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.