8  Chapter 7: Capstone Project-II - P.A.T.H.F.I.N.D.E.R.

8.1 Picking and Assembly Task Handler For Industrial Navigation, Dynamics, and Energy Reduction

8.1.1 Introduction: Your Mission

Welcome to your second capstone challenge. This project shifts our focus from space exploration to the heart of modern manufacturing: industrial automation.

You are the lead robotics engineer for a “smart factory” cell. Your task is to program a stationary robotic arm to perform a pick-and-place operation. The arm must pick up parts from a designated pickup zone and place them into one of two assembly jigs, navigating around a fixed obstacle in its workspace. The factory’s goal is to minimize the energy consumption of the arm for a complete cycle, subject to a strict cycle time limit to maintain production throughput.

This project will test your ability to model a robot’s workspace, analyze its gripper dynamics, plan collision-free paths using PDEs, and optimize its motion for energy efficiency.

8.1.2 Project Structure & Core Modules

This project follows a five-module structure, applying the course concepts to this new industrial scenario.

8.1.3 Module 1: Workspace & Kinematics

(Concepts from: Lab 1 - Linear Algebra & Coordinate Geometry)

Before the arm can move, we must define its physical capabilities and its environment.

  • Task 1: The Workspace. Create a 2D top-down map of the robot’s workspace. Define the coordinates for the arm’s base, a “Pickup Zone,” two “Assembly Jigs” (A and B), and a fixed “Obstacle” (e.g., a pillar or another piece of machinery).

  • Task 2: Inverse Kinematics. Model a 2-link planar arm. Instead of forward kinematics, your primary task is to implement a simple Inverse Kinematics function. Given a target (x, y) coordinate in the workspace, this function must calculate the required joint angles \((\theta_1, \theta_2)\) for the arm to reach it.

  • Task 3: Reachability Check. Use your inverse kinematics function to create a can_reach(target_pos) helper function. This function will be crucial to confirm that the pickup and assembly zones are physically within the arm’s reach.

  • Deliverable: A Python script (kinematics_v2.py) with functions for inverse kinematics. Include a plot showing the arm’s configuration when reaching for each of the key locations (pickup and jigs).

8.1.4 Module 2: Gripper Dynamics & Actuation

(Concepts from: Lab 3 & 4 - Laplace Transforms & ODEs)

The arm’s gripper is a dynamic system. We must model its closing time to ensure it securely grasps parts.

  • Task 1: System Modeling. Model the gripper’s closing mechanism as a second-order system. The differential equation for the gripper’s finger position, \(x(t)\), in response to a “close” command is: \[ m \ddot{x}(t) + c \dot{x}(t) + k x(t) = F_{motor} \] where \(F_{motor}\) is a constant force applied by the actuator. Use parameters: Mass \(m=0.1\), Damping \(c=5\), Spring-like resistance \(k=50\), and Motor Force \(F_{motor}=50\).
  • Task 2: Laplace Analysis.
    • Assume the “close” command is a step input of force at \(t=0\).
    • Use the Laplace Transform method in sympy to solve this ODE for \(x(t)\), with initial conditions \(x(0)=0\) and \(\dot{x}(0)=0\).
  • Task 3: Constraint Discovery - Gripper Settling Time.
    • Plot the step response \(x(t)\).
    • Determine the settling time: the time it takes for the gripper to close and stabilize. This gripper_settling_time is the minimum time the robot must wait after issuing a “grasp” command before it can start moving the part.
  • Deliverable: A Python script (dynamics_v2.py) that solves the gripper ODE and plots its response. Clearly state the calculated gripper_settling_time.

8.1.5 Module 3: Obstacle-Aware Path Planning

(Concepts from: Lab 2 - Partial Differential Equations)

To move safely, the arm must navigate around the obstacle. We will implement a classic robotics algorithm called Potential Field Path Planning.

  • Task 1: The Potential Field. The workspace is discretized into a grid. The path is found by solving Laplace’s Equation over this grid: \[ \nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0 \]

    • Boundary Conditions: Set the potential \(\phi\) to a high value at the obstacle’s boundary (e.g., \(\phi=1\)) and a low value at the target’s location (e.g., \(\phi=0\)).

  • Task 2: Numerical PDE Solution. Implement an iterative solver (like the Jacobi method) to solve for the potential \(\phi(x,y)\) at every point on the grid. This creates a smooth “potential surface” where the target is a low point and obstacles are high points.

  • Task 3: Path Generation. Write a function that finds a path from a start point to the target by using gradient descent. From any point on the grid, the next point on the path is in the direction of the steepest descent (the negative gradient, \(-\nabla\phi\)).

  • Deliverable: A Python script (path_planner.py) that computes a potential field for a given target and generates a collision-free path for the arm’s end-effector. Include a heatmap visualization of the potential field with the generated path overlaid.

8.1.6 Module 4: Energy & Time Optimization

(Concepts from: Lab 5 - Optimization)

The factory wants to minimize electricity costs while meeting production quotas. This requires optimizing the arm’s movement speed.

  • Task 1: Define Motion Profiles. The arm can move along the path from Module 3 at two different speeds:
    • Slow & Precise: Low energy cost (20 J/s), speed = 0.5 m/s.
    • Fast & Inaccurate: High energy cost (70 J/s), speed = 1.5 m/s.
  • Task 2: LP Formulation. For a complete pick-and-place cycle (Pickup Zone -> Assembly Jig A), formulate a linear program to find the optimal motion profile.
    • Decision Variables: Let \(t_{slow}\) be the time spent moving at slow speed, and \(t_{fast}\) be the time spent moving at fast speed.
    • Objective Function: Minimize total energy: \(Z = 20 t_{slow} + 70 t_{fast}\).
    • Constraints:
      1. Total Distance Constraint: The distance covered must equal the path length: \(0.5 t_{slow} + 1.5 t_{fast} = \text{Total Path Length}\). This is an equality constraint.
      2. Total Time Constraint: The total time for the cycle must be less than or equal to a factory-imposed limit (e.g., 15 seconds): \[ t_{\text{slow}} + t_{\text{fast}} + \text{gripper settling time} \leq 15. \]
  • Deliverable: A Python script (optimizer_v2.py) using scipy.optimize.linprog that calculates the optimal time to spend at each speed to meet the deadline with minimum energy.

8.1.7 Module 5: Integrated Factory Cell Simulation

This is the final deliverable, combining all modules into a cohesive visualization.

  • Task: Create a live “Factory Dashboard” animation using matplotlib.animation.
    1. Workspace View (Main Plot): A top-down 2D plot showing the workspace, obstacle, pickup/jig locations, and the robotic arm. The arm should animate its movement along the path generated by the potential field planner.
    2. Potential Field View (Subplot 1): A heatmap of the potential field from Module 3. An overlaying marker should show the arm’s current position on the gradient.
    3. System Status (Subplot 2): Text readouts for Cycle Time, Energy Consumed, and Current Action (e.g., “Moving to Pickup”, “Grasping Part”, “Moving to Jig A”).
    4. Gripper Dynamics (Subplot 3): When the action is “Grasping Part,” this plot shows the live gripper closing response from Module 2.
  • Final Deliverable: A single, executable Python script main_v2.py that runs the entire factory arm simulation, demonstrating one full, optimized pick-and-place cycle.