Complete 3-D dynamic coverage in energy-constrained multi-UAV sensor networks

Abstract

This paper considers dynamic coverage control of multiple power-constrained agents subject to 3D rigid body kinematics. The agents are deployed to patrol a domain until the entire space has reached a satisfactory level of coverage. This is achieved through the gathering of information by a forward-facing sensor footprint, modelled as an anisotropic spherical sector. Coverage and collision avoidance guarantees are met by a hybrid controller consisting of four operating modes: local coverage, global coverage, waypoint scan and subdomain transfer. Energy-aware methods are encoded into the global coverage state to shift the bulk of spatial redistribution onto less constrained agents. Additionally, a novel domain partitioning strategy is used that directs individual agents to explore within concentric hemispherical shells around a centralized charging station. This results in flight paths that are guaranteed to terminate at the charging station in the limit that agent batteries expire. The efficacy of this algorithm is presented through experimental trials with three agents in an indoor environment. Simulations are provided for ten agents.

Appendix

1.1 Automaton

The automaton is described by the following entities (Lygeros 2004):

• a set of discrete states: $Z_{i}=\{\zeta _{i0},\zeta _{i1},\zeta _{i2},\zeta _{i3},\zeta _{i4}\}$ which represent local coverage, global coverage, subdomain transfer, and waypoint scan modes (1) and (2) respectively,

• a set of continuous states: $\; \tilde{q}_i=\{x_i,y_i,z_i,\varPhi _i,\varTheta _i,\varPsi _i\}$ representing the position and orientation of agent i in Cartestian coordinates and 3-2-1 Euler angles respectively,

• a vector field:

  $$\begin{aligned} f(\zeta _{i0},\tilde{q}_i)&=\mathcal {R} \left[ \acute{\acute{u_i}}^{loc} \; \acute{\acute{v_i}}^{loc} \; \acute{\acute{w_i}}^{loc} \; 0 \; 0 \; \acute{\acute{s_i}}^{loc} \right] ^{T}, \\ f(\zeta _{i1},\tilde{q}_i)&=\mathcal {R} \left[ u_{i}^{glo} \; v_{i}^{glo} \; w_{i}^{glo} \; 0 \; 0 \; s_{i}^{glo} \right] ^{T}, \\ f(\zeta _{i2},\tilde{q}_i)&=\mathcal {R} \left[ u_{i}^{sub} \; v_{i}^{sub} \; w_{i}^{sub} \; 0 \; 0 \; s_{i}^{sub} \right] ^{T}, \\ f(\zeta _{i3},\tilde{q}_i)&=\mathcal {R} \left[ u_{i}^{wps} \; v_{i}^{wps} \; w_{i}^{wps} \; 0 \; 0 \; 0 \right] ^{T}, \\ f(\zeta _{i4},\tilde{q}_i)&=\mathcal {R} \left[ 0 \; 0 \; 0 \; 0 \; 0 \; s_{i}^{wps} \right] ^{T}, \end{aligned}$$

  where $\mathcal {R}= \begin{bmatrix} \mathcal {R}_1&0 \\ 0&\mathcal {R}_2 \end{bmatrix}$ and $\dot{\tilde{q}}_i=f(\zeta _{ik},\tilde{q}_i)$ is the control input for agent i in state $\zeta _{ik}$. Note that $\zeta _{i0}$, $\zeta _{i1}$ and $\zeta _{i2}$ command both translation and yaw while $\zeta _{i3}$ only commands translation and $\zeta _{i4}$ only commands yaw. This is because $\zeta _{i3}$ commands that agent i transition between the sub-waypoints of waypoint scan mode while $\zeta _{i4}$ commands the $360^{\circ }$ yaw sweep at each sub-waypoint.

