Altimeter with calibration

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First Claim
1. A dead reckoning altimeter, comprising:
 a computer system comprising;
a processor;
an estimator stored on a memory coupled with said processor configured to receive an input representing a forward speed, a yaw angle rate, a forward acceleration and external altitude information, said estimator configured to continuously estimate a continuous error in said acceleration due to an accelerometer bias without requiring an initial estimate of said accelerometer bias; and
an output converter stored on said memory coupled with said processor and with said estimator configured to use said forward speed, said yaw angle rate, said forward acceleration and said continuous estimated accelerometer bias to calculate using a hidden operation, a realtime change in altitude.
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Abstract
An altitude dead reckoning system using measurements of forward speed, a yaw angle rate and external altitude information in order to estimate an error in the acceleration due to an accelerometer bias; and an output converter configured to use the forward speed, yaw angel rate, acceleration and estimated accelerometer bias to calculate a change in altitude.
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20 Claims
 1. A dead reckoning altimeter, comprising:
a computer system comprising; a processor; an estimator stored on a memory coupled with said processor configured to receive an input representing a forward speed, a yaw angle rate, a forward acceleration and external altitude information, said estimator configured to continuously estimate a continuous error in said acceleration due to an accelerometer bias without requiring an initial estimate of said accelerometer bias; and an output converter stored on said memory coupled with said processor and with said estimator configured to use said forward speed, said yaw angle rate, said forward acceleration and said continuous estimated accelerometer bias to calculate using a hidden operation, a realtime change in altitude.  View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10)
 11. A computer implemented method for determining change in altitude, compromising:
receiving an input comprising a forward speed, a yaw angle rate, a forward acceleration and external altitude information at a memory for continuously estimating a continuous error in said acceleration due to an accelerometer bias without using an initial estimate of said accelerometer bias; and using said forward speed, said yaw angle rate, said forward acceleration and said estimated accelerometer bias for determining using a hidden operation of a processor, a realtime change in altitude.  View Dependent Claims (12, 13, 14, 15, 16, 17, 18, 19, 20)
1 Specification
This application is a continuation in part of pending application Ser. No. 12/157,519 by Loomis filed Jun. 11, 2008 now U.S. Pat. No. 7,856,336 assigned to the same assignee.
1. Field of the Invention
The present disclosure relates generally to dead reckoning systems and more particularly to a dead reckoning altimeter apparatus using a speedometer and a forwardlooking accelerometer for measuring changes in altitude. The present disclosure also relates particularly to an inclinometer apparatus using a speedometer and a forwardlooking accelerometer for measuring inclination angle.
2. Description of the Background Art
Dead reckoning (DR) is the process of estimating one'"'"'s current position based upon a previously determined position and advancing that position based upon measured speed, direction and/or acceleration. The DR begins with an initial known position, or fix. The fix can be determined using ranging, triangulation or map matching. It is common to use radio signals for ranging from the global navigation satellite system (GNSS) for establishing an initial position fix from which to start dead reckoning.
Dead reckoning speed can be measured by many methods. Before modern instrumentation, DR speed was determined aboard ship by throwing a wood float, called a log, overboard and counting the knots on a line tied to the float that passed a sailor'"'"'s hand in a sandglass measured time as the ship moved forward through the water. More modern ships use engine rpm, automatic logs for measuring water speed, or bottom looking Doppler sonar. Road vehicles typically measure speed by measuring revolution rates of their wheels. Road vehicles can also use engine rpm and Doppler sonar or radar for speed measurement. The horizontal direction can be measured with a magnetic or flux gate compass. Dead reckoning direction can also be determined by integrating the rate of change of angles sensed by an angular rate sensor. An angular rate sensor is sometimes referred to as a gyro. Inertial systems that integrate directional linear accelerations can be used for dead reckoning, especially for aircraft.
