Linear Regression Kalman FilteringBased on Hyperspherical Deterministic Sampling

Gerhard Kurz and Uwe D. Hanebeck

Abstract— Nonlinear filtering based on Gaussian densitiesis commonly performed using so-called Linear RegressionKalman Filters (LRKFs). These filters rely on sample-basedapproximations of Gaussian densities. We propose a novelsampling scheme that is based on decomposing the problemof sampling a multivariate Gaussian into sampling a Chidistribution and sampling uniformly on the surface of ahypersphere. The proposed sampling scheme has significantadvantages compared to existing methods because it producesa user-selectable number of samples with uniform, nonnegativeweights and it does not require any numerical optimization.We evaluate the novel method in simulations and providecomparisons to multiple state-of-the-art approaches.

I. INTRODUCTION

Nonlinear estimation has been of interest for a long timeand work in this area dates back to the 1960s, when theExtended Kalman Filter (EKF) was proposed [1]. Since then,numerous approaches have been investigated. An importantclass of methods are the so-called Linear Regression KalmanFilters (LRKFs) [2], which use sample-based statisticallinearization in conjunction with the standard Kalman filter [3].LRKFs, in particular the Unscented Kalman Filter (UKF) [4],have gained a lot of popularity due to their simplicity andthe ability to provide good results in many scenarios at areasonable computational cost. The key distinction betweendifferent LRKFs is the way they choose the set of samplesused for the statistical linearization.

In this paper, we present a novel sampling scheme that canbe used within the LRKF framework. The key idea is splittingthe approximation of a multivariate standard Gaussian in Rdinto approximation of the direction as well as the lengthof the vectors. The direction can be sampled according to auniform distribution on the hypersphere and the length can besampled according to a Chi distribution1. For this purpose, weuse an equal area partitioning approach on the hypersphere inconjunction with a one-dimensional Chi distribution samplingbased on minimization of the squared L2 distance of thecumulative distribution functions. This idea is illustrated inFig. 1. Our method creates a layered structure, where thesamples are placed on nested hyperspheres.

The high-degree Cubature Kalman Filter (CKF) [5] alsorelies on similar ideas. It uses numerical integration rulesthat are based on splitting the integral into a spherical anda radial part. For the third-degree CKF [6], all samples arelocated on a sphere, and for the fifth-degree CKF, all samples—except for a sample at the origin—are located on a sphere.

The authors are with the Intelligent Sensor-Actuator-Systems Laboratory(ISAS), Institute for Anthropomatics and Robotics, Karlsruhe Instituteof Technology (KIT), Germany. E-mail: gerhard.kurz@kit.edu,uwe.hanebeck@ieee.org

1The original version of this paper mistakenly stated that the lengthfollows a Gaussian distribution. This is the revised version of the paper,where we have corrected this issue and rerun all simulations using the correctdistribution.

Fig. 1: Idea of the proposed deterministic sampling scheme.A uniform sampling on the sphere (red balls) is computedusing equal area partitioning. Samples of one-dimensional Chidistribution (green balls) are placed on the lines emanatingfrom the center of the sphere through the spherical samples.

The downside compared to our proposed method is that thenumber of samples is fixed by the dimension and the cubaturerule. Also, for the fifth-degree rule, negative weights cannotbe avoided for dimensions d ≥ 5, which can cause problemsthat we will discuss later.

Furthermore, the proposed approach is somewhat similar tothe filter by Huber et al. [7], which samples a one-dimensionalGaussian distribution and replicates it on each axis ofthe coordinate system to obtain samples for a multivariateGaussian distribution. One of the key advantages of theproposed approach is that the sample locations are not limitedto the axes of the coordinate system, which leads to muchbetter results.

Another related approach is the randomized UKF by Strakaet al. [8]. It takes multiple UKF sample sets and performs arandom rotation as well as a random scaling on each. Whilethis approach also relies on the separation of the directionfrom the length of the sample vectors, it is inherently nonde-terministic, which makes it difficult to generate reproducibleresults. The quality of the approximation can vary stronglydepending on the chosen random values. Another problem isthe presence of negative weights.

