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The main task is to implement a function that will fuse the laser scan data into the occupancy grid map. You will learn the Bayessian update of the grid map and the practical limitations of its usage. Further you will learn how to create a dataset with sensoric data to speed-up your workflow.

Due date | October 29, 2023, 23:59 PST |

Deadline | January 13, 2024, 23:59 PST |

Points | 3 |

Label in BRUTE | t1d-map |

Files to submit | archive with `HexapodExplorer.py` file |

Resources | B4M36UIR_t1_resource_pack |

In class `HexapodExplorer.py`

implement the `fuse_laser_scan`

function. The purpose of the function is to fuse new data into the occupancy grid map as the robot traverses the environment. The laser scan measurements shall be fused to the grid using the Bayesian update described in Lab04 - Grid-based Path Planning.
The input parameters of the function are:

`grid_map`

- OccupancyGrid message - grid map representation`laser_scan`

- LaserScan message - contains the current laser scan data perceived by the robot.`odometry`

- Odometry message -`pose.position.x`

,`pose.position.y`

and`pose.orientation.quaternion`

encodes the current robot absolute position in the environment

The function returns:

- the OccupancyGrid message with the grid map fused with the new laser scan data, when the laser scan or the odometry data is invalid, the function just mirrors the grid map on the output

The `fuse_laser_scan`

function in the `HexapodExplorer.py`

class has a following prescription:

def fuse_laser_scan(self, grid_map, laser_scan, odometry): """ Method to fuse the laser scan data sampled by the robot with a given odometry into the probabilistic occupancy grid map Args: grid_map: OccupancyGrid - gridmap to fuse the laser scan to laser_scan: LaserScan - laser scan perceived by the robot odometry: Odometry - perceived odometry of the robot Returns: grid_map_update: OccupancyGrid - gridmap updated with the laser scan data """

In class

`HexapodExplorer.py`

you can change whatever you want. In evaluation, the given interfaces are fixed and the evaluation script is fixed.
The recommended approach for the occupancy map building using Bayesian approach is described in Lab03 - Grid-based Path Planning.

The `OccupancyGrid`

represents a probabilistic representation of the word. In particular the variable “OccupancyGrid.data” holds the probabilities of individual states to be occupied. The map in the `OccupancyGrid`

message is further parameterized by the metadata.

`resolution`

- size of the single cell in the real (simulated) world`width`

- width of the map`height`

- height of the map`origin`

- the real-world pose of the cell (0,0) in the map.`data`

- the actual probabilistic map data (in row-major order)

Hence the correct approach to the fusion of the laser scan data into the occupancy grid map is as follows

**Project the laser scan points to x,y plane with respect to the robot heading**

$\qquad\bullet\,$ Detailed description and guidance for the step (click to view)

* The laser scan data are represented as an array of distance measurements where each distance measurement $d_i$ is adjacent to a given scan angle $\theta_i$ with respect to the robot heading. Hence, the projection $\mathbf{p}_i$ of each scanned point is given as: $$\mathbf{p}_i = (x_i,y_i)^T = (d_i \cos(\theta_i), d_i \sin(\theta_i))^T$$

The laser scan is in relative coordinates w.r.t. the hexapod base where the x axis corresponds to the heading of the robot and y axis is perpendicular to the robot heading.

You can use some plotting function, e.g.,

`plt.scatter(x,y)`

from `matplotlib`

library to help you debug the mathematical operations and visualize the projected point cloud similar to the sample codes in `LaserScan`

class.
**Compensate for the robot odometry**- Compensate for the heading of the robot by rotating the scanned points based on the current odometry
- Compensate for the position of the robot by offsetting the scanned points to the robot's coordinates

$\qquad\bullet\,$ Detailed description and guidance for the step (click to view)

- For each projected scan point $\mathbf{p}_i$, the odometry compensated point $\mathbf{o}_i$ is calculated according to the equation: $$\mathbf{o}_i = \mathbf{R}\cdot \mathbf{p}_i + \mathbf{T},$$ where $\mathbf{R}$ is the odometry orientation component represented by the rotation matrix (i.e. the orientation of the robot) and $\mathbf{T}$ is the odometry translational component (i.e. the position of the robot)

**Transfer the points from the world coordinates to the map coordinates**(compensate for the map offset and resolution)

$\qquad\bullet\,$ Detailed description and guidance for the step (click to view)

- The map is in map coordinates that are given by the map origin $\mathbf{x}_\mathrm{origin}$ (encoded in Pose message
`grid_map.origin`

