**3.1 Accelerometers as biosensors of knee joint motion**

Measurements of physical movement have long been done focused on gait analysis. Recently, various types of advanced measurement technology are used in the field of sports science. Among them, sensors and measurement systems thought to be applicable to measurements of lower extremity motion, including sensors for pendulums described in 3.2- 3.3, may be listed as follows.

a. Electrogoniometer (commonly called potentiometer)

This is a fixed rotation axis sensor that uses a rotating variable resistor. The rotation angle is detected as an electrical potential proportional to it. It has high reliability. On the other hand, it is unsuitable for measurement of high-speed movement, because large torque is required to drive contact points and they are abrasive.

b. Magnetic-type goniometer

24 Advanced Topics in Measurements

In normal subjects, *f*e, *f*γd, *f*γs have rather small values and *f*i has rather large value, so that the α-motoneuron does not fire and no reflex occurs. Consequently, the knee joint motion at pendulum test becomes a free oscillation. On the contrary, in subjects having injuries of central nervous system, more than one of *f*e, *f*γd, *f*γs have rather large values and/or *f*i has rather small value, so that the α-motoneuron fires and the stretch reflex occurs in the knee joint. Consequently, the knee joint motion is forced to disturb from free oscillation by the

The forced oscillation is classified into two types (William, 1998). One is a forced oscillation that is caused by stretch velocity component included in the afferent signal from the muscle spindle to the α-motoneuron. The value of the contractile force induced by such a component becomes maximum at the time when stretch velocity of the quadriceps femoris muscle reaches about maximum value. We call the reflex caused by such a component phasic reflex. The other is a forced oscillation that is caused by displacement component in the afferent signal from the muscle spindle. The contractile force in this case has maximum value at the time when the displacement of the muscle is about maximum. We call the reflex

static γ-motoneuron α-motoneuron

*fγs*

*fe fγd*

α-fiber

As mentioned above, two types of reflexes can occur in the reflex arc composed of spindle, GIa fiber, presynaptic inhibition, α-motoneuron, α fiber and muscle. Evaluation of these reflexes therefore requires consideration of not only the size of the reflex but also the timing of their generation. Naturally, therefore, measurements of knee joint motion used in

*fi*

presynaptic inhibition

tract. reticulospinalis

Gla fiber

muscle spindle

nuclear chain

intrafusal muscle fiber

nuclear bag intrafusal muscle fiber

contractile force. In the following, we call such an oscillation forced oscillation.

caused by such a component tonic reflex.

spinal cord

dynamic γ-motoneuron dynamic γ-fiber

tract. vestibulospinalis

static γ-fiber

extrafusal muscle fiber

analyzing these reflexes demand high accuracy.

Fig. 5. Reflex arc.

This is a fixed rotation sensor with multiple magnetic pole and magnetic elements arranged along its circumference. It detects an electrical potential proportional to the rotation angle. It has high reliability and high accuracy. It requires a little torque since it is a non-contact-type device, and has no wearable parts.

c. Distribution constant-type electrogoniometer ("flexible goniometer")

This sensor was developed for angle measurements of complex joints (Nicol, 1989). It is not affected by movement of the joint axes, with the basic axis and movement axis set on either end of a bar-shaped resistor that changes electrical resistance with changes in shape. The angle between the two axes is measured as change in the resistance value. It has both rather large non-linearity and hysteresis.

d. Marking point measurement (or motion capture system)

Many marking points are attached to the surface of the subject's body, and images are made while the subject is moving (Fong et al., 2011). The subject is completely unrestricted. The angles at multiple points can be measured simultaneously. Its application is limited to experimental use for reasons of large filming space requirement, low time resolution, large scale of the system, etc.

e. Accelerometer

This sensor detects the movement of an object along a single axis as an acceleration signal, using a built-in strain gauge or similar element set. It is applicable to detection of accelerations in a wide range of fields, and various types have been developed from perspectives such as model type, accuracy, and stability. It does not restrict the movement of subjects, because a sensor only needs to be attached to one side of a joint even for joint movement measurements. It can also measure angle and angular acceleration simultaneously.

f. Gyroscope

Ultra-small devices have been developed using the Coriolis force and piezoelectricity based on micro-electro-mechanical systems (MEMS) technology (Tong & Granat, 1999). Currently, however, stability and reliability remain problematic.

