**2. Material and methods**

First, this section summarizes the features of the proposed human body model and describes the corresponding experimental set-up and process. Second, a preliminary calculation defines the centers of mass and centers of pressure of the model, and also develops the relation between their local and global components: these variables are known as diagnostic tools in rehabilitation and physical ergonomics [28–30], and useful for the present model analysis. Third, the theoretical investigation will develop both kinematic and dynamical analyzes related to this model, and both analyzes will be applied to the STS.

### **2.1. Model features and hypotheses**

The proposed human body model is composed of 28 position sensors (Fig. 2), defining 13 rigid bodies: the head, both upper arms, both lower arms, the trunk, the pelvis, both thighs, both shanks, and both feet. Each of the 13 bodies is defined by three position sensors, in order to know the three-dimensional configuration of each body. Further, these bodies are linked by spherical joints corresponding to 12 anatomical landmarks (referring to [31]): the C7 vertebra, both shoulders (acromioclavicular joints), both elbow joint centers, the sacrum, both greater trochanters, both knee joint centers, both lateral heads of the malleolus. Consequently, the system is fully described by a total of *13 (bodies)* × *6 variables - 12* × *3 spherical joint constraints = 42 generalized coordinates*, representing the 42 degrees of freedom of the model. As shown in Fig. 1, the inverse dynamical model provides the column vector *Q* of joint forces and torques, using three sets of inputs:


2 Biomechanics

Usually, the functional strength and the disability level during STS are evaluated by calculating the total forces of hip and knee extensors [1] and the center of mass (COM) accelerations [19]. Nevertheless, it is known that determining with accuracy the kinematics (including the COM) and dynamics (including the joint forces and torques) in the human body is still a great challenge in biomechanical modeling [20]. Consequently, the aim of the present study consists in presenting a rigorous methodology for the non-invasive assessment of joint efforts and the associated kinematic variables during STS movement. This method is based on a three-dimensional dynamical inverse model of the human body. Like other classical *dynamical inverse* analyzes [21–25] in biomechanics of motion, the model proposed here [18] uses measurements of external interactions (forces **F***ext* and torques **M***ext*) between the body and its environment, and also measurements of the system configuration *xexp*. The corresponding joint coordinates *q* are numerically determined by a kinematic identification process, and the corresponding velocities *q*˙ and accelerations *q*¨ are presently estimated from the *q*, using a numerical derivative. Finally, the model provides the joint interactions with the

use of a symbolically generated recursive Newton-Euler formalism [26, 27].

efforts.

compared.

**2. Material and methods**

**2.1. Model features and hypotheses**

**Figure 1.** Principle of the inverse dynamical model: from the experiment to the vector *Q* of the joint

This model is applied to experiments of STS : the subject, initially seated, is asked to get up without moving the feet, and without arm or hand contact with the environment or with any part of the body. In this paper, both postural behaviors of slow and fast STS are analyzed and

First, this section summarizes the features of the proposed human body model and describes the corresponding experimental set-up and process. Second, a preliminary calculation defines the centers of mass and centers of pressure of the model, and also develops the relation between their local and global components: these variables are known as diagnostic tools in rehabilitation and physical ergonomics [28–30], and useful for the present model analysis. Third, the theoretical investigation will develop both kinematic and dynamical analyzes

The proposed human body model is composed of 28 position sensors (Fig. 2), defining 13 rigid bodies: the head, both upper arms, both lower arms, the trunk, the pelvis, both thighs, both shanks, and both feet. Each of the 13 bodies is defined by three position sensors, in order to

related to this model, and both analyzes will be applied to the STS.

3. The joint coordinates, velocities and accelerations.

**Figure 2.** Human body model, featuring the 28 optokinetic sensors that define the 13 articulated rigid bodies, each defined by three points, articulated via 12 spherical joints.