• A set of initial states:

  $$\begin{aligned}&\{\zeta _{i0}\} \times \{\tilde{q}_{i} \in \mathbb {R}^6 \mid ||p_i ||\le \bar{R}_\mathcal {D} \wedge \, \varPhi _{i} \in \left( -\pi ,+\pi \right) \nonumber \\&\quad \wedge \, \varTheta _{i} \in \left( \frac{-\pi }{2},\frac{+\pi }{2}\right) \, \wedge \, \varPsi _{i} \in \left( -\pi ,+\pi \right) \} \end{aligned}$$

  encoding that each agent begin in local coverage mode inside of the domain at an orientation within the traditional Euler angle range,

• a domain:

  $$\begin{aligned}&Dom \left( \zeta _{i0}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \left( i=1 \wedge p_i \in \mathcal {D} \right) \\&\quad \vee \left( i \in \{2,\ldots ,N-1\} \wedge ||p_i ||\le \bar{R}_{i-1}\right) \\&\quad \vee \left( i=i_{cr} \wedge ||p_i ||\le \bar{R}_{cr}\right) \},\\&Dom \left( \zeta _{i1}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \left( i=1 \wedge p_i \in \mathcal {D} \right) \\&\quad \vee \left( i \in \{2,\ldots ,N-1\} \wedge ||p_i ||\le \bar{R}_{i-1}\right) \\&\quad \vee \left( i=i_{cr} \wedge ||p_i ||\le \bar{R}_{cr}\right) \},\\&Dom \left( \zeta _{i2}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \left( i=1 \wedge p_i \in \mathcal {D} \right) \\&\quad \vee \left( i \in \{2,\ldots ,N-1\} \wedge ||p_i ||> \bar{R}_{i-1}\right) \\&\quad \vee \left( i=i_{cr} \wedge ||p_i ||> \bar{R}_{cr}\right) \},\\&Dom \left( \zeta _{i3}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \left( i=1 \wedge p_i \in \mathcal {D} \right) \\&\quad \vee \left( i \in \{2,\ldots ,N-1\} \wedge ||p_i ||\le \bar{R}_{i-1}\right) \\&\quad \vee \left( i=i_{cr} \wedge ||p_i ||\le \bar{R}_{cr}\right) \},\\&Dom \left( \zeta _{i4}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \left( i=1 \wedge p_i \in \mathcal {D} \right) \\&\quad \vee \left( i \in \{2,\ldots ,N-1\} \wedge ||p_i ||\le \bar{R}_{i-1}\right) \\&\quad \vee \left( i=i_{cr} \wedge ||p_i ||\le \bar{R}_{cr}\right) \} \end{aligned}$$

  where four of the five discrete states are defined over the set of continuous states for which agent i is inside of its prescribed subdomain $\mathcal {D}^j$. $\zeta _{i3}$ (subdomain transfer mode) is defined for agent i outside of its prescribed subdomain,

• A set of edges:

  $$\begin{aligned} E= & {} \{\left( \zeta _{i0},\zeta _{i1}\right) ,\left( \zeta _{i1},\zeta _{i0}\right) , \left( \zeta _{i1},\zeta _{i2}\right) ,\left( \zeta _{i0},\zeta _{i2}\right) , \left( \zeta _{i2},\zeta _{i0}\right) \,\\&\left( \zeta _{i1},\zeta _{i3}\right) ,\left( \zeta _{i3},\zeta _{i1}\right) , \left( \zeta _{i3},\zeta _{i2}\right) ,\left( \zeta _{i3},\zeta _{i0}\right) , \left( \zeta _{i3},\zeta _{i4}\right) ,\\&\left( \zeta _{i4},\zeta _{i3}\right) ,\left( \zeta _{i4},\zeta _{i2}\right) \}, \end{aligned}$$

  each of which is a transition that may be triggered by:

• A guard condition:

  $$\begin{aligned}&G\left( \zeta _{i0},\zeta _{i1}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \big (|\hat{e}_i(t)|<\varepsilon _1\big ) \\&\quad \wedge \big (\left( i=1 \wedge p_i \in \mathcal {D}\right) \vee \left( i \in \{2,\ldots ,N-1\} \right. \\&\quad \wedge \left. ||p_i ||\le \bar{R}_{i-1}\right) \vee \left( i=i_{cr} \wedge ||p_i ||\le \bar{R}_{cr}\right) \big )\} \end{aligned}$$

  which transitions agent i from local to global coverage when the agent’s contribution to the rate of change of the global coverage error drops below some threshold (the coverage rate is insufficient) so long as the agent operates within its prescribed subdomain,

  $$\begin{aligned}&G\left( \zeta _{i1},\zeta _{i0}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \big (||p_i - p_{\star ,i} ||<\varepsilon _2\big ) \wedge \\&\quad \big (||\varPsi _i - \varPsi _{\star ,i} ||<\varepsilon _3 \big ) \wedge \big (|\hat{e}_i(t)| \ge \varepsilon _1\big ) \wedge \\&\quad \big (\left( i=1 \wedge p_i \in \mathcal {D}\right) \vee \left( i \in \{2,\ldots ,N-1\}\right. \\&\qquad \left. \wedge ||p_i ||\le \bar{R}_{i-1}\right) \vee \\&\quad \left( i=i_{cr} \wedge ||p_i ||\le \bar{R}_{cr}\right) \big )\} \end{aligned}$$

  which transitions agent i from global to local coverage when the desired position and orientation has been achieved to within some threshold, the coverage rate is sufficient, and the agent operates within its prescribed subdomain,

  $$\begin{aligned}&G\left( \zeta _{i1},\zeta _{i2}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \\&\quad \left( i \in \{2,\ldots ,N-1\} \wedge ||p_i ||> \bar{R}_{i-1}\right) \vee \\&\quad \left( i=i_{cr} \wedge ||p_i ||> \bar{R}_{cr}\right) \} \end{aligned}$$

  which transitions agent i from global coverage to subdomain transfer mode when the agent is not within its prescribed subdomain,

  $$\begin{aligned}&G\left( \zeta _{i0},\zeta _{i2}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \\&\quad \left( i \in \{2,\ldots ,N-1\} \wedge ||p_i ||> \bar{R}_{i-1}\right) \vee \\&\quad \left( i=i_{cr} \wedge ||p_i ||> \bar{R}_{cr}\right) \} \end{aligned}$$

  which transitions agent i from local coverage to subdomain transfer mode when the agent is not within its prescribed subdomain,

  $$\begin{aligned}&G\left( \zeta _{i2},\zeta _{i0}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \left( i=1 \wedge p_i \in \mathcal {D} \right) \vee \\&\quad \left( i \in \{2,\ldots ,N-1\} \wedge ||p_i ||\le \bar{R}_{i-1}\right) \vee \\&\quad \left( i=i_{cr} \wedge ||p_i ||\le \bar{R}_{cr}\right) \} \end{aligned}$$

  which transitions agent i from subdomain transfer mode to local coverage mode when the agent is within its prescribed subdomain,

  $$\begin{aligned}&G\left( \zeta _{i1},\zeta _{i3}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \big (||p_i - p_{\star ,i} ||<\varepsilon _2\big ) \wedge \\&\quad \big (||\varPsi _i - \varPsi _{\star ,i} ||<\varepsilon _3 \big ) \wedge \big (|\hat{e}_i(t)|<\varepsilon _1\big ) \wedge \\&\quad \big (\left( i=1 \wedge p_i \in \mathcal {D}\right) \vee \left( i \in \{2,\ldots ,N-1\} \right. \\&\qquad \left. \wedge ||p_i ||\le \bar{R}_{i-1}\right) \vee \\&\quad \left( i=i_{cr} \wedge ||p_i ||\le \bar{R}_{cr}\right) \big )\} \end{aligned}$$

  which transitions agent i from global coverage to waypoint scan mode (1) when the desired position and orientation has been achieved to within some threshold, the coverage rate is insufficient, and the agent operates within its prescribed subdomain,

