Robust DSP integrator for accelerometer signals

• US 7,130,764 B2
• Filed: 04/05/2004
• Issued: 10/31/2006
• 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.
• Primary Examiner(s): Bui, Bryan
• Assistant Examiner(s): KUNDU, SUJOY K
• Application Number:
  US10/818,182
• Publication Number:
  US 20040243346A1
• Time in Patent Office: 939 Days
• Field of Search: 702/182, 702/150, 700/44
• 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 the function and at least one second-derivative value of the function, the system comprising:

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

means 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,

wherein the predicted value of the function is a position value and the second-derivative value of the function is an acceleration value and the function is expressed as;

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

wherein the terms expressed generally as c_p^(b) represent the p^th coefficient associated with a term of the b^th derivative, T is a sampling period, H equals the number of previous 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, m is a numerical value within a range of from 1 to M, x(t) is the predicted value of the function at time t, and x″

(t−

hT) represents a second derivative of the function taken with respect to time and evaluated at time t−

hT.

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

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

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

5. The system according to claim 1, wherein the coefficients c_p^(b) that will produce a minimal amount of error minimize a value of the integral:

$\frac{1}{2<br><br>T}<br><br>{\int <br><br>}_{-2<br><br>\mathrm{TW}}^{2<br><br>\mathrm{TW}}<br><br>{<br><br>1-\left(\underset{k=1}{\overset{M}{\sum <br><br>}}<br><br><br><br>c_k^{\left(0\right)}<br><br>{\mathrm{e<br><br>}}^{-\mathrm{i<br><br>}<br><br><br><br>\mathrm{\pi <br><br>}<br><br><br><br>\mathrm{ku}}+{\mathrm{\pi <br><br>}}^2<br><br>\underset{h=0}{\overset{H}{\sum <br><br>}}<br><br><br><br>c_h^{\left(2\right)}<br><br>\left(-u^2\right)<br><br>{\mathrm{e<br><br>}}^{-\mathrm{i<br><br>}<br><br><br><br>\mathrm{\pi <br><br>}<br><br><br><br>\mathrm{hu}}\right)<br><br>}^2<br><br>d<br><br>u$

wherein, W is an upper-limit frequency of the signal transmitted by the sensor, and k is an integer from 1 to M.

6. The system according to claim 5, wherein the coefficients are computed from a linear system of the form:

$\begin{array}{c}\left[\begin{array}{ccccc}s<br><br>\left(0\right)& s<br><br>\left(\mathrm{\tau <br><br>}\right)& s_2<br><br>\left(\mathrm{\tau <br><br>}\right)& s_2<br><br>\left(0\right)& s_2<br><br>\left(\mathrm{\tau <br><br>}\right)\\ s<br><br>\left(\mathrm{\tau <br><br>}\right)& s<br><br>\left(0\right)& s_2<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)& s_2<br><br>\left(\mathrm{\tau <br><br>}\right)& s_2<br><br>\left(0\right)\\ s_2<br><br>\left(\mathrm{\tau <br><br>}\right)& s_2<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(0\right)& s_4<br><br>\left(\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)\\ s_2<br><br>\left(0\right)& s_2<br><br>\left(\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(\mathrm{\tau <br><br>}\right)\\ s_2<br><br>\left(\mathrm{\tau <br><br>}\right)& s_2<br><br>\left(0\right)& s_4<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(0\right)\end{array}\right]<br><br>\left[\begin{array}{c}{c}_1^{\left(0\right)}\\ c_2^{\left(0\right)}\\ c_0^{\left(2\right)}\\ c_1^{\left(2\right)}\\ c_2^{\left(2\right)}\end{array}\right]=\left[\begin{array}{c}s<br><br>\left(\mathrm{\tau <br><br>}\right)\\ s<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)\\ s_2<br><br>\left(0\right)\\ s_2<br><br>\left(\mathrm{\tau <br><br>}\right)\\ s_2<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)\end{array}\right]\\ \mathrm{wherein}\\ s<br><br>\left(\mathrm{\tau <br><br>}\right)=\sin <br><br><br><br>c<br><br>\left(\mathrm{\tau <br><br>}\right)\\ s_2<br><br>\left(\mathrm{\tau <br><br>}\right)=-{\mathrm{\tau <br><br>}}^2<br><br>s^{\mathrm{″<br><br>}}<br><br>\left(\mathrm{\tau <br><br>}\right)\\ s_4<br><br>\left(\mathrm{\tau <br><br>}\right)=-{\mathrm{\tau <br><br>}}^4<br><br>s^{\left(4\right)}<br><br>\left(\mathrm{\tau <br><br>}\right)\end{array}$

further wherein τ

=2TW, and the parenthetical superscript represents a derivative number.

