The term

Typically, inertial measurements are made to have estimates of current position and velocity in real time. The set of equations used to compute the position and velocity estimates out of the actual measurements depends greatly on the application in hand. Equations for a general navigation application are presented, for example, in [

In terms of the simplified equations, we basically need to solve the following problem: given

Now, from the theory of ordinary differential equations (ODEs), we know that we need exactly two independent constraints to unambiguously solve a 2nd order ODE (

and one of the constraints

find

Notice that there is a natural reason why problems (

The key assumption of our approach is the following:

The case where we only know the velocity at both ends is, however, special and deserves some attention. According to the discussion above, we do not have a unique solution for this kind of a BVP although it fits the definition of our model BVP (

The overall scope of this paper is to show how to treat inertial navigation problems that are naturally (or knowingly) posed as a BVP of the form (

This paper is organized as follows: in Section

Let us first motivate the chosen approach by means of examples: an application suitable to our approach is ski jumping, which has been the prime motivation of this study [

It turns out that especially

For a more general view, even a GPS-assisted inertial navigation system can be considered as a series of separate navigation periods with given boundary values rather than a single event with additional constraints given at certain time instances. This is an example of an IVP, which can be posed as a BVP.

There are two main classes of numerical methods for BVP's. One class includes so-called shooting methods and the other class methods of weighted residuals such as FEM [

In many applications, inertial measurements are not the only source of information. In practice, however, the number or the type of these additional constraints—combined with the different kinds of a solution method—does not suggest the use of the traditional filtering (see Section

Sensor errors are modeled as constant errors. While the behavior of a certain sensor error is in real life a stochastic process, one is usually able to fairly model it at least momentarily as a constant error. A suitable mathematical tool to characterize this is the

Considering consumer grade sensors, causes of the most significant errors are usually known (e.g., bias and scale factor error, both changing from turn-on to turn-on [

In particular, the sensor noise is

Additional constraints are treated as “exact”. That is, the overall error in the additional constraints is assumed to be smaller than the error caused by the simplification of the navigation equations.

When additional constraints are available, problems resembling the ones considered here have previously been resolved using the means of

Comparing the fixed interval smoothing technique to the proposed method, in addition to the previously mentioned points, the most significant differences are as follows.

As fixed interval smoothing is run in both directions, the filter needs two process models, which can be problematic [

In the filtering approach, each additional constraint is assumed to be attached to a single time instance [

Fixed interval smoothing is a two-phase method requiring numerous computations per time step [

Due to the significant differences in these two approaches, we will in this context concentrate on the proposed method. Obviously, there are situations where the two methods could both be used. Comparison of the methods in such a situation is interesting, although not addressed in this paper.

Finally, recall that problem (

The main goal of this section is to form a linear system of equations of the form

Now, let us rephrase problem (

While (

Lowest order basis functions

Given the basis functions

Time derivative of the lowest order basis functions

In total, we are now in the position to discretize equation (

Now that we have discretized the problem, we are getting closer to the equation

Our next task is to compute the vector

Basis function

Finally, let us consider how to apply the different types of boundary values into (

The other possibility for the boundary values is to have

We have now means to assembly an equation of the form

From the position data

In this section, we will generalize the method described in the previous section to the 3D case. For this, we introduce a coordinate transformation matrix

In this paper, we will assume that an estimate of

Now, following from (

for the

With the equations derived in this section, it is possible to solve the 3D BVP using the presented FEM method. The solution of this more general problem can also be found with linear time complexity, as in the 1D case.

So far we have constructed a method to compute position and velocity data with certain boundary conditions. We will now propose a way to exploit a number of additional constraints concerning the velocity or the position information to estimate sensor errors. For this, let us now precisely define the term “additional constraint”:

In this section, we will use constraints systematically to enhance accuracy. To make the concepts presented in this section clear, let us first present the equations for the 1D case.

Let us first assume that

In general, all linear constraints can be represented in the form

These constraints could be exploited just by computing the linear least squares problem constructed by adding these additional equations to (

In typical situations, bias offset (i.e., the sensor shows nonzero output when no forces are acting upon it) and scaling factor are the two terms which have to beknown very accurately in order to have any realistic position or velocity estimate as a solution. Because of the run-to-run variations of these errors, they cannot be assumed to be constant between two separate events. Thus, they should be treated as unknowns. Other typical errors are caused by misalignment of the axes, changing temperature and drifting bias.

