SCIENCE CHINA Information Sciences, Volume 62 , Issue 1 : 012202(2019) https://doi.org/10.1007/s11432-017-9306-x

## FVO: floor vision aided odometry

Yu KANG 1,2,*,
• AcceptedNov 6, 2017
• PublishedAug 15, 2018
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### Acknowledgment

This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61725304, 61673361). The authors also gratefully acknowledge the support from Youth Top-notch Talent Support Program, 1000-talent Youth Program and Youth Yangtze River Scholar.

### Supplement

Appendix

Interval arithmetic

This section exhibits some fundamentals in interval arithmetic that may be involved in this paper.

An interval $\mathcal{A}=[a_1,a_2]$ is a set of real numbers denoted by \begin{align}&[a_1,a_2]=\{x\in\mathbb{R}:a_1\leq x\leq a_2\}, \tag{31} \end{align} where $a_1=-\infty$ and $a_2=+\infty$ are allowed, with $\mathrm{mid}(\mathcal{A})=\frac{a_1+a_2}{2}$ and $\mathrm{rad}(\mathcal{A})=\frac{a_2-a_1}{2}$ denoting its midpoint and radius, respectively.

The four basic arithmetic operations are as follows.

Addition. $\mathcal{A}+\mathcal{B}=[a_1+b_1,a_2+b_2]$;

Substraction. $\mathcal{A}-\mathcal{B}=[a_1-b_2,a_2-b_1]$;

Multiplication. $\mathcal{A}\cdot\mathcal{B}=[ \min\{a_1\cdot~b_1,~a_1\cdot~b_2,~a_2~\cdot~b_1,~a_2\cdot~b_2~\},\, \max\{a_1\cdot~b_1,~a_1\cdot~b_2,~a_2~\cdot~b_1,~a_2\cdot~b_2~\}]$;

Division. $\mathcal{A}/\mathcal{B}=[a_1,a_2]\cdot[{1}/{b_2},{1}/{b_1}]$textrm if $0\notin[b_1,b_2]$.

If $b_2<a_1$ or $a_2<b_1$, the intersection of two intervals $\mathcal{A}=[a_1,a_2]$ and $\mathcal{B}=[b_1,b_2]$ is empty, that is, \begin{align}&\mathcal{A}\cap\mathcal{B}=\emptyset. \tag{32} \end{align} Otherwise, we have \begin{align}&\mathcal{A}\cap\mathcal{B}=\left\{x:x\in\mathcal{A} \textrm{ and }\mathcal{B}\right\}=[\max\left\{a_1,b_1\right\},\min\left\{a_2,b_2\right\}]. \tag{33} \end{align} The function of an interval $\mathcal{A}=\left[a_1,a_2\right]$ is defined by \begin{align}&f(\mathcal{A})=\left\{f(x):x\in\mathcal{A}\right\}, \tag{34} \end{align} which is still an interval. For the monotonic functions, such as an exponential or logarithmic function, we have \begin{align}&f(\mathcal{A})=[\min\left\{f(a_1),f(a_2)\right\},\max\left\{f(a_1),f(a_2)\right\}]. \tag{35} \end{align} For a sine function, a piecewise monotonic function with critical points at $n\pi+\frac{\pi}{2}$, where $n\in\mathbb{Z}$, we have \begin{align}&\sin(\mathcal{A})=\left\{ \begin{array}{lll} [\min\left\{\sin(a_1),\sin(a_2)\right\},\max\left\{\sin(a_1),\sin(a_2)\right\}], & \textrm{if }n_2-n_1=0, \\ \left[\min \{\sin(a_1),\sin(a_2) \},+1\right], & \textrm{if }n_2-n_1=1 \textrm{ and }n_1\textrm{ is even}, \\ \left[-1,\max\left\{\sin(a_1),\sin(a_2)\right\}\right], & \textrm{if }n_2-n_1=1 \textrm{ and }n_1\textrm{ is odd}, \\ \left[-1,+1\right], & \textrm{if }n_2-n_1\geq2, \end{array} \right. \tag{36} \end{align} where $n_1=\lfloor~\frac{a_1+\frac{\pi}{2}}{\pi}~\rfloor$ and $n_2=\lfloor~\frac{a_2+\frac{\pi}{2}}{\pi}~\rfloor$ are two integers. Similarly, we have \begin{align}&\cos(\mathcal{A})=\left\{ \begin{array}{lll} \left[\min\left\{\cos(a_1),\cos(a_2)\right\},\max\left\{\cos(a_1),\cos(a_2)\right\}\right], & \textrm{if }n_2-n_1=0, \\ \left[\min\left\{\cos(a_1),\cos(a_2)\right\},+1\right], & \textrm{if }n_2-n_1=1 \textrm{ and }n_1\textrm{ is odd}, \\ \left[-1,\max\left\{\cos(a_1),\cos(a_2)\right\}\right], & \textrm{if }n_2-n_1=1 \textrm{ and }n_1\textrm{ is even}, \\ \left[-1,+1\right], & \textrm{if }n_2-n_1\geq2, \end{array} \right. \tag{37} \end{align} where $n_1=\lfloor~\frac{a_1}{\pi}~\rfloor$ and $n_2=\lfloor~\frac{a_2}{\pi}~\rfloor$ are also two integers.