  $$\begin{aligned}&G\left( \zeta _{i3},\zeta _{i1}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid wp>13 \wedge \\&\quad \big (|\hat{e}_i(t)|<\varepsilon _1\big ) \wedge \big (\left( i=1 \wedge p_i \in \mathcal {D} \right) \vee \\&\quad \left( i \in \{2,\ldots ,N-1\} \wedge ||p_i ||\le \bar{R}_{i-1}\right) \vee \\&\quad \left( i=i_{cr} \wedge ||p_i ||\le \bar{R}_{cr}\right) \big )\} \end{aligned}$$

  which transitions agent i from waypoint scan mode (1) to global coverage when all 13 subwaypoints have been scanned, the coverage rate is insufficient, and the agent operates within its prescribed subdomain,

  $$\begin{aligned}&G\left( \zeta _{i3},\zeta _{i2}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \\&\quad \left( i \in \{2,\ldots ,N-1\} \wedge ||p_i ||> \bar{R}_{i-1}\right) \vee \\&\quad \left( i=i_{cr} \wedge ||p_i ||> \bar{R}_{cr}\right) \} \end{aligned}$$

  which transitions agent i from waypoint scan mode (1) to subdomain transfer mode when the agent is not within its prescribed subdomain,

  $$\begin{aligned}&G\left( \zeta _{i3},\zeta _{i0}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid wp>13 \wedge \\&\quad \big (|\hat{e}_i(t)| \ge \varepsilon _1\big ) \wedge \big (\left( i=1 \wedge p_i \in \mathcal {D} \right) \vee \\&\quad \left( i \in \{2,\ldots ,N-1\} \wedge ||p_i ||\le \bar{R}_{i-1}\right) \vee \\&\quad \left( i=i_{cr} \wedge ||p_i ||\le \bar{R}_{cr}\right) \big )\} \end{aligned}$$

  which transitions agent i from waypoint scan mode (1) to local coverage when all 13 sub-waypoints have been scanned, the coverage rate is sufficient, and the agent operates within its prescribed subdomain,

  $$\begin{aligned}&G\left( \zeta _{i3},\zeta _{i4}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid ||p_i - p_{\star ,i} ||<\varepsilon _2 \wedge \\&\quad \big (\left( i=1 \wedge p_i \in \mathcal {D}\right) \vee \left( i \in \{2,\ldots ,N-1\} \right. \\&\qquad \left. \wedge ||p_i ||\le \bar{R}_{i-1}\right) \vee \\&\quad \left( i=i_{cr} \wedge ||p_i ||\le \bar{R}_{cr}\right) \big )\} \end{aligned}$$

  which transitions agent i from waypoint scan mode (1) to waypoint scan mode (2) when the agent has converged upon a sub-waypoint position to within some threshold and the agent operates within its prescribed subdomain,

  $$\begin{aligned}&G\left( \zeta _{i4},\zeta _{i3}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \varPsi _{i}=\varPsi _{i}(t_{G(\zeta _{i3},\zeta _{i4})})-d\varPsi \wedge \\&\quad \big (\left( i=1 \wedge p_i \in \mathcal {D}\right) \vee \left( i \in \{2,\ldots ,N-1\} \right. \\&\qquad \left. \wedge ||p_i ||\le \bar{R}_{i-1}\right) \vee \\&\quad \left( i=i_{cr} \wedge ||p_i ||\le \bar{R}_{cr}\right) \big )\} \end{aligned}$$

  which transitions agent i from waypoint scan mode (2) to waypoint scan mode (1) when the agent has completed a $360^{\circ }$ yaw sweep at one of the 13 sub-waypoints and operates within its prescribed subdomain

  $$\begin{aligned}&G\left( \zeta _{i4},\zeta _{i2}\right) =\{\tilde{q}_i \in \mathbb {R}^6 \mid \\&\quad \left( i \in \{2,\ldots ,N-1\} \wedge ||p_i ||> \bar{R}_{i-1}\right) \vee \\&\quad \left( i=i_{cr} \wedge ||p_i ||> \bar{R}_{cr}\right) \} \end{aligned}$$

  which transitions agent i from waypoint scan mode (2) to subdomain transfer mode when the agent is not within its prescribed subdomain,

• A reset map: $R=\{\emptyset \}$ which is empty and included for the sake of completeness.