To summarize, the requirements for knee joint motion measurement systems suitable for the pendulum test in clinical practice include: (1) sufficient accuracy; (2) low susceptibility to effects from the motion of the knee joint axis; (3) no restriction of the knee joint when worn; (4) ability to be attached simply and stably; and (5) ability to obtain waveforms of angle,

Precise Measurement System

*L*<sup>1</sup> *θ*

acceleration

given as follows.

In addition, for the angular velocity

obtained from the following equation.

*L*2

accelerometer, influenced by the gravity acceleration.

A

accelerometer 1 (acceleration *α*1)

knee joint are obtained in equations (4) and (5), respectively.

accelerometer 2 (acceleration *α*2)

for Knee Joint Motion During the Pendulum Test Using Two Linear Accelerometers 27

For both equations (2) and (3), the first term on the right side is angular acceleration from the pendulum motion, and the second term is the sensing direction component of the

accelerometer 1

(a) (b)

Fig. 6. Rotary motion detection by two linear accelerometers. (a) Accelerometer-bar with two

When the first term on the right side disappears from both equations, the above-mentioned sensing direction component *g*sin*θ* of the acceleration due to gravity and the angle *θ* of the

When the second term on the right side disappears from equations (2) and (3), the angular

of equation (5) and temporal integration of values on the right side of equation (6) can be

From the above, according to the proposed method, waveforms for angle, angular velocity, and angular acceleration that are unaffected by the acceleration due to gravity and synchronized are obtained with the addition of a single differentiation or integration.

When evaluating the performance of the knee joint motion measurement system constructed in accordance with the principles described in 3.2, error can arise from the movement of the

of the pendulum motion unaffected by the acceleration due to gravity is

accelerometer 2

*θ*

bar

(4)

(5)

(6)

(7)

, the temporal differential of values on the right side

linear accelerometers; (b) Attachment of accelerometer-bar on the lower leg.

**3.3 Evaluation of the measurement system in the laboratory** 

**3.3.1 Generation of simple pendulum motion** 

angular velocity, and angular acceleration simply and with high accuracy. In the light of the above, the following conclusions may be reached with regard to the suitability of these sensors or measurement systems.

First, potentiometers are the most basic kind of angle sensor, and they have been used by Vodovnik et al. (1984), Lin & Rymer (1991), and others in studies of the pendulum test. However, when measuring knee joint angle using one potentiometer, accurate measurements cannot be made because of the axis of motion of the knee joint. Furthermore, it is not easy to attach and maintain the axis of the potentiometer in alignment with the rotation axis of the knee joint, and knee joint movement is restricted. Moreover, when seeking angular velocity and angular acceleration, one must depend on the differential, which is problematic in terms of accuracy. Magnetic goniometers perform well as angle meters, but they have the same problems as potentiometers with respect to the motion of the knee joint axis. Flexible goniometers have good properties with respect to the motion of the knee joint axis and ease of use, but the sensor itself has inadequate accuracy. Moreover, for the optical motion capture system that measures score, it is expected that the angle of knee joint motion (in some cases, angular velocity) will be detected faithfully with no contact mode, but the construction of the apparatus is too large for measurements of knee joint angle only with the body at rest, making it difficult to apply clinically. In recent years, many types of small and lightweight gyroscopes have been developed, and they have many features, such as ease of attachment, that make them suitable for measuring knee joint motion. However, stability and reliability are lacking in ultra-small types. In addition, the values detected are basically limited to angular velocity or one of the angles.

From the above, one can conclude that accelerometers fulfill nearly all of the preceding requirements, and, overall, they are the best option.

#### **3.2 Principle of the knee joint motion measurement system using two accelerometers**

We developed a method that can detect knee joint angle and angular acceleration simultaneously using two linear accelerometers in accordance with the conclusions stated in 3.1 (Kusuhara et al., 2011).

The fundamental configuration for the detection of knee joint pendulum motion is shown in Fig. 6(a). Accelerometers 1 and 2 are fixed on an accelerometer mounting bar separated by a certain distance (*L*1, *L*2) from point A on the rotation axis. The sensing direction of the accelerometer is the direction orthogonal to the bar on the paper. At this time, the direction of sensor attachment must be accurately fixed. However, attachment of the bar when measuring knee joint motion only needs to be fixed freely in a position within the plane of rotation of the knee joint and along the fibula as shown in Fig. 6(b). The lower leg is lifted until the bar reaches a certain angle *θ* (left on paper), and the pendulum motion is generated by letting the leg drop freely.

