# A Kamm’s Circle-Based Potential Risk Estimation Scheme in the Local Dynamic Map Computation Enhanced by Binary Decision Diagrams

^{*}

## Abstract

**:**

## 1. Introduction

- Layer 1: Contains permanent static information. It is a map database that preferably contains detailed road map information with application to advanced driver assistance systems (ADAS).
- Layer 2: This layer is an extension of layer 1. It includes quasi-static information, e.g., traffic signs, trees, and buildings.
- Layer 3: LDM stores temporary information for a particular region in this layer, e.g., traffic jams, weather conditions, and traffic signals.
- Layer 4: Contains temporary information about dynamic or highly dynamic objects, e.g., moving vehicles and pedestrians.

- The vehicle’s future geographical occupancy over time as a feature in the LDM.
- A extended method of data representation for a vehicle’s geographical occupancy information using a BDD.
- Possible algebraic operations between the exchanged BDDs can confirm the possibility of future interaction, which is consistent with the C-ITS nature of data sharing.
- Ways of data insertion and database operations for vehicle properties in the linked-list-based BDD running on the PostgreSQL database-based LDM.

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. Geohash

#### 3.2. Boolean Function and Reduced Ordered Binary Decision Diagrams (ROBDD)

#### 3.2.1. Boolean Function

#### 3.2.2. Reduced Ordered Binary Decision Diagrams (ROBDD)

- $(Q,{v}_{0},E)$ is a rooted directed acyclic graph. Q is a finite set of nodes. ${v}_{0}$ is the root node and $E\subset Q\times Q$. Each non-leaf node has its successors, namely low and high.
- V is a finite set of Boolean variables.
- < is a total order on $V\cup \{0,1\}$.
- L is a mapping satisfying the following conditions:
- –
- Leafs are mapped to 0 and 1 and non-leaf nodes are mapped to V.
- –
- If (v,v’) $\in E$ then L(v) < L(v’).

- Merge all zero and one nodes to a single unit of zero and one node.
- Merge any isomorphic nodes, i.e., if $l\left(x\right)=l\left(y\right)$ and $h\left(x\right)=h\left(y\right)$ then merge these nodes into one and point all incoming nodes to any one of them. Here l and h represent the low and high child of any given node of a graph.
- Eliminate any node that has two children nodes as isomorphic.

#### 3.3. Geohash Set as a BDD

- BDD representation of a unit Geohash: A Geohash is a unique symbolic representation of all the points available within the given area on the earth. For each character in Geohash, 32 values (English letters except “a”, “i”, “l”, “o”, and decimal system digits 0–9) are possible to use, and, therefore, in our model, five Boolean variables (${2}^{5}=32$) were applied correspondingly (Figure 1). Consequently, five nodes in a BDD were used to represent the corresponding Boolean variables for a binary representation of a given character in a Geohash. For a given Geohash, each character had five corresponding nodes in the BDD. For a Geohash of 10 characters/levels, 50 nodes were needed for corresponding bits, plus two extra nodes representing zero (false) and one (true) leaf node in a BDD. (for experiments, the vehicle was assumed to be within a Geohash, having a distance of 4872 m (north to south) and 3955 m (east to west); hence, a five-level BDD with 25 nodes served the purpose), i.e., the first five levels of Geohash did not change in our setting. Every corresponding node, low or high, has its values depending on the Boolean function represented. Therefore, to represent a single Geohash using BDD corresponding binary string ends at one (1) node of a BDD, and all other binary strings end in zero (0) (Figure 3).
- BDD representation of a set of Geohash: A synthesis of BDD was applied (we borrowed the term “synthesis" from [28]) to represent a set of Geohashes in a single BDD. In the BDD synthesis, BDDs were built for complex sets/functions representing Geohash locations (e.g., BDD for function f can combine with function g to represent BDD for $f\phantom{\rule{0.166667em}{0ex}}AND\phantom{\rule{0.166667em}{0ex}}g$, $f\phantom{\rule{0.166667em}{0ex}}OR\phantom{\rule{0.166667em}{0ex}}g$, $NOT\phantom{\rule{0.166667em}{0ex}}f$, $f\phantom{\rule{0.166667em}{0ex}}XOR\phantom{\rule{0.166667em}{0ex}}g$). Corresponding set interpretations were necessary for a given BDD representing f and g sets (here Geohash sets) of the above synthesis operations. The $apply$ method in [23] was introduced to achieve the following operations:
- (a)
- $f\phantom{\rule{0.166667em}{0ex}}OR\phantom{\rule{0.166667em}{0ex}}g$ is the set union operation. $f\phantom{\rule{0.166667em}{0ex}}\cup \phantom{\rule{0.166667em}{0ex}}g\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\{\alpha \mid \alpha \in f\phantom{\rule{0.166667em}{0ex}}or\phantom{\rule{0.166667em}{0ex}}\alpha \in g\}$
- (b)
- $f\phantom{\rule{0.166667em}{0ex}}AND\phantom{\rule{0.166667em}{0ex}}g$ is the set intersection operation. $f\phantom{\rule{0.166667em}{0ex}}\cap \phantom{\rule{0.166667em}{0ex}}g\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\{\alpha \mid \alpha \in f\phantom{\rule{0.166667em}{0ex}}and\phantom{\rule{0.166667em}{0ex}}\alpha \in g\}$
- (c)
- $f\phantom{\rule{0.166667em}{0ex}}XOR\phantom{\rule{0.166667em}{0ex}}g$ is the set symmetric difference operation. $f\phantom{\rule{0.166667em}{0ex}}\oplus \phantom{\rule{0.166667em}{0ex}}g\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}(\phantom{\rule{0.166667em}{0ex}}f\backslash g)\phantom{\rule{0.166667em}{0ex}}\cup (\phantom{\rule{0.166667em}{0ex}}f\backslash g)\phantom{\rule{0.166667em}{0ex}}$