Other LRKFs include the Gaussian Hermite KalmanFilter [9], [12], a filter based on a quadrature scheme thatrequires exponentially many samples with respect to thedimension. Furthermore, there is the Smart Sampling KalmanFilter (S2KF) [10], which is based on placing the samplesaccording to an optimality criterion that minimizes a distancefunction between the Gaussian distribution and the samples. Asimilar approach with additional moment constraints is takenin [13]. A downside of these methods is that a computationallyexpensive numerical optimization has to be performed toobtain the sample set. Thus, the optimization is usually

Sampling Scheme Number of Samples Weights Deterministic NumericalOptimization

UKF [4] 2d+ 1 neg. weights optionala yes noRUKF [8] multiple of 2d+ 1 neg. weights no no3rd degree CKF [6] 2d equal yes no5th degree CKF [5] 2d2 + 1 neg. weightsb yes noGHKF [9] md nonequal, positive yes noS2KF [10] arbitrary ≥ 2d equal yesc yesd

proposed arbitrary ≥ 2d equal yes no

a uniform weights are al-ways possible

b for dimensions ≥ 5c depending on initializationd an implementation of the

numerical optimizationprocedure is availableonline [11]

TABLE I: Overview of LRKFs.

performed offline for a standard normal distribution and thesamples are transformed to the current distribution using aMahalanobis transformation [14, Sec. 3.3]. This makes itdifficult to change the number of samples or dimensions atruntime. An overview of some of the most popular approachesis given in Table I.

In general, a sampling scheme for an LRKF has certaindesirable properties. First of all, we want to maintain the firstand second moment of the Gaussian distribution to ensurethat the LRKF is equivalent to the optimal Kalman filterfor linear systems. Matching the first two moments meansthat a Gaussian can be converted to samples and vice versawithout losing any information, which is why all LRKFsconsidered here fulfill this property. It may be desirableto maintain higher moments of the Gaussian distribution,because Gaussian quadrature [15, Sec. 2.7] rules guaranteeoptimality of certain moments for polynomial systems up toa predetermined degree. In practice, however, it may be moreimportant to have a fairly even distribution of the samplessuch that the space is well covered and the shape of theGaussian distribution is well matched. It is also beneficial ifthe number of samples can be adjusted to perform a trade-offbetween computational effort and accuracy. Unfortunately,many approaches only allow a fixed number of samples or achoice that is limited to very coarse steps. Another issue isthe computational effort needed to obtain the samples. Someapproaches use very costly numerical optimization methodsthat have to be performed offline, in particular for manysamples or dimensions. Moreover, we would like the sampleset to be deterministic, so results are easily reproducible andthe accuracy of the estimate does not depend on the choiceof random numbers.

Finally, it is advantageous if the weights of the samplesare as uniform as possible because in this case, all samplescontribute equally to the result. In particular, samples withnegative weights should be avoided as they can result incovariance matrices that are not positive definite, as weillustrate in the following example.

Example 1 (Negative Weights). Consider a standard normaldistribution N (0, 1), which we want to propagate throughthe simple yet nonlinear function f(x) = (x− 0.2)2.

For this purpose, we use an LRKF with negative weights,say, the UKF with negative weight γ1 = −1 (W0 = −1according to the notation used in [4, Sec. IV-A]). This yieldsthe sample locations

[0,−√

2/2,√

2/2] ≈ [0,−0.7071, 0.7071]

and weights [γ1, γ2, γ3] = [−1, 1, 1]. The propagated samples

are at locations

[r1, r2, r3] = [0.0400, 0.8228, 0.2572]

and their weights stay the same. We obtain the sample mean

µ =∑Ni=1 γi · ri = 1.04 ,

which is identical to the analytic solution. However, thecovariance is

C =∑Ni=1 γi · (ri − µ)(ri − µ)T

= −1 · 1 + 1 · 0.0472 + 1 · 0.6128 = −0.34 < 0 ,

i.e., not positive definite. As a result, future filtering stepsbased on this covariance are impossible. A common practicalsolution is to omit steps that lead to non-positive-definitecovariance matrices, which is clearly suboptimal.

