Optimization-Based Trajectory Generation: CHOMP & STOMP
CHOMP and STOMP are smart computer methods that help robots plan smooth, safe, and energy-efficient paths—like teaching a robotic arm how to move around obstacles without crashing or wasting power.
⚠️ Why It Matters
📘 Definition
CHOMP (Covariant Hamiltonian Optimization for Motion Planning) and STOMP (Stochastic Trajectory Optimization for Motion Planning) are gradient-based and sampling-based trajectory optimization algorithms, respectively, that generate dynamically feasible, collision-free robot trajectories by minimizing cost functionals over continuous-time state-action spaces while respecting kinematic, dynamic, and environmental constraints. Both operate on discretized trajectory representations but differ fundamentally in their optimization paradigms: CHOMP uses covariant gradient descent with implicit obstacle avoidance via penalty terms, whereas STOMP employs stochastic exploration of trajectory perturbations followed by weighted averaging to converge to low-cost solutions.
🎨 Concept Diagram
AI-generated illustration for visual understanding
💡 Engineering Insight
Never treat CHOMP/STOMP as 'black-box path planners'—they are *trajectory shapers* whose behavior is dominated by cost function design and initialization quality. In production systems, >70% of failed deployments trace back to mismatched λ_coll/λ_smooth ratios or using coarse SDFs that alias thin fixtures (e.g., pneumatic tubing or safety light curtains) as free space. Always validate the SDF against physical probe scans before optimization.
📖 Detailed Explanation
STOMP takes a fundamentally different approach: instead of descending a gradient, it generates dozens of noisy trajectory variants, evaluates each under the full cost function (including non-differentiable terms like binary collision checks), then constructs a new trajectory as a weighted average—giving higher weight to lower-cost variants. This makes STOMP naturally resilient to local minima and capable of handling discontinuous costs, but at higher computational cost per iteration.
Advanced implementations integrate system identification data: e.g., CHOMP’s Hessian can be preconditioned using measured motor torque–velocity curves, and STOMP’s perturbation covariance can be learned from historical failure modes (e.g., over-perturbing near narrow doorways). Real-world deployment also demands warm-start strategies—using previous day’s optimal trajectory as initialization—to guarantee sub-100 ms replanning for human-robot collaboration, meeting ISO/TS 15066 power-and-force limiting requirements.
🔄 Engineering Workflow
📋 Decision Guide
| Rock/Field Condition | Recommended Design Action |
|---|---|
| High-precision assembly (±0.1 mm repeatability required) | Use CHOMP with high λ_smooth (≥3.0), T = 4–6 s, resolution ≥120, and SDF grid resolution ≤2 mm |
| Cluttered unstructured workcell with moving humans (collab cell) | Prefer STOMP with adaptive λ_coll ramping, online replanning window ≤1.0 s, and safety-aware perturbation covariance |
| Heavy-payload palletizing (>50 kg) with servo-limited torque bandwidth | Constrain CHOMP’s Hessian approximation to first-order dynamics; limit max joint acceleration to ≤1.2 rad/s²; use time-optimal warm-start initialization |
📊 Key Properties & Parameters
Trajectory Resolution
50–200 waypoints for 3–10 s industrial motionsNumber of discrete time steps (waypoints) used to represent the continuous-time trajectory.
Too few waypoints cause discretization error and constraint violation; too many increase computation time and memory pressure on real-time controllers.
Collision Cost Weight (λ_coll)
1e2 – 1e5 (dimensionless, problem-dependent)Scalar multiplier applied to the sum of signed-distance-field (SDF) penalties for all robot links in proximity to obstacles.
Insufficient weighting leads to near-collision or penetration; excessive weighting distorts trajectory shape and degrades smoothness or feasibility.
Smoothness Cost Weight (λ_smooth)
1e−2 – 10 (N·s²/m or equivalent in normalized units)Scalar multiplier applied to the integral of squared joint acceleration (or jerk) over time, penalizing aggressive motion.
Low values yield jerky, vibration-prone motion risking payload slippage or vision misalignment; high values overly constrain dynamics, increasing cycle time.
Time Horizon (T)
1.5 – 8.0 s for pick-and-place; up to 30 s for multi-stage assemblyTotal duration over which the trajectory is optimized, from start to goal configuration.
Overly short horizons force physically infeasible accelerations; overly long horizons inflate computation and may embed unnecessary conservatism or local minima.
📐 Key Formulas
CHOMP Total Cost
J(ξ) = λ_smooth ∫₀ᵀ ||ä(t)||² dt + λ_coll Σₗ ∫₀ᵀ max(0, −ϕₗ(q(t)))² dt + λ_goal ||q(T) − q_goal||²Weighted sum of smoothness, collision avoidance, and goal accuracy penalties over trajectory ξ = {q(t)}
| Symbol | Name | Unit | Description |
|---|---|---|---|
| J | Total Cost | dimensionless | Cost functional evaluating trajectory quality |
| ξ | Trajectory | dimensionless | Time-parameterized joint configuration trajectory q(t) |
| λ_smooth | Smoothness Weight | dimensionless | Weighting factor for smoothness penalty |
| ä(t) | Joint Jerk | rad/s³ or m/s³ | Second derivative of joint acceleration (i.e., third derivative of position) |
| T | Trajectory Duration | s | Final time of the trajectory |
| λ_coll | Collision Weight | dimensionless | Weighting factor for collision avoidance penalty |
| ϕₗ(q(t)) | Signed Distance Function for Obstacle l | m | Signed distance from robot configuration q(t) to obstacle l |
| q(t) | Joint Configuration | rad or m | Robot configuration as function of time |
| λ_goal | Goal Weight | dimensionless | Weighting factor for goal accuracy penalty |
| q(T) | Final Configuration | rad or m | Robot configuration at final time T |
| q_goal | Goal Configuration | rad or m | Desired robot configuration |
STOMP Perturbation Variance
Σ_k = diag([σ₁², …, σₙ²]) where σᵢ = α · |äᵢ|_max · (t_{k+1} − t_k)Time-adaptive covariance matrix for Gaussian perturbations applied to waypoint k
| Symbol | Name | Unit | Description |
|---|---|---|---|
| Σ_k | Perturbation Covariance Matrix at Waypoint k | m²/s⁴ | Diagonal covariance matrix for Gaussian perturbations at waypoint k |
| σᵢ | Standard Deviation of Perturbation for Dimension i | m/s² | Time-adaptive standard deviation for the i-th dimension based on maximum absolute acceleration |
| α | Adaptation Gain | s² | Scaling factor that converts acceleration magnitude to perturbation variance |
| |äᵢ|_max | Maximum Absolute Acceleration in Dimension i | m/s² | Peak absolute acceleration observed or predicted along the i-th axis |
| t_{k+1} − t_k | Time Interval Between Waypoints k and k+1 | s | Duration between consecutive waypoints in the trajectory |
🏭 Engineering Example
BMW Plant Leipzig – Body-in-White Robotic Seam-Guiding Cell
N/A🏗️ Applications
- Robotic bin-picking in mixed-part logistics
- Autonomous mobile manipulator navigation in dynamic warehouses
- Precision tool-path generation for robotic deburring
🔧 Calculate This
⚡📋 Real Project Case
Palletizing Robot Path Optimization for High-Speed E-Commerce Fulfillment
Automated fulfillment center serving Amazon Prime logistics in Ohio