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• Figure 1

(Color online) Top-view schematic of mobile robot on a planar floor. The meshed yellow rectangles represent the robot's wheels and the blue polygon its body. The red non-consecutive arc represents the trajectory.

• Figure 2

(Color online) (a) Relation between floor vision and robot's heading and location. The purple grid represents tile joints. The rectangle with black non-consecutive edges represents the camera's field of view. The robot's headings fall into four different quadrants, respectively, but they have the same floor vision. (b) The schematic of Case 1.

• Figure 3

(Color online) Feature extracting process of floor vision. The green orthogonal lines in the sixth picture are the detected lines that almost coincide with the tile joints.

• Figure 4

(Color online) Test results of heading estimation of FVO. The curve “GT" represents the ground truths of the robot heading obtained from the off-board camera. The curve “Odo" represents the odometric heading estimation. The curves “FVO", “FVO$-$", and “FVO+" represent the midpoint, lower bound, and upper bound of the heading estimation of FVO, respectively.

• Figure 5

(Color online) Test results of location estimation of FVO. The curve “GT" represents the ground truths of the robot's location obtained from the off-board camera. The curve “Odo" represents the odometric location estimation. The green rectangular boxes represent the location estimation of FVO, while the curve “FVO" represents the midpoint of the location estimation of FVO.