II. KEY IDEA

In the following, we will show how to combine samplesform a Chi distribution and uniform hyperspherical samples toobtain a sampling scheme for multivariate Gaussian densities.Consider a multivariate standard Gaussian distribution in ddimensions, which can be reformulated as

N (x; 0, I) =1

(2π)d/2exp

(−xTx/2

)=

1

(2π)d/2exp

(−‖x‖2/2

)=

1

(2π)(d−1)/2N (‖x‖; 0, 1) .

It can be seen that this distribution is uniform distributionwith respect to the direction x

‖x‖ . Note that the direction doesnot appear in the equation due to uniformity.

The distribution of the norm r := ‖x‖ ≥ 0 can be derivedaccording to

f(r) =

∫{x∈Rd:‖x‖=r}

N (x; 0, I) dx

=1

(2π)d/2exp

(−r2/2

) ∫{x∈Rd:‖x‖=r}

dx

=1

(2π)d/2exp

(−r2/2

) 2πd/2

Γ(d/2)rd−1

=rd−1 exp

(−r2/2

)2d/2−1Γ(d/2)

,

where we use the fact that the surface of a (d−1)-dimensionalsphere in d dimensional space is 2πd/2

Γ(d/2)rd−1 with Gamma

function Γ(·). As can be seen, the resulting density coincideswith a Chi distribution with d degrees of freedom.

As a result, we can sample the Chi distribution on ‖x‖and the uniform distribution on x

‖x‖ separately (using the

-5 0 5

x

0

0.1

0.2

0.3

0.4

f(x)

(a) Gaussian pdf.

-5 0 5

x

0

0.5

1

F(x

)

(b) Gaussian cdf.

0 1 2 3 4 5

x

0

0.2

0.4

f(x)

(c) Chi pdf.

0 1 2 3 4 5

x

0

0.5

1

F(x

)

(d) Chi cdf.

Fig. 2: Example for a 1D deterministic approximation of a Gaussian distribution and a Chi distribution for d = 3 degrees offreedom.

techniques presented in Sec. IV and Sec. III) and then obtainsamples for the multivariate Gaussian by considering theirCartesian product.

To obtain our sample set, we scale each hypersphericalsample si with each of the Chi distribution samples βjaccording to

{r1, . . . , rN} = {si · βj |1 ≤ i ≤M, 1 ≤ j ≤ L} ,

where M is the number of spherical samples and L is thenumber of Gaussian samples or layers. Hence, each Gaussiansample creates one layer of samples. The weights of thecombined samples are obtained by multiplying the weightsof the individual samples. In our case, we obtain uniformweights γ1 = · · · = γN = 1

M ·L for all samples.Samples for a non-standard Gaussian distribution N (µ,C)

can be obtained by performing a Mahalanobis transformation,i.e, rtransformed

j = µ+√C · rj for 1 ≤ j ≤ N , where

√C can

be computed using the Cholesky decomposition.

III. DETERMINISTIC HYPERSPHERICAL SAMPLING

In this section, we focus on the problem of computing aset of samples uniformly distributed on the surface of theunit hypersphere Sd−1 = {x ∈ Rd : ‖x‖ = 1}, i.e., theset of vectors with Euclidean norm 1. We seek to obtain asubset {s1, . . . , sM} ⊂ Sd−1 of M ∈ N vectors that coversthe surface of the unit hypersphere evenly. While uniformrandom sampling on the surface of the unit hypersphere isvery easy, deterministic sampling can be quite tricky. Thisproblem has been studied quite extensively, especially forthe case d = 3, i.e., the sphere in R3 [16]. It arises in manyapplications, sometimes in slight variations as far as themeasure of uniformity is concerned. For instance, in physics,the Thompson problem [17], [18, Sec. 18.7] considers thequestion of distributing electrons on the surface of a spheresuch that the Coulomb energy is minimized. Another relatedexample is the Tammes problem in botany, which considersthe distribution of pores on pollen grains [18, Sec. 18.9].Optimal quantization, though usually on real vector spacesrather than the sphere, is also a closely related issue [19],[20]. Many approaches for the spherical problem can befound in literature, e.g., [21], [16], and also some for thehyperspherical case [22], [23].

