## Abstract

This paper develops novel covariance and square-root factor formulations of a consider-neglect Kalman filter for navigation applications. The proposed filter partitions system parameters into three distinct categories: those to be estimated by the filter, those whose contribution to the system are considered without being explicitly estimated, and those with sufficiently low effect on the system such that their contribution can be neglected altogether. Discussion on appropriate selection of parameters to be considered and neglected is provided with specific attention given to descent-to-landing navigation. Monte Carlo simulations and analysis are performed to assess the performance of the developed square-root consider-neglect filter in a descent-to-landing navigation scenario.

## 1 INTRODUCTION

Navigation filters, most often comprised of some variation of a minimum mean-square error (MMSE) estimator such as the Kalman filter, require intelligently developed system models for successful and accurate estimation. Precise modeling of a system can include dependencies on a multitude of parameters, especially in the context of spacecraft navigation. Sensor biases and other types of corruption parameters can factor heavily into the model of a particular dynamical or observational system. Incorporating an ever-increasing list of uncertain elements into a system model naturally increases the number of parameters that can be estimated. When several sensor and dynamics models are required for a given scenario, the dimension of the navigation filter’s state vector quickly grows to a cumbersome size. To further compound the problem, many of the estimated parameters may only be weakly observable, resulting in ineffective estimation. For practical implementations, a trade-off arises between including significant parameters to ensure accurate modeling while maintaining computational efficiency by omitting insignificant parameters in some way. Failure to incorporate certain elements can result in significant model-mismatch and subsequent poor filter performance; estimating the contribution of multitudes of parameters, however, can be prohibitively expensive for high-frequency, real-time applications, such as spacecraft navigation.

Two general approaches exist for handling uncertain parameters beyond directly estimating them. The obvious and seemingly simplest approach is to neglect the parameters altogether. In this approach, the contributions of the neglected parameters are not modeled in the filter in any way, even though they are known to influence the true system; as such, it is clear that there is an associated loss of both precision and accuracy. In the case of weakly observable parameters with only a minor effect on the system, neglecting parameters can be sufficient. In applications where accurate solutions are necessary, however, the selection of the neglected parameters must be handled with great care. The more cautious approach is to instead consider the effects of the parameters without directly estimating them, as originally outlined in the Schmidt-Kalman filter (Schmidt, 1966). A more in-depth treatment of the Schmidt-Kalman filter (also called the consider Kalman filter) as the optimal minimum variance solution is given in Jazwinski (1970) for linear measurements, with an alternative formulation provided in Tapley, Schutz, and Born (2004).

Consider filters are well-established, and considerable effort has been put forth in examining formulations and developing implementations of such filters. For example, Woodbury and Junkins (2010) compares the performance of the two consider filter approaches in Jazwinski (1970) and Tapley et al. (2004) and provides conditions under which the two are equivalent. Recursive implementations of the consider filter are investigated in Zanetti and D’Souza (2013), including methods for generating the consider filter by manipulation of the traditional Kalman filter. In lieu of the standard linearization-based approach, unscented (Lisano, 2006) and quadrature (Zanetti & DeMars, 2013) formulations have been proposed for processing nonlinear measurements or handling nonlinear dynamics within the consider filter framework. All of the aforementioned approaches operate on an estimated state and a representation of uncertainty that is typically either the covariance matrix or a factorized representation of the covariance matrix. To facilitate consider filters that operate on probability density functions, McCabe and DeMars (2017) develops an approach based on the Chapman-Komolgorov equation and Bayes’ rule to enable the use of consider parameters in the Gaussian sum filter.

A central issue in implementing filters that operate on the covariance matrix is maintaining the inherent properties of the covariance matrix throughout the filter recursion. The covariance matrix is, by definition, a symmetric, positive definite matrix. Ensuring symmetry is straightforward and can be accomplished in a brute-force manner; guaranteeing that the covariance matrix is positive definite, on the other hand, is more nuanced. One approach that helps to mitigate the loss of positive definiteness is operating on a factorized form of the covariance matrix. This technique was pioneered by Potter (1963) and expanded upon and refined by Bellantoni and Dodge (1967), Andrews (1968), Schmidt (1970), Carlson (1973), Choe and Tapley (1975), Tapley and Choe (1976), Bierman (1976), and Thornton (1976), among others. Owing to its superior numerical properties, Potter’s approach was implemented in the Apollo navigation filter (Battin & Levine, 1970). Bierman’s approach, which uses the so-called UD or UDU factorization, is used to avoid any possible numerical instability in the covariance matrix for Orion’s prelaunch navigation system for both Exploration Flight Test 1 and for the planned Exploration Mission 1 (now called Artemis 1) (Zanetti et al., 2017). The aforementioned factorization-based approaches have also been applied to consider filters using a Cholesky factorization and hyperbolic Householder reflections (McCabe & DeMars, 2018) as well as the UDU factorization (Zanetti & D’Souza, 2013). All of the aforementioned approaches provide a means to the same end: increased numerical stability of the filter for a moderate increase in computational cost.

The present work investigates the combination of consider formulations and factorization-based formulations of recursive filters. As alluded to previously, the complete set of parameters that comprise a system can be separated into those that are to be estimated, those whose effects are to be accounted for without explicitly estimating them, and those whose effects are to be ignored altogether in the filter. The developed filter provides a straightforward mechanism for assigning each state/parameter to one of the three aforementioned sets without requiring any special reconfiguration, thus leading to a highly modular framework for investigating the influence of the considered and neglected parameters on the estimation performance. In this way, a large trade space of available filter configurations is immediately opened up for easy testing and comparison. The developed approach is not intended for onboard application, but rather for rapid design, testing, and analysis to inform the selection of which parameters are estimated, considered, and neglected.

