🎓 Lesson 8
D5
Time-Optimal Path Parameterization Under Torque Limits
Finding the fastest way to move a robotic arm along a planned path without exceeding its motor torque limits.
🎯 Learning Objectives
- ✓ Calculate feasible velocity and acceleration bounds along a given path using robot inertia and torque limits
- ✓ Apply the numerical integration method (e.g., TOPP-RA) to generate time-optimal parameterizations for 2-DOF planar manipulators
- ✓ Analyze how path curvature and joint inertia influence achievable motion time under torque saturation
- ✓ Design a safe, time-optimal trajectory segment for a mining robotic drill arm subject to peak torque constraints
📖 Why This Matters
In automated drilling, mucking, and inspection robotics used in underground and open-pit mines, moving too fast risks stalling motors, damaging gearboxes, or causing positional overshoot—leading to safety hazards and equipment downtime. Time-optimal parameterization ensures robotic systems complete critical tasks (e.g., positioning a blast-hole drill head) as quickly as physically possible *without* violating motor capabilities—directly impacting cycle time, productivity, and equipment longevity.
📘 Core Principles
Time-optimal path parameterization transforms a geometric path q(s) (parameterized by arc-length s) into a time-parameterized trajectory q(t), where s = s(t). The key insight is that torque τ_i(q, ṡ, s̈) must satisfy |τ_i| ≤ τ_max,i for all i and all t. Using the path-parameterized dynamics model, this yields inequality constraints on s̈(t) — the path acceleration — of the form: s̈ ∈ [s̈_min(s, ṡ), s̈_max(s, ṡ)]. These bounds define a 'feasible acceleration envelope' over the (s, ṡ)-phase plane. The time-optimal solution follows the 'bang-bang' principle: accelerate at s̈_max until reaching a switching curve, then decelerate at s̈_min — analogous to minimum-time control in double-integrator systems, but generalized for nonlinear, coupled dynamics.
📐 Feasible Acceleration Bound (Single-Joint Approximation)
For educational scaffolding, we begin with a simplified single-joint rotary link under constant torque limit. This reveals the structure before generalizing to multi-DOF systems. The bound arises from τ = I(q)·s̈·(dq/ds)² + Coriolis & gravity terms; neglecting non-acceleration terms gives the dominant constraint.
💡 Worked Example
Problem: A robotic drill arm’s first joint has effective inertia I_eff = 45 kg·m² along a path segment where dq/ds = 0.8 rad/m. Motor torque limit τ_max = 180 N·m. Gravity and Coriolis effects contribute ~25 N·m offset (constant). What is the maximum feasible path acceleration s̈ at this configuration?
1.
Step 1: Net available torque for acceleration = τ_max − |τ_offset| = 180 − 25 = 155 N·m
2.
Step 2: Apply τ_acc = I_eff · (dq/ds)² · s̈ → solve for s̈ = τ_acc / [I_eff · (dq/ds)²]
3.
Step 3: Compute denominator: 45 × (0.8)² = 45 × 0.64 = 28.8 kg·m²/m²
4.
Step 4: s̈ = 155 / 28.8 ≈ 5.38 m/s² (units: since s is in meters, s̈ is m/s²)
Answer:
The maximum feasible path acceleration is 5.38 m/s², which falls within the typical range of 3–8 m/s² for medium-duty mining manipulators.
🏗️ Real-World Application
At BHP’s Olympic Dam autonomous drill fleet, robotic boom arms execute pre-planned hole trajectories in variable-grade ore zones. When transitioning between vertical and inclined holes, path curvature changes abruptly—increasing required joint torques. Without time-optimal parameterization, fixed-velocity profiles caused repeated torque-limit alarms and missed hole tolerances ±2°. Implementing TOPP-RA (Time-Optimal Path Parameterization via Reachability Analysis) reduced average per-hole positioning time by 22% while eliminating torque violations—validated via ROS2+Gazebo co-simulation and field data from Parker Hannifin servo drives (model D1VW020HNJ).
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