Simultaneous localization and mapping for aerial vehicles: a 3-D sensor-based GAS filter

Abstract

This paper presents the design, analysis, and experimental validation of a globally asymptotically stable (GAS) filter for simultaneous localization and mapping (SLAM) with application to unmanned aerial vehicles. The main contributions of this paper are the results of global convergence and stability for SLAM in tridimensional (3-D) environments. The SLAM problem is formulated in a sensor-based framework and modified in such a way that the structure may be regarded as linear time-varying for observability purposes, from which a Kalman filter with GAS error dynamics follows naturally. The proposed solution includes the estimation of both body-fixed linear velocity and rate gyro measurement biases. Experimental results from several runs, using an instrumented quadrotor equipped with a RGB-D camera, are included in the paper to illustrate the performance of the algorithm under realistic conditions.

Appendix: Proof of Theorem 4

Consider again system (6) and recall that it is related by a Lyapunov transformation to system (4). Thus, the uniform complete observability of the pair $\left( {\varvec{\mathcal {A}}}(t,{{\mathbf{y}}}(t)),{\varvec{\mathcal {C}}}(t)\right)$ implies the uniform complete observability of the pair $\left( {{\mathbf{A}}}(t,{{\mathbf{y}}}(t)),{{\mathbf{C}}}\right)$. Hence, the proof will continue with the transformed system.

A pair $\left( {\varvec{\mathcal {A}}}(t,{{\mathbf{y}}}(t)),{\varvec{\mathcal {C}}}(t)\right)$ is uniformly completely observable if and only if there exist positive constants $\delta$ and $\alpha$ such that for all $t\ge t_0$ and for all unit vectors ${{\mathbf{c}}}$ the quadratic form ${{\mathbf{c}}}^T{\varvec{\mathcal {W}}}(t,t+\delta ){{\mathbf{c}}}$ is greater than or equal to $\alpha$, i.e., if and only if

meaning that, in contrast with the observability definition used in Theorem 1, the Gramian must have uniform bounds at all times. The proof follows by exhaustion, by analysing the quadratic form in the previous expression for all the possible cases of unit vectors ${{\mathbf{c}}}$ for all time. It is recalled from (8) and (9) that ${{\mathbf{c}}}^T{\varvec{\mathcal {W}}}(t,t+\delta ){{\mathbf{c}}}=\int _{t}^{t+\delta }\left\| {{\mathbf{f}}}(\tau ,t)\right\| ^2d\tau$, with

$$\begin{aligned} \left\| {{\mathbf{f}}}(\tau ,t)\right\| ^2 = \sum \limits _{i=1}^{N_O}\left\| {{\mathbf{c}}}_i-{{\mathbf{f}}}_{v_i}(\tau ,t)\right\| ^2, \end{aligned}$$

(25)

where

$$\begin{aligned} {{\mathbf{f}}}_{v_i}(\tau ,t):=\int \limits _{t}^{\tau }{{\mathbf{R}}}_m(\sigma _i)\left( {{\mathbf{c}}}_v+{{\mathbf{S}}}[{{\mathbf{p}}}_i(\sigma _i)]{{\mathbf{c}}}_b)\right) d\sigma _i. \end{aligned}$$

In order to be able to proceed, Batista et al. (2011a, Proposition 4.2) is needed. It states that if it is possible to find a positive constant $\beta$ such that $\Vert \frac{\partial ^i}{\partial \tau ^i}{{\mathbf{f}}}(\tau ,t)\Vert \ge \beta$ then there exists a $\gamma >0$ such that $\Vert {{\mathbf{f}}}(t,t+\delta )\Vert \ge \gamma$ as long as $\frac{\partial ^{j}}{\partial \tau ^{j}}{{\mathbf{f}}}(\tau ,t)|_{\tau =t}=0$ for all $j<i$ and the norm of the $(i+1)$-th derivative is upper bounded. It is possible to see that this applies to ${{\mathbf{c}}}^T{\varvec{\mathcal {W}}}(t,t+\delta ){{\mathbf{c}}}$, and therefore it states conditions upon which the quadratic form in analysis is lower bounded if (25) is lower bounded, uniformly in time. Then, it suffices to prove that $\left\| {{\mathbf{f}}}(\tau ,t)\right\| \ge \alpha _1$ with $\tau \in {\mathcal {T}}_\delta$ and $\alpha _1>0$ for every possible ${{\mathbf{c}}}$, provided that the conditions of Batista et al.  (2011a, Proposition 4.2) are satisfied uniformly in time.