To illustrate the kind of trade space afforded by the modular framework, a series of simulations is presented. In the process of tuning and analyzing a particular filter architecture, it is oftentimes of interest to determine which states are appropriate to consider or neglect and to assess the effect of such selections on the overall filter performance. Beginning with the estimation of a full set of filter states, previous error budget and sensitivity analysis results (Ward, Fritsch, Helmuth, DeMars, & McCabe, 2019) are used to inform selection of certain consider or neglect states in a lunar descent-to-landing navigation scenario. While the results for different choices of consider and neglect states are scrutinized for changes in filter response to different events in the trajectory, the simulation is designed and presented to be broadly illustrative of the testing capabilities afforded by the proposed consider-neglect framework; namely, the ability to provide a means to rapidly change the desired estimated, considered, and neglected states without incurring significant changes to the filter itself and reducing the burden of comprehensive analysis.

The rest of this work is organized as follows. Section 2 establishes the notation used in this paper and presents the usual formulation of an extended consider Kalman filter before proceeding to develop the novel contribution of the consider-neglect filter. Covariance and square-root forms of the developed consider-neglect filter are provided, and implementation considerations are discussed. Section 3 briefly reviews a lunar descent-to-landing simulation and the dynamics and sensor models that are used to test the developed consider-neglect filter. Section 4 presents the results of four simulation configurations that investigate different selections of which parameters are estimated, considered, and neglected, including motivation based on previous works for how to allocate parameters to one of the categories. Performance comparisons between the different configurations are provided to analyze the observed changes in performance that occur. Finally, Section 5 provides concluding remarks.

## 2 CONSIDER-NEGLECT FILTER

### 2.1 Notation

Throughout this work, lowercase bold symbols, such as * a* and

*, are used to represent vectors, uppercase bold symbols, such as*

**γ***and*

**A***are used to represent matrices, and non-bold symbols, such as*

**Γ***a*and Γ, are used to represent scalars. Superscript “

*T*” is used to designate the transpose, and superscript “−1” designates the inverse. A vector of zeros that is length

*d*is denoted by

**O**

_{d}, a matrix of zeros that has

*d*

_{1}rows and

*d*

_{2}columns is denoted by

**O**

_{d1×d2}, a vector of ones that is length

*d*is denoted by

**1**

_{d}, and the

*d*-dimensional identity matrix is

**I**_{d}.

For some random vector * a*, with dimension

*d*

_{a}at time

*t*

_{k}, the mean and covariance are defined in terms of the expected values as

and the cross-covariance of * a* with another random vector

*at time*

**b***t*

_{k}is

### 2.2 System definition

Consider a discrete-time, nonlinear system of the form

1a

1b

where the states are represented by **x** and the measurements are * z*, with subscript

*k*denoting time

*t*

_{k}. The functions describing the nonlinear system are

**f**(⋅, ⋅), which represents the dynamics governing the evolution of the state, and

**h**(⋅, ⋅), which represents the mapping of states into measurements. Both the dynamics and measurements are subjected to zero-mean, white noise sequences, which are represented by

**q**

_{k−1}and

**v**_{k}for the dynamics and the measurements, respectively.

In general, the system state, **x**, is comprised of elements that are to be estimated, denoted by * x*, elements that are not estimated but whose contributions are included, denoted by

*, and elements that are not estimated and whose contributions are not included, denoted by*

**c***. Thus, the system state at*

**n***t*

_{k}can be expressed as

2

Commonly, the elements of * x* are called the estimated states (or simply states), the elements of

*are called the consider parameters (or considered parameters), and the elements of*

**c***are called the neglect parameters (or neglected parameters). For distinction with respect to the estimated states,*

**n****x**is referred to as the total state or concatenated state. A similar partitioning is applied to the total process noise

**q**

_{k−1}, such that

With the partitioning of the total state and the total process noise, the dynamical system in Equation (1a) can be re-formulated as

3a

3b

3c

3d

where the dynamical evolution of the considered parameters is given by the nonlinear function **g**(⋅, ⋅) and the dynamical evolution of the neglected parameters is given by the nonlinear function * ℓ*(⋅, ⋅), with respective zero-mean, white noise sequences

**u**_{k−1}and

**w**_{k−1}. It is important to acknowledge an assumption that is made in going from Equation (1a) to Equations (3). Implicit in Equation (1a), there is no restriction enforced in terms of the dynamical influence of the estimated states and neglected parameters on the considered parameters or of the estimated states and considered parameters on the neglected parameters. Equation (3b), however, makes the assumption that the dynamics of the considered parameters are independent of the estimated states and the neglected parameters, and Equation (3c) makes the assumption that the dynamics of the neglected parameters are independent of the estimated states and the considered parameters.

### 2.3 Minimum mean square error formulation

Given the dynamical and observational system represented by Equations (3), the goal is to formulate a linear MMSE filter (or, more precisely, an approximate linear MMSE filter) to recursively estimate the state, **x**_{k}, while considering the effects of **c**_{k} and neglecting the effects of **n**_{k}, using information from the measurements, **z**_{k}. As is common with other MMSE filters, i.e., Kalman-type filters, the process is separated into a prediction stage (also referred to as the predictor) and a correction stage (also referred to as the corrector), where the prediction stage propagates mean and uncertainty between times of received measurements, and the correction stage updates the mean and uncertainty using the received measurements. The uncertainty can either be the covariance matrix or a factorized representation of the covariance matrix.