$\hat{e}_i(t)$, defined in (12), evolves as a function of the time history of the continuous states and essentially represents how well the local coverage protocol is proceeding. $\varepsilon _1$ is a guard value for $\hat{e}_i(t)$, while $\varepsilon _2$ and $\varepsilon _3$ are guard values for position and yaw proximity with respect to the desired values, $p_{\star ,i}$ and $\varPsi _{\star ,i}$, respectively. wp is an index variable for a set of sub-waypoints which defined in Sect. 3.4. $t_{G(\zeta _{i3},\zeta _{i4})}$ is the most recent time at which $G(\zeta _{i3},\zeta _{i4})$ was satisfied. $t_{G(\zeta _{i3},\zeta _{i4})}$ is of relevance because satisfaction of $G(\zeta _{i3},\zeta _{i4})$ requires an agent to yaw around $360^{\circ }$ and resume its previous orientation at $t_{G(\zeta _{i3},\zeta _{i4})}$. The variables $\bar{R}_{i-1}$ and $\bar{R}_{cr}$ are the radii of subdomain partitions defined in Sect. 3.1.3 along with $i_{cr}$. The control laws presented in the vector field definition are derived in the body of the paper.

Note that the case in which $i=1$ is included as an exit guard condition from $\zeta _{i2}$ to $\zeta _{i0}$ despite no defined transition for $i=1$ to $\zeta _{i2}$. This is due to the fact that agent $i_{cr}$ may converge upon $\mathcal {O}$ in the state $\zeta _{i2}$ and then become instantaneously reindexed as $i=1$ upon battery exchange.

1.2 Differentiation under the integral sign

We apply the Reynold’s transport theorem (9), a 3-D generalization of the Leibniz integral rule, to (8) in order to compute its time derivative (10). Therefore, it is a necessary condition that both $h\left( C^{\star }-Q\left( t,\tilde{p}\right) \right)$, i.e. the integrand of (8), and $\frac{\partial }{\partial t}h\left( C^{\star }-Q\left( t,\tilde{p}\right) \right)$ are continuous over t and $\tilde{p}$.

Lemma 1

$h\left( C^{\star }-Q\left( t,\tilde{p}\right) \right)$ is continuous in both t and $\tilde{p}$.

Proof

It is a well known result that if two functions are continuous in t and $\tilde{p}$, then their difference, product and the maximum of the two functions are all continuous in t and $\tilde{p}$ (Kaczor and Nowak 2003). Note that 0 and $C^{\star }$ are both constants, and thus continuous in any argument. An assumption of the continuity of $Q\left( t,\tilde{p}\right)$ implies continuity of the difference $C^{\star }-Q\left( t,\tilde{p}\right)$ which implies continuity of $\max \{0,C^{\star }-Q\}$. Continuity of products implies that $h\left( C^{\star }-Q\left( t,\tilde{p}\right) \right) =\max \{0,C^{\star }-Q\}^3$ is continuous in t and $\tilde{p}$. It remains is to verify that continuity of $Q\left( t,\tilde{p}\right)$ is a valid assumption.

Continuity of $Q\left( t,\tilde{p}\right)$ in t implies that $\forall \epsilon >0$, $\exists \delta >0$ such that $|Q\left( t\pm \delta ,\tilde{p}\right) -Q\left( t,\tilde{p}\right) |<\epsilon$. Note the definition of $Q\left( t,\tilde{p}\right)$ in (6) and (7) and that $S_i\left( \tilde{q}_i(\tau ),{\tilde{p}}\right)$ is upper-bounded by 1 and lower-bounded by 0, regardless of its arguments, $\forall i \in \{1,\ldots ,N\}$. The boundedness of the integrand in (6) implies that the existence of an arbitrarily small $\delta$ is guaranteed. Continuity in t holds.