#### 3.4. Reachable Positions by a Vehicle over Time t

#### Reach/Reachable Sets and Abstraction

- Reach Set—The set of states x at time t for which sequence of control inputs ${u}_{0},{u}_{1},\cdots ,{u}_{t-1}$ exists from the initial states ${x}_{0}\in {X}_{0}$ are known as reach set $R({X}_{0},t)$ [31].
- Reachable Set—Reachable set at time t is the union of all the reach sets $\le t$$$\overline{\mathrm{R}}({X}_{0},t)={\cup}_{s\le t}R({X}_{0},t)$$
- Abstraction—For a model (refer to definition in [29]) M of a given vehicle, abstraction was defined as the model ${M}_{i}$ if the reachable set of the abstraction contains the reachable set of the model M (Figure 6).
**Figure 6.**Abstraction of a model contains all reachable states which are reachable by the original model. Here states reachable by all abstraction models ${M}_{1},{M}_{2},\cdots ,{M}_{i}$ contain reachable states by a vehicle model M.$$\forall t>0:R(M,t)\subseteq R({M}_{i},t)$$

- $c\left(t\right)$ is a position of a vehicle at time t.
- ${s}_{x}\left(0\right)\phantom{\rule{0.166667em}{0ex}}and\phantom{\rule{0.166667em}{0ex}}{s}_{y}\left(0\right)$ is the position of the vehicle at time t = 0.
- ${v}_{x}\left(0\right)\phantom{\rule{0.166667em}{0ex}}and\phantom{\rule{0.166667em}{0ex}}{v}_{y}\left(0\right)$ is the velocities in the x and y directions of the vehicle at time t = 0.
- $r\left(t\right)$ is the radius of a Kamm’s/traction circle at time t.
- ${a}_{max}$ is the maximum acceleration possible of a given vehicle.

#### 3.5. BDD for Geohash Set Enclosing Kamm’s Circle

- Reduce: Give reduced BDD in its canonical form.
- Apply: Perform synthesis operation between two BDDs. ${f}_{1}\phantom{\rule{0.166667em}{0ex}}<op>{f}_{2}$.
- Satisfy-One: Returns any one element in ${S}_{f}$, where ${S}_{f}$ is the set of all Geohash represented by a given BDD.
- Satisfy-All: Output ${S}_{f}$. All Geohashes, a given BDD, satisfy.

Algorithm 1: Algorithm to find neighboring Geohash BDD. |

1 Input: inpGeo -Geohash BDD. h in {west, east, null}, v in {south, north, null}.2 Output: Neighbour in east, west, north, south, north-west, north-east, south-east, south-west Geohash BDD. 3 S = Satisfy-One(inpGeo) 4 T = [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] 5 for i = 0 to T.$length$: 6 for j = 0 to i: 7 If $j\%2==0$ then: 8 If $h==west$ then: 9 T[j] = T[j] $and$ $not$(S[j]) 10 elif $h==east$ then: 11 T[j] = T[j] and S[j] 12 else: 13 If $v==south$ then: 14 T[j] = T[j] $and$ $not$(S[j]) 15 elif $v==north$ then: 16 T[j] = T[j] and S[j] 17 if $h!=null$: 18 T[T.$length$-1] = 1 19 if $v!=null$: 20 T[T.$length$-2] = 1 21 for i = 0 to T.$length$: 22 S[i] = S[i] $xor$ T[i] 23 return createStringtoBDD(S) |

- If ${p}_{k}<0$ then:$({x}_{k},{y}_{k})$ = $({x}_{k}+1,{y}_{k})$ and new ${p}_{k}$ is calculated as ${p}_{k+1}={p}_{k}+2{x}_{k+1}+1$
- else:$({x}_{k},{y}_{k})$ = $({x}_{k}+1,{y}_{k}-1)$ and new ${p}_{k}$ is calculated as ${p}_{k+1}={p}_{k}+2{x}_{k+1}+1-2{y}_{k+1}$

- if $p<=0$:
- Generate east BDD and union it with circle_BDD. Additionally, update the value $x\_k=x\_k+0.96$, $e\_count=e\_count+1$ and record the north limit of this BDD from the center. Finally, update the value of the decision parameter as $p=p+2\ast x\_k+1$.

