🎓 Lesson 1
D1
Getting Started with Robot Motion Planning & Trajectory Generation
Robot motion planning is figuring out a safe, efficient path for a robot to move from one point to another without hitting anything.
🎯 Learning Objectives
- ✓ Explain the difference between path planning and trajectory generation using mining robotics examples
- ✓ Design a collision-free waypoint path for an autonomous drill rig navigating a bench face
- ✓ Apply cubic polynomial interpolation to generate a smooth joint-space trajectory with zero initial/final velocity
- ✓ Analyze time-optimal trajectories under velocity and acceleration limits for a robotic muck loader arm
📖 Why This Matters
In modern mining, autonomous drilling rigs, muck loaders, and survey robots operate in dynamic, unstructured environments—steep benches, loose scree, dust-obscured visibility, and shifting blast zones. A single planning failure can cause costly downtime, equipment damage, or safety incidents. Motion planning isn’t just about 'getting from A to B'—it’s about doing so reliably, predictably, and in compliance with ISO 13849-1 safety integrity requirements for mobile machinery. This foundational skill bridges robotics theory and real-world mine site resilience.
📘 Core Principles
Motion planning begins with representation: the robot’s configuration space (C-space) encodes all possible poses (e.g., joint angles or (x,y,θ) for a mobile base). Obstacles are transformed into C-obstacles—regions in C-space where collisions occur. Planning algorithms then search this space: sampling-based methods (like RRT*) efficiently handle high-dimensional, nonconvex spaces common in articulated arms; grid-based methods (A*) suit 2D/2.5D mobile platforms on digital terrain models (DTMs). Trajectory generation adds temporal dimension—converting geometric paths into time-parameterized curves respecting actuator limits (max torque, joint velocity ≤ 2.5 rad/s, acceleration ≤ 1.2 m/s²), often using polynomials, splines, or numerical optimization.
📐 Cubic Polynomial Trajectory Segment
Used to generate smooth, continuous position, velocity, and acceleration profiles between two waypoints with specified boundary conditions. Ensures zero jerk at endpoints for actuator-friendly motion—critical for precision drilling and payload stability.
💡 Worked Example
Problem: A robotic drill arm must move joint θ from θ₀ = 0 rad to θ_f = π/2 rad in T = 3 s, starting and ending at rest (velocity = 0). Calculate coefficients of θ(t) = a₀ + a₁t + a₂t² + a₃t³.
1.
Step 1: Apply boundary conditions: θ(0) = 0 → a₀ = 0; θ'(0) = 0 → a₁ = 0.
2.
Step 2: Apply final conditions: θ(3) = π/2 = a₂(3)² + a₃(3)³ = 9a₂ + 27a₃; θ'(3) = 0 = 2a₂(3) + 3a₃(3)² = 6a₂ + 27a₃.
3.
Step 3: Solve linear system: subtract equations → (9a₂ + 27a₃) − (6a₂ + 27a₃) = π/2 ⇒ 3a₂ = π/2 ⇒ a₂ = π/6 ≈ 0.5236; substitute → 6(π/6) + 27a₃ = 0 ⇒ π + 27a₃ = 0 ⇒ a₃ = −π/27 ≈ −0.1164.
Answer:
θ(t) = 0.5236 t² − 0.1164 t³. Max velocity occurs at t = 2 s: θ'(2) = 2(0.5236)(2) − 3(0.1164)(4) ≈ 2.094 − 1.397 = 0.697 rad/s — well below typical hydraulic arm limit of 1.8 rad/s.
🏗️ Real-World Application
At Newmont’s Boddington Mine (Western Australia), autonomous LHDs (Load-Haul-Dump vehicles) use hybrid A*-RRT planners integrated with LiDAR SLAM and 3D DTM overlays. When navigating a newly blasted muck pile, the planner first updates the occupancy grid using real-time point clouds, then computes a 5 m clearance path around unstable edges and overhead hanging walls—adhering to MineSafe™ operational envelope standards. Trajectories are regenerated every 250 ms to accommodate dynamic obstacles (e.g., falling debris detected via vibration-triggered re-planning).
✏️ Student Exercise
Given a robotic survey drone with maximum linear velocity v_max = 8 m/s and acceleration a_max = 2 m/s², calculate the minimum time required to traverse a straight 40 m path from rest to rest using time-optimal (bang-bang) control. Then, plot position, velocity, and acceleration vs. time.
🔧 Interactive Calculator
🔧 Open Robot Motion Planning & Trajectory Generation Calculator📋 Case Connection
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