.3 Rules to select the motions By looking at the condition number of the Ns +1 sub-regressors obtained for one link - or one group of links - it is then possible to systematically extract excitation features for each motion and to concatenate motions that excite all the dynamics optimally. To find the proper set of motions we proceed in two steps. (1) A threshold ε is fixed such that if ∃i ∈ [0, Ns ] such that Cond(Y Bi ) < ε, the motion is selected as a candidate for concatenation.
15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 (2) The candidates are checked to avoid similar excitation patterns. For that we calculate the variance between the different selected motions based on the condition number of the sub-regressors. We create a matrix of Ns + 1 lines each line being Cond(Y Bi ) for i = 0 to Ns . Each column represents each candidate. When several motions M1 ... Mn have similar excitation patterns the variance is low and thus n − 1 motions are arbitrarily removed from the data-set. If one motion M1 is similar to two motions M2 and M3, and M2 and M3 are different enough than M1 is removed. We finally obtain an optimal data-set with a minimum number of motions that maximize the excitation of the dynamics to estimate. 4. EXPERIMENTAL APPLICATION 4.1 Measurements of the human motions The motions are recorded by an optical motion capture system consisting in 10 cameras (Motion Analysis). We use 35 optical markers pasted on the body of the subjects at defined anatomical points to insure accuracy of inverse kinematics computations. The contact forces are measured by force-plates (Kistler). The inverse kinematics, to obtain the joint angles and their derivatives, is computed from the marker coordinates by an inhouse software Yamane and Nakamura [2003] using the model of human previously defined. 4.2 Selection of the motions used for identification In our previous work we have identified the human body inertial parameters using very simple motions such as walk, squat, side walk Venture et al. [2008b]. The results showed that some parameters were not identified by lack of excitation. To a-priori optimize the excitation we propose here to use motions from a gymnastic TV program. A pre-selection of motions is done to remove the static motions such as stretching. To facilitate the execution of smooth motions, the subjects are asked to rehearse a few times the motions, and then to perform the sequence while watching the screen showing the demonstrator. The motion sequence features a total of 29 different motions separated in 3 series. Each of the series includes arm motions, leg motions, bending, jumping, standing, walking, side-walking... 11 motions are recorded twice to finally obtain a data-set of 40 motions.
Fig. 2. Model of the kinematic structure of human with 34 DOF
991
For each of the 29 motions we compute (1)Cond(Y B ); (2) the condition number of the sub-regressors for each limb as given by Eq. 8; (3) the condition number of each of the sub-regressors Y Bi for i = 0 to 14. Example results for a selection of motions are given in Table 2 for 6 motions resulting different values of the condition number of Y B . The condition numbers for each link of the limbs are not given here for all the motions as a concern of length. Only 2 examples are given in Table 3. Table 2. Condition number for 6 motions Motion YB Y B0 Y B1 Y B2 Y BLarm Y BRarm Y BLleg Y BRleg
jump/arm 91 3 6 9 38 42 41 22
bend 1 117 7 12 8 40 40 28 27
bend 2 719 15 5 6 36 25 75 71
jump 392 9 13 10 138 107 49 70
arm 1900 33 23 10 24 60 331 223
stand 33573 283 101 142 865 555 6813 89196
Table 3. Condition number of the sub-regressor for each link Left side arm 1 jump Right side arm 1 jump
Y B3 8 20 Y B6 6 10
Y B4 6 25 Y B7 3 15
Y B5 5 11 Y B8 10 6
CY B9 17 8 Y B12 40 12
Y B10 12 5 Y B13 19 5
Y B11 12 5 Y B14 49 6
We now interpret the results presented in Table 2 from the right to the left columns. In the last column, the standing motion present a high condition number and each sub-regressor too. This is expected as none of the joint is moving. In the 6th column the motion features only upper body motion, and more particularly arms motions. The condition number of the regressor is high, however when looking at the sub-regressor we clearly see that the legs are not moving and the rest is. In the 5th column, during the jumps the arms are not moving, this is shown by a high condition number of the associated sub-regressors. In the 4th column the bending features no leg motion, this is confirm by the relatively high condition number of the associated sub-regressors. When the condition number of the regressor is very low we can also the same specificities as shown in column 2 and 3.Both motions excite all the dynamics, with the bending motion the leg are more excited compared to the arms. Motions with a high condition number would be usually dismissed, however when looking at the detailed condition number for each link given in Table 3 we see that even a motion with a high condition number presents specific excitation, here the arms and thus it shouldn’t be discarded from the data-set as it provides very specific excitation properties of the arms. As expected, the results show that from looking only at the condition number of the regressor Y B it is impossible to determine precisely the excitation properties of a motion, inducing the use of motions that do not lead to a complete excitation. With our proposed analysis of the sub-regressors, we can automatically detect which part of the kinematics is excited. Thus we can concatenate motions that excite properly all the dynamics of the system and obtain more reliable identification with an optimal data-set.
15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 As can be shown they are of same range. Similar results are obtained for the limbs.
4.3 Experimental results: use of different set of motions We compare the experimental identification results obtained with a "classic" selection of motion, and the proposed method.