Consider the case where there exists at least one $i\in {\mathcal {I}_O}$ such that $\left\| {{\mathbf{c}}}_i\right\| \ge \alpha _p$ for some $0<\alpha _p<1$ and no restriction is imposed on ${{\mathbf{c}}}_v$ and ${{\mathbf{c}}}_b$. Then,

$$\begin{aligned} \left\| {{\mathbf{f}}}(t,t)\right\| ^2=\sum \limits _{j=1}^{N_O}\left\| {{\mathbf{c}}}_{j}\right\| ^2\ge \left\| {{\mathbf{c}}}_i\right\| ^2\ge \alpha _p^2 \end{aligned}$$

for all t. Consider now that $\left\| {{\mathbf{c}}}_i\right\| <\alpha _p$, for all $i\in \{1,\ldots ,N_O\}$ and for some $0<\alpha _p<1$. For that purpose, let ${{\mathbf{c}}}_p$ and ${{\mathbf{f}}}_v(\tau ,t)$ be the stacking of all the ${{\mathbf{c}}}_i$ and ${{\mathbf{f}}}_{v_i}(\tau ,t)$, respectively, and note that ${{\mathbf{f}}}(\tau ,t)={{\mathbf{c}}}_p-{{\mathbf{f}}}_v(\tau ,t)$. Then, it is possible to write

$$\begin{aligned} \left\| {{\mathbf{f}}}(\tau ,t)\right\| ^2&\ge \left\| {{\mathbf{f}}}_v(\tau ,t)\right\| \left( \left\| {{\mathbf{f}}}_v(\tau ,t)\right\| -2\Vert {{\mathbf{c}}}_p\Vert \right) \\&\ge \left\| {{\mathbf{f}}}_v(\tau ,t)\right\| \left( \left\| {{\mathbf{f}}}_v(\tau ,t)\right\| -2\sqrt{N_O}\alpha _p\right) . \end{aligned}$$

Assuming that $\left\| {{\mathbf{f}}}_v(\tau ,t)\right\|$ is lower bounded at $\tau =t^*$ in ${\mathcal {T}}_\delta$ by some $\alpha _2>0$ so that $\alpha _p$ can be chosen to satisfy $\alpha _p\le \frac{\alpha _2}{4\sqrt{N_O}}$, it follows that

$$\begin{aligned} \left\| {{\mathbf{f}}}(t^*,t)\right\| ^2\ge \frac{1}{2}\alpha _2^2. \end{aligned}$$

It is necessary then to show that $\left\| {{\mathbf{f}}}_v(t^*,t)\right\| \ge \alpha _2\ge 4\sqrt{N_O}\alpha _p$ under the conditions of the theorem. In this situation ${{\mathbf{f}}}_v(\tau ,t)$ is zero for $\tau =t$, and therefore Batista et al.  (2011a, Proposition 4.2) applies once again. Thus, it is enough to show that the norm of the derivative of ${{\mathbf{f}}}_v(\tau ,t)$, given by

$$\begin{aligned} \left\| \frac{\partial }{\partial \tau }{{\mathbf{f}}}_{v}(\tau ,t)\right\| ^2=\sum _{i=1}^{N_O}\left\| {{\mathbf{c}}}_v+{{\mathbf{S}}}[{{\mathbf{p}}}_{i}(\tau )]{{\mathbf{c}}}_b\right\| ^2, \end{aligned}$$