- else:
- Generate southeast BDD and union it with circle_BDD. Additionally, update the value $x\_k=x\_k+0.96;\text{}y\_k=y\_k-0.59$ and record the north and east limit of this BDD from the center. Finally, update the value of the decision parameter as $p=p+2\ast x\_k+1-2\ast y\_k$.

Algorithm 2: Modified midpoint circle generation algorithm. |

1 Input: inpGeo—Center Geohash BDD, r—radius in meters unit.2 Output: BDD for a set of Geohashes enclosing Kamm’s circle. 3 /*Step I.*/ 4 up_count = $\lceil radius/0.59\rceil $ 5 quad1_north_limit = quad1_east_limit = [] 6 circle_BDD = inpGeo 7 BDD1 = BDD2 = BDD3 = BDD4 = inpGeo 8 x_k = y_k = 0 9 n_count = e_count = 0 10 /*Step II.*/ 11 for k = 0 to up_count: 12 BDD1 = Generate north BDD of BDDs. 13 BDD2 = Generate south BDD of BDDs. 14 y_k = y_k + 0.59 15 n_count = n_count + 1 16 $BDD1\cup BDD2\cup circle\_BDD.$/*Apply union with circle_BDD*/ 17 /*Step III.*/ 18 p = INT(ROUND(5/4) - r) 19 /*Step IV.*/ 20 while $x\_k<=y\_k$: 21 if $p<=0$: 22 BDD1 = Generate east BDD of BDD1. 23 x_k = x_k + 0.96 24 e_count = e_count + 1 25 quad1_north_limit.append(n_count) 26 $BDD1\cup circle\_BDD.$ 27 p = p + 2 * x_k + 1 28 else: 29 BDD1 = Generate south east BDD of BDD1. 30 $BDD1\cup circle\_BDD.$ 31 x_k = x_k + 0.96 32 y_k = y_k - 0.59 33 quad1_east_limit.append(e_count) 34 e_count = e_count + 1 35 n_count = n_count - 1 36 quad1_north_limit.append(n_count) 37 p = p + 2 * x_k + 1 - 2 * y_k 38 quad1_east_limit.append(x_count) 39 /*Step V.*/ 40 for w = 0 to quad1_east_limit.$length$-1: 41 Generate BDD3 and BDD4 east and west of BDD3 respectively. 42 $circle\_BDD\cup BDD3\cup BDD4$ 43 for k = 0 to quad1_north_limit[w]: 44 Generate BDD5 and BDD6 north and south of BDD3 respectively. 45 Generate BDD7 and BDD8 north and south of BDD4 respectively. 46 $circle\_BDD\cup BDD5\cup BDD6\cup BDD7\cup BDD8$ 47 /*Step VI.*/ 48 for w = quad1_east_limit.$length$-1 to 0: 49 Generate BDD3 and BDD4 east and west of BDD3 respectively. 50 $circle\_BDD\cup BDD3\cup BDD4$ 51 a = $\lceil x\_count\left[w\right]\ast \left(1.6\right)\rceil $ /*1.6, Geohash (10 level) breadth to height ratio*/ 52 if $a>=quad1\_north\_limit\left[w\right]$ then: 53 a = quad1_north_limit[w] 54 for k = 0 to a: 55 Generate BDD5 and BDD6 north and south of BDD3 respectively. 56 Generate BDD7 and BDD8 north and south of BDD4 respectively. 57 $circle\_BDD\cup BDD5\cup BDD6\cup BDD7\cup BDD8$ 58 return circle_BDD |

Algorithm 3: Midpoint circle generation algorithm. |

1 Input: r—radius of a circle, (${x}_{c}$,${y}_{c}$) center of the circle.2 Output: Squares to include on a square grid to form a circle of radius r. 3 I. First square to include ($({x}_{0},{y}_{0})=(0,r)$) 4 II. Calculate the initial value for the decision parameter. ${p}_{0}=\frac{5}{4}-r$ III. For successive values of k, $({x}_{k},{y}_{k})$ is determined as follows. If ${p}_{k}<0$ then: $({x}_{k},{y}_{k})$ = $({x}_{k}+1,{y}_{k})$ and new ${p}_{k}$ is calculated as ${p}_{k+1}={p}_{k}+2{x}_{k+1}+1$ else: $({x}_{k},{y}_{k})$ = $({x}_{k}+1,{y}_{k}-1)$ and new ${p}_{k}$ is calculated as ${p}_{k+1}={p}_{k}+2{x}_{k+1}+1-2{y}_{k+1}$ IV. Determine the symmetry points in the other seven octants. V. Repeat Steps III to IV until $x\le y$. |