) and the map resolution $\Delta_\mathrm{res}$ (`grid_map.resolution`

), i.e., each point in the map represents an area of $\Delta_\mathrm{res}\times\Delta_\mathrm{res}~\mathrm{m}$. Therefore we need to apply a similar approach to the previous step and also divide the real-world coordinates by the resolution $\Delta_\mathrm{res}$ and round it to the closest integer to obtain the grid map indices $\mathbf{m}_i$ as:$$\mathbf{m}_i = round((\mathbf{o}_i - \mathbf{x}_\mathrm{origin})/\Delta_\mathrm{res})$$ - You should also watch for any points that fall outside of the map and act accordingly. Either you need to extend the map and perhaps move its origin (harder version of this task), or simply trim the outlying points (sufficient for passing this task)

**Raytrace individual scanned points**to give you coordinates of the cells which occupancy probability should be updated

$\qquad\bullet\,$ Detailed description and guidance for the step (click to view)

- You need to obtain the list of all the grid map points to be updated (both
`free_points`

and`occupied_points`

). This should be done using the raytracing. The simplest approach for raytracing on 2D grid is the Bresenham's line algorithm:def bresenham_line(self, start, goal): """Bresenham's line algorithm Args: start: (float64, float64) - start coordinate goal: (float64, float64) - goal coordinate Returns: interlying points between the start and goal coordinate """ (x0, y0) = start (x1, y1) = goal line = [] dx = abs(x1 - x0) dy = abs(y1 - y0) x, y = x0, y0 sx = -1 if x0 > x1 else 1 sy = -1 if y0 > y1 else 1 if dx > dy: err = dx / 2.0 while x != x1: line.append((x,y)) err -= dy if err < 0: y += sy err += dx x += sx else: err = dy / 2.0 while y != y1: line.append((x,y)) err -= dx if err < 0: x += sx err += dy y += sy x = goal[0] y = goal[1] return line

- The algorithm will give you a list of all points between the
`start`

coordinate and`goal`

coordinate. Where the`start`

coordinate is for all the scan points the odometry of the robot, and`goal`

coordinate is each scan projected scan point $\mathbf{m}_i$. It is enough to store the appropriate points in the lists, e.g.:#get the position of the robot in the map coordinates odom_map = world_to_map(odometry.position) #get the laser scan points in the map coordinates laser_scan_points_map = world_to_map(laser_scan_points_world) free_points = [] occupied_points = [] for pt in laser_scan_points_map: #raytrace the points pts = bresenham_line(odom_map, pt) #save the coordinate of free space cells free_points.extend(pts) #save the coordinate of occupied cell occupied_points.append(pt)

**Update the occupancy grid using the Bayesian update and the simplified laser scan sensor model**with $\epsilon = 0$, i.e., update all the points $d$ lying between the position of the robot (given by its`odometry`

) and each of the reflection points: $(d \in [0,r))$ and update their probability (being free) and only the reflection point $(d = r)$ being occupied

$\qquad\bullet\,$ Detailed description and guidance for the step (click to view)

- In this step the Bayesian update of the map is performed according to the description in Lab03 - Grid and Graph based Path Planning.
- The Bayesian occupancy grid update is defined as:

$$ P(m_i = occupied \vert z) = \dfrac{p(z \vert m_i = occupied)P(m_i = occupied)}{p(z \vert m_i = occupied)P(m_i = occupied) + p(z \vert m_i = free)P(m_i = free)}, $$ where $P(m_i = occupied \vert z)$ is the probability of cell $m_i$ being occupied after the fusion of the sensory data; $P(m_i = occupied)$ is the previous probability of the cell being occupied and $p(z \vert m_i = occupied)$ is the model of the sensor which is usually modeled as: $$ p(z \vert m_i = occupied) = \dfrac{1 + S^z_{occupied} - S^z_{free}}{2}, $$ $$ p(z \vert m_i = free) = \dfrac{1 - S^z_{occupied} + S^z_{free}}{2} = 1 - p(z \vert m_i = occupied), $$ where $S^z_{occupied}$ and $S^z_{free}$ are the sensory models for the occupied and free space respectively.

For the simulated LIDAR sensor, we will use the below stated model $$ S^z_{occupied} = \begin{cases} 0.95 \qquad\text{for}\qquad d = r\\ 0 \qquad\text{otherwise} \end{cases}, \qquad\qquad S^z_{free} = \begin{cases} 0.95 \qquad\text{for}\qquad d \in [0,r)\\ 0 \qquad\text{otherwise} \end{cases}, $$ where $r$ is the distance of the measured point and $\epsilon$ is the precision of the sensor. I.e. it is assumed that the space between the sensor and the measured point is free and the close vicinity of the measured point given by the sensor precision is occupied. * the recommended approach is to define two functions to help you with the Bayesian update

def update_free(P_mi): """method to calculate the Bayesian update of the free cell with the current occupancy probability value P_mi Args: P_mi: float64 - current probability of the cell being occupied Returns: p_mi: float64 - updated probability of the cell being occupied """ p_mi = #TODO #never let p_mi get to 0 return p_mi def update_occupied(P_mi) """method to calculate the Bayesian update of the occupied cell with the current occupancy probability value P_mi Args: P_mi: float64 - current probability of the cell being occupied Returns: p_mi: float64 - updated probability of the cell being occupied """ p_mi = #TODO #never let p_mi get to 1 return p_mi* There are two things to watch for during this step.