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

8. The system according to claim 7, wherein the means for computing the current value of the function comprises a digital-signal processor controlled by computer-readable instructions adapted to compute the predicted value of the function.

9. The system according to claim 7, wherein the function is expressed as:

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

wherein the terms expressed generally as c_p^(b) represent the p^th coefficient associated with a term of the b^th derivative, T is a sampling period, H=the number of previous second-derivative values sensed, M is a number of previous function values, n is a discrete digital time operator, m is an integer within a range of from 1 to M, x[n] is the predicted discrete value of the digital-time-domain function at n, and x″

[n−

h] represents a second derivative of the function taken with respect to n and evaluated at time t−

hT.

10. 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 x″

(t−

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

sensing one or more second-derivative values x″

(t−

hT) of the function and transmitting a signal representing the sensed second-derivative values x″

(t−

hT), wherein each second-derivative value x″

(t−

hT) is obtained at a unique time;

storing each of the second-derivative values x″

(t−

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

mT) in a computer-readable memory;

selecting an upper-limit frequency W; and

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

$\frac{1}{2<br><br>T}<br><br>{\int <br><br>}_{-2<br><br>\mathrm{TW}}^{2<br><br>\mathrm{TW}}<br><br>{<br><br>1-\left(\underset{m=1}{\overset{M}{\sum <br><br>}}<br><br><br><br>c_k^{\left(0\right)}<br><br>{\mathrm{e<br><br>}}^{-\mathrm{i<br><br>}<br><br><br><br>\mathrm{\pi <br><br>}<br><br><br><br>\mathrm{mu}}+{\mathrm{\pi <br><br>}}^2<br><br>\underset{n=h}{\overset{H}{\sum <br><br>}}<br><br><br><br>c_n^{\left(2\right)}<br><br>\left(-u^2\right)<br><br><br><br>{\mathrm{e<br><br>}}^{-\mathrm{i<br><br>}<br><br><br><br>\mathrm{\pi <br><br>}<br><br><br><br>\mathrm{hu}}\right)<br><br>}^2<br><br>d<br><br>u$

wherein the function is expressed as;

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

wherein the 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.

11. The method according to claim 10, wherein the solving for coefficients step comprises solving a linear system of equations of the form:

$\begin{array}{c}\left[\begin{array}{ccccc}s<br><br>\left(0\right)& s<br><br>\left(\mathrm{\tau <br><br>}\right)& s_2<br><br>\left(\mathrm{\tau <br><br>}\right)& s_2<br><br>\left(0\right)& s_2<br><br>\left(\mathrm{\tau <br><br>}\right)\\ s<br><br>\left(\mathrm{\tau <br><br>}\right)& s<br><br>\left(0\right)& s_2<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)& s_2<br><br>\left(\mathrm{\tau <br><br>}\right)& s_2<br><br>\left(0\right)\\ s_2<br><br>\left(\mathrm{\tau <br><br>}\right)& s_2<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(0\right)& s_4<br><br>\left(\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)\\ s_2<br><br>\left(0\right)& s_2<br><br>\left(\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(\mathrm{\tau <br><br>}\right)\\ s_2<br><br>\left(\mathrm{\tau <br><br>}\right)& s_2<br><br>\left(0\right)& s_4<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)& s_4<br><br>\left(0\right)\end{array}\right]<br><br>\left[\begin{array}{c}{c}_1^{\left(0\right)}\\ c_2^{\left(0\right)}\\ c_0^{\left(2\right)}\\ c_1^{\left(2\right)}\\ c_2^{\left(2\right)}\end{array}\right]=\left[\begin{array}{c}s<br><br>\left(\mathrm{\tau <br><br>}\right)\\ s<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)\\ s_2<br><br>\left(0\right)\\ s_2<br><br>\left(\mathrm{\tau <br><br>}\right)\\ s_2<br><br>\left(2<br><br>\mathrm{\tau <br><br>}\right)\end{array}\right]\\ \mathrm{wherein}\\ s<br><br>\left(\mathrm{\tau <br><br>}\right)=\sin <br><br><br><br>c<br><br>\left(\mathrm{\tau <br><br>}\right)\\ s_2<br><br>\left(\mathrm{\tau <br><br>}\right)=-{\mathrm{\tau <br><br>}}^2<br><br>s^{\mathrm{″<br><br>}}<br><br>\left(\mathrm{\tau <br><br>}\right)\\ s_4<br><br>\left(\mathrm{\tau <br><br>}\right)=-{\mathrm{\tau <br><br>}}^4<br><br>s^{\left(4\right)}<br><br>\left(\mathrm{\tau <br><br>}\right)\\ \mathrm{\tau <br><br>}=2<br><br>\mathrm{TW}\end{array}$

and the parenthetical superscript represents a derivative.