Table 4. Comparison of some identification results wit literature data and measured data
• Classic method: we discard the motions having a high condition number (Cond(Y B ) > 800) regardless of the type of motion; • Proposed method: we chose the motions to use for the identification according to the presented method in section 3.3. In both cases the regressor is computed with 18 motions out of the 29 recorded motions. However the selected motions are different. With the proposed method, 6 chosen motions have a condition number of the regressor higher than 800. When considering the standard deviation computed for each parameter Venture et al. [2006], with the classic method 111 of the 128 based parameters are properly identified. With the proposed method 117 of the 128 parameters are properly identified. An increase of 5% in the number of well-identified parameters. The computation time with the proposed method is also 4% faster due to the change in the data-set length. 4.4 Experimental results: Identification results The results of the estimation (parameter value and relative standard deviation (sxr)) for a 30 year old woman are given in Fig. 3. Body
Kg
Trunk
Kg.m Kg.m2
Head
80
8
60
6
40
4
20
2
Upper-arm
Arm
Hand
1 0.5 0 MSx MSy MSz Jxx Jyy Jzz
Kg.m Kg.m2
Jyz
Thigh
Jzx Jxy MSx MSy Jxx-Jyy Jzz Jyz Jzx Jxy MSx MSy MSz Jxx Jyy Jzz
Shank
Identified 59.47 0.48 0.18 0.47 2.76 0.06 2.75 0.16 0.34 0.14
Reference scale 59.8 0.9 ± 0.5 0.7 ± 0.5 1.2 ± 0.7 2.7 ± 0.9 2.1 ± 0.9 1.8 ± 0.7 0.16 ± 0.03 0.17 ± 0.03 0.14 ± 0.02
Finally the obtained results are P validate. For that we compare the measured contact forces: K k F k (Force-plate data) with the contact forces that are computed using the identified parameters: Y B φB . For that we use motions that are not used during the identification procedure (cross validation). In addition we also compute the position of the whole-body COM from the identified parameters to check the reliability of the estimated parameters. We present the validation results in an active interface using the 3D representation of the human body based on the 34 DOF model used. Snapshots of the results are shown Fig. 4. They show that for various motions we have a good estimation of the contact forces using the identified parameters. The position of the COM moves accordingly to the body-conformation in a convincing way.
In this paper we have presented a method to identify the the human body inertial parameters as well as a method to determine systematically the data-set of motions to use for identification to optimize the obtained results. The regressor obtained from sampling the identification model along a movement is decomposed into sub-regressors according to the kinematics of the system. For each of the sub-regressors the condition number is computed. The motions leading to a sufficiently low condition number of at least one of the sub-regressors are selected as potential candidates to be used in the final data-set. To avoid using motions with similar excitation properties that will augment
M MSx MSyMSz Jxx Jyy Jzz Jyz Jzx Jxy MSx MSyMSz Jxx Jyy Jzz Jyz Jzx Jxy MSx MSyMSz Jxx Jyy Jzz Jyz Jzx Jxy
Kg.m Kg.m2
Unit Kg Kg.m2 Kg.m2 Kg.m2 Kg.m2 Kg.m2 Kg.m2 Kg.m2 Kg.m2 Kg.m2
5. CONCLUSION
0
0
Parameter M J0xx J0yy J0zz J1xx J1yy J1zz J2xx J2yy J2zz
Jyz Jzx Jxy
Foot
1 0 −1 MSx MSy MSz Jxx Jyy Jzz
Jyz
Jzx Jxy MSx MSy Jxx-Jyy Jzz Jyz Jzx Jxy MSx MSy MSz Jxx Jyy Jzz
Jyz
Jzx Jxy
COM
Parameter estimated with standard deviation <5% when adequate: Parameter estimated with standard deviation 5%<~<15% The size of the dot is proportional to the standard deviation Left Small parameter <0.05 Right Unidentified parameter: standard deviation >15%
COM
Fig. 3. Identification results. Star marks a parameter estimated with sxr < 5%. Dot marks a parameter estimated with 5% < sxr < 15%, the size of the marker is proportional to sxr. Plain blue triangle marks the small parameters. Black triangle marks a parameter that is not estimated: sxr > 15% The identified parameters can be compared with literature data found Young et al. [1983]. A sample for the abdomen (Link 0), the trunk (Link 1) and the head (Link 2) is given in Table 4.
992
COM
Estimated position of the whole-body COM from identified parameters Computed contact forces from identified parameters Measured contact forces Computed moment from identified parameters Measured moment
Fig. 4. Validation of the identification results: comparison of the force plate measurements and the force and moment computed using the identified parameters. In addition the position of the whole-body COM is given.
15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 the data-set without providing new excitation properties we propose to remove these motions. They are removed by considerations on the condition number of the sub-regressors: motion that generates similar condition numbers of sub-regressors are removed by computing the variance of the obtained system. The proposed method allows to select the most relevant motions for identification from large data-set eventually pre-recorded. The experimental application shows that with this method it is possible to select 16 motions out of 40 and to perform the identification successfully. With the proposed method, the results are such that: • We prevent from removing motions that present interesting excitation properties but leading to regressor matrix of high condition number. • We guaranty the use of an optimal data-set leading to better experimental results as the excitation properties are