Using the separated system given in Equations (3), the *a priori*, which is denoted by a superscript “−”, means for the estimated states and considered parameters are

4a

4b

where it is important to note that the neglected parameters are explicitly set equal to zero in Equation (4a) to account for the fact that their contributions are specifically neglected from consideration. In a similar manner, the covariances for the estimated states and considered parameters, including the cross-covariance between the estimated states and considered parameters, are given by

5a

5b

5c

It should be noted that, by definition, there is no propagated covariance for the neglected parameters. For the same reason, there are no *a priori* cross-covariances between the estimated states and the neglected parameters or between the considered parameters and the neglected parameters.

The *a posteriori* estimate is constructed as a linear, unbiased estimate given the prior information, as well as the received data, **z**_{k}. Before proceeding, it is important to note that the influence of certain parameters is neglected in constructing the posterior estimate, which is similar in effect to Equations (4) and (5) for obtaining the prior estimate. Since there are neglected parameters, about which no knowledge is assumed, an unbiased solution is not, strictly speaking, obtainable. As such, “unbiased estimate” refers to the fact that no additional bias is introduced via the manner in which the posterior estimate is constructed. When considered parameters are involved, they are not updated, which leads to the posterior estimates for the estimated states and considered parameters as

6a

6b

where the superscript “+” denotes a value after the incorporation of new measurement information. The linear gain, **K**_{x,k}, is selected such that the posterior mean-square error of the estimated state—and only the estimated state—is minimized. The gain that accomplishes this is

7

Given the posterior estimates in Equations (6), it directly follows from the linear update that the posterior covariance for the estimated states, the posterior cross-covariance between the estimated states and considered parameters, and the posterior covariance for the considered parameters are, respectively,

8a

8b

8c

where

Equations (4) and (5) constitute the governing equations for the predictor stage of the linear MMSE consider-neglect filter, and Equations (6)–(8) are the representative equations governing the corrector stage. The filter recursion is initialized at *t*_{k−1} = *t*_{0} with the given initial conditions , **m**_{c,k−1} = **m**_{c,0}, , , and .

Note that this filter bears a striking resemblance to the filter described in McCabe and DeMars (2018), where new formulations of consider Kalman filters are developed. The key difference in the present work is that neglected parameters are explicitly included within the dynamical and observational models, and the resulting filter explicitly excludes any influence of the neglected parameters within the filter recursion. At no point in time are any estimates or uncertainties of the neglected parameters used to determine the estimate for the estimated state or its uncertainty. To obtain practical implementations of the filter, a method for computing the expectations is required.

### 2.4 Linearization-based implementation

In this work, a linearization-based implementation of the linear MMSE consider-neglect filter is developed. The result is akin to the extended Kalman filter, except for the case where parameters are considered or explicitly neglected. The linearization-based approach approximates nonlinear functions via first-order Taylor series expansions in order to compute the requisite expectations.

For the predictor stage, let the necessary Jacobians be defined via

where it is to be understood that **α**_{k−1} can represent **x**_{k−1}, **c**_{k−1}, **n**_{k−1}, or **q**_{k−1}, leading to the Jacobians **F**_{x,k−1}, **F**_{c,k−1}, **F**_{n,k−1}, **F**_{q,k−1}, respectively. Similar relationships are implied for the Jacobians of * g*(⋅, ⋅) and of

*(⋅, ⋅). With the Jacobians available, the necessary expectations for the predictor are*

**ℓ**9a

9b

and

10a

10b

10c

For the corrector, let

be the shorthand definition for the Jacobians of * h*(⋅, ⋅, ⋅, ⋅), where it is to be understood that

**ε**_{k}can represent

**x**_{k},

**c**_{k},

**n**_{k}, or

**v**_{k}to determine the corresponding Jacobian. With these Jacobians, the necessary expectations are

11a

11b

11c

11d

and the corrector is completed via application of Equations (6)–(8).^{1}

### 2.5 Concatenated representation

The filtering recursions described in Sections 2.3 and 2.4 must be specialized for a given system description. Mechanization of the consider-neglect filter requires proper allocation of mean and covariance elements that must be individualized for any given system. It is shown in this section that it is possible to create an identical recursion to that of Section 2.4 by modifying the application of the extended Kalman filter to the system of Equations (1) under the dynamical coupling assumptions of Equations (3).

At some time, *t*_{i}, let either the prior or posterior mean and covariance of the total state be

12

for example, *i* = *k* and ♮ = + produce the posterior mean and covariance of the total state at *t*_{k}. It is important to note that, in general, there could be elements of the neglected parameters present in the estimated total state, and correlations between the neglected parameters and the estimated states or considered parameters can also exist. These elements cannot, however, be permitted to influence the estimated or considered states in any way.

Directly applying the extended Kalman filter predictor to the dynamical system of Equation (1a), the *a priori* mean and covariance of the total state at *t*_{k} are

13a

13b

where, from the dynamical coupling properties of Equations (3) and a similar shorthand notation as that used in Section 2.4 for the Jacobians,

14a

14b

It is clear from Equations (13) and (14) that the presence of neglected parameters in and uncertainties (including correlations) pertaining to the neglected parameters in can manifest contributions to the estimated states’ and considered parameters’ means, covariances, and cross-covariance.