Even with the advancement of the convenience and accuracy of the global navigation satellite system (GNSS), there continues to be a need for dead reckoning for cases when continuous GNSS fixes cannot be obtained or are noisy. Further, global navigation satellite system positioning tends to be less accurate and noisier for altitude and vertical heading angles than for horizontal positions and horizontal heading angles.
The present disclosure describes an apparatus and method for measuring changes in altitude by measuring forward motion. The present disclosure also describes an apparatus and method for determining incline angle by measuring forward motion.
One embodiment is a dead reckoning altimeter comprising an estimator configured to use a forward speed, a yaw angle rate, external altitude information and a forward acceleration in order to estimate an error in the acceleration due to an accelerometer bias without a requirement for an initial estimate of the accelerometer bias; and an output converter configured to use the forward speed, the yaw angle rate, the forward acceleration and the estimated accelerometer bias to calculate a change in altitude. In another embodiment the altimeter is configured to estimate a position offset of an accelerometer without a requirement for an initial estimate of the position offset and use the estimated position offset in a calculation of the change in altitude. In another embodiment the altimeter is configured to estimate a model altitude for the external altitude information and use the estimated model altitude to estimate one or both of the accelerometer bias and the position offset.
Another embodiment is method for determining a change in altitude having steps of using a forward speed, a yaw angle rate, external altitude information and a forward acceleration for estimating an error in the acceleration due to an accelerometer bias without a requirement for an initial estimate of the accelerometer bias and using the forward speed, the yaw angle rate, the forward acceleration and the estimated accelerometer bias for determining a change in altitude. In another embodiment the method has a step for estimating a position offset of an accelerometer without a requirement for an initial estimate, and using the estimated position offset for calculating a change in altitude. In another embodiment the method has a step for estimating a model altitude for the external altitude information and using the estimated model altitude for estimating one or both of the accelerometer bias and the position offset.
These and other embodiments and attributes of the present invention will no doubt become obvious to those of ordinary skill in the art after reading the following detailed descriptions and viewing the various drawings.
Various embodiments for carrying out the ideas of the invention will now be presented. These embodiments should be regarded as exemplary. It should be understood that it is not necessary to employ all of the details of the embodiments in order to carry out the idea of the invention. After reading the disclosure, various supersets, subsets and equivalents will no doubt become apparent to those skilled in the art. However, neither these embodiments nor these supersets, subsets and equivalents should not be regarded as limiting the idea of the invention.
The apparatus 10,12 includes a speedometer 32 and a forwardlooking linear accelerometer 34. The vehicle 14 has a turn radius line R (
The DR altimeter apparatus 10 includes a DR altitude calculator 40 (
The DR altitude calculator 40 uses the forward speed v(t) and the compensated forward acceleration α_{c}(t) for calculating an altitude change ΔH. The incline angle calculator 42 includes a speed rate calculator 55 using the speed v(t) to determine a rate of change of speed Δv/Δt versus time and then uses the speed rate Δv/Δt with the compensated forward acceleration α_{c}(t) for calculating the incline angle θ.
The acceleration compensator 50 also includes a returntoposition compensation detector 72, a restart bias detector 74, and an accelerometer bias compensator 76.
The compensation detector 72 tracks altitude changes ΔH'"'"'s between triggers, and determines an installation accelerometer bias β_{1 }for the sum of the altitude changes ΔH'"'"'s between the triggers to be zero, or for a return altitude H_{R }to be equal to a start altitude H_{1}. In order to eliminate the effect of parking place pitch angle, the vehicle 14 may be parked facing opposite directions for determinations of altitudes H_{R }and H_{1}. The accelerometer bias β_{1 }is determined that results in the zero sum altitude changes ΔH'"'"'s. The trigger can be automatic (preferably after a manual enablement) when the apparatus 10,12 senses that the horizontal position has returned, or manual when an operator knows he has returned to the same position. In order to determine the combination of the accelerometer bias β_{1}, the linear position offset L and the yaw alignment angle γ, the vehicle 14 is driven to loop back to the start position in a least once in a clockwise loop and at least once in a counterclockwise loop. These operations are illustrated in the flow charts of
The restart bias detector 74 uses the two ideas that the speed v(t) is still very nearly zero at the start of motion of the vehicle 14 and the incline angle Θ at the start of motion is the very nearly the same as the last incline angle Θ_{L }when the vehicle 14 was last in motion before stopping. Accordingly, a restart accelerometer bias β_{o }is estimated with an equation 1. In the equation 1 the acceleration α is preferably taken after acceleration compensation for yaw angle rate ω(t). The g is the constant acceleration due to gravity.