We use the equal area partitioning approach proposedby Leopardi [24], [25], which partitions the surface of thesphere into regions of equal area. The regions are chosensuch that their diameter, i.e., the largest distance between anytwo points in the region, is small. Leopardi proves an upperbound for the diameter that converges to zero if the number

of regions goes to infinity. We use the center of each region asa hyperspherical sample. The algorithm is based on recursivepartitioning of the hypersphere in d dimensions by reducingits partition to that of the hypersphere in d− 1 dimensions.A MATLAB implementation is available online [26].

Leopardi’s algorithm has a number of beneficial properties.As it is not based on numerical optimization of an optimalitycriterion, it is extremely fast, does not get stuck in localoptima, and is independent of initialization. Also, it can beapplied to an arbitrary number of dimensions d ≥ 2 and anarbitrary numbers of samples M ∈ N.

Due to the recursive construction, the resulting samplesare not spread perfectly evenly on the hypersphere, but it canbe verified empirically that the results are very close to auniform distribution. Furthermore, the bound on the diameterof the regions and the symmetry of the construction providetheoretical guarantees. Examples for the resulting partitionsand samples are shown in Fig. 4.

IV. DETERMINISTIC CHI DISTRIBUTION SAMPLING

In this section, we consider the problem of approximating aChi distribution with a set of samples. To be specific, we arelooking for the parameters w1, . . . , wL > 0 and β1, . . . , βLof a Dirac mixture

f(x;β1, . . . , βL, w1, . . . , wL) =∑L

i=1wiδ(x− βi) (1)

with L ∈ N components, where∑Li=1 wi = 1 holds. In the

following, we restrict ourselves to equally weighted samples,i.e., w1 = · · · = wL = 1

L .In [27], Schrempf et al. proposed a sampling scheme

for one-dimensional densities based on minimization of thesquared L2 distance of cumulative distribution function. Thedistance measure for densities f1(·) and f2(·) is given by

D =∫∞−∞(F1(x)− F2(x))2 dx ,

where Fi(x) =∫ x−∞ fi(t) dt is the cumulative distribution

function. If f2(·) is a Dirac mixture as defined in (1) and isassumed to have equal weights, then it can be shown thatthe optimal approximation of f1(·) given by

βi = F−11

((2i− 1)/(2L)

)(2)

according to [27, Theorem III.1].However, this approach does not, in general, preserve

any moments of the density f1(·). For symmetric densities(such as the Gaussian), the first moment is maintained, buthigher moments are not. In [7], Huber et al. introduceda constraint for the second moment to resolve this issue,

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3M=4, L=2

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3M=4, L=5

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3M=4, L=10

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3M=10, L=2

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3M=10, L=5

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3M=10, L=10

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3M=20, L=2

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3M=20, L=5

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3M=20, L=10

(a) Examples in 2D.

-2

2

0

2

M=5, L=2

2

00

-2 -2

-2

2

0

2

M=5, L=5

2

00

-2 -2

-2

2

0

2

M=5, L=10

2

00

-2 -2

-2

2

0

2

M=10, L=2

2

00

-2 -2

-2

2

0

2

M=10, L=5

2

00

-2 -2

-2

2

0

2

M=10, L=10

2

00

-2 -2

-2

2

0

2

M=20, L=2

2

00

-2 -2

-2

2

0

2

M=20, L=5

2

00

-2 -2

-2

2

0

2

M=20, L=10

2

00

-2 -2

(b) Examples in 3D.

Fig. 3: Examples of the sample sets produced by the novel sampling scheme.

which requires numerical optimization. The main problemwith this approach is that using the one-dimensional samplesin multiple dimensions will once again violate the secondmoment constraint. For this reason, our approach doesnot use any moment constraints, but instead enforces thesecond moment retroactively in n dimensions by using theMahalanobis transformation (see Sec. V).

As discussed in Sec. III, we obtain samples from thehypersphere, which correspond to rays starting at origin toa certain direction. Thus, we need to approximate the a Chidistribution for each ray by computing its inverse cdf as givenby (2). For practical implementation, the square root of theChi squared distribution inverse cdf can be used becauseimplementations of this function are more readily available(e.g., chi2inv in MATLAB).