The outputs of accelerometers 1 and 2 with respect to this pendulum motion are taken as *α*<sup>1</sup> and *α*2, respectively. *α*1 and *α*2 are given as follows.

$$\alpha\_1 = L\_1 \ddot{\theta} + g \sin \theta \tag{2}$$

$$
\alpha\_2 = L\_2 \ddot{\theta} + g \sin \theta \tag{3}
$$

Here, *g* is the gravity acceleration.

angular velocity, and angular acceleration simply and with high accuracy. In the light of the above, the following conclusions may be reached with regard to the suitability of these

First, potentiometers are the most basic kind of angle sensor, and they have been used by Vodovnik et al. (1984), Lin & Rymer (1991), and others in studies of the pendulum test. However, when measuring knee joint angle using one potentiometer, accurate measurements cannot be made because of the axis of motion of the knee joint. Furthermore, it is not easy to attach and maintain the axis of the potentiometer in alignment with the rotation axis of the knee joint, and knee joint movement is restricted. Moreover, when seeking angular velocity and angular acceleration, one must depend on the differential, which is problematic in terms of accuracy. Magnetic goniometers perform well as angle meters, but they have the same problems as potentiometers with respect to the motion of the knee joint axis. Flexible goniometers have good properties with respect to the motion of the knee joint axis and ease of use, but the sensor itself has inadequate accuracy. Moreover, for the optical motion capture system that measures score, it is expected that the angle of knee joint motion (in some cases, angular velocity) will be detected faithfully with no contact mode, but the construction of the apparatus is too large for measurements of knee joint angle only with the body at rest, making it difficult to apply clinically. In recent years, many types of small and lightweight gyroscopes have been developed, and they have many features, such as ease of attachment, that make them suitable for measuring knee joint motion. However, stability and reliability are lacking in ultra-small types. In addition, the

values detected are basically limited to angular velocity or one of the angles.

requirements, and, overall, they are the best option.

and *α*2, respectively. *α*1 and *α*2 are given as follows.

3.1 (Kusuhara et al., 2011).

by letting the leg drop freely.

Here, *g* is the gravity acceleration.

From the above, one can conclude that accelerometers fulfill nearly all of the preceding

**3.2 Principle of the knee joint motion measurement system using two accelerometers**  We developed a method that can detect knee joint angle and angular acceleration simultaneously using two linear accelerometers in accordance with the conclusions stated in

The fundamental configuration for the detection of knee joint pendulum motion is shown in Fig. 6(a). Accelerometers 1 and 2 are fixed on an accelerometer mounting bar separated by a certain distance (*L*1, *L*2) from point A on the rotation axis. The sensing direction of the accelerometer is the direction orthogonal to the bar on the paper. At this time, the direction of sensor attachment must be accurately fixed. However, attachment of the bar when measuring knee joint motion only needs to be fixed freely in a position within the plane of rotation of the knee joint and along the fibula as shown in Fig. 6(b). The lower leg is lifted until the bar reaches a certain angle *θ* (left on paper), and the pendulum motion is generated

The outputs of accelerometers 1 and 2 with respect to this pendulum motion are taken as *α*<sup>1</sup>

(2)

(3)

sensors or measurement systems.

For both equations (2) and (3), the first term on the right side is angular acceleration from the pendulum motion, and the second term is the sensing direction component of the accelerometer, influenced by the gravity acceleration.

Fig. 6. Rotary motion detection by two linear accelerometers. (a) Accelerometer-bar with two linear accelerometers; (b) Attachment of accelerometer-bar on the lower leg.

When the first term on the right side disappears from both equations, the above-mentioned sensing direction component *g*sin*θ* of the acceleration due to gravity and the angle *θ* of the knee joint are obtained in equations (4) and (5), respectively.

$$g\sin\theta = \frac{L\_2\alpha\_1 - L\_1\alpha\_2}{L\_2 - L\_1} \tag{4}$$

$$\theta = \sin^{-1} \frac{L\_2 \alpha\_1 - L\_1 \alpha\_2}{g(L\_2 - L\_1)} \tag{5}$$

When the second term on the right side disappears from equations (2) and (3), the angular acceleration of the pendulum motion unaffected by the acceleration due to gravity is given as follows.

$$
\ddot{\theta} = \frac{\alpha\_1 - \alpha\_2}{L\_1 - L\_2} \tag{6}
$$

In addition, for the angular velocity , the temporal differential of values on the right side of equation (5) and temporal integration of values on the right side of equation (6) can be obtained from the following equation.

$$
\dot{\theta} = \frac{d\theta}{dt} = \int \ddot{\theta} \, dt \tag{7}
$$

From the above, according to the proposed method, waveforms for angle, angular velocity, and angular acceleration that are unaffected by the acceleration due to gravity and synchronized are obtained with the addition of a single differentiation or integration.

