**3.1 Gyroscopes and accelerometers**

A wide range of sensors can detect or measure angular movement. These devices range from the conventional mechanical gyroscopes and optical gyroscopes, to the ones based on atomic spin.

An accelerometer measures the physical acceleration it experiences relative to freefall, not to the coordinate systems. There are many types of accelerometers. Generally, all these devices are amenable for strap-down applications. However, the accuracy ranges widely from micro-g to fractions of g due to the variety of designs. Acceleration can be internally measured without using any external references, which are needed for the measurement of velocity. This is why it is preferable to develop algorithms based on acceleration for LPS.

MEMS technology, used to form structures with dimensions in the micrometer scale, is now being employed in manufacturing state-of-the-art MEMS-based inertial sensors.

### **3.2 Performance evaluation**

Practically, the measuring errors of accelerometer and rate gyroscope result in errors during positioning. Assuming the average output from an accelerometer when it is not undergoing any movement is *b*a, the accumulated positioning error after double integration over time period *t* is *e*s(*t*)= *b*a*t*2/2. This means that the positioning error caused by a constant bias of accelerometer increases quadratically through double integration over time. Such an error is called integral drift.

Similarly, assuming the offset of the output angular velocity from true value is *b*g, the error of a gyroscope over time period *t*, we have *e*g(*t*)= *b*g*t*. This linearly increasing angular error, i.e. orientation error, can cause the error of the rotation matrix **R**(*t*). The resulting incorrect projection of acceleration signals onto the global axes causes two problems. Firstly, the acceleration is integrated in the incorrect direction, causing interaction of velocity and position between (not along) axes. Secondly, acceleration due to gravity can not be removed

With the DCM updated, it becomes possible to project the acceleration signal **a**b(*t*) from the

After the local acceleration is obtained, the velocity and the displacement of the object can be

As mentioned above, inertial positioning relies on the measurement of accelerations and angular velocities. The estimate of changes in rotated angles can be obtained through integrating the angular velocity from a gyroscope over time. The estimate of changes in velocities and positions in the l-frame can be obtained through integrating the local acceleration over time once and twice respectively. The local acceleration is the projection of

A wide range of sensors can detect or measure angular movement. These devices range from the conventional mechanical gyroscopes and optical gyroscopes, to the ones based on

An accelerometer measures the physical acceleration it experiences relative to freefall, not to the coordinate systems. There are many types of accelerometers. Generally, all these devices are amenable for strap-down applications. However, the accuracy ranges widely from micro-g to fractions of g due to the variety of designs. Acceleration can be internally measured without using any external references, which are needed for the measurement of velocity. This is why it is preferable to develop algorithms based on acceleration

MEMS technology, used to form structures with dimensions in the micrometer scale, is now

Practically, the measuring errors of accelerometer and rate gyroscope result in errors during positioning. Assuming the average output from an accelerometer when it is not undergoing any movement is *b*a, the accumulated positioning error after double integration over time period *t* is *e*s(*t*)= *b*a*t*2/2. This means that the positioning error caused by a constant bias of accelerometer increases quadratically through double integration over time. Such an error is

Similarly, assuming the offset of the output angular velocity from true value is *b*g, the error of a gyroscope over time period *t*, we have *e*g(*t*)= *b*g*t*. This linearly increasing angular error, i.e. orientation error, can cause the error of the rotation matrix **R**(*t*). The resulting incorrect projection of acceleration signals onto the global axes causes two problems. Firstly, the acceleration is integrated in the incorrect direction, causing interaction of velocity and position between (not along) axes. Secondly, acceleration due to gravity can not be removed

being employed in manufacturing state-of-the-art MEMS-based inertial sensors.

(*t*) in the l-frame:

1 b **a Ra** () () () *ttt* (8)

accelerometers in the b-frame into acceleration **a**<sup>l</sup>

**3. Inertial sensors** 

atomic spin.

for LPS.

**3.2 Performance evaluation** 

called integral drift.

**3.1 Gyroscopes and accelerometers** 

determined according to (1) and (2) respectively to locate the object.

body acceleration, measured from accelerometers, onto the l-frame.

completely. The residue acceleration originated from gravity will become a "bias" to the true acceleration due to movement of the object.

In the strap-down navigation algorithm, the acceleration due to gravity, **g**, is deducted from the globally vertical acceleration signal before integration. When there are angular errors, there are tilt errors. Here a tilt error refers to the angle between the estimated vertical direction and true vertical direction. The tilt error *e* in radian will cause the projection of gravity onto the horizontal axes, resulting in a component of the acceleration due to gravity with magnitude: *e*a h=g×sin*e*. This component can be treated as a residual bias due to gravity, remaining in the globally horizontal acceleration signals. In the mean time, in globally vertical axis, there is a residual bias of magnitude: *e*<sup>a</sup> v=g×(1-cos*e*). Fortunately this problem is much less severe because for small *e*, we have *e*<sup>a</sup> <sup>v</sup>≈0. Therefore, a small tilt error will mainly cause positioning error in the globally horizontal plane.

In some cases, Such as human walk, the mean absolute acceleration measured is much smaller than the magnitude of gravity. As a contrast, a tilt error of 0.05° can cause a component of the acceleration due to gravity with magnitude near g/1000. This residue bias can cause a positioning error of 15.4 meters after only a minute of integration, or error of 0.49 meter after only 10 seconds (as demonstrated by the red dashed line in Fig. 3). Therefore, gyroscope errors, which propagate in the positioning algorithm, are critical errors affecting the accuracy of pedestrian tracking. Before the development of the algorithm presented in this chapter, it was believed that positioning with data from inertial sensors was not possible due to the quadratic growth of errors caused by sensor drift during double integration.