• Table 1   Transformation rules from $\theta$, $\vartheta$, and $r$ to $x$ and $y$
 Case Rule 2*Case 1: $\theta\in\left(0,\frac{\pi}{2}\right]$ $x\in\{E_xi+d|_{\vartheta\geq0}+L\cos\theta:i=0,1,2,\ldots\}$ $y\in\{E_yi-d|_{\vartheta<0}+L\sin\theta:i=0,1,2,\ldots\}$ 2*Case 2: $\theta\in\left(\frac{\pi}{2},\pi\right]$ $x\in\{E_xi+d|_{\vartheta<0}+L\cos\theta:i=0,1,2,\ldots\}$ $y\in\{E_yi+d|_{\vartheta\geq0}+L\sin\theta:i=0,1,2,\ldots\}$ 2*Case 3: $\theta\in\left(\pi,\frac{3\pi}{2}\right]$ $x\in\{E_xi-d|_{\vartheta\geq0}+L\cos\theta:i=0,1,2,\ldots\}$ $y\in\{E_yi+d|_{\vartheta<0}+L\sin\theta:i=0,1,2,\ldots\}$ 2*Case 4: $\theta\in\left(\frac{3\pi}{2},2\pi\right]$ $x\in\{E_xi-d|_{\vartheta<0}+L\cos\theta:i=0,1,2,\ldots\}$ $y\in\{E_yi-d|_{\vartheta\geq0}+L\sin\theta:i=0,1,2,\ldots\}$
• Table 2   Calculation of the interval of floor visual location measurement
Case Intervals
2* Case 1: $\widehat{\varTheta}_t\subseteq\left(0,\frac{\pi}{2}\right]$
$\mathcal{X}_{t}^{(c)}=\left\{E_xi+\bigcap_{j\in\mathcal{I}_{y,t}}\mathcal{D}_{t,j}^{(c)}+L\cos\widehat{\varTheta}_t:i=0,1,2,\ldots\right\}$, where $\mathcal{I}_{y,t}=\left\{i:\min\left\{\varPhi_{t,i}^{(c)}\right\}\geq0,i\in\mathcal{I}_{\theta,t}\right\}~~$
$\mathcal{Y}_{t}^{(c)}=\left\{E_yi-\bigcap_{j\in\mathcal{I}_{x,t}}\mathcal{D}_{t,j}^{(c)}+L\sin\widehat{\varTheta}_t:i=0,1,2,\ldots\right\}$, where $\mathcal{I}_{x,t}=\left\{i:\max\left\{\varPhi_{t,i}^{(c)}\right\}<0,i\in\mathcal{I}_{\theta,t}\right\}$
2* Case 2: $\widehat{\varTheta}_t\subseteq\left(\frac{\pi}{2},\pi\right]$
$\mathcal{X}_{t}^{(c)}=\left\{E_xi+\bigcap_{j\in\mathcal{I}_{y,t}}\mathcal{D}_{t,j}^{(c)}+L\cos\widehat{\varTheta}_t:i=0,1,2,\ldots\right\}$, where $\mathcal{I}_{y,t}=\left\{i:\max\left\{\varPhi_{t,i}^{(c)}\right\}<0,i\in\mathcal{I}_{\theta,t}\right\}~~$
$\mathcal{Y}_{t}^{(c)}=\left\{E_yi+\bigcap_{j\in\mathcal{I}_{x,t}}\mathcal{D}_{t,j}^{(c)}+L\sin\widehat{\varTheta}_t:i=0,1,2,\ldots\right\}$, where $\mathcal{I}_{x,t}=\left\{i:\min\left\{\varPhi_{t,i}^{(c)}\right\}\geq0,i\in\mathcal{I}_{\theta,t}\right\}$
2* Case 3: $\widehat{\varTheta}_t\subseteq\left(\pi,\frac{3\pi}{2}\right]$
$\mathcal{X}_{t}^{(c)}=\left\{E_xi-\bigcap_{j\in\mathcal{I}_{y,t}}\mathcal{D}_{t,j}^{(c)}+L\cos\widehat{\varTheta}_t:i=0,1,2,\ldots\right\}$, where $\mathcal{I}_{y,t}=\left\{i:\min\left\{\varPhi_{t,i}^{(c)}\right\}\geq0,i\in\mathcal{I}_{\theta,t}\right\}~~$
$\mathcal{Y}_{t}^{(c)}=\left\{E_yi+\bigcap_{j\in\mathcal{I}_{x,t}}\mathcal{D}_{t,j}^{(c)}+L\sin\widehat{\varTheta}_t:i=0,1,2,\ldots\right\}$, where $\mathcal{I}_{x,t}=\left\{i:\max\left\{\varPhi_{t,i}^{(c)}\right\}<0,i\in\mathcal{I}_{\theta,t}\right\}$
2* Case 4: $\widehat{\varTheta}_t\subseteq\left(\frac{3\pi}{2},2\pi\right]$
$\mathcal{X}_{t}^{(c)}=\left\{E_xi-\bigcap_{j\in\mathcal{I}_{y,t}}\mathcal{D}_{t,j}^{(c)}+L\cos\widehat{\varTheta}_t:i=0,1,2,\ldots\right\}$, where $\mathcal{I}_{y,t}=\left\{i:\max\left\{\varPhi_{t,i}^{(c)}\right\}<0,i\in\mathcal{I}_{\theta,t}\right\}~~$
$\mathcal{Y}_{t}^{(c)}=\left\{E_yi-\bigcap_{j\in\mathcal{I}_{x,t}}\mathcal{D}_{t,j}^{(c)}+L\sin\widehat{\varTheta}_t:i=0,1,2,\ldots\right\}$, where $\mathcal{I}_{x,t}=\left\{i:\min\left\{\varPhi_{t,i}^{(c)}\right\}\geq0,i\in\mathcal{I}_{\theta,t}\right\}$
• Table 3   Error statistics of FVO
 Variable Method 1st period 2nd period 3rd period 2*$\theta$ (deg) Odometry 1.93 7.16 19.65 FVO 1.88 2.24 2.83 2*$x$ (mm) Odometry 34.25 74.28 259.56 FVO 6.06 5.43 5.22 2*$y$ (mm) Odometry 47.80 66.22 373.17 FVO 5.34 6.34 6.61

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