optimized. In our experiments it was at least 5%. • The method is based on motion capture and contact forces measurements from external, painless sensors. They permit to perform all types of motion with no limitation. • We chose numerically thus systematically motions according to their excitation properties. The identification can thus be implemented for health monitoring or rehabilitation when a large amount of motion data is available. The parameters are optimally identified. This method allows to measure subject-specific parameters with simple motions of gymnastics. The results can be used instead of the usually used literature data for further applications such as enhancing gait analysis results, or developing personalized health monitoring. REFERENCES K. Ayusawa, G. Venture, and Y. Nakamura. Inertial parameters identifiability of humanoid robot based on the baselink equation of motion. In Proc. of the Robotics and mechatronics Conference, Nagano, Japan, volume to be published, [In Japanese], 2008. A.L. Betker, T. Szturm, and Z. Moussavi. Center of mass approximation during walking as a function of trunk and swing leg acceleration. In Proc. of the 28th IEEE EMBS Annual Int. Conf., New York City, USA, pages 3435–3438, 2006. C.K. Cheng. Segment inertial properties of chinese adults determined from magnetic resonance imaging. Clinical biomechanics, 15:559–566, 2000. J. Durkin and J. Dowling. Analysis of body segment parameter differences between four human populations and the estimation errors of four popular mathematical models. J. of Biomedical Eng., 125:515–522, 2003. M. Gautier. Numerical calculation of the base inertial parameters. J. of Robotic Systems, 8(4):485–506, 1991. M. Gautier and W. Khalil. Exciting trajectories for the identification of base inertial parameters of robots. Int. J. of Robotic Research, 11(4):363–375, 1992. J. Han, A.L. Betker, T. Szturm, and Z. Moussavi. Estimation of the center of body mass during forward stepping using body aceleration. In Proc. of the 28th IEEE EMBS Annual Int. Conf., New York City, USA, pages 4564–4567, 2006. R.K. Jensen. Estimation of the biomechanical properties of three body types using a photogrammetric method. J.Biomechanics, 11:349–358, 1978. H. Kawasaki, Y. Beniya, and K. Kanzaki. Minimum dynamics parameters of tree structure robot models. In Int. Conf. of In-
993
dustrial Electronics, Control and Instrumentation, volume 2, pages 1100–1105, 1991. W. Khalil and F. Bennis. Symbolic calculation of the base inertial parameters of closed-loop robots. The Int. J. of Robotics Research, 14(2):112–128, 1995. W. Khalil and E. Dombre. Modeling, identification and control of robots. Hermès Penton, London-U.K, 2002. G. Liu, K. Iagnemma, S. Dubowsky, and G. Morel. A base force/torque sensor approach to robot manipulator inertial parameter estimation. In Proc. of the IEEE Int. Conf. on Robotics and Automation, pages 3316–3321, 1998. H. Mayeda, K. Osuka, and A. Kangawa. A new identification method for serial manipulator arms. In Proc. IFAC 9th World ˚ Congress, page 2429U2434, 1984. Y. Nakamura and K. Yamane. Dynamics computation of structure-varying kinematic chains and its application to human figures. IEEE Trans. on Robotics and Automation, 16(2):124–134, 2000. Kyung-Jo Park. Fourier-based optimal excitation trajectories for the identification of robots. Robotica, 24:625–633, 2006. D.J. Pearsall and J.G. Reid. Inertial properties of the human trunk of males determined from magnetic resonance imaging. Annals of Biomedical Engineering, 22:692–706, 1994. F. Pfeiffer and J. Holzl. Parameter identification for industrial robots. Proc. of the IEEE Int. Conf. on Robotics and Automation, 2:1468–1476, May 1995. C. Pressé and M. Gautier. New criteria of exciting trajectories for robot identification. In Proc. IEEE Int. Conf. on Robotics and Automation, pages 907–912, 1993. V.A. Sujan and S. Dubowsky. An optimal information method for mobile manipulator dynamic parameter identification. IEEE/ASME Trans. on Mechatronics, 8(2):215 – 225, 2003. J. Swevers, C. Ganseman, D. Bilgin, J. De Schutter, and H. Van Brussel. Optimal robot excitation and identification. IEEE Trans. on Robotics and Automation, 13(5):730–740, 1997. G. Venture, P.J. Ripert, W. Khalil, M. Gautier, and P. Bodson. Modeling and identification of passenger car dynamics using robotics formalism. IEEE Trans. on Intelligent Transportation Systems, 7(3):349–359, 2006. G. Venture, K. Ayusawa, and Y. Nakamura. Dynamics identification of humanoid systems. In Proc. CISM-IFToMM Symp. on Robot Design, Dynamics, and Control (ROMANSY), pages 301–308, 2008a. G. Venture, K. Ayusawa, and Y. Nakamura. Motion capture based identification of human inertial parameters. In Proc. IEEE/EMBS Int. Conf. on Eng. in Medicine and Biology (to be published), pages 4575–4578, 2008b. G. Venture, K. Ayusawa, and Y. Nakamura. Identification of legged robots from base-link dynamics. IEEE Trans. on Robotics, under review. H. West, E. Papadopoulos, S. Dubowsky, and H. Cheah. A method for estimating the mass properties of a manipulator bymeasuring the reaction moments at its base. In Proc. of the IEEE Int. Conf. on Robotics and Automation, volume 3, pages 1510–1516, 1989. K. Yamane and Y. Nakamura. Natural motion animation through constraining and deconstraining at will. IEEE Trans. on Visualization and Computer Graphics, 9(3):352– 360, 2003. J.W. Young, R.F. Chandler, and C.C. Snow. Anthropometric and mass distribution characteristics of the adult female. Technical Report FAA-AM-83-16, US Air Force, 1983.