An example of the results can be seen in Fig. 2.

V. MOMENT CORRECTION

As the samples obtained this way do not necessarilyhave exactly the mean and covariance of a standard normaldistribution, we subtract the actual mean

µ := E(x) =∑Ni=1 γiri

from each sample. Then, we apply the Mahalanobis transfor-mation to correct the covariance as is done in the S2KF [10].For this purpose, we compute the sample covariance

C := E(xxT ) =∑Ni=1 γirir

Ti

and obtain its Cholesky decomposition, i.e., we obtain anupper triangular matrix R such that RT ·R = C. Then, wemultiply each sample from the left with R−1, which yieldssamples with covariance I. Examples of the resulting sample

sets in 2D and 3D for different values of L and M are shownin Fig. 3.

VI. FILTER

We can use the samples derived above in the standardLRKF framework [2], [10, Sec. 2].

A. Prediction Step, Time Update

The system model is given by

xk+1 = ak(xk, wk) ,

where wk ∼ N (0,Cwk ) is non-additive zero-mean Gaussian

process noise. To handle the non-additive noise, we samplefrom joint density of state and noise

[xk, wk]T ∼ N ([xek, 0]T ,diag(Cek,C

wk )) .

Based on the samples [xk,1, wk,1]T , . . . , [xk,N , wk,N ]T withweights γ1, . . . , γN , we can obtain mean and covariance ofthe predicted density according to

xpk+1 ≈∑Ni=1 γiak(xk,i, wk,i) ,

Cpk+1 ≈

∑Ni=1 γi(ak(xk,i, wk,i)− x

pk+1)

· (ak(xk,i, wk,i)− xpk+1)T .

For additive noise, we only need to sample from the statedensity and the equations can be simplified [28, Sec. 2.4.4].

B. Correction Step, Measurement Update

We assume a measurement model

yk

= hk(xk, vk) ,

N = 10 N = 20 N = 50 N = 100 N = 500

Fig. 4: Equal area partitioning on the sphere S2.

where vk ∼ N (0,Cvk) is non-additive zero-mean Gaussian

measurement noise. Once again, we sample from the jointdensity of state and noise

[xk, vk]T ∼ N ([xpk, 0]T ,diag(Cpk,C

vk)) .

Then, we compute the expected measurement

yk≈∑Ni=1 γihk(xk,i, vk,i)

and its covariance

Cyk ≈

∑Ni=1 γi(hk(xk,i, vk,i)− yk) · (hk(xk,i, vk,i)− yk)T

as well as the cross-covariance of state and measurement

Cx,yk ≈

∑Ni=1 γi(xk,i − x

pk) · (hk(xk,i, vk,i)− yk)T .

Using the actual measurement yk, we perform the Kalman

update to obtain the mean and covariance of the estimatedstate according to

xek = xpk + Cx,yk · (Cy

k)−1 · (yk− y

k) ,

Cek = Cp

k −Cx,yk · (Cy

k)−1 · (Cx,yk )T .

If the noise is additive, it is sufficient to sample from the statedensity and we can simplify the equations [28, Sec. 2.4.4].

VII. EVALUATION

We evaluate the proposed approach in comparison withLRKFs found in literature. For this purpose, we rely onthe implementations available in the Nonlinear EstimationToolbox [11] for MATLAB.

A. Sample AnalysisBefore we evaluate the novel approach in a filtering

scenario, we take a close look at the sample placementand perform a comparison to other sampling schemes.Examples of samples produced by different methods for atwo-dimensional scenario can be seen in Fig. 5. The S2KFproduces the most homogeneous samples in this case.

To visualize the behavior of the sampling schemes in higherdimensions, we consider the distance of each sample to theorigin. Although this does not capture all aspects of thesampling schemes, it helps to understand the radial component.The results for the proposed filter, the S2KF and the RUKFare shown in Fig. 6 and Fig. 7. We do not depict the UKF andthe CKF as their samples are exactly located on a hypersphere(except for a single sample at the origin). It can be seen thatthe samples of the proposed filter are not exactly located onhyperspheres due to the covariance correction (see Sec. V).In the 5D case, the S2KF also creates a structure resemblingmultiple hyperspheres in some cases, whereas the RUKF is

quite random. While the different behaviors of the consideredsampling schemes are highly interesting, it is not obviouswhich choice of samples yields the best results in a filteringapplication, so we will look at this aspect in Sec. VII-B.