A few characteristics and assumptions must be formulated about these three sets of inputs:


using finite differences. Considering the joint kinematics, we are aware that more adequate joint models could be used: in particular, previous studies [34, 35] have developed more complex three-dimensional joints for the knee and the shoulder. The present model has been implemented with spherical joints but will be extended to include more involved joints in the future. Further, the results of the kinematic analysis for this experiment show that the spherical joints considered here sufficiently fit the considered motion (see Section 3.1).

presently performed in order to discuss the repeatability of the data and results for several

Methodology for the Assessment of Joint Eff orts During Sit to Stand Movement 139

At the beginning of each test, the subject is seated as shown in Fig. 3. Then the subject is asked

• the subject do not move feet, in order to obtain a good repeatability of the initial and final

• the subject has neither arm nor hand contact with the environment or the rest of the body.

Two behaviors of STS are analyzed and compared in this paper, in order to compare a "slow" and a "fast" STS. For both tests, the time evolution of the body motion permits the definition

1. The initial phase: the subject is seated, it is assumed that the subject is at an equilibrium state, i.e. the subject is only performing forces necessary to maintain his initial posture. 2. The transient phase, composed of two sub-phases: a first transient sub-phase when the subject begins to get up and the subject thighs are still in contact with the seat; a second

3. The final phase: the subject maintains his standing-up position; it is assumed that this is

This section defines the centers of mass and centers of pressure of the proposed model, and

The position of the *global center of mass* (GCOM) of the human body can be written as follows:

*<sup>i</sup>*=<sup>1</sup> *mi*

∑<sup>13</sup> *<sup>i</sup>*=<sup>1</sup> *mi*

(i=1,. . . ,13); the values of −−→*OMi* are estimated from the human body configuration and the

Remember that the integration of the platform force data provides more accurate values of the GCOM variations [37], which are used as diagnostic tools in rehabilitation and physical ergonomics. However, the GCOM calculated by this method is equal to the actual GCOM plus one undetermined constant value. Further, it was shown for instance that the differences

−−→*OMi*

*th* member (i=1,. . . ,13); the values of *mi* are estimated from the inertia

(1)

*th* body member

−−→*OM* <sup>=</sup> <sup>∑</sup><sup>13</sup>

• −−→*OMi* is the position vector of the *local center of mass* LCOM of the *<sup>i</sup>*

transient sub-phase, when the subject continues to get up without seat contact.

also develops the relation between their local and global components [29, 30].

the second equilibrium state of the subject during the test.

**2.3. Center of mass and center of pressure**

to get up from the seat. During the whole experiment, the observers check that:

subjects and several behaviors of STS.

body configurations;

of three phases:

*2.3.1. Centers of mass*

inertia tables of de Leva [31];

• *mi* is the mass of the *i*

tables.

where

### **2.2. Experimental set-up and procedure**

Let us consider the system reference frame [ ˆ*I*], located at a fixed point *O* on the laboratory floor (Fig. 3). In this reference frame, the motion measurement set-up consists of optokinetic sensors and six infra-red cameras (*Elite* <sup>−</sup> *BTSTM*), that estimate the coordinate vectors −→*OXexp*,*<sup>n</sup>* = [ˆ*I*] �*xexp*,*n*) of the joint reference points, i.e. of the optokinetic sensors. Further, the interaction measurement set-up consists of two force platforms at the feet contact and one force platform at the seat, for the determination of the horizontal and vertical interaction forces **F***ext* = [ ˆ*I*] �*Fext* and torques **M***ext* = [ ˆ*I*] �*Mext* between the body and these platforms. The three independent platforms are composed of four force sensors [36], designed by our laboratory, and located at the edges of these platforms. The device provides a total number of *3 platforms* × *4 force sensors* × *3 force components = 36 force components*. All data are sampled at 100 Hz, using an adaptive low-pass numerical filter (implemented by *Elite* <sup>−</sup> *BTSTM*).

