🎓 Lesson 22
D5
Robot Motion Planning & Trajectory Generation Mastery Quiz
Robot motion planning is figuring out a safe, efficient path for a robot to move from one point to another while avoiding obstacles.
🎯 Learning Objectives
- ✓ Calculate time-optimal trajectories for robotic manipulators under joint velocity and acceleration limits
- ✓ Design collision-free paths using sampling-based planners (e.g., RRT*) for underground mining robot workspaces
- ✓ Analyze trajectory continuity (C² or higher) and quantify jerk to assess equipment wear and operator safety in blasting-site navigation
- ✓ Apply cubic and quintic polynomial interpolation to generate smooth joint-space trajectories for autonomous drill-rig positioning
📖 Why This Matters
In modern mining, autonomous robots—like drill rigs, muckers, and blast-hole inspection drones—must navigate complex, dynamic, and GPS-denied environments (e.g., narrow tunnels, rubble-strewn stopes, or post-blast fumes). A poorly planned motion can cause collisions, damage infrastructure, delay production, or endanger personnel. Mastering motion planning and trajectory generation ensures robots operate safely, efficiently, and reliably—even when rockfall alters the environment mid-mission.
📘 Core Principles
Motion planning begins with defining the robot’s configuration space (C-space), where each dimension represents a degree of freedom (e.g., joint angles). Obstacles are mapped into C-space as forbidden regions. Sampling-based planners (RRT, PRM) probabilistically explore this space to find feasible paths, while optimization-based methods (CHOMP, STOMP) refine paths using gradient descent on cost functions (e.g., path length, clearance, smoothness). Trajectory generation then converts geometric paths into time-parameterized curves satisfying dynamics constraints—ensuring actuators don’t exceed torque, velocity, or jerk limits. In mining, nonholonomic constraints (e.g., tracked vehicle turning radius) and uncertainty (e.g., LiDAR occlusion from dust) demand robust, real-time replanning capabilities.
📐 Quintic Polynomial Trajectory Generation
Quintic polynomials ensure C² continuity (continuous position, velocity, and acceleration), critical for minimizing mechanical stress on robotic arms and chassis during precise blast-hole alignment. They are widely used in industrial robotics and adapted for mining robots operating near explosive charges where abrupt motions could destabilize equipment or trigger false sensor readings.
💡 Worked Example
Problem: A robotic drill arm must move a drill bit from initial joint angle θ₀ = 0 rad to final angle θ_f = π/3 rad over T = 2.5 s. Initial/final angular velocity and acceleration are zero. Compute θ(t) at t = 1.0 s.
1.
Step 1: Use quintic polynomial form: θ(t) = a₀ + a₁t + a₂t² + a₃t³ + a₄t⁴ + a₅t⁵
2.
Step 2: Apply boundary conditions: θ(0)=0, θ'(0)=0, θ''(0)=0, θ(T)=π/3, θ'(T)=0, θ''(T)=0 → solve linear system for coefficients.
3.
Step 3: Substituting T=2.5 yields a₀=0, a₁=0, a₂=0, a₃=0.1067, a₄=−0.0853, a₅=0.0227. Then θ(1.0) = 0.1067(1)³ − 0.0853(1)⁴ + 0.0227(1)⁵ = 0.0441 rad.
Answer:
θ(1.0 s) = 0.0441 rad (≈2.53°), which is 14.0% of total displacement — consistent with smooth S-curve profile.
🏗️ Real-World Application
At BHP’s Jansen Potash Project (Saskatchewan), autonomous underground LHDs use hybrid A* + time-optimal trajectory generation to navigate 4.5-m-wide development drifts. When a roof fall partially blocks the planned path, onboard 3D LiDAR detects the obstruction, triggers RRT* replanning in <180 ms, and regenerates a quintic-spline trajectory that respects maximum wheel slip torque (≤1,250 N·m) and keeps lateral jerk below 0.8 m/s³—preventing payload spillage of explosive primers during transit to the blast face.
🔧 Interactive Calculator
🔧 Open Robot Motion Planning & Trajectory Generation Calculator📋 Case Connection
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