Furthermore, we investigate the expected squared distanceof a Gaussian-distributed random vector from the closestsample. Intuitively, the closer this vector is located to a samplepoint, the better the nonlinear mapping will be approximated.For x ∼ N (0, I), we consider

E(min1≤i≤N ‖x− ri‖22

).

This expectation value can be computed for a given sampleset using numerical integration. It corresponds to the squareddistortion measure used in [20, Sec. 2.1]. The results forthe 2D case are shown in Fig. 8. It can be seen that theS2KF performs very well because it spreads the samples veryevenly. The proposed approach can achieve similar resultsfor a sufficient number of layers L. It can also be observedthat increasing the number of layers only pays off when thetotal number of samples N is sufficiently high.

B. Filter Comparison

We evaluate the novel filter in scenarios with differentnumbers of dimensions. As the measurement update istypically the more challenging part, we assume a linearrandom walk system model

xk+1 = xk + wk ,

where wk ∼ N (0, 10−2 · I) is additive zero-mean systemnoise. The measurement model is given by

h(xk) = xk +

sin(c · xk,2)

...sin(c · xk,d)sin(c · xk,1)

+ vk ,

where vk ∼ N (0, 10−4 · I) is zero-mean measurement noise.We choose the constant parameter c = 10. This model isinteresting because the behavior strongly depends on theuncertainty at the current time step. Also, it can be usedin an arbitrary number of dimensions. The true initial stateis given by x0 = 0 and the initial estimate is given byxe0 ∼ N (0, 10−2 · I).

In our evaluation, we compare the proposed approach to theEKF, the UKF, the 5th degree CKF, the S2KF, the randomizedUKF, and the GHKF (with m = 3 points in each dimension).For the proposed approach, the S2KF, and the RUKF, weused N = 2d2 + 1 samples. Furthermore, we use L = 5

-2 0 2-3

-2

-1

0

1

2

3

-2 0 2-3

-2

-1

0

1

2

3

-2 0 2-3

-2

-1

0

1

2

3

-2 0 2-3

-2

-1

0

1

2

3

-2 0 2-3

-2

-1

0

1

2

3

-2 0 2-3

-2

-1

0

1

2

3

UKF CKF (5th degree) GHKF (m = 3) RUKF S2KF proposed

Fig. 5: Comparison of sampling schemes in 2D. The weights are illustrated using the size of the samples and negativeweights are shown in red.

proposed (L = 5) S2KF RUKF

Fig. 6: Norm of all samples in 5D for different number of samples N .

proposed (L = 5) S2KF RUKF

Fig. 7: Norm of all samples for different number of dimensions and N = 500.

0 50 100 150 200

number of samples

0

0.1

0.2

0.3

0.4

0.5

0.6

expe

cted

dis

tanc

e to

clo

sest

sam

ple

proposed 1 layerproposed 2 layersproposed 3 layersproposed 4 layersproposed 5 layersS2KFUKFCKF

Fig. 8: Expected distance to the closest sample in 2D. Notethat the number of samples for the UKF and CKF are fixed.

layers in our method, which has empirically been found towork quite well as long as N is large enough.

We performed the evaluation in 2D, 3D, and 5D for 50time steps each. The results of 100 Monte Carlo runs areshown in Fig. 9. We use the RMSE averaged across all runsas the evaluation criterion. The novel method achieves asimilar estimation accuracy as the S2KF and RUKF in theconsidered scenario, while being deterministic and avoidingslow precomputation. These three methods clearly outperform

the EKF and UKF. The CKF works quite well in 2D and3D, but is a lot worse in 5D because the negative weightscauses frequent failures of the update step. The RUKF alsoexhibits occasional failures, but in the considered scenarios,the effect on its performance seems limited.

