kinematic sensors.pdf

7/28/2019 Kinematic Sensors.pdf

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Karol J ODonovan, Ph.D. Thesis University of Limerick, 2007

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Kinematic Sensors

1 The Accelerometer

As its name implies, an accelerometer measures acceleration. Due to the presence of

gravitational acceleration an accelerometer detects acceleration even when stationary and

actually measures zero acceleration when in freefall. Considering this an accelerometer may

be described as a sensor that measures deviations from freefall.

A uni-axial accelerometer measures the sum of the inertial acceleration component and

gravitational acceleration component acting along its measuring axis. Fig. 1 shows a uni-axial

accelerometer attached to a segment, with an inertial acceleration of Ia .

Fig. 1 Uni-axial accelerometer attached to a segment

Eq. (1) is a mathematical model which relates the total measured acceleration, xa , of a single

axis accelerometer to the accelerations acting on the device.


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)cos()cos( gaa Ix (1)

where

xa is the magnitude of the acceleration vector component acting along the measuring axis

of the accelerometer

Ia is the magnitude of the inertial acceleration of the segment

is the inclination of the measuring axis with respect to the inertial acceleration vector

g is the magnitude of the gravitational acceleration

is the inclination of the measuring axis with respect to the vertical axis (gravity vector)

A bi-axial accelerometer is formed by two right angled measuring axes. A 2-dimensional

acceleration vector is obtained from a bi-axial accelerometer. The acceleration measured by a

bi-axial accelerometer written in array format is given by Eq. (2).

yyI

xxI

y

x

ga

ga

a

aa

coscos

coscos

(2)

Fig. 2 Bi-axial accelerometer attached to a segment under static conditions


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Fig. 2 shows a bi-axial accelerometer attached to a segment, under static conditions. Under

static conditions there is no inertial acceleration component and the measured acceleration

vector for a bi-axial accelerometer is given by Eq. (3).

sin

cos

g

g

a

aa

y

x (3)

Under static conditions a bi-axial accelerometer can thus be used to determine the inclination

of a segment with respect to the gravity vector using Eq. (4).

x

y

a

a1

tan (4)

A tri-axial accelerometer is formed by three orthogonal uni-axial accelerometers. A 3-

dimensional acceleration vector is obtained from a tri-axial accelerometer. The acceleration

measured by a tri-axial accelerometer written in array format is given by Eq. (5).

zzI

yyI

xxI

z

y

x

ga

ga

ga

a

a

a

a

coscos

coscos

coscos

(5)

2 The Rate Gyroscope

A rate gyroscope measures angular velocity. A uni-axial rate gyroscope measures the angular

velocity acting along its measuring axis. Fig. 3 shows a uni-axial gyroscope attached to a

rotating plate, rotating with angular velocity .


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Fig. 3 Uni-axial rate gyroscope attached to a rotating plate

Eq. (6) is a mathematical model which relates the measured angular velocity of the single axis

rate gyroscope to the angular velocity acting on the device.

cosx (6)

where

x is the magnitude of the angular velocity vector component along the measuring axis of

the rate gyroscope

is the inclination of the measuring axis with respect to the angular velocity vector

is the magnitude of the angular velocity acting on the sensor

A tri-axial rate gyroscope is formed by three orthogonal uni-axial rate gyroscopes. A 3-

dimensional angular velocity vector is obtained from a tri-axial rate gyroscope. The angular

velocity measured by a tri-axial rate gyroscope written in array format is given by Eq. (7).

z

y

x

z

y

x

cos

cos

cos

(7)


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3 The Magnetometer

A magnetometer is a sensor which is used to measure the direction and/or the strength of a

magnetic field. A uni-axial magnetometer measures the magnetic field vector acting along its

measuring axis. Fig. 4 shows a uni-axial magnetometer attached to a body segment.

Fig. 4 Uni-axial magnetometer attached to a body segment

Eq. 8 is a mathematical model which relates the measured angular velocity of the single axis

rate gyroscope to the angular velocity acting on the device.

cosmmx (8)

where

xm is the magnitude of the magnetic field vector component along the measuring axis of

the magnetometer

is the inclination of the magnetometer measuring axis with respect to the magnetic

field vector


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A tri-axial rate magnetometer is formed by three orthogonal uni-axial magnetometers. A 3-

dimensional magnetic field vector is obtained from a tri-axial magnetometer. The magnetic

field measured by a tri-axial magnetometer written in array format is given by Eq. (9).

z

y

x

z

y

x

m

m

m

m

m

m

m

cos

cos

cos

(9)

4 The Kinematic Sensor Signal

A standard definition for the description of a uni-axial kinematic sensor outputy, as a function

of the vector componentx directed along the sensor sensitivity axis is given by Eq. (10).

bkxy (10)

where

k is the sensor scale factor

b is the sensor offset bias

Ideal three-axis kinematic sensors are composed of three mutually orthogonal uni-axial

sensors. Misalignment may occur due to sensor axes not being exactly mutually orthogonal

and also the fact that the actual sensitivity axis of each sensor may not match exactly the

assumed sensitivity axis when the sensor unit is placed in casing. The presence of

misalignment makes it necessary to describe the actual orientation of the sensitivity axis of

each sensor with respect to the assumed orientation of the sensitivity axis in the form of a

misalignment matrix R. A standard definition for the description of a tri-axial kinematic

sensor output vector y , as a function of the measured component vector x is given by Eq.

(11) [1].


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bxRKy (11)

where

z

y

x

y

y

y

y is the tri-axial sensor output vector

z

y

x

x

x

x

x is the tri-axial measured component vector

z

y

x

g

b

b

b

b is the sensor offset bias

z

y

x

k

k

k

K

00

00

00

is the diagonal matrix of the scale factors of the three sensor axes

zzyzxz

zyyyxy

zxyxxx

rrr

rrr

rrr

R

'''

'''

'''

is the misalignment matrix describing the actual sensitivity axis

with respect to the assumed sensitivity axis

References

[1] Ferraris F, Grimaldi U, and Parvis M. Procedure for effortless in-field calibration of

three-axis rate gyros and accelerometers. Sensors and Materials 1995; 7: 311-30.

kinematic sensors.pdf

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