β_{0}=α−g sin θ_{L} 1)
The accelerometer bias β may change rapidly when the apparatus 10,12 is warming up after being in an unpowered state. There may be a long warm up period before the accelerometer bias β is stable enough to be accurately calibrated. This problem can be mitigated according to the equation 1 by determining the restart accelerometer bias β_{0 }using an assumption that the incline angle θ_{L }immediately at start of motion has not changed from the incline angle θ_{L }that was last calculated for the last motion before stopping and assuming that the speed v(t) immediate at the start of motion is zero.
The accelerometer bias compensator 76 uses the installation bias β_{1 }and/or the restart bias β_{o }and/or the bias β determined by comparisons with external navigation information for compensating the measured acceleration α_{m}(t). A Kalman filter 80 (
A small vertical misalignment angle of the accelerometer 34 acts with gravity g to change the measurement of acceleration α_{m}(t) by a nearconstant gravity bias term of g×sin(vertical misalignment angle). This gravity bias term effectively adds (or subtracts) with the accelerometer bias β that is determined and compensated.
The speedometer 32, the accelerometer 34, and the yaw rate sensor 52 provide speed v(t), measured acceleration α_{m}(t), and yaw angle rate ω(t) to the Kalman filter 80. The Kalman filter 80 filters the differences between the noisy and/or discontinuous external altitude fixes H_{EXT}'"'"'s at its input and its dead reckoning altitudes H'"'"'s at its output, where the dead reckoning altitude H is determined by accumulating altitude changes ΔH'"'"'s. The filtered differences are used in feedback loops to provide a calibrated version of the accelerometer bias β. The Kalman filter 80 may be stored on a tangible medium as computerreadable instructions for directing a computer device, such as the apparatus 10,12 to carry out the instructions.
The filter 80 uses whatever information is available that might include, but is not limited to, accelerometer bias β, altitude changes ΔH'"'"'s, speed v(t) from the speedometer 32 (or distance ΔS from the speedometer 32 from which the speed v(t) can be computed from ΔS/Δt), the measured forward acceleration α_{m}(t) (or a partially compensated acceleration or a fully compensated acceleration α_{c}(t)), and positioning navigation information from the external sources 82 including but not limited to the external altitude H_{EXT}. The Kalman filter 80 uses this information for computing three dimensional position including altitude H, three dimensional velocity, and three dimensional heading including incline angle θ.
The Kalman filter 80 uses several navigation inputs of varying continuity and accuracy for providing continuously updated best estimates for 3D heading, 3D position and 3D velocity. The navigation inputs may include, but are not limited to, barometric pressure, GNSS satellite pseudoranges and Dopplers from the GNSS receiver 84 map matching for latitude, longitude and external altitude H_{EXT}, map matching for heading, the yaw rate sensor 52 that may be a gyro for measuring the yaw angle rate ω(t), the speed v(t) or distance S measurements from the speedometer 32, and the forward acceleration α_{m}(t) measurements by the accelerometer 34. The GNSS receiver 84 can be a Global Positioning System (GPS) receiver.