VIII. CONCLUSION

We have presented a novel sampling scheme based on acombination of Chi distribution sampling and uniform hyper-spherical sampling. This approach has significant advantagescompared to state-of-the-art methods, namely, no negativeweights, a flexible number of components, no randomness,and no need for costly numerical optimization.

Based on this sampling scheme, we have proposed anew filter based on the LRKF-principle. We have evaluatedthe novel filter in multiple simulations and shown that itsperformance is clearly superior to the EKF, UKF, CKF, andGHKF. It is comparable to the RUFK and the S2KF, whileavoiding their disadvantages, namely the nondeterminismand negative weights of the RUKF and the expensiveprecomputation of the S2KF.

0 10 20 30 40 50

time step

0

0.5

1

1.5

2

2.5

3

RM

SE

ove

r al

l run

s

proposedUKFEKFS2KFRUKFCKFGHKF

(a) RMSE in 2D.

0 10 20 30 40 50

time step

0

0.5

1

1.5

2

2.5

3

RM

SE

ove

r al

l run

s

proposedUKFEKFS2KFRUKFCKFGHKF

(b) RMSE in 3D.

0 10 20 30 40 50

time step

0

0.5

1

1.5

2

2.5

3

RM

SE

ove

r al

l run

s

proposedUKFEKFS2KFRUKFCKFGHKF

(c) RMSE in 5D.

0 10 20 30 40 50

time step

0

2

4

6

8

10

faile

d m

easu

rem

ent u

pdat

es (

perc

ent)

proposedUKFEKFS2KFRUKFCKFGHKF

(d) Failures in 2D.

0 10 20 30 40 50

time step

0

2

4

6

8

faile

d m

easu

rem

ent u

pdat

es (

perc

ent)

proposedUKFEKFS2KFRUKFCKFGHKF

(e) Failures in 3D.

0 10 20 30 40 50

time step

0

20

40

60

80

100

faile

d m

easu

rem

ent u

pdat

es (

perc

ent)

proposedUKFEKFS2KFRUKFCKFGHKF

(f) Failures in 5D.

Fig. 9: Evaluation results.

ACKNOWLEDGMENT

The authors thank Jannik Steinbring for his support. Thiswork was supported by the German Research Foundation(DFG) under grant HA 3789/13-1.

REFERENCES

[1] G. L. Smith, S. F. Schmidt, and L. A. McGee, Application of StatisticalFilter Theory to the Optimal Estimation of Position and Velocityon Board a Circumlunar Vehicle. National Aeronautics and SpaceAdministration, 1962.

[2] T. Lefebvre, H. Bruyninckx, and J. D. Schuller, “Comment on ”A NewMethod for the Nonlinear Transformation of Means and Covariancesin Filters and Estimators” [with Authors’ Reply],” IEEE Transactionson Automatic Control, vol. 47, no. 8, pp. 1406–1409, Aug. 2002.

[3] R. E. Kalman, “A New Approach to Linear Filtering and PredictionProblems,” Transactions of the ASME Journal of Basic Engineering,vol. 82, pp. 35–45, 1960.

[4] S. J. Julier and J. K. Uhlmann, “Unscented Filtering and NonlinearEstimation,” Proceedings of the IEEE, vol. 92, no. 3, pp. 401–422,Mar. 2004.

[5] B. Jia, M. Xin, and Y. Cheng, “High-degree Cubature Kalman Filter,”Automatica, vol. 49, no. 2, pp. 510–518, 2013.

[6] I. Arasaratnam and S. Haykin, “Cubature Kalman Filters,” IEEETransactions on Automatic Control, vol. 54, no. 6, pp. 1254–1269,2009.

[7] M. F. Huber and U. D. Hanebeck, “Gaussian Filter based on Determin-istic Sampling for High Quality Nonlinear Estimation,” in Proceedingsof the 17th IFAC World Congress (IFAC 2008), vol. 17, no. 2, Seoul,Republic of Korea, Jul. 2008.

[8] O. Straka, J. Dunik, and M. Simandl, “Randomized Unscented KalmanFilter in Target Tracking,” in 15th International Conference onInformation Fusion (Fusion 2012), 2012, pp. 503–510.