#### **3.3 Evaluation of the measurement system in the laboratory**

#### **3.3.1 Generation of simple pendulum motion**

When evaluating the performance of the knee joint motion measurement system constructed in accordance with the principles described in 3.2, error can arise from the movement of the

Precise Measurement System

was 0.992 deg.

−8

−8

−40

Fig. 10. Angle waveforms (*θ* and *θ*R).

−20

0

angle (deg)

20

40

−4

0

acceleration (m/s2

)

4

8

−4

0

acceleration (m/s2

)

4

8

for Knee Joint Motion During the Pendulum Test Using Two Linear Accelerometers 29

included in *α*1 and *α*2 are large enough that they cannot be ignored. In addition, as understood from the example in Fig. 10, the *θ* and *θ*R waveforms are in excellent agreement. Moreover, the component of acceleration due to gravity remaining in angle waveform *θ* is small enough to be indistinguishable from noise (see *error*θ). The correlation coefficient of *θ* and *θ*R for 10 periods, including the 2 periods shown in this figure, was 0.999, and RMSE

0 0.511.5 2

0 0.511.5 2

0 0.511.5 2

θ<sup>R</sup> *error*<sup>θ</sup>

Fig. 8. Output waveforms of linear accelerometers (*α*1 and *α*2).

Fig. 9. Gravity acceleration components (*g*sin*θ*) in *α*1 and *α*2.

θ

α<sup>1</sup> α<sup>2</sup>

time (s)

time (s)

time (s)

knee joint axis, imperfect attachment of the bar when evaluation is done, etc. This makes it difficult to accurately grasp the performance of the instrumentation body unit. In the following, therefore, we evaluate the instrumentation body unit by generating pendulum motion in a simulation.

The prototype performance evaluation system made for this purpose is shown in Fig. 7. The reference angle gauge is a high-accuracy, non-contact type, rotation angle gauge (CP-45H, Midori Precisions, Japan) used for comparison and evaluation of detector performance in the proposed method. An aluminum bar corresponds to the bar in Fig. 6(a), to which a weight is attached midway to make the period of the pendulum about the same as the lower leg. The fulcrum point A is set on the rotation axis of the reference angle gauge as in the figure for accurate comparison of the detection results. Accelerometers 1 and 2 (AS-2GA, Kyowa Electronic Instruments, Japan) are located in positions separated by only *L*1 (60 cm) and *L*2 (15 cm), respectively, from the rotation axis on the aluminum bar. Acceleration *α*1 and *α*2 detected by the accelerometers are input to a computer via matching amplifier and an A/D converter (PCD-300B, Kyowa Electronic Instruments). The output of the other rotation angle gauge is input to a computer via an A/D converter.

Fig. 7. Construction of performance evaluation system.

Here, the aluminum bar of the apparatus in the figure was moved as a rigid pendulum, and performance was evaluated from the results of simultaneous measurements of the pendulum motion with the detector of the proposed method and the reference angle gauge.

#### **3.3.2 Results of evaluation**

Pendulum motion was induced by freely dropping the aluminum bar after tilting it to about 40 deg. This pendulum motion had damped oscillation of a sinusoidal waveform with a period of 1.14 s, nearly the same as knee joint motion.

Output waveform examples of accelerometers *α*1 and *α*2 when the amplitude of damped oscillation is about 30 deg are shown in Fig. 8. *α*1 and *α*2 are in opposite phases because, with *α*1, acceleration from pendulum motion is greatly affected by acceleration due to gravity, whereas the opposite is true with *α*2.

