**3. Results**

8 Biomechanics

 *)* Optimal? Symbolic

*f cost(tk )* *End Yes*

*No*

*Next iteration*

Equ. (4)

*Q* = *f*(*q*, *q*˙, *q*¨, *Fext*, *Mext*, *g*) (5) = *M*(*q*)*q*¨ + *G*(*q*, *q*˙, *Fext*, *Mext*, *g*) (6)

Cost function

*x mod,n(q(tk ))*

**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*¨

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

• *q* (42 × 1) is the vector of the human body joint coordinates, i.e. successively the three angular coordinates for each of the 13 members (*3 (translations for the first member LCOM position) + 13 (members)* × *3 (angular coordinates) = 42 components*); the three angular coordinates per member represent the spherical joint; let us note that three translations per joint have been introduced and locked in order to permit the joint force calculations

• *Fext* and *Mext* (42 × 1) are the three-dimensional components of the global external forces

dynamical model (Equation 5), a semi-direct dynamical model (Equation 6) :

• *q*˙ and *q*¨ (42 × 1) are the joint velocities and accelerations, respectively;

without interfering with the model [26, 27];

and torques applied to each of the body members;

equations

*x exp,n(tk )*

Mean distances between joints

> l i *q(tk*

accuracy.

where

*2.4.2. Dynamical analysis*

• *g* (1 × 3) is the gravity;

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


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

### **3.1. Kinematic analysis**

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

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

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 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.

sacrum (Fig. 8), i.e. from the pelvis member to the trunk member, and also at the right

−0.2 −0.1 0 0.1 0.2

Slow

**Figure 7.** Trajectory of the GCOM during the slow (green) and fast (blue) getting-up motions.

Slow

GCOM antero−posterior positions (m)

0

Fast Slow

1

Angle (°)

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> −30

**Figure 8.** Time evolution of the three consecutive angular coordinates *R*3, *R*<sup>1</sup> and *R*2, respectively, at the

On the basis of the reference frame defined in Fig. 3, the dynamical analysis provides the time evolution of the global joint torques (Fig. 10) and forces (Fig. 11), for the slow and fast

Time (s)

2

3

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> −1

Fast

Methodology for the Assessment of Joint Eff orts During Sit to Stand Movement 145

Time (s)

Slow

Fast

elbow (Fig. 9), i.e. from the upper arm to the lower arm.

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> −6

Time (s)

Fast

−20

sacrum: comparison of the slow (green) and fast (blue) getting-up motions.

−10

Angle (°)

0

10

−4 −2 0 2 4

**3.2. Dynamical analysis**

behaviors.

Angle (°)

GCOM vertical positions (m)

**Figure 5.** Fast STS: kinematic fit for four time instant *tk*; superimposition of the fitted model coordinates *xexp*,*n*(*tk*) (in blue) and the experimental model coordinates *xmod*,*n*(*q*(*tk*)) (in red).

**Figure 6.** Error analysis: time evolution of the global relative error between *xmod*,*n*(*q*(*tk*)) and *xexp*,*n*(*tk*), during the slow (green) and fast (blue) getting-up motions.

2. As an example, the model joint kinematics are compared during the slow a fast motions, for two joints : the consecutive angular coordinates *R*3, *R*<sup>1</sup> and *R*<sup>2</sup> are described at the sacrum (Fig. 8), i.e. from the pelvis member to the trunk member, and also at the right elbow (Fig. 9), i.e. from the upper arm to the lower arm.

**Figure 7.** Trajectory of the GCOM during the slow (green) and fast (blue) getting-up motions.

**Figure 8.** Time evolution of the three consecutive angular coordinates *R*3, *R*<sup>1</sup> and *R*2, respectively, at the sacrum: comparison of the slow (green) and fast (blue) getting-up motions.

### **3.2. Dynamical analysis**

10 Biomechanics

−0.5

−0.5

Relative error (%)

during the slow (green) and fast (blue) getting-up motions.

Time = 1.74 sec

0

*xexp*,*n*(*tk*) (in blue) and the experimental model coordinates *xmod*,*n*(*q*(*tk*)) (in red).

