Corrigendum to Determination of an Optimal Return-Path on Road Attributes for Mobile Robot Recharging

Abstract

This corrigendum is offered as a means to correct errors in “Determination of an Optimal Return-path on Road Attributes for Mobile Robot Recharging” (Fei Liu, Shan Liang, Xiaodong Xian, International Journal of Advanced Robotic Systems, vol.8, no.5, pp. 83–92, 2011). The error is due to the fact that we reckoned without taking into account the energy consumed by the sensors on the robot. Please see the corrections below.

Page 85–86

(1) In column 2, the last paragraph should read:

(c) The cost that the robot will pay for passing each segment includes two parts: energy consumption c_e and the influence of vibration on the robot body c_b that describes the probability of the robot's equipment failure. Actually, c_e includes two parts, the part that the sensors on the robot will consume, and secondly, the energy used for driving the motor. Here, we use c_s to describe the first part and use ĉ_e to describe the second part. Therefore, $c_{e} = {\hat{c}}_{e} + c_{s}$ (1)

Then the cost C can be described as $C = (c_{e}, c_{b})$ (2)

Page 86

(1)

In column 2, the chapter title “4.2.1 Mathematical Model of c_e “ should read: “4.2.1 Mathematical Model of ĉ_e”.

(2)

In column 2, the paragraphs after Equation (9) and before Equation (11), all c_e should be replaced by ĉ_e.

(3)

In column 2, Equation (10) should be ${\hat{c}}_{e} = λ W_{P_{m l}}$ (3)

(4)

In column 2, Equation (11) should be ${\hat{c}}_{e} = r_{e} p_{m l}$ (4)

Page 87

(1) In column 2, Equation (20) should be

${\hat{c}}_{e} =(Δ r_{e_r} p_{m r} \cos (p_{m g}) + Δ r_{e_g} \sin (p_{m g})) p_{m l}$ (5)

Page 88

(1) In column 1, after Equation (25) in chapter 4.2.2, the following should be added:

4.2.3 Mathematical Model of c_s

First, we define r_s as the energy consumption rate, which describes how much energy the sensor will consume when the robot walks for a unit distance. The value of r_s is always stable, so for any road segment, for passing distance p_ml , we have the energy consumption $c_{s} = r_{s} p_{m l}$ (6)

(2) In column 2, chapter title “4.2.3 Computing the Idle Time” should be replaced by “4.2.4 Computing the Idle Time”.

Page 89

(1) In column 1, after the first paragraph in chapter 5.1, the following should be added: $(a) r_{s}$

Here, we do not measure r_s directly, on the contrary, we first measure the consumption rate for unit time, which can be termed τ_s (unit: V/ s). If V_start (unit: V ) is the voltage value when the robot is just powered on, and after a period of time t , the value decreases to ^Vend (unit: V), we get $τ_{s} = \frac{V_{s t a r t} - V_{e n d}}{t}$ (7)

We assume that the robot walking for distance p_ml at average speed v̄ (unit: m/s), then $r_{s} = \frac{τ_{s}}{\bar{v}}$ (8)

In our experiment, we set V_start = 12.7V, V_end = 11.3V, v̄ = 0.75m/s, then it is measured that t = 4.5h, therefore, r_s is measure to be $r_{s} = \frac{12.7 V - 11.3 V}{0.75 m / s \times 4.5 h \times 3600 s / h} \approx 0.1152 V / m$ (9)

(2) Equation (38) should be $r_{e} = \frac{V_{s t a r t} - V_{e n d}}{\bar{v} \bar{t}} - r_{s}$ (10)

(3) Equation (39) should be $\begin{array}{l} r_{e_μ_{f}} = \frac{12.7 V - 11.3 V}{0.75 m / s \times 2.67 h \times 3600 s / h} - 0.1152 \times 10^{- 3} V / m \\ \approx 0.0790 \times 10^{- 3} V / m \end{array}$ (11)

(4) Equation (40) should be $\begin{array}{l} r_{e_μ_{r}} = \frac{12.7 V - 11.3 V}{0.75 m / s \times 2.07 h \times 3600 s / h} - 0.1152 \times 10^{- 3} V / m \\ \approx 0.1353 \times 10^{- 3} V / m \end{array}$ (12)

(5) Equation (41) should be $μ_{r} = \frac{r_{e_μ_{r}}}{r_{e_μ_{f}}} μ_{f} = \frac{0.1353 \times 10^{- 3}}{0.0790 \times 10^{- 3}} μ_{f} \approx 1.7127 μ_{f}$ (13)

(6) Equation (42) should be $\begin{array}{l} r_{e_μ_{f} φ_{g}} \approx 0.2034 \times 10^{- 3} V / m - 0.1152 \times 10^{- 3} V / m \\ = 0.0882 \times 10^{- 3} V / m \end{array}$ (14)

Page 90

(1) Equation (45) should be $η = \frac{r_{b_μ_{r}} μ_{f}}{r_{b_μ_{f}} μ_{r}} = \frac{0.5698 \times 10^{- 5}}{0.2645 \times 10^{- 5} \times 1.7127} \approx 1.2578$ (15)

(2) In the penultimate paragraph of column 2, the following corrections should be made:

As listed in Table 2, the cost C_RD_0n is (0.0520,0.8693×10⁻³) and the idle time T_{IDLE_} R_D₀ is 346.834s.

Table 2.

The optimal return-path to each dock D_h(h = 0, 1, 2) as well as the constitution of each R_Dh (h = 0, 1, 2), the total path length L_T, the cost and the idle time.

D_h	D ₀	D ₁	D ₀₂
RD _h	{P_SRO, P₂₁,P₀}	{P_SRP, P₁₉,P₉}	{P_SRP, P₁₆,P₁₇}
L_T/m	262.1	282.9	246.4
(^C_e, ^C_b)	(0.0520, 0.8693 ×10⁻³)	(0.0544, 2.4780 × 10⁻³)	(0.0529, 1.1827 × 10⁻³)
T_IDLE /S	351.361	390.160	372.055

(3) Equation (50) should be $Δ T_{I D L E} = 366.547 s - 346.834 s = 19.713 s$ (16)

Page 91

(1) Table 2 should be replaced by

(2) In column 2, in the paragraph below Table 2, the following corrections should be made:

Statistically, if the robot returns to recharge four times a day on average and δT_IDLE = 20.694s is taken as an example, (3) Equation (51) should be $T_{S A V E} = \frac{19.71 s \times 4 \times 360}{3600 s / h} = 7.88 h$ (17)

(4) Equation (52) should be $N = \frac{T_{S A V E}}{T_{M T T R}} = \frac{7.88 h}{4 h} = 1.97$ (18)