**Figure 3.** Experimental set-up, related to the system reference frame [ˆ*I*], located at a fixed point *O* on the laboratory floor. −→*OXexp*,*<sup>n</sup>* represents the coordinate vectors of the optokinetic sensors.

The experiments were performed by one person related to our laboratory, who gave his informed consent to perform the experiments. Note that further experiments of STS are presently performed in order to discuss the repeatability of the data and results for several subjects and several behaviors of STS.

At the beginning of each test, the subject is seated as shown in Fig. 3. Then the subject is asked to get up from the seat. During the whole experiment, the observers check that:


Two behaviors of STS are analyzed and compared in this paper, in order to compare a "slow" and a "fast" STS. For both tests, the time evolution of the body motion permits the definition of three phases:


### **2.3. Center of mass and center of pressure**

This section defines the centers of mass and centers of pressure of the proposed model, and also develops the relation between their local and global components [29, 30].

### *2.3.1. Centers of mass*

The position of the *global center of mass* (GCOM) of the human body can be written as follows:

$$
\overrightarrow{OM} = \frac{\sum\_{i=1}^{13} m\_i \overrightarrow{OM}\_i}{\sum\_{i=1}^{13} m\_i} \tag{1}
$$

where

4 Biomechanics

Let us consider the system reference frame [ ˆ*I*], located at a fixed point *O* on the laboratory floor (Fig. 3). In this reference frame, the motion measurement set-up consists of optokinetic sensors and six infra-red cameras (*Elite* <sup>−</sup> *BTSTM*), that estimate the coordinate vectors −→*OXexp*,*<sup>n</sup>* = [ˆ*I*]

the interaction measurement set-up consists of two force platforms at the feet contact and one force platform at the seat, for the determination of the horizontal and vertical interaction

The three independent platforms are composed of four force sensors [36], designed by our laboratory, and located at the edges of these platforms. The device provides a total number of *3 platforms* × *4 force sensors* × *3 force components = 36 force components*. All data are sampled at 100 Hz, using an adaptive low-pass numerical filter (implemented by *Elite* <sup>−</sup> *BTSTM*).

**Figure 3.** Experimental set-up, related to the system reference frame [ˆ*I*], located at a fixed point *O* on the

The experiments were performed by one person related to our laboratory, who gave his informed consent to perform the experiments. Note that further experiments of STS are

laboratory floor. −→*OXexp*,*<sup>n</sup>* represents the coordinate vectors of the optokinetic sensors.

�*xexp*,*n*) of the joint reference points, i.e. of the optokinetic sensors. Further,

�*Mext* between the body and these platforms.

3.1).

forces **F***ext* = [ ˆ*I*]

**2.2. Experimental set-up and procedure**

�*Fext* and torques **M***ext* = [ ˆ*I*]

using finite differences. Considering the joint kinematics, we are aware that more adequate joint models could be used: in particular, previous studies [34, 35] have developed more complex three-dimensional joints for the knee and the shoulder. The present model has been implemented with spherical joints but will be extended to include more involved joints in the future. Further, the results of the kinematic analysis for this experiment show that the spherical joints considered here sufficiently fit the considered motion (see Section


Remember that the integration of the platform force data provides more accurate values of the GCOM variations [37], which are used as diagnostic tools in rehabilitation and physical ergonomics. However, the GCOM calculated by this method is equal to the actual GCOM plus one undetermined constant value. Further, it was shown for instance that the differences between the GCOM estimated by these two methods are less than 0.3% height in all 3 components for able bodied subjects [37]. Consequently, the upper definition of the GCOM is preferred to estimate the actual GCOM value of the present human body model.

### *2.3.2. Centers of pressure*

For each force platform, the *local center of pressure* (LCOP) components, related to the system referential point *O*, can be determined from the platform force data, using the following definition:

$$
\overrightarrow{OP}\_{\text{j}} = (X\_{P\_{\text{j}}}, Y\_{P\_{\text{j}}}, Z\_{P\_{\text{j}}}) = \left( -\frac{M\_{Y\_{\text{j}}}}{R\_{P\_{\text{j}z}}}, \frac{M\_{X\_{\text{j}}}}{R\_{P\_{\text{j}z}}}, H\_{\text{j}} \right) \tag{2}
$$

**2.4. Theoretical investigation**

positions *xexp*,*n*.

physiological reasons.