0.5

Time = 1.04 sec

−0.5 0

−0.5 0

0.5

0.5

0 0.5 1 1.5

0 0.5 1 1.5

0

0.5

−0.5

−0.5

Time = 2.09 sec

0

0.5

Time = 1.39 sec

−0.5 0

−0.5 0

0.5

0.5

Slow

0 0.5 1 1.5

**Figure 5.** Fast STS: kinematic fit for four time instant *tk*; superimposition of the fitted model coordinates

Fast

0 1 2 3 4 5

Time (s)

**Figure 6.** Error analysis: time evolution of the global relative error between *xmod*,*n*(*q*(*tk*)) and *xexp*,*n*(*tk*),

2. As an example, the model joint kinematics are compared during the slow a fast motions, for two joints : the consecutive angular coordinates *R*3, *R*<sup>1</sup> and *R*<sup>2</sup> are described at the

0 0.5 1 1.5

0

0.5

On the basis of the reference frame defined in Fig. 3, the dynamical analysis provides the time evolution of the global joint torques (Fig. 10) and forces (Fig. 11), for the slow and fast behaviors.

0 1 2 3 4 5

Right knee

0 1 2 3 4 5

Right head of malleolus

0 1 2 3 4 5

Time (s)

Sacrum

0 1 2 3 4 5

Right shoulder

0 1 2 3 4 5

Right elbow

0 1 2 3 4 5

Time (s)

joint, during the slow (green) and fast (blue) getting-up motions.

0 1 2 3 4 5

Left knee

0 1 2 3 4 5

Left head of malleolus

0 1 2 3 4 5

Time (s)

C7 vertebra

0 1 2 3 4 5

Left shoulder

0 1 2 3 4 5

Left elbow

0 1 2 3 4 5

Time (s)

Left greater trochanter

Methodology for the Assessment of Joint Eff orts During Sit to Stand Movement 147

0

0

0

0

0

2

Torque (Nm)

**Figure 10.** Time evolution of the global joint torques: superposition of the global joint torques at each

4

5

Torque (Nm)

10

Torque (Nm)

1

Torque (Nm)

2 x 10−6

50

100

200

Torque (Nm)

Torque (Nm)

400

Right greater trochanter

0

0

0

0

2

Torque (Nm)

4

5

Torque (Nm)

10

Torque (Nm)

2

Torque (Nm)

4 x 10−6

50

100

Torque (Nm)

Torque (Nm)

**Figure 9.** Time evolution of the three consecutive angular coordinates *R*3, *R*<sup>1</sup> and *R*2, respectively, at the elbow: comparison of the slow (green) and fast (blue) getting-up motions.

Furthermore, body segment animations have been developed in order to show the evolution of the joint positions, the corresponding global joint torques, and also the local and global centers of mass and pressure. Samples of this animation are presented in Fig. 12, at four time instants *tk* during the fast STS behavior.

### **4. Discussion and conclusion**

This section presents the benefits and limitations of this methodology, and also the perspectives for future studies.

### **4.1. Benefits and limitations**

The present inverse dynamical model of the human body coupled with a kinematic identification of the model configurations (Fig. 1) is proposed as an accurate method to estimate the joint efforts in dynamical contexts, as presented from Fig. 1. Nevertheless, three main limitations of the present inverse dynamical model must be discussed.

1. *The geometrical limitation, due to the use of spherical joints :* The results of the kinematic analysis for this experiment show that the spherical joints considered here sufficiently fit the considered motion, with *xmod*,*n*(*q*(*tk*)) errors corresponding to a mean absolute error inferior to 3.9mm in each direction at each joint. However, using previous investigation results, the present model will be extended to include more involved joints in the future, particularly for the knees [34] and the shoulders [35].

12 Biomechanics

100

150

Angle (°)

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> −200

**Figure 9.** Time evolution of the three consecutive angular coordinates *R*3, *R*<sup>1</sup> and *R*2, respectively, at the

Furthermore, body segment animations have been developed in order to show the evolution of the joint positions, the corresponding global joint torques, and also the local and global centers of mass and pressure. Samples of this animation are presented in Fig. 12, at four time

This section presents the benefits and limitations of this methodology, and also the

The present inverse dynamical model of the human body coupled with a kinematic identification of the model configurations (Fig. 1) is proposed as an accurate method to estimate the joint efforts in dynamical contexts, as presented from Fig. 1. Nevertheless, three

1. *The geometrical limitation, due to the use of spherical joints :* The results of the kinematic analysis for this experiment show that the spherical joints considered here sufficiently fit the considered motion, with *xmod*,*n*(*q*(*tk*)) errors corresponding to a mean absolute error inferior to 3.9mm in each direction at each joint. However, using previous investigation results, the present model will be extended to include more involved joints in the future,

main limitations of the present inverse dynamical model must be discussed.

particularly for the knees [34] and the shoulders [35].