The *g*sin*θ* waveform obtained from equation (4) and the angle *θ* waveform obtained from equation (5) using these waveforms are shown in Figs. 9 and 10, respectively. In Fig. 10, the reference angle gauge output *θ*R (broken line) and the *error*θ between *θ* and *θ*R (thin solid line) are added. From Figs. 8 and 9 it is seen that the values for the *g*sin*θ* component

knee joint axis, imperfect attachment of the bar when evaluation is done, etc. This makes it difficult to accurately grasp the performance of the instrumentation body unit. In the following, therefore, we evaluate the instrumentation body unit by generating pendulum

The prototype performance evaluation system made for this purpose is shown in Fig. 7. The reference angle gauge is a high-accuracy, non-contact type, rotation angle gauge (CP-45H, Midori Precisions, Japan) used for comparison and evaluation of detector performance in the proposed method. An aluminum bar corresponds to the bar in Fig. 6(a), to which a weight is attached midway to make the period of the pendulum about the same as the lower leg. The fulcrum point A is set on the rotation axis of the reference angle gauge as in the figure for accurate comparison of the detection results. Accelerometers 1 and 2 (AS-2GA, Kyowa Electronic Instruments, Japan) are located in positions separated by only *L*1 (60 cm) and *L*2 (15 cm), respectively, from the rotation axis on the aluminum bar. Acceleration *α*1 and *α*2 detected by the accelerometers are input to a computer via matching amplifier and an A/D converter (PCD-300B, Kyowa Electronic Instruments). The output of the other rotation

amplifier

goniometer

amplifier

Here, the aluminum bar of the apparatus in the figure was moved as a rigid pendulum, and performance was evaluated from the results of simultaneous measurements of the pendulum motion with the detector of the proposed method and the reference angle gauge.

Pendulum motion was induced by freely dropping the aluminum bar after tilting it to about 40 deg. This pendulum motion had damped oscillation of a sinusoidal waveform with a

Output waveform examples of accelerometers *α*1 and *α*2 when the amplitude of damped oscillation is about 30 deg are shown in Fig. 8. *α*1 and *α*2 are in opposite phases because, with *α*1, acceleration from pendulum motion is greatly affected by acceleration due to gravity,

The *g*sin*θ* waveform obtained from equation (4) and the angle *θ* waveform obtained from equation (5) using these waveforms are shown in Figs. 9 and 10, respectively. In Fig. 10, the reference angle gauge output *θ*R (broken line) and the *error*θ between *θ* and *θ*R (thin solid line) are added. From Figs. 8 and 9 it is seen that the values for the *g*sin*θ* component

A/D convertor

laptop computer

angle gauge is input to a computer via an A/D converter.

*L* (angle *θ*R) 2=15cm

A

*α*2

*α*1

*θ*

Fig. 7. Construction of performance evaluation system.

period of 1.14 s, nearly the same as knee joint motion.

*a*2

*a*1

*L*1=60cm

**3.3.2 Results of evaluation** 

whereas the opposite is true with *α*2.

motion in a simulation.

included in *α*1 and *α*2 are large enough that they cannot be ignored. In addition, as understood from the example in Fig. 10, the *θ* and *θ*R waveforms are in excellent agreement. Moreover, the component of acceleration due to gravity remaining in angle waveform *θ* is small enough to be indistinguishable from noise (see *error*θ). The correlation coefficient of *θ* and *θ*R for 10 periods, including the 2 periods shown in this figure, was 0.999, and RMSE was 0.992 deg.

Fig. 8. Output waveforms of linear accelerometers (*α*1 and *α*2).

Fig. 9. Gravity acceleration components (*g*sin*θ*) in *α*1 and *α*2.

Fig. 10. Angle waveforms (*θ* and *θ*R).

Precise Measurement System

effect on knee joint motion.

about 50 deg.

twice was also confirmed.

**3.4 Estimation for the accuracy of the proposed system** 

for Knee Joint Motion During the Pendulum Test Using Two Linear Accelerometers 31

had a similar correlation coefficient and RMSE of about 30% smaller. Attaching the aluminum bar to the subjects is easier than these sensors. The next subsection discusses the

For the principles given in 3.2, when using this knee joint motion measurement system created for the pendulum test, there is the problem of axis of motion mentioned in 2.1, in addition to the unique aspects of biological measurements, such as the state of attachment of the aluminum bar to the knee joint and slight changes of posture by the subject during the test. Therefore, one would predict that the decrease in accuracy due to these factors cannot be ignored. However, we have found no method that can directly and precisely evaluate the decrease in accuracy resulting from these factors. In this study, therefore, the following indirect method was used to evaluate the decrease in accuracy when measuring subjects. First, the subject sitting in a chair for measurement and the evaluation system used in 3.3 were arranged as shown in Fig. 13. The chair and subject seen on the plane of the paper are located just in front of the evaluation system. The height of the evaluation system from the floor and the left-right positions seen on the plane of the paper were adjusted so that the rotation axis of the evaluation system and the rotation axis of the knee were on about the same line. The aluminum bar was fixed to the lower leg with rubber bands as shown in the

figure so that the motion of the knee joint would be restricted as little as possible.

accelerometer 2

accelerometer 1

Fig. 13. Evaluation system for knee joint motion detector.