Identification of Human Mass Properties From Motion ⋆ Gentiane Venture ∗ Ko Ayusawa ∗ Yoshihiko Nakamura ∗ ∗
The University of Tokyo, 7-3-1 Hongo Bunkyo-ku 113-8656 Tokyo, Japan, (e-mail: [email protected])
Abstract: We experimentally identify the human body dynamics from motion capture data and contact force measurements. We make use of the 6 base-link equations to write the identification model. We have shown that the obtained system is sufficient to estimate the whole base inertial parameters. The method is computationally efficient; and allows to identify the pure inertial parameters without transmission, friction considerations. However finding persistent exciting trajectories is a drawback of the method. In this paper we put emphasis in describing persistent exciting trajectories that are suitable to obtain accurate results. We propose a method to automatically select the set of motions used for the identification in order to optimize the computation time and the accuracy of the identified parameters. For the experiments we make use of gymnastic motions. Each of the 3 subjects -of different body conformations- perform 29 types of motions. The identification method and the automatic motions selection method are both applied to obtain the inertial parameters of one subject. Keywords: Parameter Identification, Excitation, Least-squares Identification, Dynamic Modelling, Human-body, Biomedical 1. INTRODUCTION
Dynamics Identification
The mass properties of the human body are crucial in biomechanics studies, in gait studies and for some medical applications: orthopedics, neurology, musculoskeletal disorders. Gait stability is often examined using the trajectory of the wholebody center of mass (COM). To monitor the variation of muscle mass during hospitalization, rehabilitation, the measurement of the inertia and the position of the COM of each link of the body is a key-data to refine the diagnosis and to provide personalized health-care. The recent portable technologies allow to develop systems to estimate in-vivo the position of the global COM Betker et al. [2006], Han et al. [2006]. However the estimation of the moment of inertia is usually computed by an interpolation of literature data Young et al. [1983], Durkin and Dowling [2003]. Which are obtained by inaccurate techniques such as photogrammetry Jensen [1978] or by expensive and time-consuming techniques based on 3D imaging (CT-scan or MRI) and 3D modeling interpolations Pearsall and Reid [1994], Cheng [2000]. Consequently, there is a pressing need to develop reliable and robust method to estimate the mass properties of the human body. In this paper we present a methodology to estimate the inertial parameters of humans from motion and contact force measurements. In section II we present the identification method that is applied to the human body. In section III we propose a method to optimize the selection of exciting trajectories, in order to guaranty reliable and fast computations. In section IV we apply the proposed scheme experimentally to identify the mass properties of young woman. We compare the identification results obtained with different sets of motions. The results obtained with the proposed method are validate and compared with literature data. ⋆ This research is supported by the Special Coordination Funds for Promoting Science and Technology: "IRT Foundation to Support Man and Aging Society".
978-3-902661-47-0/09/$20.00 © 2009 IFAC
988
Fk
Force Sensing
φB
?
YB . .. q, q,q. q
Motion otion Sensing S
Fig. 1. Principle of the identification of bipeds from base-link dynamics 2. IDENTIFICATION OF HUMAN DYNAMICS FROM BASE-LINK EQUATIONS 2.1 Robotics background In robotics, the identification of the inertial parameters of a system is a well-known problem Sujan and Dubowsky [2003], Liu et al. [1998] and general formalisms and methodologies that apply to a wide range of dynamic systems are available Khalil and Dombre [2002], Swevers et al. [1997]. Generally methods implies the measurements of joint torques, however a few method using the contact forces information have been developed West et al. [1989]. We have proposed to identify the mass properties of mobile legged systems using only the dynamics of the base link Venture et al. [2008a]. In this paper we use this method that applies to humanoid robots to identify the human inertial parameters. It allows to identify the wholebody inertial parameters making use of a motion capture studio and force-plates as it is shown in Fig. 1. 2.2 Formulation The inverse dynamics of a legged system can be written in a linear form with respect to the inertial parameters Venture et al.
10.3182/20090706-3-FR-2004.0163
15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 [2008b], Khalil and Dombre [2002] as shown in Eq.1, where the inertial parameters are grouped in the vector φ and the joint information is grouped in the regressor Y . The upper part of the equation describes the 6 DOF motion of the base-link in the 3D space. The lower part of the equation describes the motion of the N bodies of the various kinematic chains constituting the whole system. Usually the base-link is chosen as the lower torso, the chains are the limbs, the upper-torso, the head. where:
X Nc Y1 0 K k1 φ= + Fk Y2 K k2 τ
(1)
k=1
Y1 is the regressor, which is function of the Y2 ¨ system joint angles θ, velocity θ˙ and acceleration θ, and of the vector of generalized coordinates q0 and its derivatives; • φ is the vector of inertial parameters to estimate such that: T (2) φ = φT0 φT1 ·· φTN
• Y =
- φj is the vector of standard parameters for each link Bj (j = 0 to N ), such that: φj = [ mi msi,x msi,y msi,z Ii,xx (3) Ii,yy Ii,zz Ii,yz Ii,zx Ii,xy ]T - mj is the mass, - Ij,xx , Ij,yy , Ij,zz , Ij,yz , Ij,zx , Ij,xy are the 6 independent components of the inertia matrix I j , - msj,x , msj,y , msj,z are the first moments components of the vector msj
Not all of the 10N parameters are used to describe the dynamics. The necessary parameters are called base parameters φB They are the only identifiable parameters Khalil and Dombre [2002]. They are computed from the standard parameters φ Gautier [1991], Kawasaki et al. [1991], Khalil and Bennis [1995] to obtain the minimal identification model Eq.4. Y B φB =
Y B1 Y B2
X Nc 0 K k1 φB = + Fk K k2 τ
(4)
k=1
Considering the upper part of Eq. 4 we obtain the reduced system given by Eq. 5 that is no longer function of the joint torque τ . Eq. 5 is as a classical identification problem . The sampled system is solved by the least squares method. Y B1 φB =
Nc X
K k1 F k
(5)
k=1
In the following, to simplify notations we will use E to denote the contribution of the external forces. E=
Nc X
K k1 F k
k=1
2.3 Advantages of the method We have mathematically demonstrated in Venture et al. [under review], Ayusawa et al. [2008], that the structural identifiability of the base parameters is maintained by showing that dim(col(Y B1 )) = dim(col(Y B )). Using the identification
989
Table 1. Human body model name of joint waist neck left shoulder left elbow left wrist right shoulder right elbow right wrist left hip left knee left ankle right hip right knee right ankle
type of joint free spherical spherical spherical revolute spherical spherical revolute spherical spherical revolute spherical spherical revolute spherical
DOF 6 3 3 3 1 3 3 1 3 3 1 3 3 1 3
link lower torso upper torso head left upper arm left arm left hand right upper arm right arm right hand left thigh left shank left foot right thigh right shank right foot
index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
