Robust DSP integrator for accelerometer signals

• US 20040243346A1
• Filed: 04/05/2004
• Published: 12/02/2004
• Est. Priority Date: 04/05/2003
• Status: Active Grant

• Current Assignee: University of Akron
• Original Assignee: University of Akron
• Inventors: Wu, Yan, Mugler, Dale H.
• Granted Patent: US 7,130,764 B2
• Time in Patent Office: Days
• US Class Current: 702/182
• CPC Class Codes: G06F 17/10 Complex mathematical operations function generation by table look-up G06F1/03; evaluation of elementary functions by calculation G06F7/544

Abstract

A system and method for determining a predicted value of a function from at least one previous value of said function and at least one second-derivative value of the function. The system includes a sensor provided to sense a second-derivative value of the function and transmit an oscillatory signal representing the sensed second-derivative value, a computer-readable memory in communication with the sensor to store the previous value of the function and the second-derivative value, and a feature for computing the predicted value of the function using an integration algorithm including a coefficient of integration that is dependant upon at least a frequency of the signal.

Claims

1. A system for determining a predicted value of a function from at least one previous value of said function and at least one second-derivative value of said function, the system comprising:

a sensor provided to sense a second-derivative value of said function and transmit an oscillatory signal representing said sensed second-derivative value;

a computer-readable memory in communication with said sensor to store said previous value of said function and said second-derivative value; and

means for computing said predicted value of said function using an integration algorithm including a coefficient of integration that is dependant upon at least a frequency of said signal.

2. The system according to claim 1, wherein said coefficient of integration is determined from at least said frequency of said signal and a sampling period of said sensor.

3. The system according to claim 1 further comprising a filter to remove a portion of said signal that exists at a predetermined frequency range.

4. The system according to claim 3, wherein said frequency of said signal used to determine said coefficient of integration is a cutoff frequency of said filter.

5. The system according to claim 1, wherein said function is expressed as:

6. The system according to claim 5, wherein said coefficients c_p^(b)that will produce a minimal amount of error minimize a value of the integral:

7. The system according to claim 6, wherein said coefficients are computed from a linear system of the form:

8. The system according to claim 1, wherein said signal transmitted by said sensor is converted to a digital signal.

9. The system according to claim 8, wherein said means for computing said current value of said function comprises a digital-signal processor controlled by computer-readable instructions adapted to compute said predicted value of said function.

10. The system according to claim 8, wherein said function is expressed as:

11. The system according to claim 1, wherein said predicted value of said function is a position value and said second-derivative value of said function is an acceleration value.

12. A method for calculating a predicted function value x(t) from at least one previous function value x(t−

mT) and at least one second-derivative value xⁿ(t−

hT) of said function, the method comprising the steps of;

sensing one or more second-derivative values xⁿ(t−

hT) of said function and transmitting a signal representing said sensed second-derivative values xⁿ(t−

hT), wherein each second-derivative value xⁿ(t−

hT) is obtained at a unique time;

storing each of said second-derivative values xⁿ(t−

hT) and said function values x(t), x(t−

mT) in a computer-readable memory;

selecting an upper-limit frequency W;

solving for integration coefficients to be used in said function by minimizing the integral;

$\frac{1}{2<br><br>T}<br><br>{\int <br><br>}_{-2<br><br>\mathrm{TW}}^{2<br><br>\mathrm{TW}}<br><br>{<br><br>1-\left(\underset{m=1}{\overset{M}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>c_k^{\left(0\right)}<br><br>{\mathrm{<br><br>}}^{-\mathrm{<br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\pi <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{mu}}+{\mathrm{\pi <br><br>}}^2<br><br>\underset{n=h}{\overset{H}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>c_n^{\left(2\right)}<br><br>\left(-u^2\right)<br><br>\text{\hspace{1em}<br><br>}<br><br>{\mathrm{<br><br>}}^{-\mathrm{<br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\pi <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{hu}}\right)<br><br>}^2<br><br><br><br>u$

wherein said function is expressed as;