$Q\left( t,\tilde{p}\right)$ may be rewritten in terms of its infinite series right-handed Riemann sum:

$$\begin{aligned} Q\left( t,\tilde{p}\right) =\sum _{i=1}^{N}\lim _{n\rightarrow \infty }\sum _{\ell =1}^{n}S_i\bigl (\tilde{q}_i\left( \ell tn^{-1} \right) ,\tilde{p}\bigr )\left( tn^{-1}\right) . \end{aligned}$$

(58)

We may now show continuity of $S_i\bigl (\tilde{q}_i\left( \ell tn^{-1} \right) ,\tilde{p}\bigr ),$ $\forall \tilde{p} \in \mathcal {D}$. Note from (3) and (5) that $S_i\bigl (\tilde{q}_i\left( \ell tn^{-1} \right) ,\tilde{p}\bigr )$ is defined over all of $\mathcal {D}$ but takes nonzero values only within $\mathcal {S}_i$, as parametrized by continuous constraint functions $C_{ki}, \forall k \in \{1,2,3\}$, and is defined as zero along the boundary $\partial \mathcal {S}_i$. As $S_i\bigl (\tilde{q}_i\left( \ell tn^{-1} \right) ,\tilde{p}\bigr )\rightarrow 0$ as $\tilde{p}\rightarrow \partial \mathcal {S}_i$ from any direction in $\mathbb {R}^3$, we have that $S_i\bigl (\tilde{q}_i\left( \ell tn^{-1} \right) ,\tilde{p}\bigr )$ is continuous $\forall \tilde{p} \in \mathcal {D}$. This continuity holds through summation in (58) as well. This concludes the proof. $\square$

Lemma 2

$\frac{\partial }{\partial t}h\left( C^{\star }-Q\left( t,\tilde{p}\right) \right)$ is continuous in both t and $\tilde{p}$.

Proof

In Lemma 1 it is shown that $Q\left( t,\tilde{p}\right)$ and thus $\max \{0,C^{\star }-Q\left( t,\tilde{p}\right) \}^3$ are continuous in t and $\tilde{p}$. Taking the time derivative we have that $\frac{\partial }{\partial t}h\left( C^{\star }-Q\left( t,\tilde{p}\right) \right) =\left( -3\max \{0,C^{\star }-Q\left( t,\tilde{p}\right) \}^2\right) \left( \frac{\partial Q}{\partial t}\right)$ is continuous in the first multiplicative term. It remains to show that $\frac{\partial Q}{\partial t}$ is continuous in t and $\tilde{p}$. Recall from (6) and (7) that:

$$\begin{aligned} Q\left( t,\tilde{p}\right) =\sum _{i=1}^{N}\int _0^{t}S_i\left( \tilde{q}_i \left( \tau \right) ,\tilde{p}\right) d\tau . \end{aligned}$$

(59)

It is shown in Lemma 1 that $S_i\left( \tilde{q}_i\left( \tau \right) ,\tilde{p}\right)$ is continuous in $\tilde{p}$. Continuity in t is guaranteed as the state of $\mathcal {S}_i$ is determined by the position and orientation of agent i for which discontinuity is physically impossible. Thus (59) meets all necessary conditions for differentiation via the first fundamental theorem of calculus:

$$\begin{aligned} \frac{\partial Q\left( t,\tilde{p}\right) }{\partial t}=\sum _{i=1}^{N}\frac{\partial }{\partial t}\int _{0}^{t}S_i\left( \tilde{q}_i\left( \tau \right) , \tilde{p}\right) d\tau =\sum _{i=1}^{N}S_i\left( \tilde{q}_i \left( \tau \right) ,\tilde{p}\right) . \end{aligned}$$

which is continuous in t and $\tilde{p}$ as verified above and in Lemma 1. This concludes the proof. $\square$