*2.4.1. Kinematic analysis*

optimization process;

time instant *tk*+1.

where

written at each time instant *tk* as follows:

provided by the experimental set-up.

variable size of the bodies would be irrelevant.

*fcost*(*tk*) =

• the index *n* = 1, . . . , 28 indicates the optokinetic sensor;

obtained from the *q*(*tk*), using the forward kinematic model;

28 ∑ *n*=1

The theoretical investigation of the model is developed in two steps :


Methodology for the Assessment of Joint Eff orts During Sit to Stand Movement 141


The joint coordinates *q* are numerically determined by an identification process that estimates the joint coordinates of the multibody model that best fit the experimental joint positions *xexp*,*n*. As proposed by Ref. [20], the optimization problem can be formulated as a nonlinear least-square problem applied for each body configuration, at each time instant *tk*, *k* = 1, . . . , *T*, where *T* is the last time sample of each test. Consequently, the cost function *fcost*(*tk*) can be

• *q*(*tk*) is the joint coordinate vector at the time instant *tk*, and is the variable of the

• *xmod*,*n*(*q*(*tk*)) is the cartesian coordinate of the *nth* optokinetic sensor at the time instant *tk*,

• *xexp*,*n*(*tk*) is the cartesian coordinate of the *nth* optokinetic sensor at the time instant *tk*,

Fig. 4 schematically outlines the identification process, which involves two consecutive steps:

member, using the experimental joint cartesian coordinates *xexp*,*n*(*tk*). The reason is that the approach is based on a multibody model, composed of rigid bodies, for which a

2. The model joint cartesian coordinates *xmod*,*<sup>n</sup>* are given by a forward kinematic model using the *li* distances and an initial value (set to zero) of the joint coordinates *q*(*tk*) that we want to determine. The cost function of this least-square optimization is defined as the sum of the square components of the absolute error vector between *xexp*,*n*(*tk*) and *xmod*,*n*(*q*(*tk*)) of the *n* optokinetic sensors at the time instant *tk*. In order to improve the process convergence, the optimal value of *xmod*,*n*(*q*(*tk*)) becomes the initial condition of the next iteration at the

1. A pre-process calculates the mean distances *li* between the joints for each of the *i*


<sup>2</sup> (4)

*t h* body

where


The *global center of pressure* (GCOP) [29, 30] is defined as the weighted sum of the LCOP on every contact platform. Its expression related to the system referential point *O* is given by :

$$
\overrightarrow{OP} = \frac{\sum\_{j=1}^{3} \ R\_{P\_j} \cdot \overrightarrow{OP}\_j}{\sum\_{j=1}^{3} \ R\_{P\_j}} \tag{3}
$$

where, for the platforms from j = 1 to 3 (i.e. j = 1 for the left foot platform, j = 2 for the right foot platform and j = 3 for the seat platform) :


Let us note that −→*OPi* and *RPj* are totally estimated from the platform force data. In particular, during the second part of the transient phase and the final phase, when there is no contact between the subject thighs and the seat, *RP*<sup>3</sup> <sup>=</sup> 0 and −→*OP* does not take into consideration *OP*3, which is undetermined from Equation (2).

Finally, both centers of mass and centers of pressure will be presented in the 'Results' Section, because these are useful in rehabilitation and physical ergonomics. However, only the LCOMs, estimated from the system configuration and the tables of inertia, are essential for the implementation of the musculoskeletal analysis presented in Fig. 1.

