Convergence Order in Trajectory Estimation by Piecewise-Cubics and Exponential Parameterization

. This paper discusses the problem of estimating the trajectory of the unknown curve γ from the sequence of m + 1 interpolation points Q m = { γ ( t i ) } mi =0 in arbitrary Euclidean space E n . The respective knots T m = { t i } mi =0 (in ascending order) are assumed to be unknown. Such Q m is coined reduced data . In our setting, a piecewise-cubic Lagrange interpolation ˆ γ 3 : [0 , ˆ T ] → E n is applied to ﬁt Q m . Here, the missing knots T m are replaced by their estimates ˆ T m = { ˆ t i } mi =0 in accordance with the exponential parameterization. The latter is controlled by a single parameter λ ∈ [0 , 1]. This work analyzes the intrinsic asymptotics in approximating γ by ˆ γ 3 based on the exponential parameterization and Q m . The multiple goals are achieved. Firstly, the existing result established for λ = 1 (i.e. for the cumulative chord parameterization) is extended to the remaining cases reparameterizable to the domain of γ are formulated. Lastly, the motivation for using the exponential parameterization with λ ∈ [0 , 1) is also outlined.

Having chosen an interpolation schemeγ : [0,T ] → E n and some substitutes {t i } m i=0 of the respective missing knots {t i } m i=0 , the question of the intrinsic asymptotic order α in γ approximation byγ arises naturally. Recall first the basic definition (see e.g. [1], [6] or [19]): The following subclass of (1.1) is here of particular relevance (see [9]): Definition 2. The sampling {t i } m i=0 is more-or-less uniform if for some constants 0 < K l ≤ K u and sufficiently large m: K l m ≤ t i − t i−1 ≤ Ku m holds for all i = 1, 2, . . . , m. Equivalently, the last two inequalities can be replaced by satisfied for some β ∈ (0, 1] and sufficiently large m. Recall now the following (see e.g. [1] or [6]): We say that f δm is of order O(δ α m ) (denoted as f δm = O(δ α m )), if there is a constant K > 0 such that, for someδ > 0 the inequality |f δm (t)| ≤ Kδ α m holds for all δ m ∈ (0,δ), uniformly over [0, T ]. In case of vector-valued functions F δm : [0, T ] → E n by F δm = O(δ α m ) it is understood that F δm = O(δ α m ). This paper deals with a special family of discrete exponential parametetrizations designed to estimate the missing knots {t i } m i=0 (see e.g. [14]) according to:t 0,λ = 0, t i+1,λ =t i,λ + q i+1 − q i λ , for arbitrary λ ∈ [0, 1] and i = 0, 1, 2, . . . , m − 1. In order to preserve the ascending order ofT λ = {t i,λ } m i=0 an extra constraint q i+1 = q i is imposed here on the admitted class of reduced data Q m . Visibly, the case of λ = 0 Theorem 1. Let γ be a C 3 ([0, T ]) regular curve in E n sampled more-or-less uniformly (see Definition 2). Assume that Q m forms reduced data with the unknown knots estimated by (1.3). Then, uniformly over [0, T ]: and for either uniform {t i } m i=0 = iT /m or λ = 1 we have: Moreover, if the mapping ψ 2,i is a reparameterization of I i intoÎ i (e.g. when {t i } m i=0 is uniform or for λ = 1) then, in both (1.4) and (1.5), a phraseγ 2,i can be replaced byγ 2,i .
Note that if λ = 1 the condition (1.2) can be replaced by the weaker assumption imposed on samplings T i.e. by the condition (1.1). The asymptotics from Theorem 1 is also proved to be sharp (see [9]) understood as: Definition 4. For a given interpolation schemeγ based on reduced data Q m and some estimatesT of the unknown knots T (and subject to some selected mapping ψ : [0, T ] → [0,T ]) we say that asymptotics γ −γ • ψ = O(δ α m ) over [0, T ] is sharp within the prescribed family of curves γ ∈ J and family of samplings T ∈ K, if for some γ ∈ J and some sampling T ∈ K, there existt ∈ [0, T ] and some positive constant K such that (γ • ψ)(t) − γ(t) = Kδ α m + O(δ η m ), where η > α.