[9] K. Ito and K. Xiong, “Gaussian Filters for Nonlinear Filtering Problems,”IEEE Transactions on Automatic Control, vol. 45, no. 5, pp. 910–927,May 2000.

[10] J. Steinbring, M. Pander, and U. D. Hanebeck, “The Smart SamplingKalman Filter with Symmetric Samples,” Journal of Advances inInformation Fusion, vol. 11, no. 1, pp. 71–90, Jun. 2016.

[11] J. Steinbring, “Nonlinear Estimation Toolbox,” 2015. [Online].Available: https://bitbucket.org/nonlinearestimation/toolbox

[13] U. D. Hanebeck, “Sample Set Design for Nonlinear Kalman Filtersviewed as a Moment Problem,” in Proceedings of the 17th InternationalConference on Information Fusion (Fusion 2014), Salamanca, Spain,Jul. 2014.

[12] Z. Wang and Y. Li, “Cross Approximation-based Quadrature Filter,”in 2016 IEEE 55th Conference on Decision and Control (CDC), LasVegas, NV, USA, 2016.

[14] W. Hardle and L. Simar, Applied Multivariate Statistical Analysis,2nd ed. Berlin, Germany: Springer, 2007.

[15] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration,2nd ed., ser. Compute Science and Applied Mathematics. San Diego,California, USA: Academic Press, Inc., 1984.

[16] E. B. Saff and A. B. Kuijlaars, “Distributing Many Points on a Sphere,”The Mathematical Intelligencer, vol. 19, no. 1, pp. 5–11, 1997.

[17] J. J. Thomson, “On the Structure of the Atom: An Investigation ofthe Stability and Periods of Oscillation of a Number of CorpusclesArranged at Equal Intervals around the Circumference of a Circle;with Application of the Results to the Theory of Atomic Structure,”Philosophical Magazine Series 6, vol. 7, no. 39, pp. 237–265, 1904.

[18] T. Aste and D. Weaire, The Pursuit of Perfect Packing, 2nd ed. Taylor& Francis, 2008.

[19] Q. Du, V. Faber, and M. Gunzburger, “Centroidal Voronoi Tessellations:Applications and Algorithms,” SIAM review, vol. 41, no. 4, pp. 637–676,1999.

[20] G. Pages and J. Printems, “Optimal Quadratic Quantization forNumerics: The Gaussian Case,” Monte Carlo Methods and Applications,vol. 9, no. 2, pp. 135–165, 2003.

[21] A. Katanforoush and M. Shahshahani, “Distributing Points on theSphere, I,” Experimental Mathematics, vol. 12, no. 2, pp. 199–209,2003.

[22] H. Peng and Y. Yu, “Project Report for CS59000 OPT: Optimizationon the Surface of the (Hyper)-Sphere,” Tech. Rep., May 2012.

[23] F.-F. Henrich and K. Obermayer, “Active Learning by SphericalSubdivision,” Journal of Machine Learning Research, vol. 9, no. Jan,pp. 105–130, 2008.

[24] P. Leopardi, “A Partition of the Unit Sphere into Regions of Equal Areaand Small Diameter,” Electronic Transactions on Numerical Analysis,vol. 25, no. 12, pp. 309–327, 2006.

[25] ——, “Distributing Points on the Sphere: Partitions, Separation,Quadrature and Energy,” Ph.D. dissertation, University of New SouthWales, Apr. 2007.

[26] ——, “Recursive Zonal Equal Area Sphere Partitioning Toolbox 1.10,”http://eqsp.sourceforge.net/, Jun. 2005.

[27] O. C. Schrempf, D. Brunn, and U. D. Hanebeck, “Density Approxima-tion Based on Dirac Mixtures with Regard to Nonlinear Estimation andFiltering,” in Proceedings of the 2006 IEEE Conference on Decisionand Control (CDC 2006), San Diego, California, USA, Dec. 2006.

[28] I. Gilitschenski, “Deterministic Sampling for Nonlinear Dynamic StateEstimation,” Ph.D. dissertation, Karlsruhe Institute of Technology,Intelligent Sensor-Actuator-Systems Laboratory, 2015.