Next, the pendulum test was done by freely dropping the lower leg after it had been lifted

goniometer

rubber band

Angle waveforms obtained in this way are shown in Fig. 14 (a). *θ* and *θ*R are the angle measured with the present method and the angle measured with the reference angle gauge, respectively. It is clear from this figure that the agreement is so close that it is difficult to distinguish the two angles. To examine the angle detection accuracy with this method in greater detail, Fig. 15 shows a window display of one section of the waveform in Fig. 14 (a). The correlation coefficient in this part was 1.000, and RMSE was 0.672 deg. The RMSE value, compared with the value for simple pendulum motion (Fig. 10), was 1.84-fold, equivalent to 1.94% with respect to the maximum amplitude (34.7 deg) of *θ*R. Fig. 14 (b) shows the angular acceleration waveforms measured with this method. Good agreement between this waveform and the waveform when the reference angle gauge output was differentiated

The detection results for angular acceleration were as follows. The angular acceleration obtained by substituting acceleration waveforms *α*1 and *α*2 from Fig. 8 into equation (6), and the angular acceleration R obtained by twice differentiating *θ*R in Fig. 10, are shown in Fig. 11. The solid and broken lines are and R, respectively, and the thin solid line is the *error* θ of the two. Noise is superimposed in R obtained with a differential, but there is good agreement between the two. The correlation coefficient of and R and RMSE for 10 periods, including the 2 periods shown in Fig. 11, was 0.998 and 0.749 rad/s2, respectively.

Next, let us look at the angular velocity waveform. The angular velocity waveform 1 obtained by differentiating angle waveform (*θ*) in Fig. 10 (solid line) and the waveform 2 obtained by integrating angular acceleration waveform in Fig. 11 (broken line) are shown in Fig. 12. There is good agreement between the two, although this is due partly to the fact that these values were obtained in a simulation trial done in a laboratory with little noise.

Fig. 11. Angular acceleration waveforms ( and R).

Fig. 12. Angular velocity waveform ( 1 and 2).

The accuracy of the above-mentioned knee joint motion measurement system itself was obtained using two accelerometers of the same type purchased with no special conditions. This accuracy, when compared with the accuracy of detecting uniaxial arm motion with a gyroscope, goniometer, and potentiometer (correlation coefficients (0.9997-0.9999) and RMSE (1.37-1.47 deg (Furuse et al., 2005)) for similarity between these measured values), had a similar correlation coefficient and RMSE of about 30% smaller. Attaching the aluminum bar to the subjects is easier than these sensors. The next subsection discusses the effect on knee joint motion.

#### **3.4 Estimation for the accuracy of the proposed system**

30 Advanced Topics in Measurements

obtained by substituting acceleration waveforms *α*1 and *α*2 from Fig. 8 into equation (6), and

periods, including the 2 periods shown in Fig. 11, was 0.998 and 0.749 rad/s2, respectively. Next, let us look at the angular velocity waveform. The angular velocity waveform

obtained by differentiating angle waveform (*θ*) in Fig. 10 (solid line) and the waveform

0 0.511.5 2

 and 

0 0.511.5 2

 1 and 

2).

The accuracy of the above-mentioned knee joint motion measurement system itself was obtained using two accelerometers of the same type purchased with no special conditions. This accuracy, when compared with the accuracy of detecting uniaxial arm motion with a gyroscope, goniometer, and potentiometer (correlation coefficients (0.9997-0.9999) and RMSE (1.37-1.47 deg (Furuse et al., 2005)) for similarity between these measured values),

R).

θ<sup>1</sup>

・

θ<sup>2</sup>

・

in Fig. 12. There is good agreement between the two, although this is due partly to the fact that these values were obtained in a simulation trial done in a laboratory with little noise.