Internal or hidden operation of the Kalman filter 80 provides accelerometer bias β calibration that is used for compensating the measured acceleration α_{m}(t), and correcting and/or smoothing the heading, position and velocity outputs. The Kalman filter 80 operates in a similar manner to the Kalman filter described in U.S. Pat. No. 5,416,712 by Geier et al. for a “position and velocity estimation system for adaptive weighting of GPS and dead reckoning information”, the teachings of which are incorporated by reference in this application. Further understanding the filtering technology of the Kalman filter 80 is provided by Elliot Kaplan and Christopher Hegarty in “Understanding GPS: principles and applications”, 2nd edition, published by Artech House, Inc. of Norwood, Mass., copyright 2006; ISBN 1580538940. Chapter 9.3 on sensor integration in land vehicles, written by Geier et al., is especially instructive.
An altitude change ΔH is calculated in a step 116 from the speed v(t) and the compensated acceleration α_{c}(t). In a step 118 external positioning information such as positions, pseudoranges, altitudes, Dopplers and headings are received from external positioning sources 82. In a step 122 the altitude changes ΔH are accumulated to provide a DR altitude H. For continuous operation the DR altimeter 10 accumulates a sequence of altitude changes ΔH'"'"'s to the last previous DR altitude H for providing a continuous sequence of DR altitudes H'"'"'s. In a step 124 the DR altitude H is filtered based on the external positioning information with Kalman filtering techniques for calculating three dimensional position including the altitude H, three dimensional velocity and three dimensional heading including the incline angle Θ. In a step 126 the accelerometer bias β is recalculated with Kalman filtering techniques using the DR altitude H and the external positioning information then applied for updating the accelerometer bias β used for providing the compensated acceleration α_{c}(t). The steps 124 and 126 are normally performed together with states of the Kalman filtering technique.
The rate of change of speed Δv/Δt is calculated in a step 132 from the speed v(t) and time. In a step 134 the incline angle Θ is calculated from the compensated acceleration α_{c}(t) and the rate of change of speed Δv/Δt using an equation 4 below.
The apparatus 10,12 is powered down in a dormant or turned off state for an arbitrary period of time. In a step 154 the apparatus 10,12 is turned on and starts motion. In a step 155 the apparatus 10,12 operates for measuring the acceleration α_{m}(t) and compensating the measured acceleration α_{m}(t) for yaw angle rate ω(t) in order to provide the acceleration α in the equation 1 above. In a step 156 the restart accelerometer bias β_{o }is calculated in the equation 1 from the incline angle θ_{L}.
In a simple case a driver issues a trigger to the apparatus 10,12 when the vehicle 14 parked and then drives in a loop back to the same parked location. The method makes the assumption that the return loop altitude H_{R }is the same as the start altitude H_{1 }when it detects that the apparatus 10,12 has returned to the same horizontal position or is triggered a second time. In a step 208 the apparatus 10,12 determines the accelerometer bias β that causes the sum of the altitude changes ΔH'"'"'s to be zero or determines the accelerometer bias β that cause the start altitude H_{1 }and return loop altitude H_{R }to be equal.
The vehicle 14 is slowly driven in a step 308 to return to the same parking space and stopped facing in the opposite direction. A second acceleration α_{m}(t) is measured. This measured acceleration α_{m}(t) will be small because the vehicle 14 is not moving. In a step 314 the difference between the first and second measured accelerations α_{m}(t) is used to distinguish between the accelerometer bias β_{1 }and an effect from gravity g for a parking incline angle of the ground 25.