model given by Eq. 4 or the identification model given by Eq. 5 leads to similarly identify the base parameters. However when using Eq.4 it is necessary to have a measurement of the base-link position and orientation, the joint coordinates for each joint, the contact forces and the joint torque for each joint. It is also necessary to take into account the visco-elastic properties of the joints if there are some Khalil and Dombre [2002], Venture et al. [2006]. And eventually to use a model of the actuation to obtain the joint torques: friction model for humanoids, muscle model for humans, which are still open problems. In conventional methods the identified inertial parameters are thus contaminated by the inaccuracies in the respective models. These issues are eliminated when using the base-link approach given in Eq. 5, as it is neither a function of the joint torque τ nor the visco-elasticities as shown in Fig. 1. The inertial parameters are then estimated straightforwardly. Nevertheless, as there is no local measurements of the joint torque a drawback of the base-link approach resides in the design of exciting motions. 2.4 Application to the human body Definition of a 34 DOF human model We consider a model of the human body made of 34 DOF and 15 rigid links Nakamura and Yamane [2000], as described in Table 1. It represents the most important DOF that are used in daily activities such as locomotion, grasping. In addition to the 34 DOF of its kinematics 6 DOF are used to define the generalized coordinates. Identification model and notations This kinematic leads to a 128 × 1 vector of base parameters φB given by Eq. 6. h iT (6) φB = φTB0 φTB1 ... φTB14
such that: [Mi M Si,x M Si,y M Si,z Ji,xx (i = 0) Ji,yy Ji,zz Ji,yz Ji,zx Ji,xy ]T [M Si,x M Si,y Ji,xx − Ji,yy Ji,zz revolute φBi = Ji,yz Ji,zx Ji,xy ]T [M Si,x M Si,y M Si,z Ji,xx Ji,yy spherical Ji,zz Ji,yz Ji,zx Ji,xy ]T where:
- Mi is the base parameter of link i representing the sum of the masses of links that are lower in the chain: Mi−1 = mi−1 + Mi ,
15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 - M Si is the base parameter of link i representing the sum of the first moment of inertia, for i from 30 to 1: M S i−1 = msi−1 + Mi i−1 pi + i−1 r i , - Ji is the base parameter of link i representing the inertia, for i from 30 to 1: J i−1 = I i−1 + Mi [i−1 pi ×]T [i−1 pi ×] + [i−1 pi ×]T [i−1 r i ×] + [i−1 r i ×]T [i−1 pi ×] + Ji,yy i−1 Ri U i−1 RTi i−1 Ri is the rotation matrix from the frame attached to link i − 1 to the frame attached to link i, - i−1 pi is the translational vector from the frame attached to link i − 1 to the frame attached to link i, - i−1 r i and U defined as follow: " # " # 0 1 0 0 i−1 i−1 0 ri = Ri ,U = 0 1 0 M S i−1,z 0 0 0
-
It corresponds to a regressor with 128 columns; it is decomposed in 15 sub-regressors (one for each link) as shown in Eq. 7, where each sub-regressor has as many columns as its attached link has base parameters. [ Y Blink0 Y Blink1 . . . Y Blink14 ] (7) As it is sometimes valuable to look also at the excitation properties at the limb level we also define sub-regressors for each limb as given in Eq 8. Y BLarm Y BRarm Y BLleg Y BRleg
= = = =
[ Y Blink3 Y Blink4 Y Blink5 ] [ Y Blink6 Y Blink7 Y Blink8 ] [ Y Blink9 Y Blink10 Y Blink11 ] [ Y Blink12 Y Blink13 Y Blink14 ]
(8)
a-posteriori computation of global excitation criteria Venture et al. [2006]. To avoid massive, time-consuming computations it often requires to closely looking at each motion to extract the excitation properties, it thus prevents to use systematic selection of motions and automatic identification. Hence, identification of the dynamics of complex systems is often regarded as a cumbersome task and is consequently seldom utilized. Usually exciting the whole dynamics with only one motion is impossible. Using the linearity in the parameters to estimate of the identification model, it is common to use a vertical concatenation of M different prescribed motions that each excites specific dynamics to create the regressor Mayeda et al. [1984], Venture et al. [2006], as shown in Eq. 9. Y Bmotion1 E motion1 Y Bmotion2 E motion2 φB = .. .. . . Y BmotionM E motionM
(9)
The actual excitation properties of the motion are verified aposteriori by looking at the condition number of each individual regressor and at the resulting regressor Venture et al. [2006]. However with complex systems it is difficult to interpret directly the obtained values of the condition number of the regressor and to understand the actual excitation properties in detail as two regressors, obtained with two different motions, can have the same condition number as it will be detailed in section IV. In that case, to limit the computation time it is necessary to closely look at the executed motions to define their excitation properties and chose the appropriate motions to concatenate. Thus systematic identification is not possible and post-processing is a manual, time-consuming task. 3.2 Proposed method outlines
3. EXCITING TRAJECTORIES FOR OPTIMAL IDENTIFICATION RESULTS 3.1 State of the art Once modeling and measuring issues are solved, it is widely recognized that the remaining issue resides in appropriately choosing the motions along which the identification model is sampled Gautier and Khalil [1992], Pfeiffer and Holzl [1995], Swevers et al. [1997]. To accurately solve the identification problem adequately chosen motions that excite the dynamics must be used. These motions are generally named persistent exciting trajectories Gautier and Khalil [1992] or optimal excitation trajectories Swevers et al. [1997]. For a robot they are prescribed according the kinematics and the joint limits using statistical information Swevers et al. [1997], linear algebra computations Pressé and Gautier [1993] or Fourier decomposition Park [2006]. Then, the designed motions are executed by the robot, the joint and the torque information are measured to feed the identification model to compute the inertial parameters. Nevertheless, these methods developed for manipulators are difficult to utilize when: (1) the number of degrees of freedom (DOF) of the system is large, as methods require large-scale non-linear optimizations, which complexity increases with the number of DOF; (2) the generation of precise prescribed motions not feasible; as it is the case for human identification. The exciting motions used for identification are thus chosen according to a detailed knowledge of the system, common sense, and
990
We propose an approach to select systematically the persistent exciting trajectories from a set of recorded data. The method is based on the computation of the regressor from the base-link dynamics and the linearity in the inertial parameters that allows to concatenate several motions into one regressor. Usually the motions are chosen from considerations on the condition number of the regressor and from a-priori excitation properties. This can not stand when the system is complex and when exciting motions can not be properly designed. To optimize the calculation procedure the motions to be concatenated are automatically selected without looking at the motion itself. It consists in looking not only at the condition number of the regressor, but also at the condition number of sub-matrices, called sub-regressors, obtained from the columns of the regressor and given by Eq. 7 and Eq. 8. 3.3 Rules to select the motions By looking at the condition number of the Ns +1 sub-regressors obtained for one link - or one group of links - it is then possible to systematically extract excitation features for each motion and to concatenate motions that excite all the dynamics optimally. To find the proper set of motions we proceed in two steps. (1) A threshold ε is fixed such that if ∃i ∈ [0, Ns ] such that Cond(Y Bi ) < ε, the motion is selected as a candidate for concatenation.