(26)

is lower bounded by some $\alpha _3>0$ for a $\tau \in {\mathcal {T}}_\delta$. Then, there exists an $\alpha _2$ that bounds $\left\| {{\mathbf{f}}}_v(\tau ,t)\right\|$ below for some $t^*\in {\mathcal {T}}_\delta$ and $\alpha _p$ can be chosen accordingly. Under the restriction $\left\| {{\mathbf{c}}}_i\right\| <\alpha _p$ for all $i\in {\mathcal {I}_O}$, there exist three possibilities for ${{\mathbf{c}}}$, depending on ${{\mathbf{c}}}_v$ and ${{\mathbf{c}}}_b$. The first case is set by $\left\| {{\mathbf{c}}}_v\right\| \ge \alpha _v$ and $\left\| {{\mathbf{c}}}_b\right\| <\alpha _b$ for some $\alpha _v$ and $\alpha _b$ in the interval (0, 1). It is possible to write

$$\begin{aligned} \left\| \frac{\partial }{\partial \tau }{{\mathbf{f}}}_{v}(\tau ,t)\right\| ^2&\ge \sum _{i=1}^{N_O}\Vert {{\mathbf{c}}}_v\Vert \left( \Vert {{\mathbf{c}}}_v\Vert -2\Vert {{\mathbf{p}}}_{i}(\tau )\Vert \Vert {{\mathbf{c}}}_b\Vert \right) \\&\ge N_O\alpha _v\left( \alpha _v-2P_M\alpha _b \right) \\&\ge \frac{N_O}{2}\alpha _v^2, \end{aligned}$$

where Assumption 2 was employed, and $\alpha _v$ was chosen so that it satisfies $\alpha _b\le \frac{\alpha _v}{4P_M}$. The second case, where $\left\| {{\mathbf{c}}}_v\right\| < \alpha _v$ for some $0<\alpha _v<1$ and $\left\| {{\mathbf{c}}}_b\right\| \ge \alpha _b$ for some $0<\alpha _b<1$, and the third case, where $\left\| {{\mathbf{c}}}_v\right\| \ge \alpha _v$ and $\left\| {{\mathbf{c}}}_b\right\| \ge \alpha _b$, can be analysed together. Consider then that $\left\| {{\mathbf{c}}}_b\right\| \ge \alpha _b$ and that $0\le \Vert {{\mathbf{c}}}_v\Vert \le 1$, in which case the conditions of the theorem must be addressed separately. The first condition of the theorem states that there are at least three landmarks ($N_O\ge 3$) that are sufficiently away from collinearity, i.e., the plane defined by the vectors that unite each pair of landmarks is well-defined uniformly in time and sufficiently away from degenerating into a line. This can be illustrated by taking the cross product between each vector that defines the plane and ${{\mathbf{c}}}_b$, summing the norms of the results, and noting that the lowest possible value for this sum occurs when ${{\mathbf{c}}}_b$ is collinear with the largest of these vectors, say, ${{\mathbf{c}}}_b=\pm \frac{\Vert {{\mathbf{c}}}_b\Vert }{\Vert {{\mathbf{p}}}_{1}(t_1)-{{\mathbf{p}}}_{3}(t_1)\Vert }({{\mathbf{p}}}_{1}(t_1)-{{\mathbf{p}}}_{3}(t_1))$. This worst case can be substituted in the conditions of the theorem to yield

$$\begin{aligned} \Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)-{{\mathbf{p}}}_{2}(t_1)]{{\mathbf{c}}}_b\Vert \ge \frac{\alpha _b\alpha _*}{2P_M} \end{aligned}$$

where Assumption 2 was used to show that $\Vert {{\mathbf{p}}}_{1}(t_1)-{{\mathbf{p}}}_{3}(t_1)\Vert \le 2P_M$. This means that

$$\begin{aligned} \Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)-{{\mathbf{p}}}_{2}(t_1)]{{\mathbf{c}}}_b\Vert +\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)-{{\mathbf{p}}}_{3}(t_1)]{{\mathbf{c}}}_b\Vert \ge \frac{\alpha _b\alpha _*}{2P_M} \end{aligned}$$

for any ${{\mathbf{c}}}_b$ with norm greater than $0<\alpha _b<1$. It is a matter of algebraic manipulation to obtain

$$\begin{aligned}&\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v-({{\mathbf{S}}}[{{\mathbf{p}}}_{2}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v)\Vert \\&\quad +\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v-({{\mathbf{S}}}[{{\mathbf{p}}}_{3}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v)\Vert \ge \frac{\alpha _b\alpha _*}{2P_M} \end{aligned}$$

from the previous expression, which can be further manipulated using the triangle inequality to yield

$$\begin{aligned}&2\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert + \Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{2}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert \nonumber \\&\quad +\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{3}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert \ge \frac{\alpha _b\alpha _*}{2P_M}. \end{aligned}$$