- First, it is necessary to correctly set the cell values given the row-major order of
`OccupancyGrid`

representation. HINT code:#construct the 2D ggrid data = grid_map.data.reshape(grid_map_update.height, grid_map_update.width) #fill in the cell probability values #e.g.: data[y,x] = update_free(data[y,x]) #serialize the data back (!watch for the correct width and height settings if you are doing the harder assignment) grid_map.data = data.flatten()

- Second, the probability values shall never reach 0 or 1!!! The effect of letting this happen is inability of the map to dynamically add/delete obstacles as it is demonstrated in the following example:
- Correct behavior

- Wrong behavior

Note, when using the sensory model as it is described in Lab03 - Grid-based Path Planning, once the grid cell probability is set to 0, it stays 0, hence, all the nice features of the probabilistic mapping vanishes. Therefore it is recommended to use a different sensory model with:
$$
S^z_{occupied} = \begin{cases} 0.95 \qquad\text{for}\qquad d = r\\ 0 \qquad\text{otherwise} \end{cases},
$$
$$
S^z_{free} = \begin{cases} 0.95 \qquad\text{for}\qquad d \in [0,r)\\ 0 \qquad\text{otherwise} \end{cases},
$$

Note that in the evaluation script of the

`t1d-map`

task the grid map is predefined with the size $100\times100$ points and correct map origin of $(-5,-5, 0)^T$ and orientation $(0,0,0,1)$ which is prepared exactly for seamless finishing of the `t1f-map`

assignment; however, for the increased difficulty that will pay of during the work on the Project assignment you are encouraged to generalize your solution to allow for dynamicaly growing map.
The evaluation focus on the ability of the robot to do the correct probabilistic update. You can choose between two variants: simple one, that initializes the `gridmap`

to a sufficient size at the beginning of the evaluation, and more advanced version which initializes the empty `gridmap`

and it is your responsibility to correctly handle the size changes of the map. The hard variant has no effect on scoring of the task T1d-map - Map building; however, it will be beneficial in the forthcoming Semestral project on mobile robot exploration.

The code can be evaluated using the following script (also attached as `t1d-map.py`

).

#!/usr/bin/env python3 # -*- coding: utf-8 -*- import matplotlib.pyplot as plt import sys import os import math import numpy as np sys.path.append('hexapod_robot') sys.path.append('hexapod_explorer') #import hexapod robot and explorer import HexapodRobot as hexapod import HexapodExplorer as explorer #import communication messages from messages import * #switch to select between the simple variant or the hard variant SIMPLE_VARIANT = True if __name__=="__main__": robot = hexapod.HexapodRobot(0) explor = explorer.HexapodExplorer() #turn on the robot robot.turn_on() #start navigation thread robot.start_navigation() #assign goal for navigation goals = [ Pose(Vector3(3.5,3.5,0),Quaternion(1,0,0,0)), Pose(Vector3(0.0,0.0,0),Quaternion(1,0,0,0)), ] #prepare the online plot plt.ion() fig, ax = plt.subplots() #prepare the gridmap gridmap = OccupancyGrid() gridmap.resolution = 0.1 if SIMPLE_VARIANT: gridmap.width = 100 gridmap.height = 100 gridmap.origin = Pose(Vector3(-5.0,-5.0,0.0), Quaternion(1,0,0,0)) gridmap.data = 0.5*np.ones((gridmap.height*gridmap.width)) #go from goal to goal for goal in goals: robot.goto_reactive(goal) while robot.navigation_goal is not None: plt.cla() #get the current laser scan and odometry and fuse them to the map gridmap = explor.fuse_laser_scan(gridmap, robot.laser_scan_, robot.odometry_) #plot the map gridmap.plot(ax) plt.xlabel('x[m]') plt.ylabel('y[m]') #plt.axis('square') plt.show() plt.pause(0.1) robot.stop_navigation() robot.turn_off()The expected output is the visually correct map of the environment even in presence of dynamic events.

courses/uir/hw/t1d-map.txt · Last modified: 2023/10/19 11:23 by pragrmi1