The apparatus 10,12 in a step 316 uses the just calculated accelerometer bias β_{1 }and preselected estimates of position offset L and yaw alignment angle γ for determining a start altitude H_{1}. In a step 318 the vehicle 14 is rapidly driven in a clockwise (or counterclockwise) loop back to the parking space. The driving must be fast enough to cause a yaw angle rate ω(t) similar to the yaw angle rates ω(t)'"'"'s that will be encountered in operation. In a step 322 a first return to position (RTP) altitude H_{R1 }is determined. In a step 324 the vehicle 14 is expeditiously driven in the opposite direction loop back to the parking space. The driving must be fast enough to cause a yaw angle rate ω(t) similar to the yaw angle rates ω(t)'"'"'s that will be encountered in operation. In a step 326 a second return to position (RTP) altitude H_{R2 }is determined. In the step 328 the effective installation position offset L and yaw alignment angle γ are calculated from the start altitude H_{1}, first RTP altitude H_{R1 }and second RTP altitude H_{R2 }using equations 2 and 3 below. The step 328 determines the accelerometer bias β_{1}, the position offset L and the yaw alignment angle γ for the sum of the altitude changes ΔH'"'"'s to be zero between the start altitude H_{1 }and the first loop altitude H_{R1 }and between the first loop altitude H_{R1 }and the second loop altitude H_{R2}; or determines the accelerometer bias β_{1}, the position offset L and the yaw alignment angle γ that equalizes the start altitude H_{1}, the first loop altitude H_{R1 }and the second loop altitude H_{R2}.
The following section shows the operation of the apparatus 10,12 for a calculation of altitude change ΔH and incline angle Θ. The equation 2 shows a calculation of altitude change ΔH based on compensated acceleration α_{c}(t) and speed v(t) for a measurement time ΔT and the gravity acceleration constant g.
The equation 3 shows the compensated acceleration α_{c}(t) as a function of measured acceleration α_{m}(t), accelerometer bias β, positional yaw rate error ω^{2}(t)×L, and an alignment yaw rate error ω(t)×v(t)×γ.
α_{c}(t)=α_{m}(t)+β−ω^{2}(t)L−ω(t)v(t)γ 3)
The positional yaw rate error is the same for either left or right turns. The alignment yaw rate error is equal and opposite for left and right turns. The equation 4 shows the incline angle Θ as a function compensated acceleration α_{c}(t) and rate of change of speed versus time Δv/Δt.
Θ=Sin^{−1}{[(α_{c}(t)−Δv/Δt]/g} 4)
An embodiment may improve the performance of a vehicle navigation apparatus that comprises a GPS receiver, yaw rate gyro or heading gyro, transmission shaft or wheel speed measurement device, and optionally a mapmatch capability. An embodiment may add a forward direction linear accelerometer and several accelerometer compensation algorithms to improve the precision of this apparatus to preferably within a meter or two, without a requirement for differential GPS measurements; and to provide continuous and smooth altitudes and incline angles without a requirement for continuous or smooth GPS measurements.
It is wellknown that GPS latitude and longitude measurements are correlated to errors in altitude. By improving knowledge of altitude, the latitude and longitude knowledge may be improved through these correlations. In obstructed view situations, the GPS velocity may be noisy, or the geometry (DOP) of the GPS satellite signals may be poor, or only three GPS satellite signals may be available. In such cases the improved altitude measurement makes a significant improvement in latitude and longitude measurements.
Direct knowledge of the altitude that may be better than GPS accuracy is used in combination with a mapmatching data base that includes altitude information. Using precise altitude or altitude change information, a mapmatch algorithm can quickly determine whether the vehicle is on one of two parallel tracks at a highway offramp, one of which is rising or falling and one of which is not. Using GPS and heading gyro alone may not produce the required accuracy to make such a determination until a substantial distance has been traveled, to the point that the two possible paths are separated horizontally in the mapmatch data base by a distance commensurate with the GPS accuracy. Knowledge of whether the vehicle has left the highway is critical in determining routing information promptly. Direct knowledge of altitude can also establish the vertical location of the vehicle when in a multifloor parking structure with no GPS coverage or imprecise GPSbased altitudes.
To compute altitude change ΔH accurately, we calibrate the accelerometer bias β which changes with time, and the position offset L which is usually constant with time. The calibration would also calibrate any position offset L that changed with time. It has been noted above that a DR altitude is determined by accumulating ΔH'"'"'s onto a prior determination of altitude. The calibration eliminates the need for the installation accelerometer bias β_{1 }and the restart accelerometer bias β_{0}, that are described above. This calibration eliminates all need to make any initial estimate or have any prior knowledge or any separate measurement for the accelerometer bias β or the accelerometer position offset L. Those skilled in the art know that the idea of forward acceleration as described herein is sometimes called proper acceleration or specific force.