To rectify any appearance of neglected parameters, define state and noise suppression vectors, respectively, as

such that 𝖘_{(n)} ◦ **m**_{x,k−1} suppresses the appearance of the neglected parameters, where “◦” denotes the Hadamard (element-wise) product. Note that the subscript of the suppression vectors designates the elements that are to be suppressed. Based on the suppression vectors, let 𝕱_{(n)} ∈ ℝ^{dx × dx} and 𝕱_{(w)} ∈ ℝ^{dx × dq} be corresponding suppression matrices that are constructed such that each row is the transpose of 𝖘_{(n)} and 𝖞_{(w)}, respectively. The modified predictor is then

15a

15b

The suppression of the neglected parameters from the input into the dynamical system in Equation (15a) leads to the same propagation equations for the estimates as those given in Equations (9). Jacobian suppression, which is accomplished via (𝕱_{(n)} ◦ 𝐅_{x,k−1}) and (𝕱_{(w)} ◦ 𝐅_{q,k−1}), effectively sets **F**_{n,k−1} = **O**_{dx×dn}, **L**_{n,k−1} = **O**_{dn×dn}, and **L**_{w,k−1} = **O**_{dn×dw}. Under these conditions, it is straightforward to show that Equation (15b) yields

16a

16b

16c

16d

16e

16f

which produces exactly the covariance and cross-covariance propagation equations presented in Section 2.4. In particular, Equations (16a), (16b), and (16d) are identical to Equations (10). Meanwhile, Equations (16c), (16e), and (16f), which all evaluate to zero matrices, reflect the fact that any covariance or cross-covariance terms involving the neglected parameters are not estimated in this approach. It is important to note that the suppressed total state, i.e., 𝖘_{(n)} ◦**m**_{x,k−1}, must be used when evaluating the Jacobians in Equations (14) to ensure that the neglected parameters do not have any influence.

Now, consider the corrector of the EKF. Given the prior from Equations (13) along with measurements in the form of Equation (1b), the posterior mean and covariance of the total state are obtained as

17a

17b

where the Joseph form of the covariance update is employed, the linear gain is

18

the required expectations to determine the posterior mean and covariance are

and from the shorthand notation previously used, the Jacobian of **h**(⋅, ⋅) with respect to **x**_{k−1} is

19

The posterior mean and covariance of Equations (17) represent updates to the estimates and covariances (as well as cross-covariances) for all of the estimated states, considered parameters, and neglected parameters. As such, the natural outputs of Equations (17) do not conform with the recursion described in Sections 2.3 and 2.4, and modifications are required.

The first modification deals with the neglected parameters. Let 𝕳_{(n)} ∈ ℝ^{dz×dx} be a suppression matrix in which each row is the transpose of 𝖘_{(n)}. The second modification deals with both the considered and neglected parameters. By definition, neither of these sets of parameters is updated. As such, define a gain suppression vector

and let the gain suppression matrix, given by 𝕲(*c*,*n*) ∈ ℝ^{dx × dz}, be constructed such that each column is 𝖘(*c*,*n*). The modified corrector is then

20a

20b

where

21a

21b

21c

and the linear gain of Equation (18) is used. The application of the gain suppression matrix is similar to the observation in Zanetti and D’Souza (2013) that the consider filter can be implemented via a full-state filter by setting the appropriate elements of the linear gain to zero. Furthermore, the modification of the gain via the suppression matrix requires the Joseph form of the covariance update to be used.

The suppression of the neglected parameters from the input of Equation (21a) leads to the same predicted measurement as Equation (11a). The effect of (𝕳_{(n)} ◦ **H**_{x,k}) is to set **H**_{n,k−1} = **O**_{dz×dn}. Noting that, from the definition of **x**_{k} in Equation (2),

it follows that Equations (21b) and (21c) produce

which are observed to be the same as Equations (11b)–(11d), except that now exists. It is important to note that is not guaranteed to be zero simply by the act of suppressing the Jacobian. Given the *a priori* relationships of Equations (16), it should be the case that and . To fully ensure that the neglected parameters do not present any influence in the corrector, gain suppression is used. Gain suppression, which is enacted via (𝕲(*c*,*n*) ◦**K**_{x,k}), has the effect of modifying the linear gain of Equation (18) to yield

In other words, under Jacobian and gain suppression modifications, the same mean update equations of Equations (6) are obtained by Equation (20a), no update to the neglected parameters occurs, and the same linear gain of Equation (7) is obtained. This gain suppression process also ensures that there is no effect of the neglected parameters.

While the preceding arguments establish the equivalence of the modified corrector’s mean update to those in Equations (6), a similar equivalence needs to be established for the covariance update. As such, the Jacobian and gain suppression matrices, as well as the prior covariance in the form of Equation (12) with *i* = *k* and ♮ = −, upon substitution into Equation (20b), permit it to be established that the posterior elements of the covariance are

22a

22b

22c

22d

22e

22f

Equations (22a), (22b), and (22d) can be algebraically manipulated to arrive at equivalent results to the covariance update given by Equations (8). Interestingly, posterior correlations between the estimated states and neglected parameters appear through any prior correlations between the estimated states and neglected parameters or through any prior correlations between the considered parameters and neglected parameters. The Jacobian and gain suppression mechanisms within the corrector stage do not prevent this from occurring. Recalling Equations (16c) and (16e), however, establishes that these correlations are prevented from arising from the Jacobian suppression mechanisms within the predictor stage. It is also important to note, from Equations (16c), (16e), and (16f), that , , and . As a consequence, evaluation of Equations (22c), (22e), and (22f) directly yields , , and .

In summary, the consider-neglect filter’s predictor is described by Equations (15), where the necessary Jacobians are given in Equations (14), and the consider-neglect filter’s corrector is described by Equations (20), where the necessary expectations are given in Equations (21), the necessary Jacobians are given in Equation (19), and the linear gain of Equation (18) is used. The filter recursion is initialized at *t*_{k−1} = *t*_{0} with the given initial conditions , **m**_{c,k−1} = **m**_{c,0}, **m**_{n,k−1} = **m**_{n,0}, , , , , , and , which are appropriately inserted into and in accordance with the structure in Equations (12).