Main results and motivation
We emphasize now the main contributions and the motivation of this paper: 1. This paper extends the analysis of approximation error inγ 3,i •ψ 3,i −γ for the remaining λ ∈ [0, 1) -the case λ = 1 is covered in [17]. In fact, as proved in the next section an analogous sharp deceleration in convergence orders claimed by Theorem 1 appears also for r = 3 and λ ∈ [0, 1]. Before formulating the main result we adopt a similar notation for the adjusted cubicsγ 3,i : R → E n satisfyingγ 3,i (t i+j ) = q i+j andγ 3,i | [ti,ti+3] =γ 3,i . As previously if ψ 3,i maps J i intoĴ i thenγ 3,i • ψ 3,i =γ 3,i • ψ 3,i . The latter holds e.g. if ψ 3,i defines a genuine reparameterization of J i intoĴ i (i.e.ψ 3,i > 0).
The following result is proved in this paper (see Section 3): Theorem 2. Let γ be a C 4 ([0, T ]) regular curve in E n sampled more-or-less uniformly (see Definition 2). Suppose that Q m defines reduced data with the missing knots T = {t i } m i=0 compensated according to (1.3). Then the following holds (uniformly over [0, T ]): In addition, for either T u = { iT m } m i=0 or λ = 1 (for λ = 1 -see [17]) we obtain: Moreover, if additionally the mapping ψ 3,i is asymptotically a reparameterization of J i intoĴ i (e.g. when {t i } m i=0 is uniform or for λ = 1) then, in both (2.1) and (2.2), the curveγ 3,i can be replaced byγ 3,i .
Recently a similar linear convergence order as in (2.1) is established in [7] for the so-called modified Hermite interpolantγ H ∈ C 1 combined with the exponential parameterization (1.3) and λ ∈ [0, 1) (the case of λ = 1 was covered in [6] or [8]). The proof used in [7] exploits the final claim of Theorem 2.
m ) asymptotics for the second component in ρ follows directly from (2.1). The linear asymptotics determining the first componentγ H • φ −γ 3 • ψ 3 results upon conducting more advanced argument (as compared to Section 3) -see [7].
2. The convergence orders established in Theorem 2 are justified analytically as sharp in accordance with Definition 4 -see Section 5.
3. Section 7 supplements a) and b) with the numerical tests conducted in Mathematica confirming experimentally the sharpness of the asymptotics determined in (2.1) and (2.2). 4. In Section 6 (see Example 3) it is also demonstrated with the aid of symbolic computation performed in Mathematica that more-or-less uniformity (1.2) cannot be dropped in Theorem 2. Remarkably, by [6] or [17] the case of λ = 1 does not require more-or-less uniformity (1.2). Here, merely the general class of admissible samplings (1.1) is sufficient to ascertain (1.5) or (2.2). Noticeably, for the uniform sampling T u the coefficient β = 1 as introduced in (1.2). 5. Sufficient conditions enforcing ψ 3,i to be a genuine reparameterization of J i intoĴ i are additionally formulated in the closing part of Section 4 (they are visualized in Figure 1). In general, the question of ψ : [0, T ] → [0,T ] rendering a reparameterization is vital e.g. for estimating the length d(γ) = T 0 γ(t) dt of γ with d(γ) = T 0 γ (t) dt (see e.g. [2] and [6]). However, in certain circumstances, the non-parameterization cases (yielding loops in trajectory pathγ • ψ over some segments J i ) may also be desirable for some specific applications like robot or airborne flying devise trajectory planning (e.g. for inspecting electrical poles by drones).