15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 (2) The candidates are checked to avoid similar excitation patterns. For that we calculate the variance between the different selected motions based on the condition number of the sub-regressors. We create a matrix of Ns + 1 lines each line being Cond(Y Bi ) for i = 0 to Ns . Each column represents each candidate. When several motions M1 ... Mn have similar excitation patterns the variance is low and thus n − 1 motions are arbitrarily removed from the data-set. If one motion M1 is similar to two motions M2 and M3, and M2 and M3 are different enough than M1 is removed. We finally obtain an optimal data-set with a minimum number of motions that maximize the excitation of the dynamics to estimate. 4. EXPERIMENTAL APPLICATION 4.1 Measurements of the human motions The motions are recorded by an optical motion capture system consisting in 10 cameras (Motion Analysis). We use 35 optical markers pasted on the body of the subjects at defined anatomical points to insure accuracy of inverse kinematics computations. The contact forces are measured by force-plates (Kistler). The inverse kinematics, to obtain the joint angles and their derivatives, is computed from the marker coordinates by an inhouse software Yamane and Nakamura [2003] using the model of human previously defined. 4.2 Selection of the motions used for identification In our previous work we have identified the human body inertial parameters using very simple motions such as walk, squat, side walk Venture et al. [2008b]. The results showed that some parameters were not identified by lack of excitation. To a-priori optimize the excitation we propose here to use motions from a gymnastic TV program. A pre-selection of motions is done to remove the static motions such as stretching. To facilitate the execution of smooth motions, the subjects are asked to rehearse a few times the motions, and then to perform the sequence while watching the screen showing the demonstrator. The motion sequence features a total of 29 different motions separated in 3 series. Each of the series includes arm motions, leg motions, bending, jumping, standing, walking, side-walking... 11 motions are recorded twice to finally obtain a data-set of 40 motions.
Fig. 2. Model of the kinematic structure of human with 34 DOF
991
For each of the 29 motions we compute (1)Cond(Y B ); (2) the condition number of the sub-regressors for each limb as given by Eq. 8; (3) the condition number of each of the sub-regressors Y Bi for i = 0 to 14. Example results for a selection of motions are given in Table 2 for 6 motions resulting different values of the condition number of Y B . The condition numbers for each link of the limbs are not given here for all the motions as a concern of length. Only 2 examples are given in Table 3. Table 2. Condition number for 6 motions Motion YB Y B0 Y B1 Y B2 Y BLarm Y BRarm Y BLleg Y BRleg
jump/arm 91 3 6 9 38 42 41 22
bend 1 117 7 12 8 40 40 28 27
bend 2 719 15 5 6 36 25 75 71
jump 392 9 13 10 138 107 49 70
arm 1900 33 23 10 24 60 331 223
stand 33573 283 101 142 865 555 6813 89196
Table 3. Condition number of the sub-regressor for each link Left side arm 1 jump Right side arm 1 jump
Y B3 8 20 Y B6 6 10
Y B4 6 25 Y B7 3 15
Y B5 5 11 Y B8 10 6
CY B9 17 8 Y B12 40 12
Y B10 12 5 Y B13 19 5
Y B11 12 5 Y B14 49 6
We now interpret the results presented in Table 2 from the right to the left columns. In the last column, the standing motion present a high condition number and each sub-regressor too. This is expected as none of the joint is moving. In the 6th column the motion features only upper body motion, and more particularly arms motions. The condition number of the regressor is high, however when looking at the sub-regressor we clearly see that the legs are not moving and the rest is. In the 5th column, during the jumps the arms are not moving, this is shown by a high condition number of the associated sub-regressors. In the 4th column the bending features no leg motion, this is confirm by the relatively high condition number of the associated sub-regressors. When the condition number of the regressor is very low we can also the same specificities as shown in column 2 and 3.Both motions excite all the dynamics, with the bending motion the leg are more excited compared to the arms. Motions with a high condition number would be usually dismissed, however when looking at the detailed condition number for each link given in Table 3 we see that even a motion with a high condition number presents specific excitation, here the arms and thus it shouldn’t be discarded from the data-set as it provides very specific excitation properties of the arms. As expected, the results show that from looking only at the condition number of the regressor Y B it is impossible to determine precisely the excitation properties of a motion, inducing the use of motions that do not lead to a complete excitation. With our proposed analysis of the sub-regressors, we can automatically detect which part of the kinematics is excited. Thus we can concatenate motions that excite properly all the dynamics of the system and obtain more reliable identification with an optimal data-set.
