Solution: LiDAR Sensors

Introduction

LiDAR (Light Detection and Ranging) is a sensing technology that measures distance by emitting laser light and measuring the time taken for reflected light to return.

Working Principle

Laser pulse → Object
              ↓
        Reflected pulse
              ↓
           LiDAR sensor
              ↓
        Time measurement
              ↓
        Distance calculation

Distance is approximately calculated as:

$$d=\frac{c\Delta t}{2}$$

where:

• $d$ = distance to the object
• $c$ = speed of light
• $\Delta t$ = round-trip travel time
• factor 2 accounts for outgoing and returning paths.

A scanning LiDAR rotates or electronically scans the beam to obtain measurements over many directions.

Functions

1. Distance measurement
2. Obstacle detection
3. Environment mapping
4. Localization support
5. Object detection and tracking

Usage in Autonomous Robots

LiDAR is commonly used with SLAM (Simultaneous Localization and Mapping). The robot compares LiDAR observations with its map to estimate its position while building or updating the environment map.

Applications

• Autonomous mobile robots
• Warehouse robots
• Self-driving systems
• Industrial AGVs/AMRs
• Obstacle avoidance
• 2D/3D mapping
• Building inspection
• Surveying

Advantages

• High distance-measurement accuracy
• Works in low-light conditions
• Provides dense spatial measurements
• Useful for mapping and localization

Limitations

• Higher cost than some simple sensors
• Performance can be affected by highly reflective, transparent or absorbing surfaces
• Moving parts may be present in mechanical scanning LiDAR
• Outdoor operation can be affected by environmental conditions

Conclusion

LiDAR is an important sensor for autonomous navigation because it provides accurate geometric information about the surrounding environment for mapping, localization and obstacle avoidance.

Solution: Path vs Trajectory and Path Planning

Path

A path is the geometric route followed by the robot from an initial position to a goal position. It specifies where the robot should go but does not necessarily specify the time required to reach each point.

Trajectory

A trajectory is a time-parameterized path. It specifies both position and how that position changes with time, including velocity and often acceleration.

Feature	Path	Trajectory
Describes	Geometric route	Route + time evolution
Time included	No	Yes
Velocity	Not necessarily defined	Defined
Acceleration	Not necessarily defined	Usually considered
Example	Shortest obstacle-free route	Same route traversed at specified speed

Path Planning

Path planning determines a collision-free route from a start state to a goal state while considering obstacles and robot constraints.

Example: A* Algorithm

A* is a graph-search algorithm that finds a low-cost route using:

$$f(n)=g(n)+h(n)$$

where:

• $g(n)$ = cost from start to node $n$
• $h(n)$ = estimated cost from $n$ to goal
• $f(n)$ = estimated total cost

Working

1. Represent the environment as a graph/grid.
2. Put the starting position in the open list.
3. Calculate $f(n)$ for candidate nodes.
4. Select the node with the lowest estimated cost.
5. Expand its neighbors while avoiding obstacles.
6. Repeat until the goal is reached.
7. Trace parent nodes backward to obtain the path.

Advantages of A*

• Complete for appropriate finite graphs
• Finds an optimal path when the heuristic is admissible
• Widely used in grid-based mobile robot navigation

Conclusion

A path specifies where the robot moves, while a trajectory specifies where and when it moves. A* is a common path-planning method for finding an efficient obstacle-free route.

Solution: Robot Singularity and Mapping

Robot Singularity

A singularity is a robot configuration in which the manipulator loses effective degrees of freedom or the Jacobian matrix becomes rank-deficient.

At a singular configuration, some desired Cartesian motions may require extremely large joint velocities or may become impossible.

Effects

• Loss of a direction of motion
• Very high joint velocities near the singularity
• Reduced controllability
• Possible path-tracking problems
• Increased mechanical/control stress

Example

For an articulated robot, a wrist singularity can occur when two wrist axes become aligned, making two rotational directions indistinguishable.

Mapping in Autonomous Navigation

Mapping is the process of constructing a representation of the robot's environment using sensor data such as LiDAR, depth cameras or sonar.

Sensors → Sensor Data → Mapping/SLAM → Environment Map
                         ↑
                      Robot Pose

Importance

1. Obstacle representation — identifies free and occupied regions.
2. Path planning — planners use the map to find safe routes.
3. Localization — the robot compares sensor observations with the map to estimate its position.
4. Navigation — enables movement toward goals while avoiding obstacles.
5. Exploration — allows robots to build maps of previously unknown environments.

Occupancy Grid

A common 2D representation divides the environment into cells classified approximately as:

• occupied
• free
• unknown

Conclusion

Singularity concerns the robot manipulator's kinematic limitations, whereas mapping provides the environmental information required for autonomous navigation, localization and path planning.