(27)

There are two possible conclusions to draw from this inequality depending on the values of $\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{2}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert$ and $\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{3}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert$. If either is greater than or equal to some $\alpha _{p_1}>0$, (26) is lower bounded by $\alpha _{p_1}$ at $\tau =t_1$, as intended. On the other hand, if both are smaller than $\alpha _{p_1}>0$, then (27) leads to

$$\begin{aligned} \left. \left\| \frac{\partial }{\partial \tau }{{\mathbf{f}}}_{v}(\tau ,t)\right\| \right| _{\tau =t_1}&\ge \Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert \\&\ge \frac{\alpha _b \alpha _*}{4P_M}-\alpha _{p_1}\ge \frac{\alpha _b\alpha _*}{8P_M} \end{aligned}$$

if $\alpha _{p_1}$ is chosen so that $\alpha _{p_1}\le \frac{\alpha _b \alpha _*}{8P_M}$. The second condition can be treated in similar terms. Consider, without loss of generality that the landmark set that is considered is $\{{{\mathbf{p}}}_{1}(t_1),{{\mathbf{p}}}_{2}(t_1),{{\mathbf{p}}}_{1}(t_2)\}$. Using the same reasoning used for the analysis of the first conditions, it is possible to write

$$\begin{aligned}&\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v-({{\mathbf{S}}}[{{\mathbf{p}}}_{2}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v)\Vert \\&\quad +\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v-({{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_2)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v)\Vert \ge \frac{\alpha _b\alpha _*}{2P_M}, \end{aligned}$$

which can also be manipulated to obtain a lower bound to the sum of the norms, yielding

$$\begin{aligned} 2\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert + \Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{2}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert \nonumber \\ +\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_2)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert \ge \frac{\alpha _b\alpha _*}{2P_M}. \end{aligned}$$

(28)

Once again, the analysis depends on the behaviour of $\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert$ and $\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{2}(\tau )]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert$. If either is greater than or equal to some $\alpha _{p_1}>0$, then, for $\tau =t_1$, $\left\| \frac{\partial }{\partial \tau }{{\mathbf{f}}}_{v}(\tau ,t)\right\| \ge \alpha _{p_1}$. Conversely, if both are smaller than $\alpha _{p_1}$, then from (28) it is possible to write

$$\begin{aligned} \left. \left\| \frac{\partial }{\partial \tau }{{\mathbf{f}}}_{v}(\tau ,t)\right\| \right| _{\tau =t_1}&\ge \Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert \\&\ge \frac{\alpha _b\alpha _*}{4P_M}-\alpha _{p_1}\ge \frac{\alpha _b\alpha _*}{8P_M}, \end{aligned}$$

for the previous choice of $\alpha _{p_1}$. Finally, for the third condition, the set of landmarks considered is $\{{{\mathbf{p}}}_{1}(t_1),{{\mathbf{p}}}_{1}(t_2),{{\mathbf{p}}}_{1}(t_3)\}$. In the same line of thought followed previously, one knows that

$$\begin{aligned}&\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v-({{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_2)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v)\Vert \\&\quad +\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v-({{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_3)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v)\Vert \ge \frac{\alpha _b\alpha _*}{2P_M}. \end{aligned}$$

Applying the triangle inequality to the previous expression as was done when manipulating the similar expressions for the first two conditions yields

$$\begin{aligned}&2\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_1)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert +\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_2)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert \nonumber \\&\quad +\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(t_3)]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert \ge \frac{\alpha _b\alpha _*}{2P_M}. \end{aligned}$$

(29)