15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 As can be shown they are of same range. Similar results are obtained for the limbs.
4.3 Experimental results: use of different set of motions We compare the experimental identification results obtained with a "classic" selection of motion, and the proposed method.
Table 4. Comparison of some identification results wit literature data and measured data
• Classic method: we discard the motions having a high condition number (Cond(Y B ) > 800) regardless of the type of motion; • Proposed method: we chose the motions to use for the identification according to the presented method in section 3.3. In both cases the regressor is computed with 18 motions out of the 29 recorded motions. However the selected motions are different. With the proposed method, 6 chosen motions have a condition number of the regressor higher than 800. When considering the standard deviation computed for each parameter Venture et al. [2006], with the classic method 111 of the 128 based parameters are properly identified. With the proposed method 117 of the 128 parameters are properly identified. An increase of 5% in the number of well-identified parameters. The computation time with the proposed method is also 4% faster due to the change in the data-set length. 4.4 Experimental results: Identification results The results of the estimation (parameter value and relative standard deviation (sxr)) for a 30 year old woman are given in Fig. 3. Body
Kg
Trunk
Kg.m Kg.m2
Head
80
8
60
6
40
4
20
2
Upper-arm
Arm
Hand
1 0.5 0 MSx MSy MSz Jxx Jyy Jzz
Kg.m Kg.m2
Jyz
Thigh
Jzx Jxy MSx MSy Jxx-Jyy Jzz Jyz Jzx Jxy MSx MSy MSz Jxx Jyy Jzz
Shank
Identified 59.47 0.48 0.18 0.47 2.76 0.06 2.75 0.16 0.34 0.14
Reference scale 59.8 0.9 ± 0.5 0.7 ± 0.5 1.2 ± 0.7 2.7 ± 0.9 2.1 ± 0.9 1.8 ± 0.7 0.16 ± 0.03 0.17 ± 0.03 0.14 ± 0.02
Finally the obtained results are P validate. For that we compare the measured contact forces: K k F k (Force-plate data) with the contact forces that are computed using the identified parameters: Y B φB . For that we use motions that are not used during the identification procedure (cross validation). In addition we also compute the position of the whole-body COM from the identified parameters to check the reliability of the estimated parameters. We present the validation results in an active interface using the 3D representation of the human body based on the 34 DOF model used. Snapshots of the results are shown Fig. 4. They show that for various motions we have a good estimation of the contact forces using the identified parameters. The position of the COM moves accordingly to the body-conformation in a convincing way.
In this paper we have presented a method to identify the the human body inertial parameters as well as a method to determine systematically the data-set of motions to use for identification to optimize the obtained results. The regressor obtained from sampling the identification model along a movement is decomposed into sub-regressors according to the kinematics of the system. For each of the sub-regressors the condition number is computed. The motions leading to a sufficiently low condition number of at least one of the sub-regressors are selected as potential candidates to be used in the final data-set. To avoid using motions with similar excitation properties that will augment
M MSx MSyMSz Jxx Jyy Jzz Jyz Jzx Jxy MSx MSyMSz Jxx Jyy Jzz Jyz Jzx Jxy MSx MSyMSz Jxx Jyy Jzz Jyz Jzx Jxy
Kg.m Kg.m2
Unit Kg Kg.m2 Kg.m2 Kg.m2 Kg.m2 Kg.m2 Kg.m2 Kg.m2 Kg.m2 Kg.m2
5. CONCLUSION
0
0
Parameter M J0xx J0yy J0zz J1xx J1yy J1zz J2xx J2yy J2zz
Jyz Jzx Jxy
Foot
1 0 −1 MSx MSy MSz Jxx Jyy Jzz
Jyz
Jzx Jxy MSx MSy Jxx-Jyy Jzz Jyz Jzx Jxy MSx MSy MSz Jxx Jyy Jzz
Jyz
Jzx Jxy
COM
Parameter estimated with standard deviation <5% when adequate: Parameter estimated with standard deviation 5%<~<15% The size of the dot is proportional to the standard deviation Left Small parameter <0.05 Right Unidentified parameter: standard deviation >15%
COM
Fig. 3. Identification results. Star marks a parameter estimated with sxr < 5%. Dot marks a parameter estimated with 5% < sxr < 15%, the size of the marker is proportional to sxr. Plain blue triangle marks the small parameters. Black triangle marks a parameter that is not estimated: sxr > 15% The identified parameters can be compared with literature data found Young et al. [1983]. A sample for the abdomen (Link 0), the trunk (Link 1) and the head (Link 2) is given in Table 4.
992
COM
Estimated position of the whole-body COM from identified parameters Computed contact forces from identified parameters Measured contact forces Computed moment from identified parameters Measured moment
Fig. 4. Validation of the identification results: comparison of the force plate measurements and the force and moment computed using the identified parameters. In addition the position of the whole-body COM is given.