1.3 A supporting lemma

Lemma 3

Under the assumption that $x \le y$ and that $x\ge 2$, one may conclude that:

$$\begin{aligned} \frac{x^{3}}{x-1}\le \frac{y^3}{y-1}. \end{aligned}$$

Proof

The assumptions directly provide the following:

$$\begin{aligned} 2 \le x\le y. \end{aligned}$$

Subtract 1:

$$\begin{aligned} 1 \le x-1 \le y-1, \end{aligned}$$

invert all terms, and then rearrange into the equivalent statement:

$$\begin{aligned} \frac{1}{y-1} \le \frac{1}{x-1} \le 1. \end{aligned}$$

We may refer to $\{\frac{1}{y-1},\frac{1}{x-1}\}$ and $\{\frac{1}{x-1},1\}$ as similarly ordered sets. Chebyshev’s Sum Inequality Theorem as presented in Hardy et al. (1952) may be applied to these similarly ordered sets to yield:

$$\begin{aligned} \frac{1}{\left( y-1\right) \left( x-1\right) }+\frac{1}{x-1} \ge \frac{1}{2}\left( \frac{1}{y-1}+\frac{1}{x-1}\right) \left( \frac{1}{x-1} + 1 \right) . \end{aligned}$$

Now simplify and rearrange as follows:

$$\begin{aligned}&\frac{1}{\left( y-1\right) \left( x-1\right) }+\frac{1}{x-1} \ge \frac{1}{2}\left( \frac{1}{y-1}+\frac{1}{x-1}\right) \left( \frac{x}{x-1} \right) ,\\&\frac{1}{\left( y-1\right) \left( x-1\right) }+\frac{1}{x-1} \ge \frac{1}{2}\left( \frac{x}{\left( y-1\right) \left( x-1\right) } +\frac{x}{\left( x-1\right) ^{2}}\right) ,\\&\frac{1}{\left( y-1\right) \left( x-1\right) }+\frac{1}{x-1} \ge \frac{\frac{1}{2}x}{\left( y-1\right) \left( x-1\right) }+\frac{\frac{1}{2}x}{\left( x-1\right) ^{2}},\\&\frac{1}{x-1} \ge \frac{\frac{1}{2}x-1}{\left( y-1\right) \left( x-1\right) }+\frac{\frac{1}{2}x}{\left( x-1\right) ^{2}}. \end{aligned}$$

Multiply by $2x^{2}\left( x-1\right)$:

$$\begin{aligned} 2x^{2} \ge \frac{x^{3}-2x^{2}}{y-1}+\frac{x^{3}}{x-1}, \end{aligned}$$

and rearrange to form:

$$\begin{aligned} \frac{x^{3}}{x-1} \le 2x^{2}-\frac{x^{3}-2x^{2}}{y-1}. \end{aligned}$$

(60)

Thus, the proof will be completed by verifying that the right hand side of (60) is less than or equal to $\frac{y^{3}}{y-1}$ under the assumptions provided by the lemma. Noting the fact that:

$$\begin{aligned} 2x^{2}-\frac{x^{3}-2x^{2}}{y-1} = \frac{2x^{2}y-x^{3}}{y-1}, \end{aligned}$$

this is equivalent to verifying that $2x^{2}y-x^{3}\le y^{3}$ for $y\ge x$. This is straight forward:

$$\begin{aligned}&\left( x-y\right) ^{2} \ge 0,\\&x^{2}+y^{2}-2xy \ge 0,\\&x^{3}+xy^{2}-2x^{2}y \ge 0, \end{aligned}$$

and thus for $y \ge x$:

$$\begin{aligned} 2x^{2}y \le x^{3} + xy^{2} \le x^{3}+y^{3}. \end{aligned}$$

This verifies the condition $2x^{2}y-x^{3} \le y^{3}$. Therefore, Eq. (60) may be rewritten as:

$$\begin{aligned}&\frac{x^{3}}{x-1} \le 2x^{2}-\frac{x^{3}-2x^{2}}{y-1} \le \frac{y^{3}}{y-1},\\&\frac{x^{3}}{x-1} \le \frac{y^{3}}{y-1}. \end{aligned}$$

This concludes the proof. $\square$