### **2.4. Theoretical investigation**

The theoretical investigation of the model is developed in two steps :



### *2.4.1. Kinematic analysis*

The joint coordinates *q* are numerically determined by an identification process that estimates the joint coordinates of the multibody model that best fit the experimental joint positions *xexp*,*n*. As proposed by Ref. [20], the optimization problem can be formulated as a nonlinear least-square problem applied for each body configuration, at each time instant *tk*, *k* = 1, . . . , *T*, where *T* is the last time sample of each test. Consequently, the cost function *fcost*(*tk*) can be written at each time instant *tk* as follows:

$$f\_{\rm cost}(t\_k) = \sum\_{n=1}^{28} |\mathbf{x}\_{mod,n}(q(t\_k)) - \mathbf{x}\_{\rm exp,n}(t\_k)|^2 \tag{4}$$

where

6 Biomechanics

between the GCOM estimated by these two methods are less than 0.3% height in all 3 components for able bodied subjects [37]. Consequently, the upper definition of the GCOM is

For each force platform, the *local center of pressure* (LCOP) components, related to the system referential point *O*, can be determined from the platform force data, using the following

) =

• the index j indicates the platform: j = 1, 2 or 3 for the left foot platform, the right foot

• *MXj* and *MYj* are anterior-posterior and lateral components, respectively, of the resulting

The *global center of pressure* (GCOP) [29, 30] is defined as the weighted sum of the LCOP on every contact platform. Its expression related to the system referential point *O* is given by :

*<sup>j</sup>*=<sup>1</sup> *RPj* ·

∑3 *<sup>j</sup>*=<sup>1</sup> *RPj*

where, for the platforms from j = 1 to 3 (i.e. j = 1 for the left foot platform, j = 2 for the right

*th* platform.

Let us note that −→*OPi* and *RPj* are totally estimated from the platform force data. In particular, during the second part of the transient phase and the final phase, when there is no contact between the subject thighs and the seat, *RP*<sup>3</sup> <sup>=</sup> 0 and −→*OP* does not take into consideration *OP*3,

Finally, both centers of mass and centers of pressure will be presented in the 'Results' Section, because these are useful in rehabilitation and physical ergonomics. However, only the LCOMs, estimated from the system configuration and the tables of inertia, are essential for

the implementation of the musculoskeletal analysis presented in Fig. 1.

−→*OPj*

*th* platform;

<sup>−</sup> *MYj RPj*,*<sup>z</sup>* , *MXj RPj*,*<sup>z</sup>* , *Hj*

*th* platform;

*th* platform; *Hj* is the assumed to be constant during the

(2)

(3)

preferred to estimate the actual GCOM value of the present human body model.

,*YPj* , *ZPj*

*th* platform, related to the reference *O*;

−→*OP* <sup>=</sup> <sup>∑</sup><sup>3</sup>

−→*OPj* = (*XPj*

platform or the seat platform, respectively; • *RPj*,*<sup>z</sup>* is the vertical component of the force on the *j*

*2.3.2. Centers of pressure*

moment on the *j*

experiment.

• *Hj* is the measured height of the *j*

foot platform and j = 3 for the seat platform) :

• −→*OPj* is the vector of position of the LCOP on the *<sup>j</sup>*

• the index j indicates the platform;

• *RPj* is the global force data on the *j*

which is undetermined from Equation (2).

definition:

where


Fig. 4 schematically outlines the identification process, which involves two consecutive steps:


• *M*(*q*) (42 × 42) is the positive-definite symmetric mass matrix;

**3. Results**

the external forces and torques and gravity applied to the system.

In this section, the model is applied to two behaviors of STS, as follows :

two behaviors of STS, defined as a *slow* and a *fast* motion, respectively.

slow and fast motions, respectively.

*mexfunction* from *MATLABTM* to *C*++.

**3.1. Kinematic analysis**

on Ref. [40], and a few samples are described in this section.