## 4. Experiment

## 5. Results

- Task1—getLaneletId (to get the lanelet id and data corresponding to an ego vehicle).
- Task2—getVehicleInAdjacentLanelet (to retrieve data of all vehicles (other than ego) present in the ego vehicle’s current lanelet or its adjacent lanelets).
- Task3—averageNoOfVehicles (to retrieve the number of vehicles present around an ego vehicle for a given scenario).

## 6. Discussions

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Geohash follows an alternate sequence of space-filling curves. Characters binary representation determining latitude X bits and longitude Y bits cross bit by bit.

**Figure 2.**(

**a**) Binary decision tree representation for a given set has a fixed size and is large compared to BDD representation. (

**b**) Binary decision diagram representation for a given function has compact representation.

**Figure 4.**(

**a**–

**c**) BDD OR operation is equivalent to set union operation. (

**d**–

**f**) BDD AND operation is equivalent to set intersection operation. (

**g**–

**i**) BDD XOR operation is equivalent to a set symmetric difference operation.

**Figure 5.**BDD for a set of 701 Geohashes. (Interconnection between the 25th–50th nodes is shown for brevity.)

**Figure 7.**Overall force is limited to ${F}_{f}$. (

**a**) An increase in longitudinal force limits the lateral force. (

**b**) An increase in lateral force limits the longitudinal force.

**Figure 10.**(Scenario-1) An example of an intersection center (geographical data from OSM [36] are illustrated as a superimposed background image).

**Figure 11.**(Scenario-2) A city road scenario (geographical data from OSM [36] are illustrated as superimposed background image).

**Figure 13.**(

**a**) Projected reachable Geohash for the vehicle at $t\in \{0.3,0.7,1.2\}$ s. (

**b**) Projected reachable Geohash for the vehicles at $t\in \{0.4,0.8,1.2\}$ s. (

**c**) Flow chart to avoid collision using BDD in the LDM setup.

**Figure 15.**Time in milliseconds for operations (get ego vehicle lanelet id, get vehicles ids in adjacent lanelets of ego vehicle, average number of vehicles in adjacent lanelets, BDD intersection operation with adjacent vehicles for collision avoidance).

**Figure 16.**Time in milliseconds for operations (get ego vehicle lanelet id, get vehicles ids in adjacent lanelets of ego vehicle, average of vehicles in adjacent lanelets, collision risk warning algorithm from Shimada et al.).

#Label in Geohash | Distance in North and South (m) | Distance in East and West (m) | A Geohash Example |
---|---|---|---|

1 | 4,989,600 | 4,050,000 | w |

2 | 623,700 | 1,012,500 | wy |

3 | 155,925 | 126,562.5 | wyh |

4 | 19,490.625 | 31,640.625 | wyhb |

5 | 4872.65625 | 3955.07813 | wyhby |

6 | 609.082031 | 988.769531 | wyhby3 |

7 | 152.270508 | 123.596191 | wyhby3k |

8 | 19.0338135 | 30.8990479 | wyhby3kf |

9 | 4.75845337 | 3.86238098 | wyhby3kf5 |

10 | 0.59480667 | 0.96559525 | wyhby3kf5f |

11 | 0.14870167 | 0.12069941 | wyhby3kf5fs |

12 | 0.01858771 (≈ 1.86 (cm)) | 0.03017485 (≈ 3.02 [cm]) | wyhby3kf5fst |

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**MDPI and ACS Style**

Kumar, A.; Wagatsuma, H.
A Kamm’s Circle-Based Potential Risk Estimation Scheme in the Local Dynamic Map Computation Enhanced by Binary Decision Diagrams. *Sensors* **2022**, *22*, 7253.
https://doi.org/10.3390/s22197253

**AMA Style**

Kumar A, Wagatsuma H.
A Kamm’s Circle-Based Potential Risk Estimation Scheme in the Local Dynamic Map Computation Enhanced by Binary Decision Diagrams. *Sensors*. 2022; 22(19):7253.
https://doi.org/10.3390/s22197253

**Chicago/Turabian Style**

Kumar, Arvind, and Hiroaki Wagatsuma.
2022. "A Kamm’s Circle-Based Potential Risk Estimation Scheme in the Local Dynamic Map Computation Enhanced by Binary Decision Diagrams" *Sensors* 22, no. 19: 7253.
https://doi.org/10.3390/s22197253