θ

‥

good agreement between the two. The correlation coefficient of

and

were as follows. The angular acceleration

R, respectively, and the thin solid line is the

 and 

R obtained with a differential, but there is

in Fig. 11 (broken line) are shown

R obtained by twice differentiating *θ*R in Fig. 10, are shown in Fig.

*error*<sup>θ</sup>

‥

1

2

R and RMSE for 10

time (s)

time (s)

The detection results for angular acceleration

of the two. Noise is superimposed in

obtained by integrating angular acceleration waveform

the angular acceleration

−30 −20

−6

Fig. 12. Angular velocity waveform (

−4

−2

0

angular velocity (rad/s)

2

4

6

−10

angular acceleration (rad/s2

)

0

θ<sup>R</sup>

Fig. 11. Angular acceleration waveforms (

‥

10 20

30

*error* θ

11. The solid and broken lines are

For the principles given in 3.2, when using this knee joint motion measurement system created for the pendulum test, there is the problem of axis of motion mentioned in 2.1, in addition to the unique aspects of biological measurements, such as the state of attachment of the aluminum bar to the knee joint and slight changes of posture by the subject during the test. Therefore, one would predict that the decrease in accuracy due to these factors cannot be ignored. However, we have found no method that can directly and precisely evaluate the decrease in accuracy resulting from these factors. In this study, therefore, the following indirect method was used to evaluate the decrease in accuracy when measuring subjects.

First, the subject sitting in a chair for measurement and the evaluation system used in 3.3 were arranged as shown in Fig. 13. The chair and subject seen on the plane of the paper are located just in front of the evaluation system. The height of the evaluation system from the floor and the left-right positions seen on the plane of the paper were adjusted so that the rotation axis of the evaluation system and the rotation axis of the knee were on about the same line. The aluminum bar was fixed to the lower leg with rubber bands as shown in the figure so that the motion of the knee joint would be restricted as little as possible.

Next, the pendulum test was done by freely dropping the lower leg after it had been lifted about 50 deg.

Fig. 13. Evaluation system for knee joint motion detector.

Angle waveforms obtained in this way are shown in Fig. 14 (a). *θ* and *θ*R are the angle measured with the present method and the angle measured with the reference angle gauge, respectively. It is clear from this figure that the agreement is so close that it is difficult to distinguish the two angles. To examine the angle detection accuracy with this method in greater detail, Fig. 15 shows a window display of one section of the waveform in Fig. 14 (a). The correlation coefficient in this part was 1.000, and RMSE was 0.672 deg. The RMSE value, compared with the value for simple pendulum motion (Fig. 10), was 1.84-fold, equivalent to 1.94% with respect to the maximum amplitude (34.7 deg) of *θ*R. Fig. 14 (b) shows the angular acceleration waveforms measured with this method. Good agreement between this waveform and the waveform when the reference angle gauge output was differentiated twice was also confirmed.

Precise Measurement System

instrumentation body unit.

the aluminum bar upward.

have high phasic reflex.

**4.1 Examples of waveforms measured** 

**measured** 

for Knee Joint Motion During the Pendulum Test Using Two Linear Accelerometers 33

The evaluation system used for application to the pendulum test is the same as the system used in the waveform measurements in Fig. 10. Therefore, the difference in RMSE for the waveforms in Fig. 10 and the waveforms in Fig. 15 is thought to have been produced by the difference in damped oscillation that is generated artificially and damped oscillation that occurs in the biological body, by whether or not there was positional displacement or distortion of the aluminum bar with shaking of the lower leg, or by the rotation axis movement described in 2.1. This result means that, when the fulcrum point of the aluminum bar is nearly matched to the knee joint rotation axis, both of RMSEs for the angles with this method and with the angle gauge worsen by only about 0.3 deg compared with the

Finally, let us briefly consider the decrease in accuracy with the addition of the aluminum bar. When the aluminum bar (85 g) and two accelerometers (15 g × 2) are added, the lower leg mass of an average normal subject (about 3 kg) increases by roughly 4%. However, in the accuracy evaluation results mentioned up to this point, the descriptions have shown that there is almost no effect. Even so, the effect on measured knee joint motion in subjects cannot be ignored. When the center of the gravity of the lower leg changes with the addition of the aluminum bar and accelerometers, the moment of inertia changes in proportion to the square of the distance to the rotation axis, affecting the period of the oscillation and the

bar and two accelerometers on the knee joint motion in an average normal subject is an increase of about 3% in the time for one period and a decrease of about 6% in the damping coefficient in the results of rough trial calculations. When more precise measurements are required, the increase in the moment of inertia can be suppressed, and the influence on the period and the damping coefficient can be decreased, by shifting the attachment position of