Time (s)

Slow Fast

200

250

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>50</sup>

Time (s)

Fast

Slow

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> −50

Slow

Fast

Time (s)

Angle (°)

−150 −100 −50 0 50 100

elbow: comparison of the slow (green) and fast (blue) getting-up motions.

instants *tk* during the fast STS behavior.

**4. Discussion and conclusion**

perspectives for future studies.

**4.1. Benefits and limitations**

Angle (°)

**Figure 10.** Time evolution of the global joint torques: superposition of the global joint torques at each joint, during the slow (green) and fast (blue) getting-up motions.

−0.5

−0.5

Time = 1.75 sec

0

0.5

**Figure 12.** Samples of the fast STS, at four time instants *tk* : the fitted model (in blue), featuring the global torques (in black) at each model joint, and also the GCOM (red point) and the GCOP (green star).

accelerations *q*¨, and thus introduce errors in the estimation of the internal efforts.

3. *The dynamical limitation, due to the approximation of the body inertia parameters :* The body

inertia parameters, i.e. the masses *mi*, moments of inertia *Ii* and center of mass positions −−→*OMi* of the *<sup>i</sup>*

Consequently, the errors in the estimated internal efforts *Q* increase if the corresponding body member accelerations increase. This is the reason why the present model is only proposed for rather small dynamics, such as the STS experiment, walking experiments or other motions without significant impact. Further, let us remember that previous investigations [33] showed that the non-invasive body parameter identifications during the

*th* body member (i = 1,. . . ,13) are approximated, using inertia tables [31].

2. *The kinematic limitation, due to the rigid multibody system assumption :* Like other classical *dynamical inverse* analyzes [21–25] in biomechanics of motion, the proposed model is composed of linked rigid bodies. However, in reality, the body is not composed of a set of rigid bodies. Rather, each body member consists of a rigid part (bone), and a non-rigid part (skin, muscle, ligament, tendon, connective tissue, and other soft tissue structures) [38]: during any motion, the skeletal structures of the body experience accelerations, whereas the soft tissue motion is delayed, due to damped vibrations of the member. Consequently, the errors in the optimized joint coordinates *q* may introduce errors in the velocities *q*˙ and

Time = 1.05 sec

−0.5 0

−0.5 0

0.5

0.5

0 0.5 1 1.5

0 0.5 1 1.5

0

0.5

−0.5

−0.5

Time = 2.10 sec

0

0.5

Time = 1.40 sec

−0.5 0

Methodology for the Assessment of Joint Eff orts During Sit to Stand Movement 149

−0.5 0

0.5

0.5

0 0.5 1 1.5

0 0.5 1 1.5

0

0.5

**Figure 11.** Time evolution of the joint forces: superposition of the three components of joint forces at each joint, during the slow (green) and fast (blue) getting-up motions.

14 Biomechanics

Force (N)

Force (N)

0

0

0

0

0

0

20

Force (N)

**Figure 11.** Time evolution of the joint forces: superposition of the three components of joint forces at

40

50

100

50

Force (N)

Force (N)

100

200

Force (N)

400

1000

2000

500

1000

0 1 2 3 4 5

Left knee

0 1 2 3 4 5

Left head of malleolus

0 1 2 3 4 5

Time (s)

C7 vertebra

0 1 2 3 4 5

Left shoulder

0 1 2 3 4 5

Left elbow

0 1 2 3 4 5

Time (s)

Left greater trochanter

0 1 2 3 4 5

Right knee

0 1 2 3 4 5

Right head of malleolus

0 1 2 3 4 5

Time (s)

Sacrum

0 1 2 3 4 5

Right shoulder

0 1 2 3 4 5

Right elbow

0 1 2 3 4 5

Time (s)

each joint, during the slow (green) and fast (blue) getting-up motions.

Right greater trochanter

0

0

0

0

20

Force (N)

40

50

100

200

Force (N)

Force (N)

Force (N)

400

500

Force (N)

1000

Force (N)

**Figure 12.** Samples of the fast STS, at four time instants *tk* : the fitted model (in blue), featuring the global torques (in black) at each model joint, and also the GCOM (red point) and the GCOP (green star).


motions are presently inappropriate to the human body dynamics, because the resulting body parameters present large errors due to experimental errors in the input data, such as the body configuration, external force and torque measurements.

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29 - Jul 1.

### **4.2. Perspectives**

Finally, in the context of the hardness to perform efforts, the perspectives of this research is to quantify with a satisfying accuracy the main joint and muscle efforts of subjects in different dynamical contexts, and to apply the model to:

