# Occlusion-Aware Path Planning to Promote Infrared Positioning Accuracy for Autonomous Driving in a Warehouse

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## Abstract

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## 1. Introduction

#### 1.1. Related Work

#### 1.2. Contributions

#### 1.3. Organization

## 2. Problem Statement

**z**to its footprint, thus $fp(z(s))\subset {\mathsf{\chi}}_{\mathrm{free}}$ represents the collision-avoidance constraints [45].

## 3. A Four-Layer Path Planning Method

#### 3.1. Layer One: Search an A* Path

#### 3.2. Layer Two: Generate a Reference Line

_{FE}equidistant waypoints ${\mathrm{W}}_{\mathrm{grrs}}=\left\{({\mathrm{x}}_{i}^{\mathrm{grrs}},{\mathrm{y}}_{i}^{\mathrm{grrs}})|i=1,\dots ,{\mathrm{N}}_{\mathrm{FE}}\right\}$. A reference line is generated by driving a virtual vehicle to track the waypoints. This process can be described as a trajectory planning-oriented OCP:

_{W}denotes the vehicle wheelbase (Figure 3). The boundary constraints $\underset{\_}{z}\le z(t)\le \overline{z}$ and $\underset{\_}{u}\le u(t)\le \overline{u}$ are defined as

_{FE}waypoints $\left\{({\mathrm{x}}_{i}^{\mathrm{grrs}},{\mathrm{y}}_{i}^{\mathrm{grrs}})|i=1,\dots ,{\mathrm{N}}_{\mathrm{FE}}\right\}$ in a sequence.

#### 3.3. Layer Three: Search for a DP Path in the Frenet Frame

_{start}, l

_{start}) and (s

_{end}, l

_{end}), respectively (Figure 4). The second step is to generate (N

_{S}+ 1) equidistant skeleton points along the reference line ranging from (s

_{start}, 0) to (s

_{end}, 0). A normal line is drawn along each skeleton point, which is orthogonal to the reference line. Along each normal line, N

_{L}candidate grids are sampled (Figure 4), which range in an interval around the skeleton point. For the nominal line passing through the last skeleton point located at (s

_{end}, 0), N

_{L}is set to 1 and the only candidate grid left is specified as the goal (s

_{end}, l

_{end}).

_{collision}, travel efficiency cost J

_{efficiency}, smoothness cost J

_{smoothness}, and positioning-related cost J

_{positioning}. The collision cost penalizes the case that the ego vehicle collides with the surrounding cargoes when driving along the concerned path segment. J

_{collision}is set to a large value (e.g., ${10}^{20}$) if a collision occurs, otherwise, J

_{collision}is set to 0. The efficiency cost J

_{efficiency}is written as the length of the concerned path segment because this term encourages the ego vehicle to travel across a short distance. Suppose the parent of Node1 is Node0, J

_{smoothness}is defined as $\left|(\mathrm{Node}0.\mathrm{config}-\mathrm{Node}1.\mathrm{config})\times (\mathrm{Node}1.\mathrm{config}-\mathrm{Node}2.\mathrm{config})\right|.$ Intuitively speaking, the smoothness cost J

_{smoothness}penalizes the case that the heading direction changes from Node0, Node1, to Node2. J

_{positioning}penalizes the case that the ego vehicle stays in the positioning-poor regions for a long distance. Furthermore, N

_{sample}waypoints $\left\{({\mathrm{s}}_{i}^{\mathrm{wp}},{\mathrm{l}}_{i}^{\mathrm{wp}})|i=1,\dots ,{N}_{\mathrm{sample}}\right\}$ are evenly sampled along the concerned line segment from Node1 to Node2. Regarding the ith sampled waypoint $({\mathrm{s}}_{i}^{\mathrm{wp}},{\mathrm{l}}_{i}^{\mathrm{wp}})$, the corresponding coordinate value in the Cartesian frame is identified as $({\mathrm{x}}_{i}^{\mathrm{wp}},{\mathrm{y}}_{i}^{\mathrm{wp}})$ via frame conversion. Suppose the infrared emitter is installed at the height of h onto the ego vehicle, one may draw a line from the 3D point $({\mathrm{x}}_{i}^{\mathrm{wp}},{\mathrm{y}}_{i}^{\mathrm{wp}},\mathrm{h})$ to each infrared receiver and then check if the line is occluded by cargoes in the warehouse. If there is no occlusion, then the receiver is regarded as valid (Figure 5). If the total number of valid infrared receivers is larger than three, then the concerned waypoint $({\mathrm{s}}_{i}^{\mathrm{wp}},{\mathrm{l}}_{i}^{\mathrm{wp}})$ is regarded as valid. J

_{positioning}measures the rate of valid waypoints along the concerned line segment from Node1 to Node2.

Algorithm 1. Path planning via DP search. |

Input: Reference line, scenario layout, location of cargoes;Output: A path $\Lambda =\left\{({\mathrm{s}}_{i}^{\mathrm{dp}},{\mathrm{l}}_{i}^{\mathrm{dp}})|i=0,\dots ,{\mathrm{N}}_{\mathrm{S}}\right\}$;1. InitializeNodes(); 2. $\mathrm{Set}\mathrm{Node}(0,1).\mathrm{config}=({x}_{\mathrm{start}},{y}_{\mathrm{start}})$; 3. For each $j\in \left\{1,\dots ,{\mathrm{N}}_{\mathrm{L}}\right\}$, do4. $\mathrm{Set}\mathrm{Node}(1,j).\mathrm{parent}=\mathrm{Node}(0,1)$; 5. $\mathrm{Identify}\mathrm{Node}(1,j).\mathrm{config}$; 6. $\mathrm{Set}\mathrm{Node}(1,j).\mathrm{cost}=MeasureCost\left(\mathrm{Node}(0,1),\hspace{0.17em}\mathrm{Node}(1,j)\right)$; 7. End for8. For each $i\in \left\{1,\dots ,{\mathrm{N}}_{\mathrm{S}}-2\right\}$, do 9. For each $j\in \left\{1,\dots ,{\mathrm{N}}_{\mathrm{L}}\right\}$, do 10. For each $k\in \left\{1,\dots ,{\mathrm{N}}_{\mathrm{L}}\right\}$, do11. $\mathrm{Identify}\mathrm{Node}(i+1,k).\mathrm{config}$; 12. $\mathrm{cost}\_\mathrm{candidate}=MeasureCost\left(\mathrm{Node}(i,j),\hspace{0.17em}\mathrm{Node}(i+1,k)\right)$; 13. If $\mathrm{Node}(i+1,k).\mathrm{cost}>\mathrm{Node}(i,j).\mathrm{cost}+\mathrm{cos}\mathrm{t}\_\mathrm{candidate}$, then14. $\mathrm{Node}(i+1,k).\mathrm{parent}=\mathrm{Node}(i,j)$; 15. $\mathrm{Node}(i+1,k).\mathrm{cos}\mathrm{t}=\mathrm{cos}\mathrm{t}\_\mathrm{candidate}$; 16. End if17. End for18. End for19. End for20. $\mathrm{opti}\_\mathrm{id}=\underset{j=1,\dots ,{\mathrm{N}}_{\mathrm{L}}}{\mathrm{arg}\mathrm{min}}\left(\mathrm{Node}({\mathrm{N}}_{\mathrm{S}}-1,j).\mathrm{cost}+MeasureCost\left(\mathrm{Node}({\mathrm{N}}_{\mathrm{S}}-1,j),\hspace{0.17em}\mathrm{Node}({\mathrm{N}}_{\mathrm{S}},1)\right)\right)$; 21. $\Lambda =TraceBack\left(\mathrm{opti}\_\mathrm{id}\right)$; 22. Return. |

#### 3.4. Layer Four: Optimize a Curvature-Continuous Path

_{S}+ 1) waypoints. Since N

_{S}is not large, the derived path is quite coarse. This section is focused on how to refine the coarse path via numerical optimization within the Frenet frame. In this work, path refinement is performed via Baidu Apollo EM planner [20], which involves implementing path-velocity decomposition in an iterative loop before an optimum (rather than sub-optimum) is finally derived. Since there are only static obstacles, the EM planner is degraded as a run-once path planning method, the details of which are given as follows.

_{1}, w

_{2}, w

_{3}, and w

_{4}are weighting parameters, and ${l}_{\mathrm{DP}}(s)$ denotes the coarse path derived by DP in layer three. An OCP is formed by combining (6), (7), (10), and (11). The discretized version of this OCP is a QP, which is easily solved using a QP solver, such as osqp [51] and qpOASES [52]. The resultant path, after being converted back to the Cartesian frame, may be infeasible if its curvature exceeds the allowable bounds. The infeasibility is inevitable because the vehicle kinematics cannot be accurately modeled within the Frenet frame [44]. As a remedy for this, we check the resultant path for violations of curvature limits in the Cartesian frame; if an infeasible solution is derived, w

_{1}is set smaller before the QP problem is solved again. This process continues until a curvature-feasible path is derived.

## 4. Simulation Results and Discussion

#### 4.1. Simulation Setup

#### 4.2. On the Efficacy of the Proposed Planner

#### 4.3. On the Occlusion Awareness of the Proposed Planner

_{positioning}in layer three. When the cost term J

_{positioning}is discarded, the rate of good positioning distance along the entire path is 86.5% and 92.5% in the aforementioned two typical simulation cases, respectively. By contrast, with the cost term included, the rate grows to 97.0% and 96.5%. This comparative result clearly shows that our proposed planner can efficiently reduce the positioning inaccuracy caused by occlusions.

#### 4.4. On the Closed-Loop Tractability of the Proposed Planner

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**A warehouse workspace with poor positioning regions. Note that there are no kinematically feasible paths if the ego vehicle follows the global route because the goal lies in the poor positioning region.

**Figure 5.**Schematics for the infrared receiver validation check. Five blue infrared beams together with corresponding receivers are valid because they are not occluded by the cargoes.

**Figure 7.**Schematics for the formulation of collision-avoidance constraints in EM planner (zoom in to see more clearly).

**Figure 8.**Path planning results of typical simulation cases in side view and bird-eye view: (

**a**) Case 1; (

**b**) Case 2 (zoom in to see more clearly).

**Figure 9.**Path planning performance, w.r.t. collision avoidance, and kinematic feasibility: (

**a**) Case 1; (

**b**) Case 2.

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**MDPI and ACS Style**

Li, B.; Tang, S.; Zhang, Y.; Zhong, X.
Occlusion-Aware Path Planning to Promote Infrared Positioning Accuracy for Autonomous Driving in a Warehouse. *Electronics* **2021**, *10*, 3093.
https://doi.org/10.3390/electronics10243093

**AMA Style**

Li B, Tang S, Zhang Y, Zhong X.
Occlusion-Aware Path Planning to Promote Infrared Positioning Accuracy for Autonomous Driving in a Warehouse. *Electronics*. 2021; 10(24):3093.
https://doi.org/10.3390/electronics10243093

**Chicago/Turabian Style**

Li, Bai, Shiqi Tang, Youmin Zhang, and Xiang Zhong.
2021. "Occlusion-Aware Path Planning to Promote Infrared Positioning Accuracy for Autonomous Driving in a Warehouse" *Electronics* 10, no. 24: 3093.
https://doi.org/10.3390/electronics10243093