The proof of Theorem 2
In this section the proof of Theorem 2 is given. In doing so recall first the Hadamard's Lemma (see e.g. [16]): ] → E n be of class C l , where l ≥ 1 and assume that f (t 0 ) = 0, for some t 0 ∈ (a, b). There exists a C l−1 function g : . If function f (t) has multiple zeros t 0 ≤ t 1 ≤ ... ≤ t k with k + 1 ≤ l, then Hadamard's Lemma applied k + 1 times yields: where h is C l−(k+1) and h = O(d k+1 f /dt k+1 ). From now on, to justify Theorem 2, the abbreviated notation for both cubics ψ 3,i = ψ i andγ 3,i =γ i is used.
Proof. The proof of Theorem 2 admits an arbitrary λ ∈ [0, 1) and any more-or-less uniform sampling. In contrast, the special case λ = 1 (omitted here as already justified in [17]) extends to all admissible samplings (1.1) but is still restricted to γ ∈ C 4 . Noticeably, for λ = 1 each cubic ψ i defines a genuine reparameterization of J i intoĴ i -see [6] or [17]. Define now the following "error function" Combining f i (t i+j ) = 0 (for j = 0, 1, 2, 3) with Hadamard's Lemma yields (see here (3.1)): Sinceγ i and ψ i are cubics the chain rule applied to (3.2) results in: with the respective derivatives over t ort = ψ i (t) expressed by either dotted or apostrophed notations, respectively. Coupling the latter with (3.2) and γ ∈ C 4 defined over a compact set [0, T ] leads to: We pass now to the determination of the respective asymptotics for all contributing terms appearing in (3.4).

Estimation of derivatives ofγ i
We estimate now the derivatives ofγ i : [t i , t i+3 ] → E n present in (3.2) and (3.3). Again Newton's interpolation formula (see e.g. [1]) applied toγ i yields (for anyt ∈ R): Hencě As previously, the orders ofγ need to be examined. In doing so, observe first that: Taylor's expansion applied to γ yields: Since γ(t) = 1 (and thus γ(t)|γ(t) = 0) we arrive at: The last step exploits (1 + x) . Hence by (3.18), (3.19) and (3.20) we obtaiň In the latter the asymptotic positivity of both factors from (3.20) is used as for the exponential parameterization (1.3) with q i+1 = q i we havet l −t s > 0 for l > s.
The special case of uniform samplings addressed by (2.2) follows once (3.16), (3.32) and (3.33) are incorporated into (3.4) resulting in (uniformly over J i ): which in turn yields: The proof is thus completed.

Sufficient conditions on reparameterization
In this section several sufficient conditions for the cubic ψ i to yield a genuine reparameterization are formulated. The case of λ = 1 renders ψ i as a parameterization for arbitrary admissible sampling (1.1) as proved in [17]. The importance ofψ i > 0 is outlined in Section 2 -see item e). Noticeably to enforce ψ i as a reparameterization it suffices to let the quadraticψ i (t) = a i t 2 + b i t + c i > 0 over each J i . The latter follows if e.g. (see Figure 1): Clearly for a given collection of Q m and T , the testing ofψ i > 0 over different J i can vary between constraints (i) − (iv) (or between any other ones). Assuming that we admit the subfamily of more-or-less uniform samplings (1.2) satisfying 0 < β 0 ≤ β ≤ 1 (with some β 0 fixed) the conditions (4.1) can be expressed in terms of β 0 . A full treatment of solving (4.1) for arbitrary λ ∈ [0, 1) exceeds the scope and page limit for this paper. Some hints can be found in [10], where the parameterization issue for piecewise-quadratic Lagrange interpolationγ 2 based on Q m and exponential parameterization (1.3) is thoroughly studied.