In this section, we will deal with spastic patients as the subject of the analysis. Such patients

The knee joint motion of a normal subject was measured in the pendulum test using the measurement system shown in Fig. 6 in section 3. Fig.16 shows examples of the waveforms measured. In the figure, (a) and (b) show the angle waveform and angular acceleration waveform, respectively. There is absolutely no restriction on motion of the lower leg, since only two accelerometers were attached to it. It is difficult to estimate the error in this measurement result quantitatively, but the measurement was probably made with about the same accuracy as obtained in the investigation in the preceding section. The collapse of the waveform that appears in the early stage of oscillation is noise produced by the state of contact between the hand of the investigator and the lower leg in the instant when the lifted leg was released. If this portion is eliminated, the angle waveform and angular acceleration waveform have typical damped oscillation with nearly the same periods, although the phases differ. These waveforms are the free oscillations mentioned in subsection 2.1. Both

**4. Analysis of knee joint motion using the simulator with waveforms** 

, the effect of the aluminum

damping coefficient in the pendulum motion. In both *θ* and

Fig. 14. Angle and angular acceleration waveforms measured from a normal subject by pendulum test. (a) Angle; (b) Angular acceleration.

Fig. 15. A window display of the angle waveforms in the Fig.14(a).

From the above, the accuracy of the knee joint motion measurement system based on the present method was assured to be comparable to that of a reference angle gauge when applied to the pendulum test.

This knee joint motion measurement system is also thought to have sufficiently high accuracy to be applicable to the pendulum test for the following reasons.

θ θ<sup>R</sup>

0 2 4 6 810 12

(a)

0 2 4 6 810 12

(b)

0 0.511.5 2

From the above, the accuracy of the knee joint motion measurement system based on the present method was assured to be comparable to that of a reference angle gauge when

This knee joint motion measurement system is also thought to have sufficiently high

<sup>θ</sup><sup>R</sup> *error*<sup>θ</sup>

Fig. 14. Angle and angular acceleration waveforms measured from a normal subject by

time (s)

time (s)

time (s)

−60 −40 −20

> −30 −20 −10

−40

applied to the pendulum test.

−20

0

angle (deg)

20

40

pendulum test. (a) Angle; (b) Angular acceleration.

θ

Fig. 15. A window display of the angle waveforms in the Fig.14(a).

accuracy to be applicable to the pendulum test for the following reasons.

0

angular acceleration (rad/s2

)

10 20 30

0

angle (deg)

20 40 60 The evaluation system used for application to the pendulum test is the same as the system used in the waveform measurements in Fig. 10. Therefore, the difference in RMSE for the waveforms in Fig. 10 and the waveforms in Fig. 15 is thought to have been produced by the difference in damped oscillation that is generated artificially and damped oscillation that occurs in the biological body, by whether or not there was positional displacement or distortion of the aluminum bar with shaking of the lower leg, or by the rotation axis movement described in 2.1. This result means that, when the fulcrum point of the aluminum bar is nearly matched to the knee joint rotation axis, both of RMSEs for the angles with this method and with the angle gauge worsen by only about 0.3 deg compared with the instrumentation body unit.

Finally, let us briefly consider the decrease in accuracy with the addition of the aluminum bar. When the aluminum bar (85 g) and two accelerometers (15 g × 2) are added, the lower leg mass of an average normal subject (about 3 kg) increases by roughly 4%. However, in the accuracy evaluation results mentioned up to this point, the descriptions have shown that there is almost no effect. Even so, the effect on measured knee joint motion in subjects cannot be ignored. When the center of the gravity of the lower leg changes with the addition of the aluminum bar and accelerometers, the moment of inertia changes in proportion to the square of the distance to the rotation axis, affecting the period of the oscillation and the damping coefficient in the pendulum motion. In both *θ* and , the effect of the aluminum bar and two accelerometers on the knee joint motion in an average normal subject is an increase of about 3% in the time for one period and a decrease of about 6% in the damping coefficient in the results of rough trial calculations. When more precise measurements are required, the increase in the moment of inertia can be suppressed, and the influence on the period and the damping coefficient can be decreased, by shifting the attachment position of the aluminum bar upward.