If $\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(\tau )]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert \ge \alpha _{p_1}$ for $\tau =t_2$, $t_3$ or both, then (26) is bounded by $\alpha _{p_1}$ in one or both of these instants. However, if $\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(\tau )]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert <\alpha _{p_1}$ for all $\tau \in {\mathcal {T}}_\delta$, $\tau \ne t_1$ and nothing is imposed for $\tau =t_1$, (29) leads to

$$\begin{aligned} \left. \left\| \frac{\partial }{\partial \tau }{{\mathbf{f}}}_{v}(\tau ,t)\right\| ^2\right| _{\tau =t_1}\ge \frac{\alpha _b\alpha _*}{4P_M}-\alpha _{p_1}\ge \frac{\alpha _b\alpha _*}{8P_M} \end{aligned}$$

where the fact that $\left\| \frac{\partial }{\partial \tau }{{\mathbf{f}}}_{v}(\tau ,t)\right\| =\Vert {{\mathbf{S}}}[{{\mathbf{p}}}_{1}(\tau )]{{\mathbf{c}}}_b+{{\mathbf{c}}}_v\Vert$ was used, and $\alpha _{p_1}$ was chosen so that $\alpha _{p_1}\le \frac{\alpha _b\alpha _*}{8P_M}$. In these three conditions, positive lower bounds were found for $\left\| \frac{\partial }{\partial \tau }{{\mathbf{f}}}_{v}(\tau ,t)\right\|$, which, from Batista et al. (2011a, Proposition 4.2) implies that ${{\mathbf{f}}}_{v}(\tau ,t)$ is also lower bounded and, therefore, ${{\mathbf{f}}}(\tau ,t)$ has a lower bound too within the possibilities studied. All possible cases are now enumerated, hence the sufficiency of the conditions of Theorem 4 for the pair $\left( {\varvec{\mathcal {A}}}(t,{{\mathbf{y}}}(t)),{\varvec{\mathcal {C}}}(t)\right)$ to be uniformly completely observable is proved. Because (5) is a Lyapunov transformation, the uniform complete observability of the transformed pair implies uniform complete observability of the $\left( {{\mathbf{A}}}(t,{{\mathbf{y}}}(t)),{{\mathbf{C}}}\right)$, and thus the proof of sufficiency of the conditions of the theorem is concluded.

The proof of the necessity of the conditions of the theorem follows by contraposition. The hypothesis that the conditions of the theorem do not hold is considered, and it is shown that this implies that the pair $\left( {\varvec{\mathcal {A}}}(t),{\varvec{\mathcal {C}}}(t)\right)$ is not uniformly completely observable.

Consider then that the conditions of the theorem do not hold. The negation of these conditions means that all the landmarks in the set ${\mathcal {I}_O}$ are not sufficiently away from collinearity. Consider the alternative definition for a landmark, given by

$$\begin{aligned} {{\mathbf{p}}}_{i}(t)=p_{d_i}(t){{\mathbf{d}}}+p_{d^{\perp _1}_i}(t){{\mathbf{d}}}^{\perp _1}+p_{d^{\perp _2}_i}(t){{\mathbf{d}}}^{\perp _2}+{{\mathbf{p}}}_{1}(t_0), \end{aligned}$$

where $\{{{\mathbf{d}}},{{\mathbf{d}}}^{\perp _1},{{\mathbf{d}}}^{\perp _2}\}\in {\mathbb {R}}^{3}$ are orthonormal vectors that span the planes defined by all the combinations of three landmarks in ${\mathcal {I}_O}$, and $p_{d_i}(t),p_{d^{\perp _1}_i}(t),p_{d^{\perp _2}_i}(t)\in {\mathbb {R}}^{3}$. The negated conditions imply that, either there are no landmarks which immediately renders the pair unobservable or, considering, without loss of generality, that ${{\mathbf{d}}}$ is the predominant direction of landmarks, there are at least two directions ${{\mathbf{d}}}^{\perp 1}$ and ${{\mathbf{d}}}^{\perp 2}$ along which the vector ${{\mathbf{p}}}_{i}(t)-{{\mathbf{p}}}_{1}(t_0)$ is bounded and as small as wanted for some $t\ge t_0$, i.e.,

$$\begin{aligned} \underset{\beta >0}{\forall }\quad \underset{i\in {\mathcal {I}_O}}{\forall }\quad \underset{t\ge t_0}{\exists }:\quad |p_{d^{\perp _1}_i}(t)|<\beta \wedge |p_{d^{\perp _2}_i}(t)|<\beta . \end{aligned}$$