As a practical example used in the example 1 later in Section

Let us now replace the right-hand side of (

In general, a problem of linear constraints used to model linear sensor errors can be stated as

Now, since

In the case where

Let us now consider the use of additional constraints in a 3D case. Using the notions form the previous section and (

In addition to the sensor errors discussed in the 1D example, it is now also possible to model errors like attitude errors, unknown value of the acceleration due to the gravity, and cross-correlation of different sensors. A particularly useful method is to treat the acceleration due to the gravity as an unknown three-dimensional vector, which is then subtracted from the specific force measurements given in the global coordinates. This reduces the systems sensitivity to initial attitude errors.

As an example, let us now derive the equations for sensor error of (

To demonstrate the use of the proposed method, an example using readily available consumer grade accelerometers [

Measurement setup of example 1.

The aim of this example was not to get position and velocity as accurately as possible, but to compare different solution methods in a situation where one needs to get reasonable position and velocity estimates regardless of the fact that the measurement contains significant errors. The angular velocity of the rate table was set to follow function illustrated in the Figure

Angular velocity of the rate table.

In first test, the rate table was set to repeat exactly the function represented in Figure

The raw accelerometer data was mapped to accelerations with the scale factor given by the manufacturer. From this acceleration data, position and velocity were computed by a number of different methods:

“Traditional” IVP (double integration with trapezoid rule),

FEM with Dirichlet boundary conditions (BCs),

FEM with Dirichlet BCs combined with additional Neumann BCs to supply the error model (

In methods I and II, the bias error of the sensor was estimated by averaging the output while the sensor was at rest. This was done in order to make method I (and to some extent, method II) comparable to the method III by reduction of the large bias error. This is rarely possible in general and only method III can be used to reliably detect any remaining bias (example 2 in Section

Ideal (thick line) and measured acceleration (thin line).

Table

Comparison of different methods computing object's velocity and position. “I“ stands for time-stepping, “II“ for FEM, and “III“ for FEM with sensor error modeling.

Test 1 | Test 2 | |||||
---|---|---|---|---|---|---|

Vel. error [m/s] | ||||||

I | ||||||

II | ||||||

III | ||||||

Pos. error [m] | ||||||

I | ||||||

II | ||||||

III |

As seen in Table

Figures

Velocity estimates given by methods I (thin black line) and II (thin gray line) compared to the ideal velocity (thick line).

Position estimates given by methods I (thin black line) and II (thin gray line) compared to the ideal position (thick solid line).

This example considers the computation of the velocity and position of a ski jumper during a single jump. As compared to the previous example, this is a more realistic and general inertial navigation problem with six degrees of freedom. Computation of the attitude of the object is based on the data given by three standard consumer grade gyroscopes [

the location of the jumper at five points evenly spread on the inrun hill (in Figure

the trajectory of the jumper after the landing, which should coincide with the linearization of the landing hill (in Figure

Computed two-dimensional trajectories of two independent events (thin gray and thin black lines) along with the known profile of the hill (thick black line).

In Figure

Figure

Vertical velocity (

Given that two independent measurement systems show the same variations in the velocity at the same locations, it is evident that there is actually only a negligible amount of stochastic errors present. Instead, the small variations are caused by deterministic sources, namely, in this particular application the uneven inrun hill.

Unfortunately, the estimates cannot be compared with a reference trajectory, because such data are not available. Thus, we cannot give the exact amount of error present in the position and velocity estimates. We do however claim that the achieved accuracy is something one does not typically expect from consumer grade inertial sensors.

The work was motivated by applications, where it is natural to encounter BVPs instead of IVPs. In many cases, it is also possible to formulate an IVP as a BVP, given that the results are not required in real time.

Finite element method is utilized to solve inertial navigation problems formulated as BVPs. As a result, we get a linear system of equations for the position estimates, whose solution can be found with linear time complexity. It is demonstrated that solving a BVP rather than an “equivalent” IVP gives more accurate results.

For further accuracy enhancements, an efficient way of combining inertial measurements with possible additional constraints is created. This gives us a possibility to model constant sensor errors, known to limit the achievable accuracy of the system. While the error model significantly enhances the accuracy of the system, it is kept computationally simple and easily adoptable.

In practice, the accuracy improvements allow us to exploit inertial sensors of certain performance level in more challenging applications. For this, it is necessary to see that the concept of inertial navigation does not invariably imply an IVP, but a BVP as well. Then, the use of FEM will provide an efficient way to compute position and velocity estimates not prone to the accumulation of errors.

In larger scale, the current paper serves as an introduction to the idea of formulating inertial navigation problems as BVPs. As a consequence, further studies are needed to address problems to which the presented tools do not provide an obvious solution. These include, for example, stochastic errors, reliability of the possible additional constraints (as compared to the accuracy of the IMU), and coupling of position and attitude errors.