Necessity of more-or-less uniformity
We demonstrate now that more-or-less-uniformity for (2.1) (with λ ∈ [0, 1)) is essential and cannot be omitted in Theorem 2. A simple inspection shows that the admissible sampling (6.1) does not satisfy more-or-less uniformity (1.2) (with δ m → 0). As previously we simplify the notation used for t i+j (with j = 0, 1, 2, 3) by setting i = 0. The asymptotics of h q (t) = (γ • ψ)(t) − γ q (t) is examined over the sub-interval J 0 = [−δ m , δ m ]. The detailed symbolic computations used in Example 3 are conducted in Mathematica Notebook stored under the URL link [21].
Example 3. Consider now the following non-more-or-less uniform sampling: The exponential parameterization (1.3) applied to {γ q (t i )} 3 i=0 and (6.1) yields (asymptotically): The corresponding divided differences for a cubic ψ : J 0 → R are equal to: Hence latter yields: The evaluation of (6.6) att = (δ m /2) with Factor [ψ[δ/2, λ]] results in: Since γ q andγ coincide at interpolation points {q i+j } 3 j=0 we have γ(t 1 ) = 0, Hence by (6.2) the respective divided differences forγ are equal: Consequently we obtain: which in turn leads to (λ 1 = 1 − λ): Incorporating (6.2), (6.7), (6.8), (6.9) and the latter into Newton's interpolation formula yields for , 0 , (6.10) wheret = ψ(t). Setting now e.g. λ = 0 in (6.10) and using Mathematica Simplify to h q (t) leads to (see Mathematica Notebook under the URL link [21]): Visibly the latter is of the sharp order: not coinciding with the linear asymptotics from Theorem 2 established for any λ ∈ [0, 1) and arbitrary more-or-less uniform sampling. Such deceleration effect is due to the fact that sampling (6.1) does not satisfy (1.2). A similar sharp asymptotics with the slowest term K λ δ −2 m (with K λ = 0) follows upon substituting any λ ∈ [0, 1) into (6.10) and applying Mathematica Simplify function. Due to the page limitation the Mathematica Notebook is attached (see ( [21])) so that the latter can be verified upon performing symbolic computation for any fixed λ ∈ [0, 1). Note also that feeding λ = 1 into (6.10) results in: which yields an exact quartic convergence order in trajectory estimation at t = (δ m /2). The latter justifies sharpness of Theorem 2 claimed also for λ = 1 and the general class of admissible samplings. Evidently, though (6.1) does not fulfill more-or-less uniformity it still complies however with (1.1).

Experiments
We verify now numerically the sharpness of the asymptotics established in Theorem 2 for some regular 2D and 3D curves. All tests are carried out in Mathematica 10.0 (see [22]) and resort to either uniform or more-or-less uniform samplings (with t i ∈ [0, 1] and for i = 0, 1, . . . , m) defined according to: i/m, for i even, is calculated, where m min and m max are sufficiently large fixed constants. Next a linear regression yielding a function y(x) =ᾱ(λ)x + b is applied to {(log(m), − log(E m ))} mmax m=mmin . Mathematica built-in function LinearModelFit extracts a coefficientᾱ(λ) ≈ α(λ). The tests conducted here use the following three different C ∞ regular curves: a spiral γ sp and a cubic γ o in E 2 and a Steinmetz curve γ st in E 3 .

Conclusions
The main results and the motivation of this work are fully listed in Section 2 (see items a)-f)). The principal findings can be summarized as follows: Section 3 proves Theorem 2 which extends Theorem 1 holding merely for λ = 1. The combination of the latter yields a surprising abrupt discontinuity in convergence orders while estimating γ withγ 3 • ψ 3 . Here a fast quartic order α(1) = 4 drops to the linear one α(λ) = 1 holding for any λ ∈ [0, 1) incorporated into the exponential parameterization (1.3).
Section 5 justifies the sharpness of (2.1) and (2.2) with the aid of non-trivial analytic and symbolic computations in accordance with Definition 4. Section 6 justifies the necessity of more-or-less uniformity (2) in proving Theorem 2. The analytic argument combined with symbolic computation is employed. Additionally, the case of λ = 1 relies merely on a general class of admissible samplings (1.1) as also confirmed herein.
Section 7 verifies independently with the aid of numerical tests performed in Mathematica the sharpness of the asymptotics established in Theorem 2.
Section 2 specifies the main motivation standing behind this paper including desirable parameterization or non-parametrization cases of ψ 3 . The related literature in the context of specific applications to fit Q m in conjunction with exponential parametrization is also listed.
Future work may involve C 2 cubics splines (see e.g. [1]) and (1.3) as an extension of C 0 piecewise-cubic non-parametric interpolation discussed in this paper. The case of C 1 modified Hermite interpolation is covered in [7] or [8].