The external altitude H_{EXT }from the external altitude source 82 is described below as a measured altitude h_{m}. A rate of change h′_{m }(altitude rate) of measured altitude h_{m }may be derived as a change Δh_{m }in the measured altitude h_{m }for a time Δt or may be directly available from a Doppler signal measurement in a vertical direction or from a GNSS vertical Doppler measurement from the GNSS receiver 84. The external altitude information used by the calibrators 600, 610, 620 may be the measured altitude h_{m }or altitude rate h′_{m }or both.
The estimator 601 can be implemented with a one state Kalman filter described as follows. The one state Kalman filter has a state for a recursive estimate β_{e }of the accelerometer bias β. A propagator 603 for the Kalman filter and an updater 604 for the Kalman filter act together to make recursive estimates for the state. The propagator 603 receives control inputs for forward acceleration, yaw angel rate and forward speed and the updater 604 receives a measurement input for an external altitude rate. The propagator 603 processes the information in the control inputs and the state of the estimate of the accelerometer bias β according to a dynamic model equation in order to propagate the state. The updater 604 processes the information in the measurement input and the state of estimate of the accelerometer bias β according to a measurement model equation in order to update the state. The effect is that the propagator 603 and the updater 604 interconnect with each other to make recursive estimates for the accelerometer bias β.
The output converter 602 processes the information in the control inputs and the state of the estimate of the accelerometer bias β, according to output model equations to provide outputs for change in altitude ΔH and incline angle Θ.
Control Inputs

 forward acceleration a_{m}; noise v_{a }with noise variance q_{a }
 yaw angle rate ω
 speed v or distances and times for change ΔS in distance for a change Δt in time (may be determined from backwards differenced speedometer).
Measurement Inputs
altitude rate h′_{m}=(Δh/Δt)_{m}; noise v′_{h′} with noise variance r′_{h }(h′_{m }is an altitude rate received from an external positioning source or derived by differencing altitudes h_{m }received from an external positioning source with respect to Δt or ΔS).
States for One State Kalman Filter

 accelerometer bias β(t); varies slowly with time; variation modeled as a Marko process w_{β} with noise variance q_{β}
Dynamic Equations (State Propagation from Control Inputs)
Δβ/Δt=0+w_{β} (5)
Measurement Equations (G=Local Gravitational Acceleration)
h′_{m}=v(a_{m}+β_{e}−ω^{2}L_{0}−Δv/Δt+v_{a})/G+v′_{h} (7A)
Output Conversion Equations
 accelerometer bias β(t); varies slowly with time; variation modeled as a Marko process w_{β} with noise variance q_{β}
ΔH from reference time point to t_{0 }current time t_{1}.
ΔH={∫v(t)(a_{m}(t)+β(t)−ω^{2}(t)L_{0})dt−½[v_{1}^{2}−v_{0}^{2}]}/G (8)
where the integral is computed from time t_{0 }to t_{1}.
The Instantaneous Inclination is also Available as
Θ=arc sin (ΔH/Δs) or arc sin (ΔH/(v Δt)) (9)
over short distances Δs or short time periods Δt.)
The estimator 611 can be implemented with a two state Kalman filter described as follows. The two state Kalman filter has a state for a recursive estimate β_{e }of the accelerometer bias β and a state for a recursive estimate L_{e }of the position offset L. A propagator 613 for the Kalman filter and an updater 614 for the Kalman filter act together to make the recursive estimates for the states. The propagator 613 receives control inputs for forward acceleration, yaw angel rate and forward speed and the updater 614 receives a measurement input for an external altitude rate. The propagator 613 processes the information in the control inputs and the states of the estimates of the accelerometer bias β and the position offset L according to dynamic model equations in order to propagate the states. The updater 614 processes the information in the measurement inputs and the states of estimates of the accelerometer bias β and the position offset L according to a measurement model equation in order to update the states. The effect is that the propagator 613 and the updater 614 interconnect with each other to make recursive estimates for the accelerometer bias β and the position offset L.
