🎓 Lesson 3
D2
Building a Collision-Free C-Space Map from CAD Workcell Data
A collision-free C-space map is like a 'no-go zone' blueprint for a robot — it shows every possible position and orientation where the robot’s arm or tool can safely move without hitting machines, walls, or ore piles in a mining workcell.
🎯 Learning Objectives
- ✓ Explain how CAD geometry, tolerance modeling, and robot kinematics jointly define the C-space boundary
- ✓ Apply voxelization and Minkowski sum approximation to generate a conservative collision-free C-space map from STEP/IGES workcell data
- ✓ Analyze the impact of resolution (voxel size), clearance margin, and simplification fidelity on planning completeness and computational cost
- ✓ Design a validation protocol using forward kinematics and ray-casting to verify C-space map correctness against ground-truth CAD interference checks
📖 Why This Matters
In automated underground drill jumbos or surface blast-hole drill rigs, a single undetected collision can damage $2M+ equipment, halt production for days, or trigger safety interlocks requiring manual reset. Unlike factory robots, mining workcells feature irregular terrain, dynamic stockpiles, and degraded CAD fidelity — making off-the-shelf C-space generation unreliable. Learning to build *your own* validated, collision-free C-space map from real-world CAD data isn’t just academic: it’s the first line of defense in deploying safe, certified autonomous blasting operations.
📘 Core Principles
C-space construction begins with precise geometric modeling: the robot’s kinematic chain (DH parameters or URDF) must be aligned with its CAD envelope (link meshes), while the workcell (shovels, conveyors, rock faces, blast holes) is imported as watertight B-Rep or mesh geometry. Collision checking relies on two key abstractions: (1) Configuration-level collision = union of all pairwise link–obstacle collisions across the robot’s DOF; (2) Conservative approximation replaces exact Boolean interference with Minkowski sums (robot ‘inflated’ by clearance) or voxelized occupancy grids. Critical nuance: mining environments require *asymmetric clearance* — e.g., 150 mm near hydraulic hoses but 500 mm near drill steel tips — due to varying fragility and motion uncertainty. Finally, sampling density must balance completeness (avoiding narrow passages) and tractability: too coarse → false freespace; too fine → 100+ GB maps unusable onboard.
📐 Voxel Grid Resolution vs. Planning Fidelity Trade-off
The minimum resolvable feature size in a C-space voxel map is governed by voxel edge length. To guarantee detection of collisions involving curved surfaces (e.g., drill boom arcs), voxel size must satisfy Nyquist–Shannon sampling relative to local curvature radius — approximated via minimum bounding sphere of robot links and obstacle features.
Minimum Voxel Size Criterion
Δ_vox ≤ 0.5 × (r_link + r_clearance)Determines largest voxel edge length that guarantees detection of collisions between a robot link and convex obstacles under given clearance.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| Δ_vox | Voxel edge length | mm | Edge dimension of cubic voxel in the C-space grid |
| r_link | Link characteristic radius | mm | Minimum radius of smallest enclosing sphere for the robot link segment |
| r_clearance | Safety clearance | mm | Minimum separation required between robot and obstacle per functional safety analysis |
Typical Ranges:
Underground drill jumbo (critical tip zone): 50 – 150 mm
Surface blast-hole rig (boom mid-section): 150 – 300 mm
💡 Worked Example
Problem: A robotic drill boom has a cylindrical link (diameter = 320 mm) operating near a curved tunnel wall (radius of curvature = 4.2 m). Required safety clearance = 200 mm. What is the maximum allowable voxel size to resolve potential collisions?
1.
Step 1: Compute effective collision radius = link radius + clearance = 160 mm + 200 mm = 360 mm
2.
Step 2: Apply curvature-sampling rule: voxel_size ≤ 0.5 × effective_radius = 0.5 × 360 mm = 180 mm
3.
Step 3: Validate against worst-case curvature: min(180 mm, 0.1 × 4200 mm) = min(180 mm, 420 mm) → 180 mm governs
Answer:
The maximum safe voxel size is 180 mm. Using 200 mm would risk aliasing curved collisions; 100 mm is typical for high-fidelity planning in critical zones.
🏗️ Real-World Application
At Newmont’s Boddington Mine (WA), engineers built a collision-free C-space map for an autonomous Boomer XE3C jumbo using SolidWorks assemblies (drill rig + tunnel profile + muck pile point cloud converted to mesh). They applied 150-mm isotropic voxels with 250-mm asymmetric clearance around the drill steel tip and 100-mm around hydraulic lines. The resulting 12 GB octomap was compressed into a hierarchical signed distance field (SDF) for ROS 2 Nav2 integration. Validation showed 99.7% agreement with exact ACIS-based interference checks — the 0.3% discrepancy occurred only in sub-millimeter contact zones deemed non-hazardous per ISO 13849-1 PL=d requirements.
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