### 2.6 Square-root implementation

Following the developments presented in Ward et al. (2019), the conversion of the consider-neglect filter developed in Section 2.5 to a factorized representation is straightforward. The covariance factorization method chosen in this work is the square-root factor represented by a lower-triangular Cholesky factor of the covariance matrix. The *a priori* mean and square-root factor of the total state at *t*_{k} are given by

where is used to represent a lower-triangular Cholesky factor of the corresponding matrix (with *a* designating the variable of interest, *i* denoting the time index, and ♮ representing whether the quantity is prior to or after an update) and qr{⋅} represents a QR decomposition.^{2} The *a posteriori* mean and square-root factor of the total state at *t*_{k} are given by

where the required expectations are

and the linear gain is given by

The filter recursion is initialized at *t*_{k−1} = *t*_{0} with the initial conditions and .

### 2.7 Other considerations

It is important to note that the elements of the total state may not always be cleanly separated, as they are in Equation (2); that is, while each element of **x** belongs to only one of * x*,

*, or*

**c***, they are not likely to be consecutive elements of*

**n****x**. For instance, consider a system with five elements in

**x**. In one case, the first two elements are to be estimated, the fourth element is to be considered, and the third and fifth elements are to be neglected. In another case, the first and fourth elements are to be estimated, the second and fifth elements are to be considered, and the third element is to be neglected. Neither of these cases conforms to the structure of Equation (2), but that is easily resolved by properly selecting 𝖘

_{(n)}, 𝖞

_{(w)}, and 𝖘(

*c*,

*n*). That is, it is only required that all elements of 𝖘

_{(n)}that appear in the indices associated with neglected parameters are zero, all elements of 𝖞

_{(w)}that appear in the indices associated with the noise driving the neglected parameters are zero, and that all elements of 𝖘(

*c*,

*n*) that appear in the indices associated with the considered and neglected parameters are zero. The mechanics of the presented square-root consider-neglect filter then handle the rest without further modification. In this way, a large trade space of available filter configurations is immediately opened up for easy testing and comparison.

When testing various filter configurations, it is common to employ additional modifications, such as incorporating additional process noise in the predictor stage of the filter (which is commonly referred to as “tuning noise”) and supplementing the corrector stage with underweighting factors. These additional elements are easily handled with the square-root implementation of the developed consider-neglect filter. In fact, the method presented in Ward et al. (2019) for handling these modifications is directly applicable. As such, explicit inclusion of such alterations is omitted from the present discussion for brevity.

Another element that warrants discussion for practical filter implementations is the estimation of attitude. Due to the inherently attractive properties of attitude quaternions, they are ubiquitously used in applications involving the estimation of vehicle attitude. The attitude error, on the other hand, is typically represented in a three-parameter space. Treatment of attitude estimation in this way is generally referred to as the multiplicative extended Kalman filter (MEKF). A complete discussion of the MEKF is beyond the scope of this work, and the reader is referred to Crassidis and Junkins (2011) and Markley and Crassidis (2014) for more details concerning the MEKF. Representing the vehicle attitude with an attitude quaternion and the error in the attitude using a three-parameter representation of attitude, such as the rotation vector, generally means that the dimension of the “full state” is larger (by one element) than the dimension of the “error state,” which dictates the size of the uncertainty representation. For the consider-neglect filter of this work, this means that 𝖘_{(n)} will have two definitions: one for the full state and one for the error state. Thus, 𝖘_{(n)} is first defined based on the indexing of the error state to generate 𝕱_{(n)} and 𝕳_{(n)}. Then, 𝖘_{(n)} is redefined with the indexing in terms of the full state. For the noise suppression elements, 𝖞_{(w)} and 𝕱_{(w)} are based on indexing of the error state. Similarly, for the gain suppression elements, 𝖘(*c*,*n*) and 𝕲(*c*,*n*) are defined with indexing in terms of the error state.

## 3 MODELING AND SIMULATION

With the filter architecture and framework for considering and neglecting various states established, candidate parameters to be neglected or considered must be determined. In this work, it is assumed that the proposed filter is processing inertial measurement unit (IMU), star camera, terrain camera, slant-range, and slant-speed measurements to estimate the position, velocity, attitude, and sensor-related parameters. The sensor models are described in detail in Ward et al. (2019). Each sensor is subjected to bias and noise; the accelerometers and gyros that comprise the IMU are additionally subjected to scale factor, misalignment, and nonorthogonality in accordance with the LN200^{3} basis for the IMU model and advanced IMU error modeling practices (Flenniken, Wall, & Bevly, 2005).

Previous research conducted with a similar filter design regarding the error contribution of different states suggests several appropriate choices for the parameters that should be considered and/or neglected (Ward et al., 2019). Specifically, error budget analysis indicates that the states that consistently contribute the least to the filter uncertainty include a majority of the IMU corruption parameters, i.e., gyro scale factor, gyro and accelerometer misalignment, and gyro and accelerometer nonorthogonality. With these prior results in mind, the aforementioned IMU corruption parameters, as well as the accelerometer scale factor, are selected to be considered and/or neglected in the filter while the gyro and accelerometer biases remain in the estimated state.

The consider-neglect filter is tested using four simulations of a lunar descent-to-landing scenario. The first simulation involves full estimation of all states (position; velocity; attitude; accelerometer bias, scale factor, misalignment, and nonorthogonality; gyro bias, scale factor, misalignment, and nonorthogonality; star camera bias; terrain camera bias; slant-range bias; and slant-speed bias) in order to establish a baseline performance for comparison. The second simulation considers the effects of gyro and accelerometer scale factor, misalignment, and nonorthogonality, while estimating all remaining parameters. To also examine the performance of the filter when neglecting parameters, the third simulation considers the IMU scale factors and neglects IMU misalignment and nonorthogonality parameters. The final simulation is identical to the third in consider/neglect parameter selection, but investigates a means to mitigate the performance degradation incurred from the effects of neglecting parameters.