(30)

The proof follows by showing that this implies that the pair in analysis is not uniformly completely observable, which can be stated as

For that purpose, the expansion of ${{\mathbf{c}}}^T{\varvec{\mathcal {W}}}(t,t+\delta ){{\mathbf{c}}}$ is considered for some unit ${{\mathbf{c}}}$, yielding the following expression that can be obtained by substituting (25) into (8),

$$\begin{aligned}&{{\mathbf{c}}}^T{\varvec{\mathcal {W}}}(t,t+\delta ){{\mathbf{c}}} =\\&\quad \int \limits _{t}^{t+\delta }\sum _{i=1}^{N_O}\left\| {{\mathbf{c}}}_i- \int \limits _{t}^{\tau }{{\mathbf{R}}}_m(\sigma _i)\left( {{\mathbf{c}}}_v+{{\mathbf{S}}}[{{\mathbf{p}}}_{i}(\sigma _i)]{{\mathbf{c}}}_b\right) d\sigma _i\right\| ^2d\tau . \end{aligned}$$

In the following steps, a particular ${{\mathbf{c}}}$ is chosen and the negated condition (30) is used. Considering that ${{\mathbf{c}}}_i={{\mathbf{0}}}$ for all $i\in {\mathcal {I}_O}$, and using the Cauchy–Schwartz inequality in the previous expression leads to

$$\begin{aligned} {{\mathbf{c}}}^T{\varvec{\mathcal {W}}}(t,t+\delta ){{\mathbf{c}}}\le \sum _{i=1}^{N_O}\int \limits _{t}^{t+\delta }\int \limits _{t}^{\tau }\left\| {{\mathbf{c}}}_v-{{\mathbf{S}}}[{{\mathbf{c}}}_b]{{\mathbf{p}}}_{i}(\sigma )\right\| ^2d\sigma d\tau . \end{aligned}$$

Recall that ${{\mathbf{R}}}_m(t)$ maintains the norm, which is why it is omitted in the previous expression. Let a be a real constant, and choose ${{\mathbf{c}}}_b=a{{\mathbf{d}}}$ and ${{\mathbf{c}}}_v=a{{\mathbf{S}}}[{{\mathbf{d}}}]{{\mathbf{p}}}_{1}(t_0)$ such that $\Vert {{\mathbf{c}}}\Vert =1$. This yields

$$\begin{aligned}&{{\mathbf{c}}}^T{\varvec{\mathcal {W}}}(t,t+\delta ){{\mathbf{c}}}\\&\quad \le a^2\sum _{i=1}^{N_O}\int \limits _{t}^{t+\delta }\int \limits _{t}^{\tau }\left\| {{\mathbf{S}}}[{{\mathbf{d}}}]\left( p_{d^{\perp _1}_i}(\sigma ){{\mathbf{d}}}^{\perp 1}+p_{d^{\perp _2}_i}(\sigma ){{\mathbf{d}}}^{\perp 2}\right) \right\| ^2d\sigma d\tau \\&\quad \le a^2\sum _{i=1}^{N_O}\int \limits _{t}^{t+\delta }\int \limits _{t}^{\tau }\left( |p_{d^{\perp _1}_i}(\sigma )|+|p_{d^{\perp _2}_i}(\sigma )|\right) ^2d\sigma d\tau \\&\quad \le a^2N_O\int \limits _{t}^{t+\delta }\int \limits _{t}^{\tau }(2\beta )^2d\sigma d\tau , \end{aligned}$$

which can be rewritten as

as $\beta :=\frac{1}{a\delta }\sqrt{\frac{\epsilon }{2N_O}}$. With this step it was shown that the transformed system cannot be uniformly completely observable if the conditions of the theorem do not apply, which, along with the fact that the Lyapunov transformation preserves the observability properties of the system, leads to the necessity of the conditions for the uniform complete observability of the nonlinear system (4), regarded as LTV, and thus concluding the proof.