12. The method according to claim 10, wherein the predicted function value x(t) is a position value and the second-derivative value x″

(t−

hT) of the function is an acceleration value.

13. The method according to claim 10, wherein the step of obtaining an upper-limit frequency W comprises the steps of:

Filtering the signal representing the sensed second-derivative values x″

(t−

hT) with a filter; and

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

14. The method according to claim 13, wherein the step of filtering the signal comprises the step of filtering the signal with at least one of a low-pass filter, a band-pass filter, a high pass filter.

15. The method according to claim 10, wherein the step of sensing the second-derivative values x″

(t−

hT) comprises the steps of;

periodically sensing the second-derivative values x″

(t−

hT) at regular intervals to establish a generally uniform sampling period T between sensing successive second-derivative values x″

(t=hT); and

sensing the second-derivative values x″

(t−

hT) at a suitable frequency to minimize aliasing.

16. 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 x″

[n−

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

sensing one or more second-derivative values x″

[n−

h] of the function and transmitting a signal representing the sensed second-derivative values x″

[n−

h], wherein each second-derivative value x″

[n−

h] is obtained at a unique time;

storing each of the second-derivative values x″

[n−

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

m] in a computer-readable memory;

selecting an upper-limit frequency W; and

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

$\frac{1}{2<br><br>T}<br><br>{\int <br><br>}_{-2<br><br>\mathrm{TW}}^{2<br><br>\mathrm{TW}}<br><br>{<br><br>1-\left(\underset{m=1}{\overset{M}{\sum <br><br>}}<br><br><br><br>c_m^{\left(0\right)}<br><br>{\mathrm{e<br><br>}}^{-\mathrm{i<br><br>}<br><br><br><br>\mathrm{\pi <br><br>}<br><br><br><br>\mathrm{mu}}+{\mathrm{\pi <br><br>}}^2<br><br>\underset{n=h}{\overset{H}{\sum <br><br>}}<br><br><br><br>c_h^{\left(2\right)}<br><br>\left(-u^2\right)<br><br><br><br>{\mathrm{e<br><br>}}^{-\mathrm{i<br><br>}<br><br><br><br>\mathrm{\pi <br><br>}<br><br><br><br>\mathrm{hu}}\right)<br><br>}^2<br><br>d<br><br>u$

wherein the function is expressed as;

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

wherein the 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.

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

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

means 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,

wherein the predicted value of the function is a position value and the second-derivative value of the function is an acceleration value and the function is expressed as;

$x<br><br>\left(t\right)=\underset{m=1}{\overset{M}{\sum <br><br>}}<br><br>c_m^{\left(0\right)}<br><br>x<br><br>\left(t-\mathrm{mT}\right)+T^2<br><br>\underset{h=0}{\overset{H}{\sum <br><br>}}<br><br>c_h^{\left(2\right)}<br><br>x^n<br><br>\left(t-\mathrm{hT}\right)$

wherein the terms expressed generally as c_p^(b) represent the p^th coefficient associated with a term of the b^th derivative, T is a sampling period, H equals the number of previous 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, m is a numerical value within a range of from 1 to M, x(t) is the predicted value of the function at time t, and x″

(t−

hT) represents a second derivative of the function taken with respect to time and evaluated at time t−

hT,

wherein the signal transmitted by the sensor is converted to a digital signal and the function is expressed as;

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

wherein the terms expressed generally as c_p^(b) represent the p^th coefficient associated with a term of the b^th derivative, T is a sampling period, H equals the number of previous second-derivative values sensed, M is a number of previous function values, n is a discrete digital time operator, m is an integer within a range of from 1 to M, x[n] is the predicted discrete value of the digital-time-domain function at n, and x″

[n−

h] represents a second derivative of the function taken with respect to n and evaluated at time t−

hT.