The trajectory used in all simulations is a lunar descent-to-landing trajectory beginning at a 50-km altitude and traveling northeast to terminate just before touchdown near Beer Crater. The vehicle uses IMU, star camera, terrain camera, slant-range, and slant-speed measurements interspersed along the trajectory. Details of the sensor schedules and other specifications are given in Table 1. The sensor operation windows are illustrated in Figure 1 over a period of almost 33 min, from the beginning to the end of the simulated trajectory. Starting at 24-min mission elapsed time (MET), the terminal descent phase of the trajectory begins and is comprised of several attitude maneuvers. The vehicle attitude during the terminal descent phase is given in Figure 2 and is augmented with the sensor operation windows to illustrate what kinds of data are available through different challenging elements of the trajectory. There are three sets of attitude maneuvers shown in Figure 2 that can be challenging for a navigation filter. The first, as mentioned previously, occurs at 24-min MET. The combination of the relatively short time period over which this maneuver occurs and the lack of star camera measurements due to vehicle jitter when thrusting makes this a very difficult estimation hurdle for the filter to overcome. The second set of maneuvers occurs between 30- and 31-min MET, and the final set noted here occurs after 32-min MET. The smaller magnitude and longer duration of these latter two maneuvers makes them less challenging in comparison to the first maneuver, but their effects can still negatively alter filter performance, if not handled carefully.

## 4 RESULTS AND DISCUSSION

For each of the four configurations of the consider-neglect filter considered in this work, a Monte Carlo (MC) simulation of 250 trials is used to determine the sample statistics of the estimator from the actual estimation errors. In addition, the average filter uncertainty is computed from the collection of filter uncertainties obtained from the MC simulation.

### 4.1 Nominal simulation

Results of the baseline (nominal) configuration, in which all of the states and corruption parameters are estimated, are given in Figures 3–5 for the position, velocity, and attitude, respectively. The MC error and average filter uncertainties (±3σ) are given from the beginning of the terrain camera operation window to the end of the simulation. Prior to the activation of the terrain camera, the position and velocity uncertainties grow unabated in the absence of external measurements related to either set of states. At the beginning of the simulation, the attitude uncertainty immediately contracts from its initial value to the steady state seen in Figure 5, and this continues up to 24-min MET. The results given in Figures 3–5 are therefore truncated to focus on the descent phases in which the vast majority of the estimation challenges occur.

The position results in Figure 3 show the expected, rapid convergence of uncertainty just after the terrain camera is activated (shortly before 10-min MET) with a minor amount of over-convergence observed most prominently in the *z*-axis (right panel). The filter then settles to a desirable, slightly cautious estimation through to the star camera shutoff just before 24-min MET. That is, the average filter 3σ interval is somewhat larger than the corresponding MC interval, indicating that the filter is cautious or less precise in its solution than it could be. After the end of terrain camera operations, inclusion of slant-range measurements is able to mostly maintain the estimate through several small attitude maneuvers until the end of the simulation.

Results for the velocity states in Figure 4 show similar trends to their corresponding position states. The initial convergence of uncertainty at the beginning of terrain camera operations is comparatively less dramatic due to the fact that any reduction in velocity uncertainty must be done through correlations between states. After the last star camera measurement is processed at approximately 23.75-min MET, all three velocity states experience a rapid growth in uncertainty, due to the loss of the precise attitude estimate. Once the slant-speed operation window begins after 30-min MET, the estimates quickly converge once again down to their final steady-state values.

Due to the processing of very precise star camera measurements, the attitude results in Figure 5 show consistent (i.e., equivalency of MC and average filter statistics), but unremarkable, steady-state behavior until the vehicle begins thrusting just before 24-min MET. At this point, a large and rapid attitude maneuver causes a significant growth in the uncertainty in all three attitude channels. For several minutes after the maneuver, the filter exhibits somewhat smug estimation, especially in the vehicle roll (left panel). That is, the MC error statistics are not fully captured by the average filter uncertainties, indicating the filter is smug, or overconfident in its estimation. The inclusion of slant-range measurements produces a more conservative estimate and decent convergence, especially in the yaw (right panel) and pitch (center panel) channels.

The results for the remaining states, consisting of the aforementioned IMU corruption parameters and sensor biases, are largely agnostic to the chosen consider and neglect states in the following simulations. The obvious exception is the lack of any update to the uncertainty for the considered or neglected IMU corruption parameters; however, even in the nominal case, there is little improvement to these estimates. As there are no noteworthy differences to discuss, in the interest of conciseness, results for states other than position, velocity, and attitude are not presented.

### 4.2 Consider simulation

Performance results for the second simulation, where the IMU scale factors, misalignments, and nonorthogonalities are considered instead of estimated, are given in Figures 6–8 for the position, velocity, and attitude, respectively. All three sets of estimated states look visually identical to their counterparts in Figures 3–5. The lack of significant change in the results corroborates the findings in Ward et al. (2019): the IMU scale factors, misalignments, and nonorthogonalities are weakly observable and have a minimal effect on the filter performance in terms of position, velocity, and attitude estimation.

