Autonomous Warehouse Robot Navigation

Problem Description

Imagine an autonomous robot tasked with navigating a warehouse to pick and deliver items. The robot operates in a grid-like layout with the following characteristics:

• The warehouse has designated pick-up locations, drop-off zones, and obstacles (e.g., shelves, walls).
• The robot can move in four directions: up, down, left, or right.
• The robot has a battery that depletes with movement and must be recharged periodically.
• The goal is to minimize the total time or energy cost of completing a series of tasks (e.g., delivering items from pick-up to drop-off locations).

MDP Formulation

1. State Space (S)

A state s is represented as a tuple (x, y, b), where:

• x, y: The robot's grid coordinates.
• b: The robot's battery level (discrete levels from 0 to full).

Example state: (3, 4, 2) means the robot is at grid cell (3, 4) with a battery level of 2.

2. Action Space (A)

• Actions include:
• Moving: {Up, Down, Left, Right}.
• Recharging: {Recharge} (only at a charging station).
• Example action: "Up" moves the robot up one grid cell.

3. Transition Probabilities (P)

• Moving to a neighboring cell has a high probability of success (e.g., 0.9) but might fail due to unforeseen obstacles (e.g., 0.1, resulting in staying in the same cell).
• If the battery level is 0, the robot cannot move until recharged.

4. Rewards (R)

• Picking up or delivering an item gives a positive reward (e.g., +10).
• Moving costs a small negative reward (e.g., -1 per move, representing time or energy cost).
• Running out of battery incurs a large penalty (e.g., -50).
• Recharging has no reward but enables further actions.

5. Policy (π)

A policy maps each state to an action, specifying what the robot should do at each location and battery level.

6. Objective

Find an optimal policy π* that minimizes the total cost (time or energy) while ensuring all tasks are completed.

Example Dynamics

1. States

Initial state: (1, 1, 5) (robot starts at (1, 1) with a battery level of 5).

Goal states: Completing a delivery task (e.g., reaching a drop-off zone).

2. Actions and Transitions

Action: "Right" from (1, 1, 5).

Transition:

• With 90% probability: Moves to (1, 2, 4) (battery decreases by 1).
• With 10% probability: Stays at (1, 1, 4) (battery still decreases).

3. Rewards

• Reward for moving: -1.
• Reward for successful delivery: +10.
• Penalty for running out of battery: -50.

MDP Solution

1. Value Iteration or Policy Iteration

Use value iteration to compute the optimal value function V(s) and policy π(s).

At each state, decide whether to move toward the goal, recharge, or take another action.

2. Resulting Optimal Policy

• For high battery levels, move directly to the task location.
• For low battery levels, move to the nearest charging station.
• Avoid unnecessary detours to minimize the cost.

Applications

• Logistics Optimization: Managing fleets of warehouse robots to minimize energy and time. Dynamic task allocation for multiple robots.
• Path Planning: Ensuring collision-free, energy-efficient paths in complex environments.
• Reinforcement Learning: Training robots to autonomously discover optimal navigation policies through trial and error.

Literature Survey paper.