15th IFAC SYSID (SYSID 2009) Saint-Malo, France, July 6-8, 2009 the data-set without providing new excitation properties we propose to remove these motions. They are removed by considerations on the condition number of the sub-regressors: motion that generates similar condition numbers of sub-regressors are removed by computing the variance of the obtained system. The proposed method allows to select the most relevant motions for identification from large data-set eventually pre-recorded. The experimental application shows that with this method it is possible to select 16 motions out of 40 and to perform the identification successfully. With the proposed method, the results are such that: • We prevent from removing motions that present interesting excitation properties but leading to regressor matrix of high condition number. • We guaranty the use of an optimal data-set leading to better experimental results as the excitation properties are optimized. In our experiments it was at least 5%. • The method is based on motion capture and contact forces measurements from external, painless sensors. They permit to perform all types of motion with no limitation. • We chose numerically thus systematically motions according to their excitation properties. The identification can thus be implemented for health monitoring or rehabilitation when a large amount of motion data is available. The parameters are optimally identified. This method allows to measure subject-specific parameters with simple motions of gymnastics. The results can be used instead of the usually used literature data for further applications such as enhancing gait analysis results, or developing personalized health monitoring. REFERENCES K. Ayusawa, G. Venture, and Y. Nakamura. Inertial parameters identifiability of humanoid robot based on the baselink equation of motion. In Proc. of the Robotics and mechatronics Conference, Nagano, Japan, volume to be published, [In Japanese], 2008. A.L. Betker, T. Szturm, and Z. Moussavi. Center of mass approximation during walking as a function of trunk and swing leg acceleration. In Proc. of the 28th IEEE EMBS Annual Int. Conf., New York City, USA, pages 3435–3438, 2006. C.K. Cheng. Segment inertial properties of chinese adults determined from magnetic resonance imaging. Clinical biomechanics, 15:559–566, 2000. J. Durkin and J. Dowling. Analysis of body segment parameter differences between four human populations and the estimation errors of four popular mathematical models. J. of Biomedical Eng., 125:515–522, 2003. M. Gautier. Numerical calculation of the base inertial parameters. J. of Robotic Systems, 8(4):485–506, 1991. M. Gautier and W. Khalil. Exciting trajectories for the identification of base inertial parameters of robots. Int. J. of Robotic Research, 11(4):363–375, 1992. J. Han, A.L. Betker, T. Szturm, and Z. Moussavi. Estimation of the center of body mass during forward stepping using body aceleration. In Proc. of the 28th IEEE EMBS Annual Int. Conf., New York City, USA, pages 4564–4567, 2006. R.K. Jensen. Estimation of the biomechanical properties of three body types using a photogrammetric method. J.Biomechanics, 11:349–358, 1978. H. Kawasaki, Y. Beniya, and K. Kanzaki. Minimum dynamics parameters of tree structure robot models. In Int. Conf. of In-
993
dustrial Electronics, Control and Instrumentation, volume 2, pages 1100–1105, 1991. W. Khalil and F. Bennis. Symbolic calculation of the base inertial parameters of closed-loop robots. The Int. J. of Robotics Research, 14(2):112–128, 1995. W. Khalil and E. Dombre. Modeling, identification and control of robots. Hermès Penton, London-U.K, 2002. G. Liu, K. Iagnemma, S. Dubowsky, and G. Morel. A base force/torque sensor approach to robot manipulator inertial parameter estimation. In Proc. of the IEEE Int. Conf. on Robotics and Automation, pages 3316–3321, 1998. H. Mayeda, K. Osuka, and A. Kangawa. A new identification method for serial manipulator arms. In Proc. IFAC 9th World ˚ Congress, page 2429U2434, 1984. Y. Nakamura and K. Yamane. Dynamics computation of structure-varying kinematic chains and its application to human figures. IEEE Trans. on Robotics and Automation, 16(2):124–134, 2000. Kyung-Jo Park. Fourier-based optimal excitation trajectories for the identification of robots. Robotica, 24:625–633, 2006. D.J. Pearsall and J.G. Reid. Inertial properties of the human trunk of males determined from magnetic resonance imaging. Annals of Biomedical Engineering, 22:692–706, 1994. F. Pfeiffer and J. Holzl. Parameter identification for industrial robots. Proc. of the IEEE Int. Conf. on Robotics and Automation, 2:1468–1476, May 1995. C. Pressé and M. Gautier. New criteria of exciting trajectories for robot identification. In Proc. IEEE Int. Conf. on Robotics and Automation, pages 907–912, 1993. V.A. Sujan and S. Dubowsky. An optimal information method for mobile manipulator dynamic parameter identification. IEEE/ASME Trans. on Mechatronics, 8(2):215 – 225, 2003. J. Swevers, C. Ganseman, D. Bilgin, J. De Schutter, and H. Van Brussel. Optimal robot excitation and identification. IEEE Trans. on Robotics and Automation, 13(5):730–740, 1997. G. Venture, P.J. Ripert, W. Khalil, M. Gautier, and P. Bodson. Modeling and identification of passenger car dynamics using robotics formalism. IEEE Trans. on Intelligent Transportation Systems, 7(3):349–359, 2006. G. Venture, K. Ayusawa, and Y. Nakamura. Dynamics identification of humanoid systems. In Proc. CISM-IFToMM Symp. on Robot Design, Dynamics, and Control (ROMANSY), pages 301–308, 2008a. G. Venture, K. Ayusawa, and Y. Nakamura. Motion capture based identification of human inertial parameters. In Proc. IEEE/EMBS Int. Conf. on Eng. in Medicine and Biology (to be published), pages 4575–4578, 2008b. G. Venture, K. Ayusawa, and Y. Nakamura. Identification of legged robots from base-link dynamics. IEEE Trans. on Robotics, under review. H. West, E. Papadopoulos, S. Dubowsky, and H. Cheah. A method for estimating the mass properties of a manipulator bymeasuring the reaction moments at its base. In Proc. of the IEEE Int. Conf. on Robotics and Automation, volume 3, pages 1510–1516, 1989. K. Yamane and Y. Nakamura. Natural motion animation through constraining and deconstraining at will. IEEE Trans. on Visualization and Computer Graphics, 9(3):352– 360, 2003. J.W. Young, R.F. Chandler, and C.C. Snow. Anthropometric and mass distribution characteristics of the adult female. Technical Report FAA-AM-83-16, US Air Force, 1983.