The position results in Figure 6 show the same slight overconvergence at the beginning of the terrain camera operation window, followed by a gradual reduction of the uncertainty down to a desirable, slightly cautious steady state. The velocity states in Figure 7 show oscillating performance between smug and cautious behavior, depending on the velocity channel and time period. Finally, the attitude states in Figure 8 exhibit the same estimation trends as the nominal simulation with similar, slightly smug behavior for a short period of time after the star camera is deactivated.

### 4.3 Unmodified consider/neglect simulation

The first of the two simulations with both considered and neglected parameters uses no techniques for ameliorating the estimation quality reduction caused by neglecting the effects of IMU misalignments and nonorthogonalities. Position, velocity, and attitude results for the filter average and MC error ±3σ are shown in Figures 9–11. The position results with both considered and neglected parameters in Figure 9 look visually identical to those of the nominal and consider-only simulations in Figures 3 and 6, respectively. The same can be said of the velocity and attitude results before the terminal descent phase begins; as such, Figures 9–11 start just before the large attitude maneuver at 24-min MET.

Results for the velocity states with both considered and neglected parameters are nearly indistinguishable from their nominal (Figure 4) and consider-only (Figure 7) counterparts for the most part. Some loss of estimation quality can be seen after the large attitude maneuver when the velocity uncertainty experiences rapid growth. After the maneuver, the filter becomes smug again, most significantly in the *x*- and *z*-axis (left and right panels of Figure 10, respectively). The performance degradation seen in the velocity states at this time is due in part to neglecting some accelerometer states, as well as correlations with attitude.

The largest penalties for neglecting the effects of the IMU parameters, specifically in the gyro, are seen in the attitude results in Figure 11. The attitude maneuver at the beginning of the terminal descent phase causes a massive loss of estimation quality, especially in the roll and yaw channels (left and right panels, respectively). Even with the later inclusion of slant-range and slant-speed measurements, the filter never truly manages to recover the estimate once the star camera is deactivated.

### 4.4 Consider/neglect with modifications

In order to address the consider-neglect filter performance degradation in the presence of challenging maneuvers, an appropriate means to artificially increase the filter uncertainty must be determined. Further examination of the error budget analysis in Ward et al. (2019) indicates that during terminal descent, one of the largest contributors to the overall filter uncertainty is the gyro process noise. In much the same way that the gyro and accelerometer consider and neglect parameters are selected due to their relatively low overall contribution, the gyro process noise is chosen to mitigate the performance loss due to its high contribution during the most challenging descent phase. As one of the main factors in choosing to neglect certain parameters is to reduce overall complexity, the approach for improving the estimation quality must not significantly increase the computational burden of the filter. To that end, a static multiplication factor is chosen to artificially boost the gyro process noise. To ensure the correction of the comparatively smaller loss in estimation quality in the velocity states, the accelerometer process noise is also subjected to the same multiplication factor as its gyro counterpart.

Multiplication factors of 1.5, 1.75, 2, 3, and 10 are tested to find an appropriate selection that results in more consistent estimation overall without drastically over-inflating the uncertainties in the process. Results for a static multiplication factor of two in the IMU process noise are shown in Figures 12–14 for the terminal descent phase. Filter performance in all states is once again visually indistinguishable from the previous three simulations until the attitude maneuver at 24-min MET. The modified process noise filter’s estimation trends during the terminal descent phase show significant differences compared to the consider-neglect filter without additional noise from Section 4.3.

Comparison of the velocity results with modified process noise in Figure 13 to the same states in the filter without additional noise given in Figure 10 shows a desirable return to more consistent estimation when the process noise is increased, especially in the *z*-axis velocity (right panel). With modified process noise, the filter provides a more consistent estimate at the cost of moderately increasing the overall uncertainty magnitudes. For the velocity states in the modified process noise filter, the maximum 3σ values are approximately 60, 50, and 60 cm/s in the *x*-, *y*-, and *z*-axis velocities, respectively; the same values for the filter without modified process noise are 46, 44, and 45 cm/s.

The most significant disparity in performance is seen in the attitude, which is shown in Figure 14. Although the filter does not quite manage to return to perfectly consistent estimation with the modified process noise, the slightly smug estimation in the roll and yaw channels (left and right panels, respectively) from 24- to 25-min MET is a vast improvement over the previous results of Figure 11. This recovery of estimation consistency comes at the cost of an overall increase in the uncertainty. The roll, pitch, and yaw 3σ values with modified process noise reach maximums of approximately 400, 300, and 300 arc seconds, respectively, while the corresponding values for the filter without modified process noise are around 250, 200, and 200 arc seconds.

### 4.5 Performance comparison

In order to better understand the differences between the four simulations presented in Sections 4.1–4.4, a more direct comparison of the filter estimation performance is necessary. Figure 15 provides the ratio of the filter average standard deviation (the dashed lines in Figures 3–14) to the MC error standard deviation (solid lines in Figures 3–14) for the nominal (blue), consider only (orange), unmodified consider/neglect (green), and consider-neglect with modified process noise (dark red). Ratios greater than one (green shading) indicate cautious filter performance, while values less than one (red shading) denote overconfident estimation. Ratio values at unity denote equivalency in the MC error and average filter statistics, i.e., consistency in the filter estimation. The performance ratios for all four simulations are visually indistinguishable until the beginning of the terminal descent phase; with this in mind, Figure 15 only presents results for the last ten minutes of the trajectory.