$x<br><br>\left(t\right)=\underset{m=1}{\overset{M}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>c_m^{\left(0\right)}<br><br>x<br><br>\left(t-\mathrm{mT}\right)+T^2<br><br>\underset{h=0}{\overset{H}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>c_n^{\left(2\right)}<br><br>x^{\mathrm{″<br><br>}}<br><br>\left(t-\mathrm{hT}\right)$

wherein said integration coefficients c_p^(b)modify the p^th term of the b^th derivative, further wherein;

t is a time value;

T is a sampling period;

H is a number of second-derivative values sensed;

M is a number of previous function values;

h is a numerical value within a range of from 0 to H; and

m is a numerical value within a range from 1 to M.

13. The method according to claim 12, wherein said solving for coefficients step comprises solving a linear system of equations of the form:

14. The method according to claim 12, wherein said predicted function value x(t) is a position value and said second-derivative value xⁿ(t−

hT) of said function is an acceleration value.

15. The method according to claim 12, wherein said step of obtaining an upper-limit frequency W comprises the steps of:

filtering said signal representing said sensed second-derivative values xⁿ(t−

hT) with a filter; and

assigning a cutoff frequency of said filter as said upper-limit frequency W.

16. The method according to claim 15, wherein said step of filtering said signal comprises the step of filtering said signal with at least one of a low-pass filter, a band-pass filter, a high pass filter.

17. The method according to claim 12, wherein said step of sensing said second-derivative values xⁿ(t−

hT) comprises the steps of;

periodically sensing said second-derivative values xⁿ(t−

hT) at regular intervals to establish a generally uniform sampling period T between sensing successive second-derivative values xⁿ(t−

hT); and

sensing said second-derivative values xⁿ(t−

hT) at a suitable frequency to minimize aliasing.

18. A method for calculating a predicted function value x[n] from at least one previous function value x[n−

m] and at least one second-derivative value xⁿ[n−

h] of said function, the method comprising the steps of;

sensing one or more second-derivative values xⁿ[n−

h] of said function and transmitting a signal representing said sensed second-derivative values xⁿ[n−

h], wherein each second-derivative value xⁿ[n−

h] is obtained at a unique time;

storing each of said second-derivative values xⁿ[n−

h] and said function values x[n], x[n−

m] in a computer-readable memory;

selecting an upper-limit frequency W;

solving for integration coefficients to be used in said function by minimizing the integral;

$\frac{1}{2<br><br>T}<br><br>{\int <br><br>}_{-2<br><br>\mathrm{TW}}^{2<br><br>\mathrm{TW}}<br><br>{<br><br>1-\left(\underset{m=1}{\overset{M}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>c_m^{\left(0\right)}<br><br>{\mathrm{<br><br>}}^{-\mathrm{<br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\pi <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{mu}}+{\mathrm{\pi <br><br>}}^2<br><br>\underset{n=h}{\overset{H}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>c_h^{\left(2\right)}<br><br>\left(-u^2\right)<br><br>\text{\hspace{1em}<br><br>}<br><br>{\mathrm{<br><br>}}^{-\mathrm{<br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\pi <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{hu}}\right)<br><br>}^2<br><br><br><br>u$

wherein said function is expressed as;

$x<br><br>\left[n\right]=\underset{m=1}{\overset{M}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>c_m^{\left(0\right)}<br><br>x<br><br>\left[n-m\right]+T^2<br><br>\underset{h=0}{\overset{H}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>c_h^{\left(2\right)}<br><br>x^{\mathrm{″<br><br>}}<br><br>\left[n-h\right]$

wherein said integration coefficients c_p^(b) modify the p^th term of the b^th derivative, further wherein;

n is a discrete digital time;

T is a sampling period;

H is a number of second-derivative values sensed;

M is a number of previous function values;

h is an integer within a range of from 0 to H; and

m is an integer within a range from 1 to M.