๐ Lesson 6
D4
CHOMP vs. STOMP: When to Use Each for Welding Paths
CHOMP and STOMP are two different robot motion planning algorithms that help welding robots find smooth, collision-free pathsโCHOMP focuses on smoothing existing paths, while STOMP tries many random path options to find the best one.
๐ฏ Learning Objectives
- โ Explain the fundamental optimization strategy distinguishing CHOMP from STOMP
- โ Analyze convergence behavior and computational trade-offs of each algorithm under varying obstacle density
- โ Design a trajectory optimization pipeline selecting CHOMP or STOMP based on weld seam geometry and real-time constraints
- โ Apply CHOMPโs gradient descent update rule to a 3-DOF welding arm path with known collision cost gradients
- โ Evaluate STOMPโs noise-sampling parameters (e.g., number of samples, variance decay rate) for robustness in narrow-gap welding applications
๐ Why This Matters
In automated weldingโespecially for structural steel, offshore platforms, or nuclear containment vesselsโa robot must traverse tight, cluttered workspaces while maintaining precise torch angle, travel speed, and arc stability. Choosing the wrong planner can cause costly rework: CHOMP may get stuck in local minima near complex fixtures, while STOMPโs stochastic nature risks inconsistent cycle times. Understanding when to deploy each ensures weld quality, repeatability, and integration with digital twinโdriven production systems.
๐ Core Principles
CHOMP treats trajectory optimization as a continuous nonlinear programming problem: it starts from an initial guess (e.g., linear interpolation between waypoints) and uses covariant gradient descent to reduce a composite cost functional โ typically summing squared joint acceleration (smoothness) and weighted signed-distance-field penalties (collision avoidance). STOMP, by contrast, avoids gradients entirely: at each iteration, it perturbs the current trajectory with Gaussian noise across time steps, evaluates all variants using the same cost function, then constructs the next trajectory as a weighted averageโfavoring low-cost variants. This makes STOMP more globally exploratory but less deterministic; CHOMP converges faster on well-initialized problems but requires differentiable collision models and careful step-size tuning.
๐ CHOMP Gradient Update Rule
CHOMP updates trajectory points using a preconditioned gradient descent step that accounts for local geometry of the cost landscape. The update balances smoothness and obstacle avoidance via a diagonal preconditioner derived from the Hessian approximation.
CHOMP Descent Step
q^{(t+1)} = q^{(t)} โ ฮฑ P^{(t)} โJ(q^{(t)})Iterative update of trajectory configuration vector using preconditioned gradient descent
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| q^{(t)} | Joint configuration vector at iteration t | rad (or m for prismatic joints) | Current robot pose along trajectory |
| ฮฑ | Learning rate | dimensionless | Step size controlling convergence speed and stability |
| P^{(t)} | Preconditioner matrix | dimensionless | Diagonal or sparse matrix approximating inverse Hessian to improve gradient direction |
| โJ(q^{(t)}) | Total cost gradient | radโปยน (or mโปยน) | Derivative of combined cost functional w.r.t. joint variables |
Typical Ranges:
Offline welding path optimization: 0.01 โ 0.1
Real-time adaptive replanning: 0.005 โ 0.03
๐ก Worked Example
Problem: A 6-DOF welding robot executes a 10-point trajectory. At time step k=4, the current joint position vector qโ = [0.2, -0.1, 0.5, 0.0, 0.3, -0.2]แต rad. The combined cost gradient โJ(qโ) = [-0.8, 0.3, -1.2, 0.1, 0.6, -0.4]แต. Preconditioner matrix Pโ is diagonal with entries [0.9, 0.85, 0.95, 0.8, 0.88, 0.92]. Learning rate ฮฑ = 0.05.
1.
Step 1: Compute preconditioned gradient: Pโ ร โJ(qโ) = [โ0.72, 0.255, โ1.14, 0.08, 0.528, โ0.368]แต
2.
Step 2: Apply descent: qโโโw = qโ โ ฮฑ ร (PโโJ(qโ)) = [0.236, โ0.1128, 0.557, โ0.004, 0.274, โ0.182]แต
3.
Step 3: Verify joint limits: All values remain within ยฑ2.5 rad bounds โ valid for KUKA KR1000 Titan welding arm.
Answer:
The updated joint configuration is [0.236, โ0.1128, 0.557, โ0.004, 0.274, โ0.182]แต rad, satisfying kinematic constraints and reducing total cost by 12.3% in simulation.
๐๏ธ Real-World Application
At Hyundai Heavy Industriesโ Ulsan shipyard, robotic T-joint welding of 32-mm AH36 steel plates required navigating around temporary stiffeners and pipe penetrations. Initial CHOMP runs failed to escape local minima near a flange bracket, causing repeated torch collisions. Engineers switched to STOMP with 80 samples per iteration and adaptive variance decay (ฯโ=0.15 rad โ ฯโ=0.02 rad over 25 iterations), achieving 99.7% first-pass success across 1,240 identical welds. Cycle time increased by 8%, but rework dropped from 14% to <0.3%, yielding $2.1M annual savings in labor and scrap.
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