Initializing an inertial sensor using soft constraints and penalty functions
View Patent ↗An initialization method for an inertial sensor that estimates starting orientation and velocity without requiring the sensor to start at rest or in a well-known location or orientation. Initialization uses patterns of motion encoded as a set of soft constraints that are expected to hold approximately during an initialization period. Penalty metrics are defined to measure the deviation of calculated motion trajectories from the soft constraints. Differential equations of motion for an inertial sensor are solved with the initial conditions as variables; the initial conditions that minimize the penalty metrics are used as estimates for the actual initial conditions of the sensor. Soft constraints and penalty metrics for a specific application are chosen based on the types of motion patterns expected for this application. Illustrative cases include applications with relatively little movement during initialization, and applications with approximately periodic motion during initialization.
1. A method for determining an estimate of the initial orientation Q 0 and initial velocity v 0 of an inertial sensor, comprising the steps:
collecting sample data from an inertial sensor over a capture time interval [t i , t f ] that includes an activity of interest, said sample data including accelerometer data s(t) and rate gyro data w(t);
selecting an initialization interval [t 0 , t n ]⊂[t i , t f ] within said capture time interval;
defining a set of soft constraints {C i } on the position r(t) and velocity v(t) of said inertial sensor during said initialization interval, where each C i ⊂{r:[t 0 , t n ]→ 3 }×{v: [t 0 , t n ]→ 3 }, where said soft constraints represent approximations of typical patterns of position and velocity trajectories for said activity of interest during said initialization interval;
defining penalty metrics m j (r(t), v(t)) to quantify the deviation of calculated trajectories of said position r(t) and said velocity v(t) from said soft constraints {C i }; and
calculating estimates of the initial orientation Q 0 =Q(t 0 ) and initial velocity v 0 =v(t 0 ) of said inertial sensor at the start of said initialization interval to minimize said penalty metrics when said initial orientation and initial velocity are used as initial conditions to integrate said sample data into orientation, position and velocity over said initialization interval via the coupled differential equations
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2. The method of claim 1 , further comprising:
including among said soft constraints a requirement that the change in position between the start and end of said initialization interval be zero, r(t n )=r(t 0 );
setting said estimate of initial velocity to zero, v(t 0 )=0; and
including among said penalty metrics the absolute value of the difference in position between the end of said initialization interval and the start of said initialization interval, m(r(t),v(t))=|r(t n )−r(t 0 )|.
3. The method of claim 1 , further comprising:
including among soft constraints a requirement that the position and velocity of said inertial sensor be periodic during said initialization interval;
calculating frequency domain signals for each axis of the rate gyro by applying a discrete Fourier transform to the corresponding axis of the rate gyro data time series during said initialization interval,
Ω x ( f )=DFT( w x ( t )),Ω y ( f )=DFT( w y ( t )),Ω z ( f )=DFT( w z ( t );
selecting the dominant frequency f max as the frequency with the largest transform magnitude across the 3 axes, f max =argmax(max(|Ω x (f)|, |Ω y (f)|, |Ω z (f)|)), and calculating the dominant period T from said dominant frequency as T=1/f max ;
including among said penalty metrics a velocity penalty metric that is the absolute value of the change in velocity over one period of motion, m v (r(t), v(t))=|v(t+T)−v(t)|, and a position penalty metric that is the absolute value of the change in position over one period of motion, m r (r(t), v(t))=|r(t+T)−r(t)|;
selecting a designated period within said initialization interval, [t 1 , t 1 +T]⊂[t 0 , t n ], for calculation of said estimates of initial velocity and initial orientation;
calculating said estimate of initial orientation by minimizing said velocity penalty metric over said designated period; and
calculating said estimate of initial velocity by minimizing said position penalty metric over said designated period, using said estimate of initial orientation.