The output converter 612 processes the information in the control inputs and the states of the estimates of the accelerometer bias β and the position offset L according to output model equations to provide outputs for change in altitude ΔH and incline angle Θ.
Control Inputs

 forward acceleration a_{m}; noise v_{a }with noise variance q_{a }
 yaw angle rate ω; noise v_{ω} with noise variance q_{ω}
 forward speed v or change ΔS in distance for a change e Δt in time (as determined from backwards differenced odometer); noise v_{v }with noise variance q_{v }
Measurement Inputs
altitude rate h′_{m}=(Δh/Δt)_{m}; noise v_{h′} with noise variance r′_{h }(h′_{m }is an altitude rate received from an external positioning source or derived by differencing altitudes h_{m }received from an external positioning source with respect to Δt or ΔS).
States for Two State Kalman Filter

 accelerometer bias β(t); varies slowly with time; variation modeled as a Marko process w_{β} with noise variance q_{β}
 accelerometer position offset L_{e}, constant
Dynamic Model (State Propagation from Control Inputs)
Δβ/Δt=0+w_{β} (5)
ΔL/Δt=0 (10)
Measurement Model (G=Local Gravitational Acceleration)
h′_{m}=v(a_{m}+β−ω^{2}L−Δv/Δt)/G+v_{h′} (7A)
Output Conversion Equations
ΔH from reference time point to t_{0 }current time t_{1}.
ΔH={∫v(t)(a_{m}(t)+β(t)−ω^{2}(t)L)dt−½[v_{1}^{2}−v_{0}^{2}]}/G (8)
where the integral is computed from time t_{0 }to t_{1}.
The Instantaneous Inclination is also Available as
Θ=arc sin (ΔH/Δs) or arc sin (ΔH/(v Δt)) (9)
over short distances Δs or short time periods Δt.)
Observability of the states is guaranteed from the history of h′_{m }measurements, provided that there is sufficient movement of the vehicle 14.
An error in the estimate of β will cause the calculated h′_{m }to be biased by an amount proportional to the product of the error in the estimate and the speed. The value of β requires forward motion to be observable, and becomes more observable as more distance is traveled while h′_{m }measurements are made, and the error in the estimate will decrease.
An error in the estimate of L will cause the calculated h′_{m }to be biased by an amount proportional to the product of the error in the estimate and the quantity ω^{2}*v. The value of L requires left or right turns to be observable, and becomes more observable as more turns accumulate (especially highrate turns) while h′_{m }measurements are made, and the error in the estimate will decrease.
The estimator 621 can be implemented with a three state Kalman filter described as follows. The three state Kalman filter has a state for a recursive estimate β_{e }of the accelerometer bias β, a state for a recursive estimate L_{e }of the position offset L, and a state for a recursive estimate Z_{e }of a model altitude Z. A propagator 623 for the Kalman filter and an updater 624 for the Kalman filter act together to make the recursive estimates for the states. The propagator 623 receives control inputs for forward acceleration, yaw angel rate and forward speed and the updater 624 receives measurement inputs for an external altitude and optionally an external altitude rate. The propagator 623 processes the information in the control inputs and the states of the estimates of the accelerometer bias β, the position offset L, and the model altitude Z according to dynamic model equations in order to propagate the states. The updater 624 processes the information in the measurement inputs and the states of estimates of the accelerometer bias β, the position offset L, and the model altitude Z according to measurement model equations in order to update the states. The effect is that the propagator 623 and the updater 624 interconnect with each other to make recursive estimates for the accelerometer bias β, the position offset L, and the model altitude Z. The state of the model altitude Z may be viewed as a catalyst for the calibration of the accelerometer bias β and the position offset L.