• *G*(*q*, *q*˙, *Fext*, *Mext*, *g*) (42 × 1) is the dynamical vector containing the gyroscopic, centrifuged and three-dimensional terms resulting from the system configurations, velocities, and also

Methodology for the Assessment of Joint Eff orts During Sit to Stand Movement 143

• At each time instant *tk*, the kinematic optimization problem provides the human body joint coordinates *q*(*tk*) that best fit the experimental joint positions *xexp*,*n*(*tk*). From these results, a error analysis of the fitted model and a short joint kinematic analysis are developed for

• The inverse dynamical model provides the vector *Q* of the joint forces and torques for the

Furthermore, segment animations have been developed in order to present the kinematics and dynamics results on the model in a convenient manner. These animations are available

In terms of CPU time performance, the kinematic identification process, using *MATLABTM* on a Pentium IV 530, 3 GHz processor, requires ca. 30 CPU seconds per 100 experimental samples, i.e. per second of studied motion. Further, the data reconstruction for the animation requires ca. 25 CPU seconds per second of studied motion. Consequently, the total optimization and display process requires ca. 55 CPU seconds per second of studied motion, i.e. in practice, this approximately requires 11 minutes for 10 seconds of motion data recording. Finally, let us note that the identification process time was reduced by 60% using a

At each time instant *tk*, the model joint cartesian coordinates *xmod*,*n*(*q*(*tk*)) of one behavior (here, the fast motion) can be recalculated in order to build the fitted model (blue in Fig. 5). This fitted model, using purely rigid bodies, can be compared to the purely experimental

Further, an error analysis provides the global relative errors between *xmod*,*n*(*q*(*tk*)) and *xexp*,*n*(*tk*) for the two behaviors of STS, in percentage of the corresponding *xexp*,*n*(*tk*) at each time instant *tk* (Fig. 6). For the fast motion (resp. the low motion), the maximal value of the global relative error is equal to 11.46% (resp. 8.27%) of the corresponding *xexp*,*n*(*tk*), and the mean value of the global relative error is equal to 0.31% (resp. 0.33%), corresponding to a mean absolute error equivalent to 3.8mm (resp. 3.9mm) in each direction at each joint. In both

model (red in Fig. 5), based on the experimental joint cartesian coordinates *xexp*,*n*(*tk*).

cases, the error peaks occur during the transient phase of the motion. Finally, selected results of joint kinematics are presented as follows :

1. The GCOM trajectories are presented (Fig. 7) during the slow a fast motions.

**Figure 4.** Identification process for a body configuration, at a time instant *tk*, as proposed by Ref. [20].

The corresponding velocities *q*˙ and accelerations *q*¨ are presently derived from the joint coordinates *q*(*tk*) and approximated by finite differences. The noise in *q*(*tk*) could be a significant source of error in the *q*˙ and *q*¨ estimations, and thus in the dynamical analysis. Consequently, an optimization of *q*˙ and *q*¨ will probably be suggested in the future. Nevertheless, the fact that the *xexp*,*n* are measured using an adaptive low-pass numerical filter and that the *q*(*tk*) are obtained using a kinematic optimization largely improves the *q*˙ and *q*¨ accuracy.

### *2.4.2. Dynamical analysis*

As proposed by Ref. [39], the system dynamical equations are obtained from a Newton-Euler formalism [26, 27]: this algorithm provides the vector *Q* of internal interaction torques and forces at the joints for any configuration of the multibody system, in the form of an inverse dynamical model (Equation 5), a semi-direct dynamical model (Equation 6) :

$$Q = f(\boldsymbol{q}\_{\prime}\boldsymbol{\dot{q}}\_{\prime}\boldsymbol{\ddot{q}}\_{\prime}\boldsymbol{F}\_{\mathrm{ext}\prime}\boldsymbol{M}\_{\mathrm{ext}\prime\prime}\boldsymbol{g})\tag{5}$$

$$\ddot{q} = M(q)\ddot{q} + G(q, \dot{q}, F\_{\text{ext}\prime} M\_{\text{ext}\prime} g) \tag{6}$$

where