As expected, the largest discrepancies in performance across the four filter configurations occur in the attitude ratios (bottom plot). The unmodified consider-neglect filter shows a significant loss in estimation quality (that is, the MC error statistic interval is significantly larger than that of the average filter) after the large attitude maneuver at 24-min MET and slowly improves thereafter, but never returns to consistent estimation performance. The consider/neglect filter with modified process noise presents significantly improved performance (i.e., ratio values closer to unity) compared to the same filter without modifications, but also fails to match the performance of the nominal and consider only filters. Larger values of the process noise multiplication factor could return the consider/neglect filter to more consistent performance; but as seen in Figures 12–14, increasing the process noise comes at the cost of an overall decrease in estimation precision. The nominal and consider-only simulations exhibit largely indistinguishable performance in attitude estimation, with both displaying slightly cautious filter behavior throughout the last half of the terminal descent phase.

The velocity estimation ratio (middle plot) for all four filter configurations shows somewhat smug behavior during the first few minutes of terminal descent. For the two consider-neglect filters, the estimation quality suffers further after two small attitude maneuvers between 25- and 26-min MET. However, the modified process noise again mitigates the effects of neglecting the IMU parameters, as the consider-neglect filter with the modification displays less smug behavior than its counterpart without modification during this time period. The inclusion of inflated process noise also allows the filter to return to mostly desirable performance trends through the last half of the terminal descent phase. The nominal and consider-only filters exhibit similar cautious velocity estimation behavior throughout most of the last seven minutes of the trajectory.

Comparison of the position ratios across all four filter configurations shows similar trends in estimation quality. Since the IMU corruption parameters are not directly tied to the position states, the effects of considering or neglecting them are seen less prominently in the position estimation. All four configurations exhibit smug behavior during the last minute of the descent, and the assumed cause of this performance degradation is two-fold. The relatively low number of Monte Carlo trials used in this analysis could contribute to the inconsistent estimation; with a larger sample size of 1,000 trials, a similar filter architecture to the one used in the nominal simulation has been shown (Ward et al., 2019) to produce consistent to slightly cautious behavior. The second root cause of the loss of quality in the position estimate is the slant-range sensor. Previous analysis of the topographic slant-range model (Ward & DeMars, 2018) used in this work indicates that the sensor exhibits sensitivity to large attitude uncertainties, especially at altitudes lower than the grid spacing of the underlying topographic model (in this case, approximately 60 m). Since the star camera in this work is deactivated earlier along the trajectory than in previous research, the attitude uncertainty growth is larger and creates more sensitivity in the slant-range sensor model.

Overall, the results of the nominal simulation indicate that the filter is capable of producing an appropriately cautious estimate for the majority of the simulation. While the generally desirable filter performance is not invariant in the presence of difficult estimation scenarios, such as large, rapid convergence or significant attitude maneuvers, the filter demonstrates its ability to recover from such challenges without significant loss of estimation quality. Results for the consider IMU configuration indicate that there is not a drastic change in the filter behavior when several IMU corruption parameters are considered instead of estimated. Comparison between the two consider/neglect configurations indicates that, although the filter without additional process noise ultimately produces a more precise estimate (in terms of the magnitude of the 3σ interval), the result is meaningless if the filter cannot consistently and accurately estimate its own uncertainty; therefore, the modified process noise filter is preferable in this case.

## 5 CONCLUSION

A novel method for incorporating consider and neglect states in square-root navigation filter analysis and development is presented. In increasingly high-fidelity simulations, the number of parameters that must be estimated grows and consequently, so does the associated computational burden. Decreasing the complexity of the navigation filter can be achieved through appropriate selection of parameters to consider or outright neglect in the filter. By building upon previous error budget analysis of the filter architecture used in this work, candidate parameters to consider and neglect are selected and tested in a descent-to-landing simulation. Results show that the filter performance is largely unchanged when the scale factor, misalignment, and nonorthogonality corruptions in the inertial measurement unit gyro and accelerometer are considered instead of estimated. A combination of considered scale factors and neglected misalignment/nonorthogonality parameters is also tested. While the resulting consider-neglect filter is found to produce largely indistinguishable performance from the full estimation filter during the initial descent phase of the trajectory, large and rapid attitude maneuvers during terminal descent prove insurmountable without the now unavailable information from the neglected inertial measurement unit parameters. Mitigating the performance loss when neglecting certain parameters is achieved by artificially inflating the process noise. Results for the modified consider-neglect filter show improvement in filter consistency over that of its unmodified counterpart, but do not fully recover performance seen by the full estimation or consider-only filters. Overall, the presented method and analysis of consider/neglect parameter incorporation provides a means to reduce the computational burden on a navigation filter while also providing insights into the effects of several selections for consider/neglect parameters and performance loss mitigation techniques.

## HOW TO CITE THIS ARTICLE

DeMars KJ,Ward KC. Modular framework for implementation and analysis of recursive filters with considered and neglected parameters. *NAVIGATION*. 2020;67:843–863. https://doi.org/10.1002/navi.388

## Footnotes

↵∗ Associate Professor, Department of Aerospace Engineering

↵† Graduate Student, Department of Mechanical and Aerospace Engineering

**Funding information**National Aeronautics and Space Administration, Grant/Award Numbers: 80NSSC19M0208, NNX16AF11A

↵1 While the results are very similar to those in McCabe and DeMars (2018), there are a few typographical errors in McCabe and DeMars (2018) that are corrected here.

↵2 The manner in which qr{⋅} is used in this work conforms with the “economy form” of Matlab’s qr routine in conjunction with lower-triangular Cholesky factors. It is worth noting that Matlab naturally works with upper-triangular Cholesky factors in its qr routine; the transposing of terms, therefore, facilitates lower-triangular Cholesky factors.

↵3 https://www.novatel.com/assets/Documents/Papers/LN200.pdf

- Received January 3, 2020.
- Revision received April 29, 2020.
- Accepted June 15, 2020.

- Copyright © 2020 Institute of Navigation

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.