The output converter 622 processes the information in control inputs and the states of the estimates of the accelerometer bias β and the position offset L according to output model equations to provide outputs for change in altitude ΔH and incline angle Θ.
Control Inputs

 forward acceleration a_{m}; sensor noise v_{a }with noise variance q_{a }
 yaw angle rate ω; sensor noise v_{ω} with noise variance q_{ω}
 forward speed v (as measured by a speedometer or from backwards differenced odometer (ΔS traveled in distance for a change Δt in time); sensor noise v_{v }with noise variance q_{v }
States for Three State Kalman Filter
model altitude Z (altitude h_{m }of external positioning source smoothed by Kalman Filter)
 accelerometer bias β(t); varies slowly with time; variation modeled as a Marko process w_{β} with noise variance q_{β}
 accelerometer position offset L, constant
Measurement Inputs  altitude h_{m}; noise v_{h }with noise variance r_{h }(h_{m }is an altitude received from an external positioning source)
 altitude rate h′_{m}; optional is available; sensor noise v_{h′} with noise variance r_{h′} (h′_{m }is an altitude rate received from an external positioning source)
Dynamic Model (State Propagation From Control Inputs, G=Local Gravitational Acceleration)
Δβ/Δt=0+w_{β} (5)
ΔL/Δt=0 (10)
ΔZ/Δt=v(a_{m}+β−ω^{2}L−Δv/Δt+v_{a})/G (11)
Measurement Model
h_{m}=Z+v_{h}, (7B)
Optional Measurement Model (Use Both h_{m }and h′_{m }if h′_{m }is Available)
h′_{m}=v(a_{m}+β−ω^{2}L−Δv/Δt)/G+v_{h′} (7A)
h_{m}=Z+v_{h} (7B)
Output Conversion Equations
The change in altitude ΔH from reference time point t_{0 }to current time t_{1 }is computed as a function of the control inputs where the integral is computed from time t_{0 }to t_{1}.
ΔH={∫v(t) (a_{m}(t)+β(t)−ω^{2}(t)L)dt−½[v_{1}^{2}−v_{0}^{2}]}/G (8)
The Instantaneous Inclination is also Available as
Θ=arc sin (ΔH/Δs) or arc sin (ΔH/(v Δt)) (9)
over short distances Δs or short time periods Δt.)
Observability of the states is guaranteed from the history of h_{m }measurements, provided that there is sufficient movement of the vehicle.
The state Z is directly measured by h_{m }measurements, and is thus observable.
An error in the estimate of β will cause the estimated Z to drift relative to the measurements h_{m }by an amount proportional to the product of the error in the estimate of β and the distance traveled. The value of β requires forward motion to be observable, and becomes more observable as more distance is traveled while h_{m }measurements are made.
An error in the estimate of L will cause the estimated Z to drift relative to the measurements h_{m }by an amount proportional to the product of the error in the estimate of L and the accumulation of the product of v*w^{2}. The value of L requires turns to be observable, and becomes more observable as more turns accumulate (especially highrate turns) while h_{m }measurements are made.
In general, the external measurements of the altitude information h_{m }and/or h′_{m }are intermittent, i.e. not continuously available for the measurement inputs. For example the external altitude information h_{m }and/or h′_{m }derived with the GNSS receiver 84 is not available during time periods when the GNSS signals are blocked. The updater 604,614,624 does not update estimates of the states β or L when it does not receive new measurement input information for h_{m }and/or h′_{m}. However, the propagator 603,613,623 continues to propagate the state estimates to the output converter 602,612,622. The output converter 602,612,622 uses the continuous sensor measurements being received in the control inputs for velocity v, acceleration α_{m }and yaw angle rate ω^{2 }and the propagated state estimates to provide continuous outputs of the change in altitude ΔH and/or the incline angle Θ.
Although the disclosure describes details of embodiments, it is to be understood that this disclosure is not to be interpreted as limiting. Accordingly, it is intended that the claims, written below be interpreted as covering the present invention'"'"'s, spirit, scope and limitations.